The Sun : a Guide to Stellar Physics. [1 ed.] 9780128143353, 0128143355

Table of contents :
Front Cover
The Sun as a Guide to Stellar Physics
The Sun as a Guide to Stellar Physics
Copyright
Contents
List of Contributors
Preface
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Discoveries and Concepts: The Sun's Role in Astrophysics
1. THE SOLAR CONSTANT
2. THE SUN'S CHEMICAL COMPOSITION
2.1 SPECTROSCOPIC METHODS
2.2 MODELING OF THE SUN'S ATMOSPHERE
2.3 SETTLING OF LIGHT ELEMENTS
3. INTERNAL STRUCTURE AND HELIOSEISMOLOGY
3.1 DETECTION OF OSCILLATORY PATTERN
3.2 INTERPRETATION OF SOLAR OSCILLATIONS
4. THE MAGNETIC SUN AND ITS VARIABILITY
4.1 SOLAR CYCLE
4.2 MAGNETIC FIELDS 4.3 INTERNAL STRUCTURE AND LOCATION OF THE MAGNETIC DYNAMO5. THE SOLAR CORONA AND WIND
5.1 THE TEMPERATURE OF THE CORONA
5.2 THE SHAPE OF THE CORONA
5.3 THE SOLAR WIND
6. EARTH-SUN CONNECTION
6.1 AURORA AND GEOMAGNETIC STORMS
6.2 THE CARRINGTON EVENT
6.3 SOLAR FLARES, X-RAYS AND ENERGETIC PARTICLES
6.4 RECONNECTION OF MAGNETIC FIELDS
6.5 CORONAL MASS EJECTIONS
7. TESTING TWO CONCEPTS
7.1 NEUTRINO OSCILLATIONS IN THE SUN
7.2 TESTING GENERAL RELATIVITY
8. CONCLUDING REMARKS
ACKNOWLEDGMENTS
REFERENCES
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Stellar and Solar Chromospheres and Attendant Phenomena
1. INTRODUCTION 2. WHY CHROMOSPHERES EXIST2.1 STELLAR CONVECTION ZONES
2.2 THE SOLAR CHROMOSPHERE
2.3 STELLAR CHROMOSPHERES
2.4 WHY ARE CHROMOSPHERES SO THICK?
2.5 THE WILSON-BAPPU EFFECT
3. THE ROTATION-AGE-ACTIVITY CONNECTION
3.1 BACKGROUND
3.2 POST-SKUMANICH LAW INSIGHTS INTO THE ROTATION-AGE-ACTIVITY CONNECTION
3.3 THEORY BEHIND THE SKUMANICH LAW
4. STELLAR ACTIVITY CYCLES
REFERENCES
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The Sun's Atmosphere
1. INTRODUCTION
2. OBSERVATIONS OF THE SOLAR ATMOSPHERE
3. THE SOLAR SPECTRUM
4. PHYSICS OF THE PHOTOSPHERE/CHROMOSPHERE
4.1 ONE-DIMENSIONAL MODELS
4.2 THREE-DIMENSIONAL MODELS 5. PHYSICS OF THE CHROMOSPHERE/CORONA5.1 CORONAL EMISSION AND MAGNETIC STRUCTURE
5.2 BASIC CONSIDERATIONS OF THE ENERGETICS
5.3 HEATING PROCESSES AND MODERN MODELS
5.4 CONNECTION TO THE LOW ATMOSPHERE
ACKNOWLEDGMENTS
REFERENCES
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Helioseismic Inferences on the Internal Structure and Dynamics of the Sun
1. INTRODUCTION
2. THEORETICAL BACKGROUND
2.1 EQUATIONS GOVERNING SOLAR OSCILLATIONS
2.2 PROPERTIES OF SOLAR OSCILLATIONS
2.3 RELATING FREQUENCY CHANGES TO CHANGES IN STRUCTURE
2.4 EFFECTS OF ROTATION
2.5 INVERSIONS
3. INFERENCES ABOUT SOLAR STRUCTURE
3.1 BASIC RESULTS 3.2 BASE OF THE SOLAR CONVECTION ZONE3.3 THE QUESTION OF DIFFUSION
3.4 CONVECTION ZONE HELIUM ABUNDANCE
3.5 THE ISSUE OF SOLAR COMPOSITION
4. INFERENCES ON SOLAR DYNAMICS
4.1 PROPERTIES OF THE TACHOCLINE
5. HELIOSEISMIC INFERENCES ON THE SOLAR CYCLE
5.1 CHANGES TO GLOBAL-MODE FREQUENCIES AND MODE PARAMETERS
5.2 ZONAL FLOWS
5.3 MERIDIONAL FLOWS
5.4 THE 1.3-YEAR PERIODICITIES NEAR THE TACHOCLINE
5.5 CHANGES IN EVEN-ORDER A COEFFICIENTS
6. SEISMIC STUDIES OF OTHER STARS
ACKNOWLEDGMENTS
REFERENCES
5 --
Atmospheric structure, Non-Equilibrium Thermodynamics and Magnetism 5.1 --
Spectroscopy and Atomic Physics

Citation preview

CHAPTER

Discoveries and Concepts: The Sun’s Role in Astrophysics

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Jack B. Zirker1, Oddbjørn Engvold2 National Solar Observatory, Sunspot, NM, United States1; Rosseland Centre for Solar Physics, Institute of Theoretical Astrophysics, University of Oslo, Oslo, Norway2

CHAPTER OUTLINE 1. The Solar Constant ................................................................................................. 2 2. The Sun’s Chemical Composition............................................................................. 3 2.1 Spectroscopic Methods............................................................................ 3 2.2 Modeling of the Sun’s Atmosphere............................................................ 4 2.3 Settling of Light Elements........................................................................ 5 3. Internal Structure and Helioseismology ................................................................... 6 3.1 Detection of Oscillatory Pattern ................................................................ 6 3.2 Interpretation of Solar Oscillations............................................................ 7 4. The Magnetic Sun and Its Variability ....................................................................... 8 4.1 Solar Cycle ............................................................................................. 9 4.2 Magnetic Fields ...................................................................................... 9 4.3 Internal Structure and Location of the Magnetic Dynamo .......................... 10 5. The Solar Corona and Wind .................................................................................. 11 5.1 The Temperature of the Corona............................................................... 11 5.2 The Shape of the Corona........................................................................ 13 5.3 The Solar Wind ..................................................................................... 13 6. EartheSun Connection.......................................................................................... 14 6.1 Aurora and Geomagnetic Storms............................................................. 14 6.2 The Carrington Event ............................................................................. 15 6.3 Solar Flares, X-Rays and Energetic Particles ............................................ 16 6.4 Reconnection of Magnetic Fields ............................................................ 17 6.5 Coronal Mass Ejections.......................................................................... 18 7. Testing Two Concepts ........................................................................................... 19 7.1 Neutrino Oscillations in the Sun ............................................................. 19 7.2 Testing General Relativity ...................................................................... 22 8. Concluding Remarks............................................................................................. 23 Acknowledgments ..................................................................................................... 24 References ............................................................................................................... 24 The Sun as a Guide to Stellar Physics. https://doi.org/10.1016/B978-0-12-814334-6.00001-7 Copyright © 2019 Elsevier Inc. All rights reserved.

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CHAPTER 1 Discoveries and Concepts: The Sun’s Role in Astrophysics

1. THE SOLAR CONSTANT The amount of energy the Earth receives from the Sun is critically important to astronomers, physicists, and meteorologists. This constant is defined as the flux of energy (in watts/m2) above the Earth’s atmosphere, at the mean distance of the Earth from the Sun’s surface. The constant includes all electromagnetic radiation summed over all wavelengths. The theory of stellar evolution predicts that the luminosity of the Sun changes only very slowly, over billions of years. The question is, what is its current value? The main difficulty in determining the constant from measurements at the Earth’s surface is the correction for the absorption of the atmosphere. In 1838, French physicist C. Pouillet obtained a value of 1.228 kW/m2, which (perhaps by chance) was close to the best modern value. S.P Langley determined a value of 2.9 kW/m2 at the top of Mount Whitney in 1884, in strong discord with Pouillet. C.G. Abbot, who followed Langley as the director of the Smithsonian Astrophysical Observatory in 1907, spent 40 years in search of a reliable estimate. He established observing stations at high dry locations such as Mount Wilson; Bassour, Algeria; and Calama, Chile. His best estimates (1.318e1.548 kW/m2) were obtained with balloon sondes, some of which reached an altitude of 25 km. Abbot was convinced the Sun actually varied by such an amount within a few years. Measurements of extreme precision became possible with the use of satellites and with the development of a sensitive detector, the Active Cavity Radiometer Irradiation Monitor (ACRIM). Richard C. Willson, a physicist at the National Aeronautics and Space Administration’s (NASA’s) Jet Propulsion Laboratory, was principally responsible for its development. The first series of measurements was made during the flight of the Solar Maximum Mission (1980e89). It showed a distinct decrease of about 0.06% (from 1366.5 to 1365.8 W/m2 in 1980e85 and a return to 1366.6 W/m2 in 1986e89) with a day-to-day “noise” of about 0.3%. This noise was actually the response to the appearance and disappearance of sunspots. In a tour de force, Woodard and Hudson (1983) analyzed the first 10 months of ACRIM data and extracted 5-min oscillations of low degree (long horizontal wavelength). Frequencies, amplitudes, and line widths were obtained for individual pulsations. This result was a tribute to the precision and stability of ACRIM 1. A succession of satellites carried improved versions of ACRIM detectors, and with extensive calibration and cross-comparisons, an 11-year record of genuine variations was pieced together (Fig. 1.1, from Foukal et al., 2006). The solar luminosity varies in step with the sunspot cycle (Willson and Hudson, 1991). The reasons for this correlation will be addressed in Chapter 8.

2. The Sun’s Chemical Composition

FIGURE 1.1 Active cavity radiometer irradiation monitor measurements of solar constant during 11year sunspot cycles (Foukal et al., 2006).

2. THE SUN’S CHEMICAL COMPOSITION The chemical abundance of the Sun is a fundamental yardstick in astronomy. Knowing the Sun’s chemical composition became essential for discovering energy generation in the Sun and stars. The final breakthrough came in 1936 with the discovery by Hans Bethe, Charles Crichfield, and Carl Friedrich von Weiza¨cker of nuclear reactions taking place under the extreme pressure and temperatures in the core of the Sun (Foukal, 2004).

2.1 SPECTROSCOPIC METHODS Spectral observations of the solar photosphere are currently possible and available with very high spectral resolution and signal-to-noise ratio because of the great brightness of the source, allowing the profiles of a multitude of weak or blended absorption lines to be measured accurately. Element abundances of essentially all astronomical objects are referenced to the solar composition and basically every process involving the Sun and stars depend on their compositions. The abundance of elements in the Sun has become more extensively and reliably known than in any other star. The German optician Joseph von Fraunhofer was the first to observe and describe the multitude of dark lines in the emission spectrum of the Sun. He designated the principal absorption features with the letters A through K, and weaker lines with

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lowercase letters. Physicist Gustav Kirchoff, also from Germany, realized that the dark lines corresponded to the emission lines that he and his colleague Robert Bunsen observed in emission from heated gases. Kirchhoff concluded that the lines on the spectrum of the Sun were dark because they resulted from absorption by cooler layers of gas in the Sun’s atmosphere above hotter layers where the continuous emission spectrum originated. Kirchhoff’s formulated the following three laws that enabled solar scientists to exploit the potential of spectrometry in chemical analysis of the Sun and subsequently in stars: (1) A solid, liquid, or dense gas excited to emit light will radiate at all wavelengths and thus produce a continuous spectrum; (2) a low-density gas excited to emit light will do so at specific wavelengths, and this produces an emission spectrum; and (3) if light composing a continuous spectrum passes through a cool, low-density gas, the result will be an absorption spectrum. The emission spectra of elements, which could be vaporized by the Bunsen burner, were examined and compared with solar absorption line spectra. This became a truly fundamental astrophysical tool and a breakthrough in the science of astronomy. Kirchoff and Bunsen discovered lines from cesium and rubidium in the Sun. Swiss mathematician and physicist Johann Jakob Balmer observed the visible line spectrum of hydrogen and determined its wavelengths. The dominant ˚ , is referred to by astronomers as Ha red Fraunhofer line C, at wavelength 6563 A of the Balmer series. At a solar eclipse in India in 1868, French astronomer Pierre-Jules Janssen recorded the emission spectrum of a solar prominence, which contained a yellow ˚ , which had not yet been seen in laboratory spectra. line (Fraunhofer’s D3) at 5875 A This led Janssen and his contemporaries to conclude that it must represent a purely solar element, which soon was named helium after helios, the Greek word for “Sun.” In 1895, Swedish chemists Cleve and Langlet could confirm the presence of terrestrial helium gas coming out of a uranium ore called cleveite. The availability of spectral data and an understanding of the origin of the absorption lines stimulated development of analytical techniques to determine the constitution and structure of the solar atmosphere, including its chemical composition.

2.2 MODELING OF THE SUN’S ATMOSPHERE Before around 1940, calculations of solar spectral lines were based on the “SchustereSchwarzschild” model of the atmosphere, in which the photosphere radiated a continuous spectrum and was overlaid by a cooler layer that resulted in pure absorption. This crude approximation, which was most appropriate to use for strong resonance lines, was often applied in combination with the so-called curve of growth technique developed by Dutch astronomer Marcel Minnaert and collaborators C. Slob and G.F.W. Mulders in the early 1930s (Goldberg et al., 1960). The curve of growth is a graph showing how the equivalent width of an absorption line, or the radiance of an emission line, increases with the number of atoms producing the line and depends on the oscillator strength of the transition.

2. The Sun’s Chemical Composition

The MilneeEddington model is considerably more sophisticated. Here, the condition for spectral line formation, i.e., the ratio of the emission coefficient to the absorption coefficient, which is denoted the line source function, may vary with optical depth in the atmosphere. However, solar and stellar abundance determinations are only as accurate as the modeling ingredients. The most recent determinations of the solar chemical composition are based on the use of state-of-the art threedimensional atmospheric modeling and the calculation of spectral line formation, which also accounts for departures from local thermodynamic equilibrium (Asplund et al., 2009). A comprehensive listing of element abundances in the solar photosphere and in meteorites is provided by Nicolas Grevesse and Jacques Sauval (1998).

2.3 SETTLING OF LIGHT ELEMENTS When the solar abundances of lithium, beryllium, and boron are compared with their abundances in carbonaceous chondrite meteorites, in younger stars, and in the interstellar medium, it is found that the current solar lithium abundance is about a factor of 160 lower than in the primordial material, whereas the abundances of beryllium and boron are about normal. The variations in abundance of light elements with stellar age is associated with the existence of a subsurface convective layer in solar-type stars. The core region where nuclear fusion takes place is followed by the radiative zone out to 70% of the radius where the energy is transported outward by radiation whereas the remaining outer layer is the convection zone. A layer of thickness 0.02 Rʘ between the base of the convection zone and the top of the radiative zone is termed the solar tachocline (Elliot and Gough, 1999). This layer occurs because the inner radiative region rotates as a solid body while the convection zone rotates faster at the solar equator than near the poles. Because lithium burns at about 2.4
106K whereas beryllium requires 3.5
106K, its surface abundance is considerably affected, because the surface convection zone reaches down to the dynamic tachocline layer, at a temperature around 2
106K, where some exchange of material with the radiative zone takes place. This process also explains the observed increased lithium depletion in cooler, low-mass stars, which are expected to have deeper convection zones than the Sun (Vauclaire, 1998). An additional settling of elements will also result from the migration of elements through the interface between the convection zone and the radiative zone. A 10% reduction in helium abundance relative to hydrogen from the solar surface downward to the tachocline has been demonstrated from helioseismic studies and is explained as element migration (Grevesse and Sauval, 1998). This effect is active in both the Sun and stars.

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3. INTERNAL STRUCTURE AND HELIOSEISMOLOGY 3.1 DETECTION OF OSCILLATORY PATTERN In 1960, Robert Leighton, professor of physics at Caltech, and his students (R. F. Noyes and G. W. Simon) discovered a cellular pattern of vertical oscillations of the solar surface. Within a decade, the discovery would lead to the development of a new tool for solar physics (helioseismology) and the exploration of the solar interior. Leighton had joined the faculty in 1949 and had built a reputation as an inventive experimentalist, a keen researcher, and a fine teacher. His main scientific interest had been the decay products of cosmic rays, but in the late 1950s, he turned his attention to solar physics, specifically to the velocity fields of the solar surface. To pursue his objective, Leighton modified an existing instrument, the spectroheliograph at Mount Wilson’s 65-foot solar tower (Leighton et al., 1962). George Ellery Hale (and independently, Henri-Alexandre Deslandres) had invented this instrument in the 1890s. It was designed to form an image of the solar surface in monochromatic light and record it as a photograph. The instrument contained two narrow slits. The first slit selected a narrow strip on the solar image and passed its white light to a second slit, which was the entrance to a monochromator. This device contained a prism or a diffraction grating that dispersed the white light and isolated a chosen spectrum line. The light from the monochromator was focused on the photographic plate. As the spectroheliograph was driven slowly across the solar image, it recorded a monochromatic image of the solar surface. To measure vertical velocities, Leighton introduced a beam splitter just behind the exit of the monochromator and installed two glass blocks, one for each beam. The blocks could be tilted by equal amounts in opposite directions, thus shifting one beam to the red wing and the other beam to the blue wing of a symmetrical ˚ . A Doppler shift of the line thus increased the brightness line such as Ca 6103 A in one wing and decreased the brightness in the other wing. The two monochromatic beams were recorded on a photographic plate as the spectroheliograph moved across the solar image. After the scan was completed, the red channel on the photograph was subtracted from the blue channel by the use of a clever photographic technique. This brightness difference is proportional to the Doppler shift, or equally, the velocity of the solar surface and was recorded on a new plate. Thus, the velocity at each point in a strip of the solar image (within the length of the scanning slit) was presented as a brightness pattern. Bright elements are rising; dark elements are receding. Successive scans were made from north to south and in reverse over a period of many minutes. When the results were examined, two global cellular patterns emerged from the data.

3. Internal Structure and Helioseismology

In large cells were detected typically 16,000-km-diameter, horizontal flows from center to boundary that persisted for several hours. The root mean square (rms) speed of the flow depended on the height of formation of the spectral line: for ˚ and 1.8 km/s for Ca II 8542 A ˚ . The similarity to example, 0.4 km/s for Ca 6103 A the flows in photospheric granulation suggested the name “super granulation.” A second pattern of smaller cells (on the order of 2000 km) was also found. The researchers expected that the vertical velocity in such small cells would vary randomly in time. To their surprise, they found instead that cell velocity was “quasioscillatory” with a unique period of 296 s, a mean amplitude of 0.4 km/s, and a lifetime of at least three periods. Moreover, the cell brightness varied in phase with the velocity and with nearly the same period: bright plasma was rising and dark plasma was receding. The cell diameters increased from 1700 to 3500 km with increasing height above the surface.

3.2 INTERPRETATION OF SOLAR OSCILLATIONS The authors proposed several possible explanations for the oscillations but cautioned that more and better observations would be needed to choose one. They were persuaded, however, that the oscillations were determined only on “local properties of the solar atmosphere.” During the following decade, a variety of explanations were proposed for the oscillations, but none was definitive. However, Roger Ulrich, a postdoctoral student at the University of California Los Angeles, proposed in 1970 that the oscillations were the surface manifestations of a three-dimensional system of resonant acoustic waves that were trapped below the surface (Ulrich, 1970). Leibacher and Stein (1971) proposed a similar explanation. Standing acoustic waves in a three-dimensional cavity are distinguished by their horizontal and vertical wavelengths and by their frequencies. In the Sun, the observed horizontal wavelengths of the 5-min oscillations (around 2000 km) are small compared with the solar radius, so that a plane parallel geometry is a useful approximation. The vertical wavelengths however, are determined by the steep temperature and ionization gradients below the surface. Therefore, to construct a realistic model of the standing wave system, Ulrich needed an adequate model of the deep layer below the surface. Fortunately, he had calculated just such a model for his doctoral dissertation. Ulrich predicted that acoustic power at the surface would be observed primarily at particular combinations of horizontal wavelength (or wave number Kh) and frequency of oscillation u. In other words, power exists only along curved lines in the Kh e u diagram (Fig. 1.2). He wrote that previous observations of the oscillations were not long enough and did not cover a sufficient area to resolve the curved lines, and he specified the necessary limits. Deubner (1975) carried out the definitive observations and so confirmed Ulrich’s theory. The rest, as they say, was history.

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FIGURE 1.2 Observations of low-wavenumber nonradial eigenmodes of the Sun. The curved lines are theoretical (Deubner, 1975; Ando and Osaki, 1977).

4. THE MAGNETIC SUN AND ITS VARIABILITY The Sun is far from a static state. The so-called “quiet” Sun that may be described by relatively restricted, simple solar and stellar models is subjected to a variety of nonstationary active processes that represent the multitude of features and characteristics of solar as well as stellar activity. The variable Sun and subsequently its cyclic variability became known initially as a result of the GermaneDutch spectacle maker Hans Lipperhey’s invention of telescopes in 1608. As a result of this new invention, Galileo Galilei and a handful of contemporary scientists realized in the following years that the solar surface was blemished with dark spots and their associated bright faculae that came and went. Trackable spots on the Sun’s surface informed the quick intuitive mind of Galilei and the pertinent observer and Jesuit priest Christoph Scheiner that the Sun rotates and that its axis of rotation is tilted close to 7 degrees relative to the normal to the ecliptic plane, a discovery that led to the end of the geocentric model.

4. The Magnetic Sun and Its Variability

Even the Sun’s differential rotation was noticed by these very early scientists (Engvold and Zirker, 2016).

4.1 SOLAR CYCLE The 11-year solar activity cycle was first noticed by Heinrich Schwabe in 1843, who patiently recorded the number of sunspots over 17 years. This cycle, usually referred to as the Schwabe cycle, is the most prominent variability in the sunspot-number series. Rudolf Wolf of the Zu¨rich observatory collected observations of sunspots from the 1600s onward and introduced the index known as the Zu¨rich Wolf Sunspot number Rz, which was generally used in following years: Rz ¼ kð10 g þ nÞ where g is the number of sunspot groups, n is the number of individual sunspots, and k is constant correction factor that brings each observer to a common scale. Solar activity in all of its manifestations is dominated by the 11-year Schwabe cycle, but it has a variable length of 9e14 years for individual cycles. Hoyt and Schatten (1998) derived also a more robust series of sunspot activity indices, which is based on the more easily identified sunspot groups and excluded the number of individual spots. German astronomer Gustav Spo¨rer noted that observations of the Sun in several decades close to 1700 revealed few sunspots. Later studies by Hoyt et al. (1994) confirmed that the Sun was well-observed during the extended period from about 1645 until 1715 and showed few spots, which J. Eddy (1976) referred as the Maunder minimum, in recognition of the impressive contribution to studies of sunspot variability by the solar astronomers Annie and Walter Maunder. The following period, from 1795 to 1823, which had a remarkably low sunspot index, he termed the Dalton minimum. Early observations by Carrington and Spo¨rer showed that the locations of spots migrated toward the equator throughout the cycle; these were followed up by Maunders, who visualized this phenomenon in a timeelatitude histogram, which is referred to as the “butterfly diagram” (Maunder, 1904).

4.2 MAGNETIC FIELDS George Ellery Hale used the powerful Snow Telescope at the Mount Wilson Observatory when he noticed Zeeman split lines in spectral observations of sunspots, and he argued that they must be magnetic in origin (Hale, 1908). Sunspots became the first astronomical objects known to harbor magnetic fields. Father and son Harold and Horace Babcock invented the magnetograph around 1950, which enabled mapping of distribution, strength of the order of 1 G, and polarity of magnetic fields over the entire solar surface. The magnetic role of all aspects of solar activity were realized and settled (Babcock and Babcock, 1955).

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After the remarkable discoveries of sunspots, the Sun’s activity cycle, the peculiar timeelatitude pattern, and the overall magnetic link, Hale et al. (1919) showed that spots often emerged in bipolar pairs oriented roughly eastewest and that most westward or “leading” spots in the northern hemisphere have the same magnetic polarity during a cycle. Similarly, most of those in the southern hemisphere have the opposite polarity. Hale’s empirical rule provided fundamental clues about the interaction of emerging magnetic fields with the Sun’s differential rotation then thought to give rise to the spots and their distribution on the solar surface. In fact, the solar surface contains two types of magnetically active regions. There are polar and equatorial areas of the Sun, both of which are dominated by magnetic fields and structures. Work by Grotrian and Ku¨nzel (1950) showed that the polar and equatorial fluxes are comparable in magnitude. Babcock and Livingston (1958) found that the time of maximum of the polar fields was delayed by 3 years after sunspot minimum. Sheeley (2008) used solar faculae visible on white-light images as proxies of magnetic fluxes from a much longer time period and confirmed that polar magnetic fluxes undergo a cyclic variation, disappearing at sunspot maximum and appearing in large numbers around sunspot minimum. These results serve to shed light on the poleward migration of solar magnetic fields. The cyclic variations in intensity and distribution of magnetic flux on the solar surface demonstrate that the magnetic cycle is actually 22 years. The maxima of individual Schwabe cycles vary considerably and it is usual to distinguish among three long-lasting episodes containing around a dozen cycles, i.e., Grand maxima, Grand minima, and episodes of regular variations (de Jager et al., 2016). A long-term trend in the Schwabe cycle amplitude is called the secular Gleissberg cycle, with the mean period of about 90 years, or rather, a modulation in the cycle envelope with a varying timescale of 60e120 years (Gleissberg, 1971; Ogurtsov et al., 2002). The Sun’s proximity and resulting traceable effects of its variable activity on the Earth, such as 14C in tree rings and 10Be in polar ice, have enabled reconstructions of solar activity on multimillennial timescales (Solanki et al., 2004; Usoskin et al., 2016). According to these reconstructions, the level of solar activity during the past 70 years is exceptional, and the previous period of equally high activity occurred more than 8000 years ago. Such reconstructed data from the previous 11,000 years show numerous activity minima of duration ranging from 50 to 150 years.

4.3 INTERNAL STRUCTURE AND LOCATION OF THE MAGNETIC DYNAMO Our knowledge of the Sun’s interior is founded solely on theoretical models based on assumptions about physical conditions and processes that are likely to prevail there. The models were later successfully confirmed via helioseismology and the measured neutrino flux from the Sun’s inner core (Bahcall and Ulrich, 1988).

5. The Solar Corona and Wind

Eugene Parker showed how isolated toroidal magnetic flux tubes could rise from the depth of the tachocline layer (see Section 2.3) by magnetic buoyancy through the convection zone and form sunspots where they break through the solar surface (Parker, 1955). His work stimulated a search for the origin of solar magnetic fields. The Solar Dynamo action is discussed in detail in Chapter 7. Recent progress in our understanding of the solar magnetic dynamo and the nature of the solar tachocline, have stimulated further investigation of the origin of the variable period of the Schwabe cycles and of the episodes of changing cycle amplitudes. The unusual long-lasting minimum following the previous Schwabe cycle #23 in solar activity, which is being referred to as a transitional period, has inspired further studies of long term variations in the solar tachocline (de Jager et al., 2016).

5. THE SOLAR CORONA AND WIND 5.1 THE TEMPERATURE OF THE CORONA In the centuries preceding the invention of the telescope, astronomers in Babylonia and China might have noted the appearance of a faint ring of light around the Sun during a total eclipse. They may have speculated on the source of the light. Was it some fluke of the air? Was it attached to the Moon or the Sun? Not until the total eclipse of May 22, 1724 did an Italian astronomer, Giacomo Filippo Maraldi, realize that the ring of light was a part of the Sun, because it did not follow the motion of the Moon. Progress in understanding the nature of this “corona” had to await the invention of the telescope (around 1600) and the spectroscope (around 1814, by Joseph Fraunhofer). Fraunhofer found hundreds of dark lines in the spectrum of sunlight, which were later identified as absorptions by chemical elements in the solar atmosphere. The earliest spectra of the corona, taken at total eclipse, showed these dark Fraunhofer lines. This suggested that coronal light was simply scattered photospheric light. Then, at the total eclipse of Aug. 7, 1869, Charles Augustus Young and William Harkness independently discovered a bright line (brighter than the surrounding con˚ . More bright lines were discovered at the 1879 tinuum) at a wavelength of 5303 A eclipse and in later eclipses. Their wavelengths corresponded to no known element. Therefore, the observers postulated the existence of a new element, coronium. The spectrum of the corona was the subject of vigorous debate until at least 1918, however (Perrine, 1918). Some astronomers claimed to observe the Fraunhofer lines in the coronal spectrum whereas others observed only a smooth continuum devoid of lines. The issue is critical. A spectrum with lines would imply that the corona is composed of small particles that scatter photospheric light. A spectrum without lines would suggest that the corona is composed of incandescent gas that emits a continuum.

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Part of the confusion arose from the fact that the outer and inner corona have different spectra. In 1934, Grotrian separated the two components of the coronal light using eclipse spectra. He discovered that the K component (K for Kontinuum) is polarized and decreases in intensity rapidly with increasing distance from the Sun. He postulated that it is caused by the scattering of free electrons. The F component (F for Fraunhofer) contains dark lines and falls off slowly with the distance from the Sun. It is caused by diffraction by solid dust particles along the line of sight to the Sun. Near the Sun, the dust evaporates. Thus, the inner corona is predominantly K and the outer corona is predominantly F. Grotrian thought he found a clue to the temperature of the electrons that produce the K corona (Grotrian, 1933). When he examined the spectrum taken at the total eclipse of 1929, he discovered that it lacked the dark lines almost entirely. However, at the wavelengths at which two exceptionally broad and dark lines of ionized cal˚ ), a weak deprescium appeared in the spectrum of the solar disk (3933 and 3358 A ˚ wide. sion of the continuum appeared. It was only a few percent deep and over 100 A If confirmed, it would suggest that the particles that scatter photospheric light broaden the calcium lines almost to the point of extinction by virtue of their Doppler effect. That would imply a very high temperature for the K corona. Unfortunately, observations at later eclipses failed to confirm Grotrian’s broad, shallow depression. The precision of his measurements have also been called into question (Menzel and Pasachoff, 1968). Grotrian was not deterred, however,. He was about to make a crucial connection. In 1939, he received preliminary measurements of the energy levels of atoms that had lost 10 or more valence electrons. Bengt Edle´n, a Swedish physicist, had determined these levels from the extreme UV spectra of excited atoms. He forwarded these data to Grotrian in preparation for a report on the nebular phases of novae at the Paris meeting on novae. Grotrian noticed that the separation of two energy levels of Fe X (nine times ˚ . Simiionized iron) corresponded to the wavelength of the coronal red line at 6374 A larly, the separation of two levels of Fe XI corresponded to the wavelength of the ˚. coronal line at 7892 A According to P. Swings (1943), Edle´n became deeply interested in Grotrian’s remark. From his unpublished measurements of the spectra of Ca XII and XIII, ˚ . Assuming he found coincidences with two faint coronal lines at 3328 and 4086 A these four identifications to be correct, he predicted the forbidden lines of Fe XIII, XIV, Ni XII, and others. He found more coincidences too remarkable to be caused by pure chance. The coronal spectrum problem was solved! These highly ionized atoms could only be formed in a plasma at temperatures of 1e3 million K. Confirmation of the high temperature of the corona was not long in coming. On Sep. 29, 1949, Herbert Friedman and his colleagues at the US Naval Research Laboratory launched a V-2 rocket that carried an x-ray photon counter to an altitude of ˚ were detected above 87 km and UV 150 km (Friedman et al., 1951). X-rays of 8 A ˚ radiation around 1200 and 1500 A above 70 and 95 km, respectively. But what were

5. The Solar Corona and Wind

the physical implications? What could heat the corona to such temperatures? Surely, the mechanism had to be nonthermal because heat does not flow from low to high temperatures. This question would challenge solar astronomers for the next 70 years (Zirker and Engvold, 2017). The relative abundances of different stages of ionization of an element could be predicted later with the development of the nonlocal thermodynamic equilibrium theory. Stellar astrophysics has benefitted considerably from the application of the theory, as applied to atmospheres of extremely low particle density, the “coronal approximation.” The theory has been developed further to cover multilevel atoms and radiative transfer in denser atmospheres.

5.2 THE SHAPE OF THE CORONA Before the invention of photography, observers at total eclipse could only draw a quick sketch of the corona or commit its shape to memory. The early daguerreotypes were too slow to record the extensive plumes that can be seen at totality. Despite these handicaps, French astronomer Jules Janssen noticed a change in the shape of the corona between the eclipses of 1871 and 1878. It was initially round and later enhanced mainly at the solar equator. He realized that the corona changed shape in step with the 11-year sunspot cycle that Schwabe had discovered in 1843: round in 1871 at sunspot maximum and equatorial in 1878 at minimum. This result would later suggest a magnetic framework for the structure of the corona.

5.3 THE SOLAR WIND Ludwig Biermann (Max Planck Institute fu¨r Naturforschung) was a theoretical astrophysicist who made important contributions to the theory of stellar convection, stellar interiors, comet nuclei, interstellar magnetic fields, and plasma physics. Around 1951, he noticed that the tails of comets always pointed away from the Sun while orbiting the Sun (Biermann, 1951). That observation led him to postulate a radial streaming of particles from the Sun. In subsequent articles, he estimated speeds of 500e1500 km/s and densities at the orbit of Earth of 500 to 105 particles per cm3. In 1958, Eugene Parker developed a gas-dynamic theory to explain Biermann’s estimates. He showed how a hot corona must expand at supersonic speeds into interplanetary space. An isothermal corona of 2 million K would reach the Earth at a speed of 500 km/s. Moreover, the radial flow of ions would draw a weak dipole magnetic field into an Archimedes spiral, as seen from interplanetary space. Parker’s theory was met with considerable skepticism at first but was vindicated by the detection of the solar wind by the Soviet satellite Luna I in 1959 and by the US Mariner II en route to Venus, in 1962. The study of the wind has grown into a major subdiscipline within solar physics.

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6. EARTHeSUN CONNECTION 6.1 AURORA AND GEOMAGNETIC STORMS The auroral polar light displays in the northern and southern hemispheres are now seen as the most dramatic visual feature of a whole new science referred to as space weather (cf. Chapter 10). Our current understanding of the central aspects of Earthe Sun connections is based on tireless efforts by many scientists over 250 years. The connections include several components: coronal mass ejections (CMEs), the solar wind, flare particle and x-ray emissions, geomagnetic storms, and auroras. Northern lights, i.e., the Aurora Borealis, are one of nature’s most spectacular light phenomena that can be observed with the naked eye. Through millennia, northern lights have triggered the human imagination, curiosity, and fear, as reflected in a number of mythologies from those of Nordic areas where the northern lights occur most frequently to those of North Dakota Indians and that of Tiberius Caesar Augustus in Rome, where the aurora shows up occasionally. The 17th century French scientist Pierre Gassendi applied the name “aurora” to the northern lights after Aurora, the Goddess of Dawn in Roman mythology, which has become the commonly used name for the northern lights. Figure 1.3 shows an aurora display observed from the Svalbard archipelago.

FIGURE 1.3 An aurora display observed from the European Incoherent Scatter Scientific Association (EISCAT) antenna site in the Svalbard archipelago on Nov. 10, 2010. The characteristic upper red emission from above 200 km results from the 6300A˚ line of atomic oxygen whereas the 100e200 km region is dominated by the green line at 5577 A˚, also from oxygen. Credit: Dr. Nja˚l Gulbrandsen, University of Tromsø, Norway.

6. EartheSun Connection

Swedish physicists Anders Celsius and Olof Hiorter started systematic observations in the 1740s with magnetic needles. They were able to confirm a strong correlation between aurora events and geomagnetic fluctuations. As solar activity resumed after the Maunder minimum (1645e1715), some intense auroras were observed at midlatitudes. In 1733, French geophysicist and astronomer JeanJacques d’Ortous de Mairan noticed an apparent link between sunspots and auroras and suggested that auroral light could result from solar fluid impinging upon the Earth’s atmosphere. German amateur astronomer Samuel Heinrich Schwabe, who first publicly suggested the existence of the sunspot cycle, followed with an additional discovery in 1843 when he noticed a correlation between aurora and geomagnetic activity and the number of sunspots. A few years later, Scottish geophysicist J.A. Broun found that geomagnetic storms had a tendency to recur after 27 days, a time close to the rotation period of the Sun seen from the Earth. These clues, indicating that activity on the Sun somehow influences the Earth’s magnetic field were further strengthened by a dramatic event in 1859.

6.2 THE CARRINGTON EVENT A white light solar flare within a huge sunspot that was observed and recorded by British astronomers Richard Carrington and Richard Hodgson on Sep. 1, 1859 was followed by a powerful geomagnetic storm the next day. They saw two patches of very intense light. Carrington immediately thought that his equipment had malfunctioned but soon realized that he saw a real solar feature (Carrington, 1859). The storm is now referred to as the Carrington event. The storm caused interruptions of telegraph systems, which were sensitive to strong geomagnetic signals, throughout the world for several hours. The storm was followed by sparkling bright aurora that could be seen by people around the world and was visible even at latitudes of Italy, England, and France. It is well-known today that clouds of particles from very strong solar eruptions penetrate deeper into the geomagnetic magnetic fields and thus cause geomagnetic storms and aurora at lower latitudes than normal. Solar storms of the same magnitude occurring today can cause life-threatening power outages, satellite damage, communication failures, and navigation. A very large X15-class solar flare on Mar. 6, 1989 resulted in a geomagnetic storm on the following Mar. 9, with disturbing consequences on Earth. The storm began with extremely intense auroras at the poles. Satellites in polar orbits lost control for several hours and Geostationary Operational Environmental Satellite weather communications were interrupted. Strong fluctuations in the Earth’s magnetic field resulted in serious electric power failure in Quebec, Canada. An eruption comparable in strength to the 1859 Carrington event took place on Jul. 23, 2012 but it missed hitting the Earth that time. Norwegian physicist Kristian Birkeland was the first to claim that charged particles from the Sun could trigger the aurora. In 1896, he presented his theory that

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the northern lights result from electric charged particles from the Sun being deflected by Earth’s magnetic field and pulled down toward the poles, where they collide with the atmosphere and create this magical light. This is essentially the theory of the aurora today. One of Birkeland’s experiments was based on his magnetized terrella, which consisted of a small model of the globe containing an electromagnet in a vacuumsealed chamber simulating the Earth. Using the electromagnet, he could create a magnetic field around the terrella mimicking the Earth’s magnetic field. The atmosphere was like a layer of fluorescent paint that would give off light when it was struck by charged particles. Birkeland’s theory of the aurora remained dismissed by mainstream astrophysicists long after his death in 1917. It took over 60 years before Birkeland’s theory could be confirmed, when NASA’s Mariner II spacecraft, on its way to Venus in 1962, measured the presence of ionized gas with speeds up to 300e700 km/s, i.e., the solar wind.

6.3 SOLAR FLARES, X-RAYS AND ENERGETIC PARTICLES Solar flares are sudden releases of energy in the solar atmosphere. They were first detected and studied as chromospheric outbursts in the light of Ha by observers such as G.E. Hale, H.A. Deslandres, and M.A. Ellison. They learned that flares occur in regions of intense magnetic field that are located among groups of sunspots. Their frequency varied in step with the 11-year sunspot cycle. Their area and intensity ranged over several orders of magnitude and the total amount of energy released could reach up to 1025 J. An impulsive release, within a few minutes, could be followed by a slow release over several hours. Evidence of a terrestrial response to most flares continued to accumulate. G.E. Hale (1931) listed a dozen especially large flares that were followed in a day or two by geomagnetic storms. That suggested to him that flares can emit streams of charged particles. S.E. Forbush (1946), pioneer observer of galactic cosmic rays, detected sudden decreases in cosmic ray intensity after some flares. This was later interpreted as resulting from CMEs that swept away the protective geomagnetic field. In addition, he measured sporadic increases of giga-electron volt protons after very energetic flares. Rocket observations of flares in the 1950s revealed the emission of soft x-rays (about 1 kiloelectron-volt [keV]) that caused ionospheric fadeouts. Subsequent observations from satellites have shown that large eruptive flares can emit radiation from radio wavelengths to gamma rays and particle emission up to 1000 megaelectron volts (MeV). In fact, up to half of the total energy released may be in the form of energetic charged particles. In so-called “proton” events, the initial flare brightening is detected in 10e100 keV x-rays (electron bremsstrahlung) and type III decimeter bursts. After a few minutes, the products of the CMEs are detected. These include 10- to 100MeV gamma rays and up to 600-MeV protons and helium nuclei. A fraction of these

6. EartheSun Connection

very energetic protons may penetrate the Earth’s geomagnetic field, enhance the ionosphere at 50- to 80-km altitudes, and reach ground level. Researchers soon agreed that the huge amounts of flare energy could be derived only from the energy stored in strong nonpotential magnetic fields. But what mechanism could account for the rapid conversion of energy? R.G. Giovanelli (1948), an Australian physicist, proposed the reconnection of twisted magnetic fields.

6.4 RECONNECTION OF MAGNETIC FIELDS Reconnection of magnetic fields is an important process in astrophysics. It is thought to occur in the Sun, in the geomagnetic field, and in the magnetic dynamo. It is observed in laboratory plasmas and specifically in controlled fusion experiments. The process involves a flow of plasma and embedded field toward a neutral point, where the magnetic field strength vanishes and field lines can be cut and reconfigured, with the release of kinetic, thermal, and accelerated particle energy. In 1958, Peter Sweet (University of London Observatory) proposed a model in which two bipolar sunspot groups collide, forcing their magnetic fields to contact at a neutral point. The subsequent development depends on the conductivity of the solar plasma at that point. In a perfectly conducting plasma, no merging of fields is possible. In plasma with a small but finite electrical resistance, opposite polarity field lines can cancel and release copious amounts of energy. Sweet presented a theory for the development of the contact region, which he visualized as a thin linear current sheet of finite length (Fig. 1.4A). Plasma and

FIGURE 1.4 Sweet’s reconnection model (Sweet, 1958). (A) Embedded field lines converge from left and right on neutral line N (B) The magnetic field strength and polarity change abruptly across the current sheet and reconfigure at X and Y to form U-shaped lines that retract, pulling plasma toward the top and bottom as in (C).

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embedded field lines with opposing directions approach the sheet from the left and right sides at a slow speed that is determined by the rate of cancellation of field lines, which in turn is fixed by the rate of diffusion across the thin sheet. The crucial transformation occurs at the ends of the sheet at point X, where the original field lines are reconfigured to form U-shaped lines. These in turn are pulled rapidly away from the ends by their magnetic tension. A heuristic hydrodynamical model (Fig. 1.4) was used to describe this flow. The speed of outflow could approach the Alfve´n speed. In principle, a steady state could be reached as long as the supply of plasma and field was maintained. Sweet adopted a plausible chromospheric temperature (104K), sheet length 4 (10 km), and field strength (103 Gauss). He calculated that the total energy released could reach 1033 erg in a flare lifetime of 104 s, which he thought reasonable. Moreover, the electric field in the current sheet seemed sufficient to account for the acceleration of charged ions. Stimulated by Sweet’s theory, E. N. Parker (1957) used dimensional arguments to reach similar conclusions, and the theory became known as the SweeteParker theory. Actually, Sweet’s flare model was too slow by a factor of 103 or more to accord with observations, and he sparked an intense effort to improve on it. Parker (1963) showed that the Sweet mechanism is efficient only when oppositely directed field lines are exactly aligned. In the following decades, theorists have explored a variety of possible models of reconnection in the context of flares (Petchek, 1964; Sturrock, 1968), but many details remain unresolved. A current sheet is predicted to be only a few meters thick and perhaps some 100 km long, far below the resolution of current telescopes. However, after a flare, observations of the reconfiguration of large-scale fields are seen as compelling evidence for reconnection. Flare observers therefore often apply some form of reconnection theory to analyze their observations (Shibata and Magara, 2011; Vilmer, 2012). The frontier in reconnection theory is the extension to three dimensions. Magnetic reconnection is described in Chapter 7.

6.5 CORONAL MASS EJECTIONS Erupting prominences provided the first observed evidence of expulsion of mass into the higher coronal regions. Edison Pettit collected observations of prominences from a number of observatories (Meudon, Arcetri, Kodaikanal, Zurich, and Yerkes) between 1919 and 1931. He found that the upward rise of eruptive prominences started slowly but was followed by notably rapidly increasing velocities (Pettit, 1932). The coronal response to solar eruptions was finally explored with instruments in space. Coronagraphs on board the orbiting OSO-7 Satellite (Tousey, 1973) and on Skylab (Gosling et al., 1974) during its nearly 8-month mission enabled unprecedented studies of the evolution of the outer solar corona. A slit-less spectrograph on board Skylab recorded emission at extreme UV and UV wavelengths, which

7. Testing Two Concepts

enabled observations of coronal structures in a range of temperatures from 104 to 106K and revealed the thermal variations in the dynamic coronal responses to large flares and filament eruptions. Further instruments such as the Extreme-ultraviolet Imaging Telescope onboard the Solar and Heliospheric Observatory allowed for more detailed studies, from their initiation at the Sun out to their arrival at 1 AU. See Chapters 12.1 and 12.2 for further details and discussions on solar instruments. The corona responds to flares and erupting filaments with sudden expulsions of magnetic flux and dense clouds of plasma into interplanetary space (Munro et al., 1979). These eruptions are termed CMEs, which are distinctly different from the continuous outflows of the solar wind. The events are observable in white light owing to Thomson scattering of photospheric light by the coronal electrons in the ejected mass. The OSO-7 series showed violent CMEs occurring every couple of days during sunspot minimum and several times a day during sunspot maximum (Gopalswamy, 2016). Munro et al. (1979) claimed that eruptive prominences are rarely if ever seen without an accompanying mass ejection. From CME observations obtained during the first Skylab mission in 1973e1974, near the minimum of the activity cycle, they suggest that 40% are associated with flares and that 70% of the recorded CMEs were associated with erupting prominences both with and without flares. Whether the CMEs is a cause or effect of other activities remains a challenging issue in the interpretation of observations and theoretical modeling of dynamic coronal events. The simplest form of CMEs is composed of a leading edge followed by a dark cavity and a bright core, which also can contain the remains of the erupting filaments. The mass in coronal ejection events is largely coronal matter being swept up on its way outward (Poland and Munro, 1976). The high-energy particles associated with CMEs may strongly affect planetary environments (Gosling et al. 1991). One realized soon that interactions between CMEs and interplanetary magnetic fields is a major cause of large magnetic storms (Gopalswamy et al., 2000). For detailed discussions of these issues, including models of mass ejections, we refer to Chapter 10.

7. TESTING TWO CONCEPTS 7.1 NEUTRINO OSCILLATIONS IN THE SUN The Sun has had a central role in confirming an exotic process in elementary particle physics, namely the changes in identifying a neutrino as it propagates. In 1931, Wolfgang Pauli, a German theoretical physicist, postulated the existence of an unknown elementary particle to account for the missing momentum in radioactive beta decay events. This hypothetical particle had neither mass nor charge but moved at the speed of light. It would interact with dense matter so weakly that it could pass through the Earth without being deflected. His Italian American colleague, Enrico Fermi, named it the “neutrino,” or little neutron.

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The neutrino was also invoked in the quest to explain the source of the Sun’s radiated energy. In 1938, Hans Bethe and Charles Critchfield constructed a chain of nuclear reactions that converts four protons into a helium nucleus, with the emission of one neutrino and 26.7 MeV of energy in gamma rays. Their work would earn them the Nobel Prize in Physics for 1967, only after the existence of the neutrino was proven. Twenty-five years would pass before this elusive particle would be detected. Clyde Cowan and Frederick Reines (Los Alamos National Laboratory) realized in 1951 that a nuclear reactor would emit intense fluxes of antineutrinos, which differ from ordinary neutrinos only by the direction of a kind of spin. (Every elementary particle has a twin with the same mass but different electrical charge or spin.) If they could trap antineutrinos, they would prove the existence of neutrinos. Their experiment in 1956 at the reactor at Savanna, Georgia, was successful: electron neutrinos are real particles. With this confirmation, the path was open to test the validity of the Bethee Critchfield theory of solar energy production. In 1963, Ray Davis and John Bahcall (Brookhaven National Laboratory) made plans for a critical experiment. Davis was the experimentalist and Bahcall the theorist. They planned to count solar antineutrinos and compare theory and observation. Davis used a 100,000-gallon tank of tetrachloroethylene (a common drycleaning fluid) as his antineutrino detector. To avoid false signals from cosmic rays, Davis set his tank 1500 m deep in the Homestake Gold Mine in South Dakota. A passing neutrino could convert an atom of chlorine-37 to a radioactive argon-37 atom, with a vanishingly small probability. Davis invented an exquisitely sensitive technique to collect argon-37 atoms about every 2 months by flushing his tank with helium. He was able to detect single argon atoms. Using the best model of the solar interior and combining it with the BC theory, Bahcall calculated the rate of neutrino captures Davis should see if the BC theory was valid. He predicted about two captures per week. Davis ran his experiment for 5 years until he had sufficient captures to compare with theory. His preliminary results in 1968 showed a rate one-third as large as theory predicted. Where were the missing neutrinos? His result provoked a serious crisis in stellar and nuclear physics. Over the following two decades, Davis scrutinized his procedures and refined his estimates of possible experimental error. Bahcall examined the possible sources of error in his calculations. These included uncertainties in nuclear reaction cross-sections and in the theoretical models of the solar interior. Nothing was large enough to account for a factor of three discrepancy. Davis continued his experiment until 1984. Meanwhile, several discoveries were made elsewhere that bore on the solar neutrino problem. A second type of neutrino, associated with the muon particle, was discovered in 1962 by three scientists at CERN. They bombarded a target with a powerful proton beam to produce measurable muon neutrinos. A third type of low-mass particle, the Tau particle, was detected in 1978 by the Stanford Linear Accelerator Center and its associated neutrino in 2000 by CERN.

7. Testing Two Concepts

When the first results from the Homestake experiment were published, Bruno Pontocorvo and Vladimir Gribov proposed a radical solution to the solar neutrino problem: neutrinos might oscillate among the three “flavors”: electron, muon, and tauon. (This was possible in the weird world of quantum mechanics.) If solar neutrinos arrived at Earth as either muon or tauon neutrinos, Davis’s tank would not detect them. Particle physicists were skeptical and there was no way to test the idea. In any case, a more plausible solution had to be explored. The production of electron neutrinos is extremely sensitive to the temperature distribution in the solar interior. Could the uncertainty in the temperature account for the missing neutrinos? Helioseismology is the technique of using the observed vibrations of the solar surface to determine the properties of the solar interior. There are two ways to go about this. In the forward method, one computes a solar model of temperatures and densities and predicts the oscillation frequencies of many different modes. Then one compares the predicted and observed frequencies and modifies the model until they match. In the inverse method, one uses the fact that acoustic modes of different frequency are refracted (internally reflected) at different distances from the Sun’s center. Therefore, it is possible to combine modes to sample the speed of sound (and hence, temperature) at a chosen depth in the Sun. If the empirical and model temperatures disagree, the model has to be modified. As the quality and duration of observations steadily improved during the past two decades, agreement between empirical and computational sound speed distributions matched to within 0.1% through most of the Sun’s depth. This precision eliminates the possibility that temperature uncertainty is the cause of the solar neutrino problem. That conclusion lent support to Pontocorvo’s idea that neutrinos oscillate among three types as they propagate from the Sun’s center to the Earth. The only way to test the idea was to count the different types. Indeed, that is what two neutrino observatories have done: one in Sudbury, Canada, and the other in Kamioka, Japan. The Sudbury Neutrino Observatory counts only electron neutrinos whereas the Japanese neutrino observatory counts the sum of all three types. From the difference in counts, one could determine the fraction of muon and tauon neutrinos. It turned out that only a third of all neutrinos arriving from the Sun are electron neutrinos; the other two-thirds arrive as muon or tauon neutrinos. Physicists could breathe a sigh of relief. The missing neutrinos have been found; the Bethe-Critchfield theory of energy production is valid! However, a new challenge has arisen. The standard theory of elementary particle physics assumed that neutrinos have no rest mass, whereas the observed oscillation of flavors requires that they do indeed have mass. The challenge is to determine their masses. The difficulty is that each flavor does not have a permanent mass as it moves through a material body such as the Sun. Instead, each flavor has a superposition of three absolute masses, with different probabilities. As of 2016, it was known that the sum of the three masses is less than 106 of the electron mass.

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7.2 TESTING GENERAL RELATIVITY In 1916, Albert Einstein published his General Theory, which describes gravity as a curvature of space in the presence of a material body. Relatively few scientists were able to follow his argument, whereas those who did awaited experimental confirmation. By 1918, Einstein’s prediction of the advance of the perihelion of Mercury by 43 arcsec per century convinced some astronomers, but the theory was still in doubt. Therefore, Einstein proposed another test of his theory. He encouraged astronomers to measure the deflection of starlight in the vicinity of the Sun at a total solar eclipse. He predicted that a ray of starlight that grazed the edge of the eclipsed Sun would be deflected by precisely 1.76 arcsec. He also predicted the deflections at different distances from the Sun. Einstein was evidently unaware of the earlier work of Johann Georg von Soldner, a German astronomer. In 1804, von Soldner used Isaac Newton’s formula for the gravitational attraction of two bodies to calculate the deflection of a ray of starlight near the Sun. In this estimate, von Soldner assumed, as did Newton, that light consists of a stream of tiny corpuscles that have mass. von Soldner obtained a deflection angle of 0.875 arcsec. To carry out Einstein’s suggested test of his theory, an astronomer would photograph the positions of as many stars as possible during an eclipse. Then, a few months later, when the Sun was no longer present in the same field of stars, he would photograph the same stars with the same telescope. By subtracting one photograph from another, he could determine the deflections and compare them with Einstein’s predictions. A team from the Lick Observatory was the first to attempt this test. They observed the total eclipse of Jun. 1918 in Washington State, but the clouds allowed them to see only a few stars. The test was inconclusive. Therefore, the British astronomers resolved to try their luck at the eclipse of May 29, 1919. They sent one expedition to Sobral, Brazil, and another one to the island of Principe, in the Gulf of Guinea. Arthur Eddington, who would become one of the foremost theorists of the century, joined the team in Principe as an observational astronomer. The eclipse was partly cloudy at Principe. After returning to Oxford, Eddington could measure the positions of only four stars, but he derived a deflection of 1.61 arcsec at the Sun’s edge. The Sobral team had better luck. They found seven stars on their photographic plates. After measurement and interpolation, they obtained a deflection at the Sun’s edge of 1.98 arcsec, compared with the prediction of 1.76 arcsec. The British concluded that a deflection does occur at the edge of the Sun, in an amount compatible with Einstein’s prediction, but with a relatively large random error of 0.3 arcsec. They urged their colleagues to repeat this critical test. Over the next 40 years, a dozen measurements were obtained at total eclipses, with varying random errors. It became clear that systematic errors introduced by a

8. Concluding Remarks

host of external factors were larger than random errors and very difficult to evaluate. A major effort was made at the 7-min eclipse in 1973 by a team from Princeton and the University of Texas. That team obtained a value of 1.66  0.18 arcsec at the Sun’s edge, in close agreement with Einstein’s prediction. Moreover, their measured deflections of 40 stars, at different distances from the Sun, followed Einstein’s curve, but with huge scatter. That was where the issue stood when radio astronomers entered the game, around 1969. They had used interferometers with long baselines to measure the position of radio sources with millliarcsec precision. In particular, they measured the positions of two quasars that are very close to each other in the sky and pass near the Sun during the year. By 1972, theory and observations agreed to within 3%. In 1974 and 1976 the two scientists A. Fomalont and R. Sramek (1975) were able to observe three sources that lie in a straight line in the sky. They obtained 1.761  0.016 arcsec, or within 1% of Einstein’s number. The latest measurement (2009) was made by Fomalont and colleagues using the Very Long Baseline Interferometer, a 5000-mile chain of radio telescopes between Europe and the United States. They achieved agreement between theory and observation of less than 1%. Fomalont projects that this uncertainty can soon be reduced by a factor of four. The bending of light rays by a massive object has produced some surprising images and has developed into a subtopic in astronomy: gravitational lensing. In 1979, three astronomers discovered a double image of a distant quasar in visible light. It turned out that a nearby galaxy was acting as a lens for the rays from the quasar. Subsequently, examples were found of complete rings, or a pattern of arcs. Background stars or galaxies can act as sources of light or radio waves that pass by nearby massive galaxies and seem to vary in brightness. Searches for such events have uncovered the imprint of dark matter.

8. CONCLUDING REMARKS The focus of this chapter has been on breakthroughs in studies of the Sun that have stimulated stellar physics. Stellar global oscillations and stellar activity cycles are two examples of this impact. Olin C. Wilson’s initial survey of variations in stellar activity from observations of Ca II H and K lines (Wilson, 1963) opened a momentous new field of astrophysics and a comprehensive literature, as reviewed by Vidotto et al. (2014), Testa et al. (2015), and others. Conversely, stellar studies have given essential context to the behavior of the Sun. Relationships between large-scale surface magnetic fields, stellar age, and rotation are well-documented. Empirical relations between stellar rotation periods and the length of corresponding activity periods shed light on the dynamo actions in the Sun and sun-like stars. We look forward to further fruitful cross-fertilization.

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ACKNOWLEDGMENTS The authors are grateful for comments and suggestions from Aad van Ballegooijen, Sara F. Martin, and Jean-Claude Vial in their preparation of this chapter.

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CHAPTER

Stellar and Solar Chromospheres and Attendant Phenomena

2 Thomas R. Ayres

University of Colorado, 389-UCB (CASA), Boulder, CO, United States

CHAPTER OUTLINE 1. Introduction ......................................................................................................... 27 2. Why Chromospheres Exist..................................................................................... 28 2.1 Stellar Convection Zones........................................................................ 28 2.2 The Solar Chromosphere ........................................................................ 29 2.3 Stellar Chromospheres........................................................................... 31 2.4 Why Are Chromospheres So Thick? ......................................................... 32 2.5 The WilsoneBappu Effect ...................................................................... 43 3. The RotationeAgeeActivity Connection ................................................................. 45 3.1 Background .......................................................................................... 45 3.2 PosteSkumanich Law Insights Into the RotationeAgeeActivity Connection ........................................................................................... 46 3.3 Theory Behind the Skumanich Law ......................................................... 50 4. Stellar Activity Cycles .......................................................................................... 52 References ............................................................................................................... 56

1. INTRODUCTION One might surmise that the physics of the stars and that of the Sun must be intimately connected. After all, physics is physics and the Sun is a star. However, any common ground fails at a granular level for the simple reason that sets the Sun apart from all the other stars: it is possible to observe the solar surface (and the interior, thanks to helioseismology) in remarkable detail with almost arbitrarily high temporal, spatial, and spectral resolution, free of interstellar absorption. Furthermore, as a singular object of attention, the Sun has inspired long-term records of various phenomena such as the enigmatic sunspot cycle, for which detailed surface maps of the dark spots (initially drawings, more recently digital images) extend back four centuries or more. Meanwhile, the stars are so distant that observational limitations dictate more superficial examinations, certainly spatially unresolved, except in a few special cases. Also, the stars are so numerous and diverse The Sun as a Guide to Stellar Physics. https://doi.org/10.1016/B978-0-12-814334-6.00002-9 Copyright © 2019 Elsevier Inc. All rights reserved.

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in size, mass, temperature, luminosity, chemical composition, age, etc., that statistical considerations must come into play. The wide but shallow view of the stars lends itself to equally superficial physical modeling. Thus, there is little to be learned about the fine details of solar phenomena from the necessarily more global examinations of the stars. That said, studies of the stars can help our understanding of the Sun in a number of important ways. We have learned much from our privileged solar vantage point but still must appeal to the broader collection of sunlike stars to understand pivotal aspects of the solar condition. For example, is the Sun typical of G-type dwarfs? Or is the Sun special in some way that has allowed life to develop and flourish on one of its planets? What might the Sun have been like in its youth, when the solar system was still forming? Will normal evolutionary changes in our star someday threaten planetary habitability? Are there infrequent events on the Sun that might affect our technology-laden civilization right now, for which there is little or no historical experience because these episodes are rare? Observations of stars can help answer such questions. This is the playing field of the Solar-Stellar Connection laid out by Leo Goldberg a half-century ago in a prestigious Russell Lecture before the American Astronomical Society. There is one more point to be made in favor of the stellar viewpoint: the necessarily macroscopic views of stars potentially can act as a filter to isolate important global phenomena that might be glossed over in the microscopic details of solar observations: a forest view despite the trees.

2. WHY CHROMOSPHERES EXIST 2.1 STELLAR CONVECTION ZONES The chromosphere story begins with convection. The presence of an outer convective envelope is the main characteristic that distinguishes late-type (cool) stars from their early type (hot) cousins. Convection zones make their appearance at the early F spectral types, for which the effective temperature, Teff, is about 7000K; increase in thickness through the solar spectral types (G2 V: Teff z 5770K), down to the Ktypes (Teff z 4500K); then into the cooler M’s (Teff z 3500K), which become fully convective by mid-M. Convection plays several key roles in defining the physical state of the stellar outer atmosphere (sequentially outward from the stellar surface): photosphere (6500
4000K [at solar T
eff]), chromosphere (5000e8000K), transition zone (TZ)  (104
105K), and corona TT 106 K : 1. Convective turbulence, and collapse of convective granules, together produce acoustic noise, which powers internal seismic waves and leaks into the chromosphere, inducing shocks and other dynamical effects. 2. Convection transforms the solid body rotation of a star into differential zonal flows (the Sun’s equator rotates almost 40% faster than the poles). Differential rotation, in turn, has a central role in the (a-U) dynamo generation of strong

2. Why Chromospheres Exist

magnetic fields in the stellar interior, at the interface between the convective and radiative zones. The internal magnetic flux ropes occasionally become buoyant, rise through the convective envelope, then erupt into the surface layers as active regions, often showing a great deal of complexity. These dynamic magnetic ecosystems cause heating in the chromosphere, and transient events such as flares and coronal mass ejections. 3. Convective turbulence also can directly produce (by the a-a effect) small-scale magnetic fields in the near-surface layers, which can bubble up into the photosphere as ephemeral bipoles and other types of disorganized field. 4. Chaotic flows associated with the overturning convection cells kinematically buffet the photospheric footpoints of small-scale magnetic flux tubes, launching various types of Alfve´n waves and magnetosonic disturbances that can dissipate heat higher up. Organized larger-scale horizontal convective flows can sweep up the small-scale field and collect it into discrete surface patterns called the supergranulation network. There, flux concentrations of opposite polarity are driven against one another by turbulence in the subduction lanes of the largescale pattern. Reconnection heating can occur as a consequence, sometimes ejecting high-speed gas plumes. Given the myriad roles of convection, it is unsurprising that solar-like highenergy activity is confined mainly to late-type stars.

2.2 THE SOLAR CHROMOSPHERE ˚ K resoFig. 2.1 depicts a full-disk filtergram of the Sun taken in the Ca II 3933 A nance line during a period of moderate solar activity. The K line brightens in magnetically disturbed areas where local chromospheric heating is elevated. The image illustrates several of the magnetic-related features mentioned earlier, specifically their chromospheric counterparts. Numerous sunspots, some organized in groups, appear as small, dark, roughly circular patches (dim in visible light; apparently also in chromospheric Ca II emission). Surrounding the dark spots are halos of bright Ca II plage. These are extensive, moderately magnetic areas (z100 G, spatially averaged) initially associated with emerging sunspots, but often persist long after the magnetically intense spots (z1000 G) have decayed. Further from the active regions are additional Ca II bright mottles organized in the lacy pattern of the supergranulation network. At a finer spatial scale, one also would see a peppering of Ca II bright points in the network cell interiors, which represent transient chromospheric excitation mainly by shock waves. ˚ filtergrams have a significantly different As illustrated in Fig. 2.2, Ha 6563 A appearance. This is partly because the formation of the subordinate hydrogen line has a more complex relationship to plasma properties owing to the difficulty of populating the 10 eV lower level of the transition and partly because, as the lightest element, hydrogen is the most affected by thermal Doppler broadening. The Ha feature can be seen in relative absorption or emission, depending on local physical conditions and also at what velocity displacement the line profile is sampled

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FIGURE 2.1 Ca II K-line (3933 A˚) full-disk filtergram of solar chromosphere. Smaller dark areas are sunspot umbrae; surrounding lighter patches are chromospheric plage. Away from the spot groups, the brighter, lacy patterns are part of the supergranulation network. Not visible in this global image are small-scale transient Ca II bright points found in the internetwork regions. From the Big Bear Solar Observatory.

(primarily owing to the Doppler broadening effect). Unlike Ca II images, which are dominated by collections of small-scale bright features near the lower-altitude footpoints of magnetic structures, the Ha pictures are composed of light and dark horizontal striations of various lengths, sometimes organized in radially spoked rosettes, all of which trace the higher altitude extensions of the magnetic field, forming the complex topology of the chromospheric canopy. A common feature of full-disk Ha filtergrams are long, dark, narrow curvilinear features called filaments (prominences when seen in emission at the limb). These are unusually cool (z104K) and dense plasma structures, in the otherwise hot (z106K) tenuous corona, suspended high above the surface inside long-lived magnetic flux ropes. In fact, prominence-like structures are inferred to exist on a few fast-rotating, active stars such as the K dwarf AB Doradus (Collier Cameron et al., 1990) and the G giant FK Comae Berenices (Huenemoerder et al., 1993). The signature is Doppler-shifted components in Ha that are synchronized with the rotation period,

2. Why Chromospheres Exist

FIGURE 2.2 High-resolution Ha (6563 A˚) filtergram of solar chromosphere. Scales are in arcseconds. From the Swedish Solar Observatory.

but whose projected velocities exceed that of the stellar surface by as much as a factor of two, which suggests a physical extension to perhaps a stellar radius above the photosphere. The fast rotation of the host stars makes high-altitude prominence-like structures especially easy to detect spectroscopically. Nevertheless, such coronal condensations probably are common among cool stars.

2.3 STELLAR CHROMOSPHERES Like the solar counterpart, stellar chromospheres are known principally from optical emission lines, especially Ca II H and K and Ha (the latter usually is seen in absorption, although emission is common in more active stars). As shown in Fig. 2.3, the Ca II emission cores sit at the bottoms of broad, deep photospheric absorption fea˚ wavelength interval in stars such tures, which completely dominate the 3900-4000 A as the Sun. In fact, thanks to favorable atomic physics, the Ca II resonance doublet is the strongest feature in the visible solar spectrum, even through calcium is not particularly abundant (compared with iron, for instance). In contrast, the most prominent hydrogen line in the visible, Ha, is much weaker, despite the overwhelming abundance of hydrogen, now owing to unfavorable atomic physics (the 10 eV lower level of Ha is only weakly populated in a cool atmosphere). The chromospheric Ca II emission cores are doubly-reversed (M-shaped). They barely are discernible in low-activity G dwarfs such as the Sun but strengthen in active dwarfs (or solar plage regions). In late-type stars, the Ca II emission features broaden systematically with increasing absolute visual luminosity over a remarkable

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FIGURE 2.3 Ca II H and K lines in the solar spectrum, dominating the violet region. Inset shows blowup of K-line core in quiet Sun (lower, thicker curve) and three plage regions (thinner, higher curves): x-axis is wavelength displacement from line center in A˚. Note that the full widths at half maximum (FWHM) (delimited by vertical dashed lines) are similar; the K1 minimum features become broader with increasing activity (stronger K-line emission) but the K2 peak separations become perhaps slightly narrower. From the National Solar Observatory Fourier transform spectrometer archive.

15 stellar magnitudes. This is called the Wilson-Bappu effect (WBE), named after its codiscoverers (Wilson and Bappu, 1957); it is illustrated in Fig. 2.4.

2.4 WHY ARE CHROMOSPHERES SO THICK? The chromospheric temperature inversion exists because nonradiative heating above the photosphere disrupts the classical radiative equilibrium (RE) stratification. A remarkable characteristic of chromospheres that begs explanation is why they are so thick. Semiempirical solar models (Vernazza et al., 1981) indicated that the layer is about 1500 km in extent, roughly 10 pressure scale heights, three times thicker than the photosphere. Empirical measurements, especially at total eclipses, have proposed even larger extents. As we will see subsequently, this key structural property of the chromosphere can be addressed even without full knowledge of the possible heating mechanisms. How Is the Chromosphere Cooled? The chromosphere is an ideal environment to explore the balance, or lack of balance, between heating and cooling processes. The reason is that although the heating mechanisms might be varied and complex, the cooling is straightforward: creation of photons by collisional interactions and then their escape from the chromosphere. This radiative cooling dominates in the

2. Why Chromospheres Exist

FIGURE 2.4 Left: stacked photographic spectra of 18 representative late-type stars (higher intensities are darker) in the Ca II region. Star 3 has narrow H and K emissions, whereas star 15 has noticeably broader, doubly reversed profiles (see schematic tracing of reversal shapes in inset). Right: stellar WilsoneBappu effect: full width at half maximum intensity (W0, expressed in equivalent velocity units) of the Ca II K-line emission core increases systematically with absolute visual magnitude. Adapted from Wilson, O.C., Bappu, V.M.K., 1957. Astrophys. J. 125, 661.

low-density layers because the gas is optically thin, so local convection is inhibited; while the temperature gradients are relatively mild, so heat conduction is suppressed (except at the steep TZ interface at the top of the chromosphere). A lucky consequence of radiative cooling control is that, in principle, solar observers can record the escaping chromospheric radiation directly, so it is a simple matter conceptually to count all the radiative losses to determine the cooling, and thus also the incident heating. However, given the observational challenges to record often subtle chromospheric emissions across a wide range of species and wavelengths, a more practical way to carry out the radiative cooling inventory is to calculate the individual contributions using a chromospheric model, especially when it is “calibrated” against the more robust chromospheric signatures (Mg II and Ca II). By this approach, Anderson and Athay (1989) estimated chromospheric radiative losses for the quiet Sun of 1.4  107 erg/cm2 s. The authors found that Fe II was responsible for about half the total line cooling (Mg II and Ca II accounted for much of the rest), and that the energy dissipation was nearly constant across the chromospheric temperature plateau, 6000e8000K.

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The chromospheric energy loss is only about 0.02% of the solar bolometric sur2 10 face flux, F 1 bol ¼ 6  10 erg=cm s. The corresponding fractions for the TZ and corona are down by additional orders of magnitude. In this light, the outer atmospheric heating might seem trivial. Nevertheless, one must keep in mind that it is source of the Sun’s high-energy activity that feeds Earth-affecting space weather. Chromospheric Structure. In the early-1960s, Thomas and Athay (1961) argued that the chromospheric temperature rise resulted in part from a shift in the radiative cooling from the negative hydrogen ion, H
(dominant visible and near-infrared [IR] broadband opacity) to hydrogen bound-bound transitions (Lya, Ha) and bound-free continua (Balmer and Paschen), which can be excited at higher temperatures than H
. In the late-1970s, T. Ayres (1979) pointed out that the H
continuum cooling and that of important chromospheric resonance lines all are similar in magnitude and all have the same dependence on the electron density. So, it might be that the electron density itself was the key to understanding the chromospheric temperature inversion. In fact, solar and stellar semiempirical models available at the time almost invariably had remarkably constant electron densities [ne(cm
3)] in the chromospheric layers despite the typical factor of 104 exponential decline of the hydrogen density [protons þ neutrals: nH(cm
3)] over the same height range. The Ionization Valve Effect. A simple analytical framework was proposed by Ayres (1979) to understand the basic foundations of chromospheric structure. The underlying mechanism later was called the ionization valve effect. Here is how it works. Photospheric temperatures of a cool star decrease monotonically outward in RE, eventually stabilizing at a boundary value (T z 0.8 Teff z 4850K for the Sun) in the optically thin layers. At these temperatures, hydrogen is almost completely neutral, but a soft lower boundary on the ionization fraction, ne/nH z 10
4, is maintained by the easily ionized metals (such as iron). The atmosphere can remain in balance at these low temperatures because both the radiative heating and cooling are controlled by mirror-image processes in the same species (H
). However, if there is extra, nonradiative energy deposition (e.g., acoustic shocks, magnetosonic waves, magnetic reconnection, and so forth) in these high layers, the gas cannot rid itself of the additional heat in its baseline low ionization state. The reason is that the radiative cooling (per gram of material) is proportional to the electron density, and thus must follow the precipitous outward hydrostatic decline of the hydrogen density (because at low temperatures, ne z 10
4 nH). Given the outwardly falling radiative cooling, any increased energy input will cause a thermal instability, forcing the temperatures to rise above the RE boundary value, until enhanced ionization boosts the collisionally induced thermal emission. This tipping point defines the initial chromospheric temperature rise. Above that point, increasing ionization can keep up with a more or less heightindependent extra heating over many pressure scale heights, even despite the rapid outward decline of nH, because when hydrogen begins to ionize (T > 5000K), there are lots of electrons available (i.e., ne/nH can increase four orders of magnitude from

2. Why Chromospheres Exist

the metal-dominated limit 10
4 at and below about 4500K; up to w1 near 8000K, where hydrogen is fully stripped). This region of slowly rising temperatures but rapidly increasing ionization is the middle chromosphere. Once the hydrogen is almost completely ionized, however, the gas has run out of new cooling electrons and no longer can compensate for extra heating by slowly increasing the temperatures outward. Rather, a catastrophic thermal instability must ensue, imposing a sharp temperature rise (104 / 106 K) at the top of the thick, nearly isothermal (6000e8000K), middle chromosphere. This second inversion is the chromosphere-corona TZ. Now, let us consider the scenario more quantitatively, piece by piece. Low-Temperature Metal Ionization. Near the top of the photosphere (T < 5000K), hydrogen is essentially neutral (owing to its high 13.6 eV ionization potential) and electrons come mainly from the big-three easily ionized metals: Mg, Si, and Fe, each about 3.5  10
5 by number relative to hydrogen. Between 3000 and 5000K, the ionization fractions of these species are close to unity, so: ne z ðAMg þ ASi þ AFe ÞnH z 1:2  10
4 Ae nH , where the A’s are abundances relative to hydrogen, and Ae is a relative metallicity factor introduced for the general 
stellar case Ae1 h1 . H
Cooling and Heating. In a normal late-type atmosphere, the broadband visible and near-IR opacity (i.e., that which controls the bolometric flow of radiative energy through the photosphere) is dominated by the loosely bound (0.75 eV) negative hydrogen ion, H
. Under such conditions, in the Sun, temperatures fall steadily outward from about 6500K in the deep photosphere ðsH
z 1Þ until they reach a boundary value of about 4850K roughly 500 km above sH
z 1. (The continuum optical depth unity surface could be called the see level, the deepest layers visible to an external observer.) H
cools the gas by associative attachment (analogous to recombination for ions): a free electron interacts with the polarization electric field of a neutral hydrogen atom and can be captured into the single weakly bound state of the quasimolecule. The kinetic energy of the captured electron plus the binding energy, 0.75 eV, is liberated as a photon (with wavelength, l ( 1.6 m). The packet of energy carried away by the photon represents a net drain from the local “thermal pool,” i.e., cooling. Radiative heating mainly is by the complementary process, photodetachment of H
: a sufficiently energetic photon (below the dissociation limit at 1.6 m) ejects the loosely bound electron. The freed hot electron then thermalizes its energy by collisions with other particles in the gas, thereby heating the plasma. In the solar outer photosphere, near the radiative boundary temperature (4850K), the H
heating and cooling are in balance. On either side of the equilibrium temperature, the net heating (or cooling) scales as DT=T relative to the (large) absolute H
cooling. Consequently, the net H
cooling is significantly reduced near the boundary temperature, but on the other hand, the substantial H
heating term is largely canceled.

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Radiative Cooling by Lines. The most significant chromospheric cooling lines ˚ , Ca II H and K at 3950 A ˚ , and a series of strong tranare Mg II h and k at 2800 A ˚ interval. Of secondary sitions of singly ionized iron, mainly in the 2400-2600 A ˚ importance are H I 1215 A Lya (primarily at the top of the chromosphere), Ha, ˚ ). and the Ca II subordinate IR triplet (near 8500 A All of the parent species are the dominant ionization stages of the respective elements under chromospheric conditions. Mg II and Ca II have low-lying excited levels easily populated by collisions and connected to the ground state by pairs of strong near-UV resonance transitions. Mg II is more important than Ca II because magnesium is 15 times more abundant. Fe II has generally weaker resonance transitions than Mg II, but more of them; and like Mg II, is important owing to the large abundance of iron. The resonance line cooling works differently from the H
associative attachment described earlier, but follows the same principle: a collisional interaction creates a photon, which then escapes from the region, carrying away the energy of formation. In this case, the interaction is an inelastic collision between a fast free electron and the ion, which promotes one of the bound electrons into a higher orbital at the expense of most of the kinetic energy of the incident electron. Electrons are the dominant colliding partners, because the light particles have faster thermal speeds than the heavier hydrogen atoms and thus higher collision frequencies. Resonance transitions such as Ca II H and K are characterized by very fast spontaneous radiative decays (Einstein A-values typically of order 108 s
1). Consequently, an initial inelastic collisional excitation usually is followed by a spontaneous decay, producing a photon. If the newly minted photon travels a long distance, the energy it carries away would represent a net drain locally, thus cooling. However, the most common situation at low densities is scattering: a resonance transition initially is photo-excited and then almost immediately decays radiatively. The new photon is essentially identical to that absorbed in all respects (except perhaps the direction of flight). The main way in which a scattered photon is destroyed is when the radiative decay that normally follows the photo-excitation is short-circuited by an inelastic collisional de-excitation. When the collisional-deactivation rates are slow, as at low density, an absorbed photon is less likely to be removed from the radiation field after each photoexcitation, and thus ultimately might escape the atmosphere after a long journey of many scatterings. On the other hand, when deactivation rates are fast, as at high density, an absorbed photon more likely would be thermalized by a downward collision after each photo-excitation. In the high-collision-rate limit, photons are able to travel only a single optical depth, on average, before being destroyed. The multiple scattering process is characterized by a thermalization depth, L, the average number of vertical line-center optical depths a newly created photon can traverse before destruction. When collision rates are high, L z 1. However, when collision rates are low, L could be very large, thousands or many thousands of optical depths for chromospheric resonance lines.

2. Why Chromospheres Exist

Several aspects of the scattering process control L, especially frequency redistributions. This effect causes photons to random-walk not only in physical space but also diffuse through the line profile in frequency. This diffusion is important because photons escape more readily in the transparent line wings than in the opaque line core. Such nuances make the calculation of thermalization depths and associated escape probabilities tricky. Regardless, resonance lines of abundant species are capable of removing collisionally-created photons efficiently from deep in the chromosphere, and thus can be potent radiative coolants. When a species is emitting within a thermalization depth of the surface, we call the transition effectively optically thin, or simply effectively thin. The collision-induced radiative cooling can be expressed as a differential increment to the local energy flux, F , with respect to height. For our purposes, the most convenient height scale is the mass column density m(g/cm2), the mass of gas in a cm2 column above a given altitude. In the solar atmosphere, m decreases two orders of magnitude from 4 g/cm2 at sH
z 1 to 0.03 g/cm2 at sH
z 10
4 (top of the photosphere); then another four orders of magnitude through the chromosphere itcooling  self. In these variables, the line cooling can be written as dF [ dm. For strong resonance lines under chromospheric conditions, the radiative cooling is directly proportional to the upward collisional rate because virtually every excitation results in emission of a photon;  but  with a correction for the fraction able to escape the region:

dF cooling nl Clu [ dm z w r

hc llu

erg=s g; where nl is the population of the

lower level; Clu is the upward collision rate; llu is the transition wavelength (parenthetical term is the photon energy); w is a cooling escape fraction that accounts for the collisionally created photons that actually cool the gas; and r is the matter density (g/cm3). The latter factor converts the expression from physical height, z, to mass column density, m (because dm ¼ r dz). (A useful relation for later: r z 1.4 mH nH for a 10% helium abundance by number; mH is the mass of the hydrogen atom.) If the species is the dominant ionization stage of the element, and most of the population resides in the ground state (normally the case), then: nl z Ael nH, where Ael is the abundance of the element by number relative to hydrogen. In the effectively thin layers, the minimum value of w likely is z½, representing the photons scattering in the outward hemisphere, which ultimately escape the atmosphere. Note that w / 0 in the effectively thick limit, where all the photons are thermalized more or less locally. The collision rate can be written as Clu ¼ ne Ulu ðT=5000Þ
1=2 e
hc=lkT s
1 . 
pffiffiffiffiffiffiffiffiffiffi  Here, Ulu is the collision strength, which can be evaluated as glu ðTÞ gl 5000 , where glu(T) is the factor tabulated in, for example, the Chianti Atomic Database, and gl is the statistical weight of the lower level. For ions, glu(T) has a slow logarithmic dependence on temperature (Burgess and Tully, 1992).

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CHAPTER 2 Stellar and Solar Chromospheres and Attendant Phenomena

Mg II h and k provide a significant fraction of the total resonance line cooling in the solar chromosphere, perhaps as much as 30%, so are a useful bellwether for the total effect. Substituting parameters appropriate for these lines from Chianti into the previous equations, and taking w z 1=2, yields: cooling

dF Mg II dm

h i e e e e erg=s g z 2:2  10
3 An z 6:3  10þ1 ðT=5000Þ
1=2 e
51390=T An ðT ¼ 5000KÞ:

Here, the magnesium abundance was assumed to scale as the stellar metallicity. A similar calculation for Ca II H and K yields a coefficient of 1.5  10
3 (5000K). Important Fe II is more complicated, given its numerous strong UV transitions. An estimate can be obtained by considering the ground state and four lowest excited levels, and the associated resonance transitions to higher levels. Assuming purecollisional population ratios and the Chianti collision strengths yields a numerical factor of 3.6  10
3 (5000K), comparable to Mg II þ Ca II (i.e., Fe II accounts for about half the resonance line total, as concluded by Anderson and Athay (1989) from their numerical modeling). Instabilities and Stabilities. When nonradiative heating is present at the top of the photosphere, there should be a point (call it m*), where the extra heating begins to overwhelm the ability of the gas to cool itself at low temperature. At that tipping point, the temperatures must break upward from the boundary value because of the mismatch between heating and cooling. Beyond the initial inversion, the radiative cooling can tap into the ionization valve to maintain quasistability with only slowly increasing temperatures, independent of the outward decline in the hydrogen density, until all the bound electrons are exhausted. A one-dimensional semiempirical reference model of the solar chromosphere in Fig. 2.5 illustrates some of these basic structural features. In the middle photosphere, where the metal resonance lines are effectively thick and thus poor coolants, the H
net cooling ðwðDT=TÞ ne Þ still can balance a significant nonradiative heat input with only a small DT as long as the hydrogen density is high enough (noting again that ne z 10
4 nH). Thus, there could be, and probably is, substantial nonradiative heating in the dense middle photosphere, which nevertheless might not be obvious because the temperature profile still could be close to that expected in RE (Ayres, 1975). However, as the hydrogen density continues to fall outward, and so too the radiative cooling, there will come a point at which H
no longer can keep up with extra heating at small DT (and low ionization). Generic Nonradiative Heating Rate. Because the radiative cooling is proportional to ne, regardless of whether H
or lines dominate, and the collective cooling must balance the nonradiative input, the constancy of the electron density in

2. Why Chromospheres Exist

FIGURE 2.5 Lower panel: VAL-C’ quiet Sun reference model of Maltby et al. (1986). Thick, warm chromospheric layer, above Tmin region at top of photosphere, is terminated at left by steep rise to million-degree corona, through narrow transition zone. Critical mass column density, m*, described in the text, is located just above Tmin. Height zero is continuum optical depth unity at 5000 A˚ (the “see level”). Dashed curve represents the scattering source function (in equivalent temperatures) for the Ca II K line. The source function breaks downward from the temperature profile above Tmin as Ca II photons begin to escape from the open boundary. mL marked on the upper mass column density scale indicates the thermalization depth mentioned in the text. Different intensity features in the Ca II profile (Fig. 2.4: K1, K2, and K3) are mappings of corresponding features in the source function, as indicated. Upper panel: Hydrogen and electron densities for the model, showing almost constant ne in the chromosphere, as well as the ne z 10
4 nH behavior in the photosphere where hydrogen is neutral and the electrons come from singly ionized metals.

semiempirical models suggests that the extra heating (per gram) is relatively constant with altitude above the base of the chromosphere (here designated m ), as noted by Anderson and Athay (1989) from their model simulations.

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CHAPTER 2 Stellar and Solar Chromospheres and Attendant Phenomena

With this assumption, the heating rate at m* (or indeed anywhere in the chromoheating sphere) can be estimated as dFdm ¼ const ¼ F tot m
1  erg/s g, where F tot is the total extra heat deposited in the chromosphere (z1.4  107 erg/cm2 s for the quiet Sun, as mentioned earlier). We could generalize this expression to other stars if we knew how the total heating varied with surface gravity, effective temperature, and activity level. We can estimate these trends by considering the chromospheric fluxes of different types of stars (noting again that the radiative emissions are proxies for the nonradiative heating). Bolometrically Normalized Fluxes. A useful way to compare chromospheric emission levels of different stars is to divide a cooling line flux, say fMg II, by the bolometric flux (total over all wavelengths) fbol (both quantities in erg/cm2 s at Earth). The stellar bolometric flux can be calculated from a formula based on, say, the Johnson visual V-band, with a bolometric correction that depends mainly on the stellar effective temperature. The fMg II/fbol index tells us what fraction of the total stellar energy budget has been diverted to produce the chromospheric Mg II emission, at least the part that escapes to be detected remotely, so it is something like an efficiency factor (and typically small, zfew  10
5). The fMg II/fbol ratio has the great virtue that it is independent of stellar sizes and distances, and thus is less biased in comparisons of different types of stars, such as dwarfs and giants. The fMg II/fbol ratio is equivalent to LMg II/Lbol (and often written that way). Another useful equivalence is 4 ; where F is the surface flux (erg/cm2 s at the star). F Mg II s Teff Stellar Mg II Emission Trends.We now consider the Mg II emissions of stars to gain insight into how the total chromospheric heating, F tot , might vary with stellar parameters. Mg II is superior to Ca II (or Fe II) for this purpose, because the near-UV Mg II core emissions show larger intensity contrasts than the often barely visible Ca II (or Fe II) reversals. A valuable stellar Mg II inventory was provided by the venerable International Ultraviolet Explorer satellite (1978e96). These measurements can be summarized as follows: 1. At any given spectral type, there is a wide range of LMg II/Lbol ratios extending upward from a soft lower limit, often called the basal flux. 2. The spread of LMg II/Lbol values above the basal limit appears to be controlled by a hidden property of stars, which we might call their activeness (presumably related to elevated magnetic activity). 3. Mg II time series of individual stars show variability at the rotation period and also on longer time scales associated with stellar analogs of the decadal sunspot cycle. Nevertheless, the variability amplitude generally is much smaller than the LMg II/Lbol spread between inactive and active stars. 4. The basal flux boundary, albeit fuzzy, displays decline with decreasing  a modest 4 wT þ2  2 (Linsky and effective temperature, something like F Mg II s Teff eff Ayres, 1978).

2. Why Chromospheres Exist

þ6  2 The item 4 dependence implies F Mg II wTeff along the basal flux trend. In other words, the baseline Mg II chromospheric surface flux, powered by some combination of acoustic shocks and the supergranulation network, falls off strongly with decreasing stellar effective temperature, perhaps even a little faster than expected from a strict proportionality with the bolometric surface flux. Given the empirical Mg II trend, and the absolute chromospheric cooling for the quiet Sun, the total inferred stellar heating can be estimated as: e Teþ6  2 erg=cm2 s; where Teeff is the stellar effective temperaF tot w1:4  107 F eff 1 e is an activeness factor to account for deture in solar units (Teff ¼ 5770K), and F e z 1 for a low-activity star such viations from the basal heating law. For example, F e as the Sun, whereas F z 10 for a solar plage region, or a young, active G dwarf. The (depth-independent) heating rate now can be written as: dF heating z 1:4  107 F e Teþ6  2 m
1 .  eff dm

Thickness of the Chromosphere. In equilibrium, the local cooling must balance the local heating (radiative plus nonradiative) throughout the chromosphere, in particular at the base, where temperatures are low enough that metal-dominated ionization still holds. For the sake of argument, suppose that the radiative cooling at m*, evaluated at a chromospheric base temperature of z5000K, is the Fe II plus Ca II effectively thin result (Mg II is the most likely of the three to be effectively thick). This acknowledges that the H
cooling essentially cancels the photospheric radiative heating at that temperature, so we can safely ignore both contributions in dF cooling the energy equation. The Fe II þ Ca II cooling rate is tot ðm Þ z 5:1  10
3 Ae n dm

e

erg/s g (5000K).The leading numerical factor shows only a mild increase over the range 5000e8000K, so the metal cooling per gram depends mainly on the electron density. At low temperatures, as described earlier, the electron density is proportional to the total hydrogen density, nH. The latter, in turn, is related to the mass column density, m, through the hydrostatic equilibrium condition: g m ¼ 1.1 nH k T, where g is the stellar surface gravity (g1 ¼ 2:74  104 cm=s2 ) and k is Boltzmann’s constant. The left-hand side represents the weight of the mass column and the right-hand side is the gas pressure at the base of the column. The latter invokes the perfect gas law, a 10% helium abundance by number, and negligible ionization. Solving for the hydrogen density, introducing the result into the metal-ionization, then substituting cooling 2 tot for ne in the cooling function yields: dFdm ðm Þ z 2:2  1010 Ae e gm erg=s g;  where e g ¼ g g1 accounts for nonsolar surface gravities. Here, the metallicity factor enters squared: one power for the line cooling and the other for the lowtemperature metal ionization. Equating the heating and cooling expressions at the
1 þ12
1 þ3  1 e ge 2 Te tipping point yields: m z 0:025Ae F g=cm2 . Note that the chromoeff spheric thickness goes as the inverse square-root of the key structural parameter, surface gravity. In other words, a lower-gravity atmosphere, with its lower densities,

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CHAPTER 2 Stellar and Solar Chromospheres and Attendant Phenomena

must push the chromospheric inversion inward in mass column density to reach a depth where there are enough electrons to fuel the metal-ionization dominated cooling. Together with the other dependencies, the relation says that chromospheres become thicker in lower gravity stars (e.g., giants), metal-poor objects, more active stars, and with increasing effective temperature. In the semiempirical quiet Sun model VAL-C0 of Maltby et al. (1986), the column mass density at the base and initial rise of the chromosphere is about 0.01e0.05 g/cm2 (for the height range 710
530 km [5000
4400K]). The modelized location of the solar inversion agrees reasonably well with the numerical coefficient in the m scaling law, which is encouraging. Mean Electron Density of the Chromosphere. Substituting the m relation into the expression for nH, then that result into the low-temperature metal ionization (again for T z 5000K) yields a second scaling law: þ12 þ1 þ3 1  11
3 e ge 2 Te ne w1:1  10 F cm . Note that the metallicity factor has canceled eff out. This relation says that a thin chromosphere (small m*, typically high g) must have a larger mean electron density to radiate away the same total extra heat input as a thick chromosphere (large m , typically low g). In addition, the mean electron density rises with increasing activeness. However, both ne and m* depend on the activeness factor in the same way: a more active atmosphere is thicker and has a higher electron density. Curiously, m depends strongly on metallicity (inversely), whereas ne does not at all, at least when metal ions dominate the radiative cooling. This says that a metalpoor chromosphere will be thicker but will have the same mean electron density compared with a solar-abundance chromosphere of the same gravity and activity. The numerical coefficient compares well with the ne z 1.0
1.7  1011 cm
3 found above the Tmin in the VAL-C0 model (1.1  1011 at 5000K). Also, the model C0 electron density is relatively constant in the chromospheric layers, whereas the hydrogen density falls outward about four orders of magnitude (see Fig. 2.5). The near-constancy of ne over the same height range where nH is dropping precipitously supports the basic premise of the ionization valve effect: It is all about the electrons. A semiempirical chromospheric model for the red giant star Aldebaran (a Tauri: K5 III; Teff ¼ 3920K, log g ¼ 1.25, solar metallicity), published by McMurry (1999), has log m ¼
0.5 g/cm2 and a chromospheric electron density of 1
10  108 cm
3, with an average of about 8  108 cm
3 in the hotter layers (6000e8000K). Over the same range in mass column, the hydrogen density drops the familiar four orders of magnitude. For these stellar parameters, the scaling laws predict log m ¼
0.5 g/cm2 and ne ¼ 9  108 cm
3 ; again, not bad agreement. An older model chromosphere for the metal-deficient red giant Arcturus (a Boo¨tis; K1 III; Teff z 4250K; log g z 1:7; log Ae z
0:5), published by Ayres and Linsky (1975), has log m z þ 0.25 and an electron density of 0.5
1.7  109 (5000e8000K). For the given stellar parameters, the scaling laws predict: log m z
0.15 and ne ¼ 1:9  109 . Compared with the roughly similar

2. Why Chromospheres Exist

red giant a Tau, aside from metallicity, the a Boo chromosphere is thicker but has about the same electron density. Again, the agreement with the scaling laws is encouraging.

2.5 THE WILSONeBAPPU EFFECT An indirect way to test the chromospheric scaling laws is to assess their impact on a b g d e ge Te ; where W0 is the WBE mentioned earlier. Observations suggest W0 wAe F eff the FWHM of the Ca II K (or Mg II k) emission core. Consensus empirical values for the power law indices are: a z 0, b z 0,
0.23  g 
0.20, and 1.3  d  1.7 (Linsky, 1999). The emission width thus displays, remarkably, almost no sensitivity to metallicity or activeness. The dependence on effective temperature is mild, noting that Teff varies only a factor of two from cool M-types to the warm F-types. The main sensitivity is the inverse dependence on surface gravity, because g decreases three or four orders of magnitude from dwarfs to supergiants. Despite the weak sensitivity of W0 to activeness, Ca II (and Mg II) profiles of solar plage and active G dwarfs reveal that e , whereas, counterintuthe base of the emission profile broadens with increasing F itively, the separation of the peaks at the top of the profile seems to narrow, leaving the FWHM more or less unchanged (Ayres, 1979). Now, we make a crucial assumption: the edges of the emission core just inside the K1 minimum features are controlled by the Lorentzian wings of the line profile. In the wings, the opacity depends quadratically on the wavelength shift, Dl, from line center: f(Dl) w Dl
2. In the central Doppler core, on the other hand, the 2 2 dependence is exponential, fðDlÞwe
Dl =DlD (here, Dl is the Doppler width, D

related to local thermal and turbulent velocities). The Lorentzian assumption is contrary to early attempts to understand the WBE, which assumed that the emission features form in the Doppler core; consequently, the width would be dictated by chromospheric velocity fields (e.g., Hoyle and Wilson, 1958). This is a reasonable expectation in principle, but in practice the implied velocities (already near-sonic in the Sun) would quickly become supersonic in giants and supergiants (e.g., Fig. 2.4, noting that the chromospheric sound speed is only about 7 km/s). Such extreme dynamics would be difficult to sustain in the face of strong dissipation by shocks. In that respect, Lorentzian control of the outer profile seems physically more appealing. To proceed further, we can apply one of the EddingtoneBarbier relations, based on analytical solutions of the radiation transport equation. It says that at a given wavelength in the line profile, f(Dl), one can “see” down to the atmospheric depths (and their associated thermal emission) corresponding to s z 1 at that wavelength. Thus, as one tunes through the line profile from the transparent far wings inward to the opaque core, the optical depth unity surface sweeps upward through the atmosphere, encountering progressively lower temperature layers and thus lower intensities, ultimately passing through the chromospheric temperature inversion, which then maps onto the profile as the initial K1 / K2 emission rise.

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CHAPTER 2 Stellar and Solar Chromospheres and Attendant Phenomena

þ1

þ1

Making these assumptions, it is easy to show that DlK1 wACa2 m 2 . In other words, as the chromosphere thickens, the base of the emission reversal moves further from line center. The specific square root relationship works only for species, such as Ca II and Mg II, that are the dominant ionization stages at chromospheric temperatures. Then, the column density of (ground-state) absorbers is directly proportional to the column density of hydrogen, NH(cm
2), and thus also the mass column (m z 1.4 mHNH). 3 1 e þ14 ge
14 Teþ2  2 ; assuming Substituting the scaling law for m yields: DlK1 wF eff e which then cancels that factor that the calcium abundance follows the metallicity, A; in m . This relation shows a similar mild inverse dependence on surface gravity as the consensus index for W0, a weak sensitivity to activeness, no dependence on metallicity, and a positive moderate dependence on effective temperature. A similar scaling relation can be worked out for the separation of the K2 emission peaks. The formulation is more complicated because the scattering thermalization depth, L, mentioned earlier, comes into play. The key is that the latter tends to scale inversely as the electron density, Lwn
1 e . This has a significant impact on the DlK2

e
4 ge
14 Te
2 2 xþ12 ; where the new quantity x is a relation, which turns out to be F eff typical Doppler velocity in the upper chromosphere. Note that DlK2 has the same dependence on surface gravity as DlK1 and likewise none on metallicity, but opposite sensitivities to the other two parameters, especially activeness. This says that DlK2 also widens with decreasing surface gravity, like DlK1 , but narrows with increasing activity, contrary to DlK1 . This matches the counterintuitive behavior seen in plage profiles of Ca II. Because the WBE width falls roughly midway between DlK1 and DlK2 , the e and Teeff suggests that W0 will be less sensitive contrary behavior with respect to F to these parameters, with exponents intermediate, perhaps even close to zero (as is e empirically). Also, because DlK and DlK display no sensitivity to the case for F 1 2 metallicity, it stands to reason that the WBE width would not either, as observed. In short, despite the enormous simplifications of the analytical arguments, the remarkable agreement of the scaling law predictions with the general properties of the WBE supports the idea that chromospheres owe their peculiar properties, especially their great thickness, mainly to the ionization valve effect rather than, say, to the specific heating mechanism(s) or the particular cooling opacities. It used to be thought that the WBE was caused by a dramatic increase in atmospheric turbulence, going from dwarfs to giants, which would greatly enhance the Doppler broadening of the Ca II cores in the luminous stars. Now, we have seen an alternative view in which the Ca II width is controlled by the thickness of the chromosphere, which in turn adjusts itself to the impact of nonradiative heating according to a pressure-dependent instability in the low-temperature cooling. In this sense, the WBE is a barometer, not a tachometer. 1

3

1

3. The RotationeAgeeActivity Connection

3. THE ROTATIONeAGEeACTIVITY CONNECTION 3.1 BACKGROUND About 50 years ago (a year before Goldberg’s pivotal lecture on the solarestellar connection), A. Skumanich, a noted solar physicist at the High Altitude Observatory in Boulder, Colorado, published a letter in the Astrophysical Journal announcing the discovery of a rotationeageeactivity connection among latetype dwarf stars of solar type. This remarkable note, a scant two-plus pages in length, became one of the most cited and celebrated in the cool-star literature. The premise was simple: Rotational velocities of sunlike stars in the few (at the time) well-studied galactic clusters of known age, and the Sun itself, were seen to decline as the square root of time on the main sequence. This correlation famously became known as the Skumanich law. In parallel, there was a fading of stellar chromospheric activity, as measured by the Ca II HK emissions, as well as a decrease in the lithium abundance, a secondary marker for stellar age. (The primordial light element is slowly burned by low-energy nuclear reactions as material in the stellar envelope is circulated through the hot base of the convective zone, creating a kind of chemonuclear clock.) All of this was contained in a single diagram, reproduced in Fig. 2.6. The close connection between stellar rotation, age, and activity is fundamental. Cool stars such as the Sun normally begin their lives rotating rapidly, thanks to the swirling Keplerian accretion disks in which they are born. These new stars display intense magnetic effects owing to internal dynamo action, driven by the fast spin. The magnetic activity in turn inspires strong coronal ˚ ) emissions of multiply soft x-rays (106
107 K); far-UV (FUV) (1000e1700 A 5 ionized species such as C IV (few  10 K), and very bright H I Lya (2  104 K), but also extending down to chromospheric temperatures ( 20 d), showing a shallower slope of Pcyc versus Prot. Some stars displayed double periods and thus could sit on both branches. A schematic rendition of the Bo¨hm-Vitense (2007) version of the diagram is illustrated in Fig. 2.11. The remarkable aspect is that the cycle and spin timescales differ by roughly two orders of magnitude, yet enough of a connection exists to impose some apparent order. As noted by Bo¨hm-Vitense, the number of rotational cycles per starspot period appears to be constant on the two branches, but it differs by a factor of about six between them. The Sun sits in the middle of the diagram, not especially close to either branch. This raised the possibility that the Sun is unique in some important respect that would cause it to occupy its isolated position. The two circled points refer to the a Cen stars (based on their apparent X-ray cycles). Curiously, coronally weak a Cen A falls on the active branch whereas its x-ray brighter companion sits squarely in the middle of the inactive branch (although in a region populated mainly by later-type dwarfs, so perhaps it is unsurprising that a Cen B is found there).

FIGURE 2.11 Spot cycle duration (Pcyc) versus rotational period (Prot). Circled A and B symbols refer to Alpha Centauri AB. Shaded areas are the active (left) and inactive (right) branches from the original publication. Symbols are values aggregated from several additional sources. Adapted from Bo¨hm-Vitense, E., 2007. Astrophys. J. 657, 486.

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CHAPTER 2 Stellar and Solar Chromospheres and Attendant Phenomena

A more recent reevaluation of the older measurements, including newer observations and additional objects, by R. Egeland in his doctoral thesis (Egeland 2017) suggests that the previously coherent branches dissolve into fuzzier blobs of points, now making the Sun less of an outlier. The filling-in of the diagram is disappointing in the sense that the previous structure offered a possible avenue to a theoretical description. In fact, the superficial lack of correlation between cycle duration and rotation period for many of the stars seems to suggest that there is a hidden parameter yet to be identified that might unify the behavior across the range of stellar types, something like what the Rossby description did to collapse the color dependence of the broader rotationeactivity connection. Alternatively, the dynamo might be a low-Q oscillator such that a coherent period can be maintained only for a few dozen cycles, after which a new modulation engages, longer or shorter than before. In that situation, our view of the solar cycle could be biased by the limited historical time frame, and thus our perception that it is an entirely regular phenomenon might be unfounded. Unfortunately, the study of long-term stellar activity cycles benefits little from better instrumentation, or a larger telescope, but rather time itself is the adversary. Thankfully, Olin Wilson showed keen insight in starting the Mount Wilson HK program a half-century ago. Although crude by modern standards, the historical measurements nevertheless were scalable to future monitoring efforts using the flux ratio strategy he devised, achieving the long-term continuity that is at the heart of the hunt for stellar cycles. It is sobering to realize that alternative programs, exploiting different observational techniques or following vastly expanded stellar samples, started today would bear fruit only decades from now.

REFERENCES Anderson, L.S., Athay, R.G., 1989. Astrophys. J. 346, 1010. Ayres, T.R., 1975. Astrophys. J. 201, 799. Ayres, T.R., 1979. Astrophys. J. 228, 509. Ayres, T.R., 1997. JGR (Planets) 102, 1641. Ayres, T.R., Linsky, J.L., 1975. Astrophys. J. 200, 660. Baliunas, S.L., Donahue, R.A., Soon, W.H., et al., 1995. Astrophys. J. 438, 269. Barnes, S.A., 2003. Astrophys. J. 586, 464. Bo¨hm-Vitense, E., 2007. Astrophys. J. 657, 486. Burgess, A., Tully, J.A., 1992. Astron. Astrophys 254, 436. Chianti Atomic Database. http://www.chiantidatabase.org. Collier Cameron, A., Duncan, D.K., Ehrenfreund, P., et al., 1990. MNRAS 247, 415. Duncan, D.K., Vaughan, A.H., Wilson, O.C., et al., 1991. Astrophys. J. Suppl. 76, 383. Egeland, R., 2017. (Ph.D. thesis). Montana State University, Bozeman, Montana, USA. Egeland, R., Soon, W., Baliunas, S., et al., 2017. Astrophys. J. 835, 25. FISM database. http://www.lasp.colorado.edu/lisird/data/fism/. Frazier, E.N., 1970. Sol. Phys. 14, 89. Hoyle, F., Wilson, O.C., 1958. Astrophys. J. 128, 604.

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Huenemoerder, D.P., Ramsey, L.W., Buzasi, D.L., et al., 1993. Astrophys. J. 404, 316. Linsky, J.L., 1999. Astrophys. J. 525, 776. Linsky, J.L., Ayres, T.R., 1978. Astrophys. J. 220, 619. Lockwood, G.W., Skiff, B.A., Henry, G.W., et al., 2007. Astrophys. J. Suppl. 171, 260. Maltby, P., Avrett, E.H., Carlsson, M., et al., 1986. Astrophys. J. 306, 284. Matt, S.P., MacGregor, K.B., Pinsonneault, M.H., et al., 2012. Astrophys. J. 754, L26. McMurry, A.D., 1999. MNRAS 302, 37. Noyes, R.W., Hartmann, L.W., Baliunas, S.L., et al., 1984. Astrophys. J. 279, 763. Parker, E.N., 1966. Astrophys. J. 143, 32. Radick, R.R., Lockwood, G.W., Skiff, B.A., et al., 1998. Astrophys. J. 118, 239. Rebull, L.M., Stauffer, J.R., Hillenbrand, L.A., et al., 2017. Astrophys. J. 839, 92. Saar, S.H., Brandenburg, A., 1999. Astrophys. J. 524, 295. Schatzman, E., 1962. Ann. Astrophys. 25, 18. Skumanich, A., 1972. Astrophys. J. 171, 565. Skumanich, A., Smyth, C., Frazier, E.N., 1975. Astrophys. J. 200, 747. Stauffer, J.R., Hartmann, L.W., 1987. Astrophys. J. 318, 337. Stauffer, J., Rebull, L., Bouvier, J., et al., 2016. Astron. J. 152, 115. Thomas, R.N., Athay, R.G., 1961. Physics of the Solar Chromosphere. Interscience, New York (Chapter 5). Vernazza, J.E., Avrett, E.H., Loeser, R., 1981. Astrophys. J. Suppl. 45, 635. Vilhu, O., Rucinski, S.M., 1983. Astron. Astrophys. 127, 5. Weber, E.J., Davis Jr., L., 1967. Astrophys. J. 148, 217. Wilson, O.C., Bappu, V.M.K., 1957. Astrophys. J. 125, 661. Wilson, O., Vaughan, A., Kraft, R., et al., 1981. Sky Telesc. 62, 312. Wood, B.E., Linsky, J.L., 1998. Astrophys. J. 492, 788. Wood, B.E., Mu¨ller, H.-R., Zank, G.P., et al., 2002. Astrophys. J. 574, 412.

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3

Alexander I. Shapiro1, Hardi Peter1, Sami K. Solanki1, 2 Max-Planck-Institut fu¨r Sonnensystemforschung, Go¨ttingen, Germany1; School of Space Research, Kyung Hee University, Yongin, Korea2

CHAPTER OUTLINE 1. 2. 3. 4.

Introduction ......................................................................................................... 59 Observations of the Solar Atmosphere ................................................................... 60 The Solar Spectrum.............................................................................................. 61 Physics of the Photosphere/Chromosphere............................................................. 65 4.1 One-Dimensional Models ....................................................................... 65 4.2 Three-Dimensional Models ..................................................................... 70 5. Physics of the Chromosphere/Corona..................................................................... 73 5.1 Coronal Emission and Magnetic Structure ............................................... 74 5.2 Basic Considerations of the Energetics.................................................... 76 5.3 Heating Processes and Modern Models ................................................... 77 5.4 Connection to the Low Atmosphere ......................................................... 78 Acknowledgments ..................................................................................................... 79 References ............................................................................................................... 79

1. INTRODUCTION The solar atmosphere is usually defined as the outer, directly observable part of the Sun (in contrast to the deeper solar layers, which can be probed only indirectly with the help of helioseismology (see Chapter 4). In other words, the solar atmosphere gives birth to photons that leave the Sun in the form of sunlight. The solar atmosphere has a sharp lower edge in which the continuum radiation in the green part of the spectrum is emitted. Sometimes it is more precisely defined as the location at which optical depth unity is reached at 500 nm (s500 ¼ 1). If photon escape is used as the definition, however, a deeper level should be used, such that almost no photons emitted below it make it into space. An optical depth of 10 is such a value. The two are separated by roughly 100 km on average, a tiny distance compared with the solar radius of roughly 7  105 km (which for the earth-based observer corresponds to the visible angular radius of about 1000 arcsec). By chance, the solar surface roughly coincides with the depth at which hydrogen ionization The Sun as a Guide to Stellar Physics. https://doi.org/10.1016/B978-0-12-814334-6.00003-0 Copyright © 2019 Elsevier Inc. All rights reserved.

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starts to set in (as the temperature rapidly rises with depth). The upper edge of the solar atmosphere is much more ambiguous to define. It extends out to a distance of several million kilometers, where it smoothly transitions into the solar wind, which is itself sometimes considered to be part of the solar atmosphere. The solar atmosphere is traditionally divided into four layers, although considering these to be layers is now known to be only a very rough approximation. The deepest and coolest of these layers (with a temperature between approximately 6000e7000K at s500 ¼ 1 and 4000e5000K at the top) is the solar photosphere, which emits most of its radiation as visible wavelengths. The photosphere is close to being in hydrostatical equilibrium with a pressure scale height of around 100 km. Consequently, solar matter quickly becomes opaque and the total vertical extension of the photosphere is just about 500 km (i.e., less than 0.1% of solar radius). Above the photosphere lies the chromosphere, a layer with an average thickness of about 1500 km, where the temperature gradually rises to 10,000K in a spatially averaged sense. The chromosphere can be observed in the far UV, at millimeter wavelengths, and in the cores of the strong spectral lines at wavelengths in between. Because the temperature in the chromosphere is within a factor of 2 to that in the photosphere, the scale height of chromospheric gas is roughly the same as that of photospheric gas. In a simple one-dimensional (1D) representation, the temperature above the chromosphere first jumps to more than 1 million K just within a 100-km-thick layer called the transition region and then flattens to form the solar corona. The transition region and corona can be observed in UV and at radio wavelengths of around 1 cm and longer X-rays. We note that such a 1D representation of the solar atmosphere is a strong simplification: the solar atmosphere is strongly structured by various processes, with convection, magnetism, oscillations, and waves being the major structuring agents. Furthermore, the solar atmosphere is highly dynamic, so that at any given point in space, the properties of the atmosphere can change strongly with time.

2. OBSERVATIONS OF THE SOLAR ATMOSPHERE In contrast to stellar physics, solar physics has been largely driven by the ability to resolve the features on the solar disk. In particular, high-resolution solar observations have made it possible to study the emergence and disappearance of solar magnetic fields directly as well as their sophisticated interaction with solar plasma. Consequently, such observations serve as a test bench for simulations of solar and stellar atmospheres. High-resolution solar observations showed that in addition to a strong stratification with depth, the solar atmosphere has a rich and complex horizontal structure. The photospheric convective flows lead to the formation of granules, which are cellular features with a mean size of about 1000 km and a lifetime of about

3. The Solar Spectrum

10e20 min. Granules cover nearly the entire solar surface (with the exception of spots), so that at every moment there are several millions of granules on the Sun. Further inhomogeneities on the solar surface are caused by the magnetic field. Large concentrations of the magnetic field form dark sunspots, which consist of two parts: central umbral regions, which are the darkest parts of the sunspots with a predominantly vertical magnetic field, and surrounding penumbral regions, which are much lighter areas with an inclined magnetic field. The ensembles of smaller magnetic elements form bright network and faculae, which are most easily seen near the solar limb. Over the last decade significant advances have been made with both groundbased (e.g., the detection of convective downflows in a sunspot penumbra with the Swedish 1-m Solar Telescope (Scharmer et al., 2011; Joshi et al., 2011a,b) and space-based (e.g., inversion of data from the Solar Optical Telescope aboard the Hinode satellite, which showed that solar internetwork consists of very inclined hG fields (Orozco Sua´rez et al., 2007)) solar observations. Another interesting strategy that allows observations with a large telescope in near-space conditions is to use balloon-borne solar observatories. Two flights of the SUNRISE (Solanki et al., 2010; Solanki et al., 2017), which is a balloon-borne solar Gregory telescope with a 1-m aperture, observing the Sun at a resolution of 50e100 km, resulted in a number of important discoveries. In particular, SUNRISE observations finally made it possible to resolve small-scale magnetic flux concentration in the quiet Sun (Lagg et al., 2010) and to study the migration and dispersal of such concentrations in intergranular lanes (Jafarzadeh et al., 2017). Fig. 3.1 presents an overview of the solar atmosphere. Plotted are images showing the same field-of-view that cover an active region, sampling different temperature regimes corresponding to the photosphere, chromosphere, and corona. One interesting detail is that although a strong magnetic field (associated with the active region) is present over a large area (upper middle panel), only a small part of this field leads to the formation of sunspots (upper left panel). At the same time, the structure of the faculae/plage (in which the plage is the chromospheric counterpart of faculae) is similar to that of the magnetic field (compare the upper right panel with the upper middle panel). The lower left panel shows the transition region whereas the lower central and right panels sample the corona at different temperatures. One can clearly see the bright loops of gas in the corona connecting opposite magnetic polarities. These loops are thought to outline magnetic field lines.

3. THE SOLAR SPECTRUM In this chapter we will focus on the disc-integrated solar spectrum. It is particularly important when considering the Sun as a star, because then no horizontal spatial information can be gleaned and only the flux as a function of wavelength or/and time is available. Interestingly, the distribution of energy in the solar spectrum is close to the spectral sensitivity of the human eye. Consequently, if not for the earth’s

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FIGURE 3.1 Overview of the solar atmosphere. The images show a 500  500-arcsec2 part of the Sun (recall that the visible solar radius is about 1000 arcsec) with active region AR12139 recorded on Aug. 17, 2014 around 14:49 UT. Whereas the dark sunspots in photospheric intensity cover only a small fraction of the field of view, the magnetogram shows enhanced magnetic flux in a much larger area (black and white show opposite polarities). The intensity in the low chromosphere as seen in 170 nm pretty much correlates with the strong magnetic field, whereas the upper layers at hotter temperatures show the transition region and coronal structures connecting opposite polarities. For the latter panels, the main ions contributing to the wavelength channels are displayed along with the rough formation temperature. Images are courtesy of the National Aeronautics and Space Administration/Solar Dynamics Observatory and the Atmospheric Imaging Assembly (AIA) and Helioseismic and Magnetic Imager (HMI) science teams.

atmosphere, the Sun would appear white to the human observer. However, because short-wavelength photons are better scattered in the atmosphere, the Sun appears to be yellow whereas the sky is blue (however, the true color of the Sun can still be glimpsed by looking at the white snow and clouds, which mix backscattered and transmitted solar photons). In addition to the continuous spectrum (first spectrally analyzed by Isaac Newton), visible sunlight contains millions of absorption lines, first noticed by William Wollaston and studied in detail by Joseph Fraunhofer and later called Fraunhofer lines. A significant milestone in solar spectroscopy was reached with the introduction of the Fourier Transform spectrometer (Brault, 1985) at the McMath-Pierce Solar Facility at the National Solar Observatory on Kitt Peak in the early 1980s (see Doerr

3. The Solar Spectrum

et al., 2016, for a historical overview of available solar spectra). Since then, several versions of the Kitt Peak spectral atlases have been released (Kurucz, 1984; Kurucz, 2005a; Wallace et al., 2011). Fig. 3.2 presents the first version of the Kitt Peak Solar Flux Atlas by Kurucz et al. (Kurucz, 1984). The large number of absorption lines is striking; it is so

FIGURE 3.2 Kitt peak solar flux atlas (Kurucz, 1984) Upper panel: an artificially created image representation. Each of the 50 slices covers 6 nm; together, they span the complete spectrum across the visual range from 400 to 700 nm. The wavelength increases from left to right along each strip, and from top to bottom. Lower panel: a graphical representation of the same atlas between 300 and 800 nm. The flux values at every wavelengths are normalized to the local continuum value. Source: The image in the upper panel as well as the caption are adapted from https://www.noao.edu/image_ gallery/html/im0600.html. Credits: N.A. Sharp, National Optical Astronomy Observatory/National Solar Observatory/Kitt Peak Fourier Transform spectrometer/Association of Universities for Research in Astronomy/National Science Foundation. The figure in the lower panel is taken from http://kurucz.harvard.edu/sun/fluxatlas2005/.

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huge that whereas current linelists used for modeling and interpreting solar spectra contain more than 100 million atomic and molecular lines, many if not most of the lines are so strongly blended that they cannot be identified (Kurucz, 2005b). Another important detail that makes solar spectra look different from spectra of hotter stars is the absence of the Balmer jump around 364.6 nm. It is practically invisible in the solar spectrum because the dominant source of the continuum opacity in the solar photosphere is the negative hydrogen ion. A small continuum jump introduced by neutral hydrogen (see, e.g., Fig. 1 from Shapiro et al., 2015) is completely masked by the Fraunhofer lines. The role of neutral hydrogen in the continuum opacity increases with the effective temperature and the Balmer jump becomes one of the most dominant features in the spectra of A stars. Interestingly, because the opacity of negative hydrogen depends only weakly on the wavelength, the solar spectrum is much more similar to the Planck’s law than spectra of hotter stars. The main deviations of the solar spectrum from the Planck’s law in the visible spectral domain are caused by the Fraunhofer lines. Fig. 3.2 identifies the main physical mechanisms responsible for the creation of the photons forming sunlight. In the UV these are scattering and thermal emissions of photons in spectral lines. The number of lines decreases toward longer wavelengths where continuum photons also increasingly escape the atmosphere. Consequently, starting from approximately 500e550 nm, most of the photons come from the continuum and are produced during the recombination of neutral hydrogen with electrons to form the negative hydrogen ion. Whereas the main source of line opacity in the solar atmosphere is atomic lines, the solar spectrum also contains several strong molecular features. The most prominent among them are the CN violet system at about 380 nm and the CH G-band at about 430 nm. Interestingly, approximately a quarter of solar brightness variability on the timescale of the 11-year magnetic activity cycle comes from the molecular lines (Shapiro et al., 2015). The contribution of molecular lines to the spectrum increases for stars cooler than the Sun, and in the spectra of sunspots and pores, comparatively dark and hence cool concentrations of magnetic flux. With a decrease in the effective temperature, first lines of carbon-based diatomic molecules become more pronounced and then (for temperatures below 4500K, i.e., in M stars and in sunspot umbrae, i.e., the darkest parts of sunspots) the main contribution to line opacity shifts to molecules composed of a-elements, in particular TiO and MgH (see Fig. 3 from de Laverny et al., 2012). The vast number of lines in the spectra of stars with near-solar temperature and cooler is an asset for detecting planets with the radial velocity (RV) method (which currently accounts for roughly half of all discovered exoplanets (http://exoplanets.eu). The number of spectral lines decreases with the effective temperature and the lines also become broader owing to the faster rotation of these stars (because they are younger than the Sun and consequently did not undergo such strong magnetic braking). Consequently, it becomes difficult to determine the Doppler shift and most of the RV exoplanets were discovered around stars later than spectral type F6 (Hatzes, 2016).

4. Physics of the Photosphere/Chromosphere

Interestingly, spectral lines below 180 nm are observed in emission whereas most of the lines above this threshold are observed in absorption. This happens because of the temperature inversion in the solar atmosphere: the temperature first decreases with height in the photosphere, reaches a minimum of about 4000e5000K, and then starts to increase in the chromosphere. The pseudocontinuum at about 180 nm is formed around the temperature minimum so that lines below 180 nm are chromospheric (and seen in emission) whereas lines above 180 nm are photospheric (and seen in absorption). Interestingly, there is also a second spectral window to observe the atmospheric layers around the temperature minimum: the opacity in the far infrared is dominated by the negative hydrogen free-free absorption, which rapidly increases with wavelength (as l2). Consequently, the chromosphere becomes optically thick at about 150 mm and similar to the UV case, this threshold separates photospheric and chromospheric photons.

4. PHYSICS OF THE PHOTOSPHERE/CHROMOSPHERE 4.1 ONE-DIMENSIONAL MODELS We start with a brief overview of simplified 1D models that describe the solar atmosphere as a plane-parallel structure. Although they are obvious simplifications, these models had an important role in understating some of the physical processes in the solar atmosphere. The 1D models can be divided into two main classes. The first class encompasses models with a temperature structure calculated assuming radiative equilibrium (hereafter RE models), i.e., assuming that the only source of energy transport is radiation. Because this is not correct close to the solar surface, they are often corrected for the transport of mechanical energy by convection, which is parameterized through the mixing length theory (Bo¨hm-Vitense, 1958) or overshooting approximation (Castelli et al., 1997). RE models can be calculated for stars with arbitrary fundamental parameters (i.e., effective temperature, metallicity, and surface gravity), and although they are gradually superseded by 3D magnetohydrodynamic approximation (MHD) models (Tremblay et al., 2013; Beeck et al., 2013), they are still routinely used in stellar modeling. The most prominent examples of RE models are ATLAS9 and ATLAS12 (Kurucz, 2005c), MARCS (Gustafsson et al., 2008), and PHOENIX (Husser et al., 2013). A strong drawback of the RE models is that they cannot describe the temperature rise in the chromosphere as well as structures of the bright magnetic features (such as network and faculae). In the solar case, such a drawback is overcome by the second class of model, namely the semiempirical models (hereafter, SE models). The atmospheric temperature structure in these models is empirically determined from the observed solar spectrum and its center-to-limb variation. Although the SE models do not directly rely on the assumption of RE, their photospheric

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stratification, in particular in the quiet Sun (as well as in sunspots) often appears to be close to RE (Fontenla et al., 1999; Rutten, 2002). Most of the SE models still used nowadays (Avrett, 1985; Fontenla et al., 1993; Fontenla et al., 2015 and references therein) stem from the work by Vernazza, Avrett, and Loeser (VAL) (Vernazza et al., 1981). They used a set of observations to create a series of models for several brightness components of the quiet Sun. For example, they used spectra of the continuum in the 135- to 168-nm range (where the transition from emission to absorption lines happens) (see Section 3) to constrain the properties of atmospheric layers around the temperature minimum region. The temperature stratifications of VAL models are plotted in Fig. 3.3. Six models that differ quantitatively from each other but are qualitatively similar are meant to represent regions with different amounts of magnetic flux, from the dark parts of

FIGURE 3.3 Temperature as a function of height for a Vernazza, Avrett, and Loeser (Vernazza et al., 1981, Avrett, 1985) set of semiempirical solar model atmospheres: a dark point within a supergranule cell interior (A), the average supergranule cell interior (B), the average quiet Sun (C), the average network (D), a bright network element (E), a very bright network elements (F). From Vernazza, J.E., Avrett, E. H., Loeser, R., 1981. ApJS 45 635. http://adsabs.harvard.edu/abs/ 1981ApJS...45..635V.

4. Physics of the Photosphere/Chromosphere

the interiors of supergranule cells in the quiet Sun to the brightest network elements (even brighter plage regions are found within active regions). Basically, these regions differ in the degrees of their coverage by small-scale magnetic flux tubes (which are a widely used description of the magnetic flux concentrations on the solar surface) (see, e.g., Solanki et al., 1993 for a detailed review) (Solanki et al., 2006). Together with the granulation, the magnetic flux tubes are the main structuring agents in the photosphere. Furthermore, they have an important role in structuring and energizing the entire solar atmosphere, providing a channel for the transport of energy from the convection zone to the outer atmosphere. Independently of the size, all magnetic flux tubes have a roughly equally strong field of 1e1.5 kilogauss (kG) (which is a mean field, i.e., averaged over their crosssection) sufficient to inhibit the convection and energy flux associated with it significantly. Consequently, the flux tubes with a diameter larger than a 1000 km are visible as dark pores and the even larger ones as sunspots. In contrast, smaller flux tubes (often called magnetic elements) form bright points, ensembles of which can be observed as network and faculae. This is because the optical depth unity surfaces within magnetic flux concentrations are found at lower layers of the Sun owing to the horizontal pressure balance with their surroundings. This lowering of the solar surface within magnetic flux concentrations is called the Wilson depression. Because near the solar surface the temperature increases rapidly downward, the flux tubes are surrounded by hot walls (Fig. 3.4) (Solanki et al., 2013). This leads to two effects that define the spectral contrasts of magnetic elements relative to the surrounding quiet regions: (1) interiors of flux tubes are heated by radiation from the hot walls, and (2) the hot walls can be directly seen when magnetic elements are observed away from the disc center. In particular, the second effect is responsible for the center-to-limb variation of contrasts of magnetic elements: they are barely distinguishable from the quiet regions when observed close to the disk center (because hot walls are not visible) and appear bright toward the limb. As a result, the network and active-region faculae are mainly visible as bright structures in the continuum at visible wavelengths (e.g., in white light) mainly close to the solar limb. The photospheric temperature stratification of six brightness components in Fig. 3.3 is produced to emulate this purely 3D effect of hot walls’ visibility within the constraints of 1D geometry. Namely, the temperature structures of all six models are similar in the lower photosphere (where continuum radiation emitted near the solar disk center is formed) and start to diverge with height, at least below the temperature minimum (where the radiation emitted away from the disk center is formed). The decrease in the photospheric temperature with height in all components can be easily understood in terms of RE: such a temperature gradient is needed to transfer the radiative energy through the photosphere. On the contrary, the increase in the temperature with height in the chromosphere (i.e., above the temperature minimum) cannot be explained under the assumption of RE: there must be a nonradiative heating mechanism. For the quiet Sun, the necessary energy is transported by acoustic waves generated in the upper convection zone and traveling upward through the

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FIGURE 3.4 The emergent intensity from a magneto-convection simulation observed from a vantage point corresponding to it being near the limb. Still from a movie by Mats Carlsson taken from:Nordlund,A ., Stein, R.F., Asplund, M., 2009.

solar atmosphere (Solanki et al., 2002; Werner et al., 2003). The amplitude of the waves increases with decreasing gas density in the chromosphere and they start to form shocks in the layers above the temperature minimum, heating them. Consequently, the temperature in the chromosphere, on average, steeply increases between the temperature minimum and layers at h z 1000 km, although the actual temperature is strongly perturbed in space and time. At these layers, the ionization of hydrogen starts. It absorbs a lot of the energy released within the chromosphere, dramatically increasing its heat capacity. Consequently, the temperature profile flattens, forming the plateau. At h z 2000 km, the number density of neutral hydrogen is no longer sufficient to absorb all of the energy from the shock waves. Furthermore, a significant amount of energy is transported to these layers from the solar corona by thermal conduction. As a result, the temperature suddenly jumps by more than 10,000K. Then the chromosphere starts to be transparent for Lyman-a radiation, which effectively cools it down again, forming the second temperature plateau at 20e30/c $ 103 K seen in the models in Fig. 3.3.

4. Physics of the Photosphere/Chromosphere

The calculations of the spectra emergent from 1D models are relatively fast, and with increasing computer capacity more and more sophisticated radiative transfer effects can be taken into account (Werner et al., 2003). Of particular importance are the effects caused by the deviations from the local thermodynamic equilibrium (for non-LTE effects, see Section 9.1 of Hubeny and Mihalas (Hubeny and Mihalas, 2014) for a detailed review). They arise when the coupling between photons and atmospheric gas (enforced by the inelastic collisions) weakens and photons created in one part of the atmosphere affect the conditions (e.g., the degree of ionization) in nearby parts. In this case, atomic and molecular populations can no longer be calculated with the SahaeBoltzmann equation and instead a system of statistical balance equations has to be solved simultaneously with the radiative transfer equation. The non-LTE effects are present when the assumption of the detailed balance, i.e., that any process in the atmosphere is exactly balanced by the inverse process, is broken. The non-LTE effects are especially strong in the chromosphere where the gas density and consequently collision rates are too small to maintain LTE. Because of the strong temperature gradient, they also have an important role in the middle and upper photosphere: UV photons emergent in the lower and hotter parts of the photosphere penetrate into the higher and cooler layers and cause excessive (relative to the one given by the SahaeBoltzmann equation) iron and other metals ionization (Short and Hauschildt, 2009). The 1D SE models describe only some of the aspects of the highly inhomogeneous and dynamical 3D solar atmosphere and are by far not as reliable for the diagnostics of atmospheric properties as modern 3D models, at least in the photosphere (Koesterke et al., 2008; Uitenbroek and Criscuoli, 2011) and more recently also in the chromosphere (de la Cruz Rodrı´guez et al., 2016; Ermolli et al., 2013). At the same time, they yield a convenient way of interpolating from a specific set of spectral measurements (e.g., emergent intensities from quiet Sun and magnetic features measured at some sparse grids of wavelengths and position angles, i.e., distances to the solar disc center) to the entire parameter space of wavelengths and position angles. Consequently, SE models are still used extensively in studies of solar irradiance variability (see, e.g., reviews by Ermolli et al. (Ermolli et al., 2013) and Solanki et al. (Solanki et al., 2013)) for providing contrasts of magnetic features relative to the quiet Sun as functions of the wavelength and position angle. Because of the lack of data, the SE approach has not been widely used for stellar atmospheres, but the main concept behind the approach proved to be useful in stellar physics as well. For example, Liseau et al. (Liseau et al., 2013) observed the spectrum of a Cen A in the far infrared wavelength range, where the solar temperature minimum can be observed (see Section 3). Similar to the solar case, they found that the brightness temperature has a minimum at around 160 mm and Tmin ¼ 3920  375K. This became the first direct measurement of a temperature minimum on a star other than the Sun.

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4.2 THREE-DIMENSIONAL MODELS The 3D models of the solar atmosphere usually rely on the MHD, which describes plasma as a fluid and combines equations of hydrodynamics (i.e., the continuity equation, the equation of motion, and the energy equation) with the induction equation for the magnetic field (see a detailed description of the MHD equations in Chapter 7). There are two main classes of 3D magnetohydrodynamic models: “idealized” and “realistic.” Idealized studies assume a fully ionized, ideal plasma and ignore radiative transfer (so that the energy is only transferred by thermal conduction and advection) (see, e.g., review by Schu¨ssler et al. (Schu¨ssler et al., 2001)). The real breakthrough has been the realistic 3D simulations of the lower solar atmosphere. Two main features of the realistic simulations are that they account for (1) partial ionization and (2) the radiative transfer of energy in 3D. The former is crucial in the upper convective zone because the ionization energy dominates the convective energy transport whereas the latter takes over from convection as the dominant energy transport mechanism in the photosphere and determines the cooling of the solar atmosphere. The effects of partial ionization are taken into account via the equation of state (EOS), which gives the main thermodynamical quantities (e.g., pressure, electron concentration, and internal energy) as functions of temperature and density. The EOS can be either ideal and rely on the Saha ionization equation and chemical equilibrium, or nonideal, e.g., include many-body effects, electron degeneracy, and corrections owing to Coulomb interaction (see the detailed discussion and intercomparison of various EOS in (Vitas and Khomenko, 2015)). The treatment of the radiative transfer is relatively simple at depths below the surface. The radiation is almost isotropic and trapped there, so that the diffusion approximation can be applied (Hubeny and Mihalas, 2014 p. 374). At the same time the photons can escape in the solar atmosphere and a more sophisticated treatment of the radiative transfer is needed. Generally, the solution of the radiative transfer is the most time-consuming part of the realistic simulations. The main numerical problem here is accounting for the millions of atomic and molecular lines that affect the structure of the solar atmosphere via the line cooling and back-warming effects (see Sections. 4.2 and 5.3 of the doctoral thesis of Alexander Vo¨gler). The most direct way to take these effects into account is to solve the radiative transfer on a fine frequency grid. This is feasible in static 1D calculations but it becomes unbearably time-consuming in 2D and 3D dynamical simulations. One way to solve this problem, which is now routinely applied in most MHD codes, is to sort all frequency points into a small number of groups (4e12 are typically used) according to the formation heights of radiation at these frequencies. Then the group-integrated intensity values are calculated assuming gray radiative transfer within each of the groups (Nordlund, 1982; Vo¨gler et al., 2005). Another important sophistication is non-LTE effects, which are especially strong in the chromosphere where collision rates drop (see Section 4.1). In particular, nonLTE effects influence the cooling rates in spectral lines and the EOS (owing to

4. Physics of the Photosphere/Chromosphere

non-Saha ionization). Over the past few years, realistic simulations started to account for various non-LTE effects, e.g., the effects of scattering in spectral lines and the continuum (Hayek et al., 2010), non-equilibrium hydrogen and helium ionization (Golding et al., 2016), and non-LTE radiative cooling in chromospheric hydrogen, magnesium, and calcium lines (Carlsson and Leenaarts, 2012). In contrast to the parameterized consideration in 1D models, 3D MHD models make it possible to simulate the structure and dynamics of the magnetic field and granulation directly, as well as the interplay between them. In particular, modern realistic simulations are capable of reproducing granules: areas of hot upflows with a typical size of 1000 km surrounded by cooler lanes of downflowing gas. Fig. 3.4 shows the granulation pattern simulated by Carlsson et al. (Carlsson et al., 2004). The figure illustrates a remarkable property of the solar surface: it has an uneven 3D structure. This can be explained by the strong sensitivity of the negative hydrogen ion concentration (which is the main source of the opacity in the lower photosphere) to temperature (see Section 3). The concentration of negative hydrogen ions is lower in the cooler downflowing lanes (owing to a drop in the free electron concentration there) than in the warmer upflowing interiors of granules. Consequently, the surface of unity optical depth is approximately 35 km higher in upflows than in downflows in nonmagnetic regions (Frutiger et al., 2000). Along the same line, concentrations of the strong magnetic field are lower than the nonmagnetic solar surface by up to 350 km owing to the Wilson depression (see Section 4.1). Of particular relevance to stellar studies is that one of the best observational constants in simulations of solar convection comes from the spatially unresolved observations of the Fraunhofer lines (see, e.g., a detailed discussion in Nordlund et al., 2009). This is because whereas convective instability stops just below the solar surface, it overshoots into the solar photosphere, affecting the shapes and widths of the Fraunhofer lines. Fig. 3.5 illustrates the high quality of the agreement between spectra computed using realistic simulations and those with observations. Namely, it shows a comparison between the Kitt Peak solar spectral atlas (see Section 4.1) and a spectrum computed with the SPINOR code (Frutiger et al., 2000) using a piece of the solar atmosphere simulated with the Max-Planck-Institut fu¨r Sonnensystemforschung/University of Chicago Radiative MHD (MURaM) (Vo¨gler et al., 2005) code. The agreement between the two spectra is remarkably good for both the UV region with numerous OH and atomic lines (upper panel) and the visible spectrum dominated by several intermediate strength Fe lines (lower panels). Naturally, the comparison between simulated and observed spectra is not limited to the solar case. Numerous studies have shown that realistic simulations of stellar atmospheres give velocity fields consistent with observed line bisectors and widths (Allende Prieto et al., 2002; Ramı´rez et al., 2009; Trampedach et al., 2013). Convective motion in the photosphere and in subphotospheric layers has a crucial role in defining the structure of the small-scale photospheric magnetic concentrations (i.e., bright points, network, and faculae). Dynamo action in the solar convective envelope (and probably in the overshoot layer at its base) produces a continuum of

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FIGURE 3.5 Excerpts from a National Solar Observatory spectral atlas of the quiet Sun at the center of the solar disc (black lines), spatially averaged synthetic spectra (red lines) for the part of the OH band plus atomic lines around 311.8 nm (upper panel) and for the spectral region around the Fe I line at 525.02 nm, often used in solar physics because of its large magnetic sensitivity (lower panel).

magnetic structures of different sizes. These rise through the visible solar surface and into the atmosphere in the form of loops. At the surface itself, they are visible as ensembles of magnetic flux tubes with mixed polarities. Convective motion quickly drags magnetic flux tubes into the intergranular lanes, and then toward the supergranule boundaries (on timescales of days) (Stein, 2012). At the same time, magnetic flux tubes are canceling (if of opposite polarity), dissolving, fragmenting, and forming again (so that Schrijver et al. (1998) estimated that the flux in the magnetic network is renewed every 40 h). Consequently, the final structure of the photospheric magnetic field is determined by the balance between the emerging field and the rate at which the field is dragged by the convective motion, with most of the magnetic flux tubes being concentrated in the supergranule boundaries. The structure of the magnetic field in the quiet Sun changes dramatically in the chromosphere. The pressure in the hotter magnetic flux tubes (see Section 4.1) decreases slower than in the cooler surroundings. Consequently, magnetic flux tubes cannot be confined by the external gas at layers above about 700e1000 km (Solanki and Hammer, 2002). As a result, large patches of an almost horizontal field known as a magnetic canopy are formed. Above this height, the solar atmosphere is completely filled by magnetic field and also energetically dominated by it, i.e., the outer atmosphere of the Sun is a magnetosphere. Simulations of the solar atmosphere have reached a level of realism that allows predictions to be made about the structure of the atmosphere of other cool stars. Unlike earlier, purely hydrodynamic simulations (Nordlund and Dravins, 1990; Freytag et al., 1996; Asplund et al., 1999), they now also include magnetic field. Fig. 3.6 shows maps of the vertically emerging bolometric intensity (upper panel) and of the vertical component of the magnetic field simulated by Beeck et al. (2015) with the MURaM code for main sequence stars of F3, G2 (i.e., solar case), and K0 spectral classes. The

5. Physics of the Chromosphere/Corona

FIGURE 3.6 Maps of the vertical bolometric intensity for Max-Planck-Institut fu¨r Sonnensystemforschung/University of Chicago Radiative magnetohydrodynamic approximation simulations of F3V, G2V, and KOV stars (upper panels) and corresponding maps of the vertical component, Bz, of the magnetic field at z ¼ 0, the average geometrical depth of the optical surface (lower panels). For improved image contrast, brightness in the upper panels saturates at the values indicated by the gray scales on the right of each panel. The initial magnetic field was unipolar, vertical with a uniform field strength of 500 G. The figure is adapted from Beeck B., Schu¨ssler M., Cameron R.H., et al. Three-dimensional simulations of nearsurface convection in main-sequence stars. III. The structure of small-scale magnetic flux concentrations. Astron. Astrophys. 2015; 581:A42. doi:10.1051/0004e6361/201525788. 1505.04739.

horizontal size of the simulations has been scaled with the expected granule sizes. One can see that independently of the spectral class, the convective motion drags the magnetic field to the convective downflows in the intergranular lanes where a field up to several kG is formed. The locations of a strong field coincide with regions of enhanced or reduced intensity. The intensity contrasts of the magnetic elements notably depend on the spectral type (decreasing toward cooler stars) (see detailed explanations of this effect in Beeck et al., (2015)).

5. PHYSICS OF THE CHROMOSPHERE/CORONA One major difference between the photosphere and the corona is found in the roles the plasma and the magnetic field have, which are characterized by the ratio of the

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energy densities of the gas and the magnetic field, the plasma-b. On average, in the photosphere (outside sunspots) b is above unity, i.e., the thermal energy dominates, whereas in the corona b is below unity and the magnetic field dominates energetically. This causes the radically different appearance of the two atmospheric regimes. In the photosphere, the (overshooting) convection structures the atmosphere, e.g., resulting in granules, and through the interplay with the magnetic field convection creates small flux tubes with field strengths of up to kG (Lagg et al., 2016). In contrast, in the corona the magnetic field is basically filling space and heated plasma is trapped along the magnetic field, forming coronal loops. These are best seen in ˚ emission from plasma at around 106K, e.g., visualized in filter images near 171 A that are dominated by emission lines from Fe IX (Fig. 3.1). Because the magnetic field dominates in the corona, the gas does not alter the magnetic field significantly, and thus the plasma outlines magnetic field lines in a way similar to that of iron fillings in a school experiment. If, in general, the gas dominates in the photosphere and the magnetic field does so in the corona, there must be an interface where they roughly balance. This is the case somewhere in the chromosphere. The exact height where this happens depends on the average magnetic flux density of the region. This transition is why this atmospheric regime is so interesting for the magnetic connection from the surface to the upper atmosphere. Besides, the chromosphere is challenging with respect to a correct description of its physics. At low temperatures (Fig. 3.3), the gas is only partially ionized, and combined with the density and collisional coupling, processes such as the Hall effect and ambipolar diffusion become important. This basically accounts for the increased slippage of the magnetic field through the plasma, or vice versa (see Section 5.4). Together with the requirement to treat chromospheric lines under non-LTE conditions (see Section 4.1), these effects make the chromosphere the most challenging regime in the solar atmosphere in terms of modeling and interpretation. Between the chromosphere and the corona lies the transition region, in which the temperature jumps from a few 10,000K to nearly a million K. In 1D models the transition region is extremely narrow, having a thickness well below a few dozen km. Its properties are mainly determined by the heat conduction from the corona downwards.

5.1 CORONAL EMISSION AND MAGNETIC STRUCTURE Coronal loops, as seen in emission originating from gas at around 1 MK (e.g., in a ˚ ), usually emerge from the magnetic concentrations in wavelength band near 171 A an active region and appear roughly semicircular with lengths of around 100 Mm and reaching heights of often 50 Mm or more. Mostly, one footpoint of these loops is in the penumbra around the dark core of the sunspot. In a sunspot group, typically the other footpoint is not, as one might expect, at the other sunspot, but found in a plage region of opposite magnetic polarity, i.e., inside the region around the other sunspot with enhanced magnetic flux density. Observations can also show plasma

5. Physics of the Chromosphere/Corona

˚ that conat higher temperatures above 7 MK, e.g., in X-rays or in a band near 94 A tains emission lines of Fe XVIII. These reveal a structure different from what is seen at around 1 MK. Here, the active region is more compact, with shorter loops mostly connecting the sunspots: hence, the term “hot core of an active region.” An overview of the appearance of the corona is shown in Fig. 3.1. We come back to a physicsbased interpretation of this appearance in Section 5.3, and for the moment discuss only some basic properties of coronal loops. To test whether the coronal emission that is visible in the form of loops really outlines the magnetic field, one can extrapolate the magnetic field from the photosphere into the upper atmosphere. Direct measurements of the coronal magnetic field are available only above the limb in coronagraphic observations (Lin et al., 2004; Raouafi, 2005), which provide only poor spatial and temporal resolution (because of the limitation of the photon flux). Thus, extrapolations are the major tool for exploring the magnetic structure of the upper atmosphere. Naturally, the magnetic field B is solenoidal, i.e., V  B ¼ 0. If we now assume that in the corona there are no (or negligible) currents, j f V  B, the magnetic field would also be irrotational, V  B ¼ 0. Consequently, B can be represented by the gradient of a scalar potential field, B ¼ VF. Hence, this particular case is called the potential magnetic field and has to satisfy DF ¼ 0. The solution of this Laplace equation in any given volume depends on the boundary conditions only. Thus, knowledge of the magnetic field at the solar surface is sufficient to determine the magnetic field throughout the whole atmosphere (e.g., assuming periodic boundary conditions in the horizontal direction and B ¼ 0 at infinity). More complex (and, it is hoped, more realistic) forms of magnetic extrapolations exist (Feng et al., 2007). Using these, one can show that indeed the coronal loops seen in observations match the field lines derived from the magnetic field from the extrapolations well (Feng et al., 2007; Wiegelmann et al., 2005). Typically, the overall appearance of coronal loops evolves on timescales of about 30 min to 1 h. This is much longer than the typical cooling time expected from the energy losses through optically thin radiation (mostly in the extreme UV) and through heat conduction back to the cooler lower atmosphere (Aschwanden, 2004). Therefore, there needs to be a continuous energy input, and the bulk of the coronal loops cannot be understood through a single short heating pulse (Warren et al., 2003). Of course, also much faster small-scale variations are seen on timescales down to a few seconds. Probably these are a direct response to small heating events (Re´gnier et al., 2014). The emission in the extreme UV is seen all the way to the apex of loops at heights of 50 Mm or more, which is almost a tenth of a solar radius. However, the barometric scale height at 1 MK in fully ionized hydrogen plasma is 50 Mm. Therefore, one would expect to see a considerable drop in the emission (which is proportional to the density squared). However, that is not the case (Aschwanden et al., 2001). Thus, these loops cannot be in hydrostatic equilibrium, but have to be overdense. This might be achieved through the flow dynamics of the loops and their temporal evolution (Mu¨ller et al., 2003), but it is not yet fully understood.

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5.2 BASIC CONSIDERATIONS OF THE ENERGETICS In the corona, the magnetic field energetically dominates the gas, and thus it has to be space filling. The field lines then basically act as a tube to define the coronal loop in which the plasma can move only along but not across the field. Therefore, as a first approximation, one can consider the problem of a coronal loop in only one dimension (along a field line). Figuratively speaking, the basic question is then why we see one field line highlighted by coronal extreme UV emission while the neighboring field line is dark. We will see that this is determined solely by the energy input. Consider a loop in equilibrium between energy input, radiative losses, and heat conduction. In a thought experiment, we now increase the energy input, and naturally the temperature rises. At high temperatures, radiative losses are smaller than the loss through heat conduction back to the Sun. So, the additional energy is conducted back to the Sun. There, in the low corona (and upper chromosphere), the plasma becomes heated and evaporates into the loop. This increases the density in the loop. At the same time, the base of the corona moves downward to higher densities (because the very top of the chromosphere was evaporated) and thus the radiative losses at the coronal base increase as well. So, the loop finds a new equilibrium between energy input, heat conduction along the loop, and radiation at its base. So essentially, increasing the energy input results in a higher temperature, but also in a higher density (and thus pressure). Therefore, heating a set of field lines increases the coronal radiation from the plasma trapped between them and they appear as a coronal loop. This equilibrium of the loop can be described mathematically through scaling laws (Rosner et al., 1978; Priest, 1982) that connect the energy flux FH into the corona and the loop length L to the resulting temperature T and pressure or number density n (Peter et al., 2012):   
2=7 T½K ¼ 1700 FH W m2 ðL½mÞ2=7 ; (3.1)    
4=7  ðL½mÞ3=7 . n cm3 ¼ 3:9  1010 FH W m2

(3.2)

One important corollary is that the heat input sets both the temperature and the density. This implies that for a given loop, one is not free to select the temperature and density independently, but (in equilibrium) they are both set by the energy input (and loop length), and hence they are not independent. The weak dependence of the temperature on the energy input is because of the high sensitivity of the heat conduction flux on the temperature. The heat conduction in a fully ionized gas is given though q f T5/2 VT (Spitzer, 1962). Therefore, increasing the temperature just slightly results in a strong increase in heat conduction. In other words, increasing the heating results in a significant enhancement of the heat conduction (carrying away most of the additional heat) and only a small amount of energy is left for only a modest temperature increase: the heat conduction acts like a thermostat.

5. Physics of the Chromosphere/Corona

The requirements for the average heating of the corona have been derived from early extreme UV and x-ray observations through the radiative losses. These gave values for the required energy flux into the upper atmosphere in the quiet Sun corona of about 100 W/m2 and some 104 W/m2 in an active region (Withbroe and Noyes, 1977). Employing these scaling laws, for a 100-Mm-long loop, this yields temperatures of about 1 and 5 MK, values that are consistent with modern observations (Landi and Feldman, 2008).

5.3 HEATING PROCESSES AND MODERN MODELS Knowing the requirement for the energy input, this still leaves the questions of where the energy is originating, how it is transported, and how it is finally dissipated. Generally, one distinguishes AC and DC heating, termed after alternating and direct currents. In both cases, the magnetic field at the coronal base (or in the photosphere) is driven by the motions in the photosphere, which essentially leads to a Poynting flux, i.e., a flux of electromagnetic energy, into the upper atmosphere. The magnitude of this flux has to match the requirements in the preceding subsection to sustain a corona. The driving (or stressing) of the magnetic field induces a perturbation of the magnetic field that propagates essentially with the Alfve´n speed, i.e., the speed of a transversal wave of the magnetic field (see Chapter 7 and Priest, 1982). If the driving is faster than the Alfve´n speed, the perturbation will be a wave, similar to a piece of rope held in one hand and moved back and forth quickly. The changes of the magnetic field go along with changing induced currents, hence the term “AC.” The driving motion can be linear or torsional and will launch a range of waves into the upper atmosphere (van Ballegooijen et al., 2011). If the driving is slower than the Alfve´n speed, one simply stresses the magnetic field. If the driving occurs in a randomized fashion (through the motion of the footpoints, e.g., owing to photospheric convection), figuratively speaking, one is braiding the magnetic field lines. The induced currents will not change in a wave fashion, and hence the term “DC.” The braiding will build up increasingly stronger currents until a (secondary) instability sets in and releases the energy in a process termed nanoflaring (Parker, 1972, 1988). Often AC and DC heating are treated as an either/or problem, but on the Sun we expect a wide range of driving timescales. Therefore, wherever there is strong DC heating, one also expects increased AC heating, and vice versa. The energy input in a model that is based on DC or AC heating alone should therefore be considered as a lower limit only, because one might expect both mechanisms to be operational. In the end, what really matters is the magnitude of the Poynting flux into the upper atmosphere used to energize the plasma. Three-dimensional MHD models have become available that solve the problem of the driving of the corona based on the motions in the photosphere in numerical experiments (Gudiksen and Nordlund, 2002). These showed that indeed, the driving at the solar surface caused a braiding-type effect that can sustain a loop-dominated million-K hot corona (Gudiksen and Nordlund, 2005) and these models match solar

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observations in many aspects (Peter et al., 2004). Such 3D models give a good understanding of the observed Doppler shifts (Peter et al., 2006; Solanki et al., 2017), show that most of the energy is dissipated low in the atmosphere in thin current sheet-like structures aligned with the magnetic field (Bingert and Peter, 2011), provide some understanding of the observed constant width of coronal loops (Peter and Bingert, 2012), and show the clear relation of the increased Poynting flux at the base of the corona with the appearance of bright loops (Chen et al., 2015), to name a few. The last result also clarifies why loops are not rooted in the center of a sunspot, the dark umbra. There, the magnetic field is so strong that it suppresses the (convective) horizontal motions in the photosphere to a large extent. Consequently, in the umbra, the Poynting flux is low, whereas it is high in the region of the penumbra (Chen et al., 2014), and the loops are primarily rooted in the penumbra if they come close to a sunspot. One key ingredient of these 3D MHD models is the inclusion of heat conduction. Only then are the temperature and density of the corona set properly (see Section 5.2). This is a prerequisite to derive the coronal emission. The emission from the corona is optically thin and dominated by emission lines from highly ionized species. In general, the excitation of these lines is caused by electron collisions, and mostly the de-excitation is the result of spontaneous emission. Spontaneous emission is much faster than collisional excitation; hence, all ions are in the ground state almost all of the time. Because of the electron collisional excitation, the emissivity is (roughly) proportional to the density squared. Mostly the assumption of ionization equilibrium, mediated by electron collisions, is justified (Chen et al., 2014), and the typical ionization stages are restricted to a narrow range in plasma temperature (typically over logT[K] z 0.2, i.e., a factor of 1.5). This is why we can consider that a narrow extreme UV band dominated by one emission line provides information about the distribution of plasma in a narrow temperature range, e.g., why the ˚ band dominated by Fe IX shows plasma at around 1 MK. 171 A

5.4 CONNECTION TO THE LOW ATMOSPHERE Of course, the corona has to be magnetically connected to (or rooted in) the photosphere. However, finding a clear correspondence between coronal intensity structures, such as loops, and features of the magnetic field in the photosphere is extremely challenging. One reason for this is the significant difference in spatial resolution of observations, which typically is a factor of five or more worse in the corona compared with in the photosphere. The workhorse of coronal observations, the AIA (Lemen et al., 2012), provides a plate scale of 0.6 arcsec per pixel and a spatial resolution of about 1.4 arcsec. On a rocket flight, the High-resolution Coronal imager could provide a few minutes’ worth of coronal images at 0.3e0.4 arcsec resolution (Cirtain et al., 2013) and the Interface Region Imaging Spectrograph (De Pontieu et al., 2014) regularly provides about 0.35 arcsec resolution, although it concentrates on the chromosphere and the transition region into the corona. These resolutions are considerably lower than those obtained in photospheric and

References

chromospheric observations achieved by the SUNRISE balloon observatory (Solanki et al., 2017) or ground-based solar observatories with apertures of up to 1.5 m that can go town to 50 km on the Sun. Despite this mismatch in spatial resolution, one can patch together observations from the photosphere, chromosphere, and corona to study the connectivity. One of the most enigmatic features in the solar atmosphere is spicules and what drives them. Reported already by Secchi in the late 19th century, they are wellinvestigated observationally, but their physics is poorly understood. In spicules, plasma is propelled upward, best seen in the emission of Ha. In one type (I), the chromospheric plasma follows a ballistic trajectory, eventually falling back to the Sun; in another type (II), they seem simply to dissolve in Ha (de Pontieu et al., 2007), which probably is a signature of heating of the plasma (Pereira et al., 2014). It has been suggested that ambipolar diffusion has a critical role (Martı´nez-Sykora et al., 2017). Then the magnetic field can emerge more efficiently from the photosphere upward (in slipping through the partially ionized plasma). This then leads to more vigorous reconnection driving the spicules. An important aspect of connecting the coronal structures to the photospheric magnetic field concerns the response of the corona to changes in the connectivity of the magnetic field deep in the photosphere and chromosphere. Observations established the important role of reconnection at height levels where the chromosphere would normally be located (Peter et al., 2014). In particular, one could track extreme UV brightenings and relate them to reconnection sites (determined by magnetic extrapolations) that were located only some 500 km above the photosphere (Chitta et al., 2017a), i.e., near the temperature minimum (see Fig. 3.3). Comparing coronal data with the highresolution photospheric magnetic field in more detail shows that loops are rooted at locations of small-scale mixed polarities (Chitta et al., 2017b). This highlights the importance of reconnection in the chromosphere for the heating the corona. Thus, besides AC and DC heating discussed in Section 5.3, this an adds a third path to coronal energization, which needs to be explored in the future.

ACKNOWLEDGMENTS This project has received funding from the European Research Council under the European Unions Horizon 2020 research and innovation program (Grant Agreement Nos. 624817 and 695075) and was supported by the BK21 Plus program through the National Research Foundation funded by the Ministry of Education of Korea.

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Helioseismic Inferences on the Internal Structure and Dynamics of the Sun

4

Sarbani Basu1, William J. Chaplin2 Yale University, Department of Astronomy, New Haven, CT, United States1; University of Birmingham, School of Physics and Astronomy, Edgbaston, United Kingdom2

CHAPTER OUTLINE 1. Introduction ......................................................................................................... 88 2. Theoretical Background........................................................................................ 89 2.1 Equations Governing Solar Oscillations ................................................... 90 2.2 Properties of Solar Oscillations ............................................................... 92 2.3 Relating Frequency Changes to Changes in Structure ............................... 94 2.4 Effects of Rotation ................................................................................ 97 2.5 Inversions............................................................................................. 98 3. Inferences About Solar Structure.........................................................................100 3.1 Basic Results...................................................................................... 101 3.2 Base of the Solar Convection Zone........................................................ 102 3.3 The Question of Diffusion .................................................................... 103 3.4 Convection Zone Helium Abundance ..................................................... 104 3.5 The Issue of Solar Composition ............................................................ 106 4. Inferences on Solar Dynamics.............................................................................107 4.1 Properties of the Tachocline................................................................. 109 5. Helioseismic Inferences on the Solar Cycle .........................................................110 5.1 Changes to Global-Mode Frequencies and Mode Parameters ................... 110 5.2 Zonal Flows ........................................................................................ 116 5.3 Meridional Flows ................................................................................. 117 5.4 The 1.3-Year Periodicities Near the Tachocline...................................... 119 5.5 Changes in Even-Order a Coefficients .................................................... 120 6. Seismic Studies of Other Stars ............................................................................120 Acknowledgments ...................................................................................................121 References .............................................................................................................121

The Sun as a Guide to Stellar Physics. https://doi.org/10.1016/B978-0-12-814334-6.00004-2 Copyright © 2019 Elsevier Inc. All rights reserved.

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CHAPTER 4 Helioseismic Inferences on the Internal Structure

1. INTRODUCTION Published in 1926, Eddington’s seminal text on the internal structures of stars (Eddington, 1926) opened with the following thoughts: At first sight it would seem that the deep interior of the Sun and stars is less accessible to scientific investigation than any other region of the universe. Our telescopes may probe farther and farther into the depths of space; but how can we ever obtain certain knowledge of that which is hidden beneath substantial barriers? What appliance can pierce through the outer layers of a star and test the conditions within?

Helioseismology has provided that answer for the Sun, and almost one century later, it has been possible to perform detailed studies of the solar interior. Eddington was certainly well aware of the diagnostic potential of resonant pulsations. In his 1927 monograph, Stars and Atoms (Eddington, 1927), he remarked: Ordinary stars must be viewed respectfully like objects in glass cases in museums; our fingers are itching to pinch them and test their resilience. Pulsating stars are like those fascinating models in the Science Museum provided with a button which can be pressed to set the machinery in motion. To be able to see the machinery of a star throbbing with activity is most instructive for the development of our knowledge.

At the time Eddington wrote that, all known pulsating stars, the so-called classical pulsators, were ones that resonated in one or at most two modes, meaning that their period data could provide information on bulk or global stellar properties only. Eddington could not have foreseen that observations of the Sun made later in the century would herald the discovery of a new class of oscillating stars, the solarlike oscillators, with the data on the Sun showing many thousands of detectable modes, the use of which would reveal an exquisitely detailed picture of the internal structure and dynamics of our host star. The discovery that small patches of the Sun’s surface showed periodic motions on a 5-min timescale was made in the early 1960s (Leighton et al., 1962). However, it was not until almost a decade later that Ulrich (1970) and Leibacher and Stein (1971) offered the correct explanation of the phenomenon: the surface manifestation of standing waves in an internal cavity. Observational confirmation of the standing-wave nature of the signal was provided by Deubner (1975) and independently by Rhodes et al. (1977), and with the discovery that some oscillations engaged the entire Sun in pulsation (Claverie et al., 1979), The picture was complete. Most solar oscillations are standing acoustic waves for which the gradient of pressure (p) is the principal restoring force and are therefore usually referred to as “p modes.” The Sun shows globally coherent oscillations on a wide range of different length scales. The modes are excited stochastically and damped intrinsically by turbulence in the outermost layers of the subsurface convection zone (CZ).

2. Theoretical Background

The stochastic excitation mechanism limits the intrinsic amplitudes of the p modes to very low values, which explains why it took so long for the oscillations to be detected (and why doing so for other sun-like stars has been a recent advance). However, the mechanism gives a rich spectrum of detectable modes, the most prominent of which are high-order overtones. In this chapter, we set out to illustrate how one may use these modes to probe the solar interior, what we have learned since the first observations of globally coherent modes were made in the 1970s, and what outstanding challenges remain (and there are many) for our understanding of the Sun’s internal structure.

2. THEORETICAL BACKGROUND Solar oscillations are labeled by three quantities to describe the three-dimensional nature of the oscillations. The radial order n describes oscillations in the radial direction, with jnj being the number of nodes along the radius. The angular dependence is described by spherical harmonics Ylm ; the degree l of the mode represents the number of nodes along the circumference and the azimuthal order m is the number of nodes along the equator. The radial order n can be positive, negative, or zero. Conventionally, positive values of n are used to represent p modes, the pressure-supported modes. Modes with n ¼ 0 are surface waves called f modes, and negative n is used to denote gravity (g) modes, i.e., waves whose restoring force is buoyancy. Observed solar modes are p and f modes. Although there have been reports of the detection of g modes in the Sun (Garcı´a et al., 2007, 2008; Garcı´a, 2010; Fossat et al., 2017), the results remain controversial. For a spherically symmetric star, all modes with the same values of n and l have the same frequency. Departures from spherical symmetry, which in the case of the Sun is caused mainly by rotation, lift this degeneracy and cause “frequency splitting,” i.e., the frequencies become m dependent. The frequency splittings are usually expressed in terms of splitting coefficients: nnlm ¼ nnl þ

jmax X

aj ðnlÞP jnl ðmÞ

(4.1)

j¼1

where nnlm ¼ unlm/2p is the frequency of a mode with a given (n, l, m), and jmax sets the total number of coefficients aj that are used in the expansion. Most available sets of solar oscillation data have jmax ¼ 36. These coefficients are known as a coefficients, which, with a suitable choice of polynomials P nl ðmÞ, can be related to ClebscheGordon coefficients. For a slow rotator such as the Sun, nnl depends only on the structure, odd-order a coefficients depend on rotation, and even-order a coefficients depend on magnetic fields, structural asphericities, and the secondorder effects of rotation. Solar structure is determined using nnl, and solar rotation using the odd-order a coefficients.

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2.1 EQUATIONS GOVERNING SOLAR OSCILLATIONS Solar oscillations can be described well using the linear adiabatic wave equations (LAWE). Linearity is easy to establish. The amplitudes of p modes are highest close to the surface, with velocity amplitudes on the order of tens of centimeters per second. The relevant velocity scale is established by the speed of sound, which in these layers is on the order of 10 km/s and thus much larger than the amplitudes. Adiabaticity is not as good an approximation; it holds well in the interior where the thermal time scale is much longer than the periods of the modes, but it breaks down in the near-surface layers where the thermal timescales are much smaller. This introduces an error in the calculated frequencies and is a part of the so-called “surface term” discussed later. There are many texts that show how the equations of stellar oscillations are derived (Aerts et al., 2010; Unno et al., 1989; Basu and Chaplin, 2017); thus, we go through only the salient points here. LAWE are derived by a linear perturbation analysis of the equations of fluid dynamics, and in the stellar case the results are then cast into a form suitable for application to a sphere. The major assumptions made are ! that there are no velocities in the system until it is perturbed (i.e., ! v ¼ d x dt, ! where x is the displacement) and that tangential gradients of equilibrium quantities do not exist (if they did, there would be horizontal flows, which would wipe out the gradients). The equilibrium quantities are only a function of radius r, whereas the perturbed quantities are a function of (r, q, f, t), in which q is the colatitude, f the longitude, and t time. An analysis of the equations shows that the perturbations can be written as x r ð! r ; tÞ ¼ xr ðrÞYlm ðq; fÞ expðiutÞ; P1 ð! r ; tÞ ¼ P1 ðrÞYlm ðq; fÞ expðiutÞ; ! ! m r ð r ; tÞ ¼ rðrÞY ðq; fÞ expðiutÞ; F ð r ; tÞ ¼ F ðrÞY m ðq; fÞ expðiutÞ 1

1

l

1

l

(4.2) where xr is the radial component of the displacement, r1 is the perturbation to density r, P1 is the perturbation to pressure P, and F1 is the perturbation to the gravitational potential F. The process yields the following equations: !   dxr 2 1 dP 1 S2l lðl þ 1Þ ¼ þ  1 P 1  2 2 F1 ; (4.3) xr þ 2 2 r G1 P dr rcs u u r dr  where, G1 is the first adiabatic index, c2s ¼ G1 P r is the squared sound speed, and S2l is the Lamb frequency, defined by: S2l ¼

lðl þ 1Þc2s . r2

(4.4)

The second equation obtained is:   dP1 1 dP dF1 P1 þ r ¼ r u2  N 2 x r þ G1 P dr dr dr

(4.5)

2. Theoretical Background

where N is BrunteVa¨isa¨la¨ or buoyancy frequency, defined as:   1 dP 1 dr  . N2 ¼ g G1 P dr r dr

(4.6)

The last equation results from the perturbation to the Poisson equation and can be expressed as:     1 d P1 rxr 2 lðl þ 1Þ 2 dF1 N þ F1 ; (4.7) r þ ¼ 4pG r 2 dr r2 dr c2s g where G is the gravitational constant. Eqs. (4.3), (4.5), and (4.7) form a set of fourthorder differential equations and constitute an eigenvalue problem with eigenvalue u. Upon solving the equations, we can determine the eigenfunctions xr(r), P1(r), and F1(r) as well as dF1/dr. Each eigenvalue is a “mode” of oscillation. The perturbation to density can be obtained from P1 using the perturbed heat equation:   r 1 dP 1 dr  P1 þ rxr r1 ¼ . (4.8) G1 P G1 P dr r dr The transverse component of the displacement vector can be written in terms of P1(r) and F1(r), and one can show that:  m  vYl c 1 vYlm c ! xh ðr; q; f; tÞ ¼ xh ðr Þ (4.9) aq þ af expð  iutÞ; sin q vf vq af are the unit vectors in the q and f directions, respectively, and: where c aq and c   1 1 (4.10) xh ðrÞ ¼ 2 P1 ðrÞ  F1 ðrÞ . ru r There is no n or m dependence in Eqs. (4.3), (4.5), or (4.7). The different eigenvalues are labeled by n. The lack of an m dependence is a result of the assumption of spherical symmetry: we need to be able to define a unique equator to define m uniquely. Because we cannot do so, all modes with a given n and l are 2l þ 1-fold degenerate, because for spherical harmonics Ylm , m can have values l.0.þ l. An important property of modes is their inertia Inl, which determines how much the frequency of a mode with a given (n, l) will change compared with the frequencies of modes with other values of n and l for a given change in structure. The mode inertia is usually defined as: Z Z R h i ! ! 3! In;l ¼ r x n;l $ x n;l d r ¼ r x2r;n;l þ lðl þ 1Þx2t;n;l rr 2 dr. (4.11) V

0

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The reason for this definition will become clear in Section 2.3. Because the normalization of eigenfunctions can be arbitrary, Inl is often explicitly defined as:  2 2 i R R h    2 r þ lðl þ 1Þ x x  r;n;l t;n;l  r dr 0 (4.12) In;l ¼ h 2  2 i ;     M xr;n;l ðRÞ þ lðl þ 1Þxt;n;l ðRÞ where R is the total radius and M the total mass.

2.2 PROPERTIES OF SOLAR OSCILLATIONS Eqs. (4.3), (4.5), and (4.7) are opaque and do not tell us much about the properties of the oscillations. A few simplifying assumptions allow us to reveal the salient properties of the modes. The first assumption is the so-called Cowling approximation, i.e., we neglect F1, the perturbation to the gravitational potential. This can be shown to be a good approximation for modes with large l or large n. This reduces the equations to: !   dxr 2 1 1 1 S2l ¼   1 P1 ; (4.13) x þ r G1 Hp r rc2s u2 dr and   dP1 1 1 ¼ r u 2  N 2 xr  P1 ; G1 Hp dr

(4.14)

where cs is the sound speed and Hp is the pressure scale height defined as HP ¼ dr/d ln P. Next, we assume that we are looking away from the center (which means that the 2/r term in Eq. (4.13) can be ignored) and the final assumption is that the eigenfunctions vary more rapidly than equilibrium quantities. This reduces the equations to:   2   Sl dxr 1 dP1 ¼ ¼ r u2  N 2 x r .  1 P1 ; and (4.15) rcs 2 u2 dr dr Eq. (4.15) implies that: d2 x r ¼ K ðr Þxðr Þ; dr 2

where;

K ðr Þ ¼

 2   Sl u2 N 2  1  1 . cs 2 u 2 u2

(4.16)

Clearly, Eq. (4.16) has oscillatory solutions when either (a) u2 < S2l and u < N2, or (b) u2 > S2l and u2 > N2. The first condition holds for g modes, and the second for p modes. The Sl and Nl profiles of a solar model are shown in Fig. 4.1. We can see that modes that satisfy condition (1) are confined to the radiative zone, and those that satisfy condition (2) cover most of the solar interior, with low-l modes penetrating deep into the Sun and high-l modes restricted to shallower layers. Fig. 4.1 is often referred to as a “propagation diagram” because it shows where 2

2. Theoretical Background

FIGURE 4.1 Propagation diagram for a solar model. The thick black line marks the BrunteVa¨isa¨la¨ frequency and the thin gray lines are the Lamb frequency for a few different values of l. The horizontal line marks the propagation cavity for a 1000-mHz mode of l ¼ 5.

modes can propagate. Because most observed solar modes are p modes, we shall concentrate on those here. The p modes have frequencies that are larger than both the Lamb and Brunte Va¨isa¨la¨ frequencies. As can be seen from Fig. 4.1, in the observed range of solar frequencies (approximately 1e5 mHz), the Lamb frequency is larger than the BrunteVa¨isa¨la¨ frequency, and thus the modes are trapped between the surface and a lower (or inner) turning point rt, defined as the points where u2 ¼ S2l , i.e., c2s ðrt Þ u2 . ¼ rt2 lðl þ 1Þ

(4.17)

This can be used to determine the depth to which a given mode can penetrate. For modes with u2 [N 2 , K(r) in Eq. (4.16) can be approximated to: KðrÞx

u2  S2l ðrÞ ; c2s ðrÞ

(4.18)

which tells us that the behavior of these p modes is basically governed by the soundspeed profile. We have thus far assumed that the modes are trapped between the surface and a lower turning point. However, how close to the surface the upper turning point ru really is depends on the frequency of a mode. To determine this, we cannot assume that the pressure (and density) scale heights vary more slowly than the

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eigenfunctions. Deubner and Gough (1984) showed that under Cowling approximation, one could recast the equations as: d2 J þ K 2 ðrÞJ ¼ 0; dr 2

(4.19)

with ! J ¼ r1=2 c2s V$ x ;

and K 2 ðrÞ ¼

  u  u2c lðl þ 1Þ N2 þ ; r2 c2s u2  1

where uc, the acoustic cutoff frequency is defined as:   cs 2 dHr 2 uc ¼ 12 . 4Hr2 dr

(4.20)

(4.21)

Hr is the density scale height. The upper turning point ru is defined as the radius where u ¼ uc. Solving Eq. (4.19) for J ¼ 0 gives the solution for f modes and their dispersion relation u2 z gk, where g is the acceleration owing to gravity and k is the wavenumber.

2.3 RELATING FREQUENCY CHANGES TO CHANGES IN STRUCTURE Eqs. (4.3), (4.5), and (4.7) are not in a form that tell us much about how frequencies respond to changes in structure. For this, we need to recast the equations; details of the process may be found in Basu and Chaplin (2017), Basu (2016), etc. In a nutshell, one begins with the perturbed momentum equation. One again assumes that the time variation is of the form exp(iut) and uses the perturbed form of the continuity and adiabatic heat equations to eliminate the perturbations in density and pressure. This results in an equation of the form: 0 1  ! 3 Z B V$ r x d r C   ! ! ! ! C u2 r x ¼ V c2s rV$ x þ VP$ x  ! g V$ r x  GrVB  @ !0  A (4.22) V ! r  r  where we have expressed the perturbation to the gravitational potential as an integral. Eq. (4.22) clearly shows that the frequencies for a mode with displacement ! eigenfunction x are related to the sound speed and density (pressure is related to density through the equation equilibrium). Eq. (4.22) is an eigenvalue ! of hydrostatic ! equation of the form L x nl ¼ u2 x nl , where L represents the differential operator in Eq. (4.22). Chandrasekhar (1964) showed that under the boundary conditions

2. Theoretical Background

r ¼ 0, P ¼ 0 at the outer boundary, Eq. (4.22) is a Hermitian eigenvalue problem, and the relevant inner product is: Z Z R

! ! 

!  ! 3! x ; h ¼ rx $ h d r ¼ 4p xr ðrÞhr ðrÞ þ L2 xt ðrÞht ðrÞ r 2 rdr. (4.23) 0

V

Thus, the frequency unl, which is the eigenvalue corresponding to eigenfunction ! x nl , can be expressed as: !  3 ! R !  2 V rxn;l $L x n;l d r . (4.24) un;l ¼ R ! !  3 V rxn;l $ x n;l d r Frequencies calculated in this manner are known as variational frequencies. The Hermitian nature of Eq. (4.22) allows us to determine how changing the internal structure will change the frequencies. Changing the internal structure results in changing the differential operator L. The process of determining changes in frequencies begins with linearizing Eq. (4.22) around a known solar model, known as the reference model, yielding an equation of the form: ! ! ! ! ðL þ dLÞ x þ d x ¼ ðu þ duÞ2 x þ d x ;

(4.25)

The variational principle tells us that under first-order perturbations, the eigenfunctions of the perturbed state can be assumed to be those of the unperturbed state, and hence Eq. (4.25) gives: Z Z ! ! !! rx dL x dV ¼ 2udu rx x dV; (4.26) or, in other words, R ! R ! ! 3! ! 3! du V r x $dL x d r V r x $dL x d r ¼ ¼  ; R ! ! 2u2 In;l u r 2u2 r x $ x d3 !

(4.27)

V

where Inl is the mode inertia defined earlier. The equation clearly shows that for a given perturbation in structure, frequencies of modes with high inertia change less than those with low inertia. The quantity dL can be obtained by perturbing Eq. (4.22):    2 ! dr 2 ! ! ! ! dL x ¼ V dcs V$ x þ d g $ x þ V c V$ x r s  ! Z (4.28) V$ dr x 3 !0 1 ! ! 2 ! þ Vrdcs V$ x þ d g V$ x  GV  ! d r ; r V !  r  r0  g , and dr are the differences in sound speed, acceleration owing to where dc2s , d! gravity, and density between the reference model and the perturbed model (or the

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Sun). The quantity d! g can, of course, be expressed in terms of dr. Substituting Eq. (4.28) into Eq. (4.27) and rearranging the terms yields an equation of the form: Z Z dui dc2s dr i i ðrÞdr. (4.29) ¼ Kc2 ;r ðrÞ 2 ðrÞdr þ Kr;c 2 ðrÞ s s r ui cs i ðrÞ are known functions of the reference model and The terms Kci 2 ;r ðrÞ and Kr;c 2 s s connect the changes in frequency to changes in sound speed and density. These two functions are thus “kernels” that connect the change in structure to the change in frequencies. Fig. 4.2 shows a few kernels for sound speed and density. Eq. (4.29) can also be written in terms of relative differences in G1 and density and their corresponding kernels. Eq. (4.29) forms the basis of helioseismic structure inversions. The frequency responses for a localized change in sound speed at two different radii are shown in Fig. 4.3. A near-surface change yields a frequency difference that is predominantly a function of frequency alone once mode inertia is taken into account. The effects of mode inertia can be removed by multiplying the frequency differences by the inertia ratio Qnl (Christensen-Dalsgaard and Berthomieu, 1991), which is given by:  Qnl ¼ Inl Iðnnl Þ; (4.30)

where Iðnnl Þ is the inertia of a radial mode of the same frequency.

(A)

(B)

FIGURE 4.2 Kernels for (A) squared sound speed and (B) density for a few modes. Note that the kernels for the higher-degree modes are restricted to the shallower layers.

2. Theoretical Background

(A)

(B)

FIGURE 4.3

 Change in frequencies in response to a localized change in sound speed with dcs2 cs2 in the form of a Gaussian with a height of 0.04 and width of 0.001R1 but at two different positions: (A) rg ¼ 1.0005R1 and (B) rg ¼ 0.75R1 . In both panels, modes with l < 15 are plotted in black; the rest is in gray. Modes up to l ¼ 50 are shown. Note that the shallow perturbation results in differences that are a smooth function of frequency; the deeper one results in a more complex change. For each l, the change can be represented as a sinusoid.

2.4 EFFECTS OF ROTATION Rotation is the primary cause of deviation from spherical symmetry in the Sun. The Sun is a slow rotator, and slow rotation can be treated as a linear perturbation that makes it easy to determine how rotation changes mode frequencies. Slow rotation in this case is one for which the maximum centrifugal force is much smaller than the force of gravity and frequency u is much larger than the rotational frequency U. This works well for the Sun, for which the ratio of centrifugal force to gravity at the equator at the solar surface is on the order of 105; the frequency of solar rotation is about 450 nHz, much lower than oscillation frequencies of a few millihertz. Unlike in the case of perturbation analysis for structure, in the presence of rotation we can no longer assume the absence of velocities. Even at equilibrium, we can ! define a velocity ! v0 ¼ U ! r . This velocity is ultimately responsible for changing frequencies of modes with different azimuthal orders m but the same n and l. The changes can again be written in a variational form: R ! !  ! V r0 x $ð v 0 $VÞ x dV du ¼ i . (4.31) !2 R   r dV x   0 V Details of how this equation is derived can be found in Aerts et al. (2010).

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For rotation that does not depend on latitude, Eq. (4.31) becomes:   2 RR 2 2 2 2 0 UðrÞ xr þ L xh  2xr xh  xh r rdr dunml ¼ m .  R R 2 2 2 2 0 xr þ L xh r rdr

(4.32)

In this case, the frequency perturbation, or as it is usually called, the frequency “splitting,” is proportional to m. The m ¼ 0 mode is not affected but the frequencies of a mode with a given (n, l) is “split” into 2l þ 1 components. Eq. (4.32) can be written in terms of kernels: Z R UðrÞKnl ðrÞdr; (4.33) dunml ¼ mbnl 0

where the kernel Knl is



 x2r þ L2 x2h  2xr xh  x2h r 2 r ; Knl ðrÞhR R  2  2 2 2 2 0 xr þ L xh  2xr xh  xh r rdr

(4.34)

and R R bnl ¼

0

 x2r þ L2 x2h  2xr xh  x2h r 2 rdr :  R R 2 2 x2 r 2 rdr þ L x r h 0

(4.35)

This definition of Knl ensures that it is unimodular and that in the case of uniform rotation, when U(r) ¼ Uu, dunml ¼ mbnl Uu ;

(4.36)

i.e., the rotational splitting is simply proportional to the rotation and depends only on bnl. In general, U h U(r, q) and thus: Z RZ p dunlm ¼ m Knlm ðr; qÞUðr; qÞdrdq; (4.37) 0

0

where now the kernels Knlm are much more complicated. Expressions for Knlm may be found in Basu and Chaplin (2017) and Aerts et al. (2010). One can also determine the kernels for the a coefficients (see Section 2) and readers are referred to Ritzwoller and Lavely (1991) and Pijpers (1997).

2.5 INVERSIONS Helioseismic studies are usually conducted through inversions of mode frequencies and frequency splittings (or splitting coefficients). Given the large number of solar oscillation frequencies that have been observed, it is relatively easy to determine the structure and dynamics of the Sun. However, some fundamental issues are at play. The usual way to use data in stellar astrophysics is so-called “forward” modeling. A model is constructed, and if the attributes of the model satisfy the observed

2. Theoretical Background

constraints, the model is considered to be a good proxy for the star; hence, the properties of the model are assumed to match the properties of the star closely. This does not work well when trying to determine solar structure. In Section 2.1, we had mentioned the surface term. The surface term is the frequency difference between the Sun and a model caused by uncertainties in modeling the near-surface regions of the Sun. The issue of adiabaticity is not the only cause. The major contribution comes from our inability to model convection properly. Conventional solar models are one-dimensional and convection is modeled using the mixing length approximation, which assumes that convective eddies have a size aHp, where a is a free parameter. Absent in these models are the dynamical effects of turbulence in the near-surface regions. There are other near-surface errors, such as those caused by using simplified models of the solar atmosphere. All of these factors introduce a frequency-dependent frequency error; any l dependence is purely the result of differences of mode inertia: mode inertia decreases as l increases. The l-independence of this term can be explained by the response of the frequencies to a near-surface perturbation that we have seen earlier in Fig. 4.3A. The surface term can be seen clearly when frequencies of solar models are compared with those of the Sun. This is shown in Fig. 4.4A. The surface term is also seen between models (Fig. 4.4B) and is usually much larger than frequency differences caused by differences in structure. As a result, instead of testing models by comparing their frequencies with those of the Sun, it is common to perform inversions where the surface term can be removed and the underlying structure revealed. Splitting coefficients of the Sun show that the Sun has both radial and differential rotation. A forward study of the problem (i.e., assuming a rotation rate, comparing the splitting coefficients and comparing with the Sun) would take too much time to do given that the form of internal dynamics cannot be guessed. As a result, inversions are commonly performed to determine the internal rotation profile of the Sun. Structure inversions start with Eq. (4.29), but with an arbitrary function of frequency added to account for the surface term, i.e., Z Z dui dc2 dr Fsurf ðuÞ i ðrÞ dr þ ¼ Kci 2 ;r ðrÞ 2s ðrÞ dr þ Kr;c ; (4.38) 2 ðrÞ s s r Ii ui cs where Fsurf is an arbitrary function of frequency that is constrained to be a slowly varying function and Ii is the mode inertia of the ith mode. There is one such equation for each observed mode. There are two common methods  of performing the inversions, i.e., determining the relative structural differences dc2s c2s and dr/r between the Sun and a model using the relative difference of their frequencies using Eq. (4.38): (1) the regularized least squares (RLS) method, and (2) the method of optimally localized averages (OLA). These methods are  complementary (Sekii, 1997). RLS aims to find the three unknown functions (dc2s c2s , dr/r and Fsurf) that give the best fit to the data while keeping the propagated errors low. OLA does not fit the data at all; the techniques involve finding linear combinations of the frequency differences such that the

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(A)

(B)

FIGURE 4.4 (A) Weighted frequency differences between the Sun and Model S of ChristensenDalsgaard et al. (1996). The weighting factor Qnl is defined in Eq. (4.30) and corrects for the fact that high-degree modes have lower mode inertial than low-degree modes, and thus their frequencies change more for the same perturbation. For l ¼ 0 modes, Qnl ¼ 1 by definition, and Qnl becomes lower as l increases. In this panel, modes of l  30 are marked in black; higher l modes are in gray. (B) Weighted frequency difference between one model constructed with the CanutoeMazzitelli formalism of convection (Canuto and Mazzitelli, 1991) and one constructed with the standard mixing-length formalism; all other inputs are the same. The only difference in the structure of these models lies in the outermost layers and we see that the effect of near-surface errors is of the same order of the frequency dependence of the frequency differences in (A).

same combination of the kernels gives localized averages of the underlying function, again keeping errors small. Details of how these techniques are implemented can be found in works such as Gough and Thompson (1991), Basu (2016), and Basu and Chaplin (2017). Rotation inversions start with Eq. (4.37) or an equivalent one for splitting coefficients. The same techniques, RLS and OLA, modified to cover two dimensions, are used for rotation inversions. Details may be found in Christensen-Dalsgaard et al. (1990), Schou (1991a,b), Schou et al. (1998), etc.

3. INFERENCES ABOUT SOLAR STRUCTURE The most easily obtained information about solar structure are the sound speed, density, and G1 profiles. The sound-speed profile and related quantities can be used to

3. Inferences About Solar Structure

determine other properties such as rb, the position of the base of the CZ, and YCZ, the CZ helium abundance.

3.1 BASIC RESULTS Fig. 4.5 shows the sound speed and density differences between several solar models and the Sun. The sound speed of the models agrees with that of the Sun to fractions of a percent and density matches to within a few percent. Converting the relative differences to solar sound speed at about 0.06R1 is 511 km/s and density is 121 g/cm3. At about 0.96R1 , the sound speed is 60 km/s and density is 0.006 g/cm3 (Basu et al., 2009). The extremely good match between the models and the Sun helped in solving the so-called “solar neutrino problem.” For about 3 decades, the detected flux of solar neutrinos was about a factor of three lower than the predicted flux. For many years, this was thought to be an issue with the solar models. However, as can be seen in Fig. 4.5, the models agree with the Sun to an amazing degree, and if in fact models are constructed to reduce the number of neutrinos, the models no longer agree with the Sun (Bahcall et al., 1998). This indicated that the problem did not lie with the solar models and was more fundamental. As it turns out, the problem lies with the standard model of particle physics, which postulates that neutrinos are massless, and hence electron type neutrinos cannot change into m neutrinos or s neutrinos. There was an additional problem: Neutrino fluxes detected by one type of detector

(A)

(B)

FIGURE 4.5 Relative difference in sound speed (A) and density (B) profiles of the Sun and three solar models. The inversion results are from Basu et al. (2009). Model S refers to the model from Christensen-Dalsgaard et al. (1996), BSB is model BSB(GS98) of Bahcall et al. (2006) and BP04 is from Bahcall et al. (2005).

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were not consistent with the neutrino fluxes detected by a different kind of detector; different detectors have different energy thresholds and are therefore sensitive to neutrinos of different energies produced by different sources. These were clues that the standard model of particle physics is not correct and that neutrinos can change flavor. The Sudbury Neutrino Observatory is sensitive to both electrontype and m-type neutrinos, and the neutrino fluxes detected by the observatory actually match the predicted neutrino fluxes (Ahmad et al., 2002), which confirms that the solution to the solar neutrino problem is a particle physics one.

3.2 BASE OF THE SOLAR CONVECTION ZONE The base of the CZ is the layer in which the temperature gradient changes from being adiabatic to radiative. For low-mass stars such as the Sun that have envelope CZs, the position, rb, of this transition depends on the total opacity of the region, which in turn depends on both the intrinsic opacity and the metallicity. A low opacity results in a shallow CZ; a high opacity results in a deep one. The sudden change in the temperature gradient between the convective and radiative regions results in an abrupt change in the sound speed; in fact, the second derivative of the sound-speed gradient becomes discontinuous. A consequence of this abrupt change is that the sound-speed difference between two models with different rb results in a sharp increase or decrease that can be used to determine rb for the reference model (or the Sun). Fig. 4.6 illustrates this point both for models and for the Sun. This difference can be calibrated to determine rb for the

(A)

(B)

FIGURE 4.6 (A) Relative sound-speed difference between a model with rb ¼ 0.713R1 and models with other values of rb. (B) Relative sound-speed difference between the Sun and models with different values of rb.

3. Inferences About Solar Structure

Sun. Using a related technique, Christensen-Dalsgaard et al. (1991) found rb ¼ 0.713  0.001R1 , a result confirmed by Basu (1998) using better data. Investigations into the latitudinal variation of rb have detected no dependence on latitude (Monteiro and Thompson, 1998; Basu and Antia, 2001a,b). The change at the base of the CZ is an acoustic glitch; it is a region where the sound speed changes over a distance that is shorter than the wavelength of the modes. Such a glitch leaves its signature on mode frequencies. In other stars, this signature is used to determine rb; however, in the case of the Sun, it is put to other use. Gough and Sekii (1993) had suggested that in the case of the Sun, the signature of the acoustic glitch can be used to determine the extent of overshoot below the CZ, because the larger an overshooting region is, the larger the size is of the acoustic glitch and consequently, the amplitude of the signature in the frequencies is larger, as well. Of course, this works only if the overshooting region in the Sun behaves like the overshooting region in solar models; in solar models, overshoot is modeled by extending the adiabatic temperature gradient into the overshooting region in the radiative zone, and this makes the first derivative of the sound speed discontinuous at the edge of the overshooting region. If this is indeed how overshooting works, analyses of the glitch signature show that overshoot in the Sun is extremely small (Christensen-Dalsgaard et al., 1995; Basu et al., 1994; Roxburgh and Vorontsov, 1994). Basu (1997) put an upper limit of 0.05Hp on the extent of overshoot. Of course, such a simple model of overshoot is unlikely to be correct, and the actual behavior of that region is still a matter of active research.

3.3 THE QUESTION OF DIFFUSION Microscopic processes that do not involve nuclear reactions can change the composition of any given layer of a star. If there is a chemical gradient, concentration diffusion will tend to make the layer chemically homogeneous. Even in a chemically homogeneous layer, heavy elements will move toward regions of higher temperature. In addition, pressure gradients cause heavy elements to move toward regions of higher pressure. In a star, pressure stratification is caused by gravity; hence, this process is usually referred to as gravitational settling. All diffusion processes in the context of stellar models, including gravitational settling, are usually referred to under the heading of “diffusion.” The high precision and accuracy of the solar sound speed and density profiles obtained from helioseismic inversions, and in particular the high precision with which the position of the solar CZ base is known, can be used to test physical processes that are relevant. For instance, Fig. 4.7 shows the sound speed and density differences between the Sun and two solar models. The models have the same input physics; however, one was constructed assuming that helium and heavy elements undergo diffusion and gravitational settling; the other did not include this effect. The differences between the models are clear: the model without diffusion gives a much larger difference with respect to the Sun. It also has a shallower CZ. Since

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FIGURE 4.7 Relative difference in the sound speed (left panel) and density (right panel) profiles of the Sun and two solar models: one that includes the diffusion and gravitational settling of helium and heavy elements and one that does not. Otherwise, the models have identical inputs.

Christensen-Dalsgaard et al. (1993) showed that inclusion of diffusion makes models much better, diffusion has been a standard ingredient of solar models.

3.4 CONVECTION ZONE HELIUM ABUNDANCE Helium has a crucial role in stellar evolution. However, in cool stars such as the Sun, it is impossible to detect the photospheric helium abundance, which is the helium abundance in the Sun’s CZ, through spectroscopy. Helioseismic data have proved effective in this regard. Hydrogen and helium ionize close to the solar surface. In the case of helium, this is the first ionization. The singly ionized helium then ionizes completely at about 0.98R1 . Helium ionization, and for that matter, ionization of any species, decreases the adiabatic index G1: the larger the amount of helium, the more pronounced is the decrease in G1 (Fig. 4.8A). Any decrease in G1 causes a decrease in the sound speed relative to what it should have been at a given pressure and density; this is used to determine the abundance of helium. The slightly localized decrease in sound speed can be amplified by calculating the dimensionless sound-speed gradient: r 2 dc2s . (4.39) GM dr The quantity W(r) is generally constant in the CZ, except in the ionization regions. The contribution of the HeII ionization zone to W(r) is much larger than the contribution from heavier elements. Sound-speed inversions can be used to WðrÞh

3. Inferences About Solar Structure

(A)

(B)

FIGURE 4.8 (A) Profile of the adiabatic index G1 in the outer regions of solar models constructed with different helium abundances. The dip around 0.98R1 is the HeII ionization zone. Note that the dip increases with the amount of helium. (B) Dimensionless sound-speed gradient W(r) (Eq. 4.39) for the Sun and models constructed with different equations of state and helium abundances. Note that the shape of the W(r) peak and the height for a given Y is different for models with different equations of state.

determine W(r) for the Sun, which can then be compared with W(r) of models constructed with different helium abundances to determine YCZ, the CZ helium abundance of the Sun. Fig. 4.8B shows W(r) for the Sun and models constructed with different helium abundances for two different equations of state, OPAL (Rogers et al., 1996; Rogers and Nayfonov, 2002) and MihalaseHummereDappen (Da¨ppen et al., 1988; Mihalas et al., 1988; Hummer and Mihalas, 1988). Different equations of state cause slightly different changes in G1 for a given helium abundance; thus, the derived value of YCZ has systematic errors caused by differences in the equation of state between the Sun and solar models. The helium can be obtained from normal inversions after converting  abundance  kernels for c2s ; r to kernels for (u h P/r, Y) under the assumption that the equation of state of the model is the same as the equation of state of the Sun. However, Basu and Antia (1995) showed that these results are more sensitive to equation-of-state effects than others that use the change in sound speed indirectly. Basu and Antia (2004) showed that the CZ helium abundance of the Sun is 0.2485  0.0034. Of course, what matters for the evolution of the Sun is not the CZ helium abundance today, but the helium abundance, Y0, with which the Sun was born.

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The current YCZ is lower than Y0 because of the diffusion and gravitational settling of helium over the lifetime of the Sun. It is possible to work backward and use current constraints to estimate what Y0 was, although there are uncertainties that depend on the parameters and physics processes used to construct solar models. If only standard solar models are used, Serenelli and Basu (2010) found that Y0 has to be 0.278  0.006; the result is independent of metallicity. If, on the other hand, nonstandard models are used with turbulent mixing below the CZ, which effectively reduce the effect of gravitation settling, then Y0 ¼ 0.273  0.006.

3.5 THE ISSUE OF SOLAR COMPOSITION It is ironic that although the Sun and its attributes set the standards for studies of other stars, the composition of the Sun is a matter of some debate. The solar heavy-element abundance Z/X affects structure, mainly through the effect of metallicity on radiative opacities. The most commonly used value of Z/X is 0.023, as determined by Grevesse and Sauval (1998). Updated analyses techniques, including the use of three-dimensional atmospheric models, brought the solar Z/X into question. Asplund et al. (2005) revised the abundances downward to Z/X ¼ 0.0165, but with further improvement, Asplund et al. (2009) increased the abundances to Z/X ¼ 0.0181. However, independent estimates using different three-dimensional atmospheric models yielded Z/X ¼ 0.0209 (Caffau et al., 2011), leaving the field even more confused. It was not merely the total metallicity that changed, but the relative abundances of different elements as well, thus changing the contribution of different elements to the total opacity of any given layer. The effect of metallicity on opacities affects solar structure, and a comparison of models constructed with different heavy-element abundances but otherwise identical physics is shown in Fig. 4.9. It is clear that models constructed with lower metallicities do not match the Sun well: they have shallow CZs and low CZ helium abundances as well. A number of possible solutions have been put forward to mitigate the effect of low abundances on the structure of models; these are discussed in Basu and Antia (2008). However, short of increasing the underlying opacities, none of the solutions works well. Opacities under solar conditions are difficult to test under laboratory conditions. Nevertheless, Bailey et al. (2015) tested iron opacities at temperatures and densities similar to the solar CZ base at the Sandia Z facility. Iron is the second most important source of opacity at the base of the CZ (the most important is oxygen), followed by neon and magnesium (Basu and Antia, 2008). The wavelengthdependent iron opacity was measured by Bailey et al. (2015) to be 30%e400% higher than predicted. The effect on the Rosseland mean opacities is lower. This has led to considerable debate about theoretical calculations of iron opacities (Iglesias and Hansen, 2017; Nahar and Pradhan, 2016; Blancard et al., 2016) and this issue has not been resolved fully. In this regard, more laboratory experiments will be immensely helpful.

4. Inferences on Solar Dynamics

FIGURE 4.9 Relative difference in the sound speed (left panel) and density (right panel) profiles of the Sun and three solar models constructed with different compositions but otherwise identical inputs. GS98 refers to the model with Grevesse and Sauval (1998) composition, AGSS09 is the model with Asplund et al. (2009) composition, and Caffauþ is the model with Caffau et al. (2011) composition.

4. INFERENCES ON SOLAR DYNAMICS Unlike in the case of solar structure, before helioseismic inversions of solar rotation, there was no model that actually predicted solar rotation from first principles. There were models that fitted the observed latitudinal differential rotation at the solar surface, but they gave wrong predictions about the internal rotation profile (Glatzmaier, 1987). Before helioseismology began to reveal the true nature of the Sun’s internal rotation in the 1980s, it was predicted from numerical models and from arguments based on the TayloreProudman theorem (Taylor, 1917; Proudman, 1916) that the rotation would be constant on cylindrical surfaces throughout the CZ, matching the observed surface rotation at the top of the CZ. What helioseismology reveals about solar rotation is different from what was predicted. It is also much more complicated (Thompson et al., 1996; Schou et al., 1998). A comprehensive review of helioseismic inferences about solar rotation may be found in Howe (2009). The mean rotation profile of the Sun is shown in Fig. 4.10. The inversion results do not go deeply into the solar interior because low-degree modes have few splittings; the inversions do not sample the highlatitude regions, either, because there few splittings sample those regions. There are four salient features of solar internal rotation: (1) The Sun does not rotate on cylinders; there is some hint of rotation on cylinders in the near-equatorial regions of the CZ, but for the most part, rotation is nearly constant on cones. (2) The interior

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FIGURE 4.10 Helioseismic inferences of the solar rotation rate as a function of radius and latitude. Only one quadrant is shown. The results are axisymmetric and also symmetric about the equator. The abscissa represents the equator and the ordinate is the axis of rotation; both are calibrated in terms of fractional radius.

rotates almost like a solid body, although the rotation rate of the core is not known well. If g modes were detected unambiguously, the rotation rate of the core could be determined. (3) There is a strong shear layer that roughly coincides with the base of the CZ and is referred to as the “tachocline” (Zahn, 1992). (4) There is a shear layer at the surface and in the CZ. The maximum of the rotation rate occurs inside the Sun. The solar rotation rate shows considerable solar-cycle variation in the form of “zonal flows” discussed further in Section 5.2. Frequency splittings and splitting coefficients carry information about dynamical phenomena that are only north-south symmetric, i.e., those that are identical in both southern and northern hemispheres. There is a class of flows that does not have this feature, the “meridional flows,” which in the case of the Sun flow from the equator to the poles in the near-surface layers. These are discussed further in Section 5.3. Solar frequency splitting and splitting coefficients can also be used to estimate the Sun’s moment of inertia and other parameters such as angular momentum. Antia et al. (2000b) determined the following: Moment of Inertia

I ¼ 7:11  1053 g cm2 ;

Angular Momentum H ¼ 1:91  1048 g cm2 s1 ;  Rotational Kinetic Energy T ¼ 2:57  1042 g cm2 s2 ; Quadrupole Moment

J2 ¼ 2:18  101 .

(4.40)

4. Inferences on Solar Dynamics

4.1 PROPERTIES OF THE TACHOCLINE Early helioseismic inversions to determine the solar internal rotation rate (Brown et al., 1989; Dziembowski et al., 1989) had hinted at the existence of the tachocline; however, the actual characterization of the tachocline had to wait for better data. The tachocline is extremely thin and the change in the rotation rate from being latitudinally differential in the CZ to solid-body as in the radiative zone occurs over a length scale that is too small to resolve in inversions. As a result, the properties of the tachocline are usually determined through a forward modeling exercise: the tachocline is modeled in terms of a position, a width, and a transition in rotation rate. Rotational frequency splittings for such a model are calculated and fitted to the data to determine the properties. In addition to the thinness of the tachocline, it is well-established that the tachocline does not completely coincide with the base of the CZ, and is actually prolate (Charbonneau et al., 1999; Basu and Antia, 2003; Antia and Basu, 2011). Antia and Basu (2011) modeled the tachocline as a two-dimensional function: 8 > dU > > if r  0:70 Uc þ > > 1 þ exp½ðr > t  rÞ=w > > > < dU if 0:70 < r  0:95 Utac ðr; qÞ ¼ Uc þ Bðr  0:7Þ þ > 1 þ exp½ðr t  rÞ=w > > > > > dU > > > : Uc þ 0:25B  Cðr  0:95Þ þ 1 þ exp½ðr  rÞ=w if r > 0:95; t (4.41)

where r is the radial distance in units of solar radius, q is the colatitude, and B ¼ B1 þ B3 P3 ðqÞ þ B5 P5 ðqÞ; dU ¼ dU1 þ dU3 P3 ðqÞ þ dU5 P5 ðqÞ; rt ¼ rd1 þ rd3 P3 ðqÞ; w ¼ w1 þ w3 P3 ðqÞ;

(4.42)

P3 ðqÞ ¼ 5cos2 q  1; P5 ðqÞ ¼ 21cos4 q  14cos2 q þ 1: The central point of the transition region in the rotation rate is given by rt and is thus the “position” of the tachocline. The half-width of the tachocline is given by the quantity w, and dU is the jump in the rotation rate across the tachocline. The properties of the tachocline, as determined by Antia and Basu (2011), are shown in Table 4.1. We show the weighted means of the Global Oscillation Network Group (GONG) and Michelson Doppler Imager (MDI) results. As can be seen from the table, the center of the tachocline is in the radiative zone at the equator, but well within the CZ at higher latitudes, giving it a prolate shape. The tachocline is also considerably thicker at high latitudes than at low latitudes. We do not show the results at 75 degrees because the GONG and MDI results are not consistent with each other at that latitude.

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Table 4.1 Properties of the Solar Tachocline Latitude

rt r/R1

w R1

dU nHz

0 15 30 45 60

0.6933  0.0010 0.6954  0.0010 0.7010  0.0010 0.7085  0.0011 0.7145  0.0014

0.0034  0.0009 0.0043  0.0007 0.0086  0.0005 0.0166  0.0012 0.0250  0.0022

18.87  0.2947 14.18  0.2181 1.65  0.2181 30.68  0.2567 68.18  0.3483

5. HELIOSEISMIC INFERENCES ON THE SOLAR CYCLE Solar p modes show temporal variations in their parameters that correlate strongly with the 11-year cycle of surface activity. Accumulated helioseismic data from dedicated ground-based networks and space missions reveal a variety of these solar-cycle related signatures, which in turn have allowed inferences to be made about subtle structural changes in the subsurface layers of the solar interior.

5.1 CHANGES TO GLOBAL-MODE FREQUENCIES AND MODE PARAMETERS The first detection of temporal, cycle-related changes in p-mode parameters was made by Woodard and Noyes (1985). They used data on low-l p modes, collected by the Active Cavity Radiometer Irradiance Monitor instrument onboard the Solar Maximum Mission satellite to uncover a systematic decrease in the frequencies of around one part in 104 between 1980 and 1984, during which time levels of solar activity declined from peak to much lower levels. These results were independently confirmed and improved (thanks to higher-quality low-l data) by Palle et al. (1989) and Elsworth et al. (1990) and then extended to medium l by Libbrecht and Woodard (1990). Examples of modern data on low-l frequency shifts, averaged over modes of different frequencies, are shown in Fig. 4.11. Also shown for reference are data for four commonly used global proxies of solar activity: the International Sunspot Number, the 10.7-cm radio flux, the Kitt Peak global magnetic field index, (BKP ) and the Ca K index. The frequencies of the modes are seen to track the changing activity, with low-l modes at the center of the p-mode spectrum increasing in frequency by about 0.4 mHz between solar minimum and maximum. Also apparent in the helioseismic frequency shifts is a “quasibiennial” signal manifesting as a quasiperiodic x0.1-mHz variation of the p-mode frequencies, superimposed on

5. Helioseismic Inferences on the Solar Cycle

2400 ≤ ν ≤ 2920 μHz

0.2

RF BKP SSN CaK

0.1

δ ν (μ Hz)

0.0 −0.1 −0.2 −0.3 −0.4 −0.5 1985

Cycle 22

1990

Cycle 23

1995

2000 Year

2005

Cycle 24

2010

2015

FIGURE 4.11 Mode frequency shifts (in mHz) from the Birmingham Solar-Oscillations Network, averaged in frequency for modes in the range 2400  n  2920 mHz and covering activity Cycles 22e24. Also plotted are four global proxies of solar activity: the International Sunspot Number (SSN), the 10.7-cm radio flux (RF), the Kitt Peak global magnetic field index (BKP) and the Ca K index (CaK).

the x0.4-mHz 11-year cycle shifts (Broomhall et al., 2009; Fletcher et al., 2010). Quasibiennial signatures are also known to manifest in global proxies of surface activity (Bazilevskaya et al., 2014). From resolved-sun data on medium-l modes, one may make surface maps showing the strength of the shifts (relative to some reference epoch or overall average) as a function of latitude and time (Howe et al., 2002). The example given here in Fig. 4.12 reveals a striking spatial correlation of the shifts with the surface magnetic fields. The frequency shift of a given mode evidently depends on the strength of that component of the surface magnetic field that has the same spherical harmonic projection on the surface. The general trend is for higher-frequency modes to experience a larger shift than their lower-frequency counterparts. This frequency dependence suggests that the perturbations responsible are located close to the solar surface (Libbrecht and Woodard, 1990). The upper boundaries of the cavities of the modes (whose radial locations show only a weak dependence with l) lie deeper in the Sun for low-frequency modes than they do for high-frequency modes. As such, higher-frequency modes are more sensitive to perturbations confined close to the surface. That the dominant perturbations are not spread appreciably in radius can be understood by considering the internal propagation of the acoustic waves. When a

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CHAPTER 4 Helioseismic Inferences on the Internal Structure

δ ν(μ Hz)

60

Latitude

112

40

20

1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0

0 2000

2005 Year

2010

2015

FIGURE 4.12 Mode frequency shifts (in mHz) as a function of time and latitude. The values come from analysis of Global Oscillation Network Group data. The contour lines indicate the surface magnetic activity.

wave reaches the lower boundary of its cavity, by definition it will be moving horizontally and its phase speed will be: unl 2pnnl R (4.43) ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffifnnl =L; kh lðl þ 1Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where kh is the horizontal wavenumber, L ¼ lðl þ 1Þ, and R the outer cavity radius (this is to a very good approximation the solar radius). The ratio nnl/L therefore relates to the location of the lower boundary of the cavity, and hence also the cavity size. The shifts do not scale with this ratio, and so the perturbations cannot be spread so widely as to span the entirety of the cavities of many of the modes. The precise nature of the dependence of the shifts on the mode inertia and mode frequency tells us something about the specific location and nature of the nearsurface perturbations. First, as l increases, so the mass engaged in pulsation Mnl h Inl M1 , where Inl is the inertia, decreases and the more sensitive a mode will be to a near-surface perturbation of a given amplitude. This behavior is apparent in Fig. 4.13. The top panel shows activity minimum-tomaximum frequency shifts as a function of frequency for modes of different degree l. At fixed frequency, modes of higher l show a larger shift than their lower-l counterparts. The shifts may be rendered independent of l by multiplying them by the inertia ratio Qnl defined earlier in Eq. (4.30). The bottom panel of Fig. 4.13 shows the shifts from the top panel after normalization by the inertia ratio. cs ¼

5. Helioseismic Inferences on the Solar Cycle

1.5

δ ν ( μHz)

1.0

l= 5 l= 65 l=125

0.5

0.0

−0.5 1000

1500

2000

2500 3000 ν ( μHz)

3500

4000

4500

1500

2000

2500 3000 ν ( μHz)

3500

4000

4500

1.2

Qnlδ ν ( μHz)

1.0 0.8 0.6 0.4 0.2 0.0 −0.2 1000

FIGURE 4.13 Top panel: Mode frequency shifts (in mHz) from activity minimum to maximum, as a function of frequency, for modes of different values of degree l. Bottom panel: Frequency shifts from the top panel after normalization by the inertia ratio Qnl. Note the resemblance of this figure to Fig. 4.3A; this indicates that the cause of the frequency changes must be confined to near-surface layers of the Sun.

The frequency dependence of these inertia-normalized shifts, dnnlQnl, may then be described as a power law in frequency, i.e., c dnnl Qnl ¼ nanl ; (4.44) Inl where the power-law index is a and c is a constant. A perturbation located within the photosphere but confined to extend no more than one pressure scale height has a x 3 (Libbrecht and Woodard, 1990). When the perturbation extends beneath the surface, the frequency dependence will be weaker and a will be smaller (Gough, 1990). Chaplin et al. (2001) found that for low-l modes a x 2 for n  2500 mHz, whereas a is approximately zero for n < 2500 mHz. RabelloSoares et al. (2008) repeated the analysis for medium-l and high-l modes, finding similar behavior. Because they had more medium- and high-l data at low

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frequencies, they were able to extend the analysis to n < 2000 mHz, where they found that a appears to change sign and become negative. The high-l modes provide important information because they are confined in the layers close to the surface where perturbations are located. These results suggest that the perturbations cannot arise solely in the photosphere. Because a is smaller for the lowerfrequency modes, the results suggest that the perturbations extend to greater depths the lower in frequency one goes. The physical mechanism responsible for the perturbations is undoubtedly associated with the magnetic activity. Magnetic fields can affect the modes directly by the action of the Lorentz force on the plasma providing an additional restoring force and an increase in frequency (and the appearance of new modes); or they can affect the modes indirectly via their impact on the physical properties of the mode cavities (which can increase or decrease the frequencies). Theoretical work (Dziembowski and Goode, 2005) suggests that indirect effects dominate, in particular changes to the near-surface stratification owing to the suppression of convection by the magnetic field. However, the direct effect is likely to be more important in deeper layers, where the fields are stronger. Helioseismic studies have revealed intriguing indications of near-surface structural changes in the lead-up to weak activity Cycle 24. That cycle proved to be significantly weaker in a variety of activity proxies and indicators (Hathaway, 2015) than the preceding several cycles of the modern Grand Maximum. Indeed, peak levels of activity as seen in the long-standing sunspot record had not been as low for about 100 years (e.g., Cycles 14 and 15). Not surprisingly, measured frequency shifts of the p modes were found to be significantly weaker than in the preceding cycles (Basu et al., 2012; Salabert et al., 2015; Tripathy et al., 2015; Howe et al., 2017a). However, more subtle changes were also revealed. Basu et al. (2012) found that the fairly close correspondence of the frequency shifts of low-l modes to the sunspot number and the 10.7-cm radio flux altered during an epoch extending from the end of Cycle 22 into the Cycle 23 maximum. Modes with frequencies below x2400 mHz saw their correlation with the proxies deteriorate first, and this was followed by a drop in correlation for higherfrequency modes. As activity levels fell in the falling phase of Cycle 23, the correlations broadly recovered for higher-frequency modes but failed to do so at lower frequencies. Basu et al. (2012) interpreted this unusual behavior in terms of a thinning of the layer of magnetic field at depths less than about 3000 km beneath the solar surface, with the shrinkage in radius sufficient to render the lower-frequency modes less sensitive to the perturbations arising from this field (recall from the earlier discussion that the outer boundaries of the cavities of lower-frequency modes lie at greater depths than for higher-frequency modes). Subsequent work (Howe et al., 2017b) that extended the study through the declining phase of Cycle 24 showed that the structural changes appeared to have persisted. Moreover, that analysis revealed a more general, subtle change of the sensitivity of the modes to changing

5. Helioseismic Inferences on the Solar Cycle

levels of magnetic activity, which the researchers speculated might be result from a change in the balance between amounts of magnetic flux tied up in ephemeral versus active regions. Predating that work had been results suggesting possible solar-cycle changes in the near-surface He II ionization zone, which lies at a depth of approximately 14,000 km. Basu and Mandel (2004) studied variations over time of secondfrequency differences computed from medium-l data, combinations that isolate signatures of abrupt structural change in the interior, like the He II zone. From their analysis, they uncovered indications of solar-cycle variations in the amplitude of the depression in the adiabatic index, G1, in the He II zone. They interpreted these variations, if genuine, as being caused by the influence of changes to the nearsurface magnetic field on the equation of state of the gas in this layer. Similar cyclic signatures were found by both Verner et al. (2006) and Ballot et al. (2006) using lowl data. Gough (2013) subsequently questioned these results. He pointed out that any changes in the He II ionization zone should also be apparent as a periodic signature in the frequency shifts, when frequency is used as the independent variable. He suggested that if the second-frequency-difference signatures were explained by changes to the near-surface magnetic field, such changes would have to be large and would have resulted in conspicuous periodicities in the basic frequency-shift data that have not been seen. Further observations and analysis are clearly needed to resolve these apparent discrepancies. Variations in global f-mode frequencies reveal information about changes in a thin layer that extends just a few megameters below the base of the photosphere. Because the f-mode frequencies depend only on the local acceleration caused by gravity, measured shifts are interpreted in terms of a few-kilometer change in the “seismic radius” (Dziembowski et al., 1998; Antia et al., 2000a). Results suggest that as activity rises, there is both expansion and contraction of layers close to the surface; however, it is not yet possible to reconcile the observations with theoretical predictions of the variations (Lefebvre et al., 2007). As we have seen, the dominant contribution to the p-mode frequency shifts is undoubtedly sourced in the superficial layers of the interior. However, there have also been claims that once these contributions are removed, subtle signatures of deeperlying changes are present. Chou and Serebryanskiy (2005) and Serebryanskiy and Chou (2005) found signatures consistent with perturbations to the sound speed at depths of 0.65e0:67 R1 , just beneath the base of the convective envelope. Baldner and Basu (2008) used a completely different analysis approach, based on principal component analysis. Having isolated the dominant near-surface perturbation (as the first component), they, too, found evidence (in the subsequent components) of much deeper-lying changes. The origins of the quasibiennial frequency variations remain unclear. However, that this biennial signature has similar amplitude in lower-frequency and higherfrequency modes suggests that its origins lie deeper than the very superficial layers responsible for the dominant 11-year shifts (Broomhall et al., 2012). Simoniello

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et al. (2013) suggested that the signal might be caused by beating between a magnetic dipole and quadrupole dynamo mode, whereas Beaudoin et al. (2016) presented results from theoretical mean-field dynamo simulations that lend support to the hypothesis that the signal is associated with a secondary dynamo process operating within the solar CZ. Finally in this section, solar-cycle variations have also been uncovered in mode parameters associated with the excitation and damping of the modes (Chaplin et al., 2000; Komm et al., 2000). The damping of the modes is observed to increase by about 15% between activity minimum and maximum, with some suggestion of the variations being peaked in size at about x3000 mHz. At the same time, mode powers are observed to decrease by about the same fractional amount. As pointed out by Chaplin et al. (2000), the relative size of these changes can be explained as being the result of variations in damping alone, with the time-averaged forcing of the modes remaining unchanged over the solar cycle. Analysis of changes to the medium-l modes shows a clear spatial correlation with the evolution of the active-region field (Komm et al., 2002). Houdek et al. (2001) noted that the high Rayleigh number of the magnetic field could modify the preferred horizontal length-scale of the convection, which in turn could affect the damping rates. Their numerical calculations of the implied changes for low-l modes show reasonable agreement with the observations.

5.2 ZONAL FLOWS The so-called surface torsional oscillations were discovered by Howard and Labonte (1980) and correspond to bands of plasma that rotate either slightly faster or slower at particular latitudes than the smooth, underlying pattern of differential rotation. These zonal (i.e., eastewest) signatures show a strong band that migrates toward the equator during the solar cycle, like sunspots and active regions. Helioseismology has revealed that these surface signatures penetrate the CZ (Howe et al., 2000a; Antia and Basu, 2000, 2001; Vorontsov et al., 2002). Moreover, the subsurface zonal flows show a strong, poleward-moving branch that appears to penetrate the entire zone. The amplitudes and phases of the signals show systematic variations with position in the zone (Howe et al., 2005). Fig. 4.14 shows helioseismic measurements of zonal flows at a target radius of 0:99 R1 . At a given latitude, the flows are seen to accelerate before the appearance of magnetic surface features. The appearance of surface activity usually coincides with equatorward flows, reaching a latitude of about 25 degrees. The poleward branch of the flows starts at about same time as the equatorward branch. Fig. 4.14 clearly bears a striking resemblance to the well-known butterfly diagram, which shows the latitudinal migration of sunspots during the solar cycle. The observed behavior of the flows as such suggests that they are the result of magnetic effects associated with the solar cycle. The action of the Lorentz force on the plasma is an obvious candidate; this was originally proposed by Schuessler (1981) and Yoshimura (1981) to explain the surface torsional oscillations. Subsequent

5. Helioseismic Inferences on the Solar Cycle

60 40

2 δ Ω /2π (nHz)

Latitude

20 0 −20 −40

1 0 −1 −2

−60 1995

2000

2005 Year

2010

2015

FIGURE 4.14 Helioseismic inferences on zonal flows at a depth of 0:99R1 : variations in the solar internal rotation, relative to the rotation at the epoch of solar minimum, as determined by analysis of Global Oscillation Network Group, Michelson Doppler Imager, and Helioseismic and Magnetic Imager observations.

studies have considered the Lorentz force, owing to the small-scale magnetic field, driving turbulent Reynolds stresses (Kueker et al., 1996) and driving from gradients of temperature caused by the changing magnetic field (Spruit, 2003). Rempel (2007) forced torsional oscillations in a mean-field differential rotation model and found that the poleward-moving branch could be explained by either the Lorentz force or thermal driving; however, in his models, the equatorward-moving branch needed a thermal source to be successfully driven. Howe et al. (2009) found that at the minimum between Cycles 23 and 24, the equatorward branch started late compared with the previous cycle. They flagged this delayed migration as a possible precursor of the delayed onset of Cycle 24. Moreover, as Cycle 24 progressed, at first it seemed that the poleward branch had failed to begin as expected; however, when Howe et al. (2013) subtracted a shorter-term average of the underlying pattern of rotation, instead of a multicycle average, the poleward flow became visible. This indicated that the underlying pattern of rotation had changed at higher latitudes between Cycles 23 and 24. This change may be related to the weak polar fields seen in Cycle 24 (Rempel, 2012). The poleward branch remained relatively weak throughout Cycle 24, but interestingly, the equatorward branch appears to have shown the opposite behavior, i.e., it has been stronger in Cycle 24 than 23 (Howe et al., 2017b).

5.3 MERIDIONAL FLOWS Meridional (i.e., northesouth) flows have an important role in flux-transport dynamos. The surface signatures can be measured using standard Doppler or

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magnetic feature tracking techniques. As per the zonal flows, helioseismology has made it possible to measure the meridional circulation beneath the solar surface, providing invaluable data for constraining dynamo models. The helioseismic signatures of the flows are extremely hard to disentangle from the global-mode frequencies because they are subtle and second-order in nature; however, they are much more straightforward to extract using local helioseismic techniques. Here, one studies the propagation of waves in local volumes beneath the surface. Unlike global-mode helioseismology, which can provide inferences in two dimensions, local helioseismology provides a three-dimensional picture of the region being probed. The first estimates of subsurface meridional circulation were made using timeedistance measurements (Giles et al., 1997), a technique that is analogous to those used in terrestrial seismology. Subsequent measurements came from ringdiagram analyses (Basu et al., 1999; Haber et al., 2002, etc.), which involve the measurement of oscillation frequencies from data on localized patches of the Sun (see Hill, 1988; Patr’n et al., 1997, for details of the technique). The current picture shows that up to latitudes of 70 degrees, the flows are poleward directed, with typical amplitudes of about 15 to 20 m/s. Further analysis suggests the flows probably occupy the entire CZ and are directed equatorward within approximately the lower third of the zone, where velocity amplitudes are likely significantly lower than at the surface (Giles, 2000). Some studies suggest that there may be more than one meridional cell present in the CZ (Haber et al., 2001; Zhao et al., 2013) Analysis of long-term observations has revealed that the residual meridional flow (i.e., the residuals left after subtracting a long-term average) shows “inflow” signatures into active latitudes (Gizon, 2004 and Zhao and Kosovichev, 2004). At latitudes on the equatorward side of active regions, the flows tend to be more poleward in nature, whereas on the poleward side, they tend to be equatorward. Just like the equatorward branch of the zonal flows, these residual meridional signatures also migrate toward the equator as the cycle progresses (Komm et al., 2015). The meridional flow is faster around times of cycle minimum and slower around times of cycle maximum (Komm et al., 1993; Basu and Antia, 2003, 2010; Hathaway and Rightmire, 2010, 2011). We show an example in Fig. 4.15. Studies also show intriguing asymmetries in these time-dependent variations (Komm et al., 2015). The meridional flow was found to be slower at Cycle 22/23 minimum than it was at Cycle 23/24 minimum; Hathaway and Rightmire (2010) noted that the faster flow at the Cycle 23/24 boundary may have contributed to weaker polar fields and also the extended minimum and weaker subsequent cycle. Indeed, Jiang et al. (2010) noted that in flux-transport dynamos the inflows will reduce the strength of the polar fields. Jiang et al. (2015) proposed that observed weak polar magnetic fields, and as a result, the weak Cycle 24, may have resulted from the emergence of low-latitude flux having the polarity opposite that expected (which then hindered growth of the polar fields).

5. Helioseismic Inferences on the Solar Cycle

(A)

(B)

FIGURE 4.15 Helioseismic inferences on the northesouth component of solar meridional flows as obtained with Michelson Doppler Imager data. Results are shown at two depths for epochs close to the minimum of Cycles 23 and 24 and the Cycle 23 maximum. (A) 0.998R1 . (B) 0.990R1 . Results are from Basu and Antia (2010).

5.4 THE 1.3-YEAR PERIODICITIES NEAR THE TACHOCLINE Claims that the rotation rate in the layers just above and below the tachocline varies on a timescale of z1.3 years (Howe et al., 2000b) remain controversial. When they were first uncovered, the changes appeared to be most prominent in the low-latitude regions just above the base of the convective envelope. At the same time, there were suggestions of variations in antiphase some z60,000 km deeper, in the outer parts of the radiative zone. The uncovered variations then all but disappeared in the midlatitude regions, whereas a periodic signal closer to 1 year was found at higher latitudes. The apparent signatures were in antiphase above and below the tachocline. If real, they might suggest angular momentum exchange between the interior and envelope, mediated by the tachocline. However, independent analyses (Antia and Basu, 2000; Basu and Antia, 2001a,b; Corbard et al., 2001) subsequently failed to confirm the results. Moreover, later analyses by Howe et al. showed that the quasiperiodic signal was no longer present after about 2001 (Howe et al., 2007) and also cast doubt on the veracity of the higher-latitude 1-year signal. Intriguingly, there are reports in the literature of quasie1.3-year periodicities in observations of sunspots and geomagnetic indices (Howe, 2006). The nature of the claimed tachocline signals remains unresolved.

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5.5 CHANGES IN EVEN-ORDER A COEFFICIENTS As mentioned, rotation is the biggest cause of deviation from spherical symmetry of the Sun. However, the magnitudes of the even-order a coefficients suggest that there are other contributions to asphericity; these could be magnetic fields, aspherical sound-speed profiles, or both. Unfortunately, even-order a coefficients cannot distinguish among these. As a result, analyses of even-order a coefficients assume that once second-order effects of rotation are subtracted, the remaining signatures in the coefficients are either the result of magnetic fields or of structural asphericities. GONG, MDI, and Helioseismic and Magnetic Imager observations show that the even-order coefficients also show solar-cycle related changes. Antia et al. (2001) assumed that the even-order a coefficients were caused by structure asphericities; they showed that the coefficients could be interpreted as being caused by a latitudinally dependent distribution of sound speed. The latitudedependent sound-speed profile showed solar cycleerelated changes at layers above 0.98R1 . Similarly, Baldner and Basu (2008) found significant latitude-dependent sound-speed changes in the active latitudes. Analysis of local helioseismic data also show substantial activity-dependent latitudinal variations in sound speed and adiabatic index G1 (Basu et al., 2007). As mentioned, the cause of the even-order a coefficients is ambiguous. Antia et al. (2000c) interpreted them as being caused by magnetic fields and showed that the signal could be explained as a 20-kG field localized at a depth of 3 Mm below the surface. Baldner et al. (2009) also analyzed the coefficients, assuming that they carry the signature of magnetic fields; they showed that at each epoch, the data could be explained as being the result of a poloidal field and a doublepeaked, near-surface toroidal field with peaks at 0.999 and 0.996R1 . The field were correlated with solar activity and the ratio of the strengths of the poloidal and toroidal fields was nearly a constant.

6. SEISMIC STUDIES OF OTHER STARS We have focused in this chapter on a detailed seismic study of the Sun. Other stars that have outer CZs also oscillate like the Sun; these stars are usually referred to as solar-like oscillators. They can also be studied using their oscillation frequencies (Brown and Gilliland, 1994). The space missions Convection, Rotation and planetary Transits and Kepler have observed tens of thousands of different types of stars, which have enabled the field of asteroseismology to flourish. See Chaplin and Miglio (2013), Aerts (2015), Gilliland (2015), and so on for reviews of the field. Asteroseismic analyses are different from helioseismic ones. For one thing, the amount of seismic data available on other stars is limited: we can usually obtain frequencies only for l ¼ 0, 1, 2, and 3 modes. For another, the number of constraints is also limited: for the Sun, we have independent estimates of mass, radius, and age. For other stars, these quantities have to be determined from the same set of data

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used to probe internal structure. However, asteroseismic data sets contain a large variety of stars at different evolutionary stages that have different seismic properties compared with main-sequence stars such as the Sun. The limited number of constraints and data available for other stars has meant that specialized techniques have had to be developed to interpret these data; Basu and Chaplin (2017) detail some of these methods. Most analysis techniques involve forward modeling, although there has been some success in inverting for the core structure of another main-sequence star (Bellinger et al., 2017). Most asteroseismic analyses have involved determining masses, radii, and ages of stars (Chaplin et al., 2014; Pinsonneault et al., 2014; Serenelli et al., 2017) and characterizing exoplanet hosts (Ballard et al., 2014; Silva Aguirre et al., 2015; Campante et al., 2015), although there has been some work dealing with detailed modeling and the characterization of stars (Silva Aguirre et al., 2017) and analyses of acoustic glitches (Verma et al., 2014; Mazumdar et al., 2014). Future space missions such as TESS and PLATO will observe thousands of other stars, including the very brightest stars in the sky.

ACKNOWLEDGMENTS The authors are grateful to Rachel Howe for help with figures.

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5 SUBCHAPTER

Spectroscopy and Atomic Physics

5.1 Philip G. Judge

National Center for Atmospheric Research, High Altitude Observatory, Boulder, CO, United States

CHAPTER OUTLINE 1. 2. 3. 4. 5.

Overview ........................................................................................................... 128 Regimes of Solar Plasmas .................................................................................. 128 Origin and Types of Atomic Transitions ................................................................ 132 Atomic Structure ................................................................................................ 133 Spectrum Formation in a Nutshell........................................................................ 138 5.1 Optically Thick Formation .................................................................... 140 5.2 Optically Thin Formation...................................................................... 140 5.3 NoneLocal Thermodynamical Equilibrium and Further Complications...... 143 6. Plasma Spectroscopy ......................................................................................... 145 7. Closing Remarks ................................................................................................ 152 References ............................................................................................................. 153

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1. OVERVIEW We review the spectroscopy of atoms embedded in plasmas, specifically in solar plasmas. The general goal of remote spectroscopic work is to discover properties of the emitting plasma, whether they are in small laboratory plasmas or clusters of galaxies. Fujimoto (2004) calls such studies “plasma spectroscopy,” as opposed to “atomic spectroscopy,” whose focus is atomic structure. In the Sun we can use spectroscopy to measure elemental abundances, plasma motions, densities, temperatures, and magnetic and electric fields. First, we place the Sun into the context of other plasmas. Then we review the origins of spectral lines and continua in unperturbed atoms, and present scalings for atomic parameters. Next, we sketch the formation of solar spectral features, relating the emergent spectrum to elementary ideas from radiative transfer, under local thermodynamical equilibrium (LTE), non-LTE, coronal, and collisional-radiative (CR) conditions. We examine the interactions of individual atoms and atomic ions with solar plasmas, stressing scaling laws and highlighting common threads. The reader can refer to reviews on plasma spectroscopy by Cooper (1966), on the interpretation of spectral intensities from laboratory and astrophysical plasmas by Gabriel and Jordan (1971), and on atomic processes in the Sun by Dufton and Kingston (1981).

2. REGIMES OF SOLAR PLASMAS The solar spectrum originates from plasma in a variety of regimes. A low-resolution UV spectrum showing emission and absorption lines and various continua is shown in Fig. 5.1.1. Based on more than a century of research using data such as these, Fig. 5.1.2 shows the range of solar plasmas in the context of astrophysical and terrestrial plasmas. In astronomy it is usual to term the visible surfaces of stars as “atmospheres,” those regions from which the bulk of the luminosity of the star is emitted. However, the photosphere and chromosphere are also partially ionized plasmas. A plasma is characterized and defined through three criteria: 1. the Debye screening length (using Gaussian units) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffi rD ¼ kTe =4pe2 ne ¼ 6:92 Te =ne cm

(5.1.1)

must be far smaller than the macroscopic size of the plasma. The plasma is quasineutral and its dynamics are controlled mostly through its own bulk selfinteractions and not with boundaries; 2. the plasma parameter L ¼ 4pne r3D is [1; and 3. the electron plasma frequency qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi (5.1.2) upe ¼ ne e2 =pme ¼ 8:98  103 ne rad=s

FIGURE 5.1.1 A “classic” figure of the solar spectrum, highlighting emission and absorption features across the solar atmosphere. Some strong lines are deep in absorption yet have emission in the core (Mg II is the case shown here; emission is not visible at this modest spectral resolution). Notice the Dn ¼ 1 transitions of H, He I, and He II, marked with asterisks, at comparable wavelengths with Dn ¼ 0 transitions of much more highly ionized species. Adapted from Scheffler, H., Elsa¨sser, H., 1974. Physik der Sterne und der Sonne.

Solar core

1025

ne cm-3

1020

LTE

metals F-D M-B

Λ

lightning arc photosphere chromosphere

1015 1010

Ionosphere

105

EBIT--> Active coronaFlares Quiet corona

PN

Magnetosphere interplanetary 1AU

100 interstellar

10-5 10-2

10-1

pinch fusion tokamak

AGN BLR

flame

nLTE

lasers

=1

15

10

Λ=

Galaxy halos

100

101

102

103

104

105

Te eV

FIGURE 5.1.2 Plasmas are identified as a function of electron temperature (1 eV h 11,605K) and electron density. Loci for plasma parameters L ¼ 1 and L ¼ 1015 are shown, in which L is the number of free electrons inside a Debye sphere. Plasma behavior requires L [ 1. Solar plasmas are identified in red; the plus sign shows weakly ionized plasmas. The dotdashed lines separate regions in which local thermodynamical equilibrium (LTE) versus non-LTE conditions prevail; the upper line is for ionization and the lower line is for a typical E1 line. The dashed line separates Fermi-Dirac from MaxwelleBoltzmann statistics for electrons. EBIT, electron beam ion trap device; PN, planetary nebulae.

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CHAPTER 5 Atmospheric structure, Non-Equilibrium Thermodynamics

exceeds collision frequencies with neutral particles. Electrostatic interactions domipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nate the electron dynamics. (Note that rD upe ¼ kTe =4p2 me ). All observable solar plasmas easily satisfy these definitions. They clump in the center of Fig. 5.1.2. They have lower densities than terrestrial plasmas and higher densities than photoionized astronomical plasmas. The spectra in Fig. 5.1.1 form in the corona (x-rays), chromosphere (Ly a), transition region (C IV), and photosphere (Mg II line wings). The mean times between particle collisions within plasmas are a matter of some delicacy, because collisions involve solutions to the Boltzmann transport equation with its associated closure problem. Assuming that departures from thermal distributions are small (Braginskii, 1965; Mihalas and Mihalas, 1984), approximate times needed for electrons and ions with charge Z  1 and ionization potential IZ1 to relax to the same temperatures are of the order  3=2 1 6 kTe see w 1:6  10 s (5.1.3) ne R rffiffiffiffiffiffi 3=2 mi kTi 1 4 7 sii w7  10 ðZ  1Þ s (5.1.4) ni mp R   1 2 mi kTe 9 4 kTi mp þ 5:4  10 sie w1:4  10 ðZ  1Þ s. (5.1.5) mp R R m i ni For neutrals (Z ¼ 1), collision times are gas-kinetic (cross-sections are z pa2a , a0 ¼ -2/2mee2 is the Bohr radius) or sometimes are dominated by resonant processes such as charge transfer. In the case of charge transfer, one must make distinctions between times for the transfer of momentum and energy. Z is the net core charge “seen” by the outermost “optical electron.” For H I Z ¼ 1; for He II Z ¼ 2, etc. Temperatures  are normalized by the Boltzmann constant k and Rydberg energy unit R ¼ e2 2a0 . In anticipation of later sections, here are time scales for spontaneous radiative transitions and “inelastic” collisions involving changes-of-state of the internal structures of atomic ions by electron impact: sE1 w108 Z 4 s; spontaneous decay of excited levels through electric dipole ðE1Þ transitions sm w10þ1 Z 8 s; metastable levels     IZ1 2 IZ1 1 9 1 exp s; ionization sion w9  10 pffiffiffiffiffi R kTe ne Te srr w2  1010 Z 2 Te0:7

1 s; radiative recombination ne

(5.1.6) (5.1.7) (5.1.8) (5.1.9)

2. Regimes of Solar Plasmas

In the Sun’s atmosphere, srr can be small or, for chromospheric ions with small Z, they can exceed 60 s, longer than dynamical times of interest (Carlsson and Stein, 2002; Judge, 2005). In any case, sion wsrr [sie [sii [see ;

(5.1.10)

and sion wsm [see [sE1 . (5.1.11)  ffiffiffiffiffi p h Another parameter is LTe ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 7  106 Te , the “thermal” de Broglie 2pme kTe wavelength of electrons. When ne L3Te  1; or ne  3  1015 Te3=2 ;

(5.1.12)

the electrons are nondegenerate and obey MaxwelleBoltzmann statistics; otherwise Fermi-Dirac statistics must be used. Fig. 5.1.2 shows a dashed line demarking regimes of classical and quantum statistics. With some notable exceptions (e.g., charge transfer) (Butler and Dalgarno, 1980), free electrons dominate the plasmaeatom interactions. The Maxwelle Boltzmann distribution for electron speed v is   3=2  m mv2 f ðvÞdv ¼ exp  (5.1.13) 4pv2 dv 2pkTe 2kTe which is normalized to unity. We use the notation Z ∞ vsðvÞf ðvÞdv hvsi ¼

(5.1.14)

0

to describe the probability that per each impacting electron, a process described by the cross-section s(v) (such as for electron impact ionization) takes place, its units are cm3/s. When atomic ions are in thermal equilibrium with a bath of electrons, two level populations q2 and q1 obey (* denotes LTE):   q2 g2 E2  E1 ¼ exp  . (5.1.15) q1 g1 kTe The equation describing the ionization state of stages of the element of interest in equilibrium is the Saha equation (which allows for the extra degrees of freedom of the free electron):   qZ 2 UZ ðTe Þ EZ  EZ1 exp  . (5.1.16) ne  ¼ 3 qZ1 LTe UZ1 ðTe Þ kTe Here, U are partition functions and IZ1 ¼ EZ  EZ1 is the ionization potential of ion Z  1. Truly isolated atoms have U ¼ ∞, a problem resolved in real plasmas via collisions and plasma microfields (Section 6). The lower dot-dashed line in Fig. 5.1.2 separates LTE from non-LTE conditions between bound levels, in which ne(Te) ¼ A z 108Z2 s, divided by

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CHAPTER 5 Atmospheric structure, Non-Equilibrium Thermodynamics

Cz104Z 2T 1/2 cm3/s, assuming that ions of charge Z  1 form near Te z 104Z2 (1 < Z < 92). This locus is meaningful only on a log-log plot. Also, we will see that for plasmas close to 2-body ionization equilibrium (equality of Eqs. 5.1.8 and 5.1.9), IZ1> and (often [) at temperatures where ion Z  1 is abundant.

3. ORIGIN AND TYPES OF ATOMIC TRANSITIONS The full intricacy of atomic structure and associated transitions is highly specialized. It is not surprising that atomic physics is complicated. One must solve “many-” and “few-” body problems that have no closed-form solutions, even in the classical limit. It draws upon mathematics of vector spaces, symmetries, special relativity, nuclear physics, and quantum electrodynamics. The purpose here is to provide some physical insight. There are many textbooks on atomic structure and spectra. A fine reference is Cowan (1981). From where do “spectral lines” originate? One of the greatest triumphs in 20th-century physics was the development of wave mechanics, based on clear experimental evidence of the need to abandon or build on classical mechanics. In mathematical language, the wavefunctions j for isolated atoms are solutions to H0 j ¼ Ej;

(5.1.17)

where H0 is the Hamiltonian. The fundamental postulates of quantum mechanics are equivalent to writing particle momenta p and energy E in the form of operators p/  iZvi ; E/iZvt ;

(5.1.18)

which operate on complex wavefunctions j(xi,t). According to Born, we are to interpret j*(xi,t)j(xi,t) as the probability of finding a particle at the given position and time. Importantly, these operators are fully compatible with Einstein and de Broglie’s arguments that E ¼ -u and p ¼ -k from experiments, implying that particles behave like waves, with f w exp i(k.r  ut). With these substitutions for p and E, Eq. (5.1.17) becomes an eigenfunction equation, the Schro¨dinger equation, the solution of which yields a set of eigenvalues and functions. The eigenstates have a definite energy E, but there is nothing about the Schro¨dinger equation that demands that the eigenvalue spectrum be discrete. It is only when we impose boundary conditions compatible with Born’s interpretation that we find discrete eigenvalues. If we demand that the wavefunction be localized by a negative potential (i.e., that it has zero amplitude at infinity), or if cyclic coordinates (such as angular variables) exist, we find that the eigenvalues are discrete. The “lines” of Fraunhofer, Bunsen, and Kirchoff are identified with the interaction between electromagnetic radiation and two of the discrete states, assuming that the Hamiltonian H0 is a good “zeroth order” approximation to the atom. The induced transition probabilities between levels (Einstein “B” coefficients) are most simply evaluated treating the radiation field classically via a small perturbation term HR added to H0. Spontaneous emission (through the “A” coefficient) was invoked by

4. Atomic Structure

Einstein to account for the phenomenon of unprovoked emission, a process also needed for compatibility with MaxwelleBoltzmann populations and Planck’s thermal equilibrium function for radiation (Eq. 5.1.27). This yields 2hn3 B21 ; (5.1.19) c2 relations obtained from the principle of detailed balance (microreversibility) for a thermal system. In other words, the two levels are coupled through a thermal radiation field (a black body). The spontaneous emission is understood physically from a treatment of the coupled and quantized electromagnetic field and atoms, in which fluctuations in the vacuum field induce spontaneous emission. Here we assume that the atomic lines have finite widths owing to natural broadening (quantum uncertainty in energy resulting from the finite lifetimes), thermal plasma motions, and other processes that contain further information on plasma conditions, but without further discussion. This summarizes the physical origin of the “boundebound” (b-b) spectral transitions. Transitions between two unbound states are called “freeefree” ( f-f ) transitions, or “Bremsstrahlung”. Finally, “boundefree” (b-f ) radiative transitions occur between bound states with n electrons to free states with n  1 bound electrons and a free electron. These processes are called photoionization, and in reverse, radiative recombination. Only b-b transitions are considered to be “line” radiation. Sharp (“autoionization”) resonances in b-f transitions are common when the free states are mixed with levels in which two electron spin-orbitals are excited. g1 B12 ¼ g2 B21 ;

A21 ¼

4. ATOMIC STRUCTURE To understand spectra, naturally one must first understand the structure of atoms and atomic ions. In general, the atomic structure and transitions are solutions to Schro¨dinger’s (or Dirac’s) equation for atoms, which are many body systems (many electrons clumped around the oppositely charged nucleus). As such, the solutions are generally complicated. We provide only a brief guide to understanding some elementary properties of atomic levels, eigenfunctions, and transitions. The case of hydrogenic ions provides valuable information because the wavefunctions are analytical. It is only useful in multielectron atomic ions to the extent that an optical electron is well-separated from the nucleus and other electrons, and in highly charged ions, ideas that will be used subsequently. The correspondence principle, namely that classical behavior be recovered as a limiting case of quantum mechanics, is also useful. States with large quantum numbers can be surprisingly well-described using classical concepts. For example, cross-sections tend to increase geometrically with the classical orbit areas, which are proportional simply to the principal quantum number n4/Z2. Electron spin, however, is an entirely quantum phenomenon. The wavefunctions are intrinsically built

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upon manifolds in both real and spin space. The implications simply have to be accepted with only limited classical analogies. Full wavefunctions for complex atoms and atomic ions can be expanded in terms of products of single electron “spin-orbitals” to form a hierarchy of configurations, terms, levels, and states, and the “good” quantum numbers (parity and total angular momentum) are readily obtained from the straightforward symmetries of these single electron wavefunctions. Thus, the reflection of the wavefunction in real space (the parity of a given state, another constant of the motion) has an eigenvalue of 1; it depends only on the orbital angular momenta li of the individual electrons through PN P ¼ ð1Þ i li ; (5.1.20) for a state described by an N-electron configuration or combination of configurations when they are mixed. P for a complex state built upon say, a 1s22p62s23p53d electron configuration, has a parity determined only by the partly filled 3p5 shell. (The s,d orbitals and even occupation numbers of p, f,. orbitals contribute only unit multipliers to P.) This is therefore an odd-parity state P ¼ 1. This picture is useful because we can then identify the kinds of transitions that are going to be strong electric dipole (E1) transitions, such as H La, because the E1 operator eri is of odd parity. Thus, strong E1 transitions connect levels with opposite parity only. Similarly, the magnetic dipole M1 (m) and E2 quadrupole (erirj) operators can lead to transitions only between levels of the same parity. Examples in solar physics include M1 transitions seen during eclipse among the ground levels of ions of iron, in the fourth row ˚ , which is the tranof the periodic table, such as the “green line” of Fe XIV at 5304 A 2 0 2 0 sition 2p P3=2 /2p P1=2 of the Al-like isoelectronic sequence (ions with the same number of electrons as Al). Other conservation laws guide our understanding of which lines might be strong and which weak. As well as constraints on parity, the “triangle” selection rules, coming from the need for photoneatom interactions to conserve angular momentum, allow one to identify “permitted” and “forbidden” transitions by inspection of quantum number J for the total angular momentum. Even the “not-so-good” quantum numbers S and L obey approximate rules, such as DS ¼ DL ¼ 0 when relativistic terms in the Hamiltonian (e.g., spin-orbit interactions) are small. This is readily understood because the E1 operator that induces the strongest line transitions operates only on the wavefunction in real space, not the component in spin space. Another important class of transitions is the “intercombination” (“IC”) transitions, E1 transitions for which DS s 0. These lines can be bright yet remain optically thin and sensitive to plasma densities in a manner different from that of E1 transitions, which permits one to explore plasma densities through joint observations of IC (or M1, E2) and E1 lines. There are other limiting cases in which further simplifying ideas can be used. Single electrons orbiting closed shells occur along the first column of the periodic table. Even in open shell cases, single electron excitations often occur between states

4. Atomic Structure

that are far from the strongly coupled electrons buried deeper in the net electrostatic potentials in the core atomic configurations. Wavefunctions that correspond almost to single electron excitations are amenable to simpler treatments. Hydrogen itself also serves as a pedagogical tool for at least two reasons. First, analytical solutions for eigenstates are available. Second, solutions for large quantum numbers, in which electrons spend time far from the nucleus, are (closely) hydrogenic in nature. For a nucleus of charge Z, hydrogenic (one-electron) eigenfunctions satisfying 1 Z H ¼  V2  (5.1.21) 2 r    yield the following scalings: EZ,n ¼ Z2/2n2 and rZ;n /n2 Z, both of which have important consequences. Electrons are Fermions. Thus, for N-electron atomic ions, antisymmetric combinations of one-electron orbitals are needed to find physically meaningful eigenfunctions of  N  X 1 2 Z 1X 1 H0 ¼  Vi  þ . (5.1.22) 2 r 2  rj j jr i i i; j i¼1 These “orbitals” comprise solutions to the purely radial part of Eq. (5.1.22), each characterized by quantum numbers n and l in the same way as for hydrogen (the Schro¨dinger equation separates into radial and angular components in both cases). The orbitals and their combination to yield atomic wavefunctions cannot be closed-form functions because the unperturbed potential, although of the form V(r), is no longer of the Coulomb form Z/r. The residual noncentral part is treated as a perturbation. Solutions are called atomic “terms” with well-defined antisymmetric products of orbitals. The solutions commute with total orbital and spin angular momentum operators, yielding well-defined orbital and spin angular momentum quantum numbers L and S, respectively. In this case, new energy levels are produced, lifting the central field’s degeneracy. For a given n, several levels exist, which can be of the same or different parity. Finally, by including the various relativistic effects (spin-orbit, etc.), the terms are split into levels through the additional terms in the Hamiltonian. Only parity and total angular momentum operators commute with the Hamiltonian so that only parity and J are good quantum numbers. This hierarchy of splitting of levels is essential to understanding real spectra that occur via transitions between atomic levels and, if external magnetic or electric fields are applied, between magnetic substates. The central-field approximation clearly works well when Z  (N  1) [ 1, i.e., the outer electron experiences a strong central field owing to the nuclear charge and incomplete screening by core electrons. On this basis, it is straightforward to find the scalings listed in Table 5.1.1. Early x-ray spectroscopy of atomic ions by Moseley helped to complete the period table of elements by allowing Z to have noninteger values as core electrons shield nuclear charge. There is a powerful theory based on “quantum defects” (Seaton, 1983), in which energy eigenvalues in complex atoms and ions are used to determine the “defect” d through EZ,n ¼ Z2/2(n  d)2.

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Table 5.1.1 Scalings of Parameters of Isolated Atomic Ions With Z, n Parameter

Example

Scaling With Z, n

Term in Hamiltonian

hri DEji (Ionization) DEji(Dn s 0) DEji(Dn ¼ 0) DEji (Fine structure, spinorbit)a

. 2s2 1 S0 /2s2 S1=2 þ e 0 2s2 1 S0 / 2s3p1 P1 0 2s2 1 S0 /2s2p1 P1 0 2s2p 3 P2 /2s2p3 P10

a0n2/Z Z2 Z2 Z/n2 Z4/n3

. Ze2/r Ze2/r e2/jrj  rij m$Bf12a2 Z 4 R

The fine structure induced by spin-orbit interactions arises from the term m$B ¼ ZmB /r3, which is z m2B Z 4 =a30 h12Z 4 a2 R in the Hamiltonian, noting that RZ 2 is the central field energy. Here, a ¼ e2/Zc is the fine structure constant; mB ¼ eZ/2mec ¼ a(ea0)/2 is the Bohr magneton. The magnetic field strengths B ¼ cv Ze2 =r experienced by electrons orbiting the nucleus are of the order 105e108 G. In  contrast, observable solar fields lie below 4000 G. R ¼ me e4 2Z2 is the Rydberg unit of energy (h13.6 eV, the ionization potential of neutral hydrogen for an infinitely heavy proton).

a

The E1 operator does not operate in spin space. If the wavefunctions used to evaluate matrix elements are pure LS-coupled states, IC (DS s 0) transitions are forbidden. Such transitions arise naturally because of mixing of LS-coupled states by relativistic effects. They are present in solar and stellar photoionized objects and their spectra, generally weaker than E1 collisions owing to smaller crosssections for excitation by electron impact (Table 5.1.1). Inspection of Tables 5.1.1 and 5.1.2 reveals several important properties of atomic ions of relevance to plasma spectroscopy, and that are reflected in spectra such as shown in Fig. 5.1.1: •

•

•

In H-like and noble gas spectra, to excite any electron requires a change in principle quantum number n and z RZ 2 of energy. Thus, the lowest excited levels of H, He, Ne and isoelectronic ions have strong lines principally at extreme UV radiation (EUV) wavelengths. La lines of H and He II are obvious in Fig. 5.1.1 at 1215 ˚ . Lines of highly ionized H and He-like ions for abundant elements (C, and 304 A N, O, Si, Fe, etc.) collect in the softex-ray region as a result (a few angstroms to a ˚ [see Fig. 5.1.3]). few tens of angstroms [see Fig. 5.1.1]) and below 100 A In ions with incomplete subshells, the lowest excited levels have the same principal quantum number as the ground level, requiring energies of only z RZ. As a result, such lines (having Dn ¼ 0) lie at much longer wavelengths and can be intense. In the Sun, lines of the Li- and Na-isoelectronic ˚, sequences are prominent: resonance lines of Ca II are close to 3933 and 3969 A ˚ ˚ ˚ ˚ Mg II 2800 A, Na I 5890 A, C IV 1550 A, O VI 1026 A, and so on (Fig. 5.1.1). ˚. In contrast, the 2s  3p transitions of O VI lie at 150 A As noted before, when close to ionization equilibrium, ions form when kTe or z RZ 2. Thus, these Dn ¼ 0 lines contribute more strongly to total radiation losses than Dn ¼ 1 lines, which is of importance in laboratory as well as astrophysical plasmas.

4. Atomic Structure

Table 5.1.2 Scalings of Transition Probabilities Parameter Type Aji/s

Y21(Te)

S cm3/s

brr cm3/s

Magnitude and Z,n,Te Dependence

Type

Example

E1, Dn s 0 E1, Dn ¼ 0 IC, Dn ¼ 0, D Ss 0 M1, Dn ¼ Dl ¼ 0 E2, Dn ¼ 0

2s2 1 S0 / 2s3p1 P1 0 2s2 1 S0 / 2s2p 1 P1 1 3 0 2 2s S0 / 2s2p P1 2p2 3 P0 / 2p2 3 P1 2p2 3 P0 / 2p2 3 P2

Two-photon

2s2 1 S0 / 2s2p 1 P0

E1 E1 (small gf ) Intercombination Forbidden e-Impact ionization

Rad recombination

0

0

2s2 1 S0 / 2s3p

1 0 P1

0 Si II 3s2 3p 2 P3=2 / 3s2p2 2 D5=2 0 2s2 1 S0 / 2s2p 3 P1 3 3 2 2 2p P0 / 2p P1 2s2 1 S0 þ e / 2s 2 S1=2 þ 2e

2s 2 S1=2 þ e / 2s2 1 S0 þ hn

109 Z4 108 Z1 100 Za, a ¼ 5  7 10 Za, a ¼ 3  12 100 Z1 10 Z 6 n2 lnTe/Z2 0.1 lnTe/Z2 1/Z2 1000/(Te Z 2)  2 pffiffiffiffiffi 1010 Te ℛ IZ   IZ1 exp  kTe 5  1011 Z 2 Te0:7

Data in this table are intended largely for transitions with modest principal quantum numbers n ( nG þ 4, say, where nG is the largest principal quantum number of the ion’s ground state. Hydrogenic values can be used for higher-n levels (refer to Fujimoto, 2004). Dependences on principal quantum number n are needed only for E1 transitions that dominate the collisions and decays of high-n levels. Values of a for radiative decays of intercombination and forbidden transitions depend on the coupling schemes. Refer to Cowan (1981). Scalings for Maxwellian-averaged collision strengths Y are from Burgess and Tully (1992). “Small gf” cases are those in which accidental cancellation occurs in E1 radial integrals.

•

•

IC and forbidden lines often have conditions in which collisional depopulation is comparable in magnitude to radiative deexcitation. The emission from such lines can become a linear function of density (compare with Eq. [5.1.31], for permitted E1 lines). The comparison of E1 and the IC and forbidden lines are thus “density-sensitive line ratios.” As Z increases along an isoelectronic sequence, the IC and F transitions increase dramatically. Indeed, by the time one reaches H- and He-like spectra, such lines are of comparable strength in plasmas, yielding singularly rich information of the plasma temperature and density in the soft x-ray region (Gabriel and Jordan, 1971; Jordan and Veck, 1982).

To highlight the rich variety of lines in the x-ray region, Fig. 5.1.3 shows lines of calcium observed during a solar flare. To see the regularities in spectra represented by the scalings in Table 5.1.2, refer to Gabriel and Jordan (1971) and Dufton and Kingston (1981).

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CHAPTER 5 Atmospheric structure, Non-Equilibrium Thermodynamics

FIGURE 5.1.3 Ca XIX and Ca XVIII (Z ¼ 19 and 18) lines from a solar flare observed with the bent crystal spectrometer on the Solar Maximum Mission satellite. The broad variety of permitted, intercombination, and forbidden transitions includes Ca XIX lines (upper arrows): w is the 0 0 /1s 2 1 S respectively; and z resonance line 1s2p 1 P 1 /1s 2 1 S 0 ; x,y are from 1s2p 3 P2;1 0 1 3 2 is the fully forbidden 1s2s S 1 /1s S 0 . The other lines (a  t) are from Ca XVIII from doubly excited 1s2p2s and 1s2p2 configurations that decay to 1s22s and 1s22p configurations. Adapted from Jordan, C., Veck, N.J., May 1982. Comparison of observed CA XIX and CA XVIII relative line intensities with current theory. Sol. Phys. 78, 125e135.

5. SPECTRUM FORMATION IN A NUTSHELL Unlike laboratory plasmas, the Sun and astronomical sources are vast. Radiation transport, with notable exceptions, must therefore be dealt with. The reader should refer to a standard radiative transfer text (e.g., Mihalas, 1978), or to Chapter 5.2, because here we only review some essential ideas. In this section, we look at sources

5. Spectrum Formation in a Nutshell

and sinks of radiative energy given by source functions S and opacity k in terms of atomic-level populations and appropriate cross-sections (which in turn determine transition probabilities), leaving until later the ingredients needed for these quantities. Conservation of radiant energy along a certain ray yields the equation of radiative transfer in terms of three macroscopic quantities: the optical depth s along the ray, the specific intensity I, and the source function S (initially dropping the subscript for frequency n): dI ¼IS (5.1.23) ds in which ds ¼ kds along an element of the ray ds. I and S depend on the ray geometry (angles through an atmosphere) and frequency. The “source function” S ¼ h/k ¼ hl in which l ¼ k1 cm is the photon mean free path and h is the emission coefficient in Hz1. S is easily understood to be the contribution to the radiant energy integrated along one photon mean free path. Given a source function S, the solution to Eq. (5.1.23) is Z 0 IðtÞ ¼ SðsÞejtsj ds (5.1.24) smax

where, in particular, the emergent (observed) intensity (at s ¼ 0): Z 0 Ið0Þ ¼ SðsÞes ds smax

(5.1.25)

When mean free paths exceed the size of the emitting plasma (optically thin plasma), it is more meaningful to work with h and z (see subsequent discussion). I(0) represents the observed intensity of radiation at each frequency n. Unlike many astronomical objects, on the Sun we can resolve the surface. I(0) is a function of position r, f on the solar disk projected onto the plane of the sky. Historically, solutions to radiation transport generally assume rotational symmetry; I depends only on the cosine m of the angle q between the local gravity vector (the direction of atmospheric stratification) and the line of sight. Thus, we can write I(0) h Inm. Next, we examine plasma at high and low densities, leading to LTE (the detailed balancing of thermal processes) and the “coronal approximation” (balancing only between binary, or two-body, collisions), respectively. The intermediate case, the CR model, is reviewed. A discussion of the collisional processes responsible for setting, along with any radiation, the state (i.e., level populations q) of the atomic ions in plasma is delayed until the next section. Along with basic atomic crosssections (oscillator strengths, photoionization cross-sections, and other quantities), these populations q determine the emergent radiation along with the transport equation.

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5.1 OPTICALLY THICK FORMATION When thick, we can approximate modest depth variations of S (such as occur in radiative equilibrium) as Sn ¼ an þ bnsnmþ., which, with lim SðsÞes /0 and keeping s/∞ only the linear term: Inm ð0Þ z an þ bn m ¼ Sn ðs ¼ mÞ;

(5.1.26)

which is the “EddingtoneBarbier relation.” If thermal processes dominate the emission and absorption of photons, such as collisions with a bath of particles at temperature T and density n, and/or the radiation field is a black body at temperature T, then Sn ¼ Sn ¼ Bn ðTÞ in which Bn is the Planck function Bn ðTÞ ¼

2hn3 1 c2 expðhn=kTÞ  1

(5.1.27)

and k ¼ k*(n,T) is the absorption coefficient in LTE (denoted by *). In LTE, calculations of k consume most of the effort needed to compute the spectrum. In a typical stellar atmosphere, one has a mix of elements, each satisfying thermal statistical distributions, through the Saha equation. LTE is sometimes valid for visible and infrared spectral lines formed in the solar photosphere, and often for submillimeter to centimeter wavelength radiation formed in the upper photosphere and chromosphere where thermal (free-free) processes dominate both S and k. Whereas the photosphere is the densest visible plasma on the Sun (Fig. 5.1.2), transport of radiation in boundfree transitions often leads to significant non-LTE effects in k, if not also in S (Bruls et al., 1992). This should be unsurprising, when we acknowledge that thinner radiation in continua, freely escaping to space, implies no thermal equilibrium. A spectral line is characterized by having more opacity at core frequencies than in the line wings. Therefore, in the line core, we see source functions from regions higher in the atmosphere than in the wings and neighboring continuum. If the temperature decreases with height, when close to LTE, we thus see an absorption line spectrum. This explains the nature of Fraunhofer’s photospheric absorption spectrum when the photospheric temperature decreases with height in radiative equilibrium. Such observations allow the photospheric temperature structure to be mapped as a function of optical depth, and height if the relationship between line opacity and height is known, through a model atmosphere. Line depths yield information on abundances, line profiles on plasma densities, motions, electric and magnetic fields, and other parameters beyond this chapter’s scope (Griem, 1964).

5.2 OPTICALLY THIN FORMATION Solar plasma above the chromosphere can often be treated as optically thin. In plasmas where s < 1, with s measured from the observer to the source, Z 0 hðsÞds; (5.1.28) Ið0Þ  Iðs1 Þ ¼ s1

5. Spectrum Formation in a Nutshell

The intensity owing to the emitting plasma is just an integral of h along the line of sight minus any “incident radiation” behind the thin layer I(s1). In the Sun, the hot, thin corona emits on top of emission from the underlying photosphere I(s1) z S(s ¼ 1) or space (I[s1] ¼ 0 when observed above the visible limb). S(s ¼ 1) is large at visible or infrared wavelengths, so that I(0) z I(s1) and the thin plasma’s radiation is negligible. However, at EUV or x-ray wavelengths, I(0) [ S(s ¼ 1) and the corona is readily visible in emission above a much darker background. Again, a spectral line is characterized by having more opacity and emissivity at core frequencies than in the wings. Therefore, in the line core, more atoms emit in the hot optically thin layer between s1 and the observer than in the wings. Eq. (5.1.28) therefore predicts an emission line spectrum on top of any background (I(s1)). This explains the nature of the coronal spectrum at EUV and x-ray wavelengths (Fig. 5.1.1). Unlike the photospheric case, we cannot probe the temperature as a function of height; instead, we can derive a particular distribution of emitting material as a function of temperature from many spectral lines, as follows. The simplest pedagogical case is where Eq. (5.1.28) applies to emission lines between two atomic levels labeled 2 and 1, integrating over frequencies where line emission is significant. Using the rate of spontaneous emission of photons given by the Einstein Ae coefficient A21, we have h ¼ hn4p21 q2 ðsÞA21 and Z 0 hn21 q2 ðsÞA21 ds. (5.1.29) Ið0Þ  Iðs1 Þ ¼ s1 4p where q2(s) is the population density of level 2. In the particular case of a two-level atomic ion in which all photon emissions originate from collisions with plasma electrons from level 1 to level 2, we find the first part of the familiar “coronal approximation,” q2 A21 ¼ q1 ne hvs12 ihq1 ne C12 ðTe Þ. those electrons with 12me v2 > hn21 contribute, so that relaxed to temperature Te, C12(Te) f exp(hn21/kTe).

(5.1.30)

Only when the electrons are fully The second part of the coronal approximation is that the fraction of all ions of the element containing levels 1 and 2 is also controlled by two-body electron collisions (impacts between ions and single electrons). The latter dominates at low densities over three-body collisions (see the upper dot-dashed line in Fig. 5.1.2), such as in the solar corona (Woolley and Allen, 1948). The relevant rates are the inverse of the times sion and srr provided earlier. Then, in a fully ionized plasma in statistical equilibrium (dynamical time scales exceeding both sion and srr), q1 fAne F1 ðTe Þ (in which A is the elemental abundance relative to hydrogen) and then Z 0 Z 2 ne G12 ðTe Þdsh xðTe ÞG12 ðTe ÞdTe (5.1.31) Ið0Þ  Iðs1 Þ ¼ s1

DTe

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CHAPTER 5 Atmospheric structure, Non-Equilibrium Thermodynamics

in which the distribution function x(Te) measures a purely solar quantity, the emission measure x(Te)dTe. x(Te)dTe is visualized by seeking all plasma between Te and Te þ dTe along the line of sight and forming the sum of the product of n2e and dsk for all matching segments k found. x(Te) is derived by inverting the integral equation from a set of emission lines, “regularized” to produce a minimally structure solution compatible with the data, as to deal with nonuniqueness characteristic of such inversions (Craig and Brown, 1976). Several lines can used to determine plasma elemental abundances following early work by Woolley and Allen (1948). The CR model of Bates et al. (1962) extends the coronal approximation to optically thin plasmas at higher densities of laboratory and some astrophysical sources. Their treatment includes the large number of bound levels of high principal quantum number, n [ 1. Bates et al. (1962) developed a heuristic argument to treat the atomic populations by identifying and tracking populations of long-lived (“metastable”) levels. Two sets of equations are developed and solved independently. One is for ionization and recombination involving only long-lived or “metastable” levels and the other is for the fully relaxed set of number densities of the many short-lived excited levels (108Z4 s), given the populations of slowly evolving (metastable) levels from the first equation. The model compresses the general problem of time evolution by using just two modes corresponding to “ionizing” and “recombining-plasma” conditions. For a modern review, see Fujimoto (2004), who provides a qualitative justification for this method. A mathematical justification is given by Judge (2005). Fujimoto (2004) examines the ionizing and recombining components, allowing one to understand the role of the vast number of high-n levels in the atomic rate equations. Above a certain principal quantum number, the “quasithermal” value nqt, collisions dominate and connect higher levels to the ground state of the ion. For all levels with n > nqt, the populations approach LTE wrt the ground state of the ion above. For levels with n < nqt the coronal approximation is valid, with the levels being fed by collisions from lower-lying metastable levels. Given the Saha and Boltzmann equations, the populations of excited levels for most plasmas (neLTe  1) (see Eq. 5.1.16) are far smaller than the number densities of electrons and ions in the metastable (e.g., ground) state. The reader should refer to Cooper (1966), Section 7, and to Fujimoto (2004), Chapters 4 and 5, for details. Using hydrogenic approximations to levels with high principal quantum numbers n[1, the critical value is roughly  n 2=17 e nqt z 95 7 (5.1.32) Z which is Eq. (4.29b) of Fujimoto (2004) with ne in cm3. For the solar atmosphere with lgne ¼ 14, 11, 8 for the photosphere, chromosphere, and corona, nqtz3, 5, 14 respectively. These values agree well with detailed calculations for the Sun’s photosphere and chromosphere (Table 17 of Vernazza et al., 1981).

5. Spectrum Formation in a Nutshell

In astrophysics, the CR model helps us to identify limits of the coronal approximation. We can often make the following simplifications: 1. Above the chromosphere, ionization balance (b-f processes) can be computed separately from the excitation (b-b). The b-f processes are slowest, roughly 1 s1 ion þ srr per second. Ionization and statistical equilibrium are valid when dynamical processes are slower; 2. As found in the coronal ionization equilibrium approximation, each ion tends to be abundant at electron temperatures such that kTe  IZ; 3. We can compute ionization balance reasonably accurately using the coronal approximation (two-body processes) up to perhaps 1011 cm3, with care (see, e.g., Fig. 7 of Cooper, 1966); and 4. We can interpret the spectra of low-lying levels (with n ( nqt) using collisional excitation only from lower metastable levels. Point 3 is not valid in the cases where dielectronic recombination (DR) is important because of the role of high-n doubly excited states whose “spectator electron” has n [ nqt (Summers, 1974).

5.3 NONeLOCAL THERMODYNAMICAL EQUILIBRIUM AND FURTHER COMPLICATIONS These cases are limits of the more general case in which particle collisions are not frequent enough to maintain LTE, yet the plasma remains optically thick in some atomic transitions of interest. These conditions plague the chromosphere and prominences, and strong lines in the transition region. In this case, one must solve equations for radiative transfer simultaneously with kinetic equations for the populations q of atomic levels belonging to several neighboring ionization stages. This coupled set of nonlinear and nonlocal equations requires iterative procedures to bring the radiation and population densities to consistency. The nonlocality is illustrated clearly in the “two-level atom,” in which the non-LTE population equations



 dq2 dq1 ¼ q2 A21 þ B21 J þ qe C21 þ q1 B12 J þ qe C12 ¼  dt dt

(5.1.33)

i become, assuming statistical equilibrium (dq dt ¼ 0, which can include derivatives for advection and diffusion) (Delache, 1967; Fontenla et al., 1990; Pietarila and Judge, 2004),

S¼

h 2hn3 1  ¼ 2  hεB þ ð1  εÞJ; q1 g2 k c 1 q2 g1

e C21 where ε ¼ A21nþn , and C21 ¼ C21(1  exp  hn/kT). e C21

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The emergent spectra reflect the nonlocal nature of the formation of the spectrum. The source function (the term in J) includes J, J ¼ L½S.

(5.1.34)

in which L[.] is an operator extending over the entire atmosphere and over all frequencies across the line profile. When more than two levels are considered, the spectra also become nonlinearly coupled because transitions become implicitly dependent on one another through the multilevel population equations (Scharmer and Carlsson, 1985). Thus, the relatively “simple” interpretation of spectra that applies in LTE or the coronal limit becomes modified by the effects of photon scattering. For the strong Fraunhofer lines, source functions depend on the conditions across almost the entire chromosphere (which can exceed 104 local photon mean free paths). The ratio of the range of influence to the (much smaller) mean free path, the “thermalization length” (Mihalas, 1978), is fundamentally important in stellar atmospheres. In general, the emergent intensity of an optically thick plasma depends on the geometry and the boundary conditions, as well as the plasma itself. As such, numerical solutions are required. Algorithms to solve the coupled nonlinear and nonlocal equations of radiative transfer and atomic populations in modern astrophysics usually fall into two categories: linearization (Scharmer and Carlsson, 1985) and preconditioning (Rybicki and Hummer, 1991, 1992). Three-dimensional codes are readily available for the statistical equilibrium problem, including the linearization code MULTI3D (Botnen and Carlsson, 1999) and the preconditioning code RH (Uitenbroek, 2000; Uitenbroek, 2001). Further departures from the non-LTE case are encountered in solar physics. For example, statistical equilibrium becomes a poor assumption when relaxation times for the populations exceed dynamical times, typically the slowest of the ionization and recombination times. These effects occur in both “quiet” as well as active and flaring regions of the chromosphere. These are best studied on a case-by-case basis, although there are some general ideas related to the “ionizing” or “recombining” conditions from the CR model (Judge, 2005; Laming and Feldman, 1992). Another example is the unambiguous evidence of high-energy tails in electron distribution functions, which are found in the form of type III radio bursts and hard x-ray flare emission (Fletcher et al., 2011). Coulomb collisions are infrequent at high energies (see(ε) f ε2), so that once they are generated, such electrons thermalize only when they penetrate to the deep chromosphere or photosphere. Table 5.1.3 summarizes these various regimes.

6. Plasma Spectroscopy

Table 5.1.3 Regimes of Spectrum Formation Regime

Conditions

Solar Example

Thermal equilibrium LTE

Detailed balance High densities

Coronal

Hot, tenuous plasma (108/cm3) optically thin, including nonequilibrium Solution of coupled RTa and SEa equations Populations evolve in time with or without RT

. Weak lines, deep photosphere Quiet solar corona

approximation Collisional-radiative

Non-LTE Nonequilibrium

Active solar corona

Strong lines chromosphere, prominences Chromosphere, transition region corona, flares

a LTE, local thermodynamical equilibrium; RT, radiative transfer; SE, statistical equilibrium. All of these regimes assume that electrons are fully relaxed to an equilibrium temperature Te.

6. PLASMA SPECTROSCOPY In contrast to isolated atoms, the wavefunctions of atoms and ions embedded in a tenuous plasma are solutions to equations such as v j (5.1.35) vt in which HP(t) is the additional term in the atomic Hamiltonian owing to the collective effect of particles and fields in the plasma. HP(t) is a function of time t because warm gases and plasmas contain vast numbers of particles, each undergoing complicated motions in the presence of external and self-generated fields. When the particles are thermalized, these motions are stochastic, and so they must be the wavefunctions of individual atoms. However, Eq. (5.1.35) is not generally solved in this fashion (nevertheless, see DeWitt and Nakayama, 1964), for it is only in a statistical sense that we need to know the evolution of the atomic wavefunctions from macroscopic objects. Instead, the quantized atoms are examined in a plasma described classically. Then we can speak of various processes that are treated as perturbations of H0, in which ensembles of atoms and atomic ions evolve within a plasma: ðH0 þ HP ðtÞÞj ¼ iZ

1. 2. 3. 4.

Lowering of ionization potential by the plasma electric microfields Stimulated emission and absorption of radiation Inelastic particle collisions with charged and neutral particles Elastic particle collisions

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The ionization potential is lowered because half of the time, thermally varying electric fields in each Debye sphere supply small amounts of energy to aid ionization processes. The infinite sums in partition functions are removed physically; the amount of the reduction is DIZ z

Ze2 2 kTe ; rD z rD 4pe2 ne

(5.1.36)

Now IZ z (Ze2/2a0), so that DIZ/IZ z 2a0/rD. The maximum quantum number pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi corresponding to this energy is rD =2a0 . This is not to be confused with the quasithermal (collisional) limit nqt or with the IngliseTeller estimate arising from line-broadening, which makes the discrete spectrum appear as a continuum at high n. Table 5.1.4 lists processes that are important in changing the states of atomic ions in plasmas, and which therefore contribute to the radiation emerging from the plasmas. Changes in the electron wavefunctions are most readily induced by radiation (processes 2, 3, 5, and 6) through E1 interactions, and by collisions with plasma electrons. Electron collisions are far more efficient than ion collisions because of a classical result: namely, that the energy exchanged between colliding particles is maximized when the particles have the same mass. Collisions with protons and ions can have significant effects under specific conditions, such as between fine structure levels or when Ti [ Te (Seaton, 1964a). One more general result is clear: that three-body recombination (process 8) is rare in most plasmas except for the densest conditions, because it requires the simultaneous presence of three particles (the atomic ion and two electrons) within a volume of order (a0/Z)3. In Fig. 5.1.2, we see that for most plasmas of interest, three-body recombination is negligible. In contrast, the inverse process, collisional ionization (5.1.7) is often dominant. The balance between ionization (5.1.7) and two-body recombinations (5.1.4) and (5.1.13) constitutes part of the coronal approximation. Surprisingly, classical arguments yield the correct asymptotic dependence on Te (kTe  IZ) and order of magnitude for (process 7) (Seaton, 1962b). The formula listed in Table 5.1.4 is in fact a semiempirical fit to cross-sections based on the classical behavior for kTe  IZ. Updates to such fits are to be found in Arnaud and Rothenflug (1985) and Arnaud and Raymond (1992) and later citations to these two publications. Collisions between bound levels with n < nqt are fundamentally different depending on whether the target atom is neutral or ionized. In the neutral case, the cross-section at threshold (12 mv2 ¼ Ej  Ei ) is zero and increases algebraically with energy up to a maximum close to 2(Ej  Ei), when it declines. In contrast, because of the Coulomb attraction of the projectile electron by the target charge, cross-sections at threshold are finite for charged targets. Yji(Te) is the Maxwellianaveraged collision strength. Yji(Te), a measure of probability, approaches zero for

Table 5.1.4 Atomic Processes in Plasmas a

Name

Processb

Example

1

Spontaneous decay

XZ,k / XZ,j þ hn

2s2p1 P1 / 2s2 1 S0 þ hn

2

Stimulated emission

XZ,k þ hn / XZ,j þ 2hn

2s2p1 P1 þ hn / 2s2 1 S0 þ 2hn

3(1)

Absorption

XZ,j þ hn / XZ,k

4(5)

Radiative recombination

XZ,k þ e / XZ1,j þ hn

2s2 1 S0 þ hn / 2s2p1 P1

5(4)

Photo-ionization

6

Stimulated recombination

0 0

0

2s2 S1=2 þ e / 2s2 1 S0 þ hn 

XZ,j þ hn / XZ þ 1,k þ e

2s2 1 S0 þ hn / 2s2 S1=2 þ e



XZ,k þ hn þ e / XZ1,j þ 2hn 



2s2 S1=2 þ e þ hn / 2s2 1 S0 þ 2hn



7(8)

Direct ionization

XZ,k þ e / XZ þ 1,j þ e þ e

2s2 1 S0 þ e / 2s2 S1=2 þ e þ e

8(7)

Three-body recombination

XZ,k þ e þ e / XZ1,j þ e

2s2 S1=2 þ e þ e / 2s2 1 S0 þ e

9(10)

Excitation (direct)

10(9)

12(11)

c

13

c

14

c

a

XZ,j þ e / XZ,k

0

2s2 1 S0 þ e / 2s2p1 P1 þ e



Deexcitation (direct)

XZ,k þ e / XZ,j

Dielectronic capture

XZ;j þ e / XZ1;k 

Autoionization Dielectronic recombination Excitation via autoionization

XZ1;k  / XZ;j þ

0

2s2p1 P1 þ e / 2s2 1 S0 þ e 2s2 S1=2 þ e / 2 pnp 1 S1

e

2pnp 1 S1 / 2s2 S1=2 þ e

XZ1;k  / XZ1;k  þ hn / XZ1;j 11 then XZ1;k  / XZ;k

þ e

þ hn0

0

2pnp 1 S1 / 2snp1 P1 þ hn / 2s2 1 S0 þ hn0 0

0 2s2 S1=2 þ e / 2pns1 P1 / 2p2 P1=2 þ e0

The column lists P(I) in which P represents the process written and I is the inverse process, where applicable. XZ,k refers to atomic species “X,” of ion stage Z (core charge) and energy level k. The energies are in alphabetical order: the energy of level XZ,j is lower than that of XZ,k. c These processes involve “two-electron” excitations. “One” and “two” electron processes mean that the atomic wavefunctions have just one spin-orbital and two spin-orbitals that have been excited, leaving behind one and two holes in the “core,” respectively. b

6. Plasma Spectroscopy

11(12)

c



147

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CHAPTER 5 Atmospheric structure, Non-Equilibrium Thermodynamics

Te / 0 for neutral targets and a finite value for ions. The collision rate per target atom is given by rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Yji ðTe Þ Yji ðTe Þ 2 2 Cji ¼ pa0 2R ne ; ¼ 8:63  106 Te1=2 ne . (5.1.37) pme kTe gj gj The upward collision rate is simply given by qi Cij ¼ qj Cji and Y is symmetric in ij. Given Eqs. (5.1.15), (5.1.29), and (5.1.30), this means that lines with different values of E2  E1 have ratios that are sensitive to Te. For neutrals, unlike the ions, the collision strengths are zero at zero temperature. The different kinds of transitions listed in Table 5.1.2, which apply to ions, are thoroughly discussed in a penetrating publication by Burgess and Tully (1992). The different temperature (i.e., energy) dependences listed in the table arise from conceptually simple physics. The E1 transitions are dominated by long-range interactions at relatively low energies; as a result, the cross-sections are very large, and the effects of doubly excited states (“autoionization resonances”) on the cross-sections are modest. These crosssections are proportional to the E1 oscillator strengths, which is the origin of the well-known “Van Regemorter” and related approximations (Van Regemorter, 1962). Generally speaking, the cross-sections for ions are much better known than for neutrals, with the exceptions of neutral H and He, which have naturally received much attention because of their astrophysical abundances. Packages such as CHIANTI (Young et al., 2016) contain the results of decades of efforts to compute, measure, and compile such data. For levels with principal quantum numbers above nqt, cross-sections become large owing to the classical orbit size and the correspondence principle. Hydrogenic results can be used. For IC transitions, a change of spin is needed that requires penetration of the ion “target,” i.e., short-range interactions. There is no E1 component; the cross-sections are therefore smaller and exchange of electrons can be important. As a result, the cross-sections drop rapidly with increasing energy as the interaction time is reduced. More seriously, the cross-sections and the Maxwellian averaged Yji(Te) become sensitive to the doubly excited states through autoionization (process 14). The resonances are particularly difficult to compute in energy, so that the IC collisional transition probabilities entail substantial uncertainties unless (rare) measurements are made. Forbidden transitions behave somewhat like IC transitions, and as noted earlier, they often have contributions from proton collisions. Radiative recombination is computed employing the principle of detailed balance, using populations from the Saha equation and with photoionizing radiation given by the Planck function. It is dominated by the spontaneous recombination component. Most of the recombination occurs to levels with low principal quantum number n. Recombination involves collisions with electrons that, in the attractive Coulomb field of the target, can excite doubly excited states. Use of the term “radiative recombination” explicitly excludes resonance structures associated with doubly excited states. Hence, the rate depends monotonically on Te, as indicated in the Table 5.1.2. Except for recombination onto a bare nucleus, separation between

6. Plasma Spectroscopy

“radiative” recombination and DR, which involves the resonances, is artificial (Nahar and Pradhan, 1992). Recombinations directly to levels with n > nqt are effectively not recombinations at all because collisions upward from n to neighboring n þ 1 dominate in determining the flux of populations from nqt up to the continuum (see Chapter 3 of Fujimoto, 2004). This leads us to discuss the last process: DR (process 13). Following a suggestion by Unso¨ld in the late 1950s, this was first studied by Seaton’s group. Burgess (1964, 1965) was able to show its importance and develop a general formula based on the Rydberg series of high-n states converging on to a doubly excited ionization limit (Fig. 5.1.4). Burgess’s work showed that discrepancies in coronal temperatures using a variety of techniques were largely resolved by inclusion of this process (Seaton, 1962a, 1964b). The process is illustrated by Fig. 5.1.4 taken from Cooper (1966). To understand the processes involved, we look at the impact of an electron with energy E on the state marked “First ionization limit.” To fix ideas, let this be the ground level of Be-like ions 1s22s2S10 at E1, and let the second limit (“first excited state of the 0 ion”) be 1s2 2s2p1 P1 at E2. Now consider the possible outcomes of varying E: •

•

• •

• •

If E > (E2  E1), the impacting electron can excite level 2 with an outgoing electron with lower energy (process 9). The rapid radiative decay of this level yields emission of a photon in the resonance lines of the Be-like ion. If E < (E2  E1), lying between the doubly excited levels, the electron may emit a photon owing to direct recombination at UV wavelengths (forming a localized recombined atomic state) or its acceleration in the charge of the Be-like ion (Bremsstrahlung, leaving the ion unperturbed and a free electron). • If E < (E2  E1) and we are at energies coincident with a doubly excited level energy (there is in principle an infinity of these levels converging onto E2), the incoming electron’s wavefunction may change from that of a continuum (Coulomb wavefunction extending to infinity) of the form 1s22s2pkl in which -k2/2m ¼ E  E1 to a local wavefunction characterized by a two-electron excitation that, in terms of spin-orbitals, will have the form 1s22s2pnl. The wavefunctions can be seen as a superposition of these two types of “single electron” wavefunctions. The wavefunction collapses to either 1s22s2pkl (autoionization: process 12) or 1s22s2pnl (dielectronic capture, process 11). If collapse occurs to 1s22s2pnl, this state will most rapidly decay (for large nl values) via the transition 1s22s2pnl / 1s22s2nl (see the line marked “radiative transition probability” in the Figure 5.1.4). This line will have a longer wavelength than the resonance line of ion Z because it occurs in the presence of the core plus the “spectator” nl electron. Such lines are called “dielectronic satellites.” Radiative decay of the nl electron to a lower orbital is termed “stabilization.” The net effect is that the electron has been captured by ion Z, leading to an excited level of ion Z  1. This is DR.

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FIGURE 5.1.4 (Left) A series of energy levels for single-electron excitations such as those found for hydrogen-like ions. They converge onto the shaded continuum (“First ionization limit”). (Right) Two electrons are excited and the energy levels shown converge onto a higher continuum. Those doubly excited levels lying between these continua are the cause of autoionization and dielectronic recombination. Adapted from Cooper, J., January 1966. Plasma spectroscopy. Rep. Prog. Phys. 29, 35e130.

The sum over all the nl terms was examined by Burgess (1965) to derive a general formula applicable to low-density plasmas. It should be clear that at finite densities, lowering of the ionization potential by plasma microfields and the important effects of upward collisions in the series of high nl levels will suppress DR. Summers (1974) addressed this problem, solving the large matrix equations needed to account for these effects in detail.

6. Plasma Spectroscopy

Using these results and the scalings of nqt with density, we can make general statements about the importance of DR in the solar atmosphere and interior. DR is unimportant below the photosphere of the Sun because there, nqt  3. In other words, all levels with nl > 3 are strongly collisionally coupled to the next ionization stage and so LTE prevails for these levels. In the chromosphere (ne z 1011 cm3, Te z 7000K), DR may be important but only for one or two levels, because nqt z 5. The corona is the optimal region of the Sun’s atmosphere for Burgess’ DR mechanism to operate because nqt > 10. The effects of DR are most important when E2  E1 is smallest, because then impacting electrons with modest energies can contribute to the DR process. Impacts of electrons on H- and He-like ions tend to have lower DR rates than those of Li-like ions for this reason. Fig. 5.1.5 shows recombination rates split into radiative and DR components, as well as a self-consistent calculation from Nahar and Pradhan (1992) for two ions with prominent lines in the solar transition region. Several lessons are immediately learned from this Fig. 5.1.5. DR is a phenomenon that peaks in energy and electron temperature, unlike radiative recombination, which decreases monotonically with temperature. This is because the states involved converge on to the first excited levels of ion Z. In the example shown, the O IV to O III DR rate peaks near 1.6  10 K, corresponding to 13.6 eV. The first excited levels

FIGURE 5.1.5 Recombination rate coefficients computed using standard approaches and a full treatment of the electron-ion collision problem (Nahar and Pradhan, 1992). All calculations apply to the zero-density limit. Radiative recombination is characterized by the monotonic drop of the coefficient with Te, dielectronic recombination peaks close to the energy of the first excited levels of the doubly excited state.

151

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connected by E1 transitions to the ground level of O IV lie near 15 eV. The second important lesson is that care is needed in using atomic data. It is seen that factors of two exist between different calculations, even without the effects of collisions in levels with n > nqt. Finally, when DR is present, it can exceed radiative recombination by orders of magnitude. Storey (1981) pointed out the importance of a type of DR efficient at low temperatures, when the first excited levels E2 are metastable levels lying close to the ground level of ion Z. These occur in Be-, C-, N-, and O-like ions in the second row of the periodic table (also, of course, in third and higher rows), which are abundant in many astrophysical plasmas. Stabilization occurs via excitation and decay of a doubly excited electron, which, in contrast to Burgess’ high temperature case, has n  5.

7. CLOSING REMARKS This chapter is necessarily just a short overview. It is hoped that the information provided a little insight and physical understanding about how spectra are formed in the solar atmosphere. The reader should refer to the reviews and textbooks referenced for a more complete understanding. Spectroscopy, with radiation transfer and polarimetry (Chapters 5.2 and 5.3 in this volume), lie at the heart of all quantitative knowledge of the Sun and stars. Essentially all interesting problems in solar physics must be addressed at some level through the building blocks outlined in this chapter, including the most important remaining unsolved problem: How does the Sun regenerate its global magnetic field every 22 years? For completeness, here is a list of useful spectroscopic resources, plus a brief assessment of the goals of the various repositories and projects: •

•

•

The National Institute of Standards and Technology provides critically assessed atomic energy levels and spectral lines. These are necessarily incomplete as critical evaluations take time to ingest. See https://physics.nist.gov/ PhysRefData/ASD/levels_form.html CHIANTI has become a workhorse for solar physicists. It endeavors to capture the latest and best energy levels, transition probabilities, collision strengths, and various tabulations of ionization equilibrium for use primarily in low-density, coronal, and transition-region plasma. The user should be aware of the need to allow for finite-density effects in ionization equilibria. CHIANTI targets UV, EUV, and x-ray spectra. See http://www.chiantidatabase.org/ HAOS-DIPER is an ingestion of CHIANTI and OPACITY and IRON PROJECT data with some allowance for finite density effects and nonequilibrium conditions. The database adopts quantum numbers for energy levels and as such can be used to approximate various parameters and modify atomic models when needed. See http://www.hao.ucar.edu/modeling/haos-diper/

References

•

•

•

• •

The OPACITY and IRON projects used state-of-the-art calculations of opacities. The data must be understood by the user as applicable to opacity calculations, and their use for spectroscopy must be handled with appropriate care. See http:// cdsweb.u-strasbg.fr/topbase/home.html The Atomic Data and Analysis Structure project is de facto a primary source for plasma spectroscopy under optically thin conditions. It is a major provider for the plasma machine and nuclear fusion communities. See http://www.adas.ac. uk/about.php For radiative transfer, LTE or otherwise, perhaps the best starting place is the code RH (“Rybicki-Hummer”) written by Uitenbroek (2000) because of its ready adaptation to one-, two-, or three-dimensional cases. Uitenbroek can be contacted for the code itself. R. Kurucz maintains a large database of atomic transition probabilities for opacity calculations. See http://kurucz.harvard.edu/linelists.html There are many other packages that are more specialized: for instance, for application to molecules and/or soft s-ray astronomy. See, e.g., https://heasarc. gsfc.nasa.gov/xanadu/xspec/or http://www.atomdb.org/

Finally, the reader is recommended not to use packages as black boxes for reasons that should be clear.

REFERENCES Arnaud, M., Raymond, J., 1992. Iron ionization and recombination rates and ionization equilibrium. ApJ 398, 394. Arnaud, M., Rothenflug, R., 1985. An Updated Evaluation of Recombination and Ionization Rates, vol. 60. A&ASS, p. 425. Bates, D.R., Kingston, A.E., McWhirter, R.W.P., 1962. Recombination between electrons and atomic ions I. Optically thin plasmas. Proc. R. Soc. London A267, 297e312. Botnen, A., Carlsson, M., 1999. Multi3D, 3D non-LTE radiative transfer. In: Miyama, S.M., Tomisaka, K., Hanawa, T. (Eds.), Numerical Astrophysics, vol. 240. Astrophysics and Space Science Library, p. 379. Braginskii, S.I., 1965. Transport processes in a plasma. Rev. Plasma Phys. 1, 205e311. Bruls, J.H.M.J., Rutten, R.J., Shchukina, N.G., 1992. The formation of helioseismology lines. I. NLTE effects in Na I and K I. AA 265, 237e256. Burgess, A., Tully, J.A., 1992. On the analysis of collisions strengths and rate coefficients. AA 254, 436e453. Burgess, A., 1964. Dielectronic recombination and the temperature of the solar corona. ApJ 139, 776. Burgess, A., 1965. A general formula for the estimation of dielectronic recombination CoEfficients in low-density plasmas. ApJ 141, 1588e1590. Butler, S.E., Dalgarno, A., 1980. Charge transfer of multiply charged ions with hydrogen and helium Landau-Zener calculations. ApJ 241, 838e843. Carlsson, M., Stein, R.F., 2002. Dynamic hydrogen ionization. ApJ 572, 626e635. Cooper, J., 1966. Plasma spectroscopy. Rep. Prog. Phys. 29, 35e130.

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Cowan, R.D., 1981. The Theory of Atomic Structure and Spectra. University of California Press, Berkeley, CA. Craig, I.J.D., Brown, J.C., 1976. Fundamental limitations of X-ray spectra as diagnostics of plasma temperature structure. AA 49, 239e250. Delache, P., 1967. Contribution a` l’e´tude de la zone de transition chromosphe`re-couronne. Annales d’Astrophysique 30, 827. DeWitt, H., Nakayama, T., 1964. A quantum statistical approach to level shift and level width in a hydrogenic plasma. J. Quant. Spectrosc. Radiat. Transfer 4 (5), 623e638. Dufton, P.L., Kingston, A.E., 1981. Atomic processes in the Sun. Adv. At. Molec. Phys. 17, 355e418. Fletcher, L., Dennis, B.R., Hudson, H.S., et al., 2011. An observational overview of solar flares. SSR 159, 19e106. Fontenla, J.M., Avrett, E.H., Loeser, R., 1990. Energy balance in the solar transition region. i. hydrostatic thermal models with ambipolar diffusion. ApJ 355, 700e718. Fujimoto, T., 2004. Plasma spectroscopy. In: International Series of Monographs in Physics, vol. 123. Oxford Science Publications, Clarendon Press, Oxford, UK. Gabriel, A.H., Jordan, C., 1971. Case Studies in Atomic Collision Physics. Chapter 4. NorthHolland, pp. 209e291. Griem, H., 1964. Plasma Spectroscopy. McGraw-Hill. Jordan, C., Veck, N.J., 1982. Comparison of observed CA XIX and CA XVIII relative line intensities with current theory. Sol. Phys. 78, 125e135. Judge, P.G., 2005. Understanding the evolution of atomic number densities with time in astrophysical plasmas. J. Quant. Spectrosc. Radiat. Transfer 92 (4), 479e510. Laming, J., Feldman, U., 1992. A burst model for line emission in the solar atmosphere. I. XUV lines of He I and He II in impulsive flares. ApJ 386, 364e370. Mihalas, D., Mihalas, B.W., 1984. Foundations of Radiation Hydrodynamics. Oxford Univ. Press, New York. Mihalas, D., 1978. Stellar Atmospheres, second ed. W. H. Freeman and Co., San Francisco. Nahar, S., Pradhan, A.K., 1992. Electron-ion recombination in the close-coupling approximation. PRL 68, 1488e1491. Pietarila, A., Judge, P.G., 2004. On the formation of the resonance lines of helium in the Sun. ApJ 606, 1239e1257. Rybicki, G.B., Hummer, D.G., 1991. An accelerated lambda iteration method for multilevel radiative transfer i. non-overlapping lines with background continuum. AA 245, 171e181. Rybicki, G.B., Hummer, D.G., 1992. An accelerated lambda iteration method for multilevel radiative transfer ii. overlapping transitions with full continuum. AA 262, 209e215. Scharmer, G.B., Carlsson, M., 1985. A new approach to multi-level non-LTE radiative transfer problems. J. Comput. Phys. 59, 56e80. Scheffler, H., Elsa¨sser, H., 1974. Physik der Sterne und der Sonne. Seaton, M.J., 1962a. The temperature of the solar corona. Observatory 928, 111e117. Seaton, M.J., 1962b. The theory of excitation and ionization by electron impact. In: Bates, D.R. (Ed.), Atomic and Molecular Processes. Academic Press, New York, p. 11. Seaton, M.J., 1964a. Excitation of coronal lines by proton impact. MNRAS 127, 191. Seaton, M.J., 1964b. The spectrum of the solar corona. Plan. Space Sci. 12, 55e74. Seaton, M.J., 1983. Quantum defect theory. Rep. Prog. Phys. 46, 167e257. Storey, P.J., 1981. Dielectronic recombination at nebular temperatures. MNRAS 195, 27Pe31P.

References

Summers, H.P., 1974. Collisional Dielectronic Recombination and Ionisation Coefficients and Ionisation Equilibria of H-like to Ar-like Ions of Elements, vol. 367. Culham Laboratory. Uitenbroek, H., 2000. The CO fundamental vibration-rotation lines in the solar spectrum. I. Imaging spectroscopy and multidimensional LTE Modeling. ApJ 531, 571e584. Uitenbroek, H., 2001. Multilevel radiative transfer with partial frequency redistribution. ApJ 557, 389e398. Van Regemorter, H., 1962. Rate of collisional excitation in stellar atmospheres. ApJ 136, 906. Vernazza, J., Avrett, E., Loeser, R., 1981. Structure of the solar chromosphere. III - models of the EUV brightness components of the quiet-Sun. APJSS 45, 635. Woolley, R.D.V.R., Allen, C.W., 1948. The coronal emission spectrum. MNRAS 108, 292e305. Young, P.R., Dere, K.P., Landi, E., et al., 2016. The CHIANTI atomic database. J. Phys. B Atom. Mol. Phys. 49 (7), 074009.

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CHAPTER

Models of Solar and Stellar Atmospheres

5.2 Petr Heinzel

Astronomical Institute, Czech Academy of Sciences, Ond
r ejov, Czech Republic

CHAPTER OUTLINE 1. Introduction ....................................................................................................... 157 2. Radiative Transfer .............................................................................................. 158 3. Two-Level Atom.................................................................................................. 161 3.1 Partially Coherent Scattering of Line Photons ........................................ 162 3.2 Two-Level Line Source Function ........................................................... 163 3.3 Solution of the Two-Level Atom Problem ............................................... 165 3.4 Radiative Heating and Cooling.............................................................. 167 4. Classical Static Atmospheres.............................................................................. 167 4.1 Basic Equations of Standard Model Atmospheres ................................... 168 4.2 Model Example ................................................................................... 171 5. Semiempirical Models........................................................................................ 172 6. Isolated Atmospheric Structures.......................................................................... 176 7. Spectral Line Synthesis ...................................................................................... 177 8. Radiation Hydrodynamics ................................................................................... 179 Acknowledgments ................................................................................................... 181 References ............................................................................................................. 181

1. INTRODUCTION Our understanding of the Sun, other stars, and celestial objects in general is almost entirely based on observations and analysis of electromagnetic radiation in a broad wavelength range, from g-rays up to the radio. This radiation travels to us through the vast cosmic space, more than 8 min from the Sun and billions of years from distant universes. What we detect by our instruments is the result of a complex interaction between the photons and matter inside the objects under study, but also other interactions along the long path to us. To understand these interactions, we have to solve a coupled, generally nonlinear problem of how radiation is transported through such media and how the media, mostly various kinds of plasmas, are affected by The Sun as a Guide to Stellar Physics. https://doi.org/10.1016/B978-0-12-814334-6.00006-6 Copyright © 2019 Elsevier Inc. All rights reserved.

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penetrating radiation. In this way, we can model the spectrum emergent from various objects; this gives us a deep insight into the structure and dynamics of, e.g., solar and stellar atmospheres. We call this the spectral diagnostics. However, the actual state of an atmosphere may strongly depend on the spectral distribution of the internal radiation field via complex atomic processes, leading, for example, to heating or cooling of the plasma by radiation, along with other processes such as the heat conduction, wave dissipation, or magnetic heating. The whole complexity of these interactions is governed by a coupled system of (magneto)-hydrodynamical equations and equations describing the transport of radiation. This is called radiation hydrodynamics (RHD) and we will briefly describe it at the end of this chapter. On the other hand, considering only the radiative transfer in a prescribed model atmosphere, we deal with a “classical” astrophysical problem that was thoroughly studied during the past century, and to a great extent, the basic concepts were developed by solar physicists. This is usually called the theory of radiative transfer, which was regarded as a synonym of solar or stellar atmospheric physics (e.g., Physik der Sternatmospha¨ren by Unso¨ld, 1938). Among the atmospheres of various stars, the solar atmosphere has the unique privilege of providing us with insight into different structural patterns and their dynamical behavior. This then serves as a guide for our understanding of various phenomena on other stars, namely late-type cool stars. One of the most prominent stellar astrophysicists, D. Mihalas, who was engaged in both solar and stellar atmospheric research, once said, “My views have been colored by my long period of residence at the High Altitude Observatory in Boulder, where I was confronted daily by the ghastly reality of the Sun as seen at high time, spatial, and spectral resolution in a wide range of spectral bands. There is probably no more sobering an experience for a stellar atmospheres modeller than a detailed inspection of solar data!” (Mihalas, 1991). The theory of stellar atmospheres is summarized in an excellent textbook by Hubeny and Mihalas (2015).

2. RADIATIVE TRANSFER Transport of radiation through a stellar atmosphere or through a specific atmospheric structure is described by the equation of radiative transfer. The unpolarized radiation field at a given position r is fully described by the specific intensity of radiation I(r, n, n, t). The energy transported by radiation in the frequency range (n, n þ dn) across an elementary area dS into a solid angle du in a time interval dt is: dE ¼ Iðr; n; n; tÞdS cos Qdudndt;

(5.2.1)

where Q is the angle between the direction of propagation n and the normal to the elementary area dS. The specific intensity provides a complete description of unpolarized radiation field from the macroscopic point of view and has the dimension ergs cm2 s1 sr1 Hz1 in cgs units. The radiation transport depends on opacity (absorption) and emissivity properties of the medium. The absorption coefficient describes the removal of energy from the radiation field by matter dE ¼ cðr; n; n; tÞIðr; n; n; tÞdSdsdudndt.

(5.2.2)

2. Radiative Transfer

The dimension of c is cm1. 1/c measures a characteristic distance a photon can travel before it is absorbed; we call it the photon mean free path. The emission coefficient describes the energy released by the material in the form of radiation dE ¼ hðr; n; n; tÞdSdsdudndt.

(5.2.3)

1

3 1

The dimension of h is ergs cm s sr1 Hz in cgs units. In thermodynamic equilibrium (TE), the microscopic detailed balance c I ¼ h holds, whereas the radiation intensity is equal to the Planck function, I ¼ B, where for a given temperature we have Bðn; TÞ ¼

2hn3 1 . c2 expðhn=kTÞ  1

(5.2.4)

Kirchhoff’s law states that h/c ¼ B. For a one-dimensional (1D) planar atmosphere in which the geometrical height is z, nz ¼ (dz/ds) ¼ cos q h m (in which q is the angle between the vertical direction and the direction of the ray), we get the standard form of the transfer equation m

dIðn; m; zÞ ¼ hðn; m; zÞ  cðn; m; zÞIðn; m; zÞ. dz

(5.2.5)

The intensity is only a function of z, n, and the directional cosine m. Dividing this equation by cn, we get m

dIn ¼ I n  Sn ; dsn

(5.2.6)

where we introduced the optical depth sn dsn h cn dz

(5.2.7)

and the source function defined as Sn h

hn . cn

(5.2.8)

Note that sn increases downward in the planar atmosphere, whereas z is the geometrical height increasing upward. This gives a minus sign in the definition of the optical depth. The formal solution of the transfer equation can be obtained as Z s2 Iðs1 ; mÞ ¼ Iðs2 ; mÞexp½ðs2  s1 Þ=m þ SðtÞexp½ðt  s1 Þ=mdt=m (5.2.9) s1

which gives us the specific intensity at optical depth s1 (here, we omitted the frequency index). The two terms have a simple interpretation: The first term is the intensity at depth s2 attenuated along the path (s2  s1)/m owing to absorption. The second term is the contribution owing to the source function at depth t, attenuated along the path to depth s1 and integrated over all depths from s1 to s2. Classical stellar atmospheres are sometimes called “semi-infinite atmospheres” because at the

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surface s1 ¼ 0 and at the bottom s2 / ∞. The emergent radiation from such an atmosphere has the intensity Z ∞ Ið0; mÞ ¼ SðtÞexpðt=mÞdt=m. (5.2.10) 0

For a given frequency, this specific intensity is a Laplace transform of the source function. Computed for a range of wavelengths (lines and continua), it represents the synthetic spectrum to be compared with the observed one. We can use it also to define the so-called contribution function: Cn ðzÞ ¼ Sn ðsn Þexpðsn =mÞcðnÞ=m ¼ hðnÞexpðsn =mÞ=m Z In ð0; mÞ ¼ Cn ðzÞdz.

(5.2.11) (5.2.12)

Cn(z) tells us how much the plasma at a given height z contributes to the emergent radiation and thus represents a useful diagnostic tool. Fig. 5.2.1 shows an example of

FIGURE 5.2.1 Contribution function for hydrogen Ha line in the quiet solar atmosphere Wavelength- and height-dependent contribution function for quiet-Sun atmospheric model VAL-C. Courtesy of J. Ka
s parova´.

3. Two-Level Atom

Cn(z) for the hydrogen Ha line as formed in a solar-type atmosphere discussed in Section 5. In this figure, we see that the line wings are formed in the photosphere whereas the line core forms higher up. Interestingly, there is a gap between these two contributions that results from low temperature around the temperature minimum region where the hydrogen is mainly in the ground state. Finally, we define the moments of the radiation field as Z 1 1 Jn ¼ In ðmÞdm (5.2.13) 2 1 Z 1 1 mIn ðmÞdm (5.2.14) Hn ¼ 2 1 Z 1 1 2 Kn ¼ m In ðmÞdm; (5.2.15) 2 1 where Jn is the mean intensity, Hn is the Eddington flux, and Kn is related to radiation pressure.

3. TWO-LEVEL ATOM Having described the phenomenological transfer equation, we now turn to details of the opacity and emissivity processes and form of the line source function. This can be well-explained using the concept of a two-level atom developed during 1960s. A largely simplified atomic model with only two bound levels has no counterpart in a real world; however, it may describe relatively well the formation of strong resonance lines that arise owing to transitions between dominantly populated atomic ground level and a nearby upper level. We will show later how real resonance lines are formed in solar and stellar atmospheres. The other class of lines, subordinate lines, refers to line transitions between two excited atomic levels. A two-level atom is also an illustrative example giving an insight into the otherwise difficult subject of nonequilibrium line-formation physics. Fig. 5.2.2 shows a drawing of all possible atomic transitions within a two-level atom. We consider two types of n2

2 B12J

A21

B21J

1

C12 C21

n1

FIGURE 5.2.2 Schematic two-level-atom The arrows show both radiative (thick blue) and collisional (thin black) transitions between atomic levels 1 and 2.

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transitions: radiative (thick arrows) and collisional (thin arrows). Atomic transitions between levels 1 and 2 are (from left to right in Fig. 5.2.2) caused by: • • • • •

absorption of a photon with energy hn (radiative excitation); spontaneous emission of a photon with energy hn (radiative deexcitation); stimulated emission of a photon with energy hn (the photon emission is triggered by an incoming identical photon); atomic excitation resulting from an atom’s inelastic collisions with surrounding free electrons; and collisional deexcitation.

Whereas the radiative processes remove or add energy from or to the radiation field, respectively, the collisional processes do the same with the thermal energy of the electron pool, with no relation to radiation. A specific process that is critically important in low-density media such as solar and stellar chromospheres is the scattering of line radiation. It is spontaneous emission “immediately” after the absorption; how “immediately” depends on the lifetime of the upper atomic state. The absorption and emission coefficients for a two-level atomic model are written as hn0 ½n1 B12 fðnÞ  n2 B21 jðnÞ (5.2.16) 4p hn0 hn ¼ n2 A21 jðnÞ; (5.2.17) 4p where B12, B21, and A21 are the Einstein coefficients for absorption, stimulated emission, and spontaneous emission, respectively; n1 and n2 are the atomic-level populations (i.e., the densities of atomic states per cubic centimeter); and n0 is the line center frequency. f(n) and j(n) are the absorption and emission profiles, respectively, normalized to unity. cn ¼

3.1 PARTIALLY COHERENT SCATTERING OF LINE PHOTONS At this point, we have to mention one important aspect of the scattering in spectral lines, which is the probability that the scattered radiation will be emitted at frequency n while the absorbed photon had a frequency n0 . The frequency and angular redistribution of photons within the line during the scattering process takes place because the atomic levels are broadened by various processes (for resonance lines, it concerns the upper level only) and because of Doppler shifts owing to atomic motions. The most general probability is described by the redistribution function R(n0 , n0 , n, n). For different cases, see Hummer (1962), Heinzel and Hubeny´ (1982) , or Hubeny and Mihalas (2015). Neglecting the angular redistribution, the line emission profile then takes the form Z ∞ Rn0 ;n Jn0 dn0 ; (5.2.18) jn ¼ J 0

3. Two-Level Atom

where J is defined subsequently by Eq. (5.2.28). This form of jn determines the frequency dependence of the line source function while assuming that no correlation between the two frequencies leads to the so-called complete redistribution or CRD, i.e., Rn0 ;n ¼ fn0 fn , which automatically gives jn h fn. A general form of the partial redistribution (PRD) for scattering in resonance lines can be expressed as (Milkey and Mihalas, 1973b) Rn0 ;n ¼ gRII þ ð1  gÞRIII g¼

(5.2.19)

P21 ; P21 þ QE

where P21 is the radiative depopulation rate and QE is the rate of elastic collisions. g is the branching ratio, i.e., the probability that the coherence in the atom’s frame (function RII) is not destroyed by collisional perturbation of the upper atomic state (function RIII). Rn0 ;n is the angle-averaged redistribution function, which is a good approximation for static media. A generalization to subordinate lines was performed in Heinzel and Hubeny´ (1982). The emergent line profiles computed with PRD may differ significantly from those computed assuming CRD; this substantially affects the spectral diagnostics, especially in the case of strong resonance lines. This was first realized in the case of the solar hydrogen Lyman a line by Milkey and Mihalas (1973a); since then, stellar atmospheric modelers were guided by such an approach. However, for subordinate lines, one can reasonably assume that the scattering is completely noncoherent, which leads to the equality of the emission and absorption profiles. The resulting line profiles depend on both the PRD scattering physics and the actual shape of the line radiation to be scattered. This can be easily understood by inspecting the scattering integral in the expression for jn. If the frequency distribution of the radiation field is flat enough over the line absorption profile, jn ¼ fn and we get the CRD case (we call this special case “natural excitation”). This shows how critical the real shape of Jn is in combination with the redistribution function. For simplicity and clarity of exposition, however, we will use in this section only the CRD approach.

3.2 TWO-LEVEL LINE SOURCE FUNCTION Now we introduce a dimensionless frequency x n  n0 xh ; DnD

(5.2.20)

in which DnD is the Doppler width of the spectral line DnD ¼ ðn0 =cÞv

(5.2.21)

 1=2 v ¼ 2kT=m þ v2t .

(5.2.22)

with the mean atomic velocity

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Here, m is the atomic mass and vt represents the microturbulent velocity. In simplest cases, the absorption profile may take the form of Doppler (Gaussian) profile  pffiffiffi p (5.2.23) fðxÞ ¼ exp x2 or the Voigt function pffiffiffi fðxÞ ¼ Hða; xÞ p;

a Hða; xÞ ¼ p

Z

∞

  exp y2

∞ ðx 

yÞ2 þ a2

dy;

(5.2.24)

where a is the damping parameter a ¼ G/(4pDnD) and G is the atomic damping parameter (natural broadening, pressure broadening). The line opacity is then written as cx ¼ cfðxÞ

(5.2.25)

and analogously for hx. The line source function takes the form S¼

n2 A21 h SL n1 B12  n2 B21

(5.2.26)

and is independent of the frequency in the case of complete redistribution. The balance in populations of atomic levels n1 and n2 is described by the statistical-equilibrium equation n1 ðR12 þ C12 Þ ¼ n2 ðR21 þ C21 Þ;

(5.2.27)

in which the R’s are the radiative rates and the C’s are the collisional rates. Furthermore, Z ∞ R12 ¼ B12 Jn fðnÞdn h B12 J (5.2.28) Z

0

R21 ¼ A21 þ B21

∞

Jn fðnÞdn h A21 þ B21 J;

(5.2.29)

0

where J is the frequency-averaged mean intensity. Moreover, C12 ¼ ne q12 ðTÞ;

(5.2.30)

where ne is the electron density and q12 is the collisional excitation rate as the function of temperature. Using relations between the Einstein coefficients B21/   B12 ¼ g1/g2 and A21 B21 ¼ 2hn30 c2 , and the relation between collisional rates (principle of detailed balance) C21/C12 ¼ (g1/g2) exp(hn0/kT), where the last expression is the Boltzmann level population distribution in thermodynamic equilibrium, we can write the line source function in a compact form: S ¼ ð1  εÞJ þ εBn0 ;

(5.2.31)

3. Two-Level Atom

where ε0 ε¼ ; 1 þ ε0

  C21 1  ehn0 =kT ε ¼ . A21 0

(5.2.32)

For hn/kT [ 1, ε may be expressed as εz

C21 . C21 þ A21

(5.2.33)

Branching parameter ε is interpreted as a destruction probability, i.e., the probability that the energy of an absorbed photon is converted into the energy of the electron pool by collisional deexcitation. This line source function has two basic parts: a scattering term and a thermal term. For ε / 1, S approaches the Planck function. This takes place in atmospheric layers with a sufficiently high plasma density so that the collisions can establish the so-called local thermodynamic equilibrium (LTE). However, contrary to strict thermodynamic equilibrium, LTE applies only to particles although the radiation field is not Planckian and comes from the formal solution of the transfer equation with S ¼ B. Situations like this can take place (e.g., in stellar photospheres). The other extreme is when ε  1; then, S is dominated by scattering and depends on the radiation field, which is unknown until the S is specified. This applies to low-density, scattering media such as solar and stellar chromospheres and causes strong nonequilibrium effects, i.e., departures from LTE. We denote that simply as non-LTE.

3.3 SOLUTION OF THE TWO-LEVEL ATOM PROBLEM In the general non-LTE case, the transfer equation becomes an integrodifferential equation because S depends on the intensity and there is no analytical solution in general. The transfer equation with the source function Eq. (5.2.31) has to be solved numerically. Various methods were designed for that over the past decades, such as the popular Feautrier method (Feautrier, 1964). One could think of a straightforward iterative method in which, starting from S¼B, the source function is iterated using Eq. (5.2.31) with an updated J from the formal solution of the transfer equation. However, this lambda iteration is notoriously extremely slow, often leading to spurious solutions. On the other hand, a practical and much-used solution is based on an ingenious way to accelerate such iterations, a method generally called accelerated lambda iterations (ALI). ALI was first demonstrated to be a useful numerical technique by solar physicist G. Scharmer (1981) and then further developed by many solar and stellar atmospheric modelers (for a comprehensive description, see Hubeny and Mihalas, 2015). Fig. 5.2.3 shows the classical solution of the twolevel atom problem by Avrett and Hummer (1965), who used yet another technique available during those pioneering times.

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FIGURE 5.2.3 Two-level-atom source function Theoretical variation of the line source function in a uniform semiinfinite stellar atmosphere, computed for various values of the parameter ε. Thermalization depths sth are indicated for the Doppler profile. Source: According to Hubeny (1997)

The line source function was computed for a semiinfinite homogeneous atmosphere with constant values of B and ε. It is displayed on a s scale corresponding to the line opacity c. At the surface, the source function is decreasing because of the escape of radiation from the outer layers of the atmosphere, reaching the surface pffiffiffi value Sð0Þ ¼ εB. At a certain optical depth called the thermalization depth, S starts to approach B, for the Doppler profile sth z 1/ε, as can be seen from the plot. We have seen from Eq. (5.2.31) that for ε / 1 the line source function approaches the Planck function, and thus we reach the LTE conditions. However, S ¼ B is achieved also deep in the low-density, small ε atmosphere, where the line formation is dominated by scattering. This takes place behind the thermalization depth. In that situation, the radiation field is also thermalized to the Planck function as in the black-body cavity from which the photons cannot freely escape (the atmosphere is already opaque there) and this thermalization does not depend on the density. Inserting J ¼ B into Eq. (5.2.31), we get S ¼ B for any value of ε.

4. Classical Static Atmospheres

3.4 RADIATIVE HEATING AND COOLING The two-level atom also demonstrates well the processes of radiative heating and cooling, which are important for modeling the temperature structure of the atmosphere, either in a classical way, as described in the next section, or in the most complex way via radiation hydrodynamics. When the absorbed photon is destroyed by a collisional deexcitation, its energy hn is converted into a thermal pool of the plasma (radiative heating). On the other hand, when the atom is excited by a collision and a photon emission follows, the energy lost by the colliding electron is converted to radiation energy of a photon and may escape from the plasma volume (radiative cooling). For line transitions, the net radiation losses are defined as     L ¼ hn0 n2 A21 þ B21 J  n1 B12 J . (5.2.34) In the case of dominant collisional excitation (e.g., in hot coronae), we get L ¼ hn0 n2 A21 .

(5.2.35)

If all absorbed photons are destroyed, the radiative heating is L ¼ hn0 n1 B12 J.

(5.2.36)

For pure scattering in a two-level atom, both radiative terms cancel and L ¼ 0 (no energy exchange between the radiation field and plasma). The loss function L can be expressed also as 

L ¼ hn0 n2 A21 r

(5.2.37)

where r ¼ 1  J S is the net radiative bracket. If J/S (detailed radiative balance), the losses go to zero.

4. CLASSICAL STATIC ATMOSPHERES In this section, we will describe two types of solar and stellar atmospheric models. First-type models are sometimes called ab initio models, which means that a set of basic equations is solved to determine all structural parameters such as temperature, gas pressure, and gas density, but also quantities such as the ionization degree of the plasma and thus the electron density. All structural parameters, as well as other quantities, are determined on a given height (depth) scale, which is usually the geometrical distance measured from a predefined zero level or using the so-called column-mass scale. Sometimes the optical-depth scale is also given: for example, s500 at the wavelength 500 nm, which in the solar case represents a standard continuum wavelength in the optical portion of the spectrum. The atmospheric model is generated in the form of a numerical table in which the individual quantities vary along the grid. The grid may consist of a few tens to a few hundreds depth (height) points. Their spatial distribution depends on the gradients of structural parameters and on variation of the line and continuum source functions. This is relatively simple

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CHAPTER 5.2 Models of Solar and Stellar Atmospheres

in 1D models, for which we have explicitly written the transfer equation. In 2D or 3D modeling, the situation becomes much more complex. The other models are called semiempirical models and we will discuss them later. As an example of classical ab initio models, we will now show 1D models in hydrostatic equilibrium, in which the temperature structure is determined using the radiative equilibrium (RE) condition. This is the simplest atmospheric model, in which the energy equation is solved with no local heating except for the radiative one. Such a situation applies to solar photosphere, and this concept was successfully applied to other stars, both cool (G, M, and K dwarfs) as well as hot early-type stellar photospheres.

4.1 BASIC EQUATIONS OF STANDARD MODEL ATMOSPHERES Next, we summarize the basic equations needed to construct 1D plane-parallel models of stellar atmospheres in radiative and hydrostatic equilibrium (Hubeny, 1997 and Hubeny and Mihalas (2015)). •

Radiative transfer equation (RTE) The actual form of radiative transfer equation (RTE) will depend on the method used to solve it. The most popular technique, as mentioned, is the Feautrier method, which uses RTE as a second-order differential equation: m2

d 2 jmn ¼ jmn  Sn ; ds2n

(5.2.38)

in which jmn h (ImþIm)/2 is the so-called Feautrier variable. One can also eliminate the angular dependence and write  d2 fnK Jn ¼ Jn  Sn ; (5.2.39) ds2n where fnK ¼ Kn =Jn

(5.2.40)

is the so-called variable Eddington factor to be iteratively updated. The ALI techniques are most often applied to solve non-LTE multilevel problems, and then Eq. (5.2.38) can be used for a formal solver between ALI iterations. The upperboundary condition for RTE states that no incident radiation enters the atmosphere from above. This is the case of the Sun, but in some stellar cases (e.g., binary stars) this can be easily modified to account for illumination from the other companion. A lower-boundary condition for semiinfinite atmospheres is usually in the form of a diffusion approximation at smax.

4. Classical Static Atmospheres

•

Hydrostatic equilibrium (HE) Using the geometrical height scale, we write dp ¼ rg; dz

(5.2.41)

where p is the total pressure, r the plasma density, and g the gravity acceleration at the stellar surface. It is common to use the column mass dm ¼ rdz and write HE as dp ¼ g. (5.2.42) dm This form has a direct solution p(m) ¼ gm þ p(0) in the case of a constant g (reasonable for 1D plane-parallel atmospheres), where p(0) is the boundary pressure. The total pressure has three components: Z 4p ∞ 1 p ¼ pgas þ prad þ pturb ¼ NkT þ Kn dn þ rv2t ; (5.2.43) c 0 2 i.e., the gas pressure, radiation pressure, and turbulent pressure. Here, N is the total particle number density, Kn is the second moment of the radiation intensity, and vt is the mean microturbulent velocity. • Radiative equilibrium (RE) RE is the simplest constraint on the atmospheric temperature structure. Its integral form states that all emissions and absorptions, integrated over the whole spectrum, must be in balance at each atmospheric depth: Z ∞ Z ∞ ðhn  cn Jn Þdn ¼ cn ðSn  Jn Þdn ¼ L ¼ 0: (5.2.44) 0

0

L is the net radiative loss function, taken as positive for radiative cooling and negative for radiative heating outside RE. Individual line or continuum transitions contribute to radiative heating or cooling, but total L ¼ 0 in RE. Equivalently, we can say that the total radiation flux (first moment of the intensity) is conserved:  Z ∞ Z ∞ d f K Jn n s 4 Hn dn ¼ dn ¼ Teff ; (5.2.45) 4p dsn 0 0 where Teff is the effective temperature of the star, which is a measure of its luminosity. For line transitions, the specific formulas have been presented in Section 3. A practical implementation of the RE constraint is not trivial even when LTE is assumed. One needs to solve the transfer equation to get the mean radiation intensity, which enters the RE condition together with the Planck function. Opacities are computed in LTE using the SahaeBoltzmann distributions. Net radiative losses in standard photospheres are simply equal to zero at all depths. However, application of this condition to temperature determination is a complex numerical task, especially when the non-LTE conditions and the line blanketing are considered.

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CHAPTER 5.2 Models of Solar and Stellar Atmospheres

•

Statistical (or kinetic) equilibrium equations (ESE) Multilevel statistical equilibrium equations (ESE) in their simplest form are straightforward generalizations of the two leveleatom ESE discussed before, i.e., X X ni ðRij þ Cij Þ ¼ nj ðRji þ Cji Þ; (5.2.46) jsi

jsi

in which Rij and Cij are the radiative and collisional rates, respectively. Contrary to the two-level atom, the multilevel form includes also the bound-free (ionization) and free-bound (recombination) transitions, i.e., transitions between the bound levels and the continuum states. ESE for all considered atomic and ionic states form a linearly dependent system, and thus one equation must be replaced by some P additional constraint, e.g., total particle number conservation for each species ni ¼ Natom . i The whole set of equations is then expressed as A n ¼ b;

(5.2.47)

where A is the full rate matrix, n the vector of level populations, and b the vector of known quantities. • Charge conservation equation The electron density is constrained by the charge conservation, which expresses the electric neutrality of the medium X ni Zi  ne ¼ 0; (5.2.48) i

in which Zi is the charge associated with the ionic state i. • Auxiliary definition equations In the section on two-level atoms, we have written explicit forms of the line absorption and emission coefficients. In the multilevel case, these coefficients contain all boundebound, boundefree, and freeefree contributions to opacity and emissivity at a given frequency. The general form of the absorption coefficient takes the form XX X     cn ¼ ni  ni ehn=kT sik ðnÞ ni  gi gj nj sij ðnÞ þ i

j>i

i

 X þ ne nk skk ðn; TÞ 1  ehn=kT þ ne se k

with sij ðnÞ ¼ ðhn0 =4pÞBij fðnÞ.

(5.2.49)

The first term is a generalization of the previously used line absorption coefficient, the second corresponds to photoionizations, the third corresponds to freeefree continuum opacity, and the last term is caused by Thomson scattering on free electrons.

4. Classical Static Atmospheres

Table 5.2.1 Summary of Structural Equations and Relevant Quantities Equation

Quantity

Notation

Radiative transfer Radiative equilibrium Hydrostatic equilibrium Statistical equilibrium Charge conservation

Mean radiation intensities Temperature Total particle density Level populations Electron density

Jn T N ni ne

s is the corresponding cross-section. Stimulated emission is included as a negative absorption. The general form of the emission coefficient then reads 2 X  3  2  XX    hn ¼ 2hn c 4 nj gi gj sij ðnÞ þ ni sik ðnÞehn=kT i

þ

X k

j>i

3

ne nk skk ðn; TÞehn=kT 5;

i

(5.2.50)

where the emission caused by Thomson scattering is included as an extra term in RTE. In Table 5.2.1, we summarize the basic equations and the corresponding plasma parameters (Hubeny, 1997). Theoretical model atmospheres based on this set of equations are parametrized by just three quantities: Teff, g, and chemical composition (usually characterized by a stellar metallicity).

4.2 MODEL EXAMPLE The solution to these coupled equations must be determined numerically and represents a complex numerical task because in the general multilevel case, we deal with a set of nonlinear and nonlocal equations. Various methods exist to treat this problem (e.g., complete linearization or preconditioning); the interested reader can consult them in the comprehensive review by Hubeny and Mihalas (2015). Apart from the hydrogen and helium lines, stellar atmospheres exhibit significant opacity in many (millions) metallic lines; this naturally enters the RE condition. Their effect on atmospheric models (i.e., the temperature structure) is called line blanketing. Its proper treatment requires the solution of RTE in many metallic lines, which in the general non-LTE case represents an extremely complicated numerical task. Fig. 5.2.4 shows several examples of atmospheric models with Teff ¼ 35,000K and log g ¼ 4, computed with all of these equations and demonstrating the effect of different approximations: the thick line is the fully blanketed non-LTE model (HeHeeCeNeOeSieFeeNi), the dashed line is the non-LTE model with light

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CHAPTER 5.2 Models of Solar and Stellar Atmospheres

FIGURE 5.2.4 Model atmosphere with Teff [35000 K and log g [4 Several solutions obtained under different assumptions. Source: Hubeny (1997)

elements only (HeHeeCeNeOeSi), the thin line is the non-LTE HeHe model, and the dotted line is the LTE HeHe model. We see that at large column-mass depths, line blanketing causes so-called back-warming, whereas the surface layers are sensitive to non-LTE effects as well to how line blanketing is treated. At the surface, line blanketing leads to more efficient radiative cooling, which results in lower temperatures compared with non-blanketed cases. An interesting feature is a rise in temperature for the non-LTE HeHe model compared with the LTE case. A detailed discussion and explanation can be found in Hubeny (1997). Perhaps the most sophisticated 1D code that treats classical stellar photospheres in HE and RE and with full non-LTE line blanketing is the code TLUSTY (Hubeny and Mihalas, 2015).

5. SEMIEMPIRICAL MODELS In these models, the temperature structure of the atmosphere is determined empirically without considering the energy equation. This largely simplifies the construction of models even if the departures from LTE lead to non-trivial solutions of other

5. Semiempirical Models

basic equations. In stellar photospheres the models in radiativeeconvective equilibrium can be constructed. However, in higher atmospheric layers such as the chromospheres, extra heating is required to account for the temperature increase with height. The mechanisms of such heating are still not well understood and are even more critical in the case of active processes such as solar and stellar flares. In case of semiempirical models, the plasma temperature at each depth is adjusted empirically by comparing the theoretically computed spectrum (synthetic spectrum) for an estimated temperature structure with the observed spectra in different spectral continua and lines. For that, it is necessary to use as many spectral data as possible, from UV to infrared or even microwaves. Because different lines (and their parts) (Fig. 5.2.1) and continua are formed at different atmospheric depths and are differently sensitive to temperature, we can map the temperature structure using the broadband spectral information. Although the semiempirical models have been sometimes criticized as unrealistic, they are widely used as reference models of the solar atmosphere and of atmospheres of some cool stars. Even fully time-dependent RHD models of solar and stellar flares use the static semiempirical models as the initial model for simulations. Semiempirical models are usually static, i.e., the equation of HE is solved along with non-LTE equations. In the solar case, the total pressure is dominated by the gas pressure and turbulent pressure. However, the latter is determined empirically with no physical prescription. The most sophisticated semiempirical models of the solar atmosphere are those of Vernazza et al. (1981), usually referred to as VAL-models. They have been constructed using the PANDORA non-LTE code of Avrett and Loeser (2008) for various brightness structures in the quiet solar atmosphere, from cell interior to bright network. Most famous is the VAL-C model, which represents the average quiet Sun. The “VAL paper” is one of the most cited articles in solar physics. We show this classical model in Fig. 5.2.5. In solar photospheric layers, we see the temperature decreasing toward the so-called temperature minimum T ¼ 4170K. This part of the atmosphere roughly corresponds to models in RE. Above that, the temperature starts to increase again, which is a typical behavior of the solar and stellar chromospheres. Finally, a steep temperature increase is shown in the chromosphere-corona transition region. The temperature reaches almost coronal values in a few hundreds of kilometers. In Fig. 5.2.5, one can identify various spectral lines and their portions plus continua as they are formed at different atmospheric depths. In the chromosphere, the line and continuum source functions may significantly depart from the Planck function, and thus the spectrum is not a straightforward indicator of the kinetic plasma temperature as it is in the photosphere, where we assume the LTE conditions. Only in the microwave range does the free-free continuum have the Planck source function, which largely simplifies empirical modeling. Among other semiempirical solar models is that by Avrett and Loeser (2008), denoted as C6. For other cool stars, such models have been also constructed. One example is shown in Fig. 5.2.6, which is a plot of the semiempirical model of M-type red-dwarf star AD Leo, which is known for its frequent flare activity. Note different temperatures and pressure scale heights of VAL-C and AD Leo

173

FIGURE 5.2.5 Classical VAL-C model of the quiet solar atmosphere Formation heights of various spectral features used to construct the semiempirical models. h is the geometrical height scale; m represents the column-mass scale. Source: Vernazza et al. (1981)

Preflare Atmosphere 7

−5

10

Temperature [K]

T (AD Leo) ρ (AD Leo)

10

6

10−6 10−7 10−8 10−9

105

10−10 10−11

104 103

10−12

−3 Gas Density [g cm ]

10

10−13 0

200 400 600 800 Height (from τ5000=1) [km]

1000

FIGURE 5.2.6 Model atmosphere of the flare star AD Leo Temperature versus atmospheric height (usually the zero height is defined at optical depth 1 in the continuum at wavelength 5000 A˚). Note the enhanced density compared with solar model VAL-C. Courtesy of A. Kowalski.

5. Semiempirical Models

models. AD Leo represents a cool star with Teff ¼ 3000K. Contrary to the solar case, the hydrogen Ha line on AD Leo is in emission. This is because its source function follows the Planck function in a high-density chromosphere up to the line-center formation region. Although the energy balance problem is not considered in semiempirical models, it is possible to evaluate the total radiation losses L owing to lines and continua at each atmospheric depth. This then gives us the amount of energy required to maintain the semiempirical temperature. The net radiative losses for VAL-C atmosphere are shown in Fig. 5.2.7.

FIGURE 5.2.7 Net radiative losses for VAL-C model Net radiative cooling rates owing to various atoms and ions. In the middle chromosphere, the CaII losses dominate whereas at the base of the transition region, the hydrogen Lyman a line is most important coolant. Positive values refer to net radiative cooling, and negative to heating. h and m scales are the same as in Fig. 5.2.5. Source: Vernazza et al. (1981)

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CHAPTER 5.2 Models of Solar and Stellar Atmospheres

In the middle chromosphere, radiative cooling is dominated by CaII lines, with a contribution from MgII, whereas at the base of the transition region the loss function is dominated by the hydrogen Lyman a line. The net losses L computed from a semiempirical model must be balanced by the same amount of nonradiative heating and cooling. Then we can try to identify the physical mechanisms that would lead to such an empirical heating profile. Although a significant effort was put into this topic, the issue of chromospheric heating is still widely debated and modeling is now focusing more on ab initio radiation-hydrodynamical simulations (see discussion in Heinzel, 2000).

6. ISOLATED ATMOSPHERIC STRUCTURES So far, we have been discussing the case of semiinfinite atmospheres. However, guided by spatially resolved solar observations, we are faced with various isolated atmospheric structures such as prominences, loops, chromospheric spicules or fibrils, and jets. They usually have chromospheric temperatures and have been observed in lines from extreme UV (EUV) to infrared, with an extension to microwaves. For a review of the prominence spectroscopy, see Labrosse et al. (2010) and Vial and Engvold (2015). A common property of these structures is that they are located above the solar surface and thus are illuminated by the disk radiation. This external irradiation is then incorporated into the boundary conditions of the transfer equation that has to be solved for such an isolated structure. The illumination is either taken from solar disk spectral observations in a broad wavelength range or it can be provided as a synthetic spectrum based on solar atmospheric models. Large-scale structures such as quiescent prominences have been initially modeled by simple 1D plasma slabs, vertically standing above the solar surface and symmetrically illuminated on both sides. These models give a reasonable agreement, especially with optical spectra. An extensive grid of 1D isothermal-isobaric models has been considered by Gouttebroze et al. (1993), who computed the hydrogen radiation properties of such 1D slabs. Later, Gouttebroze and Heinzel (2002) and Heinzel et al. (2014) extended this modeling to CaII and MgII, respectively. In the latter report, the authors also considered a nonuniform prominence temperature constrained by the condition of RE, i.e., Eq. (6.44). Taking into account all important line losses (HI, CaII, and MgII), extended and dense prominences exhibit central temperatures even below 5000K, whereas thin threads with lower pressures have temperatures that are typically observed. Vial (1982) and Paletou (1995) modeled the whole prominence as a 2D slab and (Heinzel and Anzer, 2001) used similar techniques to model vertical fine-structure threads in magneto-hydrostatic equilibrium. Then the multithread models were developed in a series of articles (for a review, see Guna´r (2014)). Structures such as flare loops (Heinzel and Shibata, 2018) or coronal mass ejections are expected on other cool stars as well, and their detailed spectroscopic study is highly desirable.

7. Spectral Line Synthesis

7. SPECTRAL LINE SYNTHESIS There is a significant difference between solar and stellar spectra. On the Sun, we can detect radiation from individual locations that is characterized by its specific intensity. On the other hand, the stellar surfaces are normally unresolved and we can detect only the flux of radiation integrated over the whole stellar disk. This complicates the situation when significant inhomogeneities are expected on stars, such as star spots or flares. However, some solar instruments in space such as EVE (Woods et al., 2012) on the Solar Dynamics Observatory (National Aeronautics and Space Administration) measure only the total solar flux; this is called a “Sun-as-a-star” observation. Using a semiempirical atmospheric model or any other model such as a snapshot from time-dependent simulations, one can synthesize the line spectrum by solving the non-LTE problem of radiative transfer. For that, various numerical codes exist, such as MULTI (M. Carlsson), MALI (P. Heinzel), or RH (H. Uitenbroek); for specific stellar cases, see Hubeny and Mihalas (2015). Once the line source function is obtained, one can map its height variation into the wavelength variation of the line specific intensity. We show this as an example of a quiet solar atmosphere in Fig. 5.2.8, in which the formation of strong MgII lines is illustrated according to Leenaarts et al. (2013). Fig. 5.2.8A and B display the line source functions computed with PRD at two wavelengths (the PRD line source function is wavelength dependent), and at the line peaks and the line center. Triangles in Fig. 5.2.8A and B show at which heights the line peaks and the line center, respectively, are formed (they indicate the optical depth unity at the respective wavelength). Fig. 5.2.8C then shows the synthetic MgII k (black) and MgII h (red) profiles. Dotted lines in Fig. 5.2.8A and B represent the Planck function for which the line profile would appear purely in emission. The line peaks are formed just above the thermalization depth where the source function starts to depart from the Planck function, whereas the line center has much lower intensity because the source function has decreased significantly at the line-center optical depth unity. Fig. 5.2.8A shows that the line source function is dominated by the scattering (i.e., by J ); MgII lines thus exhibit strong departures from LTE. To understand how S is mapped into I, one has to realize that the specific intensity I is roughly equal to the line source function S at optical depth unity at a given wavelength. Thus, moving from the line center toward the line wings, we see increasingly deeper layers and the line intensity corresponds to S at those layers (this is based on the so-called EddingtoneBarbier relation). In LTE, the source function is equal to Planck function at the local temperature B(T ), which is first decreasing with height in the photosphere and then increasing in the chromosphere. This is reflected in decreasing intensity of the line wings toward the line core and then increasing intensity in the line core forming the peaks. However, the non-LTE line source function starts to depart from B around the thermalization depth and this causes a reversal around the line center. However, in hotter and denser atmospheres, the optical depth unity in the line center can be located in layers where S is still increasing; the thermalization depth is much higher in the atmosphere. In such situation, no reversal will

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CHAPTER 5.2 Models of Solar and Stellar Atmospheres

FIGURE 5.2.8 Spectrum synthesis of the MgII lines in a quiet solar chromosphere Source functions S and line intensities I of the MgII h (red) and k (black) lines are displayed (full lines). Dotted lines represent the Planck function B; dashed ones in panel a show the mean intensity J of the radiation field. Triangles indicate the formation heights of the line peaks and the line center. Source: Leenaarts et al. (2013)

8. Radiation Hydrodynamics

be observed. The line core will go into full emission. We observe such profiles during strong solar and stellar flares. Finally, one can understand the line formation also using the concept of contribution functions, as shown in Fig. 5.2.1.

8. RADIATION HYDRODYNAMICS So far, we have described relatively simple 1D static models, in which the most challenging aspect was the non-LTE radiative transfer. However, to understand the behavior of dynamical processes in solar and stellar atmospheres, one has to combine the plasma hydrodynamics (or, more generally, magnetohydrodynamics) with fully time-dependent kinetic equilibrium and radiative transfer. This is governed by equations of RHD. However, their numerical solution, even in 1D geometry, possible only after sufficiently powerful computers were available. We next summarize the basic RHD equations used for solar and stellar atmospheric modeling in a 1D plane-parallel approximation (Heinzel, 2000). •

Equation of continuity vr vðrvÞ þ ¼ 0; vt vz

•

•

(5.2.51)

Equation of momentum balance

  vðrvÞ v rv2 vp þ ¼   rg; vt vz vz

(5.2.52)

Equation of energy balance vðreÞ vðrveÞ vv v þ þ p ¼  ðFc þ Fr þ Fh Þ; vt vz vz vz

(5.2.53)

where v 1 Fr ¼ vz 2

Z 0

∞

Z

1 1

 hnm  cnm Inm dndm

(5.2.54)

is the divergence of the radiative flux, which is the loss function L (remember that RE requires constant radiative flux through the whole atmosphere, i.e., L ¼ 0). In these equations, v represents the plasma flow velocity, r is the plasma density, p is the total pressure, g gravitational acceleration at stellar surface, and e is the internal energy (thermal plus ionization and excitation). The fluxes Fc, Fr, and Fh correspond to the conductive, radiative, and any other heat flux, respectively. In the case of a static and time-independent atmosphere that is assumed to be in RE, these equations simply reduce to those discussed in Section 4. Coupling to radiation is achieved by adding the full set of non-LTE equations.

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CHAPTER 5.2 Models of Solar and Stellar Atmospheres

•

Radiative transfer equation 1 vInm vInm þm ¼ Inm  Snm c vt vsn

•

(5.2.55)

Kinetic equilibrium equations vnik vðnik vÞ X ¼ þ ðnjk Pjik  nik Pijk Þ; vz vt jsi

(5.2.56)

where P’s are total (radiative plus collisional) rates in the multilevel system. Similarly, as in the case of standard models, we add the charge and particle conservation conditions. For a characteristic length L and hydrodynamic time scale t0, we get

1 vInm vInm ¼ ðL=cÞ=t0 . (5.2.57) m c vt vsn In standard stellar atmospheres, we have typically (L/c)  t0 and thus, the time derivative in Eq. (5.2.55) is normally neglected compared with the spatial derivative. On the other hand, the time derivative of atomic population density and the advection term on the left-hand side of Eq. (5.2.56) are both important. The latter describes the divergence of the flux of atomic-level populations, which modifies the local population density otherwise determined by the complexity of transitions between all considered atomic states. This advection term is important even in the case of stationary atmospheres with non-negligible velocity gradients. The basic coupling between the plasma hydrodynamics and the radiation field is due to the divergence of the radiation flux vFr/vz (which represents the net radiation losses) and via all terms that depend on the ionization. The radiation losses depend in a complex way on the atmospheric structure and also on how the radiation is transported. Optically thin losses are different from optically thick ones. On the other hand, timedependent ionization is sometimes called nonequilibrium ionization, which means that owing to long recombination time scales typical of low-density chromospheres, the plasma cannot reach the statistical equilibrium state. Table 5.2.2 lists the basic quantities and relevant RHD equations that determine them. Various RHD codes exist for solar and stellar atmospheric modeling. RADYN was developed by Carlsson and Stein (1995) for time-dependent simulations of 1D horizontally homogeneous solar chromosphere penetrated by acoustic waves. This code was later adapted to study solar and stellar flares (Allred et al., 2015 and references therein). Similar to RADYN is the RHD code Flarix, also used for solar flare simulations (Heinzel et al., 2016 and references therein). There was found an excellent agreement between solar flare models constructed with RADYN and Flarix. For stellar modeling, other RHD codes have been used: Hubeny (2009) and Hubeny and Mihalas (2015). There was also a significant advancement in developing fully 3D codes, e.g.the code Bifrost (Gudiksen et al., 2011). The latter code includes the magnetic

References

Table 5.2.2 Summary of Radiation Hydrodynamics Equations and Relevant Quantities Equation

Quantity

Notation

Continuity equation Momentum equation

Velocity Density or Particle density Temperature Gas pressure Radiation intensity Level populations Electron density

v r N T pg Inm or Jn ni ne

Energy equation State equation Radiative transfer Statistical equilibrium Charge conservation

fields and thus is capable of simulating highly dynamical interactions of chromospheric and transition-region plasmas and magnetic fields.

ACKNOWLEDGMENTS This work was supported by the institutional project RVO:67985815 of the Astronomical Institute of the Czech Academy of Sciences. The author acknowledges support from grants
16-18495S and 16-17586S of the Czech Science Foundation (GACR).

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Gudiksen, B.V., Carlsson, M., Hansteen, V.H., et al., July 2011. The Stellar Atmosphere Simulation Code Bifrost. Code Description and Validation. Astron. Astrophys. vol. 531, A154. https://doi.org/10.1051/0004-6361/201116520, 1105.6306. Guna´r, S., January 2014. Modelling of quiescent prominence fine structures. In: Schmieder, B., Malherbe, J.M., Wu, S.T. (Eds.), Nature of Prominences and Their Role in Space Weather, IAU Symposium, vol. 300, pp. 59e68. Heinzel, P., Anzer, U., September 2001. Prominence fine structures in a magnetic equilibrium: two-dimensional models with multilevel radiative transfer. Astron. Astrophys. 375, 1082e1090. https://doi.org/10.1051/0004-6361, 20010926. Heinzel, P., Hubeny´, I., 1982. Non-coherent scattering in subordinate lines: II. Collisional redistribution. Journal of Quantitative Spectroscopy and Radiative Transfer 27, 1e14. https://doi.org/10.1016/0022-4073(82)90039-5. Heinzel, P., Shibata, K., 2018. Can Flare Loops Contribute to the White-Light Emission of Stellar Superflares? Astrophys. J. vol. 859, 143. https://doi.org/10.3847/1538-4357/ aabe78. http://adsabs.harvard.edu/abs/2018ApJ...859..143H, Provided by the SAO/ NASA Astrophysics Data System. Heinzel, P., Ka
sparova´, J., Varady, M., et al., 2016. Numerical RHD simulations of flaring chromosphere with Flarix. In: Kosovichev, A.G., Hawley, S.L., Heinzel, P. (Eds.), Solar and Stellar Flares and Their Effects on Planets, IAU Symposium, vol. 320, pp. 233e238, 1602.00016. Heinzel, P., Vial, J.C., Anzer, U., April 2014. On the formation of Mg ii h and k lines in solar prominences. Astron. Astrophys. 564, A132. https://doi.org/10.1051/0004-6361/201322886. Heinzel, P., 2000. Models of the solar atmosphere. In: Zahn, J.P., Stavinschi, M. (Eds.), NATO ASIC Proc. 558: Advances in Solar Research at Eclipses from Ground and from Space, vol. 558, pp. 201e220. Hubeny, I., Mihalas, D., 2015. Theory of Stellar Atmospheres. Princeton University Press. Hubeny, I., 1997. Stellar atmosphers theory: an introduction. In: De Greve, J.P., Blomme, R., Hensberge, H. (Eds.), Stellar Atmospheres: Theory and Observations, Lecture Notes in Physics, vol. 497. Berlin Springer Verlag, p. 1. Hubeny, I., September 2009. From complete linearization to ALI and beyond. In: Hubeny, I., Stone, J.M., MacGregor, K., Werner, K. (Eds.), American Institute of Physics Conference Series, American Institute of Physics Conference Series, vol. 1171, pp. 3e14. Hummer, D.G., 1962. Non-coherent scattering: I. The redistribution function with Doppler broadening. MNRAS 125, 21e37. https://doi.org/10.1093/mnras/125.1.21. Labrosse, N., Heinzel, P., Vial, J.C., et al., April 2010. Physics of solar prominences: ISpectral diagnostics and non-LTE modelling. Space Science Reviews 151, 243e332. https:// doi.org/10.1007/s11214-010-9630-6, 1001.1620. Leenaarts, J., Pereira, T.M.D., Carlsson, M., et al., August 2013. the formation of IRIS diagnostics. I. A quintessential model atom of Mg II and general formation properties of the Mg II hk lines. Astrophys. J. 772, 89. https://doi.org/10.1088/0004-637X/772/2/89, 1306.0668. Mihalas, D., 1991. The quest for physical realism in stellar atmospheric modeling. In: Crivellari, L., Hubeny, I., Hummer, D.G. (Eds.), NATO Advanced Science Institutes (ASI) Series C, NATO Advanced Science Institutes (ASI) Series C, vol. 341, p. 127. Milkey, R.W., Mihalas, D., October 1973a. Calculation of the solar chromospheric La profile allowing for partial redistribution effects. Solar Phys. 32, 361e363. https://doi.org/ 10.1007/BF00154948.

References

Milkey, R.W., Mihalas, D., October 1973b. Resonance-line transfer with partial redistribution: a preliminary study of Lyman a in the solar chromosphere. Astrophys. J. 185, 709e726. https://doi.org/10.1086/152448. Paletou, F., October 1995. Two-dimensional multilevel radiative transfer with standard partial frequency redistribution in isolated solar atmospheric structures. Astron. Astrophys. 302, 587. Scharmer, G.B., October 1981. Solutions to radiative transfer problems using approximate lambda operators. Astrophys. J. 249, 720e730. https://doi.org/10.1086/159333. Unso¨ld, A., 1938. Physik der sternamospharen, MIT besonderer berucksichtigung der sonne. Vernazza, J.E., Avrett, E.H., Loeser, R., April 1981. Structure of the solar chromosphere. III models of the EUV brightness components of the quiet-sun. Astrophys. J. Suppl. Ser. 45, 635e725. https://doi.org/10.1086/190731. Vial, J.C., Engvold, O., 2015. In: Solar Prominences, Astrophysics and Space Science Library. vol. 415. https://doi.org/10.1007/978-3-319-10416-4. Vial, J.C., March 1982. Two-dimensional nonlocal thermodynamic equilibrium transfer computations of resonance lines in quiescent prominences. Astrophys. J. 254, 780e795. https://doi.org/10.1086/159789. Woods, T.N., Eparvier, F.G., Hock, R., et al., January 2012. Extreme ultraviolet variability experiment (EVE) on the solar dynamics observatory (SDO): overview of science objectives, instrument design, data products, and model developments. Solar Phys. 275, 115e143. https://doi.org/10.1007/s11207-009-9487-6.

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Spectropolarimetry and Magnetic Structures

5.3 Kiyoshi Ichimoto1, 2

Astronomical Observatory, Graduate School of Science, Kyoto University, Hida Observatory, Kurabashira Kamitakara-cho, Takayama-city, Japan1; National Astronomical Observatory of Japan, Solar-C Project, Mitaka, Japan2

CHAPTER OUTLINE 1. Polarization of Light and Physical Diagnostics With Spectropolarimetry ................185 2. Spectropolarimeter.............................................................................................189 3. Sunspots and Active Regions ..............................................................................190 4. Ubiquitous Magnetic Field in the Quiet Region Photosphere..................................193 5. Magnetic Fields in the Chromosphere..................................................................198 6. Prominences......................................................................................................199 Acknowledgments ...................................................................................................204 References .............................................................................................................204

1. POLARIZATION OF LIGHT AND PHYSICAL DIAGNOSTICS WITH SPECTROPOLARIMETRY The polarization of light originates from a nonisotropic physical state in the light source or media through which the light propagates and carries information about vectorial physical quantities such as the magnetic field, electric field, particle stream, and radiation field in astronomical bodies. Most interest regarding solar observation is in the polarization of spectral lines because they are directly related to specific physical quantities through the mechanism by which the spectral lines are polarized. By this connection, spectropolarimetry opens a window to assessing important physical quantities that govern solar activity. Table 5.3.1 summarizes the polarization mechanism of spectral lines and related physical quantities. The Zeeman effect is the most familiar tool in current solar physics for diagnosing magnetic fields on the Sun. Since the discovery of the magnetic field in sunspots by Hale (1908), the Zeeman effect has been extensively observed and

The Sun as a Guide to Stellar Physics. https://doi.org/10.1016/B978-0-12-814334-6.00007-8 Copyright © 2019 Elsevier Inc. All rights reserved.

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Table 5.3.1 Polarizing Mechanisms and Diagnostics Polarization Mechanism

Anisotropy

Diagnosis

Zeeman/Paschen-back effect Stark effect

Magnetic field

Magnetic field

Electric field

Scattering polarization

Radiation field

Hanle´ effect

Magnetic field plus anisotropic radiation Particle streams

Electric field, electron density Scatterers: chemical state and density Magnetic field

Impact polarization

Particle beam, thermal conduction

studied in various situations in the solar atmosphere. Intense efforts have been dedicated to developing state-of-the-art spectropolarimeters. The Zeeman effect arises from the interaction of the magnetic dipole moment of an atom in some quantum state with the ambient magnetic field. The atomic energy level splits into multiple sublevels depending on their magnetic quantum number, which determines the component of the magnetic dipole moment of the atom projected onto the magnetic field and the Lande´ factor (a coefficient that relates the strength of the magnetic dipole moment to the angular momentum of an atomic energy level). As a result, an atomic spectral line that arises from a transition between two Zeeman-split energy levels yields multiple components that have different polarization states, so that the composite spectral line exhibits a characteristic polarization degree that changes in wavelength across the line. Fig. 5.3.1 shows a typical profile of the Stokes parameters of a spectral line under the Zeeman effect. Circular polarization (Stokes-V, defined as the difference in intensities of right-handed and left-handed circular polarizations) is attributed to the line of sight component of the magnetic field and shows an antisymmetric profile about the line center. The linear polarizations (described either by position angle and intensity or by Stokes-Q and U, which are the difference in the intensities of orthogonal linear polarizations) are attributed to the transversal component of the magnetic field and show symmetric profiles about the line center. Using this property of the Zeeman effect, three components of the magnetic field vector can be deduced from an observation of full Stokes profiles of an atomic spectral line, whereas the orientation of the transversal magnetic field is ambiguous by 180 degrees (known as the 180-degrees Zeeman ambiguity). Table 5.3.2 shows representative spectral lines that are widely used in solar observations to measure the Zeeman effect because of their high sensitivity to the magnetic field (large Lande´ factor), together with the Lande´ factor and approximate formation height. The Hanle´ effect is an alternative tool for diagnosing the magnetic field in the solar atmosphere and has attracted growing interest. It is the physical mechanism

1. Polarization of Light and Physical Diagnostics With Spectropolarimetry

FIGURE 5.3.1 Example of synthetic Stokes profiles of the Zeeman effect: FeI 6302.5A at B ¼ 800 G; g (inclination angle between the magnetic field and line-of-sight) ¼ 45 degrees; and c (azimuthal angle of the magnetic field) ¼ 0 degrees.

Table 5.3.2 Representative Spectral Lines That Are Widely Used in Solar Spectropolarimetry Line, Wavelength (A)

Formation Height

Lande´ Factor

FeI 15648 HeI 10830

Lower photosphere Upper chromosphere, prominence Lower chromosphere Middle photosphere Temperature minimum Prominence Middle photosphere Middle photosphere Middle photosphere

3.00 1.25e2.0a

CaII 8542 FeI 6173 NaI 5896 HeI 5876 FeI 6302 FeI 6173 FeI 5250 a

1.10 2.50 1.33 0.5e2.0a 2.50 2.50 3.00

HeI D3 and 10830A have multiple components.

by which the magnetic field modifies the polarization state of atomic spectral lines that is produced by the scattering of anisotropic radiation. When atoms are illuminated by such a radiation field from the solar (or stellar) surface, the radiative excitation of the atoms produces population imbalances among the magnetic sublevels

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of the energy levels, i.e., the populations of the magnetic sublevels with different magnetic quantum numbers are different. As a result, the emission process can generate linear polarization in spectral lines. This is known as line scattering polarization. Many spectral lines with an electric dipole transition exhibit linear polarization perpendicular to the predominant direction of the illuminating radiation, so chromospheric lines emanating from the prominences on the solar limb are linearly polarized in a direction parallel to the nearby solar limb. In the presence of a magnetic field, linear polarization of the emitted photons changes direction in a statistical manner depending on and direction owing to “quantum interference” among the magnetic sublevels. The resulting linear polarization of the scattered line, as the ensemble average of the polarization of individual photons, changes direction and reduces the degree of polarization (depolarization). This is the Hanle´ effect. Although the degree of polarization decreases according to the increase in field strength, the polarization of a scattered line reaches a constant state that is determined by the direction of the magnetic field under a strong field. In such a situation, i.e., the saturation regime, the Hanle´ effect is insensitive to the field strength but sensitive only to the field direction with respect to the predominant direction of the radiation field and the line of sight. The critical field strengths at which the Hanle´ depolarization becomes significant are given by BH ¼ 1.137  10e7/geff/tlife [G] (Trujillo Bueno et al., 2006), where geff is the Lande´ factor and tlife (¼1/A in terms of Einstein’s A coefficient) is the radiative life time of the upper level of the spectral line. One finds, for example, BH w 6 G for HeI D3, 16 G for Ha and 53 G for hydrogen Lya (Sahal-Bre´chot, 1981). Beyond the field strength of w10  BH, the Hanle´ effect becomes saturated and insensitive to the field strength. The Hanle´ effect is more sensitive to weaker magnetic fields. Also, Hanle´ depolarization is not canceled out by tangled magnetic fields that are too small to be resolved. For these reasons, the Hanle´ effect has the potential to provide a diagnostic tool to assess the magnetic field in the higher atmosphere (chromosphere and corona). It is also useful in prominences where the magnetic field is weak and spectral lines are broad, and in the photosphere where one finds unresolved tangled fields. These two regions are difficult to diagnose by the Zeeman effect. Because the information on the magnetic field carried by the Hanle´ effect is the difference between the observed polarization of spectral lines and that produced by the scattering process without the magnetic field, knowledge of polarization caused by pure scattering or details of the anisotropy of the radiation field is essential for determining the magnetic field from the Hanle´ effect. This is one of the challenges of the Hanle´ diagnosis when the target of observation is illuminated by irregularly distributed radiation sources around it. The 180-degree ambiguity, i.e., the inability to determine the sign of the orientation of the magnetic field component perpendicular to the line of sight, holds in both the Hanle´ effect and the Zeeman effect.

2. Spectropolarimeter

2. SPECTROPOLARIMETER A spectropolarimeter for astronomical observations consists of a telescope that collects photons and images the object, a polarimeter to convert the polarization of light into variations of the intensity, a spectrometer to resolve the spectral feature, and photosensors to measure the intensity of the modulated intensity of light. Because all photosensors are sensitive only to the intensity of light, it is necessary to make multiple measurements of the intensity under different instrumental configurations to determine the polarization of light (Skumanich et al., 1997). Then, differences among the intensities of multiple exposures provide information about the polarization of the light from the astronomical object. The simplest way to measure the polarization is to insert six polarizers, each of which transmits only a single polarization (right- or left-handed circular polarizations, linear polarizations in azimuthal direction of 0, 45, 180, and 135 degrees) in the optical beam alternatively. However, in such a manner, the polarization of the light entering the subsequent optics, i.e., spectrometer and photosensors, changes; accordingly the measured intensity is lessened by the influence of the polarization properties of these devices. The usual way to modulate polarization is to combine a “polarization modulator” (for example, a rotating waveplate or variable retarders) to change the measured polarization state of the incident light and a subsequent “polarization analyzer” (usually a linear polarizer or a polarizing beam splitter) to transmit a particular state of the polarization. In this configuration, the polarization state of the beam in the optical path after the polarization analyzer is constant, whereas the intensity of light exiting the polarimeter changes according to the polarization of the incident light and the angle of the rotating waveplate or the retardation of the variable retarders. The role of the spectrometer is to extract the wavelength bands that exhibit the polarization signals in spectral lines. Two types of spectrometers are generally used, either a slit spectrograph or narrowband tunable filters. The former takes full Stokes profiles of spectral lines in spatial positions along a one-dimensional slit at the same time and scans the region of interest step by step to map in a two-dimensional area, whereas the latter takes two-dimensional images at once but only in a single wavelength band transmitted by a narrowband filter. The typical spectral resolutions (l/dl) of spectropolarimeters based on the spectrograph and the filter graph in current use are (1e2)  105 and (4e8)  104, respectively. The polarization sensitivity (the detection limit of the degree of polarization) achieved in current solar observations is on the order of 10e3 or higher with a spatial resolution of better than 1 arcsec. In such high-precision observations, artificial polarization induced by the optical system of the spectropolarimeters is important. For example, a telescope and subsequent optics that feed the light to the polarimeter produce spurious polarizations and cross-talk among the Stokes-I, -Q, -U, and -V of the incident light owing to their different transmissions and phase shifts for different polarization states. Such instrumental polarizations need to be carefully calibrated in high-precision polarimetry, and methods should be developed for characterizing the

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polarization of the spectropolarimeter (Ichimoto et al., 2007). Another source of error in spectropolarimetry, which is more important in ground-based observations, is the disturbance of the solar image due to the turbulence in the earth’s atmosphere, i.e., so called seeing. The fluctuation in intensity induced by the seeing during polarization modulation produces false polarization signals. One way to mitigate this error is to modulate polarization within the timescale of the seeing, which is on the order of 10 ms. Such a high-speed polarization modulation (>1 kHz) is realized by using piezoelastic devices or ferroelectric liquid crystals as the polarization modulator combined with a special charge-coupled device image sensor (e.g., Gandorfer, 1999). Another way to reduce the seeing noise is to measure both orthogonally polarized intensities simultaneously using a polarizing beam splitter as the polarization analyzer. In such a configuration, the intensities of the two beams vary in phase owing due to the seeing but in opposite phases if produced by the polarization modulation, and thus the real polarization signal can be isolated from the seeing noise. The ideal way to mitigate the noise is to locate the instrument in space, as was realized with Hinode. Although the Sun is bright, the number of photons is a limiting factor that determines the sensitivity of solar spectropolarimetry, in which high spatial, spectral, and temporal resolutions are required. Collecting a large number of photons is thus essential; this requires a large aperture solar telescope such as the DKIST.

3. SUNSPOTS AND ACTIVE REGIONS Sunspots, the most conspicuous solar manifestation of magnetic fields, yielded the first detection of a cosmic magnetic field, carried out by Hale (1908). The global structure of the magnetic field of sunspots was extensively studied in the 20th century under relatively low spatial resolution. In an isolated and round sunspot, the magnetic field is nearly vertical and has a maximum strength of 2000e4000 G at the center. Towards the outer part of the sunspot, the field strength declines and becomes more horizontal in a radial direction; it reaches an approximate elevation of 10e20 degrees from the solar surface at the edge of the sunspot. Remarkably the radial profiles of the field strength and elevation show a smooth transition from the umbra to the penumbra, which is in contrast to the sharp boundary between them, as seen in the continuum. More interesting, the umbraepenumbra boundary in stable sunspots is characterized by an invariant value of the vertical magnetic field component of 1849e1885 G, with the most probable value of 1867 G (Jurcak et al., 2018). About a half of the total magnetic flux emanates from the penumbra. The magnetic field of sunspots tends to be stronger in larger sunspots and in the darker part of the umbra. The strongest field in sunspots recorded during 1917e2004 was 6100 G (Livingston et al., 2006), whereas Okamoto and Sakurai (2018) reported an even stronger magnetic field of 6250 G in a light bridge in a large delta-type sunspot. The largest sunspot recorded in the past century had an area of about 6000 millionths of a hemisphere (observed in Mar. 1947).

3. Sunspots and Active Regions

The penumbral photosphere is dominated by a nearly horizontal radial outflow (Evershed, 1909). It was a historic puzzle that the flow appears to be across the inclined magnetic field in a penumbra, and because the magnetic field is frozen to the plasma, the flow must carry away the entire sunspot’s magnetic field in a short time. The problem was solved thanks to modern high-resolution spectropolarimetric observations. Fig. 5.3.2 shows an example of high-resolution spectropolarimetric data from a sunspot, taken by the Solar Optical Telescope (SOT) (Tsuneta et al., 2008) aboard the Hinode satellite (Kosugi et al., 2007). The vertical line on the sunspot image in the left panel shows the position of the slit of the spectrograph when the data shown on the right panel were obtained. Two Zeeman-sensitive spectral lines of FeI 6301.5A and FeI 6302.5A are in the observed spectral window. Notice that the Stokes V signal in both lines changes sign along the slit at the border of the two sunspots (upper large one and lower small one). This tells us that these two sunspots have opposite polarities. Fig. 5.3.3 shows the Stokes profiles of the spectral lines at two representative points in the sunspot. In the left panel, we recognize the typical Stokes profiles expected from the Zeeman effect, i.e., symmetric I, Q, U, and antisymmetric V profiles about the line center. They are well-reproduced by a simple theoretical model calculation, as shown by the solid curves in this plot, using a MilneeEddington atmosphere (Skumanich and Lites, 1987) in which the magnetic field and velocity are assumed to be constant along the line of sight. Individual

FIGURE 5.3.2 SOT/Hinode, 2006.12 Sunspot. The left panel shows a continuum image; the right panel shows Stokes I, Q, U, and V spectra on the slit of spectrograph, indicated by the vertical line on the left panel.

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FIGURE 5.3.3 Stokes profiles at two representative points in the sunspot on Dec. 2006.

Zeeman components are visible in Stokes-I owing to large Zeeman splitting under the strong magnetic field in the outer umbralepenumbral region of the sunspot, which results in comparable amplitudes for Stokes-V, Q, and U. On the other hand, the right panel of Fig. 5.3.3 shows Stokes profiles that significantly deviate from those expected from the Zeeman effect, i.e., Stokes-I, Q, and U profiles are significantly asymmetric and the Stokes-V profile has three lobes. Such irregular profiles cannot be reproduced by the MilneeEddington model, as shown by the solid curves in this plot. One requires strong gradients of the velocity and/or the magnetic field along the line of sight. Inversion codes needed to extract the magnetic field information from irregular Stokes profiles, taking into account whether the gradient of physical quantities along the line of sight have been developed and are ready for the community’s use (for example, SIR (Ruiz Cobo and del Toro Iniesta, 2012) and NICOLE (Socas-Navarro et al., 2000)). The profiles in the right panel of Fig. 5.3.3 can be explained by the presence of a deeply embedded strong downflow within the magnetic field tube of a polarity opposite the sunspot’s main field. If one integrates the Stokes-V signal in that panel over the wavelength, one has a nonzero net-circular polarization. Such a signal provides strong evidence of the presence of unresolved mixing of the magnetic field and plasma dynamics along the line of sight. Fig. 5.3.4 shows the distribution of the strength (left) and the elevation from the solar surface (right) of the magnetic field over the sunspot as deduced by the Milnee Eddington inversion. A remarkable feature in the penumbra is that the field elevation varies with the azimuth; there are radially elongated filaments with either nearly horizontal or more vertical fields next to each other. A detailed analysis of the

4. Ubiquitous Magnetic Field in the Quiet Region Photosphere

FIGURE 5.3.4 Inferred magnetic field strength (left) and elevation from the solar surface (right). In the left panel, white represents the field strength and white contours show the umbral and penumbral outer boundaries as seen in the continuum image. In the right panel, blue and red represent the positive and negative polarities, respectively, and green represents nearly horizontal field.

data revealed that the horizontal outflow in the penumbral photosphere is concentrated in the filaments with a nearly horizontal field (Ichimoto et al., 2007). From such observations, the Evershed flow is now understood to be a plasma flow along the magnetic field and the filamentary structure of the penumbra harboring the strong outflow is interpreted as thermal convection of plasma under the influence of the inclined strong magnetic field (Rempel, 2011). Fig. 5.3.5 shows the distribution of transversal component (perpendicular to the line of sight) of the magnetic field centered on the interface region of the two sunspots. The remarkable feature here is that the magnetic fields, which are nearly parallel to the penumbral filaments seen in the continuum intensity, are highly sheared from the potential fields, i.e., they have a large angle from straight lines connecting the two sunspots with positive and negative polarities. Such a large deviation from the potential field is evidence of the presence of strong electric currents, i.e., free energy stored in the magnetic field; it is a crucially important subject for the study of solar flares.

4. UBIQUITOUS MAGNETIC FIELD IN THE QUIET REGION PHOTOSPHERE Previously, the quiet region away from sunspots and active regions (ARs) was thought to be basically nonmagnetic. The presence of magnetic fields in quiet

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FIGURE 5.3.5 Direction and strength of the transversal component (perpendicular to line of sight) of the magnetic field overlaid on a continuum image. The smallest tick corresponds to 1 arcsec.

regions was recognized from high-sensitivity spectropolarimetric observations, i.e., from circular polarization observed in Zeeman sensitive lines formed in the photosphere with a spatial resolution of a few arcsec, preferentially in boundary regions of supergranules (network). In such observations, polarization signals were as weak as a few percent and it was thought that they represented the presence of weak magnetic fields in quiet regions. If the Zeeman splitting of a spectral line is smaller than the width of the line in the solar spectrum, the circular polarization signal is proportional to the amount of Zeeman splitting, and therefore the field strength. Such a situation is called the “weak field” regime. Because Zeeman splitting is also proportional to the Lande´ factor of the spectral lines, the expected circular polarization is also proportional to the Lande´ factor of the lines in the weak field regime. If two spectral lines have similar strength and excitation potential of the lower level of the transition, they are formed at a similar height in the solar atmosphere, and thus the ratio of the Stokes-V signal of these two lines will be nearly the same as the ratio of their Lande´ factors. However, this is not the case in actual observations. For example, two iron lines at 5250A and 5247A, which have almost the same line depth and excitation potential but different Lande´ factors of 3.0 and 2.0, respectively, show a Stokes V signal with a ratio close to unity or w1.1:1.0. This inconsistency between the theoretical prediction and the observation was interpreted to indicate that the weak field regime does not hold in the quiet Sun, i.e., the magnetic field is strong

4. Ubiquitous Magnetic Field in the Quiet Region Photosphere

but not uniformly distributed over the resolution element of the observations, i.e., kilogauss fields are concentrated in unresolved small patches. Detailed analysis inferred that such flux concentrations have a size of 100e300 km and the filling factor (the fraction of area that is filled with the kilogauss field in a resolution element) is as small as 1%e10% (Stenflo, 1973). High spatialeresolution and high-sensitivity observations have provided much rich information about the nature of magnetic fields in the quiet Sun. Fig. 5.3.6 shows a continuum image of the quiet region at disk center with the distribution of the circular polarization and linear polarization signals in FeI 6302A line (g ¼ 2.5) observed with a spatial resolution of 0.32 arcsec. It demonstrates that the quiet Sun is filled with polarization signals. Concentrations of the circular polarization in small patches are obvious and it is recognized that strong or relatively large concentrations form the network whereas numerous small patches also exist interior to the network. Fig. 5.3.7 shows the continuum image of granulation overlaid with the location of the line-of-sight component of the apparent magnetic field (Bl) by red (positive) and green (negative) contours at the level of 24 Mx cm2(*). Yellow contours are for jBlj ¼ 100 Mx cm2. Strong vertical field elements (yellow contours) are preferentially located in the intergranular lanes and are associated with small bright points. Detailed analysis of the Stokes profiles of two spectral lines with different Lande´ factors in the magnetic concentration tells us that these magnetic structures are still not spatially resolved and that the line-of-sight component of a magnetic field with a kilogauss strength is present as numerous unresolved flux tubes.

FIGURE 5.3.6 Continuum intensity (left), circular polarization (middle), and linear polarization (right) in the quiet Sun at the disk center observed with the Hinode Solar Optical Telescope in FeI 6302.5A line.

Mx cm-2 is used rather than G intentionally to infer that the field strength is the apparent one averaged over the resolution element. *Here,

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FIGURE 5.3.7 Continuum image of a quiet region at the disk center overlaid with contours of the vertical (Bl) and horizontal (Bt) magnetic field. Red and green contours show, respectively, the positive and negative Bl at 24 Mx cm2. Yellow contours are for jBlj ¼ 100 Mx cm2; blue contours are the Bt levels of 122 Mx cm2. From Lites, et al., 2008. The horizontal magnetic flux of the quiet-sun network as observed with the Hinode spectro-polarimeter. Astrophys. J. 672, 1244 (Fig. 8).

The origin of such a discrete form of the vertical magnetic field is explained by the “convective collapse” scenario proposed by Parker (1978). In this scenario, an initially uniform magnetic field is swept into the boundary of convection cells and enhanced to hectogauss (w500 Mx cm2) strength, at which the magnetic pressure is comparable to the dynamic pressure of the convective flow. The plasma inside the magnetic flux is then cooled and a strong downdraft takes place under the convectively unstable circumstance so that the flux bundle is evacuated. The magnetic field is then compressed by the ambient gas pressure to form a small magnetic flux tube with a kilogauss field. A time series of the high-resolution spectropolarimetric data confirmed a strong downward motion in flux tubes associated with a rapid

4. Ubiquitous Magnetic Field in the Quiet Region Photosphere

enhancement of the field strength (Nagata et al., 2008). Moreover, a time series of high-resolution images showed that the flux tubes are shaken back and forth by the surrounding convective motions, which excites Alfvenic waves that propagate into the corona along the magnetic field. In this picture, the magnetic flux tubes could be regarded as the channels through which the energy is transported from the photosphere into the upper atmosphere to generate the hot corona and the solar wind. The right panel of Fig. 5.3.6 shows the distribution of the linear polarization signal in a quiet region, whereas Fig. 5.3.7 shows the locations of the vertical, Bl, and the horizontal component of magnetic field, Bt, on the continuum image by colored contours. Strikingly, the quiet Sun is also covered by numerous small patches of horizontal magnetic fields. The level of the contour for the horizontal component is larger (122 Mx cm2) than that of the vertical component (24 and 100 Mx cm2). The reason for this difference is that the linear Zeeman polarization signal is second order in the Zeeman splitting whereas the circular polarization is proportional to it, so that the detection limit for the transversal (here, horizontal) component of magnetic field is higher than that of the line-of-sight (vertical) component. It makes the observation of the transversal magnetic field difficult, and the ubiquitous horizontal field in quiet Sun was discovered only in the past decades. Small patches of horizontal magnetic field are preferentially located on the edge of granulations and appear and disappear in a short time (1e10 min) and appear essentially random in location and time (Centeno et al., 2007). It suggests that their origin results from the local dynamo action at work in the convective photosphere and does not have a connection to the global magnetic field. A possible and important inference of such transient horizontal magnetic fields is that their interaction with the overlaying magnetic field yields the production of heat and plasma motions. The net Poynting flux carried by the transient horizontal magnetic fields was estimated to be comparable to that needed for coronal heating (Ishikawa and Tsuneta, 2009). It is inferred from the observations and numerical simulations that the smallest magnetic structures on the solar surface are not spatially resolved even at the current highest spatial resolution. If multiple magnetic elements exist with different polarities and/or orientations in a spatial resolution element, the Zeeman signal in spectral lines is canceled out and is undetectable. Such tangled magnetic field may be detected using the Hanle´ effect, because the Hanle´ depolarization is not canceled by an unresolved mixture of magnetic fields. A linearly polarized solar spectrum obtained near the solar limb reveals many spectral lines showing linear polarizations mostly parallel to the solar limb. Because the linear polarization spectrum of the Sun is as rich as that of the intensity spectrum, its appearance is entirely different and is referred to as the “second solar spectrum” (Stenflo and Keller, 1997). The primary source of the polarization is the coherent scattering of light coming from the photospheric surface, whose degree is modified by Hanle´ depolarization in the presence of the turbulent magnetic field. A comparison of the theoretical prediction of the scattering polarization without a magnetic field and the observed scattering polarization provides information about the magnitude of the unresolved turbulent magnetic field. Employing this approach and using multiple spectral lines (differential Hanle´

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effect), Stenflo et al. (1998) tentatively suggested field strengths in the range of 4e40 G. A few years later, Trujillo Bueno et al. (2004), using a realistic threedimensional model of the photosphere to calculate nonmagnetic reference polarization and spectropolarimetric observations in an Sr I line at 4607A, deduced that the average strength of the ubiquitous tangled magnetic field is w130 G. A remarkable point of this discovery is that the average magnetic energy density outside the flux tubes in the quiet photosphere is sufficient to balance the radiative energy losses from the chromosphere and corona.

5. MAGNETIC FIELDS IN THE CHROMOSPHERE In the photosphere, the gas pressure dominates the magnetic pressure and the kinetic energy of convection is transferred to the magnetic field in the form of magnetohydrodynamic waves or magnetic stresses (electric currents). The magnetic field in the photosphere can be determined relatively well using the Zeeman effect. On the other hand, in the chromosphere, the interface region connecting the photosphere and the corona, the magnetic pressure becomes dominant over the gas pressure. It is a region where the energy generated in the photosphere is transported to the corona and converted to heat or bulk kinetic energy by fascinating dynamical phenomena. Therefore, measurement of the magnetic field in the chromosphere is crucial for understanding the fundamental physical processes that generate the hot and dynamic outer solar atmosphere. However, the measurement is less successful than in the photosphere. The reason is that the Zeeman signal of the chromospheric lines is much weaker than that of photospheric lines because of the broadness of the spectral lines and the weakness of the magnetic field in the chromosphere. One important line preferred for measuring the chromospheric magnetic field is the HeI triplet at 10830A. The lower level of this line is a metastable ground state of the orthohelium; its population is maintained by the ionization of neutral helium by extreme UV (EUV) irradiation from the overlying corona and subsequent recombination. The HeI 10830A absorption line originates in the upper chromosphere beneath the intense corona or in dark filaments (low-lying prominences) seen on the solar disk. The polarization of HeI 10830A is generated by the joint action of scattering processes and the Zeeman and Hanle´ effects (Trujillo Bueno et al., 2002). Because the optical thickness of this line is not as large as other chromospheric lines, interpretation of its spectropolarimetric signature is simple compared with other lines for which complex radiative transfer effects need to be considered. This is a great advantage of the HeI 10830A as a diagnostic tool of the chromospheric magnetic field. A community inversion code is available to retrieve the magnetic field from the spectropolarimetric data (HAZEL, Asensio Ramos et al., 2008). A drawback of HeI 10830A is that the use of this line is limited to ARs and dark filaments (and possibly networks) because the line is very shallow in internetwork in quiet regions and in coronal holes because of the lack of EUV radiation.

6. Prominences

Close to the HeI 10830A in wavelength, a Zeeman sensitive photospheric line exists, SiI 10827A (g ¼ 1.5), which can be observed simultaneously with HeI 10830A with a simple spectrographic setup. By combining these two lines, it is possible to measure the magnetic field at two heights, i.e., the photosphere and the upper chromosphere, with a height difference of 1000e2000 km. An example of spectropolarimetric observations of a sunspot using these lines was reported by Joshi et al. (2017). They found that the umbral magnetic field strength in the upper chromosphere is lower by a factor of 1.30e1.65 compared with the photosphere, whereas the difference in the magnetic field strength between two layers steadily decreases from the sunspot center to the outer boundary of the sunspot. Beyond the outer boundary, the horizontal component of magnetic field is stronger in the chromosphere than in the photosphere. This suggests the presence of a magnetic canopy resulting from an expansion of the spot’s magnetic field with height beyond its boundary. Such canopy structures of the magnetic field can be thought to be present around the small-scale ubiquitous flux tubes in the lower chromosphere as well, which may be diagnosed using the Hanle´ effect for strong resonance lines (Faurobert, 1999). Motivated by a theoretical investigation into the Hanle´ effect of the scattering polarization of the Lya line of solar disk radiation (Trujillo Bueno et al., 2011), a novel experiment was conducted for exploring the magnetic field in the upper chromosphere and transition region using a National Aeronautics and Space Administration (NASA) sounding rocket. The Chromospheric Lyman-Alpha SpectroPolarimeter (CLASP) has provided the first successful measurement of the linear polarization in the hydrogen Lya line (1215.7A) and in the Si III line at 1206.5A near the solar limb (Kano et al., 2017; Ishikawa et al., 2017). The scattering polarization signals observed in the Lya core, Lya wing, and Si III lines were compared and the differences among them were attributed to their different sensitivities to the Hanle´ effect. CLASP provides a milestone for the future use of diagnostic techniques based on the Hanle´ effect in exploring the magnetism of the upper chromosphere and transition region (Trujillo Bueno et al., 2017).

6. PROMINENCES A prominence is a cool (w104 K) plasma (visible in Ha and other chromospheric lines) suspended in the hot (106 K) corona, which can have a size and mass that sometimes reach 100 Mm and 1015 g, respectively. To satisfy the pressure balance between the prominence and the surrounding corona, the density of prominence plasma must be higher than that of the corona by two orders of magnitude. If there is no supporting force, the prominence plasma must fall on the solar surface in a timescale of 10 min according to the solar gravitational acceleration that is about 30 times larger than that on the Earth. Therefore, the persistence of prominences (more than days) suggests the presence of a supporting force. The only candidate

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FIGURE 5.3.8 Left: Ha image showing bright prominences on the limb and dark filaments on the disk (SMART/Hida Observatory, Sep. 6, 2016). Right: Magnetograph showing the distributions of positive (green-blue) and negative (yellow-red) polarities (Helioseismic and Magnetic Imager/Solar Dynamics Observatory). White dotted line in the right panel shows the location of the dark filament between opposite polarity (green-yellow) regions.

is the magnetic Lorenz force. The close connection of prominences with the magnetic field is also suggested by the fact that prominences (or dark filaments seen on the solar disk) are always located on the magnetic neutral lines between largescale positive and negative polarity regions (Fig. 5.3.8) (see Chapter 6). Fig. 5.3.9 shows a close-up view of the two types of prominences; AR and quiescent region (QR) prominences. The former are located on magnetic neutral

FIGURE 5.3.9 Two types of prominences: active region (left) and quiescent region (right) (Hinode/Solar Optical Telescope). From JAXA/NAOJ.

6. Prominences

lines connected to ARs (see a sunspot below the prominence in the left panel of Fig. 5.3.9) and dominated by nearly horizontal thin threads whose height is usually lower than 30 Mm, whereas the latter are located along the global magnetic neutral lines in QR and dominated by vertical structures. Vertical structures seen below the AR prominence in the left panel of Fig. 5.3.9 are spicules; some of the tall ones are jets emanating from the AR in front of the prominence. Time series of highresolution images reveal that prominence plasmas are never static but exhibit highly dynamic behaviors. The horizontal threads in AR prominences show dynamic flows along their length and lateral oscillatory motions (Okamoto et al., 2007). In quiescent region (QR) prominences, numerous upward and downward motions of dark and bright blobs are persistently observed (Berger et al., 2008). Prominences occasionally experience sudden destabilization and erupt into the interplanetary space (see Fig. 5.3.10). It is known that many coronal mass ejections (CME) are associated with prominence/filament eruptions, and the erupted prominence is sometimes observed as the bright core of the CME (Gopalswamy, 2015, Howard, 2015) (see also Chapter 6). Because the CME may be the primary driver of the solar storms that affect the space environment of the Earth, understanding the mechanism of prominence eruptions and predicting their occurrence are important subjects in current research on space weather. In this regard, one needs measurements of the magnetic field of prominences and their evolution, which are crucially important. The Zeeman polarization signal in spectral lines emanating from prominences is generally small compared with that in photospheric lines because the magnetic field of prominences is relatively weak and Zeeman splitting of spectral lines is much smaller than the width of the prominence lines. It makes the measurement of magnetic fields in prominences difficult, and they are still poorly known. Early measurements of the prominence magnetic field were carried out by using the Hanle´ effect.

FIGURE 5.3.10 The largest prominence eruption ever observed, on Jun. 4, 1946. Courtesy of the High Altitude Observatory/National Center for Atmospheric Research.

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Sahal-Brechot et al. (1977) and Bommier and Sahal-Brechot (1978) had formulated the Hanle´ effect for the HeI D3 line at 5876A, which is formed in prominences and provided a theoretical framework for deriving magnetic field information from the polarization of the D3 line. The HeI D3 is a favorable line for Hanle´ diagnosis of prominences because this line is optically thin and at 5876A, incident radiation from the photosphere is spectrally flat. Using polarimetric measurements of 256 prominences on the solar limb in the HeI D3 line, after resolving the 180-degrees ambiguity statistically, Leroy et al. (1984) argued that prominences with a height larger than 30 Mm (likely quiescent prominences) preferentially have the inverse-type configuration (Kuperus and Raadu, 1974), in which the magnetic field vectors are pointing in the opposite sense with respect to the adjoining photospheric polarities and have an average field strength of 5e10 G. Those with a height smaller than 30 Mm (likely AR prominences) tend to be consistent with the normal-type configuration (Kippenhahn and Schluter, 1957; also Fig. 5.3.4 in Chapter 6 by B.C. Low for a more realistic model), i.e., parallel to the adjoining photospheric polarities, with an average field strength of w20 G. Also shown is that in both cases, the magnetic field is nearly horizontal and the mean angle between the magnetic field and the long axis of the prominence is about 20e25 degrees. Bommier et al. (1994) observed 14 prominences in HeI D3 and Ha simultaneously. By combining the Hanle´ signals in the two lines, they could determine the field vectors in those prominences and found results mostly consistent with those by Leroy (1984): 12 prominences that were located in quiet regions were found to be of the inverse polarity type and two that were located in the vicinity of ARs were of the normal polarity type. These early observations were single point measurements on each prominence. The first attempt to use both linear and circular polarization was performed by Paletou et al. (2001). They observed clear evidence in linear and circular polarization signals of HeI D3 indicative of the Hanle´ and Zeeman effects, respectively. The first spatial map of the magnetic field over a prominence using the Hanle´eZeeman effect was produced by Casini et al. (2003) using an HeI D3 line. They found that the average magnetic field in prominences was mostly horizontal and varied between 10 and 20 G. A spectropolarimetric observation of a quiescent prominence at a higher spatial resolution (w1 arcsec) was carried out by Orozco Sua´rez et al. (2014) in HeI 10830A (see Fig. 5.3.11). According to their interpretation of the data, the most probable magnetic field configuration is that the averaged magnetic field is nearly horizontal and the acute angle of the magnetic field vector with the prominence long axis is about 24 degrees, which is consistent with previous results. On the other hand, they did not find local variations in the field strength but noted that it changed globally with a weaker field (w7 G) in the prominence body than in the prominence feet, where the field strength reached w25 G. Spectropolarimetry in a single line does not allow us to determine a unique solution. Spectropolarimetry is a powerful tool for diagnosing solar magnetic fields and, it is hoped, other vectoral physical quantities in the future. The frontier for the next generation of solar scientists will be the effort to reach a higher spatial resolution

6. Prominences

FIGURE 5.3.11 Mapping of magnetic field in a prominence. From top to bottom: HeI 10830A intensity, field strength, and orientation of transversal field. From Orozco Sua´rez, D., Asensio Ramos, A., Trujillo Bueno, J., 2014. The magnetic field configuration of a solar prominence inferred from spectropolarimetric observations in the HeI 10 830 A˚ triplet. Astron. Astrophys. 566, A46 (Figs. 1 and 8).

to study elementary magnetic structures in the photosphere and higher-precision multiline spectropolarimetry to determine the three-dimensional magnetic field structure from the solar photosphere to the chromosphere as well as that associated with prominences.

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ACKNOWLEDGMENTS The author is grateful to Dr. Andrew Skumanich for valuable comments and corrections, any remaining errors are solely the author’s. The author also thanks Dr. Trujillo Bueno for valuable suggestions especially on the descriptions for the Hanle´ effect. Hinode is a Japanese mission developed and launched by ISAS/JAXA, with NAOJ as domestic partner and NASA and STFC (UK) as international partners. It is operated by these agencies in cooperation with ESA and NSC (Norway).

REFERENCES Asensio Ramos, A., Trujillo Bueno, J., Landi Degl’Innocenti, E., 2008. Advanced forward modeling and inversion of Stokes profiles resulting from the joint action of the Hanle´ and Zeeman effects. Astrophys. J. 683, 542e565. Berger, T.E., Shine, R.A., Slater, G.L., et al., 2008. Hinode SOT observations of solar quiescent prominence dynamics. Astrophys. J. 676L, 89B. Bommier, V., Landi Degl’Innocenti, E., Leroy, J.L., et al., 1994. Complete determination of the magnetic field vector and of the electron density in 14 prominences from linear polarization measurements in the HeI D3 and H-alpha lines. Sol. Phys. 154, 231e260. Bommier, V., Sahal-Brechot, S., 1978. Quantum theory of the Hanle´ effect - calculations of the stokes parameters of the D3 helium line for quiescent prominences. Astron. Astrophys. 69, 57e64. Casini, R., Lopez Ariste, A., Tomczyk, S., et al., 2003. Magnetic maps of prominences from full stokes analysis of the He I D3 line. Astrophys. J. 598, L67eL70. Centeno, R., Socas-Navarro, H., Lites, B., et al., 2007. Emergence of small-scale magnetic loops in the quiet-sun internetwork. Astrophys. J. 666, L137. Evershed, J., 1909. Radial movement in sun-spots. Mon. Not. R. Astron. Soc. 69, 454e457. Faurobert, M., 1999. Hanle´ effect of weak solar magnetic fields. In: Livingston, W.C., Ozguc, A. (Eds.), The Last Total Solar Eclipse of the Millennium in Turkey, ASP Conference Series, vol. 3, p. 108. Gandorfer, A.M., 1999. First results from ZIMPOL II. Astrophys. Space Sci. Libr. 243, 297e304. Gopalswamy, N., 2015. The dynamics of eruptive prominences. In: Vial, J.-C., Engvold, O. (Eds.), Solar Prominences, Astrophysics and Space Science Library, vol. 415, p. 381. Howard, T.A., 2015. Measuring an eruptive prominence at large distances from the sun. I. Ionization and early evolution. Astrophys. J. 806, 175. Hale, G.E., 1908. On the probable existence of a magnetic field in sun-spots. Astrophys. J. 28, 315e343. Ichimoto, K., Shine, R.A., Lites, B., et al., 2007. Fine scale structures of the evershed effect observed by the solar optical telescope aboard Hinode. PASJ 59, 593e599. Ichimoto, K., Lites, B., Elmore, D., et al., 2008. Polarization calibration of the solar optical telescope onboard Hinode. Sol. Phys. 249, 233e261. Ishikawa, R., Tsuneta, S., 2009. Comparison of transient horizontal magnetic fields in a plage region and in the quiet Sun. Astron. Astrophys. 495, 607e612.

References

Ishikawa, R., Trujillo Bueno, J., Uitenbroek, H., et al., 2017. Indication of the Hanle´ effect by comparing the scattering polarization observed by CLASP in the Lya and Si III 120.65 nm lines. Astrophys. J. 2017. Joshi, J., Lagg, A., Hirzberger, J., et al., 2017. Three-dimensional magnetic structure of a sunspot: Comparison of the photosphere and upper chromosphere. Astron. Astrophys. 604, A98. Jurcak, J., Rezaei, R., Bello Gonz alez, N., et al., 2018. The magnetic nature of umbrapenumbra boundary in sunspots. Astron. Astrophys. 611. AA, L4eL17. Kano, R., Trujillo Bueno, J., Winebarger, A., et al., 2017. Discovery of scattering polarization in the hydrogen Lya line of the solar disk radiation. Astrophys. J. 839L, 10K. Kosugi, T., Matsuzaki, K., Sakao, T., et al., 2007. The Hinode (Solar-B) mission: an overview. Sol. Phys. 243, 3e17. Kippenhahn, R., Schluter, A., 1957. Eine Theorie der solaren Filamente. Z. Astrophys. 43, 36. Kuperus, M., Raadu, M.A., 1974. The support of prominences formed in neutral sheets. Astron. Astrophys. 31, 189. Leroy, J.L., Bommier, V., Sahal-Brechot, S., 1984. New data on magnetic structure of quiescent prominences. Astron. Astrophys. 131, 33e44. Lites, B.W., Kubo, M., Socas-Navarro, H., et al., 2008. The horizaontal magnetic flux of the quiet-sun internetworks observed with the HINODE spectro-polarimeter. Astrophys. J. 672, 1237e1253. Livingston, W., Harvey, J.W., Malanushenko, O.V., et al., 2006. Sunspots with the strongest magnetic fields. Sol. Phys. 239, 41e68. Nagata, S., Tsuneta, S., Suematsu, Y., et al., 2008. Formation of solar magnetic flux tubes with kilogauss field strength induced by convective instability. Astrophys. J. 677L, 145. Okamoto, T., Tsuneta, S., Berger, T.E., et al., 2007. Coronal transverse magnetohydrodynamic waves in a solar prominence. Science 318, 1577e1580. Okamoto, T.J., Sakurai, T., 2018. Super-strong magnetic field in sunspots. Astrophys. J. 852, 16. Orozco Sua´rez, D., Asensio Ramos, A., Trujillo Bueno, J., 2014. The magnetic field configuration of a solar prominence inferred from spectropolarimetric observations in the HeI 10 ˚ triplet. Astron. Astrophys. 566, A46. 830 A Paletou, F., Lopez Ariste, A., Bommier, V., et al., 2001. Fullestokes spectropolarimetry of solar prominences. Astron. Astrophys. 375, L39eL42. Parker, E.N., 1978. Hydraulic concentration of magnetic fields in the solar photosphere. VI adiabatic cooling and concentration in downdrafts. Astrophys. J. 221, 368e377. Rempel, M., 2011. Penumbral fine structure and driving mechanisms of large-scale flows in simulated sunspots. Astrophys. J. 729, 5. Ruiz Cobo, B., del Toro Iniesta, J.C., 2012. Astrophysics Source Code Library. ASCL: 1212.008. Sahal-Brechot, S., Bommier, V., Leroy, J.L., 1977. The Hanle´ effect and the determination of magnetic fields in solar prominences. Astron. Astrophys. 29, 223. Sahal-Bre´chot, S., 1981. The Hanle´ effect applied to magnetic field diagnostics. Space Sci. Rev. 29, 391. Socas-Navarro, H., Trujillo Bueno, J., Ruiz Cobo, B., 2000. Astrophys. J. 530, 977.

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Skumanich, A., Lites, B.W., 1987. Stokes profile analysis and vector magnetic fields. I inversion of photospheric lines. Astrophys. J. 322, 473e482. Skumanich, A., Lites, B.W., Martnez Pillet, V., et al., 1997. The calibration of the advanced stokes polarimeter. Astrophys. J. 110, 357e380. Stenflo, J.O., 1973. Magnetic-field structure of the photospheric network. Sol. Phys. 32, 41e63. Stenflo, J.O., Keller, C.U., 1997. The second solar spectrum. A new window for diagnostics of the Sun. Astron. Astrophys. 321, 927e934. Stenflo, J.O., Keller, C.U., Gandorfer, A., 1998. Differential Hanle´ effect and the spatial variation of turbulent magnetic fields on the Sun. Astron. Astrophys. 329, 319e328. Trujillo Bueno, J., Landi Degl’Innocenti, E., Collados, M., et al., 2002. Selective absorption processes as the origin of puzzling spectral line polarization from the Sun. Nature 415, 403. Trujillo Bueno, J., Shchukina, N., Asensio Ramos, A., 2004. A substantial amount of hidden magnetic energy in the quiet Sun. Nature 430, 326T. Trujillo Bueno, J., Asensio Ramos, A., Shchukina, N., 2006. The Hanle´ effect in atomic and molecular lines: a new look at the Sun’s hidden magnetism. In: Casini, R., Lites, B.W. (Eds.), ASP Conference Series, vol. 358, pp. 269e292.  epa´n, J., Casini, R., 2011. The Hanle´ effect of the hydrogen Lya line for Trujillo Bueno, J., St probing the magnetism of the solar transition region. Astrophys. J. 738L, 11. Trujillo Bueno, J., Landi Degl’Innocenti, E., Belluzzi, L., 2017. The physics and diagnostic potential of ultraviolet spectropolarimetry. Space Science Reviews 210, 183. Tsuneta, S., Ichimoto, K., Katsukawa, Y., et al., 2008. The solar optical telescope for the Hinode mission: an overview. Sol. Phys. 249, 167e196.

CHAPTER

Coronal Magnetism as a Universal Phenomenon

6 B.C. Low

High Altitude Observatory, National Center for Atmospheric Research, Boulder, CO, United States

CHAPTER OUTLINE 1. Introduction .......................................................................................................207 2. The Hydromagnetic Corona .................................................................................208 3. Coronal Phenomenology .....................................................................................209 3.1 Photosphere, Chromosphere, Corona, and Solar Wind ............................. 209 3.2 Coronal Polarity Reversal: The Physical Problem .................................... 211 3.3 The Corona on February 26, 1998 ........................................................ 212 3.4 Coronal Polarity Reversal and Nature of Coronal Mass Ejections .............. 217 3.4.1 The Magnetic Hemispherical Rule and Its Dynamo Origin ................ 218 3.4.2 Coronal Outward Transport of Flux and Twist by Coronal Mass Ejections.................................................................................... 222 4. The RM [ 1, b  1 Turbulent Fluid ................................................................224 4.1 Parker Magnetostatic Theorem ............................................................. 224 4.2 Magnetic Helicity Conservation and Accumulation ................................. 228 5. Outstanding Questions and Astrophysical Implications..........................................230 Acknowledgments ...................................................................................................232 References .............................................................................................................232

1. INTRODUCTION The x-ray emitting, million-degree hot corona and its solar wind, permeated with the Sun’s external magnetic field, are the prototype of a universal astrophysical phenomenon (Parker, 1994). Most Sun-like stars in the galaxy are observed to have x-ray coronas that are presumably expanding into stellar winds. The varieties of stellar and galactic winds with their acceleration mechanisms are common astrophysical concepts. With the broader astrophysical interests in mind, this chapter presents a constructive understanding of the observed corona as a fully ionized hydromagnetic

The Sun as a Guide to Stellar Physics. https://doi.org/10.1016/B978-0-12-814334-6.00008-X Copyright © 2019 Elsevier Inc. All rights reserved.

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atmosphere (Parker, 1979; Low, 1996, 2001; Aschwanden, 2004). We focus on understanding how the voluminous corona, a near-perfect conductor of heat and electricity, keeps pace with the photosphere in the global magnetic-field reversal of the Sun in 11-year cycles: that is, the polarity of the global dipole field reverses to repeat every complete magnetic cycle of about 22 years. This chapter presents Section 2 with preliminary remarks on the hydromagnetic corona; Section 3 on coronal phenomenology describing the photospheree chromosphereecorona structure and the relationships among prominence magnetic flux ropes, flares, helmet streamers, and coronal mass ejections (CMEs) in evolutionary interplay between magnetic self-organization and coronal polarity reversal; Section 4 on making hydromagnetic sense of the observed corona with a conceptual analysis of Parker spontaneous current sheets (CSs) and generalized magnetic helicity; and Section 5 on outstanding questions and astrophysical implications. Observational data and journal references will be cited not exhaustively but as guidance in accordance with the purpose of this chapter. Physical ideas will be treated qualitatively with attention to conceptual completeness. Hereafter, the unqualified terms “energy,” “field,” “field line,” “flux,” “flux surface,” “helicity,” “polarity,” and “reconnection” in the context of discussion will mean the magnetic kind.

2. THE HYDROMAGNETIC CORONA Coronal and space physics was given a modern beginning with the concepttransforming inference of E. N. Parker (1958, 1960, 1963) that the heated, thermally highly conducting corona could not be gravitationally bound but must freely expand into the supersonic solar wind filling all interplanetary space. His inference was confirmed shortly thereafter with the in situ detection of the solar wind at 1 AU by the first generation of human-made satellites. During the many decades since, space observation of the Sun, solar wind, planets, and planetary magnetospheres blossomed. The twin spacecrafts Voyager 1 and 2, launched in 1977 by the National Aeronautic and Space Administration (NASA), have passed through the solar-wind termination shock en route into the interstellar medium. In situ observation has reached the wind’s far boundary and will soon also reach its coronal source with the NASA Parker Solar Probe launched in 2018 to descend into the Sun. With the Sun moving at about 20 km/ s relative to the stars, we have the heliocentric view of the interstellar medium streaming around the enormous solar-wind cocoon. Solar rotation deforms the wind’s open fields into Parker spirals carrying away angular momentum, azimuthal flux, and helicity, significant over astronomical timescales (Parker, 1958; Skumanich, 1972; Belcher and MacGregor, 1976; Berger and Ruzmaikin, 2000). Mass and twisted flux leave the corona in daily CMEs that can occur along all latitudes at times of global polarity reversals (Hundhausen, 1997). These magnetic structures influence the transport of outward CME/flare particles and inward (moderate-energy) galactic cosmic rays (Jokipii and Parker, 1970;

3. Coronal Phenomenology

Jokipii, Sonett, and Giampapa, 1997; Axford, Leer and Skadron, 1978; Zank et al., 1998). The global field reversal in each 11-year cycle spreads through the corona into the solar wind. The outer solar wind is thus chronologically layered into chaotic fields with an intrinsic global polarity reversing from layer to layer, about which Voyager 1 and 2 have provided only first observations. We now think of stars each actively maintaining such a cocoon of physical influence carved into the interstellar medium, of a volume orders of magnitude larger than the star itself. Space- and ground-based observations have transformed solar physics. The coronaesolar-wind system as an astrophysical prototype is today described by a wealth of data impressive in physical details and across great ranges of spatiotemporal scales (e.g., Russell, 2008; Cranmer, Hoeksema and Kohl, 2010; Pesnell, Thompson and Chamberlin, 2012; Zank et al., 2013; Shimizu, Imada and Kubo, 2018). Observing from above the terrestrial atmosphere is essential for spectroscopic study and imaging of coronal phenomena in UV/extreme UV emissions, x-rays, g-rays, and high-energy particles.

3. CORONAL PHENOMENOLOGY Thermonuclear energy from the solar core escapes at the 5  103 K photosphere through the optically thin outer atmosphere, principally as the white-light Solar Luminosity L☉ ¼ 3:9  1033 erg=s. The quiescent photosphere strongly interacts with its average field of about 10 G, turbulently convecting at about 0.5 km/s with density scale-height h ¼ 150 km ¼ 2  104 R☉ . Through this thin atmosphere, dynamogenerated magnetic fluxes emerge continually in a range of length scales from h to the 15  104 km sizes of the kilogauss sunspots. (Parker, 1955a; 1979; Lites et al., 1995). The emerged fluxes disperse into the dynamically active, global coronal field that reverses its polarity in eleven-year cycles. The observed Solar Luminosity is steady to some 0.1% (Lean 2000), showing that the dynamo and its cyclical atmospheric influence are driven by well less than 1% of the outflow of solar energy. The ratio of fluid to magnetic pressures b decreases with height from order unity in the photosphere to well below unity in the field-dominated low corona and then increases to order unity at about 2R☉ in heliocentric distance where all fields are open to interplanetary space and entrained in the solar wind under conditions of near-perfect electrical conductivity.

3.1 PHOTOSPHERE, CHROMOSPHERE, CORONA, AND SOLAR WIND Temperature increasing with height is a universal property of a ubiquitously heated, thermally conducting, radiatively cooled, optically thin, hydrostatic atmosphere. Stable stratification favors a height-decreasing density, producing overheating and overcooling, respectively, in the upper and lower reaches of the atmosphere. In a steady state, temperature increases with height to deliver a downward-conducted

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thermal flux that ensures all added heat is radiated away. A hydrostatic condition cannot be maintained rigorously in the dynamic solar atmosphere, of course, but its coupling with energy transport is a global constraint that orders the gravitationally bound part of the solar atmosphere, from the photosphere through the 104 kmthick chromosphere into the low corona. The detailed physics of the heating and radiative loss determines an increase in temperature with height past the ionization temperature of H that produces an extremely thin transition layer (z102 km) in which the fluid changes state from partial to full ionization. Denied line radiation, the corona is a poor radiator but an excellent conductor of heat and electricity. Taking its million-degree temperatures as given, a hydrostatic corona would extend radially too far out and too hot to be gravitationally bounded, which is the reason for the solar wind (Parker, 1963). Therefore, coronal temperature has a maximum in the outer corona driving thermal fluxes in the opposite directions, inward to heat the chromosphere and outward to drive the solar wind. Away from sunspots, the average 10 G photospheric field is clumped into vertical, narrow (z102 km) kilogauss fibrils by a ubiquitous downdraft instability coupled to magnetic buoyancy. These fibrils may be individually and mutually twisted, observed at the limit of current telescope resolutions to be swept to convection cell boundaries during the finite lifetimes (minutes) of cells and fibrils (Parker, 1979; Borrero et al., 2017). An encounter between a pair of nearby fibrils of equal and opposite fluxes may be observed to cancel mutually on the thin photosphere, through three possibilities. If the fibrils are connected above but low in the photosphere, the downward tension force may result in submerging the inverted-U shaped fibril. If the fibrils are connected just below the photosphere, the Ushaped fibril would rise buoyantly above the photosphere. No resistive reconnection of field is involved in both possibilities. More commonly, the two fibrils are not connected in their immediate spatial neighborhood; in this case, they may have unequal fluxes. When pressed together by a fluid-dominated flow, they may resistively reconnect, with the lower reconnected fibril submerging and the upper reconnected fibril rising to reconnect further and reconfigure over much larger length scales with the surrounding coronal fields (Martin et al., 1994; Low, 2001; Kubo et al., 2014). Such reconnections drive a quasisteady evolution of the coronal field, intermittently creating new magnetic paths for upward magnetic-twist propagation and downward mass flow back to the photosphere. This upwardedownward separation of magnetic twist and entrained mass can be dynamically more developed in the buoyant rise of a large-scale rope of twisted flux, driven by CS formation and dissipation via magnetic reconnection, as observed in sunspot formation and as demonstrated in the fundamental, three-dimensional (3D) numerical simulation of Manchester et al. (2004). Amid newly emerged sunspots, horizontal ropes of twisted flux also rise through the photosphere (Okamoto et al., 2008, 2009; Lites et al., 2010; Kuckein et al., 2012). It seems clear from these considerations that a flux system cannot be bodily pulled back below the photosphere once it has buoyantly emerged,

3. Coronal Phenomenology

self-organized, and expanded to fill a volume in the low-b, electrically highly conducting corona. Along the narrow kilogauss fibrils, magnetic twists as torsional Alfven waves drive upward jets, z200 km/s, some 106 of them at any one time, heating the chromosphere via hydromagnetic shocks to 2  104 K. These so-called spicules are gravitationally bound. Their upward mass flux, orders of magnitude larger than the 1012 g/s loss in the solar wind, largely returns back down as observed on disk as red-shifted C IV absorption everywhere.

3.2 CORONAL POLARITY REVERSAL: THE PHYSICAL PROBLEM Fig. 6.1 shows the line-of-sight (LOS) component of the magnetic field (in this chapter called the LOS field) in the photosphere, observed in 1975e2016 along the solar central meridian, as a synoptic function of latitude and year, with positive and negative fields in yellow and blue, respectively (Hathaway, 2015). At about midpoint in each 11-year cycle indicated in Fig. 6.1, a metastable global dipolar field is in place, with an equatorial principal neutral line across which the local vertical field component changes sign. This dipolar field persists until the new cycle kicks in with two “butterfly” patches of bipolar fluxes contributed by two belts of kilogauss sunspots emerging and disappearing in groups within 30 in latitude. The two wavelike sunspot belts appear and drift symmetrically from about 30 latitude to meet and wane away at the solar equator. Sunspot emergence is governed by the HaleeNicholson law: In each cycle, the common, eastewest (EW) oriented, bipolar sunspot-pairs have a statistically preferred sign for the leading polarity in the northern (N) hemisphere and the opposite preferred-sign in the southern (S) hemisphere,

FIGURE 6.1 Photospheric line-of-sight field component, along solar central meridian, 1975e2016, as a synoptic function of latitude and year, yellow/blue for positive/negative fieldcomponents, respectively (Hathaway, 2015). Black arrows indicate four dates of minimum (Min.) coronal activity with the global field dominantly dipolar (DeToma et al., 2010). The four white vertical lines identify the dates 1980, 1997, 1998, and 2003, left to right, referenced in Figs. 6.2, 6.3, and 6.7.

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with the two preferred signs changing hemispheres in the following cycle. Leading refers to the westward polarity in the direction of solar rotation. The N and S hemispheres are defined relative to the solar rotational axis. The Joy law statistically places the leading polarities in both hemispheres closer to the equator by a few-degree inclination of the sunspot pair to the exact EW direction. Both laws are explained by the dynamo and flux rope sunspot theories of Parker (1955a,b, 1979). As sunspot groups decay away, they leave behind active regions of strong fields (102 G). In each N or S hemisphere, the active-region fluxes of opposite signs migrate apart in latitude, with the poleward migrating fluxes predominantly opposite the hemisphere’s old-cycle polar field (Harvey, 1994; Hathaway, 2015). The photospheric flux distribution is highly nonaxisymmetric at this stage (Figs. 6.1 and 6.2E) but is characterized globally with changes of signs in the LOS field across five principal neutral lines encountered along a longitude, including the original equatorial neutral line of the old-cycle dipolar field. Poleward migrating new-cycle fluxes cancel and replace their respective old-cycle polar fields. New-cycle fluxes migrating equator-ward in the two hemispheres are of opposite polarities by the NicholsoneHale law, in the same NeS polarity configuration of the preexisting old-cycle fluxes. These migrating opposite-polarity fluxes mutually cancel across the equator to complete the global polarity reversal. The sunspot fields thus transform an old-cycle dipolar field in the corona, via a multipolar evolving global field, to an opposite-polarity dipolar field put in place by midcycle. The removal of coronal fields from both the polar and equatorial regions, the latter a mix of old and new cycle fluxes in the old-cycle NeS configuration, is the central problem of the global reversal process that the optically thin corona does not hide but is not straightforward to identify observationally.

3.3 THE CORONA ON FEBRUARY 26, 1998 A new solar cycle began about 1997 with complete reversals of both polar fields by 2002 and the new-cycle dipolar field in place by 2005 (Fig. 6.1). As a focus of our narrative, we describe the structures and related phenomena in the corona of Feb. 26, 1998, displayed in Fig. 6.2. Fig. 6.2A shows the total eclipse in Curacao that day under excellent seeing conditions, from the eclipse photograph archive of National Center for Atmospheric Research (NCAR)/High Altitude Observatory (HAO). The white-light corona is 106 less bright than the photosphere, rendered visible in photospheric light Thomson-scattered by free electrons into the LOS, weighted by stronger contributions from electrons located closer to the plane of the sky. The eclipse image in Fig. 6.2A is thus a map of density integrated along optically thin LOSs. Images Fig. 6.2D and E from the Michelson Doppler Imager (MDI) instruments of the Solar and Heliospheric Observatory (SOHO), respectively show the photosphere with only two isolated new-cycle sunspot groups (sg1 and sg2) and a map of the photospheric LOS field rich with extensive active regions of intense (>10 G) fields within 30 in latitude. Two regions are centered at the two sunspot groups, whereas the other regions originate from sunspot groups already decayed.

(A)

(B)

(D)

(E)

(C)

(F)

The corona on Feb. 26, 1998. (A) NCAR/HAO image of total eclipse over Curacao, displaying coronal holes ch1 and ch2. (B) Yohkoh/Soft X-Ray Telescope (SXT) image of the >4106 K corona, helmet arcades labeled P1 and P2. (C) SOHO/Extreme UV Imaging Telescope (EIT) image of the 284 A˚, 2106 K corona showing P1 to be a bright arcade with a central, narrow dark lane and P2 a broad arcade with a conspicuous dark lane. (D) SOHO/Michelson Doppler Imager (MDI) white-light photosphere. (E) SOHO/MDI photospheric line-of-sight field component; whiteeblack for positiveenegative polarities, P1 and P2 indicating approximate locations of corresponding features in (B) and (C). (F) MLSO Ha full-disk image; whiteeblack indicating emissioneabsorption. Two elongated pieces of prominence in absorption along the SOHO/EIT dark-lane of P2 are identified whereas the inconspicuous prominence along the SOHO/EIT dark-lane of P1 is not identified. In (BeF), central meridians are drawn as white vertical diameters to aid visual identifications of corresponding atmospheric features in different subfigures. Here and elsewhere, all solar images are aligned with north (N) at top.

3. Coronal Phenomenology

FIGURE 6.2

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The quasisteady white-light corona in Fig. 6.2A is structured by its still dipolar global field, frozen into the plasma under high electrical conductivity and stretched radial by the solar wind that completely dominates above about 2R☉ in heliocentric distance. The field is thus globally partially open, as illustrated geometrically by the (axisymmetric) field-line sketches in Fig. 6.7A,B and D. The wind flows out in sectors of unipolar fluxes it holds open, with the sectors separated by the so-called helmet-streamers as the building blocks of the corona. Each helmet is a lane or arcade of fields of about 1e10 G or greater in the quasistatic inner corona (z1069 H/cm3), closed over one or more photospheric neutral lines with a tension force sufficiently strong to resist the solar wind. The helmet is embedded with static plasmas relatively denser and hotter than the plasmas in its adjacent open fields along which near-hydrostatic plasmas continually heave outward to significant expansion speeds higher up and feed into the wind. The flux-surface bounding the closed-field arcade is naturally a surface of sharp density contrast. This separatrix surface bifurcates at the top of the helmet arcade into a radially extended, 2D magnetic neutral sheet separating the oppositely directed open fields along which the wind flows. A few helmet streamers can be discerned visually in the optically thin superposition of mid-latitudinal structures in Fig. 6.2A. Those arcades oriented lengthwise close to the LOS appear as a bright helmet base tapering outwardly into a narrow radial streamer. Notable are the so-called (dark) coronal holes (ch1 and

FIGURE 6.3 Three-part helmet-streamers and coronal mass ejections (CMEs). (A) HAO/NCAR/ Rhodes College total eclipse image, Feb. 16, 1980, India. (B) White-light event, Sep. 9, 1997 (Chen et al., 2000). EW-oriented helmet-streamer arcade breaking up at solar limb into CME in combined coronagraph fields of view (FOVs) of MLSO/Mark IV (12R☉) and SOHO/Large Angle Spectrometric Coronagraph 2 (LASCO2) (1.56R☉): (B-1) at UT 19:32, CME as a rising expanding cavity (Ca), bright prominence at Ca bottom, pushing sideway at helmet flank (He) and outward into still undisturbed streamer (St), (B-2) at UT 20:06, leading CME shell (Sh) rising above top of Mark IV FOV into streamer (St), (B-3) at UT 20:33, fully expanded CME with three-part structure of shell (Sh), cavity (Ca) and erupted prominence as core (Co) in LASCO2 FOV. (C) Arrow indicating Yohkoh/SXT postCME X-ray flare, Feb. 28, 1998, in progress on disk as widening bright NE arcade of newly reconnected, closed bipolar fields. (D) Arrow indicating Yohkoh/SXT post-CME X-ray flare, Jan. 24, 1992, in progress at southwest solar limb as widening bright EW arcade growing in size with newly reconnected, closed bipolar fields.

3. Coronal Phenomenology

ch2) where low-density high-speed winds flow out in diverging funnels of unipolar fluxes from around the two solar poles. The Feb. 16, 1980 corona during active polarity reversal in the HAO/NCAR/ Rhodes College eclipse image in Fig. 6.3A is in sharp contrast to the 1998 “dipolar” corona in Fig. 6.2A. With its magnetic dipole-moment essentially vanished, the 1980 corona is rimmed with helmet streamers, corroborated by the five principal sign changes of the LOS field across the 1980 latitudes in Fig. 6.1. Fig. 6.2B and C, respectively, are images of the corona at >4  106 K, from the Soft X-ray Telescope (SXT), Yohkoh Mission, and at z2  106 K, from the Extreme UV Imaging Telescope (EIT), the Solar and Heliospheric Observatory (SOHO) Mission (Takeda, 2011; Delaboudinire et al., 1995). Both images show corresponding compact, bright flares in the active regions along the two sides of the equator. The polar coronal holes, e.g., ch2, which are low in density and temperature, are dark in both images. Moderately bright, elongated x-ray structures in Fig. 6.2B are high-temperature helmet arcades weighing down over low-density ropes of twisted magnetic flux running horizontally above and along their respective neutral lines, two examples identified as P1 and P2. The P2 flux rope shows up in Fig. 6.2C as a (low-density) dark lane along its neutral line, but the dark lane is not as conspicuous in Fig. 6.2B because of its x-ray canopy of hot helmet plasmas. Within each flux rope is a lengthy prominence of partially ionized plasma two orders of magnitude denser and cooler than the surrounding corona, in the form of a vertical slab suspended in the flux rope field (Fig. 6.4B) (Low, 1994; Rust and Kumar 1994; Low and

FIGURE 6.4 Prominence flux ropes. (A) Sketch of lengthy prominences on solar disk on July 15, 1980, two polar pairs drawn black, each separating photospheric regions of opposite polarities, described in text (Leroy et al., 1983). (B) Poloidal sketch of axisymmetric corona showing (shaded) EW helmet-arcade over the equator with its arrowed field lines arching over a (unshaded) flux-rope supporting a prominence slab (P), the helical rope-field represented by closed poloidal field lines circulating around an implied rope axis. The low-density rope-interior, labelled C, appears as a dark cavity at a helmet base, as seen in the MLSO/ Mark IV white-light helmet of July 22, 2002 (C) and the Yohkoh/SXT X-ray image (D) in reverse-brightness scale. E. A Normal-type flux rope with interior poloidal field circulating opposite to its arching helmet field, distinct from the Inverse-type in (B).

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Hundhausen, 1995; Tandberg-Hanssen, 1995; Vial and Engvold, 2015; Gibson, 2018). Not all helmets contain flux ropes and not all flux ropes contain prominences, but the three-part structure of helmet, flux rope, and prominence is common. Mauna Loa Solar Observatory (MLSO) image Fig. 6.2F shows the prominences as elongated features levitating in absorption against the Ha image of the full-disk chromosphere. Flux ropes P1 and P2 are located away from strong-field active regions, containing lengthy prominences (Fig. 6.7) over relatively straight neutral lines. Such prominences are typically thick (P2). They frequently fragment and become less conspicuous (P1) as they evolve. The dispersal of the active-region strong fields over ever-larger photospheric regions leaves behind coronal remnant fields that remain highly structured well after the progenitor sunspots have decayed and after the active-region fields have fallen into the background photospheric strength of z10 G. Field strength thus increases with height into an overlying prominence where the observed vector field is generally horizontal in a range from 20 to 70 G (Leroy, 1989; Casini et al., 2003). Flux rope P2 is exemplary, extending in height with its length in an EW orientation. Such a flux rope at the solar limb reveals its characteristic oval cross-section, dark and sharply defined in Thomson-scattered white light and x-ray (see examples in Fig. 6.4C and D). In contrast, the prominences in active-region complex fields are short and narrow, located in compact low-lying flux ropes over contorted neutral lines. For most prominences, their low-density ropes are detectable on disk as depletions in radio emission (Straka et al., 1975). Lengthy prominence flux ropes do not decay away in situ but are typically ejected with gradual acceleration out of the corona as the cores of CMEs, each ejection often followed by a flare (Tandberg-Hanssen, 1995; Hundhausen, 1999; Low, 1996, 2001; Zhang and Low, 2005; Chen, 2011). The Sep. 9, 1997 CME event in the sequence in Fig. 6.3B shows an EW helmet breaking away into the solar wind, expanding self-similarly with a leading-edge speed of about 800 km/s, observed by the MLSO/Mark IV and SOHO/Large Angle Spectrometric Coronagraph 2 (LASCO2 2) coronagraphs (Chen et al., 2000). The pre-expulsion threepart structure is preserved, with the helmet, flux rope, and prominence respectively becoming the CME’s leading high-density shell, trailing low-density cavity, and dense core (Iling and Hundhausen, 1986; Gibson and Low, 1998; Gibson et al., 2010, 2018). The general properties of CMEs do not vary significantly from magnetic cycle to cycle (Burkepile & St. Cyr, 1993; Hundhausen, 1993, 1999; Webb and Howard, 1994; St. Cyr et al., 2000; Chen, 2011). Drainage of prominence mass during expulsion results in the CME core typically being an order of magnitude less massive than the leading CME shell, with the total CME mass lying in the narrow range 101416 g. CMEs travel at a broad range of speeds, 50  2  103 km/s with an average of about 450 km/s. The three-part CMEs are found at all speeds and constitute about 45% of all CMEs in each magnetic cycle. The gravitational escape speed of about 550 km/s implies that the work done against gravity is typically greater than the kinetic energy carried by the total CME mass. Magnetic energy is also carried away in the ejected flux rope. The total energy liberated in a CME expulsion is of the order of

3. Coronal Phenomenology

103032 erg, comparable to large flare-energies. CMEs are near-ideal hydromagnetic flows whereas flares are resistively dissipative processes (Kahler, 1992; Hundhausen, 1999; Low, 1994, 1996; Zhang and Low, 2005). The ejection of a prominence out of the corona opens its helmet closed field, which subsequently recloses in the low corona by resistive reconnection into a bipolar arcade field without a flux rope, producing a high-temperature (107-K) flare (Hiei et al., 1993). Well after crossing the solar limb, the EW-oriented P2 helmet in Fig. 6.2 was ejected in a CME that was not well-observed. Had P2 been ejected from the solar limb, it would have displayed a three-part morphology similar to that in Fig. 6.3B, leaving behind a well-recognized form of flare-reconnected arcade at the limb in x-ray (see an example in Fig. 6.3D, bottom). The P1 helmet with its length in a northeast (NE) orientation was ejected from the solar face on Feb. 28, 1998; its Earth-directed CME was barely detectable in white light as a halo around the coronagraph-occulted Sun. Fig. 6.3C (top) is a Yohkoh/SXT image of the P1 post-CME, reconnected-arcade x-ray flare in progress, the arcade a fraction of R☉ in length. If the CME of the NE P1 helmet had departed from the solar limb, its three-part structure would be less obvious in its oblique view in Thomsonscattered white light (Cremades and Bothmer, 2004). The helmet streamers rimming the corona of 1980 in Fig. 6.3A broke away as CMEs ejected from all latitudes during the coronal polarity reversal in progress. New helmet streamers formed and reformed, leaving as CMEs as the corona simplified magnetically to a new-cycle dipolar field by 1987 (Hundhausen, 1993) (Fig. 6.1). The magnetic cycles since 1980 were well-observed from the ground and in space, with the high data quality seen, for example, in the images in Fig. 6.7EeH. The corona may be characterized as ridding itself of fields in the old-cycle NeS polarity configuration in each magnetic cycle, not in situ or by withdrawal back into the Sun but as prominence flux ropes embedded in CMEs out into the solar wind (Zhang and Low, 2005).

3.4 CORONAL POLARITY REVERSAL AND NATURE OF CORONAL MASS EJECTIONS The key phenomena are: (1) ubiquitous heating maintaining million-degree temperatures at an estimated input of up to 107 erg s1 cm2 over the coronal base (Parker, 1988, 1994; Lu, 1995; Cargill et al., 2015); (2) impulsive flares liberating energies 103032 erg per event in 107 K plasmas and Mev particles (Kahler, 1992; Aschwanden, 2004); (3) helmets and prominence flux ropes storing significant energies (Low and Hundhausen, 1995; Chen et al., 1997); and (4) the solar wind and CMEs breaking gravitational confinement (Hundhausen, 1993, 1999; Low, 2001; Gibson and Low, 1998; Zhang and Low, 2005; Fan and Gibson, 2007). The magnetic fields are their principal source of energy. The key phenomena are individually interesting (e.g., Panesar et al., 2013; Martinez Gonzalez et al., 2015). Here we focus on their accumulative influence

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on the corona over a magnetic cycle governed by an observationally determined hydromagnetic rule that we will show has its origin in the Parker (1955b) dynamo.

3.4.1 The Magnetic Hemispherical Rule and Its Dynamo Origin

Consider the Spitzer formula for the resistivity h ¼ 4pc2/s ¼ 1012 T 3/2 of a fully ionized single-fluid plasma in cgs units, where s, c, and T denote the electrical conductivity, speed of light, and temperature, respectively. At temperature T ¼ 106 K, h ¼ 103 cm2/s showing that the corona is about as resistive as copper
at room temperature. Magnetic fields diffuse resistively on timescale td ¼ L20 h, where L0 denotes the width of a layer of flux across which the field gradient is significant. For such a layer of 100 km, comparable to the coronal mean free path, td is of the order of 3  103 years. At astronomical scales, plasma resistivity is insignificant with an important exception, when nonlinear hydromagnetic interaction produces a scale L0 / 0 within the fluid description. In other words, the otherwise electrically perfect fluid conductor can spontaneously create unlimitedly thin CSs across which field gradients become extreme enough at relevant timescales for significant resistive field diffusion, or, equivalently, resistive dissipation of the CSs. This process is a fundamental hydromagnetic property of highly conducting fluids, which we will analyze in Section 4.1 (Parker, 1972, 1991, 1994). The above simple numerical analysis makes an important point, that h-dissipation of current as a fluid process takes too long except at scales smaller than the coronal mean free path. This means that the perfect-conductivity effect of L0 / 0 inevitably proceeds to a local breakdown of the fluid description where nonideal dissipation of currents then takes place via plasma kinetic effects. Nonlinear gradient steepening is a familiar concept. The compressive, weakly viscous hydrodynamic fluid generally steepens its velocity in a finite time into a thin shock layer in which the otherwise reasonable assumption of an inviscid fluid breaks down. Shocks also form in the ideal hydromagnetic fluid, of course, but the freezing of the field in a perfect fluid conductor also creates CSs as magnetic contact discontinuities similar to the fluid contact surfaces separating distinct thermodynamic states. Whereas the physical smoothing of a discontinuity in the thermal case may be relatively benign in energy, the resistive dissipation of CSs leads to further CS formations. This is the physical reason for the ubiquitous presence in the corona of quiescent resistive heating and flare occurrences liberating energy in the broad range 102332 erg per event (Parker, 1988; Lu, 1995). Here we stay within our phenomenological considerations and simply note that the existence of quiescent and flare heating requires the ready formation of structures with sufficiently small-scale L0 for resistive dissipation to become significant. A quasisteady closed coronal field B may be regarded to be force-free, described by: ðV  BÞ  B ¼ 0;

(6.1)

V$B ¼ 0;

(6.2)

3. Coronal Phenomenology

neglecting the other body forces of the tenuous plasma (Low and Lou, 1990; Flyer et al., 2004; Lerche and Low, 2014). Free energy sufficient to drive flares and CMEs is stored as field-aligned current density that gives a twist or shear to a closed, anchored field. In contrast, twist acquired by an open field is unavoidably lost into the solar wind as Alfven waves. Many flares occur in closed fields without a CME, especially in active regions (Hundhausen, 1993). Observation supports the interpretation that these flares are transitions to a lower-energy metastable state without losing the signatures of twisted fields. The stable buildup of stored magnetic energy and retention of a part of the stored energy during a high-energy flare imply that the inevitable Parker CSs, although successful in heating the corona everywhere, do not significantly leak away the energy stored quasisteadily in largescale coronal structures. Helmet arcades P1 and P2 in Fig. 6.2B and the lengthy prominence in Fig. 6.7E are the end products of active flaring and selforganization. The longevity of magnetic twist will be identified with magnetic helicity conservation in Section 4.2. The corona is governed by a hemispherical rule observed to be unchanging from magnetic cycle to cycle, in which the twists in long-living fields are statistically lefthanded (negative twist) and right-handed (positive twist), respectively, in the NeS hemispheres relative to the solar rotational axis. This rule eliminates the possibility of twist occurring throughout the corona in random handedness. Under the rule, the dispersal of the active-region fields in each NeS hemisphere combines flux-tubes of a preferred handedness to produce self-organized flux ropes with a general additive increase in the net twist. The hemispherical rule was discovered in four classes of phenomena: the horizontal, vector fields of prominences determined by spectropolarimetry (Leroy et al., 1983, 1984; Leroy, 1989); the Ha structures of prominences (Martin et al., 1994; Martin, 2015); the sigmoidal morphology of coronal x-ray structures (Canfield et al., 1999; Low and Berger, 2003); and the vertical component of photospheric current density determined by magnetographs (Hao and Zhang, 2011). Here, we concentrate on prominence fields. The Ha prominences of July 15, 1980 are identified as sketched linear structures in Fig. 6.4A, among which two pairs are identified in solid black, respectively close to the two solar poles (Leroy et al., 1983). Each identified prominence separates plus or minus photospheric polarities with the direction of its observed axial field component along the length of the prominence arrowed at one end. Applying the right-hand rule of electromagnetism, the N and S pairs of prominences near the solar poles in Fig. 6.4A are respectively left- and right-handed in compliance with the hemispherical rule. The polar flux ropes in Fig. 6.4A have the basic field topology displayed in Fig. 6.4B showing a 2D idealized sketch of the cross-section of a flux rope levitating above and running along the solar equator, symmetric about the solar rotational axis. The unshaded region of closed field lines is the cross-section of the low-density flux rope, observable as a dark, sharply defined oval cavity, with examples in Fig. 6.4C and D in white light and x-ray (Low and Hundhausen, 1995; Gibson and Fan, 2006; Hudson et al., 1999, Gibson et al., 2010). In this so-called Inverse configuration, the

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prominence slab is located below the axis of the rope, threaded by a horizontal field pointing opposite the helmet field arched over the flux rope and anchored to the photospheric polarities as sketched (Tandberg-Hanssen, 1995). Take the field projected on the plane of the sky in Fig. 6.4B as given. The algebraic sign, positive or negative, of an implied axial field component of the rope into the plane then renders the flux rope right- or left-handedly twisted, respectively, by the right-hand rule. Although magnetic observation provides only the vector fields in the cool (104 K) prominence slabs containing field-sensitive spectral signatures, such observations suffice to detect the following defining properties of the flux rope configuration in Fig. 6.4B. Confined by the tension force of the circulating poloidal field, magnetic pressure is maximum around the rope axis where the axial field component is dominant. Therefore, the horizontal vector field in the prominence slab, suspended in the lower half of the rope, would be observed to increase with height in intensity and in alignment with the prominence length (Rust, 1967; House and Smart, 1982; Leroy, 1989; Low and Hundausen, 1995; Low and Zhang, 2004a). Most of the prominences in the higher latitudes away from the equator are of this Inverse type, governed by the hemispherical rule (Hanaoka and Sakurai, 2017). Fig. 6.4E describes a minority of low-lying prominence in the Normal configuration, generally not found in latitudes away from the equator, with the slabs threaded by horizontal fields pointing in the direction of the helmet fields arching over the flux rope (Tandberg-Hanssen, 1995; Low, 1981; Low and Zhang, 2002; 2004a). Let the

FIGURE 6.5 Magnetic handedness in magnetostatic fields. (A) Arrowed field lines of (untwisted) potential field projected over line-of-sight (LOS) field component at model coronal base displayed as solid (positive) and dashed (negative) contours; the field lines intersecting the circular boundary returning to the base outside the circular domain are shown. (B) Arrowed field lines of a globally left-handedly twisted field over the same distribution of LOS field component at model coronal base as in (A) subject to a small clockwise rotation; the field in equilibrium with plasma weight and pressure (Low, 1992). The field lines rising from the positive-polarity region to the left of the sub-figure and the two short field lines identified with fat arrows indicate the left-handedness of the global field, by the right-hand rule. (C) The mirror-reflected field solution obtained by a horizontal flip of (B), transforming the global magnetic handedness from left to right.

3. Coronal Phenomenology

axial field component perpendicular to the sketch be uniformly of the same sign. The rope and helmet fields are then oppositely twisted, which suggests that normal prominences form out of complex fields of opposite twists having migrated from the higher latitudes on two sides to meet at the equator (Tsuneta, 1996b). Fig. 6.5: displays an analytical magnetostatic field solution illustrating inverse flux ropes in realistic 3D geometry (Low, 1992, 2001). Fig. 6.5A and B respectively display the field lines of a potential field and a twisted field in static equilibrium with plasma pressure and weight, seen projected on a model coronal base fixed with the same photospheric LOS field described by solid (positive) and dashed (negative) contours. In each sub-figure showing a circular domain, field lines that intersect the artificial domain boundary have their second foot-points located external to the domain. If the coronal currents in the field solution in Fig. 6.5B are removed completely, we would obtain the untwisted potential field in Fig. 6.5A. The magnetostatic field is left-handedly twisted as shown in Fig. 6.5B, its left-handed twist simple to see in the two shorter field lines identified with fat arrows. We have rotated the field solution to tilt the principal bipolar regions downward to represent a bipolar sunspot pair located N of Equator, in accordance with the Joy law, taking up to be the solar N. A horizontal flip of Fig. 6.5B gives its mirror image shown in Fig. 6.5C, giving a corresponding bipolar sunspot pair S of Equator, tilted upward and right-handedly twisted, in compliance with the HaleeNicholson and Joy laws. Note the simple topological property that mirror reflection transforms any given field to one of the opposite handedness whereas reversing the field directions everywhere does not change the handedness of the given field. So, reversing the fields everywhere in Fig. 6.5B and C does not change their handedness. That is, these two sub-figures have captured both the HaleeNicholson law and the cycleindependence of the hemispherical rule. Suppose the flux ropes in Fig. 6.5B and C have not formed in the corona but have bodily emerged up through the photosphere. The vector fields on horizontal planes sliced across the flux rope solutions in decreasing heights carry information about the rope handedness and they map into the vector fields observed on the photosphere as a function of time during emergence. Three events of single-rope emergence have been observed in active regions 10,781 (2005, N, R), 10,953 (2007, S, L), and 10,978 (2007, S, L), denoted by the year of observation, N/S for hemispherical locations, and R/L for the observationally inferred handedness of flux rope (Okamoto et al., 2008, 2009; Lites et al., 2010; Kuckein et al., 2012). These newly emerged, active region flux ropes, described as a “sliding door effect” in the observed, evolving photospheric vector fields, are distinct from the high-latitude prominence flux ropes as end products of evolved coronal fields. The flux ropes emerging with their inherent twists complying with the hemispherical rule suggest that the rule has an interior origin (van Ballegooijen and Martens, 1990, Low, 1996, 2001). The coronal hemispherical rule is implied by the Parker dynamo (1955b, 1970). Fig. 6.6AeC are copied from the book Cosmical Magnetic Field by Parker (1979), which describe, respectively, (1) generation of toroidal flux from poloidal flux by differential rotation in the solar convection zone, (2) generation of poloidal flux

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FIGURE 6.6 Parker (1955b) dynamo and the hemispherical rule. Figures copied from Parker (1979): (A) Solar differential rotation generating azimuthal flux from poloidal flux. (B) Cyclonic convective action generating poloidal flux from azimuthal flux. (C) Dynamo product of flux rope pair, left- and right-handedly twisted, respectively, in northesouth hemispheres.

from toroidal flux by interior cyclonic convective cells, and (3) the dynamo endproducts of two twisted flux ropes, left- and right-handedly twisted in the NeS respective hemispheres. The handedness of each hemispherical dynamo flux rope is unchanged in a solar-cycle reversal of its fields in Fig. 6.6C. The sunspots emerging in each cycle thus take on the significance of transporting excess flux and magnetic twist into the corona, avoiding an untenable monotonic cycle-to-cycle accumulation of magnetic twist in the two hemispheres of the convection zone. Twist accumulation in the corona is regulated by the conditions for self-confinement of closed fields in the open atmosphere, failing which excess flux and twist are ejected into the solar wind, a process naturally identifiable with the CMEs. A significant amount of twist, which is challenging to calculate, emerging at the photosphere into open coronal fields is quasisteadily lost as Alfven waves and large-scale, rotationally-generated azimuthal flux in the solar wind (Berger and Ruzmaikin 2000). The implications of the hemispheric rule operating in the Parker dynamo merit further study beyond the classical kinematic treatment. The opposite axial fluxes of the double high-latitude flux ropes in Fig. 6.4A suggest that the end products of the dynamo may be multiple flux ropes of opposite azimuthal fluxes in each of the NeS hemispheres, all governed by the hemispherical rule.

3.4.2 Coronal Outward Transport of Flux and Twist by Coronal Mass Ejections Fig. 6.7AeD is the 2D axisymmetric interpretation of Low and Zhang (2004b) applicable to the Feb. 18, 2003 prominence-CME-flare event in Fig. 6.7EeH. A main portion of a long prominence (Fig. 6.7E) was ejected (Fig. 6.7F), its stringy form still seen clearly as the CME core (Fig. 6.7G) in the inner corona. The post-

3. Coronal Phenomenology

FIGURE 6.7 Genesis of a coronal mass ejection (CME) in a schematic axisymmetric corona and the Feb. 18, 2003 prominence-CME-flare event. (A) Initial high-latitude helmet streamer without flux rope. (B) Formation of three-part helmet streamer. (C) Field fully opened by departed CME. (D) Reformed helmet streamer without flux rope. SOHO/EIT 195 A˚, 1.6  106 K images and MLSO/Mark IV white-light image: (E) High-latitude, lengthy prominence in absorption at UT 01:13 before ejection. (F) Ejection of main portion of prominence in progress at UT 02:00. (G) Departing CME in white light in field of view 1  2R☉ at UT 02:19 showing arrowed, ejected prominence in CME cavity. (H) Post-CME flare in progress at UT 04:36 with two bright ribbons radiating post-CME flare energy from newly reclosed arcade fields.

CME flare (Fig. 6.7H) lasted for more than 8 h after the ejected prominence had left, which is usual. Fig. 6.7A describes an axisymmetric initial coronal field in the N hemisphere with two neutral lines at latitude 45 and the equator, with the field partitioned by the solar wind into the partially open geometry shown. The field lines are numerically labeled, a constant flux between any two adjacent lines. The construction is intended to illustrate a theoretical point with no particular real situation implied. To simplify physics, the equatorial bipolar field is kept unchanging as the bipolar field to the north evolves from a helmet with no flux rope in Fig. 6.7A to a helmet engorged with a flux rope containing a prominence in Fig. 6.7B. The azimuthal components of the helmet and flux rope fields are nonzero and negative by the hemispherical rule, whereas the open fields are purely poloidal. The rise of a sub-photospheric U-shaped fibril produces mutual cancelation of opposite photospheric fluxes across the neutral line, bodily transferring sheared flux threading the photosphere up into the flux rope without reconnection. Reconnection between a pair of opposite-polarity fibrils belonging to distinct flux tubes above the photosphere also cancels opposite photospheric fluxes and adds flux and shear to the coronal flux rope forming above. This flux emergence into the corona parallels

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the emergence of an active-region, large-scale flux rope treated in Section 3.4.1; the former has a direct coupling with the solar wind. The flux rope in Fig. 6.7B depends on the downward tension force of its outer layer of anchored flux and the weight of the helmet and prominence to resist expanding into the wind. Confinement is inevitably lost as the rope of unanchored flux grows at the expense of its anchored flux. The three-part helmet then pushes its way out as a CME, opening the anchored field down into the b 1 low corona, the extreme case of a fully opened field shown in Fig. 6.7C. In Fig. 6.7D, the post-CME flare partially closes the opened field to form a helmet, without a flux rope, with its top located as high as permitted by the solar wind. The process Fig. 6.7A-D describes a CME-related incremental decrease of the N polar flux during a magnetic-polarity reversal (Gopalswamy et al. 2003). Another observable consequence results from the compact size of the post-CME helmet, previously open flux-lines 14-16 having closed. The base area of the N polar corona hole, defined by its unipolar flux open into the solar wind has thus shrunk in this process. Mass loss by CMEs is only about 10% of that in the steady solar wind (Webb and Howard, 1994; Hundhausen, 1999). The role of CMEs in coronal evolution is magnetic in nature, episodically taking twisted-flux systems out of the corona. We may regard CMEs as the magnetic counterpart to the ever-present solar wind. These ejected flux systems first accumulate progressively in the corona; gravity is an important means of self-confinement. A rough estimate by A. J. Hundhausen bears repeating regarding the rate at which CMEs may remove photospheric flux (Low, 2001). A modest transfer of 1% of the photospheric flux, of any chosen sign, across a helmet base, into the prominence flux rope would take 100 CMEs to remove all of the base flux. An average latitudinal width of 30 for a helmet of arcade length 0.5R☉ has a base area about 2% of the entire solar photosphere. Therefore, 5000 CMEs can remove the entire flux of a given sign across the entire coronal base existing at any time, which implies 1.3 CMEs a day over an 11-year cycle, an observationally supportable rate.

4. THE RM [ 1, b  1 TURBULENT FLUID High electrical conductivity stores magnetic energy in the persisting current densities of force-free fields. Yet the corona is resistive in a dynamically specific manner through ubiquitous spontaneous formation of CSs, the reason for the common occurrence of resistive reconnection in the corona and elsewhere in the astrophysical universe (Tsuneta, 1996a, b, Shibata, 1999, Parker, 1972, 1979, 1994). We address how the two hydromagnetic properties may coexist.

4.1 PARKER MAGNETOSTATIC THEOREM Consider the dimensionless, resistive induction equation for a magnetic field B:   vB ¼ V  v  B  R1 m VB ; vt

(6.3)

4. The Rm [ 1, b  1 Turbulent Fluid

expressing velocity v and length in characteristic units of velocity and length (v0, L0), respectively, and scaling resistivity h in terms of the magnetic Reynolds number Rm ¼ v0 L0 =h. For most observable velocities and lengths, Rm > 108 . If h ¼ 0 rigorously (Rm / ∞), we have the induction equation for a perfect conductor: vB ¼ V  ðv  BÞ; (6.4) vt under which the motion of every fluid surface SðtÞ conserves its net flux FðSÞ, the so called frozen-in condition expressed by: Z d d FðSÞh B$dS ¼ 0; (6.5) dt dt S

where dS is the instantaneous fluid-surface element. In the perfect conductor, current densities can be induced to persist at any magnitude. A CS is the limit of a fixed finite current flowing in a layer of vanishingly small width. Fig. 6.8A and B show CS formation in 2D and 3D fields as the consequences of the frozen-in condition. We assume continuous fluid displacement with an interest in circumstances under which fluid continuity inevitably fails. In Fig. 6.8A, the vertical quadrant-pair of opposite fluxes press into the X neutral point to completely push apart the other quadrant-pair of opposite fluxes and meet along a line gap, i.e., a hole in the fluid, where the opposite fluxes form a tangential discontinuity (TD). By Ampere’s law, the TD carries a CS flowing out of the plane. We have assumed an inviscid fluid in which the TD line gap is due to a discontinuous tearing of the fluid. If viscosity is present, the TD in Fig. 6.8A is the limit of the vertical quadrant pair creasing the X-type neutral point progressively into a line.

FIGURE 6.8 Spontaneous tangential discontinuities. (A) Classic two-dimensional (2D) Cartesian formation of tangential discontinuity (TD) by opening up an X-type neutral point of a planar field into a line gap (Parker, 1994). (B) Schematics of three representative flux surfaces illustrating 3D formation of TD in hole punched into a flux surface by flux systems on either sides. (C) The Parker topological force-free field problem.

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Understood in these terms, the generalization of the 2D process to 3D is simple: Any two flux systems frozen into their embedding fluids can punch a 3D hole through a continuum layer of flux surfaces to meet and form a TD, as sketched in Fig. 6.8B. Parker (1991) has an elegant demonstration of hole creation in a flux surface of a 3D force-free field governed by Fermat’s principle, in perfect analogy to the optical exclusion of light rays from a region of high refractive index. The nature of the 3D process implies that TDs can form on any flux surface under general forcing either by fluid or field-dominated motions. The presence of a magnetic neutral point is not mandatory except in 2D systems with an ignorable coordinate (Low, 2013). Parker (1972) demonstrated a general inevitability for CSs to form in a force-free field, treating his topological boundary value problem sketched in Fig. 6.8C. On the left is a uniform field B ¼ B0 b z , threading a parallel pair of infinite, plane boundaries z ¼ 0, L, in Cartesian coordinates. By an incompressible, continuous foot-point displacement on one boundary, the uniform field is first deformed continuously into a nonequilibrium field Bs,d of some topology s, sketched in Fig. 6.8C on the right, with Bz jz¼0;L ¼ B0 . Then, taking the boundaries z ¼ 0, L to be rigid where the foot-points are now fixed, the Parker problem seeks a force-free field Bff described by Eqs. (6.1) and (6.2) subject to Bz jz¼0;L ¼ B0 , and accessible from the deformed field Bs;d by a continuous velocity v under induction Eq. (6.4). At the rigid boundaries, the electromagnetic conditions vz¼0,L ¼ 0 apply under which the topology s is unchanged. We use the textbook definition of topology: two geometric objects are topologically the same if one can be continuously deformed into the other. This definition is conceptually helpful. By specifying Bs,d given as a function of space explicitly, the topology s is implicitly defined by its realization in the given field, thus bypassing the formidable mathematical task of describing s explicitly. The Parker problem seeks a force-free field Bff that is continuously deformable into a given Bs,d. The result of the Parker (1972) demonstration can be stated as a magnetostatic theorem that not all s prescribed via the continuous field Bs,d can be found in the continuous Bff force-free solutions. The topologies of Bff are an infinite subset of measure zero, sparsely distributed in the continuum set of all admissible s, in the sense of an arbitrarily picked s having essentially zero probability of belonging to the subset; see Low and Egan (2014) for an example of sparsely distributed physical states. For most prescribed s, the solution to the Parker problem is a force-free field necessarily discontinuous, with TDs governed by Ampere’s law and force-balance. The TDs must be (1) shaped and located such that the continuum current density combined with the CSs give a field possessing the prescribed s and (2) force-free with a continuous jBj2 , an integral condition imposed by force-free Eq. (6.1). In time-dependent processes, TDs form via hole-punching in flux surfaces by the nonlinear coupling of the momentum equation with induction Eq. (6.4) (Bhattacharyya et al., 2010; Rappazzo and Parker, 2013). In the Parker static problem, that

4. The Rm [ 1, b  1 Turbulent Fluid

coupling is retained by imposing a given topology on the static force-free field (Parker, 1972, 1994; Yu, 1973; Hahm and Kulsrud, 1985; Low, 2010, 2013; Janse, Low and Parker, 2010; Boozer, 2012). Under the approximation Rm [1 the fluid has the same propensity as the perfect conductor to form TDs, until the approximation breaks down wherever current densities have grown intense enough for significant resistive dissipation and magnetic reconnection to occur. Field topology then changes, say, from s to s0 , and the ideal induction equation is restored by the intense current densities having dissipated. But the resultant s0 has essentially zero probability of being compatible with a continuous force-free field. Thus, once initiated, the Parker TDs would form interminably as a coronal field wanders in its topological space changing s via reconnection (Casini et al., 2009; Janse and Low, 2010; Judge et al., 2012). In the closed system in Fig. 6.8C (right) with the field anchored to rigid boundaries, the irrepressible TDs form and dissipate endlessly with diminishing current intensities as the available free energy decreases. Coronal fields are open systems continually energized by fresh magneic and Poynting fluxes from the photosphere. The coronal conditions of high conductivity and temperature are the reason for irrepressible CS formation and dissipation that in turn turbulently sustain the conditions (Parker 1988). The probabilistic nature of where field reconnection occurs in the tangled field underlies the statistical, sand-pile avalanche model of coronal heating (Lu, 1995). In the corona, electron thermal conduction is highly anisotropic, efficient along the field but essentially suppressed across it (Parker, 1963; Parrish and Stone, 2005; Low et al., 2012b). Resistive diffusion of field and cross-field thermal conduction are both generally weak effects; the latter is significantly weaker for b  1. Magnetic flux surfaces separating adjacent thin flux tubes act as thermal insulators whereas the pressure profile along each tube is independently established by energy balance with field-aligned thermal conduction. Pressures are naturally discontinuous across the flux surfaces, inducing a compensating discontinuity in magnetic pressure to attain a continuous total pressure for cross-tube force-balance. CSs intermittently form and dissipate at the small coronal resistivity. Thus, a weak (b < 1) pressure can impose CS formation on its dominating field, suggesting a reason for a mathematical singularity persisting in some early numerical models of the coupling of radiative energy balance with the magnetic field (Heasley and Mihalas, 1976). The high spatiotemporal observations of a NS-oriented inverse prominence at the solar limb, typically at a high latitude, show that its slab is composed of narrow (200 km) vertical filaments of cool (104 K) plasma supported by the horizontal prominence fields; see Fig. 5.3.9 image in Chapter 5.3. These filaments actively sway coherently as they descend steadily at (below free-fall) speeds, about 30 km/s, in between equally narrow lanes of hot tenuous turbulent up-flows at comparable speeds (Berger et al., 2011). An estimated 1015 g, the mass of a CME, can be drained and replenished in 1 day in a quiescent prominence (Liu et al., 2012). The dissipation of pressure-induced CSs may be the origin of the observed vertical motions across the horizontal prominence field (Low et al., 2012a, b; Low and Egan 2014).

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4.2 MAGNETIC HELICITY CONSERVATION AND ACCUMULATION Magnetic helicity is a quantitative measure of magnetic twist, a term used qualitatively in Section 3. Here we present a basic understanding of this conceptually difficult but essential measure. The solenoidal representation B¼V  A contains a free gauge, i.e.; A and A þ VG both define the same B for an arbitrary gauge function G. Any magnetic property expressed in terms of the vector potential A must be rendered gauge-independent to be physically meaningful. Classical helicity defined as Z H ¼ A$B dV (6.6) V

is gauge independent only for a field wholly contained in its domain V by virtue of zero normal component Bn(vV) h 0 at rigid wall vV and is then a constant of ideal induction Eq. (6.4) (Elssaser, 1956). Consider the simple case of B comprising two closed tubes of fluxes, F1 and F2, mutually winding a Gaussian integer N times, with the algebraic signs for rightand left-handed windings, respectively. Provided the field lines in each flux tube are closed and mutually unlinked, a direct calculation gives H ¼ 2NF1F2 describing the flux-weighted N-linkage between the two tubes, a conserved quantity. Independent of integral (6.6) and the field structure in each tube, the product NF1F2 is a constant of the induction equation because all three factors are also constants. The powerful integral (6.6) measures flux-weighted mutual linkages among all closed flux-tubes of any given field B, which is the reason for qualifying the twoflux formula for H with the condition that the two tubes contain only unlinked closed field lines. If the condition is removed, integral (6.6) would give a continuum sum of all of the flux-weighted linkages to be found. This understanding of H is basic but inadequate because in wholly contained complex 3D fields, closed field lines are exceptional whereas infinitely long field lines occur commonly in the form of a field line winding endlessly on a toroidal flux surface or filling up a finite 3D volume in the ergodic sense (Rosner et al., 1989; Low, 2013). Integral (6.6) is well-defined for all wholly contained fields, including those possessing ergodic flux, but H is generally not a sum of linkages between closed flux tubes. These conceptual issues do not arise in fields containing only finite-length field lines anchored to the boundary with Bn(vV) s 0, but, in that case, neither Gaussian linkage N nor H is meaningful. Motivated by studies of anchored coronal fields, Berger and Field (1984) discovered the relative helicity HR that has since been widely used in observational interpretation, analyses of numerical simulations, and theoretical investigations. Given a simply connected domain V with a boundary flux Bn(vV) s 0, HR is specially constructed as the difference in the classical helicities of any two admissible fields in the domain that serves as a gauge-independent global measure of the topological difference between the two fields. The boundary flux Bn(vV) defines a unique potential field Bpot in V, with Bpoth 0 for contained fields for which Bn(vV) h 0. Therefore,

4. The Rm [ 1, b  1 Turbulent Fluid

HR may be taken to be describing the relative topological difference between any given field in V and Bpot as a standard reference state, with HR reducing to H for contained fields. In time, a gauge-independent absolute helicity Habs was found for individual fields, wholly contained and anchored fields treated alike, that unifies the classical helicity H and the Berger-Field HR (Low, 2011). This has been a long ongoing theoretical development that began with the Taylor (1974) relaxation theory for the laboratory reversed-field pinch, through the first suggestion of Norman and Heyvaerts (1983) to extend the theory to astrophysics, to the discoveries of HR and Habs. The construction of Habs departed from traditional derivations by abandoning the use of the vector potential A with its encumbering free gauge (Low, 2011, 2015). Starting instead with flux conservation (6.5) as the fundamental physical property of the ideal induction equation, a continuum of conserved generalized helicity integrals was constructed, among which Habs was found as a generalization of the classical H to an anchored field. As originally applied to a wholly contained field in a low-b, Rm [1 turbulence, the Taylor (1974) theory observes that H decays less rapidly than the total field energy: Z E ¼ jBj2 dV (6.7) V

(Berger, 1984; Taylor, 1986). The interminable reconnections sustained by the Parker TDs occur intermittently in the extremely small volumes of CSs. Outside the CS events, the frozen-in condition is restored everywhere, during which helicity H is conserved, whereas energy E can change significantly through work done by the Lorentz force. Thus follows the Taylor hypothesis that the relaxed end-state for the contained field is a minimum-E force-free field subject to H ¼ H0, a conserved constant. Variational calculus then identifies this minimum-E field to be given by V  B ¼ aB, with a being a constant as an eigenvalue fixed by H0 (Woltjer, 1958). The theory successfully explains the spontaneous, turbulent formation of an outer layer of reversed axial field of a laboratory toroidal plasma that initially contained a purely unidirectional axial field (Taylor, 1974, 1986). The corresponding Woltjer minimum-E force-free state for an anchored field (Bn(vV) s 0) subject to absolute helicity Habs ¼ H0 was constructed for a finite size cylindrical domain, encountering unexpected, distinct physical and mathematical properties (Low and Fang, 2014). The Taylor theory provides a basic understanding of the longevity of magnetic fluxes and twists in evidence in the observed corona, although some fundamental issues of that understanding await future work. In particular, the mechanics of magnetic reconnection has remained an outstanding problem and the role of the generalized helicities in hydromagnetic turbulence is new and unexplored (Gonzalez and Parker, 2016; Low, 2015). Intuitively, the Rm [1 dissipation of the thin TD in Fig. 6.8A simply transfers macroscopic-sized fluxes from one flux system to another without destroying the fluxes. If a flux component perpendicular to the plane is added to the 2D field, this separate flux component merely gets partitioned and

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passed along into the reconnected horizontal quadrant pair. The field in the plane alone has no magnetic twist whereas the additional flux component introduces a signed twist between the two components. Field topology changes with reconnection whereas twists add and subtract in such a way that the sum over the domain has a conserved net helicity, H ¼ H0, for contained fields. The corresponding Woltjer minimum-E state in the corona is obviously the helmet streamer as the corona’s unit building block, generalized to include gravity and the conditions for self-confinement against the solar wind expansion. The question arises as to how large a flux rope, in terms of its engorged helicity, may be held in equilibrium by plasma weight and the tension force of the anchored field in Fig. 6.7B. In contrast, any amount of helicity can be confined within the rigid walls of a laboratory device. Ignoring the added and essential complication of the solar wind, an axisymmetric, static force-free field in unbounded space and anchored to a unit sphere has a total energy bounded above by the boundary flux distribution alone and bounded below by its helicity (Zhang, Flyer and Low, 2006). Therefore, unbounded accumulation of helicity is forbidden in any force-free field located in an open atmosphere. Excess helicity accumulated must be regularly ejected in CME flux ropes. That is, CMEs remove not just the old-cycle flux to make room for the new-cycle flux but also helicity that would otherwise accumulate monotonically under the hemispherical rule. Excessive helicity naturally accumulates in the newcycle fluxes of freshly emerged active regions, which for this reason are rich sources of CMEs, as observed.

5. OUTSTANDING QUESTIONS AND ASTROPHYSICAL IMPLICATIONS Coronal phenomenology in Section 3 largely adheres to its observational basis to keep interpretations distinct from the physical questions they pose. These questions are addressed in Section 4 within hydromagnetic theory. Irrepressible formation and dissipation of Parker CSs under conditions of high electrical and anisotropic thermal conductivity pervade the low-b corona, maintaining its million-degree temperatures and producing the flares that simplify field topologies. Although highly resistive in this dynamical manner, the Rm [1 coronal fluid does not lose its high-conductivity capability of storing considerable magnetic energy in metastable flux ropes. The generalization of helicity from the classical to the relative and absolute varieties completes the pursuit of a quantitative measure of magnetic twist in coronal fields and carries the Taylor turbulent conservation of helicity from laboratory devices over to the open atmosphere. This conservation law is the physical basis for understanding the longevity of coronal magnetic twist, its accumulation as flux ropes in helmet arcades, and final disposal in CMEs. We thus have a physical picture of the coronal polarity reversal as a processing of old and new-cycle fluxes governed

5. Outstanding Questions and Astrophysical Implications

by the hemispherical rule and regulated by the self-confinement condition for a helmet embedded with a flux rope to remain in the corona. The cycle-independent hemispherical rule as a direct consequence of the Parker dynamo extends the coronal paradigm into the convection zone. The untenable monotonic accumulations of a net helicity of an unchanging preferred sign in the respective NeS hemispheres are avoided in the convection zone by the emergence of the sunspot flux systems and helicity into the corona. The emerged helicity does not accumulate monotonically in the corona either, first stored as flux ropes and then ejected in CMEs in the solar wind. The recent spectral-polarimetric observations of large-scale flux ropes, emerging through the photosphere already twisted in compliance with the hemispherical rule, are significant when interpreted as direct evidence of the Parker dynamo. New phenomena and physics are expected with the crossings of observational frontiers. Unprecedented observations of coronal fields along optically thin LOS have directly detected coronal Alfven waves and confirmed flux rope properties (Tomczyk, 2007; Bak-Stesliska et al., 2013; Gibson, 2018). Large-aperture solar telescopes being built, notably the Daniel K. Inouye Solar Telescope at the National Solar Observatory, and future space missions, including the recently launched Parker Solar Probe, will be breaking new grounds. Outstanding physical problems are easy to pick out from Sections 3 and 4. Photospheric emergence of flux ropes, prominence fine-scale structures seen at the solar limb (Berger et al., 2011) and on the disk (Lin et al., 2005), and quantifying the Hundhausen rate of flux removal by CMEs, are some examples. The Parker TDs and concepts of helicity pose theoretical questions that are more clearly articulated than was possible 2 decades ago. The hemispherical rule and the Parker dynamo need further investigation. Should our construction fail in light of new observations and theory developments, there will be instructive and fundamental reasons to discover. The coronal processes have parallel astrophysical interests. The magnetic obstruction to gravitational collapse in star and interstellar cloud formation is a subtle problem. Interstellar clouds form by weighing down on disk-parallel, galactic magnetic fields, an effect stronger than the clouds’ self-gravity (Parker, 1966). Still, the frozen-in flux must be removed from the Parker clouds to take the process to their ultimate collapse by self-gravity. Drainage across the prominence horizontal fields, interminably driven by pressure and topologically induced CSs, may be relevant to cloud and star formation. The enormous solar wind cocoon has an influence on the local interstellar medium. A whole dipolar field is packaged in the corona during each magnetic cycle and then vaulted in a chronological layer in the outer solar wind. Does this process have astrophysical consequences for the local interstellar medium or the galaxy on astronomical time scales? The hemispherical rule in the Parker dynamo is a universal effect and it is interesting to point out that the lowest even and odd modes of the Parker (1971) galactic dynamo conform with the hemispherical rule. Is the galactic hemispherical rule observable in the galaxy and its halo (Heiles, 1998)? In their investigation of magnetic-cycle variations in coronal/chromospheric data, McIntosh et al. (2014) identified predictable properties associated

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with the overlaps of adjacent magnetic cycles. This development suggests that similar and other data from the stars may be an untapped source of information to discover stellar phenomena corresponding to those in the solar corona.

ACKNOWLEDGMENTS The author thanks Scott McIntosh, Director, for hospitality at High Altitude Observatory, Joan Burkepile and Andy Skumanich for their critical reviews of this chapter, Joan and Don Kolinski for help with figure preparation, and Roberto Casini, Chen Pengfei, Sarah Gibson, Valentin Martinez Pillet and Gene Parker for comments. The NCAR is sponsored by the US National Science Foundation. MLSO is an NCAR Facility. The Yohkoh SXT data presented were provided by the NASA-supported Yohkoh Legacy Data Archive at Montana State University. The EIT, LASCO, and MDI are instruments of the SOHO mission, an international cooperative project of the European Space Agency and NASA. Permissions are acknowledged from David Hathaway for the use of Fig. 6.1; from Solar Physics for the use of Fig. 6.4A photocopied from Leroy et al. (1983); and, from the Oxford University Press for the use of Fig. 6.6 photocopied from Cosmical Magnetic Fields (1979) and Fig. 6.8A and C photocopied from Spontaneous Current Sheets in Magnetic Fields (1994), both books authored by Eugene N. Parker.

REFERENCES Aschwanden, M.J., 2004. Physics of the Solar Corona. Springer. Axford, W.I., Leer, E., Skadron, G., 1978. The acceleration of cosmic rays by shock waves. Fifteenth International Cosmic Ray Conference. Bulgarian Academy of Sciences, p. 132. Bak-Steslicka, U., et al., 2013. The magnetic structure of solar prominence cavities: new observational signature revealed by coronal magnetometry. Astrophys. J. 770, L28. Belcher, J.W., MacGregor, K.B., 1976. Magnetic acceleration of winds from solar-type stars. Astrophys. J. 210, 498. Berger, M.A., 1984. Rigorous new limits on magnetic helicity dissipation in the corona. Geophys. Astrophys. Fluid Dynam. 30, 79. Berger, M.A., Field, G.B., 1984. The topological properties of magnetic helicity. J. Fluid. Mech. 147, 133. Berger, M.A., Ruzmaikin, A., 2000. Rate of helicity production by solar rotation. J. Geophys. Res. 105, 10481. Berger, T., Testa, P., Hillier, A., et al., 2011. Magneto-thermal convection in solar prominences. Nature 472, 197. Bhattacharyya, R., Low, B.C., Smolarkiewicz, P.K., 2010. On spontaneous formation of current sheets: untwisted magnetic fields. Phys. Plasmas 17, id112901. Boozer, A.H., 2012. Separation of magnetic field lines. Phys. Plasmas 19, id112901. Borrero, J.M., Jaffarzadeh, S., Schussler, M., et al., 2017. Solar magnetoconvection and smallscale dynamo. Recent developments in observation and simulation. Space Sci. Rev. 210, 275. Burkepile, J.T., St. Cyr, O.C., 1993. A revised and expanded catalogue of Coronal Mass Ejections observed by the Solar Maximum Mission Coronagraph. Tech. Note TN-369þSTR. Natl. Cent. for Atmos. Res, Boulder, p. 233.

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Heiles, C., 1998. Zeeman splitting opportunities and techniques at Arecibo. Astrophys. Lett. Commun. 37, 85. Hiei, E., Hundhausen, A.J., Sime, D.G., 1993. Reformation of a coronal helmet streamer by magnetic reconnection after a coronal mass ejection. Geophys. Res. Lett. 20, 2785. House, L.L., Smartt, R.N., 1982. Vector magnetic fields in prominences. I e preliminary discussion of polarimeter observations in He I D3. Sol. Phys. 80, 53. Hudson, H.S., Acton, L.W., Harvey, K.L., et al., 1999. A stable filament cavity with a hot core. Astrophys. J. 513, L83. Hundhausen, A.J., 1993. Sizes and locations of coronal mass ejections: SMM observations from 1980 and 1984e1989. J. Geophys. Res. 98, 13177. Hundhausen, A.J., 1997. Coronal Mass Ejections. In: Jokipii, J.R., Sonett, C.P., Giampapa, M.S. (Eds.), Cosmic winds and the heliosphere. University of Arizona Press, p. 259. Hundhausen, A.J., 1999. Coronal mass ejections. In: Strong, K., et al. (Eds.), The Many Faces of the Sun. Springer-Verlag, New York, p. 143. Illing, R.M.E., Hundhausen, A.J., 1986. Disruption of a coronal streamer by an eruptive prominence and coronal mass ejection. J. Geophys. Res. 91, 10951. Janse, A.A.M., Low, B.C., 2010. The topological changes of solar coronal magnetic fields. III. Reconnected field topology produced by current-sheet dissipation. Astrophys. J. 722, 1844. Janse, A.A.M., Low, B.C., Parker, E.N., 2010. Topological complexity and tangential discontinuity in magnetic fields. Phys. Plasmas 17, id092901. Jokipii, J.R., Parker, E.N., 1970. On the Convection, Diffusion, and Adiabatic Deceleration of Cosmic Rays in the Solar Wind. Astrophys. J. 160, 735. Jokipii, J.R., Sonett, C.P., Giampapa, M.S. (Eds.), 1997. Cosmic winds and the heliosphere. University of Arizona Press. Judge, P.G., Reardon, K., Cauzzi, G., 2012. Evidence for sheet-like elementary structures in the Sun’s atmosphere? Astrophys. J. 755, L11. Kahler, S.W., 1992. Solar flares and coronal mass ejections. Annu. Rev. Astron. Astrophys. 30, 113. Kubo, M., Low, B.C., Lites, B.W., 2014. Unresolved mixed polarity magnetic fields at flux cancellation sites in solar photosphere at 0.3” spatial resolution. Astrophys. J. 793, L9. Kuckein, C., Martinez Pillet, V., Centeno, R., 2012. An active region filament studied simultaneously in the chromosphere and photosphere. I. Magnetic structure. Astron. Astrophys. 539, A131. Lean, J., 2000. Evolution of the Sun’s Spectral Irradiance Since the Maunder Minimum. Geophys. Res. Lettt. 27, 2425. Lerche, I., Low, B.C., 2014. A nonlinear eigenvalue problem for self-similar spherical forcefree magnetic fields. Phys. Plasmas 21, id102902. Leroy, J.L., 1989. Observation of prominence magnetic fields. In: Priest, E.R. (Ed.), Dynamics and Structures of Quiescent Prominences. Kluwer Acad, Norwell, Mass, pp. 77e113. Leroy, J.L., Bommier, V., Sahal-Brechot, S., 1983. The magnetic field in prominences of polar crown. Sol. Phys. 83, 135. Leroy, J.L., Bommier, V., Sahal-Brechot, S., 1984. New data on the magnetic structure of quiescent prominences. Astron. Astrophys. 131, 33. Lin, Y., Engvold, O., Rouppe van der Voort, L., et al., 2005. Thin threads of solar filaments. Sol. Phys. 226, 239.

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Manchester IV, W., Gombosi, T., DeZeeuw, F.Y., 2004. Eruption of a buoyantly emergent magnetic flux rope. Astrophys. J. 610, 588. Martin, S.F., 2015. The magnetic field structure of prominences from direct and indirect observations. In: Vial, J.-C., Engvold, O. (Eds.), Solar Prominences. Springer, p. 205. Martin, S.F., Bilimoria, R., Tracadas, P.W., 1994. Magnetic field configurations basic to filament channels and filaments. In: Rutten, R.J., Schrijver, C.J. (Eds.), Solar Surface Magnetism. Kluwer Acad, Norwell, Mass, pp. 303e338. Martinez Gonzalez, M.J., et al., 2015. Spectro-Polarimetric imaging reveals helical magnetic fields in solar prominence feet. Astrophys. J. 802, id3. McIntosh, S.W., et al., 2014. Deciphering the solar magnetic activity cycle. I. On the relationship between the sunspot cycle and the evolution of small magnetic features. Astrophys. J. 792, id12. Norman, C.A., Heyvaerts, 1983. The final state of a solar flare. Astron. Astrophys. 124, L1. Okamoto, T.J., et al., 2008. Emergence of a helical flux rope under an active region prominence. Astrophys. J. 673, L215. Okamoto, T.J., et al., 2009. Prominence formation associated with an emerging helical flux rope. Astrophys. J. 697, 913. Panesar, N.K., et al., 2013. A solar tornado triggered by flares? Astron. Astrophys. 549 idA105. Parker, E.N., 1955a. The formation of sunspots from the solar toroidal field. Astrophys. J. 121, 491. Parker, E.N., 1955b. Hydromagnetic dynamo models. Astrophys. J. 122, 293. Parker, E.N., 1958. Dynamics of the interplanetary gas and magnetic fields. Astrophys. J. 128, 664. Parker, E.N., 1960. The hydrodynamic theory of solar corpuscular radiation and stellar winds. Astrophys. J. 132, 821. Parker, E.N., 1963. Interplanetary Dynamical Processes. lnterscience. Parker, E.N., 1966. The dynamical state of the interstellar gas and fields,. Astrophys. J. 145, 811. Parker, E.N., 1970. The generation of magnetic fields in astrophysical bodies. I. The dynamo equations. Astrophys. J. Jpn. 162, 665. Parker, E.N., 1971. The generation of magnetic fields in astrophysical bodies. II. The galactic field. Astrophys. J. 163, 255. Parker, E.N., 1972. Topological dissipation and the small-scale fields in turbulent gases. Astrophys. J. 174, 499. Parker, E.N., 1979. Cosmical Magnetic Fields. Oxford Univ. Press, New York. Parker, E.N., 1988. Nanoflares and the solar X-ray corona. Astrophys. J. 330, 474. Parker, E.N., 1991. The optical analogy for vector fields. Phys. Fluids B 3, 2652. Parker, E.N., 1994. Spontaneous current sheets in magnetic fields. Oxford Univ. Press, New York. Parrish, I.J., Stone, J.M., 2005. Nonlinear evolution of the magnetothermal instability in two dimensions. Astrophys. J. 633, 334. Pesnell, W.D., Thompson, B.J., Chamberlin, P.C., 2012. The Solar Dynamic Observatory (SDO). Solar Phys 275, 3. Petrie, G.J.D., Low, B.C., 2005. The dynamical consequences of spontaneous current sheets in quiescent prominences. Astrophys. J. Suppl. Series 159, 288. Rappazzo, A.F., Parker, E.N., 2013. Current sheets formation in tangled coronal magnetic fields. Astrophys. J. 773. L2.

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Rosner, R., Low, B.C., Tsinganos, K., Berger, M.A., 1989. On the relationship between the topology of magnetic field lines and flux surfaces. Geophys. Astrophys. Fluid Dyn. 48, 251. Russell, C.T., 2008. Forward: STEREO Mission. Space Sci. Rev. 136, 1. Rust, D.M., 1967. Magnetic fields in quiescent prominences. I. Observations. Astrophys. J. 150, 313. Rust, D.M., Kumar, A., 1994. Helical magnetic fields in filaments. Sol. Phys. 155, 69. Shibata, K., 1999. Evidence of magnetic reconnection in solar flares and a unified model of flares. Astrophys. Space Sci. 264, 129e144. Shimizu, T., Imada, S., Kubo, M. (Eds.), 2018. First ten years of Hinode Solar On-Orbit Observatory. Springer. Skumanich, A.P., 1972. Time scales for Ca II emission decay, rotational braking, and Lithium depletion. Astrophys. J. 171, 565. SOHO-23: Understanding a Peculiar Solar Minimum. In: Cranmer, S.R., Hoeksema, J.T., Kohl, J.L. (Eds.), ASP Conference Series 428. SOLAR WIND 13: Proceedings of the Thirteenth International Solar Wind Conference. In: Zank, G.P., et al. (Eds.), AIP Conference Proceedings 1539. St. Cyr, O.C., et al., 2000. Properties of coronal mass ejections: SOHO LASCO observations from January 1996 to June 1998. J. Geophys. Res. 105, 18169. Straka, R.M., Papagiannis, M.D., Kogut, J.A., 1975. Study of a filament with a circularly polarized beam at 3.8 cm. Sol. Phys. 45, 131. Takeda, A., 2011. Characteristics of the re-calculated Yohkoh/SXT temperature response. Sol. Phys. 273, 295. Tandberg-Hanssen, E., 1995. The Nature of Solar Prominences. Kluwer. Taylor, J.B., 1974. Relaxation of toroidal plasma and generation of reverse magnetic fields. Phys. Rev. Lett. 33, 1139. Taylor, J.B., 1986. Relaxation and magnetic reconnection in plasmas. Rev. Mod. Phys. 58, 741. Tomczyk, S., et al., 2007. Alfven Waves in the solar corona. Science 317, 1192. Tsuneta, S., 1996a. Structure and dynamics of magnetic reconnection in a solar flare. Astrophys. J. 456, 840. Tsuneta, S., 1996b. Interacting active regions in the solar corona. Astrophys. J. 456, L63. van Ballegooijen, A.A., Martens, P.C.H., 1990. Magnetic fields in quiescent prominences. Astrophys. J. 361, 283. Vial, J.-C., Engvold, O. (Eds.), 2015. Solar Prominences. Springer. Webb, D.F., Howard, R.A., 1994. The solar cycle variation of coronal mass ejections and the solar-wind mass flux. J. Geophys. Res. 99, 4201. Woltjer, L., 1958. A theorem on force-free magnetic fields. Proc. Natl. Acad. Sci. U.S.A. 44, 489. Yu, G., 1973. Hydrostatic Equilibrium of Hydromagnetic Fields. Astrophys. J. 181, 1003. Zank, G.P., et al., 1998. The radial and latitudinal dependence of the cosmic ray diffusion tensor in the heliosphere. J. Geophys. Res. 103, 2085. Zhang, M., Flyer, N., Low, B.C., 2006. Magnetic field confinement in the corona: the role of magnetic helicity accumulation. Astrophys. J. 646, 575. Zhang, M., Low, B.C., 2005. The hydromagnetic nature of solar coronal mass ejections. Annu. Rev. Astron. Astrophys. 43, 103.

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7 E.R. Priest

School of Mathematics and Statistics, University of St. Andrews, St Andrews KY16 9SS, United Kingdom

CHAPTER OUTLINE 1. Introduction .......................................................................................................240 2. Magnetohydrodynamics ......................................................................................241 2.1 Validity of Magnetohydrodynamics ........................................................ 241 2.2 The Magnetohydrodynamics Equations.................................................. 241 2.3 Induction Equation.............................................................................. 243 2.3.1 Limits Rm
1 and Rm [ 1 ........................................................ 243 2.4 Equation of Motion.............................................................................. 244 2.5 Equilibria ........................................................................................... 245 2.6 Waves ................................................................................................ 246 2.7 Magnetic Reconnection ....................................................................... 247 2.7.1 Two-Dimensional Reconnection .................................................... 247 2.7.2 Three-Dimensional Reconnection.................................................. 249 3. Dynamo Theory ..................................................................................................251 3.1 Introduction: Solar Observations and Terminology .................................. 251 3.1.1 Some Constraints From Solar Observations .................................... 252 3.1.2 Terminology and Physical Ideas.................................................... 252 3.2 A History of Dynamo Ideas ................................................................... 254 3.3 Early Turbulent Dynamos ..................................................................... 257 3.3.1 Parker’s (1955) Model................................................................. 257 3.3.2 Mean-Field Magnetohydrodynamics Theory.................................... 258 3.4 Flux-Transport Dynamos ...................................................................... 259 3.5 Tachocline Dynamos............................................................................ 262 3.5.1 Overshoot Dynamo...................................................................... 263 3.5.2 Interface Dynamo ...................................................................... 263 3.6 Global Computations ........................................................................... 264 3.7 Concluding Remarks............................................................................ 264 References .............................................................................................................266

The Sun as a Guide to Stellar Physics. https://doi.org/10.1016/B978-0-12-814334-6.00009-1 Copyright © 2019 Elsevier Inc. All rights reserved.

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1. INTRODUCTION Solar physics is crucially important for astronomy because many fundamental cosmic plasma processes can be studied in great detail on the Sun, such as magnetic turbulence, magnetic field generation, coronal heating, wind acceleration, thermal instability, flaring, eruptive dynamics, and particle acceleration. Each of these is driven by the magnetic field and great progress has been made in each. However, in none of them has a definitive understanding been reached, and so solar physics is in a vibrant state of activity as we continue to make progress. It remains a touchstone for astrophysics. A theory for the interaction between the magnetic field and plasma on the Sun is provided by magnetohydrodynamics (MHD), which represents a unification of electromagnetism and fluid mechanics. Solar material is not a normal gas but instead is plasma, in which the atoms are split into positive ions and negative electrons that can flow freely. Thus, the gas becomes electrically conducting and electric currents flow. Indeed, most of the universe is in this plasma state. The magnetic field has several effects on plasma: 1. It exerts a force, which may create structure or make the plasma move. 2. It stores energy, which may heat the corona or be released in a solar flare. 3. It acts as a thermal blanket, which protects cool prominences from the surrounding hot corona. 4. It channels fast particles, plasma, and heat. 5. It provides stability in some cases, such as a sunspot, whereas in other cases it drives instabilities, both large-scale and small-scale. 6. It supports a variety of wave modes. The MHD equations mathematically describe all of these physical effects. In the first half of this chapter (Section 2), we give an overview of MHD. We discuss its validity (Section 2.1), summarize the MHD equations (Section 2.2), and focus on the physical effects of the induction equation (Section 2.3) and equation of motion (Section 2.4). Then we discuss the nature of magnetic equilibria (Section 2.5), magnetic wave modes (Section 2.6), and the process of magnetic reconnection (Section 2.7). In the second half (Section 3), we focus on dynamo theory, starting with an introduction to relevant solar observations and terminology (Section 3.1) and a history of the development of dynamo ideas (Section 3.2). Then we describe early turbulent models (Section 3.3) and discuss flux-transport dynamo models (Section 3.4), tachocline dynamos (Section 3.5), and global computations (Section 3.6) before ending with pointers to the future (Section 3.7). This represents an extremely brief summary of huge topics; for more details and references, see, e.g., Priest (2014).

2. Magnetohydrodynamics

2. MAGNETOHYDRODYNAMICS 2.1 VALIDITY OF MAGNETOHYDRODYNAMICS In MHD, we do not deal with individual particles but regard plasma as a continuous medium that is valid when the length-scales (L0) of interest are larger than the mean-free path for collisions between particles, which is typically 3 cm in the chromosphere and 30 km in the corona. However, MHD often works well also in a collisionless plasma, where this condition is not satisfied: for instance, in the solar wind, the mean-free path is about an astronomical unit (the mean distance between Sun and Earth). MHD describes large-scale dynamics both with and without collisions. Although Vlasov kinetic theory is the most complete description of a collisionless plasma, there are several reasons why MHD is so successful in the corona and solar wind as well as the photosphere and interior of the Sun: 1. Ideal MHD embodies conservation of mass, momentum, and energy, which are universal relations in both collisional and collisionless plasmas; 2. Gyro-motion prevents particles traveling perpendicular to the magnetic field, and so the net effect may be described by MHD-like equations; 3. Often, waveeparticle interactions impede particle motion along the magnetic field; 4. In a collisionless plasma, when the E  B-drift is dominant, the drift velocity is udrift h E  B/B2, which implies an ideal Ohm’s law of the form Et þ v  B ¼ 0; 5. Summing the Lorentz force on individual particles gives a net force of F ¼ qE þ j  B; where qE
j  B when udrift
c and qE
Vp when lDebye
L0, where lDebye is the Debye length. In a collisionless plasma, there are two important modifications to the normal equations of MHD. First, the pressure is not isotropic, and so one needs a pressure tensor; second, Ohm’s law generalizes with new terms representing electron inertia, a Hall term and electron stress.

2.2 THE MAGNETOHYDRODYNAMICS EQUATIONS Maxwell’s equations of electromagnetism essentially give four equations for the curl and divergence of the magnetic induction (B) and electric field (E), where H ] B/m0 is the magnetic field, D ¼ ε0E is the electric displacement, m0 the magnetic permeability, ε is the electrical permittivity, E ¼ j/s is Ohm’s law, j is the electric current density, and s is the electrical conductivity.

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In a gas, there is also an independent set of equations of fluid mechanics for the gas density (r), velocity (v), pressure (p), and temperature (T), namely, equations of motion, mass continuity, energy, and a perfect gas law. Now, in MHD there is a remarkable coupling and unification of the electromagnetic and fluid equations. This occurs by making three assumptions: 1. We simplify Maxwell’s equations by assuming v
c, so that the displacement current ðvD=vtÞ is negligible in Ampe‘ re’s law; 2. In a moving plasma, the total electric field is E þ v  B, where E is the field acting on plasma at rest, so an extra term is added to Ohm’s law; 3. The plasma feels a Lorentz force of the form j  B, which is added to the righthand side of the equation of motion. The resulting equations can be reduced to just two main equations for v and B, namely, the equation of motion and the so-called induction equation, together with three other equations for r, p, and T. Generally, we refer to B as the “magnetic field,” although formally it is the magnetic induction. The resulting MHD equations are as follows: vB ¼ V  ðv  BÞ þ hV2 B; vt dr þ rV$v ¼ 0; dt r

dv ¼ Vp þ j  B þ rg þ F; dt


rg d p g  1 dt rg



p ¼ RrT;  ¼ V$q  Lr þ j2 s þ FH ;

(7.1) (7.2) (7.3) (7.4) (7.5)

where F represents other forces such as viscous or Coriolis and Eq. (7.5) is the form of the energy equation appropriate for the corona, in which q ¼ kVT is the heat flux vector, Lr is optically thin radiation, and FH represents extra heat sources. Also, R is the gas constant and d/dt ¼ v/vt þ v$V is the total time derivative in a frame of reference moving with an element of plasma. Details can be found in Priest (2014). These equations are coupled and determine the primary variables v, B, p, r, and T. In addition, the secondary variables j and E are given explicitly by j ¼ V  B=m;

(7.6)

E ¼ v  B þ j=s;

(7.7)

whereas B is subject to the condition V$B ¼ 0:

(7.8)

2. Magnetohydrodynamics

This last equation has the role of an initial condition for the time-dependent Eqs. (7.1)e(7.5), which form a complete set, representing nine scalar equations for nine scalar variables (namely, Bx, By, Bz, vx, vy, vz, p, r, and T), whereas the divergence of (1) shows that if V$B vanishes initially, it continues to vanish for all time.

2.3 INDUCTION EQUATION The induction equation (Eq. 7.1), in which h h ðmsÞ1 is the magnetic diffusivity, determines B once v is known. In MHD, there is a profound change of philosophy from electromagnetism, because v and B are now the primary variables, whereas j and E are secondary variables that may be calculated if required from Eqs. (7.6) and (7.7) but do not affect the basic MHD physics. v and B are determined by the equations of motion and induction, which embody the basic physics of how an MHD plasma behaves. This is different from the basic physics of electromagnetism, in which E ¼ j/s and so electric fields are driven by currents. The induction equation (Eq. 7.1) expresses the fact that the magnetic field at a fixed point in space changes in time owing to two physical processes: the transport of a magnetic field with the plasma (the first term on the right) and the diffusion of a magnetic field through the plasma (the second term). If V0 and l0 are typical velocity and length-scales, the ratio of these two terms is l0 V0 h Rm ; h which is a key dimensionless parameter known as the magnetic Reynolds number. Normally, it is much larger than unity, so magnetic diffusion is negligible and Eq. (7.7) becomes E ¼ v  B. Thus, in most of the universe, the magnetic field moves with the plasma and hangs on to its energy. Also, the electrical field is simply v  B and is not related to the current at all. For example, in an active region of the Sun, where h z 1 m2/s, l0 z 105 m and V0 z 104 m/s, we find Rm z 108. The exception to this behavior is in current singularities, where electric currents and magnetic field gradients are enormous. These form at, for example, null points or along separators (Section 2.7.2) and are locations of magnetic reconnection, where magnetic energy is released (Section 2.7).

2.3.1 Limits Rm
1 and Rm [ 1 It is only in regions where l0 is extremely small and so the electric currents jwB0 =ðm0 l0 Þ are extremely large that Rm becomes less than or on the order of unity and the second term on the right of Eq. (7.1) becomes important. However, these intense sheets or filaments of current are precisely the regions where magnetic energy conversion takes place. Consider a region where Rm
1, so that Eq. (7.1) reduces to vB ¼ hV2 B. vt

(7.9)

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Thus, B is governed by a diffusion equation, which implies that magnetic field  variations on a scale l0 diffuse away on a diffusion time sd h l20 h and with a diffusion speed vd h l0 =sd ¼ h=l0 : for example, the decay time for a sunspot, in which h ¼ 1 m2/s and l0 ¼ 106 m is 1012 s or 30,000 years. Solutions to Eq. (7.9) representing diffusion of magnetic fields in one, two, and three dimensions are discussed in Priest (2014). Consider next the opposite limit, namely, Rm [ 1, so that Eq. (7.1) reduces instead to vB ¼ V  ðv  BÞ (7.10) vt and Ohm’s law becomes E þ v  B ¼ 0. It can then be shown that the rate of change of magnetic flux through a curve C attached to the plasma vanishes. In other words, we have magnetic flux conservation. In addition, we also have magnetic field line conservation, so that two elements of plasma that are initially joined by a magnetic field line will continue to be joined by a field line at later times. We say that the field lines are “frozen to the plasma.” Plasma can move freely along magnetic field lines, but in motion perpendicular to the field lines, the plasma is dragged with the field lines or the field lines are carried with the plasma. Proofs of these results and examples are given in Priest (2014).

2.4 EQUATION OF MOTION The equation of motion dv ¼ Vp þ j  B þ rg (7.11) dt for the acceleration of plasma by a pressure gradient, a magnetic force, and gravity determines the way the plasma velocity (v) varies in time. These are the main forces operating in the solar atmosphere, although for some purposes it is important to add viscous forces (when there are strong velocity shears) or Coriolis forces (over large scales owing to solar rotation). In most of the corona, especially active regions, the magnetic force dominates the structure and behavior in a direction normal to the magnetic field. However, along the magnetic field, j  B vanishes and so the pressure gradient and gravity are important. The ratio of the sizes of the pressure gradient and gravity is p/(l0rg), which is much larger than unity when the length scale is l0 [ H, where H h p/(rg) ¼ RT/ g is the scale height, which is proportional to the temperature. Thus, the shape and position of a coronal loop in equilibrium are determined by the magnetic field, but the structure of plasma within the loop is governed by pressure and gravity, such that if the height of a loop is much less than H, gravity is unimportant. Typical values of H are 150 km in the photosphere and 100 Mm, where 1 Mm is a megametre (i.e., 106 m), in a 2 MK-degree corona, so that starting in the photosphere r

2. Magnetohydrodynamics

and moving upward in the atmosphere, the plasma pressure decreases rapidly at first and then much more slowly as the temperature reaches coronal values. The plasma pressure gradient (Vp) acts from regions of high pressure to regions of low pressure in a direction perpendicular to the curves of constant pressure (isobars). The magnetic force, on the other hand, may be rewritten after using j ¼ V  B/m, V$B ¼ 0, and a vector identity, as
2 B B j  B ¼ ðB$VÞ  V ; (7.12) m 2m which is always perpendicular to the magnetic field. The first term on the right is a magnetic tension force. It is nonzero when the field lines are curved and points toward the center of curvature such that the field lines behave like elastic bands with tension B2/m. The second term on the right has the same form as Vp and so is called a magnetic pressure force, which acts from regions of high magnetic pressure (B2/(2m)) to low magnetic pressure. The ratio of the pressure gradient to the magnetic force is the plasma beta p ; (7.13) bh 2 B =ð2mÞ so that when b
1, the magnetic force dominates. In the active region corona, b is typically 104, whereas in a photospheric flux tube it is of the order of unity. If the acceleration on the left of Eq. (7.3) is of the order of the magnetic pffiffiffiffiffiterm ffi force, v z vA hB mr, which is known as the Alfve´n speed. This is the speed to which the magnetic field tends to accelerate plasma when the magnetic force is not opposed by other forces. It has typical values of 1000 km/s in the corona and 5 km/s in a photospheric flux tube.

2.5 EQUILIBRIA If plasma velocities are much smaller than the Alfve´n speed (v
vA), the inertial term on the left of the equation of motion (3) is much smaller than the Lorentz force and so Eq. (7.3) reduces to 0 ¼ Vp þ j  B þ rg;

(7.14)

which represents magnetohydrostatic equilibrium for balance among a pressure gradient, a magnetic force, and gravity. If, furthermore, the pressure and magnetic forces are comparable and the height (h) of a structure such as a coronal loop is much smaller than the scale height, i.e., h
H h p/(rg) ¼ RT/g, the force of gravity is negligible. If also the plasma beta b h 2mp/B2
1, Eq. (7.14) approximates to j  B ¼ 0; for a so-called force-free field.

(7.15)

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A particular case is when the current (j ¼ V  B) vanishes. This may be satisfied identically by a magnetic field of the form B ¼ Vj, where the potential j satisfies Laplace’s equation, V2j ¼ 0, and we have a potential field. Calculating potential fields for the corona using the normal component of the observed magnetic field in the photosphere is extremely valuable in determining the structure and topology of the coronal magnetic field. However, to model highly sheared or twisted fields typical of solar prominences and solar flares, it is essential to solve the force-free field equation instead (Mackay and Yeates, 2012). Eq. (7.15), in which j ¼ V  B/m and V$B ¼ 0, looks disarmingly simple, but little is known in general about its properties. It implies that the electrical current ( j) is parallel to the magnetic field (B), so that V  B ¼ aB;

(7.16)

in which a is a function of position that is constant along each magnetic field line. If a is uniform, we have a so-called or “linear” force-free field. The  “constant-a”  curl of Eq. (7.16) in this case gives V2 þ a20 B ¼ 0, as a generalization of Laplace’s equation. Unfortunately, often a is far from uniform, and so linear force-free solutions are not as useful as first thought. Thus, one often resorts to numerical nonlinear force-free solutions, which are far from trivial. A detailed account of force-free solutions can be found in Priest (2014).

2.6 WAVES In a uniform gas of pressure p0 and density r0, sound waves can propagate equally in all directions at the sound speed cs ¼ ðgp0 =r0 Þ1=2 . Suppose they have a frequency u and wavenumber vector k ¼ k b k for a wave propagating in a direction b k with 2 2 2 wavenumber k. Then their dispersion relation is u ¼ k cs . The restoring force for this wave is the plasma pressure gradient. In MHD, instead of one type of wave, there are three independent wave modes, because we now have three restoring forces: a pressure gradient, a magnetic pressure gradient and a magnetic tension force. Furthermore, the modes are no longer isotropic but depend on the inclination (q) of the direction of propagation to the background uniform magnetic field. Alfve´n waves have a dispersion relation u2 ¼ k2 v2A cos2 q;

(7.17)

with magnetic tension as a restoring force. In addition, there are fast and slow magnetoacoustic waves whose restoring forces are a combination of the plasma and magnetic pressure gradients and which propagate at speeds larger and smaller, respectively, than the Alfve´n speed (vA). Their dispersion relations are    1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 1 c2s þ v2A  4c2s v2A cos2 q. (7.18) u2 k2 ¼ c2s þ v2A  2 2

2. Magnetohydrodynamics

2.7 MAGNETIC RECONNECTION 2.7.1 Two-Dimensional Reconnection The induction equation (Eq. 7.1) describes how the magnetic field changes in time owing to the transport and diffusion of field lines. The first term on the right is much larger than the second in most of the Sun, and so the magnetic field is frozen to the plasma and holds on to its energy. The exception is in singularities in which the magnetic gradient (and therefore the current) is extremely large. Such singularities can form at null points where the magnetic field vanishes, which therefore represent weak spots in the magnetic field. The singularities take the form of current sheets, which are resolved by magnetic diffusion and where the field lines break and reconnect (Fig. 7.1). The effects of this local process are: • • •

to change the topology (i.e., the magnetic connectivity) of magnetic field lines, which affects the paths of fast particles and heat along the field; to convert magnetic energy into heat, kinetic energy, and fast particle energy; and to create large electric currents, electric fields, shock waves, and turbulence, all of which may help accelerate fast particles.

If we model the magnetic field locally in a current sheet of width l by a onedimensional (1D) field By(x, t) that is diffusing through a plasma at rest according to Eq. (7.9), the field lines diffuse inward at a speed vd ¼ h/l and cancel or “annihi late” at x ¼ 0. The current sheet possesses a current jz ¼ m1 dBy dx and its width diffuses outward at the same speed while the magnetic energy is transformed into heat by ohmic heating ( j2/s). A steady state may therefore be produced if magnetic flux and plasma are brought in from large distances at a speed vd. The SweeteParker model sketched in Fig. 7.2 is an order of magnitude model for a simple diffusion region of length 2L and width 2l between oppositely directed fields of magnitude Bi. A balance between diffusion and advection implies that the speed (vi) at which the magnetic field lines are carried into the reconnection region is

A

A ve

C

2L 2Le

B

(A)

(B)

(C)

FIGURE 7.1 (a) The coming together, (b) touching and (c) reconnection of magnetic field lines when a localized diffusion region (shaded) leads to a change in magnetic connectivity of plasma elements from AB to AC.

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2l

vo Bo vi Bi 2L

FIGURE 7.2 SweeteParker model of reconnection in a current sheet of dimensions 2l and 2L, in which plasma enters at speed vi and field Bi and leaves with values vo and field Bo.

h (7.19) vi ¼ . l Conservation of mass into and out of the sheet implies (if r ¼ constant) that Lvi ¼ l vAi ;

(7.20) pffiffiffiffiffiffiffiffiffiffiffi ðmri Þ. in which plasma is expelled from the sheet at the Alfve´n speed vAi ¼ Bi The sheet width (l) may be eliminated between Eqs. (7.19) and (7.20) to give the reconnection rate (vi), which may be written in dimensionless form as Mi ¼

1 1=2 Rm

;

(7.21)

in which Mi h v/vA is the Alfve´n Mach number and Rm ¼ LvA/h. Later, Petschek put forward a much faster reconnection mechanism (Fig. 7.3) in which the diffusion region is much smaller and is surrounded by four slow-mode

FIGURE 7.3 Petschek’s reconnection model.

2. Magnetohydrodynamics

shock waves. The shock waves convert the inflowing magnetic energy into heat and kinetic energy of two outflowing hot streams of high-speed plasma. The maximum reconnection rate is Me ¼

p ; 8logRme

(7.22)

in which Me and Rme are evaluated at some large external distance from the origin. For typical solar values, the SweeteParker rate Eq. (7.21) is about 104 and so is much too slow to explain solar flares, but the Petschek rate Eq. (7.22) is typically about 102 and is fast enough. Since then, extrafast regimes of reconnection have been discovered with reconnection rates that depend on the boundary conditions and include the Petschek mechanism as a special case. They have been reproduced in numerical experiments provided the diffusion region diffusivity is enhanced, which is often expected in practice owing to current-driven microturbulence.

2.7.2 Three-Dimensional Reconnection The basic theory for reconnection in 2D is now well-understood, but we are only just beginning to investigate how it works in 3D (Priest and Forbes, 2000) and have discovered that there are key differences from 2D, as follows. First, magnetic null points, in which the magnetic field vanishes, have a completely different nearby structure in 2D and 3D. In 2D, a null point forms either an X-type or O-type structure, but in 3D the simplest field with a null at the origin and satisfying V$B ¼ 0 has components. ðBx ; By ; Bz Þ ¼ ðx; y; 2zÞ; with field lines indicated in Fig. 7.4. Two families of field lines link to the null point: an isolated spine field line, which comes in along the positive and negative z-axis toward the null; and a surface of fan field lines, which come out of the null in the xy-plane. The fan surface is a so-called separatrix surface that separates the field lines that lie above the xy-plane from those that lie below it. The second difference between 3D and 2D is that the topology is much more complex in 3D. Consider, for example, in 2D, four sources in a line on the base (the photosphere) that are situated in the order þ, , þ,  (Fig. 7.5A). Above the base, in the corona, there is an X-type null point. The field lines that thread the null point are called separatrix curves because they separate the 2D region into topologically distinct regions in the sense that the field lines immediately to the left of the null all start out at the left-most positive source and end at the left-most negative source, whereas those to the right of the null link the other two sources. Furthermore, all of the field lines below the null join the right-most positive source to the left-most negative source, whereas all of the field lines above the null join the two outermost sources. By comparison, in 3D, suppose two positive and two negative sources are placed on the plane base (photosphere), as indicated in Fig. 7.5B. Two 3D null points will

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z

Spine curve

y

x

Fan surface FIGURE 7.4 Structure of the magnetic field near a 3D null point (where the magnetic field vanishes) with components (Bx, By, Bz) ¼ (x, y, 2z). The spine is an isolated field line that approaches (or recedes from) the null point, whereas the fan is a surface of field lines that come from the null (or goes into it). Separatrix surface

Separatrix curve X-point

Separator

-

(A)

(B)

Null point

FIGURE 7.5 (A) Structure of the two-dimensional (2D) magnetic field near an X-type null point produced by four sources in a line on the base, showing the way that separatrix curves linking the X-point bound four topologically separate regions. (B) The 3D structure owing to four sources on the base, showing how separatrix surfaces linking two null points bound four topologically separate volumes. The separator is a special field line that joins the two null points and lies at the intersection of the two separatrices.

be present on the base, indicated by stars, and the fan surfaces from each of those null points form domes that curve downward and separate field lines below the dome from those above it. The two domes intersect in a special curve, called a separator, which is a field line that joins the one null point to the other.

3. Dynamo Theory

The third way in which 2D and 3D differ lies in the nature of reconnection. In 2D, reconnection can take place only at an X-point, and in Fig. 7.5A it transfers magnetic flux from two subregions to the other two subregions. In 3D, one type of reconnection, called separator reconnection, occurs in a similar way when a strong current builds up along a separator (Parnell, 2010). It has the effect of transferring magnetic flux from two three-dimensional regions to two other regions. However, in 3D, reconnection can also take place at other locations where the current can accumulate: • •

at null points by torsional spine reconnection, torsional fan reconnection, or, usually, by spine-fan reconnection; or at a so-called quasiseparator by quasiseparator reconnection (Priest and Demoulin, 1995; De´moulin et al., 1996).

In the latter case, consider the mapping of the footpoints of magnetic field lines from one part of the photosphere to another. If there is a 3D null point in the overlying atmosphere, a separatrix surface exists across which the mapping exhibits a sudden jump or discontinuity. However, when there is no null point or separatrix surface, sometimes there exist so-called quasiseparatrix surfaces across which the mapping is perfectly continuous but possesses a steep gradient. Then it transpires that large currents can again accumulate along the quasiseparator and lead to magnetic reconnection. A final important new feature in three dimensions is the existence of a topological invariant known as magnetic helicity, which includes two types: self-helicity and mutual helicity (Berger, 1984). The self-helicity is a measure of the twisting and kinking of a magnetic flux tube, whereas mutual helicity measures the linkage between different flux tubes. What happens during 3D reconnection is that the total magnetic helicity (the sum of self and mutual helicities) is conserved but it can be converted from one kind to the other: say, from mutual to self-helicity. This has been proposed as the means by which erupting flux ropes are formed in the core of coronal mass ejections (Priest and Longcope, 2017).

3. DYNAMO THEORY 3.1 INTRODUCTION: SOLAR OBSERVATIONS AND TERMINOLOGY Understanding how the Sun’s magnetic field can be generated by dynamo action cyclically is a key question regarding which huge progress has been made over several decades. In the second section of this chapter, we summarize the observational constraints and describe the terminology (Section 3.1). Then we describe the history of the subject (Section 3.2) and the early basis for turbulent dynamo theory (Section 3.3), namely, Parker’s classic 1955 article and the more systematic

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mathematical formalism of mean-field theory. This is followed with accounts of fluxtransport dynamos (Section 3.4) and tachocline dynamos (Section 3.5). Finally, some global computations are outlined (Section 3.6) and brief comments are given on future developments (Section 3.7).

3.1.1 Some Constraints From Solar Observations Many observed features of the Sun’s magnetic field need to be explained. Basic features that offer clues about how the Sun generates its magnetic field are: 1. The number of sunspots has a rough period of 11 years. Occasionally, the cycle declines to a very low level, so that few sunspots appear (such as during the Maunder minimum for 70 years in the 17th century). 2. Sunspots are restricted to two belts of latitude between typically 35 degrees, and there is a tendency for them to appear in preferred “active longitudes.” 3. The sunspot belts spread and drift towards the equator as the solar cycle progresses (Spo¨rer’s law). This shows up as a “butterfly diagram” and suggests a migration in time of dynamo activity from midlatitudes towards the equator. 4. Sunspot groups are inclined to lines of latitude (by z4 degrees); the leading sunspot is closer to the equator and the tilt increases with latitude (Joy’s law). 5. A leading spot has the same polarity as all other leaders in the same hemisphere and opposite those in the other hemisphere (Hale’s polarity laws). The polarities reverse at the start of each new 11-year cycle. 6. The magnetic fields near the two poles have dominant polarities that are opposite one another, but they reverse every 11 years near sunspot maximum.

3.1.2 Terminology and Physical Ideas 

Global Diffusion. The diffusion time (R21 h) for the decay of the global solar magnetic field is about 1010 year, which is comparable to the age of the Sun, but the Sun’s magnetic field is unlikely to be primordial, because the decay estimate for the field may well be an overestimate. Kinematic Dynamo Theory aims to construct a velocity field [v(x, y, z, t)] such that a magnetic field [B(x, y, z, t)] satisfying the induction equation (Eq. 7.1) grows and is maintained by flows against diffusion. Nonlinear Dynamo Theory asks instead whether the magnetic field can be maintained in a self-consistent manner: i.e., taking account of the back-reaction of the Lorentz force on the flow. Such a fully MHD dynamo is so difficult to set up that attention was first concentrated on the kinematic problem (Section 3.2). Slow or Fast Dynamo. A kinematic dynamo is said to be fast if its growth rate remains positive as Rm / ∞, in which Rm ¼ L0v0/h is the magnetic Reynolds number; otherwise it is said to be slow. Toroidal (Bf) and Poloidal (Bp) Fields. A key focus in solar dynamo theory concerns the way in which both the toroidal (i.e., azimuthal) and poloidal (i.e., meridional) field components are generated. A toroidal component can be naturally

3. Dynamo Theory

built up from a poloidal field by the u-effect, which is differential rotation stretching out field lines (Fig. 7.6). Alpha-Omega Dynamo. An a-u dynamo relies on both the u-effect to produce toroidal flux from poloidal flux and the a-effect (caused by, e.g., Parker’s rising cyclonic eddies) (see Section 3.3.1) to produce poloidal flux from toroidal flux (Fig. 7.7). Large-Scale or Small-Scale Dynamos. If the generated magnetic field has a scale much larger than that of the driving flow, we have a large-scale dynamo, whereas if the field is produced on scales comparable to or smaller than that of the flow, it is a small-scale dynamo. A small-scale dynamo generating a disordered field seems to be

012 3

FIGURE 7.6 The u-effect, in which poloidal flux is generated from toroidal flux by differential rotation. Light arrows show the magnetic field and dark ones the sheared velocity.

FIGURE 7.7 The a-effect, in which rising, twisting motions create poloidal flux from toroidal flux.

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highly likely whenever the fluid is turbulent and the magnetic Reynolds number is sufficiently high. Several Dynamos? Sunspots and active regions are likely to be created by a large-scale dynamo operating (at least partly) near the tachocline at the base of the convection zone, by either a tachocline dynamo (Section 3.4) or a fluxtransport dynamo (Section 3.3). It is this large-scale dynamo and general dynamo properties that we shall mainly address in this chapter. The magnetic field of the quiet Sun outside active regions (called the magnetic carpet) consists of several parts. Some flux spreads from active regions by diffusion and meridional flow. Superimposed on it is a background field largely independent of the solar cycle and arising from two sources. The first is tiny bipolar ephemeral regions whose flux is rapidly reprocessed; it is perhaps generated in a second (smallscale) dynamo that operates throughout the convection zone or in a narrow shear layer just under the solar surface on, say, supergranular scales. The second source of the background flux is the inner network field that exists inside supergranules and could perhaps be generated by a third local dynamo operating at even smaller (say, granular) scales just below the surface.

3.2 A HISTORY OF DYNAMO IDEAS The momentous discovery of strong magnetic fields in sunspots was made in 1908 by Hale. This led Larmor to ask how a rotating body such as the Sun could become a magnet, but Cowling showed in his famous theorem that a simple solution to the puzzle is impossible: a steady axisymmetric field cannot be maintained by dynamo action. Although the basic properties of the Sun’s magnetic field had been known for decades (Section 3.1.1), by the 1950s, there was no answer in sight to Cowling’s conundrum. It was realized that toroidal flux can be generated from poloidal flux in a natural way by being stretched out by differential rotation (Fig. 7.6), which is called the u-effect (after the notation (u) for vorticity or the Sun’s angular velocity). First, Parker explained how isolated toroidal magnetic flux tubes rise by magnetic buoyancy through the convection zone to give sunspots where they break through the solar surface (Fig. 7.8). Then he made the major suggestion that poloidal flux can be regenerated by the collective effect on toroidal flux of many small-scale helical flows (Fig. 7.10). He used physical heuristic arguments to propose that these turbulent cyclonic flows can be modeled by a new term (V  (aB)) in the mean induction equation in addition to differential rotation, where a is constant. He also found dynamo wave solutions that make dynamo-generated magnetic field migrate toward the equator (Parker, 1955). Next, a more systematic mathematical formalism of Parker’s dynamo ideas was developed for a turbulent flow, and it was discovered that dynamo action is indeed possible if the small-scale flows are not reflectionally (or mirror) symmetric. For example, in a rotating stratified sphere, the Coriolis force on radial motions produces

3. Dynamo Theory

photosphere

sunspot

convection zone

FIGURE 7.8 A magnetic flux tube rises through the solar surface by magnetic buoyancy and creates a pair of sunspots.

a mean helicity hv$V  vi. The resulting induction equation (Eq. 7.1) for the mean poloidal magnetic field hBi takes the form vhBi ¼ V  ½hvi  hBi þ ahBi  V  ½ðh þ bÞV  hBi; vt in which h is the magnetic diffusivity, b is a turbulent magnetic diffusivity, and ahBi is the a-effect, which can regenerate the poloidal field. Babcock (1961) and Leighton (1969) proposed a totally different qualitative scenario for regenerating a poloidal field. Bipolar sunspots have a definite tilt on average (Joy’s law), and so, when pairs of sunspots eventually decay, they suggested that magnetic flux diffuses globally in such a way that the flux in the northern hemisphere has one dominant polarity whereas that in the southern hemisphere has the opposite polarity. In other words, new poloidal flux is created from the toroidal flux that forms sunspots. The 1970s saw the development of kinematic a-u models for the Sun and other stars. It was thought that almost all sufficiently rapid and complex motions could act as kinematic dynamos. In addition, saturation of the a-effect was incorporated as a result of feedback by the Lorentz force. During that decade, it was assumed that the solar dynamo is working throughout the convection zone. However, the 1980s dealt four major blows against classical a-u dynamo models, although the u-effect was still thought to generate the toroidal field: 1. The properties of emerging fields such as latitudes of emergence and tilts of bipoles require fields of order 104e105 G (1e10 T), but it was realized that this presents two difficulties: If generated in the convection zone, they rise too rapidly by magnetic buoyancy. Also, they are too strong for helical turbulence of the expected amplitude to act on them efficiently. 2. Doubts began to be expressed about the validity of mean-field theory and the derivation of the a-effect, and therefore the ability of turbulent diffusion and the a-effect to fulfil the expectations of the models.

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3. Numerical simulations of thermally driven magnetoconvection in a spherical rotating shell did not yield solar-like patterns of migrating solar activity. 4. The measurements of rotation (U) in the convection zone from helioseismology showed that it is constant with radius (dU/dr ¼ 0) through the convection zone down to the tachocline, so that it does not have the radial increase with depth (dU/dr < 0) required by convection-zone dynamo models. The tachocline is a narrow shear layer below the base of the convection zone at r ¼ 0.713 R1 , where the angular velocity rapidly changes from being uniform in the radiative interior to being roughly constant on cones in the convection zone. As a result of these difficulties, two distinct approaches were developed to build a large-scale dynamo theory responsible for active-region flux (Fig. 7.9). The first is a tachocline dynamo working near the tachocline at the base of the convection zone (Section 3.5). One possibility is an overshoot dynamo (Spiegel and Weiss, 1980), which locates both the a- and u-effects in the overshoot region just below the base of the tachocline, while another is an interface dynamo (Parker, 1993), which separates these effects spatially and places the u-effect below the interface with the a-effect above it. The second approach is a flux-transport dynamo (Section 3.4), which develops the earlier Babcock-Leighton idea by solving the axisymmetric kinematic dynamo equations with an imposed meridional flow and an u-effect focussed near the tachocline together with an a-effect at the solar surface (Dikpati and Choudhuri, 1994). The key new aspects are diffusion of active-region flux at the solar surface and meridional flow through the convection zone. The theory has been used in different ways for solar-cycle prediction by adopting different values for the magnetic diffusivity, differential rotation, meridional circulation, poloidal flux source and alpha quenching.

α P

P,T

ω

T ω

ω

α

P

α T P T

P,T

(A) Overshoot

(B) Interface

(C) Flux Transport

FIGURE 7.9 Dynamo generation by (a)overshoot, (b)interface and (c)flux-transport models, indicating where the a- and u-effects are located and where poloidal (P) and toroidal (T) components are generated. Curly, dashed and double arrows represent transport by a dynamo wave, meridional flow and buoyancy, respectively.

3. Dynamo Theory

In the tachocline dynamo, poloidal flux is generated down in the tachocline; in the flux-transport dynamo, it is near the solar surface.

3.3 EARLY TURBULENT DYNAMOS Cowling discovered that the simplest potential configuration for a dynamo will not work. His antidynamo theorem states that: A steady axisymmetric magnetic field cannot be maintained by dynamo action.

3.3.1 Parker’s (1955) Model Parker (1955) achieved a major breakthrough in 1955 by pointing out that turbulent motions inside the convection zone (which are by nature nonaxisymmetric) may be able to sustain the Sun’s poloidal field. His heuristic ideas were later changed to a more systematic basis by the Potsdam group (see Section 3.3.2). Differential rotation may stretch out poloidal flux to create a strong toroidal component (Fig. 7.10A). This may be demonstrated mathematically by writing v ¼ vfif þ vp and B¼BfifþBp, so that the f-component of the induction equation (Eq. 7.1) becomes
 
v vBf Bf 1 f þ Rðvp $VÞ ¼ RBp $V þ h V2  2 B f ; R vt R R in which the first term on the left gives the rate of change of the toroidal field, whereas the second represents its advection with a flow. On the right-hand side, the first term shows how a shear in angular velocity (vf/R) (i.e., a differential rotation) acting on a poloidal field (Bp) can enhance the toroidal flux. Such a stretching of field lines would continue until balanced by Ohmic diffusion (the second term on the right).

Merging of eddies ω-effect

α-effect

(A)

(B)

(C)

FIGURE 7.10 Parker’s dynamo model: (A) toroidal (T) flux is generated from poloidal (P) flux by differential rotation (the u-effect); (B) two helically rising blobs twist up the toroidal field; (C) the resulting closed loops of many cyclonic eddies merge to give new large-scale (dashed) poloidal flux (the a-effect).

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Parker (1955) realized that, because the Sun is rotating, rising blobs of plasma in the convection zone would tend to act like cyclones in the Earth’s atmosphere and so would rotate owing to the Coriolis effect. These helically upflowing blobs would twist the toroidal magnetic field and produce magnetic loops with a poloidal field component (as shown in Fig. 7.10B). The toroidal field has opposite signs in the two hemispheres, but the helical motions also have opposite senses (clockwise in the northern hemisphere and anticlockwise in the south), and so Parker realized that the resulting poloidal loops have the same sense in both hemispheres. Provided the loops can reconnect efficiently with one another, they may then coalesce to create a large-scale poloidal field, as shown in Fig. 7.10C by the broken curve. He developed a clever heuristic formalism as follows for the way such helical turbulence regenerates poloidal flux from toroidal flux. The rate of generation of poloidal flux (Bp) is proportional to toroidal flux (Bf), and so Parker modeled the effect of many convection cells by writing Bp ¼ V  (Apif) and adding an electric field: Ef ¼ aBf ;

(7.23)

to the poloidal component of the induction equation (Eq. 7.1), which therefore becomes
 vAp vp 1 2 þ $VðRAp Þ ¼ aBf þ h V  2 Ap . (7.24) R vt R The name a-effect comes from the constant of proportionality (a) in the mean electric field (Ef) over many eddies. It has units of velocity and is measures the mean rotational speed of eddies. Because there are also falling motions that rotate the field in the opposite direction, there needs to be some asymmetry between up and down motions to create a net effect. The main causes of such asymmetry are stratification (because rising plasma expands while falling plasma contracts) and geometry (because plasma tends to rise at the center of a cell and fall at its boundary). Other causes are magnetic buoyancy, which aids the rising motions, and hydromagnetic inertial waves.

3.3.2 Mean-Field Magnetohydrodynamics Theory Parker’s physical idea was that the net effect of averaging many small-scale convective motions is to produce the large-scale electric field (aBf) in Eq. (7.23) and so allow regeneration of the poloidal field. A more mathematical approach to this idea was developed by Steenbeck et al (1966), by considering a small-scale fluctuating turbulent motion (v) and magnetic field (b) on a scale (l), which are statistically steady and homogeneous but not mirror-symmetric. They are superimposed on a field (B0) and flow (v0), which possess a much larger scale-length (L). After performing averages (denoted by overbars) over a scale intermediate behV  B0 , in which e h is the turbulent tween l and L and assuming v  b ¼ aB0  e diffusivity (usually [ h), the mean induction equation becomes

3. Dynamo Theory

vB0 ¼ V  ðv0  B0 Þ þ V  ðaB0 Þ þ ðh þ e hÞV2 B0 ; vt so that the effect of the turbulence is to provide an extra electric field (aB0) and enhance large-scale diffusion through the term e hV2 B0 .    b þ vp and field B ¼ Bf f bþ For an axisymmetric flow v0 ¼ RU0 f   b , where U0 is the Sun’s angular velocity and a ¼ a0 cos q, the equations V  Ap f become

 vBf Bf 1 2 þ Rðvp $VÞ h V  2 Bf ; (7.25) ¼ RðBp $VÞU0 þ ½V  ðaBp Þf þ e R vt R
 vAp vp 1 þ $VðRAp Þ ¼ aBf þ e h V2  2 A p . (7.26) R vt R These dynamo equations have been solved in a sphere, and a range of a-u dynamos have been produced, depending on the value of the dynamo number h2 , which gives the ratio of field generation to dissipation. NeD ¼ a0 U00 R31 e During the 1970s, many authors produced more sophisticated kinematic a-u dynamos so as to reproduce detailed features of the solar cycle. One approach was to impose the spatial variation of a on physical grounds; another was to model feedback of the Lorentz force on the motions by assuming a variation of a with magnetic field strength. Various functional forms for a and U0 were adopted to model flux eruption by magnetic buoyancy or time delay. However, this approach eventually outran its usefulness and the main emphasis shifted to nonlinear dynamo models that incorporate the equation of motion, especially to those that are driven near the tachocline at the base of the convection zone.

3.4 FLUX-TRANSPORT DYNAMOS Observations of the photospheric magnetic field stimulated Babcock (1961) to develop a qualitative model for the solar cycle, in which the toroidal field is produced as usual by differential rotation, but in a thin layer just below the surface. For the poloidal field, however, he proposed a new idea based on the observation that preceding sunspots in bipolar groups are closer to the equator than following spots (Joy’s law). When bipolar regions decay, he suggested some of the following flux migrates to the poles, cancels the preexisting flux, and generates a new poloidal field of opposite sign. Leighton (1969) suggested instead that the migration of poloidal flux is caused by supergranular eddy diffusion. Although turbulent dynamo theory was the mainstream approach in the 1970s and 1980s, four later developments implied that a flux-transport model better accounts for solar cycle observations:

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1. A theory for the diffusion of the radial field (Br) over the solar surface from decaying active regions was developed that reproduces observed flux distributions. Br is treated as a passive scalar that does not feed back on the flow. 2. Evidence grew for the existence of a meridional flow at the solar surface of 15e20 m/s, which carries decaying remnants of active regions poleward more effectively than supergranular diffusion. 3. Choudhuri modeled the rise of magnetic flux tubes through the convection zone under the influence of magnetic buoyancy and Coriolis forces. They found that the field strength of an initial toroidal flux tube at the base of the convection zone needs to be at least as strong as 105 G for buoyancy to overpower the Coriolis effect and enable the tube to emerge at low latitudes rather than high ones. It also gives good agreement with observed tilts of emerging bipolar regions (Joy’s law). 4. Furthermore, thin flux tubes of 105 G stored in the overshoot layer would be unstable and rise through the convection zone in less than a year. A flux-transport dynamo model, first proposed by Wang and by Dikpati and Choudhuri (1994), explains the observed surface flux and avoids the problem with early dynamo models that the helical turbulence is too weak to generate poloidal flux from a toroidal field of 105 G. It incorporates two differences from the simple BabcockeLeighton picture: that the toroidal field is produced at the tachocline rather than just below the solar surface and that the transport of poloidal flux is by meridional flow as well as diffusion. In the flux-transport dynamo (Fig. 7.11), the a and u are widely separated in space. Poloidal flux is produced at the solar surface by the BabcockeLeighton decay of tilted active regions. Then it is transported partly by diffusion and partly by meridional flow, acting as a conveyor belt, to the poles and down below the surface to the tachocline, where it is sheared as usual into a toroidal field. This toroidal field rises by buoyancy and rotates, producing new poloidal flux at the surface in addition to the poloidal flux that is being swept up from the tachocline by meridional flow. Equations similar to the standard kinematic a-u dynamo are adopted, except that a is concentrated near the solar surface (the BabcockeLeighton effect). In spherical b þ vp is assumed, in which U(r, q) is polars, an axisymmetric flow v ¼ rsinq Uðr; qÞ f the solar angular velocity, chosen to have a strong gradient at the tachocline (from helioseismology), and vp is an assumed meridional circulation. Two kinds of flux-transport dynamo have been put forward. Dikpati adopts a low magnetic diffusivity (h ¼ 5  106m2/s) in the convection zone with a diffusion time across it of 200 years, and so flux is transported downward mainly by meridional flow on a timescale of 20 years. An additional a-effect at the base of the convection zone favors an antisymmetric dynamo (with a dipolar toroidal field satisfying the Hale polarity laws). Also, the strength of the poloidal field is assumed to be proportional to the sunspot area. The resulting toroidal field (of the next sunspot cycle) has

3. Dynamo Theory

+ +++ ++ ++ ++++++ ++ ++ + ++++++ +++++ ++++++ ++++ ++ +++ +++ +++ ++ +++ ++++ ++++++ ++++++ +++++ ++++++ ++ ++ Strong differential ++ ++ +++ rotation ++ ++ ++ ++ ++ ++ ++ ++ Babcock-Leighton ++ ++++ process ++++++ + +++ Magnetic buoyancy + + + + + + + + + + Meridional circulation +++ + FIGURE 7.11 Processes involved in a flux-transport dynamo. From Priest, E.R., 2014. Magnetohydrodynamics of the Sun. Cambridge University Press, Cambridge, UK, courtesy of Cambridge University Press.

a strength that depends on the nature of the two or three previous cycles and so how much poloidal flux is brought down to be reprocessed. By comparison, Chatterjee, Choudhuri, and Jiang adopt a much higher magnetic diffusivity (h ¼ 108 m2/s) comparable with the surface value and having a convection-zone diffusion time on the order of only 10 years. This strong diffusion connects field lines across the equator and produces a dipolar field without needing an a-effect at the base of the convection zone. It suggests that the poloidal field has a random nature independent of the cycle. The resulting strength of the toroidal field in cycle n þ 1 depends on the strength of the polar field at the end of cycle n, in agreement with observations. There are various other observational consequences of the models. The dynamo period is inversely proportional to the speed of the equatorward return meridional flow at the base of the convection zone (where the diffusivity is assumed to be much smaller than the convection zone value in all models); a value of a few m/s is needed to give an 11-year cycle. By contrast with interface or classical a-u dynamos, flux-transport models are insensitive to differential rotation, the a-effect, or the diffusivity. The timing of the reversal of the polar field (namely, at sunspot maximum) is the result of the strength of the polar meridional flow at the solar surface. Furthermore, active regions in the northern hemisphere are observed predominantly to have negative magnetic helicity, whereas those in the south have the

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opposite sign. This may be explained as a natural property of a flux-transport dynamo or, alternatively, in terms of the effect of helical turbulence on a rising tube. Possible nonlinear feedback mechanisms that limit the amplitude of the magnetic field and control the amplitude of the solar cycle are back-reaction of the magnetic field on differential rotation, tilt angles of sunspot groups, cross-equator transport of magnetic flux, and converging flows toward activity belts. Flux-transport models have many strengths: the attempt to reproduce observed solar cycle amplitudes; lack of reliance on a turbulent a-effect, and so avoidance of problems associated with it; polar migration of flux from active regions; reproduction of the phase relation between polar flux and sunspots; removal of the restrictions of classical dynamo theory that the dynamo number must be negative; and the shear velocity increase with depth, because a sufficiently strong meridional flow in a fluxtransport dynamo forces equatorward propagation of the toroidal field. However, like all kinematic models, the flows in flux-transport dynamos are not determined consistently. Moreover, the flux emergence source term S(r, q, Bf) is imposed from observations rather than being determined by the model. Currently, the behavior throughout the convection zone of the meridional flow (which is prescribed in the model) is unknown. A key question is whether the polar flux at the solar surface is indeed representative of the Sun’s poloidal flux (as flux-transport dynamos assume) or whether it is a secondary consequence of poloidal flux generation in the tachocline.

3.5 TACHOCLINE DYNAMOS Interest in the concept of a tachocline dynamo began with suggestions of an overshoot dynamo (Section 3.5.1), stimulated by helioseismology observations and by Parker’s interface dynamo model (Section 3.5.2). A fully developed tachocline model has not yet been produced, but a scenario has been widely discussed and the nature of the tachocline itself has been debated. Several physical processes are involved: 1. Magnetic flux in the convection zone is transported by convection; 2. it is amplified by small-scale dynamo action; 3. it is carried down to the tachocline and held there by strong overshooting plumes or turbulent pumping; 4. strong shear in the tachocline amplifies the toroidal field; 5. which then rises into the convection zone by magnetic buoyancy or turbulent pumping; 6. the a-effect in the convection zone creates poloidal flux; 7. and the weak field is recycled; 8. while the toroidal flux erupts through the solar surface.

3. Dynamo Theory

3.5.1 Overshoot Dynamo Parker questioned the idea that the dynamo works throughout the convection zone, by showing that magnetic buoyancy would remove magnetic field from the convection zone so quickly that it would not have time to be amplified by dynamo action. This led Spiegel and Weiss (1980) to suggest that a thin overshoot region with a thickness of, say, 10 Mm, just below the bottom of the convection zone, would be much more favorable for the operation of a dynamo, because it is convectively stable and so would suppress magnetic buoyancy and hold down the magnetic field there. Convective plumes from the overlying unstable layers would overshoot and penetrate into this layer, providing turbulent motions that may drive a dynamo. This idea was later given strong support from the discovery of the tachocline in helioseismology observations as a layer of strong differential rotation where a large toroidal field would naturally be generated. In the mid-1980s, the idea of an overshoot dynamo became popular.

3.5.2 Interface Dynamo (Parker, 1993) It now seems likely that the toroidal field at the base of the convection zone is indeed 105 G, so that the turbulent flow there would not be strong enough to twist up the magnetic field lines, and the corresponding a-effect would be switched off. This led Parker to propose an “interface” dynamo, with the main shear and a-effect in two distinct regions. The mean toroidal field is created by radial differential rotation and is stored in the tachocline. The mean poloidal field, on the other hand, is located in the lower part of the convection zone above the tachocline. It is created by an a-effect produced by magnetic buoyancy instability and is then pumped down into the tachocline by convection. In Parker (1993) analysis, the dynamo takes the form of a surface wave tied to the lower surface (z ¼ 0) of the convection zone. He sets up a two-layer model, with an a-effect in the region z > 0 and a uniform shear (S0 ¼ dvy/dz) in z < 0. In z < 0 (the tachocline), the diffusivity (ht) is much smaller than in the turbulent convection-zone (hcz). Above and below z ¼ 0, he sought plane-wave solutions and imposed boundary conditions at the interface, which determine a dispersion relation for the growth-rate (ur ) and frequency (ui ) of the dynamo wave h in terms of .the ratio i (ht/hcz) of magnetic diffusivities and a dynamo number Nd ¼ aS0

h2 kx3

.

The result, in the limit when ht/hcz
1, is
2 1 ðhi =hcz Þ2 Nd2 ¼ 64 þ ur ð1 þ ur Þur ; u2i ¼ ð1 þ ur Þur . 2 The scale 1/kzr of magnetic field decay from the interface in the convection zone is on the order of the horizontal wavelength 1/kx, which is much smaller than the scale for the tachocline field. Thus, the azimuthal field in z < 0 is confined to a thin layer pressed up against the underside of z ¼ 0. Good features are the natural ways in which the radial shear generates a large toroidal field and stores the

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field by the stable stratification below the interface. However, it is not clear how flux is transported to and from the tachocline and how the a-effect operates.

3.6 GLOBAL COMPUTATIONS To perform a global computation of dynamo action in the solar interior, one needs a realistic model of convection-zone fluid dynamics, including magnetoconvection over a wide range of scales, together with differential rotation, meridional circulation, and a tachocline. A start was made by Gilman (1977), who modeled Boussinesq convection in a rotating spherical shell heated uniformly from the base. The resulting differential rotation is driven by Reynolds stresses and possesses equatorial acceleration provided the angular velocity decreases with depth (du/dr > 0). Gilman and Miller (1981) then added a magnetic field, with turbulent diffusion coefficients but no a-effect, but these pioneering models could not reproduce the butterfly diagram. Compressible models were later developed by Juri Toomre and colleagues with the anelastic spherical harmonic code. Vigorous forcing produces mean flows and coherent vortical downflows that align with the rotation axis. The resulting angular velocity is similar to helioseismology observations. Browning et al (2007) included a simple tachocline with an underlying stable layer and found that a 3000-G axisymmetric toroidal field is created in the stable layer (in contrast to the fluctuating fields that occupy the convection zone). Later, with a global large-eddy anelastic simulation, Ghizaru et al (2010) was able to produce polarity reversals with decadal periods, whereas Brun et al (2011) coupled the radiative and convection zones and Nelson described the properties of buoyant magnetic loops. Thus, full MHD global computations are able to resolve supergranulation and generate reasonable behavior for differential rotation and meridional circulation, as well as a turbulent a-effect and reversals of the magnetic fields.

3.7 CONCLUDING REMARKS Huge progress has been made in dynamo theory toward providing an explanation for the existence of the Sun’s magnetic field and its variation with the solar cycle. Nevertheless, many key problems remain to be overcome: 1. The tachocline itself needs to be understood much better. What is the detailed nature of the instabilities at work there and how is the tachocline coupled to the solar interior and convection zone? How is the observed surface meridional flow closed in the convection zone? 2. More realistic models are required for the tachocline dynamo and the fluxtransport dynamo. Which of these two approaches describes best what is actually responsible in the Sun for sunspots and active regions? Is poloidal flux carried down to the tachocline by flux pumping or by meridional flow?

3. Dynamo Theory

3.

4.

5.

6.

7.

8.

9.

Does the a-effect take place most effectively in the tachocline or near the solar surface? The rise of magnetic tubes through the convection zone and flux emergence through the solar surface need to be modeled more realistically. So far, we have simple models for the rise of thin flux tubes across the convection zone and initial attempts to model numerically the emergence through the surface. Mean-field MHD has represented a highly stimulating line of attack, but the a-u dynamo equations have not been rigorously justified for the Sun or other stars, because the conditions under which they hold do not apply. In future, a deeper theory may evolve from next-generation computations, and then perhaps some effect resembling the a-effect and its quenching will indeed be justified. Computations of an a-effect from numerical MHD turbulence experiments show that a is negligible even when small-scale dynamo action is present and a is not necessarily related to helicity. There are also difficulties with turbulent diffusion: the value of e h estimated from observed active-region dispersal is typically 109 m2/s, but in the future, will we be able to predict its value and functional form rigorously and its variation across the convection zone? It is hoped that the nature of convection over a wider range of scales and the way in which differential rotation and meridional flow are driven will be fully understood in future from massive global computations. This will lead to much better global dynamo models. Is the magnetic field in a filamentary state, and how does it interact with convection? What is the effect of flux pumping and how is it parametrized? Developing these elements of a better theory for the solar cycle should enable a more confident prediction of the strengths and dates of future solar maxima and minima. In addition to a tachocline dynamo, the details of other dynamos at work in the Sun should be worked out. Is there a dynamo that gives birth to ephemeral active regions, and where is it operating (throughout the convection zone or near the surface)? Is there a separate dynamo creating a smaller-scale inner network field and driven by subsurface shear? To what extent are dynamos on other stars scaled-up versions of what is happening on the Sun, and to what extent are they novel dynamos operating with qualitatively different physics? It is clear that dynamo theory remains one of the most highly developed and interesting branches of solar MHD. Certainly, it will remain a lively field of great interest in future, with great advances expected from helioseismology, computational modeling, and comparison with stellar activity.

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REFERENCES Babcock, H.W., 1961. The topology of the Sun’s magnetic field and the 22-year cycle. Astrophys. J. 133, 572e587. Berger, M.A., 1984. Rigorous new limits on magnetic helicity dissipation in the solar corona. Geophys. Astrophys. Fluid Dyn. 30, 79. Browning, M.K., Brun, A.S., Miesch, M.S., Toomre, J., 2007. Dynamo action in simulations of penetrative solar convection with an imposed tachocline. Astron. Nachr. 328, 1100e1103. Brun, A.S., Miesch, M.S., Toomre, J., 2011. Modeling the dynamical coupling of solar convection with the radiative interior. Astrophys. J. 742 (79), 79. De´moulin, P., Henoux, J.C., Priest, E.R., et al., 1996. Quasi-separatrix layers in solar flares. I. Method. Astron. Astrophys. 308, 643. Dikpati, M., Choudhuri, A.R., 1994. The evolution of the Sun’s poloidal field. Astron. Astrophys. 291, 975. Ghizaru, M., Charbonneau, P., Smolarkiewicz, P.K., 2010. Magnetic cycles in global largeeddy simulations of solar convection. Astrophys. J. Letts. 715, L133eL137. Gilman, P.A., 1977. Nonlinear dynamics of Boussinesq convection in a deep rotating spherical shell. I. Geophys. Astrophys. Fluid Dyn. 8, 93e135. Gilman, P.A., Miller, J., 1981. Dynamically consistent nonlinear dynamos driven by convection in a rotating spherical shell. Astrophys. J. Suppl. 246, 211e238. Leighton, R.B., 1969. A magneto-kinematic model of the solar cycle. Astrophys. J. 156, 1e26. Mackay, D., Yeates, A., 2012. The Sun’s global photospheric and coronal magnetic fields: observations and models. Living Rev. Sol. Phys. 9, 6. Parker, E.N., 1955. Hydromagnetic dynamo models. Astrophys. J. 122, 293. Parker, E.N., 1993. A solar dynamo surface wave at the interface between convection and nonuniform rotation. Astrophys. J. 408, 707. Parnell, C.E., Haynes, A.L., Galsgaard, K., 2010. The structure of magnetic separators and separator reconnection. J. Geophys. Res. 115, 2102. Priest, E.R., 2014. Magnetohydrodynamics of the Sun. Cambridge University Press, Cambridge, UK. Priest, E.R., De´moulin, P., 1995. 3D reconnection without null points. J. Geophys. Res. 100, 23443e23463. Priest, E.R., Forbes, T.G., 2000. Magnetic Reconnection: MHD Theory and Applications. Cambridge University Press, Cambridge, UK. Priest, E.R., Longcope, D.W., 2017. Flux-rope twist in eruptive flares and CMEs: due to zipper and main-phase reconnection. Sol. Phys. 292, 25. Spiegel, E.A., Weiss, N.O., 1980. Magnetic activity and variations in solar luminosity. Nature 287, 616. Steenbeck, M., Krause, F., Ra¨dler, K.H., 1966. Berechnung der mittleren lorentz-feldsta¨rke vb fu¨r ein elektrisch leitendendes medium in turbulenter, durch coriolis-kra¨fte beeinflubter bewegung. Z. Naturforsch. 21a, 369.

CHAPTER

Solar and Stellar Variability

8 Marianne Faurobert

University of Nice-Sophia Antipolis, Lagrange Laboratory, Nice, France

CHAPTER OUTLINE 1. Introduction .......................................................................................................267 2. Magnetic Activity of the Sun and Stars ................................................................269 2.1 Solar Magnetic Activity ........................................................................ 269 2.1.1 Large Scale ................................................................................ 270 2.1.2 Small Scale ................................................................................ 271 2.2 Magnetic Activity of Cool Stars ............................................................. 272 2.2.1 Observational Methods ................................................................ 272 2.2.2 Relations Among Activity, Rotation, and Rossby Number ................. 275 2.2.3 Relation Between Rotation Period and Age..................................... 278 2.2.4 Relation Between Activity and Age ................................................ 279 2.2.5 Activity of Solar Analogues ........................................................... 281 3. Irradiance Variations ..........................................................................................283 3.1 Solar Irradiance Variability ................................................................... 284 3.2 Solar Irradiance Reconstruction............................................................ 286 3.3 Stellar Irradiance Variability ................................................................. 291 4. Concluding Remarks...........................................................................................294 References .............................................................................................................295

1. INTRODUCTION If we disregard the long-term structural changes of the Sun and stars along the main sequence, their variability is mainly driven by the presence of magnetic fields originating from a dynamo action and dynamically coupled to the stellar plasma. Despite decades of effort, there is still no universally accepted model or truly predictive theory for the operation of the global solar/stellar dynamo. The connection between solar and stellar variability thus presents considerable interest to allow progress in an understanding of the relevant mechanisms. Because the Sun may be observed in much greater detail, it is widely used as a guide to modeling stellar variability. However, because magnetic activity appears in a variety of stars with different observational characteristics, they provide us with a valuable means of testing the dynamo models by varying parameters such as the age, mass, and rotation period. The Sun as a Guide to Stellar Physics. https://doi.org/10.1016/B978-0-12-814334-6.00010-8 Copyright © 2019 Elsevier Inc. All rights reserved.

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The Sun is observed continuously in great detail from ground-based telescopes and dedicated spacecrafts such as the Solar and Heliospheric Observatory1 (SOHO) or Solar Dynamic Observatory2 (SDO). Many beautiful movies stored in the galleries of their websites show how variable the Sun is on timescales of a few days to hours or even minutes. Actually, longer timescales are also welldocumented in the databases of these instruments that now contain more than 20 years of observations. The Hinode satellite3 has been providing us with detailed measurement of the solar photospheric magnetic fields at small scales with its Solar Optical Telescope (SOT) together with x-ray images and extreme UV (EUV) spectra of the corona over one solar cycle. The Sun observed as a star has also been monitored continuously from space since 1978 by various missions aimed at recording its total and spectral irradiance variations. Ground-based data allow us to go back to longer timescales. Synoptic daily photographic images of the solar photosphere and spectroheliographic images of the chromosphere in the Ca II H and K lines started more than a century ago in several observatories worldwide and are ongoing. Even longer timescales, several centuries, are reachable through archives of hand drawings from telescopic observations performed by professional astronomers. Even longer-term solar variations may be derived from indirect proxies relying on modulation of the flux of cosmic rays on the Earth by solar activity (Usoskin, 2017). As far as stars are concerned, observations of stellar magnetic cycles by systematic surveys of magnetic proxies were undertaken in the 1960s. The Mount Wilson survey of chromospheric Ca II H and K emission in operation between 1967 and 2003 yielded a wealth of fundamental discoveries on stellar magnetism (see the review by Baliunas et al., 1995). Magnetic fields on stellar surfaces may also be detected in some cases by their Zeeman effect on the intensity profiles and the polarization of spectral lines. Zeeman Doppler imaging (ZDI) techniques, based on time series of spectropolarimetric observations of rotating stars, are aimed at reconstructing the large-scale magnetic field topology of a star. ZDI surveys have brought many unexpected discoveries, such as the detection of magnetic fields in massive B and O stars without a convective envelop (see the review by Donati and Landstreet, 2009). Finally, important advances have been made in exploring stellar variability thanks to sensitive photometers onboard the Convection, Rotation, Transits Plane´taires (COROT) (Baglin et al., 2006) and Kepler (Borucki et al., 2010) missions. Aimed at detecting faint exoplanetary transits or stellar oscillations, they have also allowed a discovery of the variability of various classes of stars, in particular solar-type stars in the visible spectral domain. Apart from the study of solar and stellar magnetism, interest in characterizing and understanding stellar variability has grown up because it is an important source

1

https://sohowww.nascom.nasa.gov/. https://sdo.gsfc.nasa.gov/. 3 http://global.jaxa.jp/projects/sat/solar_b/. 2

2. Magnetic Activity of the Sun and Stars

of systematic errors and uncertainties in characterizing exoplanets. The problem of disentangling planetary and stellar activity features in transit light curves has motivated renewed collaborations among solar and stellar physicists. Finally, methods developed to reconstruct variations in the total and spectral irradiance of the Sun, which have a major impact on the Earth’s atmosphere and climate, are generalized to other stars to explore the conditions of the habitability of exoplanets. In this chapter, the broad subject of solar and stellar variability is presented in two parts. The first part is devoted to the magnetic activity of the Sun and stars; the second focuses on irradiance variations. Because the dynamo mechanisms presented in the previous chapter require the star to have a convective envelope with a shear layer, we will restrict this presentation to solar-type stars: namely, stars of spectral types F, G, and K. However, magnetic fields have also been detected in some hot B and O stars and in more and more M stars, even though they are fully convective. Short-term variability owing to instabilities of magnetic structures leading to flares, filament disruptions, or coronal mass ejections are related to the space weather issue that will be discussed in a further chapter. Thus, here we will concentrate on variability at timescales on the order of the rotation period up to the largest known timescales.

2. MAGNETIC ACTIVITY OF THE SUN AND STARS 2.1 SOLAR MAGNETIC ACTIVITY The most striking features of solar activity such as sunspots, plages, and filaments are observed at scales on the order of several tens of arcseconds up to a fraction of the solar radius. They are associated with the large-scale magnetic field of the Sun. The problem of understanding how large-scale coherent structures with a remarkable organization are generated by dynamo action driven by turbulent flows at high conductivity is still challenging. The a-effect (originating either from helical flow and/or the BabcockeLeighton flux-transport process) and the U-effect (shearing of the magnetic field by differential rotation) are believed to cause this dynamo on the Sun. This scenario relies on the mean field theory (MFT) presented in the previous chapter. A detailed discussion of the different dynamo theories and numerical simulations may also be found in Brun and Browning (Brun and Browning, 2017). Apart from large-scale magnetic structures, another type of magnetic field exists at small scales and even in the quiet phase of the solar cycle. This field is intermittent and has mixed polarity even at the resolution limits of current instruments. Thanks to the increase in the resolving power of telescopes and to the advent of space-based instruments, sub-arcsecond resolution observations of solar magnetic fields have become possible. They showed that the magnetic flux due to small-scale quiet Sun magnetic fields is three orders of magnitude larger than the magnetic flux of active regions and that the solar magnetic flux follow a power-law distribution

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over more than 5 decades (Parnell et al., 2009). Therefore, small-scale magnetic fields are potentially important for understanding many unsolved problems in solar physics, particularly solar atmospheric heating. The origin of the small-scale fields is still under debate but it seems more and more clear that a small-scale turbulent dynamo is operating in the convection zone and at the solar surface. Small-scale dynamos are ubiquitous in a broad range of turbulent flows, found in solar and stellar objects, galaxies, accretion disks, etc. The Sun offers a unique possibility of observing in detail how the turbulent small-scale dynamo is operating.

2.1.1 Large Scale Magnetic structures observed in magnetograms at the photospheric level are associated with intensity enhancements observed in emission lines formed at higher altitudes, in the chromosphere, transition region, and corona. Observations of the emission in the core of the Ca II H and K lines have had a historical role, as explained previously. These spectral features are formed in the chromosphere and are the signature of nonthermal heating, most probably of magnetic origin, as their good correlation with magnetic structures seen in the photosphere strongly suggests. This correlation is illustrated in Fig 8.1, which shows a magnetogram obtained onboard the SDO with the Helioseismic and Magnetic Imager (HMI) instrument together with images of the Sun in the coronal emission line of FeIX at 17.1 nm taken the same day with Atmospheric Imaging Assembly (AIA) and in the Ca II K3 line with the spectroheliograph at Meudon Observatory. We can observe the chromospheric and coronal bright features lying above the magnetic regions seen at the photospheric layer.

FIGURE 8.1 Left panel: Full-disk magnetogram from Helioseismic and Magnetic Imager onboard the Solar Dynamic Observatory on May 19, 2014 at maximum of solar cycle 24. Middle panel: Atmospheric Imaging Assembly image at 17.1 nm, Right panel: Ca II K3 spectroheliogram from Meudon Observatory. The three images were obtained on the same date. Source: Images downloaded from the databases https://sdo.gsfc.nasa.gov/ and http://bass2000.obspm.fr/search.php.

2. Magnetic Activity of the Sun and Stars

The Ca II K image in Fig. 8.1 also shows a network of bright cells with a typical size of 30 arcseconds covering the full solar disk. At the photospheric level, the cell boundaries are formed from a collection of small-scale magnetic patches carrying kilogauss magnetic fields that accumulates at the boundaries of the supergranulation pattern. Studies on the formation and evolution of the network were carried out using long sequences of high-resolution magnetograms and photospheric images obtained with the SOT/Narrowband Filter Imager instrument onboard the Hinode satellite (Gosic et al., 2014). They show that processes of advection and diffusion of small-scale magnetic patches that emerge inside the supergranular cells and are transported to their boundaries contribute significantly to the formation of the network.

2.1.2 Small Scale Magnetograms based on measurement of the Zeeman circular polarization in spectral lines give access to the magnetic flux density over the surface of the pixel. They are thus blind to fields with mixed polarities on smaller scales. The Hanle effect provides us with alternative diagnostics that do not cancel out in the presence of mixed-polarity fields. Observation of this effect in the quiet Sun revealed that magnetic fields on the order of approximately 100 Gauss are indeed pervading the photosphere of the quiet Sun (see the review by Stenflo, 2013). To obtain some insight into the small-scale component of solar magnetism, one needs to extract the magnetic field vector from spectropolarimetric measurements. This requires recording the line profiles of the four Stokes parameters and performing an inversion to infer the magnetic field components. The inference procedure is always based on a parametric model used for forward modeling of the profiles (see Chapter 5 of this book). Inversions of high-resolution spectropolarimetric observations of the Zeeman effect in the quiet Sun have been achieved using a MilneeEddington model in which it is assumed that the pixel surface is partially filled with a depth-independent magnetic field and that the remaining part is magnetic free (Orozco Sua´rez et al., 2010). This schematic model is probably far from the real Sun, in which the depth gradient of the magnetic field is likely and in which several magnetic components may occupy the pixel surface (Lo´pez Ariste and Sainz Dalda, 2012). Furthermore, the polarimetric signal is low over most of the quiet-Sun surface. A meaningful inversion cannot be performed at pixels where the signal is dominated by the instrumental noise, so only a small fraction of the quiet Sun could be investigated (for a critical discussion of Stokes inversions, see Stenflo, 2013). So contradictory results have been obtained regarding the statistical properties of the quiet Sun magnetic field, such as its angular distribution and the probability density function of the magnetic strength. Work by Lites et al. (2017) reexamined SOT data avoiding inversions but using the symmetry properties of the linear polarization profiles. Two populations of the small-scale magnetic field could be clearly identified, one of which shows vertical fields with the strongest polarization signals and covers a small fraction of the solar

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surface; the other one is formed by horizontal fields with weaker polarization signals that cover a large fraction of the solar surface. This component could be responsible for the Hanle effect that has been detected in the quiet Sun. The Sunrise program, a high-altitude, long-duration balloon experiment equipped with the IMaX imaging polarimeter (Martı´nez Pillet et al., 2011), explored solar magnetic phenomena at very high angular resolution, a factor of two better than the SOT telescope (0.15 arcseconds). Several programs were devoted to studying the quiet Sun magnetism. One of the many important results of the experiment is the discovery by Martı´nez Gonza´lez et al. (2012) of the continuous emergence of small-scale magnetic loops at scales between 100 and 1000 km that do not appear uniformly on the solar surface. These observations set constraints on the mechanism that give rise to these coherent small-scale structures. Possible interactions between the small-scale and global dynamo have been investigated through long-term studies of the internetwork magnetic fields along the 11-year cycle. Jin et al. (2011) and Jin and Wang (2015) used the SOHO/Michelson Doppler Imager (MDI) followed by SDO/HMI full-disk magnetograms and then Hinode magnetograms to analyze the weakest polarization signals at smaller scales. They found that the weakest flux elements, between 1016 maxwell (Mx) and 1018 Mx, do not show time variations, whereas the number and total flux of small-scale magnetic elements in the range between 1018 and 30
1018 Mx have cyclic variation anticorrelated with the sunspot number, and finally, the flux elements with flux larger than 30
1018 Mx are well-correlated with the sunspot cycle. This correlation is likely a result of the decay of sunspots that is an additional source for the strongest flux elements. Faurobert and Ricort (2015) studied the Fourier power spectra of the magneticflux spatial distribution in the internetwork at scales between 5 and 0.3 arcseconds. One advantage of Fourier analysis is that the power spectrum of the noise can be derived from the polarization measured in the continuum and removed from the observations. The power spectrum of the magnetic-flux distribution at scales between 5 and 0.3 inches in the internetwork was found to vary in anticorrelation with the sunspot cycle. Other investigations by Lites et al. (2014) and Buehler et al. (2013) using SOT spectropolarimetric measurements found no significant variations of the weakest polarization signals in the internetwork during the rising phase of solar cycle 24. Thus, this subject is still debated.

2.2 MAGNETIC ACTIVITY OF COOL STARS 2.2.1 Observational Methods 2.2.1.1 Proxies of Chromospheric and Coronal Activity The most commonly used methods for assessing the existence of stellar activity cycles have been synoptic stellar chromospheric activity observations and, in some

2. Magnetic Activity of the Sun and Stars

cases, stellar x-ray variability data. These proxies have been established thanks to observations and calibration on solar data. The Ca II HþK flux normalized by the bolometric flux is denoted by RHK, the chromospheric contribution by R0HK. From detailed measurements of the excess flux in the Ca II H and K lines in spectroscopic data and magnetic flux from simultaneous solar magnetograms, Schrijver et al. (1989) showed that R0HK is roughly proportional to the square root of the mean magnetic flux density at the surface of the Sun. However, a linear correlation between the radiance of the Ca II K line and the magnetic flux density in the network elements was derived by Skumanich et al. (1975). Activity cycles akin to the 11-year sunspot cycle have been seen for many latetype stars in their Ca II HþK line emission since Wilson (1978) initiated their systematic study. The rotation period could also be inferred from short-term modulations of the activity index owing to the presence of spots on the stellar surface. Fig. 8.2 shows some examples of R0HK time variations for some stars of the Mount Wilson HK survey. For solar and late-type stars, the x-ray flux of the coronal plasma peaks in the 0.2e2.0 keV. Solar observations show a clear relation between the magnetic flux F in solar features at the photospheric level and the coronal x-ray radiance LX. From x-ray bright points, active regions and the integrated solar disk a power-law LX xFp with P ¼ 1.13  0.05 was found to apply to 12 decades of flux (Pevtsov et al., 2003). In the case of unresolved stars, this relation provides us with another activity proxy.

FIGURE 8.2 Time variation of the activity index of the Sun and some stars of the Mount Wilson survey. (A) The Sun. Bottom panels. (B) A star with a simple cycle; (C): A star with complex cycles. Adapted from Ola´h, K., KTva´ri, Z., Petrovay, K., Soon, W., Baliunas, S., Kolla´th, Z., Vida, K., 2016. Magnetic cycles at different ages of stars. Astron. Astrophys. 590, A133. https://doi.org/10.1051/0004-6361/ 201628479. 1604.06701.

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2.2.1.2 Light Curves Modulation of the integrated visible light curve of stars owing to the presence of different magnetic features, such as spots or plages, can be recorded with unprecedented precision from space thanks to missions such as COROT and Kepler. Kepler has continuously monitored 2 105 stars for 4 years. Stellar activity has been detected and surface rotation periods could be measured for thousands of stars (Garcı´a et al., 2014). Photospheric activity proxies have been defined from the analysis of the light curve fluctuations (Mathur et al., 2014). Because variability in the light curves may have different origins and timescales (magnetic activity, convection, oscillations, and companion), the rotation period has to be taken into account in calculating a photometric magnetic proxy. The standard deviation of the whole light curve Sph and the average value of the standard deviation on k rotation periods for values of k between 1 and 30 were tested by Mathur et al. (2014) in the solar case. They used photometric data obtained by the VIRGO photometer (SPM) instrument onboard SOHO during 6000 days of observations (cycle 23 and the start of cycle 24). The Sph,k are shown to correlate well with the sunspot-based S index. In most of the following studies, Sph is identified as . The reliability of this index depends on the actual latitudinal distribution of the starspots and on the inclination angle of the star; Sph would underestimate the activity of solar-like stars with high inclination angles. Another crucial interest of the Kepler data is the measurement of acoustic oscillations in hundreds of solar-like stars. The constraints provided by asteroseismic data allows one to increase significantly the accuracy of fundamental stellar parameters such as mass, radius, effective temperature, and age, which can be derived from stellar modeling. In the near future, even tighter constraints on stellar parameters will be obtained from astrometric observations collected by the Gaia satellite (Perryman et al., 2001). This is important as far as an understanding of stellar activity mechanisms is concerned.

2.2.1.3 Zeeman Doppler Imaging In addition to these global observations of integrated stellar signals, new surveys aimed at obtaining information about the geometry of the magnetic field have been undertaken. They rely on the Zeeman effect in spectral lines, which is the only direct source of information about the strengths and topologies of stellar magnetic fields. ZDI, first introduced by Semel (1989) and Donati et al. (1989), is a powerful tomographic technique to map the large-scale distribution of stellar magnetic fields. It often uses a set of circular polarization profiles collected over one or several rotations and converts these profiles into a magnetic map of the stellar photosphere. The BCool Collaboration4 has undertaken a ZDI survey of 170 cool stars to

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2. Magnetic Activity of the Sun and Stars

determine how the field geometry and properties depend on fundamental stellar parameters; magnetic fields were detected on 67 of them (Marsden et al., 2014). Although the circular polarization signal in individual lines is within the observational noise multiline techniques (least square deconvolution) Donati et al. (1997), or multiline singular value decomposition, Carroll et al. (2012) allow to detect a polarization signature. However, most of the time the linear polarization signal remains small and hardly detectable. Tomographic inversion of the signal uses of decomposition of the magnetic structures on spherical harmonics. Only the large-scale, i.e., low-order terms with l < 6 may be recovered. A test of the method is presented in Kochukhov et al. (2017), in which simulated Sun-as-a-star spectropolarimetric observations are reconstructed from SDO/HMI solar magnetograms and photospheric images for one solar Carrington rotation. Then they are analyzed with the ZDI techniques to recover the large-scale magnetic map. Comparison of the maps of radial magnetic field given by the ZDI techniques and the original HMI data after filtering its high-order harmonic components shows qualitatively similar configurations. A quantitative analysis of the longitudinal magnetic flux in the detailed solar magnetograms shows that 95% of the flux is already contained in the low-harmonic modes with l < 6. However, less than 1% of the total magnetic energy is present in the large-scale reconstruction. The magnetic energy content at a small scale has a major role that is not recovered by ZDI based on circular polarization profiles and the mean magnetic strength is underestimated. Another difficulty of ZDI inversions is that the temperature distribution caused by spots on the stellar surface must be disentangled from the magnetic field distribution, so the magnetic reconstruction is improved if both I and V profiles are used in the inversion procedure. The average magnetic strength is also better recovered when the information on the transverse component contained in the linear polarization is available.

2.2.1.4 Interferometric Images Other techniques of direct imaging are thus welcome to confirm ZDI reconstructions. Interferometric images of the K giant star z Andromedae were obtained by Roettenbacher et al. (2016) with the six-telescope CHARA/Michigan Infrared Beam Combiner Array from observation runs in 2011 and 2013. They show cool areas on the surface, in particular a clear polar spot and lower latitude features that evolved between the two dates of observations. This technique opens up a new way for direct detection of the distribution of active regions on stellar surfaces.

2.2.2 Relations Among Activity, Rotation, and Rossby Number The aeU dynamo mechanism depends on the combined effects of differential rotation and convection. These two effects are compared by using the Rossby number, Ro ¼ Prot/sconv, in which Prot denotes the rotation period and sconv the convective turnover time. This characteristic time for the convection cannot be measured

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directly. It is derived from mixing-length models of stellar convection zones. The Rossby number defines the relative importance of the Coriolis and inertial forces in a flow. Smaller Ro values indicate greater influence of rotation upon convection. The solar value is around 2. The Mount Wilson survey observed about 100 F, G, and K stars over 36 years. About 40% of the selected stars exhibit cycles with amplitudes ranging between 35% and 10% (roughly the amplitude of the Sun’s cycle using the same measure of the disk-integrated Ca K). About 20% of the stars show no variation whereas the remaining 40% have irregular variations. When the RHK activity index is plotted against an index of their temperature, very
few stars with a (BeV) color index larger than 0.6 are found in the range of log R0HK values between 4.8 and 4.6. This gap is named for its discovers, the VaughanePreston gap. Using the Mount Wilson survey of 41 main-sequence active stars, Noyes et al. (1984) showed that the time-averaged value < R0HK > of active cool stars is actually proportional to the inverse Rossby number. However, for large values of sconv/Prot,
the star’s activity saturates at log < R0HK > x  4:2. This may be interpreted in terms of starspots filling the entire stellar surface or as saturation of the dynamo mechanism itself. For a sample of BCool stars of known mass and rotational periods, it was found that stars with a Rossby number larger than 1 have poloidal magnetic fields, whereas those with a Rossby number smaller than 1 can generate strong toroidal fields (Donati and Landstreet, 2009; See et al., 2016). Using the best available data of the Mount Wilson survey, Brandenburg et al. (1998) studied the relation between the ratio ucycl/Urot and the rotational period (ucycl ¼ 2p/Pcycl). They found a “distinct segregation of active and inactive stars into two approximately parallel bands.” The relation between cycle and rotation periods was further reexamined by Bo¨hm-Vitense (2007). She showed that two distinct sequences also appear in a Pcycl  Prot diagram. On each sequence, the ratio Pcycl/Prot is constant but with decreasing values from the active to the inactive sequence. The more active are faster-rotating, younger stars, whereas the less active ones are slowly rotating and older. However, some stars exhibit two identified activity cycles with different periods. For these stars, the shorter cycle is found on the inactive sequence, whereas the longer one is on the active sequence (Fig. 8.3). However, new surveys based on Kepler measurements of photospheric index variations show somewhat different trends. In work by Reinhold et al. (2017), light curves from 23,601 stars over 4 years of observation were analyzed. Cycles were detected in 3203 stars of this sample, with periods between 0.5 and 6 years and rotational periods between 1 and 40 days. They do not confirm the existence of two distinct sequences in the Prot  Pcyc diagram but find some trend for star gathering along the inactive branch for rotational periods between 5 and 40 days (see Fig 8.3). Because the Kepler survey does not allow detection cycle periods larger than 6 years, it is still not clear whether this result is caused by a selection effect, because a large fraction of stars on the active branch have cycle periods larger than 6 years.

FIGURE 8.3 Upper panel: Magnetic cycle period versus rotation period in solar-like stars of the Mount Wilson survey. The figure shows two sequences: a sequence of active (young) stars and sequence of (older) less active stars. Dotted vertical lines represent stars with two identified cycles. Zeeman Doppler imaging stars are plotted according to their magnetic properties. Red/blue symbols indicate poloidal/toroidal fields and polygon/star shapes indicate axisymmetric/non-axisymmetric fields. Symbol size corresponds to magnetic field energy. Lower panel: Distribution in the Prot  Pcycl diagram of 3203 active stars observed by Kepler. The straight lines show active (A), inactive (I), and short-cycle (S) branches. Numbers at the top of the figure give the numbers of stars of the sample in each column. Adapted from See, V., Jardine, M., Vidotto, A.A., Donati, J.F., Boro Saikia, S., Bouvier, J., Fares, R., Folsom, C.P., Gregory, S.G., Hussain, G., Jeffers, S.V., Marsden, S.C., Morin, J., Moutou, C., do Nascimento, J.D., Petit, P.,Waite, I.A., 2016. The connection between stellar activity cycles and magnetic field topology. MNRAS 462, 4442e4450. https://doi.org/10.1093/mnras/stw2010. 1610.03737 and Reinhold, T., Cameron, R.H., Gizon, L., 2017. Evidence for photometric activity cycles in 3203 Kepler stars. Astron. Astrophys. 603, A52. https:// doi.org/10.1051/0004-6361/201730599. 1705.03312.

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2.2.3 Relation Between Rotation Period and Age Young solar-like stars undergo important structural changes when they evolve toward the main sequence. They evolve from fully convective objects to stars with a radiative core. Their rotational velocity also changes under different physical mechanisms; on the pre-main sequence stage it is constrained by the interaction with the circumstellar disk. When the star decouples from the disk, it contracts and its rotation velocity increases. Afterward, its magnetized wind dissipates angular momentum and the star spins down. In the following, we focus on the relation between rotation and age for cool stars on the main sequence. One observes a broad distribution of initial stellar-spin periods at early ages that narrows by later times and then funnels down to a tight band at the age of the Hyades cluster (600 million years [Myr]). The reason for this convergence effect is that magnetic braking has a strong rotation rate dependence (it scales like the third power of the rotation velocity), so fast rotators are more severely braked down than slower ones. This causes the wide range of initial rotation rates to converge to a nearly unique value at ages greater than x500 Myr. A discussion on this funneling effect may be found in van Saders et al. (2018). The rotational evolution of solar-type stars older than 500 Myr was studied by Skumanich (1972). Analyzing late-type main sequence stars in the Hyades, Pleiades, and Ursa Major clusters of the Mount Wilson survey, he found that both the emission excess in Ca II lines and the rotational velocity decrease with age following the same power law in t0.5. For cool stars of higher or lower mass than the solar one, a generalization of the Skumanich power law was proposed in the form of a relation among the age, the rotational period, and the turnover time sconv of the convection in their convective envelope. This modified relation accounts for the observed trend that at a given age more massive cool stars have faster rotation than solar-mass stars. It compares well with the observed rotation periods in star clusters of the age of the Sun, as shown in Barnes et al. (2016). However, using the age and rotational periods derived by seismic methods for some Kepler stars, Metcalfe et al. (2016) showed that some stars older than the Sun have larger rotation velocity than expected from the modified Skumanich law. The rotationeage relation was then revisited through studies based on large samples of field stars observed by Kepler. In work by van Saders et al. (2018), a sample of 34,000 Kepler stars was used to analyze rotational period distribution as a function of effective temperature. The observed distribution showed a sharp upper edge at long periods that cannot be explained by standard braking models. Semiempirical models of the rotational period distribution taking into account theoretical models of stellar rotation and a stellar population model for the galaxy indicate that the observed distribution at long periods could be due to a “Rossby edge” at Ro ¼ 2.08 (that is approximately at the age of the Sun for solar-mass stars) above which long-period, high-Rossby number stars are either absent or undetected. This suggests that at higher Rossby numbers, either the magnetic braking weakens or the rotation period is undetected because stars undergo a transition with reduced

2. Magnetic Activity of the Sun and Stars

surface activity. Rotation periods inferred from asteroseismology would clarify the origin of this upper edge.

2.2.4 Relation Between Activity and Age Ola´h et al. (2016) analyzed the time variations of 29 active FeK stars of the Mount Wilson survey with time-frequency methods to extract multiple and changing cycles in the data set. Applying their analysis to the sunspot-time series from 1818 to present, they detected four varying cycles on four different timescales; the longest is the Gleissberg cycle (80e90 years) and then the Schwabe cycle (11 years) is wellrecovered, together with its half and one-third harmonic components. Mixed with the one-third component, an independent cycle that is also variable is found to vary between 3 and 4 years. This cycle may be related to the quasibiennial oscillation derived from previous studies. Among the 29 cool stars of the data set, 12 with large rotational periods exhibit simple cyclic behavior; the remaining ones with much shorter periods show complex and sometimes changing multiple cycles. Applying gyrochronology to derive the star ages, it was shown that there is a clear age division between the youngest stars with complex multiple cycles and short rotational periods and the ones with smooth cycles and longer rotational periods. The age separation between the two groups is around 2.2 billion years (Gyr). This corresponds to the VaughanePreston gap in chromospheric activity. Thus, it seems that there is a change in the dynamo around this age inducing a change in the activityerotation relationship (see the review by Fabbian et al., 2017). The results are presented in Fig. 8.4, in which the cycle periods normalized by the rotational period, the chromospheric activity indices, and the rotational periods are plotted against the gyrochronologically derived age. The age separation between complex and simple activity cycles is clearly visible, together with a change in the power law relation between activity index and age for stars younger or older than 2.2 Gyr. The position of the Sun in the diagrams follows the old-star regime well. Other surveys have been used to study the relation between activity and age on the main sequence. The x-ray flux of active stars in stellar clusters younger than 1 Gyr was analyzed by Jackson et al. (2012). They found that for very young stars below 100 Myr, the x-ray luminosity saturates, and that in the unsaturated regime there is a decrease in x-ray luminosity with age, with a power law index of 1.2. More recently, Booth et al. (2017) performed a similar study but using asteroseismic ages of stars older than 1 Gyr. They found a decrease in x-ray luminosity with age but with a steeper index of 2.8  0.72. Large surveys devoted to young-star magnetic properties were carried out with ZDI techniques. The Towards Understanding the Spin Evolution of Stars project focused on older pre-main sequence and young main sequence stars in known open clusters for an accurate age determination. The BCool project studied mainly older field stars. These surveys found a clear power law trend of decrease of the large-scale radial magnetic field with stellar rotation period and age

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FIGURE 8.4 From upper to lower panels: Logarithm of the activity indices, logarithm of the cycle period normalized by the rotational period, and rotational period as a function of age. Blue squares show stars with complex cycles, red dots show stars with simple cycle, and the position of the Sun is shown by its usual symbol. Adapted from Ola´h, K., KTva´ri, Z., Petrovay, K., Soon, W., Baliunas, S., Kolla´th, Z., Vida, K., 2016. Magnetic cycles at different ages of stars. Astron. Astrophys. 590, A133. https://doi:org/10.1051/0004-6361/ 201628479. 1604.06701.

2. Magnetic Activity of the Sun and Stars

(Roettenbacher et al., 2016). This decrease could lead to a loss of braking efficiency of magnetized stellar winds.

2.2.5 Activity of Solar Analogues In the Pcycl  Prot diagram derived from the HK survey, the Sun seems to be at an intermediate position between the two activity sequences. This raises questions about the validity of the Sun as a representative of the magnetic activity of solarlike stars (see Metcalfe et al., 2016) and has motivated various studies focused on solar analogs. Solar analogues are defined as stars with a mass between 0.9 and 1.1 solar mass, a chemical composition within 10% of the solar one, and similar temperature and bolometric luminosity as the Sun. Salabert et al. (2016) included the detection of solar-like oscillations as an selection criterion. Identification of solar analogues depends on the accuracy of the estimated fundamental stellar parameters. By combining asteroseismology with complementary high-resolution spectroscopy, Salabert et al. (2016) identified 18 new solar analogues with rotation periods between 10 and 40 days in stars observed by Kepler. They studied the properties of the photospheric and chromospheric activity of these stars in relation to the Sun. The photospheric activity proxy Sph was derived from Kepler observations collected over 1422 days (from Jun. 2009 to May 2013). The chromospheric activity proxy was measured with follow-up ground-based spectroscopic observations in 2014 and 2015 with the HERMES spectrograph on the Mercator telescope (La Palma). Fig. 8.5A shows their results regarding the relation between the photospheric activity proxy Sph and the rotational period. The solar values of Sph at maximum and minimum have been obtained from VIRGO/SPM measurements for cycle 23. This figure shows that the photospheric activity level of most of the stars in the sample (15 of 18) fall within the range of activity between the minimum and maximum of the solar cycle. A good correlation was found between the chromospheric and photospheric proxies for the stars of the sample. A selection of solar-analog stars from the Mount Wilson survey was also used for comparison. The selection was performed using the same criterion on the star effective temperature as for Kepler stars: namely, 5520 K < Teff < 6030 K. The star effective temperatures were derived by using the published B  V color index of the Tycho-2 photometry catalog obtained from HIPPARCOS. This allowed the selection of 28 solar-analogues in the Mount Wilson survey. Fig. 8.5B shows the relation between the chromospheric activity index R0HK and the rotation period for both samples and for the Sun. The Kepler and Mount Wilson samples merge well for stars with rotation periods between 10 and 40 days, where the activity level seems flat. The Sun does not occupy a peculiar position in this diagram. Its chromospheric activity level follows the overall trend of solar analogs of comparable rotational periods. The Mount Wilson stars show higher activity for rotation periods shorter than 10 days (we notice the similarity with the results shown in Fig. 8.4 for a different sample of cool stars), but the Kepler sample lacks stars in this short-period range.

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FIGURE 8.5 Upper panel: Photospheric magnetic index Sph (in parts per million [ppm]) as a function of the rotational period Prot (in days) of the 18 solar analogues observed with Kepler. The horizontal dashed lines show the solar mean activity level at maximum and minimum of 0 as a function of the the 11-year cycle. Lower panel: Chromospheric magnetic index RHK rotational period for the 18 solar-analogues in Kepler stars (black dots) and for 28 solaranalogues in the Mount Wilson survey (red squares). The solar position is represented by its usual symbol. Adapted from Salabert, D., Garcı´a, R.A., Beck, P.G., Egeland, R., Palle´, P.L., Mathur, S., Metcalfe, T.S., do Nascimento Jr., J.D., Ceillier, T., Andersen, M.F., Trivino Hage, A., 2016. Photospheric and chromospheric magnetic activity of seismic solar analogs. Observational inputs on the solar-stellar connection from Kepler and Hermes. Astron. Astrophys. 596, A31. https://doi.org/10.1051/0004e6361/201628583. 1608.01489.

3. Irradiance Variations

The Kepler sample of solar analogs contains only two stars with well-observed cyclic activity variations, KIC3241581 and KIC 10644253, over the 1422 days of Kepler measurements, with periods of 394 and 582 days, respectively. The first one has a rotation period comparable to that of the Sun and the 1.1-year cycle period could be analogous to the quasibiennial oscillation observed in solar activity proxies. The second one, with a rotation period of 11 days, falls on the inactive branch of the Pcycl  Prot diagram. In the Mount Wilson sample, the solar analogues with rotational periods similar to the Sun, between 23 and 30 days, are flat-activity stars without a cycle. Thus, it is still unclear whether the Sun is in an intermediate regime of dynamo as far as its cyclic variations are concerned.

3. IRRADIANCE VARIATIONS The magnetic variability of solar-type stars affects their radiative output. This has been well-measured for the Sun over the last four solar cycles and similar measurement have also become available for stars owing to accurate photometric programs on COROT and Kepler and to modern ground-based photometric and spectroscopic surveys. In this section, we present how the modeling of solar irradiance variations may serve as a guide to interpret stellar observations. First, we present in Fig 8.6 the

FIGURE 8.6 Comparison of the quiet-Sun solar spectral irradiance [SSI] computed from semiempirical model 1401 of Fontenla et al. (2015) with the composite observations of Thuillier et al. (2003). The spectral resolution is 1 nm. Adapted from Fontenla, J.M., Stancil, P.C., Landi, E., 2015. Solar spectral irradiance, solar activity, and the near-Ultra-violet. Astrophys. J. 809, 157. https://doi.org/10.1088/0004-637X/809/2/157.

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solar spectral irradiance (SSI) measured at the solar minimum when no magnetic features were present on the solar surface. It was obtained by merging measurements performed by different instruments in different spectral domains (see Thuillier et al., 2003). The figure also shows the computed spectrum derived from a semiempirical model of the quiet Sun by Fontenla et al. (2015). We will come back to irradiance modeling techniques in the following discussion.

3.1 SOLAR IRRADIANCE VARIABILITY The total solar irradiance (TSI) is defined as the integral over wavelength of the solar radiative flux measured outside the terrestrial atmosphere and normalized to a distance of 1 astronomical unit. The major part is emitted in the visible and in the infrared by the photosphere. TSI has been measured continuously from space since 1978, with the required accuracy to detect its small variations with the 11-year solar cycle. TSI composites obtained by merging the data sets from various instruments over four cycles show that it varies in phase with the solar cycle with a relative variation of about 0.1%. However, the solar flux is also variable on shorter and longer timescales. In the range around 5 mn, solar p-mode oscillations are detected in the solar continuum brightness; on longer timescales up to some hours, the main contribution to the irradiance variability is granular convection at the solar surface. The amplitude of this variability is small (on the order of 0.01%) because a large number of granules are present at any time on the solar surface. On the timescale of a solar rotation, the variability is higher (several tenths of a percent) because of the effects of sunspots and faculae. The most striking cycle in solar irradiance variability is the 11-year cycle corresponding to the Schwabe cycle of sunspots, as illustrated in Fig 8.7A taken from Yeo et al. (2014). The amplitude of variations on the order of 0.1% in phase with the sunspot cycle results from the network and faculae positive contributions that overcome the sunspots’ negative ones. Significant uncertainties persist over the long-term stability of the instruments in space, causing discrepancies between the TSI composites derived by the different groups (The Active Cavity Radiometer Irradiance Monitoring, Willson and Mordvinov (2003), the Institut Royal Me´te´orologique de Belgique, Mekaoui and Dewitte (2008), the Physikalisch-Meteorologisches Observatorium Davos and World Radiation Center, Fro¨hlich (2006)), and on a possible secular trend of the irradiance. The secular trend may be defined as a variation of the solar-cycle average of the irradiance or as a variation of the solar irradiance at the minimum of the cycle. However, current solar irradiance records are too short and uncertain to reveal and quantify secular changes unambiguously. Spectrally resolved observations of the solar irradiance also started in the 1970s. These observations were fragmented in time and wavelength until the launch of the Solar Radiation and Climate Experiment spacecraft in 2003 (Sparn et al., 2005). Continuous monitoring of the solar spectrum is thus even more recent than TSI monitoring. In the UV and EUV domains, important variations, up to more than a factor of two, are recorded in phase with the 11-year cycle. This contributes little

3. Irradiance Variations

FIGURE 8.7 Upper panel (A): Composite records of the total solar irradiance (TSI) from Active Cavity Radiometer Irradiance Monitoring (ACRIM) (red curve), Institut Royal Me´te´orologique de Belgique (IRMB) (green curve), and VIRGO (blue curve). The black curve shows the Spectral and Total Irradiance Reconstruction for the Satellite Era reconstruction. Vertical dashed lines mark the position of the solar minima. The measurements have been smoothened with a 181-day boxcar filter. Lower panel (B): First European Comprehensive Solar Irradiance Data Exploitation composite record of the spectral irradiance at 200.5 nm (in black) along with the individual data sets that are used to construct it. Spectral resolution is 1 nm. Adapted from Yeo, K.L., Krivova, N.A., Solanki, S.K., 2014. Solar cycle variation in solar irradiance. Space Sci. Rev. 186, 137e167. https://doi.org/10.1007/s11214-014-0061-7. 1407.4249 and Haberreiter, M., Scho¨ll, M., Dudok de Wit, T., Kretzschmar, M., Misios, S., Tourpali, K., Schmutz, W., 2017. A new observational solar irradiance composite. J. Geophys. Res. 122, 5910e5930. https://doi.org/10.1002/2016JA023492.

to the TSI variations that are mainly driven by the visible and infrared spectral domains, but UV and EUV radiation (in particular below 300 nm) has a direct influence on the upper atmosphere of the Earth, causing an increase in stratospheric ozone and related warming together with indirect dynamical effects at lower stratospheric levels that finally affects the surface climate through stratosphereetroposphere

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coupling (see the review by Haigh, 2007). Like for the TSI, composite SSI records are derived by different groups to merge observations performed at different times with different instruments. Different attempts to reconstructed SSI time series correctly reproduce recent SSI observations but differ as far as past variations on decadal timescales are concerned. Haberreiter et al. (2017), within the framework of the SOLID international collaboration (First European Comprehensive Solar Irradiance Data Exploitation5), built a composite from the data sets of 20 instruments, using a probabilistic approach to assign a weight to the different data sets according to their respective uncertainties. One result of this reconstruction is shown in Fig 8.7B for the spectral irradiance at 200.5 nm. This figure illustrates the difficulty of merging the different measurements of various spaceborne instruments. There are no direct measurements of solar irradiance on longer time series. However, the sunspot number has been recorded from the beginning of the 17th century and shows that the amplitude of the cycles varies with a period of 70e100 years (first identified by Gleissberg in 1939). Another indirect proxy is provided by geomagnetic activity records. Even longer-term solar variations may be derived from indirect proxies, such as the production of 14C and 10Be isotopes modulated by effect of the solar activity on the flux of cosmic rays on the Earth and on meteorites (see the review by Usoskin, 2017). Fig. 8.8 shows the sunspot number per decade reconstructed for the Holocene period based on the measurements of 14C in ring trees and 10Be in polar ice. We recover in particular the Maunder and Spoerer grand minima (denoted by M and S, respectively, in the figure) and the modern grand maximum of activity, a long sequence of strong cycles that the Sun experienced in the second half of the 20th century. Over a long term, the Sun spends about 70% of its time at moderate magnetic activity levels, about 15%e20% of its time in a grand minimum, and about 10%e15% in a grand maximum. A periodicity analysis yields several peaks in the range of periods between 80 and 150 years, corresponding to the frequency band of the Gleissberg cycle. Some other weak quasiperiodicities have been suggested; in particular, it seems that grand minima and grand maxima tend to cluster around highs and lows of the 2000- to 2400-year cycle called the Hallstatt cycle. Grand minima are believed to correspond to a special state of the dynamo. The commonly used MFT does not explain such incidences of low activity on extended periods, whereas dynamo models including a stochastic driver predict the intermittency of solar activity but without long-term cycles.

3.2 SOLAR IRRADIANCE RECONSTRUCTION Understanding the relation between magnetic activity and the irradiance is an active research field. Efforts made to model TSI variations currently rely on the hypothesis

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3. Irradiance Variations

FIGURE 8.8 Decadal reconstructed sunspot activity for the Holocene period. Blue circles and red stars denote grand minima and maxima, respectively. S and M letters denote the Spoerer and Maunder minima, respectively. Adapted from Usoskin, I.G., 2017. A history of solar activity over millennia. Living Rev. Sol. Phys. 14, 3. https://doi.org/10.1007/s41116-017-0006-9.

that the main drivers of solar variability are magnetic features at the solar surface and that the thermodynamical state of the quiet-Sun atmosphere is invariant. Other plausible mechanisms implying modulation of the global structure of the Sun have been proposed. Global structural changes are supported by observation of the frequencies of the acoustic p-modes that also show a well-documented solarcycle variation. The frequency shifts can be interpreted as arising from structural changes in the subsurface layers, such as changes in temperature (Kuhn et al., 1988) or the size of the acoustic cavity (Dziembowski and Goode, 2005). Changes with the solar cycle of the solar subsurface stratification were inferred by Lefebvre and Kosovichev (2005) from inversion of SOHO/MDI f-mode frequencies. A variation in the solar radius in antiphase with the solar cycle was derived from these measurements, with a lower limit of 2 km for the peak-to-peak amplitude of the variations. An upper limit of 15 km was inferred by Bush et al. (2010) from observations of one solar-cycle SOHO/MDI. Global magnetohydrodynamic (MHD) simulations of the solar convection zone achieving cyclic large-scale axisymmetric magnetic fields with polarity reversals on

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decadal timescales are presented in Cossette et al. (2013). In these simulations, the thermal convective luminosity varies in phase with the magnetic cycle. However, the simulation domain stops below the photosphere, which limits the type of comparisons that could be made with the real Sun. Theoretical investigations based on MHD simulations of the properties of the granulation in quiet solar regions in different magnetic environments were presented in Criscuoli (2013). An ambient magnetic field gives rise to two main physical effects: it inhibits convection and reduces the average opacity. The two effects have opposite consequences on the temperature gradients: The reduced opacity, which dominates at optical depths close to unity and smaller, reduces the temperature gradient; the inhibited convection, which dominates at optical depths larger than one, steepens the temperature gradient. In internetwork regions, simulations show that the temperature in the low photosphere decreases with an increase in the local magnetic field. Observational signatures of these effects on the center-to-limb variation of the continuum radiation are hardly detectable, however (for a tentative detection see Faurobert et al., 2016). Possible structural changes in the solar photosphere are likely to affect the irradiance variations on the 11 yearecycle timescale only marginally. Irradiance reconstructions based on the effect of magnetic concentrations at the solar surface actually enable the recovery of more than 90% of the observed variations on this timescale. Qualitatively, one can say that the influence of magnetic fields on the local temperature in magnetic features such as sunspots, faculae, or network patches varies strongly with the size of the magnetic region. The interior of a magnetic concentration in pressure balance with its surrounding is evacuated; the lower density creates a depression in the optical depth-unity surface. Thus, the intensity contrast in the continuum is influenced by the competing effects of magnetic suppression of convection and radiative heating from the surrounding photosphere through the walls of the depression. Sunspots and pores are dark because of the magnetic suppression of convection. However, faculae are formed by a collection of small-scale concentrations (or flux tubes) in which the optical-depth depression and lateral heating have an important role. Faculae seen far from disk center appear brighter than their surrounding as their bright side walls come into greater view. This explains why bright faculae are observed near the limb in white light. The upper layers of the photosphere within the magnetic concentrations are heated by mechanical and resistive dissipation, which enhances the intensity of spectral lines in magnetic concentrations. These qualitative arguments have to be quantified by precise computations of the radiative output of the features. There are two broad categories of solar irradiance models with different modeling approaches: “proxy” and “semiempirical.” Proxy models aim to reconstruct solar irradiance by the multivariate regression of indices of solar activity to measured solar irradiance. The activity indices serve as proxies of the effects of bright and dark magnetic structures on the radiative output of the Sun. However, this approach relies on observations of past decades (after the 1950s), which are well-covered by direct observations of solar, terrestrial, and heliospheric

3. Irradiance Variations

parameters, but they correspond to a very high level of solar activity totally dominated by the 11-year cycle without a noticeable long-term trend. Accordingly, all empirical relations based on data for this period are focused on 11-year variability and can overlook possible long-term trends. As an example, Usoskin (2017) reports on an attempt to reconstruct cosmic-ray intensity since 1610 from sunspot numbers using a (nonlinear) regression. The regression between the count rate of a neutron monitor and sunspot numbers established for the past 50 years is highly significant. Based on that, if one extrapolates the regression back in time to produce a reconstruction of cosmic-ray intensity to 1560, all cosmic-ray maxima essentially lie at the same level, from the Maunder minimum to modern times, in contradiction to what is known from other independent sources of information such as 14C isotopes measured in tree rings. This example shows that a reconstruction based on regression may miss important features when it is extrapolated far beyond the period on which it is based. A more physical modeling is derived from semiempirical, one-dimensional (1D) models of the different types of features or components (sunspot umbra and penumbra, faculae, network, and quiet Sun), together with observational records of their surface and location on the solar disk. There are five models for deriving the solar spectrum from the far-UV to the infrared (see the review by Ermolli et al., 2013): Naval Research Laboratory Solar Spectral Irradiance, Spectral and Total Irradiance Reconstruction for the Satellite Era (SATIRE-S), Solar Radiation Physical Modeling (SRPM), Observatorio Astronomico di Roma (OAR), and Code for Solar Irradiance (COSI). The reconstructions do not differ strongly in the cycle variability of TSI but they do not agree in the magnitude of the secular trend. The estimates of the increase between the Maunder minimum and the modern grand maximum differ by a factor of four to five. Thus, the TSI variations on centennial timescales, which are an important constraint for the terrestrial climate, are a strongly debated subject. Many subtle and contradictory effects come into play in the physical mechanisms that control the radiative output of the different magnetic features. Therefore, detailed radiative transfer computations of the emergent spectra are mandatory together with as realistic as possible models of the structures. 1D semiempirical models of the quiet and active Sun components of the atmosphere are extensively used. They are derived by solving equations that describe the structure and the detailed transport of radiation through the model atmosphere, reproducing the observed spectra at moderate spatial and temporal resolution (a few arcseconds and several p-mode oscillation periods). The temperature stratification is adjusted as a function of gas pressure in such a way that the computed intensities match the observed disk center intensities and center-to-limb variation in the visible and infrared. Fontenla et al. (2006) used limb darkening in the range 0.3e2.4 mm, as well as the absolute intensities and details of the spectral continua and lines in this range observed with the Precision Solar Photometric Telescope operated at Mauna Loa Solar Observatory by the High Altitude Observatory. They computed the emergent radiation from the models in full detail using atomic data from the National Institute of Standards and Technology (13,000 lines used) and molecular data from the High-Resolution Transmission

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Molecular Absorption Database (480,000 molecular lines used). Because knowledge of some of the physical processes is incomplete, it is necessary to use ad hoc parameters, e.g., to describe the effects of turbulence. The line blanketing effect caused by the absorption by thousands of spectral lines in the UV and visible domains was included in the TSI reconstruction performed by Unruh et al. (1999) followed by Shapiro et al. (2015) (Fabbian et al., 2017). Their calculations proved that the varying line blanketing has a major role in the TSI variations. The strength of spectral lines in the brightness spectrum of the Sun is known to change during the activity cycle since the pioneering Kitt Peak solar line-strength monitoring program by Livingston and Holweger (1982) measured a decrease of the equivalent-width of some spectral lines at solar maximum. Unruh et al. (1999) showed that their varying contribution is the dominant factor producing TSI variations on solar-cycle timescales, whereas changes in the continuum level contribute negligibly. More recently, Shapiro et al. (2015) found that molecular and atomic lines have an important role in all timescales from the solar rotational period to centuries. The solar brightness variations were calculated employing the solar disc area coverage of magnetic features deduced from the MDI/SOHO observations and the brightness contrasts of magnetic features relative to the quiet Sun were derived from a 1D nonelocal thermodynamic equilibrium (non-LTE) radiative transfer code as functions of disc position and wavelength. It is noticeable that in these calculations, the continuum contrast of faculae in the middle and near UV, between 200 and 400 nm, becomes negative (see Fig 8.1 of Shapiro et al., 2015). This results from the temperature structure of faculae in deep photospheric layers where the temperature is lower than in the quiet Sun according to the 1D semiempirical models used. The faculae are nevertheless brighter than the quiet Sun because the line blanketing is more important in the quiet Sun. In particular, the model indicates that roughly a quarter of the total solar irradiance variability over the 11-year cycle originates in molecular lines. The maximum of the absolute spectral brightness variability on timescales greater than a day is associated with the CN violet system between 380 and 390 nm. The reason is that the dominant part of the irradiance variability on the activity cycle originates in the 250- to 450-nm spectral domain, which is especially affected by molecular lines. However, the exact behavior of continuum variability strongly depends on the temperature structures of the quiet Sun and magnetic features in the deepest photospheric layers. These layers provide a small contribution to the emergent line spectra, so that it is difficult to constrain their temperature structures reliably in 1D semiempirical modeling, especially taking into account uncertainties in the measurements of the infrared solar irradiance that emanates from the deep photospheric layers. Yeo et al. (2017) made a step forward in TSI modeling by using realistic 3D MHD simulations of solar magnetic concentrations instead of 1D semiempirical models. The absolute level is calibrated to the TSI record from the total irradiance monitor (Kopp et al., 2004). The 3D models of faculae were obtained from three MHD simulations with the MURaM code (Rempel, 2014) initiated with a uniform magnetic field of 100, 200, and 300 G, respectively. An additional simulation

3. Irradiance Variations

performed with no magnetic field represents the quiet Sun. Synthetic bolometric images and magnetograms are then computed for each snapshot using radiative transfer codes including the effect of spectral lines. The relation between bolometric intensity at different disk positions and the magnetic flux over the faculae is then derived without calibration to observed TSI variations. HMI observations are then segmented into the quiet Sun, faculae, and sunspots and each image pixel on the solar disk is assigned the appropriate bolometric intensity by its feature type, distance to disk-center, and, in the case of faculae, the magnetogram signal. The model replicates 95% of observed variability between Apr. 2010 and Jul. 2016 over the timescales examined (days to years). This shows that TSI variability may be well-explained as a direct consequence of diurnal variation in surface coverage by faculae and sunspots. The ability to recover the TSI cyclic variations from these reconstructions based on MHD modeling of the magnetic concentrations is also proof that the radiation deficit or excess of surface magnetic features does not affect their surroundings at the photospheric level, but that energy blocked by sunspots is redistributed below in the convection zone. The convection zone acts as energy storage owing to its high thermal conductivity and slow thermal relaxation. Until now, fully consistent 3D MHD simulations of the solar magnetic concentrations are not able to model the chromospheric and coronal structures properly, so radiative output in the UV and EUV domains and their cyclic variations have not yet been accessible to these simulations. This has had a negligible impact on TSI, which mainly arises from the photospheric layers. The modeling of the UV and EUV spectrum, which has an important role in the climate of the Earth and on the habitability conditions of exoplanets, is performed mainly using semiempirical models. The irradiance variability on a centennial timescale is often described as a secular change between solar minima conditions. The magnitude of the variations and the specific physical mechanisms responsible for a possible centennial SSI variability are heavily debated (see Solanki et al., 2013). For example, in addition to strong concentrated magnetic fields, there is a weak turbulent magnetic field on the solar surface. Its effect on the rotational and 11-year irradiance variability is believed to be small, but it is unclear whether it could contribute to irradiance variability on centennial timescales.

3.3 STELLAR IRRADIANCE VARIABILITY In parallel to the Mount Wilson survey in the chromospheric Ca II H and K lines several other ground-based facilities have been running for decades gathering photometric data (Lockwood et al., 2007). Important results of these programs are recalled in Giampapa (2016b). The first is that the amplitude of the long-term brightness variations in the visible ubvy photometric bandpass is correlated with chromospheric activity and that compared with other cool stars, the Sun has somewhat too low brightness variations in the visible for its activity level. Furthermore, more active stars are darker at activity maximum and brighter at minimum, i.e., they show the

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opposite phase with activity compared with the Sun. Observations of photometric variability in solar-type stars (Giampapa, 2016a) indicate that the transition between direct and inverse correlation occurs in the range R0HK x  4:7 to  4:6, i.e., where the Vaughan & Preston gap in chromospheric activity occurs, possibly indicating a change in dynamo mode (see Section 1.2.2.2 of this Chapter). Generally speaking, the interplay between facular contrast, spot contrast, and fractional area coverage of surface features causes variations in stellar spectra, which depend on the particular wavelength and timescale being analyzed. Because the strength of the effect caused by line blanketing on irradiance variability depends on the strength of the lines themselves, the line-blanketing effect on the variability of stars with different elemental abundances (stars with lower abundances having weaker lines) and effective temperatures (cooler stars having more and stronger spectral lines) must be modeled by detailed radiative transfer calculations in a similar way as for the Sun. The cycle modulation in the far UV irradiance must be included in the development of exoplanet atmospheric models. The search for exoplanets and extraterrestrial life tends to focus on M-dwarf hosts. The low mass and luminosity of M dwarfs make these stars favorable targets for the discovery and characterization of exoplanets in their habitable zones. However, many M dwarfs are magnetically active, so their intrinsic variability may complicate the interpretation of light curves, and the variability of the stellar UV and far UV is an important issue to access the habitability conditions of a planet orbiting close to its host star. Semiempirical modeling of the temperature versus pressure distribution in the atmosphere of the M2 dwarf GJ 832 was presented by Fontenla et al. (2016). This M dwarf hosts a 0.64 MJup exoplanet with a relatively long orbital period at a mean distance from the star of a ¼ 3.56  0.28 au. Wittenmyer et al. (2014) discovered that this star also hosts a super-Earth at a mean distance, a ¼ 0.162  0.017 au, that is just inside the inner edge of the habitable zone. Similar to the Sun, the spectra of active M dwarfs show enhanced emissions in the X and EUV domains and UV emission lines that are the signature of hot chromospheres and coronae. High-resolution UV spectroscopy provides a wide range of diagnostics for computing semiempirical models of M dwarfs from the top of the photosphere through the chromosphere and into the corona that match the observed spectrum and can predict the unobserved portions of the stellar spectrum in the far UV. The Measurements of the UV Spectral Characteristics of Low Mass Exoplanetary Systems observing program (France et al., 2013) on the Hubble Space Telescope obtained the 115- to 314-nm spectra of six weakly active M dwarfs that are hosting exoplanets, including GJ 832. The stellar spectra are computed in full non-LTE using SolareStellar Radiation Physical Modeling tools. The model includes a total of 435,986 spectral lines produced by atoms and ions. Also included are the 20 most-abundant diatomic molecules and about 2 million molecular lines. We refer to Fontenla et al. (2016) and references therein for a detailed description of the techniques used for the computations.

3. Irradiance Variations

The thermal structure of the atmospheric model of GJ 832 is shown in Fig 8.9. Quantitatively, it is very different from the Sun. As a result, the spectral features of GJ 832 have fluxes different from those of the Sun. GJ 832 is nearly 104 times fainter at 200 nm but comparable to the quiet Sun in the 91- to 130-nm spectral region and much brighter than the quiet Sun in the x-ray and EUV bands (0e91 nm). This has important implications for the habitability of its close super-Earth planet.

FIGURE 8.9 (A): Semiempirical model of the M dwarf GJ 832 atmosphere compared with the quiet Sun model. (B): Comparison of the computed solar spectral irradiance below 200 nm for quiet Sun, plage regions and GJ 832. Adapted from Fontenla, J.M., Linsky, J.L., Witbrod, J., France, K., Buccino, A., Mauas, P., Vieytes, M., Walkowicz, L.M., 2016. Semi-empirical modeling of the photosphere, chromosphere, transition region, and corona of the Mdwarf host star GJ 832. Astrophys. J. 830, 154. https://doi.org/10.3847/0004-637X/830/2/154.

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4. CONCLUDING REMARKS Since the Mount Wilson HeK survey, it has been known that many cool stars show activity cycles, but COROT and Kepler photometric data, ZDI, and interferometric imaging of stellar surfaces are now providing us with powerful new tools for investigating stellar activity within a much broader context than the solar case. However, we are still at the beginning of these new observational programs and we miss the long-term view on stellar activity. The empirical relation between the rotational period and the star age, the socalled Skumanich law, has been extended to cool stars with lower and higher mass than the Sun by introducing the turnover time of convection as a parameter. However, the analysis of rotation-period distribution in large samples of Kepler stars indicates that stars may undergo a transition in magnetic activity with less efficient magnetic braking at Rossby numbers larger than 2, which is approximately at the age of the Sun for solar-mass stars. A relation between the activity level and age seems well-established. Young active stars, below 2.2 Gyr, rotate faster and tend to show irregular or multiple cycles with a higher level of activity than older stars that show a single dominant cycle (on the 36 years of the Mount Wilson survey). There is also a clear change in the power law relating the age (or rotation period) and the chromospheric activity index at the age of 2.2 Gyr. The x-ray luminosity also decreases with age and ZDI studies show a clear trend of decreases of the large-scale radial magnetic field with stellar rotation period and age that could be at the origin of a loss of the braking efficiency of magnetized stellar winds. The relation between the period of the activity cycle and the rotational period was reexamined by analyzing Kepler light curves for thousands of active stars. It shows a linear trend for rotational periods larger than 5 days corresponding to the inactive branch of the historical diagram from Bo¨hm-Vitense (2007) but with a large scatter, and its active branch is not recovered. Because a large fraction of stars on the active branch have cycle periods longer than the duration of the Kepler survey, one needs longer-term observations to come to a conclusion regarding this issue. In the previous historical diagram, the Sun seems to occupy an intermediate position between the two activity branches. The study of solar analogues in Kepler stars does not show evidence of peculiar behavior of solar activity as far as the activity level is concerned. But here again, the 4-year Kepler survey does not allow us to measure cycle periods in the range of the solar 11-year cycle, so the question of a possible “transitory” dynamo mode operating in the Sun is still open. The effect of magnetic activity on the total and spectral irradiance of the Sun and stars is a hot subject. For the case of the Sun, TSI variations on timescales of days to the 11-year cycle are well-explained by reconstructions using 3D MHD models of magnetic structures appearing at the solar surface and non-LTE radiative transfer with huge numbers of atomic and molecular spectral lines. Longer-term variations are still a matter of debate. It is still not clear from observations whether there is a secular trend of the TSI; however, this is an important issue for modeling the

References

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Lockwood, G.W., Skiff, B.A., Henry, G.W., et al., 2007. Patterns of photometric and chromospheric variation among sun-like stars: a 20 year perspective. Astrophys. J. 171, 260e303. https://doi.org/10.1086/516752. astro-ph/0703408. Lo´pez Ariste, A., Sainz Dalda, A., 2012. Scales of the magnetic fields in the quiet Sun. Astron. Astrophys. 540, A66. https://doi.org/10.1051/0004-6361/201118191. 1202.5436. Marsden, S.C., Petit, P., Jeffers, S.V., et al., BCool Collaboration, 2014. A BCool magnetic snapshot survey of solar-type stars. MNRAS 444, 3517e3536. https://doi.org/10.1093/ mnras/stu1663. 1311.3374. Martı´nez Gonza´lez, M.J., Manso Sainz, R., Asensio Ramos, A., et al., 2012. Dead calm areas in the very quiet sun. Astrophys. J. 755, 175. https://doi.org/10.1088/0004-637X/755/2/ 175. 1206.4545. ´ lvarez-Herrero, A., et al., 2011. The imaging Martı´nez Pillet, V., Del Toro Iniesta, J.C., A magnetograph eXperiment (IMaX) for the Sunrise balloon-borne solar observatory. Sol. Phys. 268, 57e102. https://doi.org/10.1007/s11207-010-9644-y. 1009.1095. Mathur, S., Salabert, D., Garcı´a, R.A., et al., 2014. Photometric magnetic-activity metrics tested with the Sun: application to Kepler M dwarfs. J. Space Weather Space Climate 4 (27), A15. https://doi.org/10.1051/swsc/2014011. 1404.3076. Mekaoui, S., Dewitte, S., 2008. Total solar irradiance measurement and modelling during cycle 23. Sol. Phys. 247, 203e216. https://doi.org/10.1007/s11207-007-9070-y. Metcalfe, T.S., Egeland, R., van Saders, J., 2016. Stellar evidence that the solar dynamo may Be in transition. Astrophys. J. Lett. 826, L2. https://doi.org/10.3847/2041-8205/826/1/L2. 1606.01926. Noyes, R.W., Hartmann, L.W., Baliunas, S.L., et al., 1984. Rotation, convection, and magnetic activity in lower main-sequence stars. Astrophys. J. 279, 763e777. https://doi.org/ 10.1086/161945. Ola´h, K., KTva´ri, Z., Petrovay, K., et al., 2016. Magnetic cycles at different ages of stars. Astron. Astrophys. 590, A133. https://doi.org/10.1051/0004-6361/201628479. 1604.06701. Orozco Sua´rez, D., Bellot Rubio, L.R., Del Toro Iniesta, J.C., 2010. Milne-Eddington inversion of the Fe I line pair at 630 nm. Astron. Astrophys. 518, A3. https://doi.org/10.1051/ 0004-6361/201014374. 1005.5013. Parnell, C.E., DeForest, C.E., Hagenaar, H.J., et al., 2009. A power-law distribution of solar magnetic fields over more than five decades in flux. Astrophys. J. 698, 75e82. https:// doi.org/10.1088/0004-637X/698/1/75. Perryman, M.A.C., de Boer, K.S., Gilmore, G., et al., 2001. GAIA: composition, formation and evolution of the Galaxy. Astron. Astrophys. 369, 339e363. https://doi.org/10.1051/ 0004-6361:20010085. astro-ph/0101235. Pevtsov, A.A., Fisher, G.H., Acton, L.W., et al., 2003. The relationship between X-ray radiance and magnetic flux. Astrophys. J. 598, 1387e1391. https://doi.org/10.1086/378944. Reinhold, T., Cameron, R.H., Gizon, L., 2017. Evidence for photometric activity cycles in 3203 Kepler stars. Astron. Astrophys. 603, A52. https://doi.org/10.1051/0004-6361/ 201730599. 1705.03312. Rempel, M., 2014. Numerical simulations of quiet sun magnetism: on the contribution from a small-scale dynamo. Astrophys. J. 789, 132. https://doi.org/10.1088/0004-637X/789/2/ 132. 1405.6814. Roettenbacher, R.M., Monnier, J.D., Korhonen, H., et al., 2016. No Sun-like dynamo on the active star z Andromedae from starspot asymmetry. Nature 533, 217e220. https://doi.org/ 10.1038/nature17444. 1709.10107.

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Salabert, D., Garcı´a, R.A., Beck, P.G., et al., 2016. Photospheric and chromospheric magnetic activity of seismic solar analogs. Observational inputs on the solar-stellar connection from Kepler and Hermes. Astron. Astrophys. 596, A31. https://doi.org/10.1051/0004e6361/ 201628583. 1608.01489. Schrijver, C.J., Cote, J., Zwaan, C., et al., 1989. Relations between the photospheric magnetic field and the emission from the outer atmospheres of cool stars. I - the solar CA II K line core emission. Astrophys. J. 337, 964e976. https://doi.org/10.1086/167168. See, V., Jardine, M., Vidotto, A.A., et al., 2016. The connection between stellar activity cycles and magnetic field topology. MNRAS 462, 4442e4450. https://doi.org/10.1093/mnras/ stw2010. 1610.03737. Semel, M., 1989. Zeeman-Doppler imaging of active stars. I - basic principles. Astron. Astrophys. 225, 456e466. Shapiro, A.I., Solanki, S.K., Krivova, N.A., et al., 2015. The role of the Fraunhofer lines in solar brightness variability. Astron. Astrophys. 581, A116. https://doi.org/10.1051/ 0004e6361/201526483. 1507.05437. Skumanich, A., Smythe, C., Frazier, E.N., 1975. On the statistical description of inhomogeneities in the quiet solar atmosphere. I - linear regression analysis and absolute calibration of multichannel observations of the Ca/þ/emission network. Astrophys. J. 200, 747e764. https://doi.org/10.1086/153846. Skumanich, A., 1972. Time scales for CA II emission decay, rotational braking, and lithium depletion. Astrophys. J. 171, 565. https://doi.org/10.1086/151310. Solanki, S.K., Krivova, N.A., Haigh, J.D., 2013. Solar irradiance variability and climate. Annu. Rev. Astron. Astrophys. 51, 311e351. https://doi.org/10.1146/annurev-astro082812-141007. 1306.2770. Sparn, T.P., Rottman, G., Woods, T.N., et al., 2005. The SORCE spacecraft and operations. Sol. Phys. 230, 71e89. https://doi.org/10.1007/s11207-005-1584-6. Stenflo, J.O., 2013. Solar magnetic fields as revealed by Stokes polarimetry. Astron. Astrophys. 21, 66. https://doi.org/10.1007/s00159-013-0066-3. 1309.5454. Thuillier, G., Herse´, M., Labs, D., et al., 2003. The solar spectral irradiance from 200 to 2400 nm as measured by the SOLSPEC spectrometer from the atlas and Eureca missions. Sol. Phys. 214, 1e22. https://doi.org/10.1023/A:1024048429145. Unruh, Y.C., Solanki, S.K., Fligge, M., 1999. The spectral dependence of facular contrast and solar irradiance variations. Astron. Astrophys. 345, 635e642. Usoskin, I.G., 2017. A history of solar activity over millennia. Living Rev. Sol. Phys. 14, 3. https://doi.org/10.1007/s41116-017-0006-9. van Saders, J.L., Pinsonneault, M.H., Barbieri, M., 2018. Forward modeling of the Kepler stellar rotation period distribution: interpreting periods from mixed and biased stellar populations. ArXiv e-prints 1803.04971. Willson, R.C., Mordvinov, A.V., 2003. Secular total solar irradiance trend during solar cycles 21-23. Geophys. Res. Lett. 30, 1199. https://doi.org/10.1029/2002GL016038. Wilson, O.C., 1978. Chromospheric variations in main-sequence stars. Astrophys. J. 226, 379e396. https://doi.org/10.1086/156618. Wittenmyer, R.A., Tuomi, M., Butler, R.P., et al., 2014. GJ 832c: a super-Earth in the habitable zone. Astrophys. J. 791, 114. https://doi.org/10.1088/0004-637X/791/2/114. 1406.5587. Yeo, K.L., Krivova, N.A., Solanki, S.K., 2014. Solar cycle variation in solar irradiance. Space Sci. Rev. 186, 137e167. https://doi.org/10.1007/s11214-014-0061-7. 1407.4249. Yeo, K.L., Solanki, S.K., Norris, C.M., et al., 2017. Solar irradiance variability is caused by the magnetic activity on the solar surface. Phys. Rev. Lett. 119 (9) https://doi.org/10.1103/ PhysRevLett.119.091102. 1709.00920.

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9

H.S. Hudson1,2, A.L. MacKinnon1 School of Physics and Astronomy, University of Glasgow, Glasgow, United Kingdom1; Space Sciences Laboratory, University of California, Berkeley, CA, United States2

CHAPTER OUTLINE 1. Introduction .......................................................................................................301 2. Nonequilibrium Plasmas .....................................................................................305 2.1 Particle Distribution Functions ............................................................. 305 2.2 Particle Acceleration ........................................................................... 307 2.3 Magnetic Reconnection ....................................................................... 307 3. Overview of Observations....................................................................................308 3.1 The Lower Solar Atmosphere ................................................................ 309 3.2 Coronal Hard X-Ray Sources................................................................. 311 3.3 Radiophysics ...................................................................................... 311 3.3.1 Meter Waves .............................................................................. 313 3.3.2 Microwaves ................................................................................ 315 3.3.3 Millimeter Waves ........................................................................ 316 3.4 Gamma-Ray Observations .................................................................... 317 3.4.1 Prompt g-Ray Emission ............................................................... 317 3.4.2 “Sustained” g-Ray Events ............................................................ 318 3.4.3 Galactic Cosmic-Ray Effects ......................................................... 319 3.5 Neutral and Relativistic Particles.......................................................... 320 3.6 Summary............................................................................................ 321 4. Solar and Stellar “Superflares” ...........................................................................322 5. Additional Topics ...............................................................................................324 5.1 Coronal Mass Ejections, Solar Energetic Particles, and Flares ................. 324 5.2 Flares, Microflares, and Nanoflares ....................................................... 325 6. Conclusions .......................................................................................................327 References .............................................................................................................328

1. INTRODUCTION Solar flares consist of sudden brightenings across a wide range of spectroscopic signatures, originating by consensus in localized plasma disruptions in the lower solar atmosphere. They generally occur in magnetic active regions (see Chapter “Solar The Sun as a Guide to Stellar Physics. https://doi.org/10.1016/B978-0-12-814334-6.00011-X Copyright © 2019 Elsevier Inc. All rights reserved.

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and Stellar Variability”). Originally observed in photospheric and chromospheric radiations, solar flares are now known to involve brightenings across the electromagnetic spectrum, from radio to x-rays and sometimes g-rays. With the advent of routine space-based observations, they are now commonly characterized by their soft x-ray (w107-K) emissions. We can distinguish the flare “impulsive phase,” lasting for tens to hundreds of seconds and characterized by rapidly fluctuating shortwavelength radiation resulting from the acceleration of nonthermal particles. A “gradual phase” follows, distinguished by coronal soft x-radiation and emissions in chromospheric lines such as Ha. The energy comes from nonpotential magnetic-field structures that develop slowly and restructure suddenly. Many other transient phenomena occur all over the Sun, punctuating the generally stable or only slowly varying large-scale structure of the corona. Coronal mass ejections (CMEs) may have closely related plasma dynamics, but unlike flares per se, their effects extend throughout the heliosphere and their effects affect the Earth in many ways (“space weather”). Stellar flares on a wide variety of objects (multiple as well as solitary stars) have great similarities observationally, so far as we can tell. Unfortunately, we cannot observe the details of stellar magnetic fields or the spatial development of the flares. However, we can study these aspects of solar flares by remote sensing down to the resolution limit and by “multi-messenger” tools such as neutral and charged particle emissions (“solar cosmic rays”). In the opposite direction of scale, the Earth’s magnetic field and its trapped particles also undergo flare-like developments. In this domain, too, we can study the plasma dynamics by remote sensing, but more important, we can make direct measurements of the dynamics of the highly excited plasma processes involved via in situ probes (e.g., via the1 four-satellite constellation.) The acceleration of charged particles to extremely high energies has a crucial role at all of these scales: Earth, Sun, and stars. Statistical equilibrium leads us to expect Maxwellian velocity distributions in the solar atmosphere, but abundant observations show that non-Maxwellian distributions (“tail populations”) occur routinely. The galactic cosmic rays represent the extreme of this process, dominating the radiation energy in interstellar space. Strong tail distributions occur dominantly in the solar wind, but dramatic departures from Maxwellians occur even in the deep, collisional solar atmosphere during flares. Nonequilibrium distributions imply intense energy release; their occurrence during flares and CMEs gives insight into the fundamental workings of these events. These events, although sporadic, can have great significance even on large spatial scales and long timescales. The historical beginning of this topic must have been the slow recognition of effects on terrestrial magnetism correlated with solar phenomena. “Corpuscular” emissions from the Sun, later called “solar cosmic rays,” were first observed in the 1940s at stations on the ground. These are now universally called “solar energetic

1

Magnetospheric Multiscale Mission.

1. Introduction

particles” (SEPs) and observed conveniently from space as well. The direct particle measurements have remote-sensing radiation counterparts: radio waves, again from the 1940s; x-rays from the 1950s; and even g-rays, first identified as clearly nuclear processes (as opposed to bremsstrahlung for the x-rays) by the presence of the spectral line at 2.223 MeV that Chupp et al. (1973) observed from the flare SOL1972-0804. This line is produced by neutron capture on protons to form deuterium nuclei and had been predicted to be observable from astronomical objects by Morrison (1958). This first true g-ray flare, observed with an instrument on the Orbiting Solar Observatory 7 (OSO-7) spacecraft, must have produced neutrons in huge numbers; furthermore, the nuclear reactions producing these neutrons (and all of the energy involved in the primary particles) must have occurred in the deep solar atmosphere. This strongly reinforced a proposed solution to the first problem of white-light flares (Carrington, 1859), namely, the source of their huge energy: it could come directly from high-energy particles. Gamma-ray observations (MeV photon energies and above) are inherently much more difficult than x-ray observations (a few to tens of kiloelectron volts). Observations of many flares from space, via their soft x-ray emission (White, 1964) immediately changed the focus of flare research from the “chromospheric flare” seen in optical emission lines such as Ha, into the corona, where dense, high-temperature plasmas could produce soft x-rays via the standard atomic processes of continuum from freeefree and freeebound transitions, as well as discrete lines from bounde bound transitions in ionized atoms, ranging up to Fe XXVI, for which the binding energy is 7.11 keV. Not only these thermal processes, but also nonthermal bremsstrahlung from still higher-energy electrons (tens to hundreds of kiloelectron volts) could also readily be detected (Peterson and Winckler, 1959; Anderson and Winckler, 1962). The presence of nonthermal electron distributions in the solar corona had already been well-known by the time hard x-rays and g-rays became detectable (Wild et al., 1963). In an important development, Neupert (1968) tied some of these observations together by noting that the flare soft x-ray time profile tended to match the time integral of the associated microwave burst. This established the fundamental flare process as the “impulsive phase” (Kane and Anderson, 1970). This produces the hot, dense coronal flare plasmas that persist for minutes to hours and whose cooling coincides with the distinctive flare emissions in chromospheric emission lines such as Ha. In understanding the physics of these (and other) solar phenomena, we often turn to the magnetosphere (“space plasma physics”) for guidance. In the near-Earth domain, we can obtain both remote-sensing and in situ observations; the latter actually detail particle distribution functions that will not become directly observable in many astronomical contexts. Globally, the system composed of ionosphere, magnetosphere, and solar wind contains many points of similarity to the solar case, and many authors have remarked on the analogous properties of magnetic storms in the Earth system and flareeCME events on the Sun. This is important with respect to the microphysics, because the in situ measurements inform us about actual

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particle distribution functions, right down to agyrotropy (fully general threedimensional [3D] distributions). From the high-energy point of view, though, the solar environment differs quantitatively by involving much more energy and much higher energy densities than in geospace. The terrestrial analog of a CME (Hughes and Sibeck, 1987), for example, probably behaves differently in deep space beyond Earth’s orbit, although CME-associated flares and the geotail may involve plasmoid dynamics. In this chapter, we provide an overview of these properties in Section 3, following some preliminaries on the physics. Stellar observations of flare activity have greatly improved but they reveal activity that may or may not reflect a solar paradigm. Equally, the stellar observations may provide clues to some forms of solar activity (Section 4). We return to these questions in Section 6. Section 5 discusses special topics: flareeCME relationships and nanoflares. Fig. 9.1 shows sketches of the two basic paradigms for solar flares: the “simple loop” configuration and the global reconnection configuration. Most flares fall into the first category, although the loops probably contain many filamentary

(A)

(B)

Turbulence acceleration region, Coronal X−ray emission

Reconnection site Energy outflows

Looptop source

Escaping particles

Thick−target footpoints

FIGURE 9.1 Two basic paradigms for solar flares. (A) A flux bundle forming a coronal loop; the multiple strands distinguish this from a “simple loop flare.” A subphotospheric electrical potential drives an unbalanced current through the corona. (B) The global reconnection scenario. Bubbles suggest the presence of turbulence (Liu et al., 2008). The Masuda et al. (1994) “above-the-loop-top” source might be identified with the lower bubble concentration. (A) Adapted from Ryutova, M., 2006. Coupling effects throughout the solar atmosphere: 2. Model of energetically open circuit. Journal of Geophysical Research (Space Physics) 111, A09102. (ADS). (B) Reproduced by permission of the American Astronomical Society).

2. Nonequilibrium Plasmas

substructures and cannot be “monolithic” as is often assumed; the global reconnection configuration serves well to describe CMEs and their near-Sun counterparts, as introduced by Hirayama (1974) in the context of prominence eruptions. Because the global reconnection models fit well to the morphology of the geoeffective “two-ribbon” CME-associated flares, most theoretical and modeling work concentrates on this geometry (the right panel in Fig. 9.1). Note that neither of these cartoons hints at a “trigger” (usually taken to be synonymous with “instability”) and that our knowledge remains far from complete in this most important area. The substorm phenomenon in the magnetosphere has many points of analogy with flare physics. In situ observations in the magnetosphere can reveal the fundamental microphysical processes, even at fundamental plasma scales of gyroradii and Debye length, which we can never probe by remote-sensing of solar plasmas. Integration of this microphysics into a full picture of flaring energy release poses problems of a multiscale character. Viewed remotely from “outside,” in its entirety, a solar flare can show us global aspects of its character that are more elusive for magnetospheric substorms, where the observer lies within the phenomenon being observed. After a brief review of the basic physics in Section 2, we identify and discuss the distinct particle populations inferred from high-energy electromagnetic radiation and also various neutral and relativistic messengers from the low corona, identifying them in terms of the two geometries shown in Fig. 9.1. These drawings provide a background reference for the emission signatures discussed in the following sections. Note that flares and CMEs originate fundamentally in closed-field structures that store energy gradually and release it suddenly, and so there is no pre-event manifestation in either scenario of processes on open fields linking into the heliosphere. Nevertheless, these processes can often connect quickly to open fields, producing the flare-associated type III bursts and to the coronal effects of the CME eruption itself, as described subsequently in Section 3.3. Relating the coronal signatures to the fundamental physics of the flareeCME process leads to many unknowns, but the present wealth of data almost across the spectrum is helping considerably. We return to this topic in Section 5.1.

2. NONEQUILIBRIUM PLASMAS 2.1 PARTICLE DISTRIBUTION FUNCTIONS At the basic level of a particle distribution function, solar plasmas should consist of isotropic Maxwellians. This will, of course, never completely be the case, given the apparent existence of steep gradients in physical parameters within the observable solar atmosphere (e.g. the apparent “loops”). Collisions tend to restore equilibrium after any transient effect such as the passage of a shock wave, and in the ionized corona, which is assumed to be hydrogenic, there is the hierarchy of timescales imposed by Coulomb collisions approximately given by

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CHAPTER 9 High-Energy Solar Physics pffiffiffiffiffiffiffiffiffiffiffiffiffiffi  see : spp : sep : 1 : me =mp : me mp . These scales are of order 0.1:4.0:200 ms in the upper (ionized) chromosphere of a standard solar atmospheric model. Such short timescales do not mean that a tail population cannot exist, though, because the collision cross-sections decrease rapidly with particle energy. Observationally, the existence of footpoint hard x-ray sources even in dense chromospheric and photospheric regions directly demonstrates the existence of important tail populations during flares. Deviations from Maxwellians certainly are directly observable everywhere in the solar wind, often following an empirical representation as a kappa distribution (Vasyliunas, 1968; Maksimovic et al., 1997). This has a core/halo property tending toward a Maxwellian at low velocities, and asymptotic to a power law at high velocities. The kappa distributions have three variables, not four as in an ad hoc powerlaw and tail distribution, and are often used to characterize observed distributions in the solar wind (where anisotropy, species differentiation, and even nongyrotropic properties may also occur). Kappa distributions may be expected on general grounds in any situation in which nonextensive statistics apply, i.e., with long-range interactions that introduce correlations between phase-space volume elements, e.g., plasma turbulence (Leubner, 2004; Treumann and Jaroschek, 2008). Some of the effects of non-Maxwellian distributions on radiative signatures have been calculated for both line intensities and effects on ionization equilibrium (Dzifcakova´, 2006; Dudı´k et al., 2017). These diagnostics often pose difficult observational problems, especially because they must be most pronounced in lesscollisional and therefore fainter structures. For high-energy particles, one typically considers the distribution functions in a test-particle sense; for example, a power-law distribution of SEPs would extend to a high-enough energy that they could be considered noninteracting with a background plasma, either Maxwellian or with a Maxwellian-like core distribution. Because the medium is magnetized, it is often convenient to describe them in terms of rigidity R ¼ p/q ¼ Br, where p is the momentum, q the charge, B ¼ jBj the magnetic induction, and r the Larmor radius. Then the measurements of a particle spectrum might be described as power laws or exponentials in energy or rigidity. The behavior of high-energy charged particles near the Sun has not received as much attention as the original theory near Earth (the “Størmer problem”). It is topically of great interest now for many reasons, specifically as a tool for understanding quiet-Sun g-ray emissions (Section 3.4.3), the cosmic-ray shadow of the Sun at TeV (1012 eV) cosmic-ray energies (Amenomori et al., 2013), other issues connected with the acceleration and propagation of SEPs, and of course, the imminent arrival of new scientific probes (Solar Probe Plus and Solar Orbiter) actually entering the inner heliosphere. Energetic particles could store energy in the coronal magnetic field, an idea proposed by Elliot (1964), via trapping in mirror geometries. The lifetime of a relativistic proton against collisional loss can be long (Hinton and Hofmann 2009); estimated as spp z 2.6/n11 h, where n11 is the plasma density in units of 1011 cm3. Normally, one thinks about energy storage in the magnetic field as the consequence of its small

2. Nonequilibrium Plasmas

departures from a potential-field representation. This energy is not directly observable, and any storage in the form of high-energy ions would also be difficult to detect. Trapped particles, in principle, can store nonpotential energy in the magnetic field without also requiring a DC current to flow through the corona from the surface of the Sun. Indeed, Harris et al. (1992) argue that such a population could not directly explain coronal heating, based on limits set on the 2.223-MeV emission from the quiet Sun. Nevertheless, Hudson et al. (2009) have shown that all three adiabatic invariants can be conserved in a solar coronal magnetic configuration, and so there is the strong probability that trapped energetic ions populate these regions at all times. These populations would go unregistered because of their weak coupling to electromagnetic signatures.

2.2 PARTICLE ACCELERATION The existence of “solar cosmic rays” and their association with solar flares established the idea that violent particle acceleration somehow could happen on a relatively passive star (Forbush, 1946). We now know of several environments in which particle acceleration has an important dynamic role in the development of coronal and chromospheric structures and around solar-system objects. Carrington’s original 1859 observation of a white-light flare, an essentially photospheric phenomenon from the point of view of radiation, turned out to have a close association with the flare hard x-ray signature characteristic of the impulsive phase, and indeed the inferred particle energies rival this dominant form of flare energy release (see references in Neidig, 1989; Hudson, 2016). What mechanisms can accelerate particles rapidly, in large intensities, and to very high energies? Fermi (1949) presented the basic idea of collisions between particles and magnetic fields as a description of the source of galactic cosmic rays. These collisions could involve multiple interactions within a turbulent medium, with a random walk in velocity space eventually producing high particle energies. Meanwhile, the ability of a cosmic plasma to sustain a parallel electric field, now amply demonstrated in the magnetosphere, provides an alternative scenario introduced by Giovanelli (1946) as a “discharge” theory. Finally, a shock wave propagating through a plasma can efficiently accelerate particles by either diffusive shock acceleration or shock drift acceleration. Nothing about some of these basic ideas precludes their operation in a dense or even essentially neutral atmosphere (think of lightning and its remarkable production of relativistic electrons, positrons, and g-rays (Parks et al., 1981, following Wilson’s, 1925 prediction). No recent and comprehensive review of these processes, as applied to solar physics, seems to exist at present (Miller et al., 1997; Zharkova et al., 2011).

2.3 MAGNETIC RECONNECTION Magnetic reconnection is clearly involved with the conversion of magnetic energy into other forms (flare and CME effects) in the sense that flux transfer between

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domains of magnetic connectivity can provide the restructuring needed to extract energy, typically in the large-scale shock waves required by the local process (Petschek, 1964). This global restructuring process frequently is called a “magnetic reconnection flare.” However, to permit this global action to take place requires the development of microphysical processes involving waveeparticle interactions and complicated (agyrotropic) distribution functions (Hesse et al., 2011) and it would be fair to say that the theory linking these two essential actions, on their different scales, remains incomplete. As mentioned in Section 2.1, one can store free energy relatively invisibly in the solar corona in the form of the magnetic field itself or even in trapped particles. Turbulence also can provide a reservoir, and in principle it also explains some of the particle acceleration (e.g., Kontar et al., 2017). In general, in a medium with low plasma b, one would expect Alfve´nic wave transport (Poynting flux) as well as energy storage (Fletcher and Hudson, 2008). We now know that major flares invariably show sudden and significant step-wise changes in the large-scale photospheric magnetic field (Wang et al., 1994; Sudol and Harvey, 2005) probably reflecting the plasma implosion conjectured to be necessary for magnetic energy release (Hudson, 2000). There is an uncomfortable theoretical relationship between the idea of magnetic reconnection in flares and the abundant particle acceleration that they require. The reason for this is that magnetohydrodynamics (MHD) theory does not offer a selfconsistent way to handle the physical development of a system in which highly nonthermal particles constitute an energetically significant component.

3. OVERVIEW OF OBSERVATIONS As noted in the Introduction, flare observations span the whole electromagnetic spectrum and also extend into the domain of “multi-messenger astrophysics,” given the detection of several kinds of ejecta. The spectral limits range from that imposed by the plasma frequency of the solar wind near Earth, some tens of kilohertz in the radio spectrum, out to about 4 GeV. This single record-holding photon was detected by the Fermi g-ray observatory, which has provided key observations at these highest energies (Ajello et al., 2014). In decreasing photon energy, other recent or current spacecraft have provided extensive coverage of solar flares. These include the Reuven Ramaty High Energy Solar Spectroscopic Imager (RHESSI)2 for hard x-rays and lower-energy g-rays (3 keVe17 MeV), the Solar Dynamics Observatory (SDO)3, with several instruments covering extreme UV (EUV), UV, and visible ranges, the Interface Region Imaging Spectrograph4, and Hinode, an orbiting solar observatory with

2

The Reuven Ramaty High-Energy Solar Spectroscopic Imager. The Solar Dynamics Observatory. 4 The Interface Region Imaging Spectrograph. 3

3. Overview of Observations

high-resolution optical capability as well as EUV and x-ray spectroscopy. These and many other orbiting instruments have provided a wealth of information in a kind of golden age for solar space astronomy, with major contributions also coming from in situ measurements and from ground-based optical and radio observatories. No chapter this brief can do full justice to this remarkable array, unfortunately. As a hint regarding the breadth of this material, though, we mention the exceptional observation of 2- to 5-MeV neutral hydrogen atoms by the Solar Terrestrial Relations Observatory (STEREO)5 spacecraft, from the flare SOL2006-12-05 (Mewaldt et al., 2009). STEREO itself uniquely has introduced stereoscopic stellar imagery to astronomy. In this section, we describe the distinct populations of nonthermal particles inferred from remote-sensing observations of the Sun, summarizing this overview in Table 9.2.

3.1 THE LOWER SOLAR ATMOSPHERE Historically, flares were first recognized via their direct visibility against the bright glare of the photosphere, a fact obviously requiring intense energy deposition. White-light flares can certainly double the intensity of the quiet photosphere, about 6  1011 erg (cm2 s)1, and so on this basis, Carrington’s original flare observation immediately established an energy scale of roughly 1032 erg for a major event. This number is a lower limit, because we still cannot extrapolate the visible continuum observations into the UV in any precise way. The Carrington flare is often referred to as an “extreme event” and defined the basis of statistical estimates of the occurrence of similarly powerful events (Love, 2012); see Chapter “Space Weather” for further details. The need for intense energy release at the footpoints of coronal magnetic loops spawned the “thick-target model,” the dominant paradigm for energy transport. In this model, the free energy stored in the coronal magnetic field propagates in the form of mildly relativistic (10- to 100-keV) particles, likely electrons, which penetrate the lower solar atmosphere and are made visible via bremsstrahlung. The efficiency of this mechanism is low (of order 104), with the inference that these electron beams must contain a large amount of energy, perhaps most of the flare’s energy requirements. The hard x-rays, and by inference the electrons themselves, have a roughly power-law distribution. The term “thick target” comes from the idea that the electrons come to a full stop in their downward motion. This simple model, inspired by the need to explain the source of the white-light flare energy (Brown, 1971; Hudson, 2016), continues to have extensive development. Nowadays, the state of the art lies in 1D radiation hydrodynamic simulations such as RADYN (Carlsson and Stein, 1997), which have developed considerably over the years (Allred et al., 2015). Nevertheless, such models remain ad hoc. Particle acceleration remains a black box; the electrodynamics of the model remains unclear because we

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The Solar-TErrestrial Relations Observatory.

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do not have good observational knowledge of the geometry or the physical parameters of the system. Impulsive-phase hard x-ray emission appears systematically in almost every major even, and has two nearly universal morphologies: the softehardesoft pattern of spectral evolution (Kane and Anderson, 1970) and the Neupert effect (Neupert, 1968). The latter has a simple explanation: the “impulsive” timescales of flare energy release, typically tens to hundreds of seconds but not wellcharacterized, are much faster than the timescales of coronal cooling, typically minutes or longer. The impulsive phase marks the epoch of strong ablation of the chromosphere, which creates the hot sources of soft x-radiation at 10  20  106 K in the corona. As for the softehardesoft pattern, this must reflect the behavior of the acceleration process, and it could suggest an origin in a turbulent volume (Benz, 1977). Its ubiquity must represent a basic clue to the nature of this process. Classical shock acceleration, for instance, would produce a spectral index dependent on the compression ratio, but this would not obviously be correlated with the number of fast particles. Fig. 9.2 shows representative spectra and images from an event well-observed by the Geostationary Operational

(A)

(B) (e) HXR 8 s 104

counts s-1

310

103

102

hhmm 1830 2006 Dec 06

1840

1850

1900

-870

-860

-850 X (arcs ecs )

-840

-830

FIGURE 9.2 Representative time series (A) (red, Geostationary Operational Environmental Satellite 1-8 A˚ soft x-rays; blue, Reuven Ramaty High Energy Solar Spectroscopic Imager 25- to 10- keV hard x-rays) and imaging (B) for the major flare SOL2006-12-06. The time-series plot illustrates the Neupert effect even for an extended “impulsive” phase lasting for many minutes but consisting of discrete bursts; the image shows the nearly exact spatial coincidence of G-band (a white-light proxy) and 25- to 100-keV hard x-ray emissions. This material appeared in Krucker, S., Hudson, H.S., Jeffrey, N.L.S., Battaglia, M., Kontar, E.P., Benz, A.O., Csillaghy, A., Lin, R.P., 2011. High-resolution imaging of solar flare ribbons and its implication on the thicktarget beam model. Astrophys. J. 739, 96. (ADS) and is reproduced by permission of the American Astronomical Society.

3. Overview of Observations

Environmental Satellite (GOES)6 in soft x-rays, RHESSI in hard x-rays, and Hinode at visible wavelengths (Krucker et al., 2011). This work suggested that the very intense energy release in the footpoint sources implies a dominant nonthermal electron pressure in the flaring plasma, approaching a plasma b of near unity or an approach to equipartition between magnetic and gas pressures after the energy release. As observations of footpoint spatial scales improve (Xu et al., 2016), this seems increasingly.

3.2 CORONAL HARD X-RAY SOURCES The idea of a second stage of particle acceleration in a solar flare (i.e., after that of the impulsive phase) got x-ray confirmation with the observations of SOL1969-0369 by Frost and Dennis (1971). Such a development was in fact known from the radio signatures (Section 3.3) of extensive late-phase coronal developments, such as the type IV emissions (Castelli and Barron, 1977; Pick and Vilmer, 2008). However, this original “FrosteDennis” hard x-ray event occurred on the invisible hemisphere of the Sun, implying that the source density (at altitudes of tens of Mm) must have been low; thus, the huge x-ray flux was surprising. Krucker et al. (2008) noted that RHESSI can frequently see coronal hard x-ray sources, many on angular scales too large to be resolved spatially, but there is a severe observational limit imposed by low source intensities and high detector backgrounds. Even an energetically significant event may escape notice. Often, we see a broadband continuum persisting for tens of minutes, attributable to gyrosynchrotron radiation from relativistic electrons entrained in relatively weak magnetic fields, and hence observed best in the meter-wave range. Fig. 9.3 shows an example that fits the FrosteDennis paradigm. Furthermore, the figure relates it to the morphology of the “sustained-ray events” (Pesce-Rollins et al., 2015; Share et al., 2017). These interesting events reflect the presence of large fluxes of highly energetic protons capable of interacting as p þ p / p0 / 2g within closed-field structures, and thus capable of emission sustained over many hours (Section 3.4.2).

3.3 RADIOPHYSICS Solar radio observations extend over many wavelength ranges, with differing areas of interest but always with sensitive diagnostic capabilities. The vastness of this domain makes general reviews difficult, but older material remains useful for basic information (Wild et al., 1963; Kundu, 1965; Bastian et al., 1998; Pick and Vilmer, 2008). Table 9.1 divides these domains crudely into three ranges: meter and longer wavelengths, centimeter waves, and millimeter waves. It identifies one each (but not all) of the remarkable new facilities that have become available. Most of these

6

Geostationary Operational Environmental Satellite.

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CHAPTER 9 High-Energy Solar Physics

(B)

(A) 1.000

100 15- 25 keV 25- 50 keV 50-100 keV 100-200 keV

0.100

0.010

1415 2695 4995 8800 15400

80 Spectral Flux

Spectral Flux

312

60

40

20

0.001

0 11:00

11:10 11:20 UT 1-September-2014

11:30

11:05

11:10 11:15 11:20 11:25 UT 1 September 2014

11:30

FIGURE 9.3 The FrosteDennis event SOL2014-09-01. (A) Hard x-ray time variations from Fermi, on a log scale. (B) The microwaves counterpart as observed by the San Vito Observatory of the Radio Solar Telescope Network, on a linear scale. The notch in the x-ray light curves at 11:22 UT is an artifact of the data representation. The microwave plot shows a peak near 11:08 UT, dominated by relatively low-frequency emission at 1415 MHz (blue); below this frequency the competing low-frequency emission, nominally a part of the type IV burst, obscures the smoothly varying FrosteDennis source. From Hudson, H.S., 2017. The Relationship Between Long-Duration Gamma-Ray Flares and Solar Cosmic Rays. ArXiv e-prints. (ADS), by permission of the author.

Table 9.1 Solar Radio Domains Wavelengths

Angular Scales

1e30 m

>1 $’$

100e1 cm

w3 $”$

1e0.03 cm

w1 $”$

Topics

Facilities

Coronal physics (types IeV, etc.) Flare loops and footpoints Photosphere and lower chromosphere

Low-Frequency Array7 Expanded Owens Valley Solar Array8 Atacama Large Millimeter/ Submillimeter Array9

facilities are general purpose rather than dedicated to solar observation, which reduces their effectiveness for solar purposes. The wavelengths accessible from ground-based observatories cover roughly 10 MHz to 1 THz in frequency. Observations from above the ionosphere can also use wavelengths as long as the plasmafrequency limit is near 1 AU, typically at about 30 kHz, but this domain (which 7

The Low-Frequency Array, headquartered in The Netherlands. The Enhanced Owens Valley Solar Array (California, United States). 9 The Atacama Large Millimeter Array (Chile). 8

3. Overview of Observations

mainly reveals processes in the outer corona and solar wind) requires spacecraft and generally remains to be explored. At longer wavelengths, coherent mechanisms dominate, especially for the many dynamic features of the solar corona. Incoherent continuum emission mechanisms work at shorter wavelengths, and the gyrosynchrotron mechanism figures most importantly in the centimeter-wave range, given that the electron Larmor frequency of kilogauss magnetic fields is about 3 GHz (or 10 cm). Free-free emission appears at all wavelengths in different situations but has limited diagnostic potential because of its lack of temperature dependence at radio wavelengths. For this reason, the fortuitous choice of 10.7 cm as a proxy index (Tapping, 2013) for solar activity has worked out well.

3.3.1 Meter Waves Plasma-frequency emission generally dominates the meter-wave signatures of solar activity. Such emissions involve microphysics that can generate electromagnetic radiation coherently, thus making the theory nonlinear and less tractable. In addition, at longer wavelengths one must deal with refraction and scattering, and so observations have relatively low spatial resolution. By compensation, the coherent mechanisms can produce high brightness temperatures (intensities) and rapid time variability, as illustrated in the representative spectrogram of Fig. 9.4.

FIGURE 9.4 Meter-decimeter spectrogram for SOL2014-09-01. This is the same event as shown for hard x-rays and microwaves in Fig. 9.3; it illustrates many of the spectral classifications possible (types II, III, and IV). This spectrogram also contains many unidentifiable features and may not be wholly representative because its parent flare was occulted. From Carley, E.P., Vilmer, N., Simo˜es, P.J.A., O´ Fearraigh, B., 2017. Estimation of a coronal mass ejection magnetic field strength using radio observations of gyrosynchrotron radiation. Astron. Astrophys. 608, A137. (ADS). Reproduced by permission of the author.

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Generally, the long-wavelength radio spectrum reveals many types of events originating in coronal phenomena, beginning with the identification of fast-drift metric bursts (type III) with outwardly propagating beams of mildly relativistic electrons (Payne-Scott et al., 1947). Similar events can occur either in conjunction with or independently of solar flares, and also as the “herringbone” pattern in radioheliograms of slow-drift (type II) bursts, in which the fundamental exciter is a coronal global wave. Many other phenomena occur, detectable mainly via plasma-frequency (coherent) radiation but also via bremsstrahlung and synchrotron radiation. In this review, we do not attempt to cover this vast area in detail, except to note its importance to the diagnostics of flare effects. Much of the literature here dates from early days, as reviewed by Bastian et al. (1998) and Pick and Vilmer (2008). Several new and powerful groundbased, general-purpose facilities (such as the Low-Frequency Array, the Murchison Widefield Array,10 and Jansky Very Large Array11) are now online at longer wavelengths and others are forthcoming; the vast improvements in bandwidth and sampling have opened up comparably large domains of parameter space for exploration. In many cases, the type iII bursts occur in coincidence with the impulsive phase of a flare, in which case the starting frequencies tend to be higher (Kane, 1981). In these cases, this implies a close association of the flare process in the impulsive phase and open field lines, suggesting that such field structures commonly occur deep within active regions, even before the flare. The tightly collimated electron beams responsible for the type iII radiation thus provide an excellent resource for the study of coronal magnetic structure (Paesold et al., 2001) as well as of beam dynamics (Reid and Ratcliffe, 2014). In contrast to the short-lived and collimated electron beams underlying the type III burst phenomenon, the type II and IV bursts reveal large structures developing after the impulsive phase. We have learned that the initiation of the disturbance creating the type II burst also occurs then (Zhang et al., 2004), as well as acceleration of the associated CME (Temmer et al., 2008). Nevertheless, the long timescales of meter-wave activity, as illustrated in the spectrogram of Fig. 9.4, clearly point to independent processes of energy release after the impulsive phase. The phenomenology associated with these large-scale coronal disturbances in an eruptive flareeCME situation is lumped into the category of the type IV burst, although there is great complexity in the signatures. The presence of unambiguous signatures of gyrosynchrotron emission in some of these events again points to the acceleration of high-energy particles (Bastian et al., 2001). Fig. 9.4 in the context of SOL2014-09-01 is an example of a FrosteDennis event as discussed in Sections 3.2 and 3.4 as well.

10

The Murchison Widefield Array (Australia). The Jansky Very Large Array (New Mexico, United States).

11

3. Overview of Observations

3.3.2 Microwaves In the centimeter-wave domain, the magnetic field becomes the dominant factor via gyrosynchrotron emission in flares and gyroresonance absorption in the “S component” (quiescent active regions). The coherent emission mechanisms become less apparent. In the quiet Sun, centimeter-wave radiation forms in the low(est) corona but exhibits spatial variations, doubtless the result of magnetic-field effects on the opacity (e.g., Selhorst et al., 2011). In active regions and particularly during flares, the opacity exhibits a strong wavelength dependence, to the extent that finite optical depth in flare emissions can extend into the millimeter-wave range in extreme cases. Fig. 9.5 illustrates the variability of the microwave domain via model dependencies for thermal (bremsstrahlung) and nonthermal (magnetic) sources. The model spectra in Fig. 9.5 refer only to the excess emission expected from magnetic fields and the basic thermal emission from freeefree transitions (bremsstrahlung) produces a Rayleigh-Jeanselike law (fn f n2) from the background

FIGURE 9.5 Model microwave spectra (fv) showing various parameter dependences for thermal (left) and nonthermal (right) (Sta¨hli et al., 1989) magnetic effects. Here DU, T, B, q and NL refer to source solid angle, temperature, magnetic intensity, observing angle, and nonthermal particle column density, respectively; d is the spectral index of the emitting electron flux. Arrows point to the directions in which the parameters alter the shape of the gyroresonance peak.

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atmosphere, or at higher frequencies an optically thin component (fn w constant) owing to intense heating and ionization in flares. Generally, the microwave range provides the most significant diagnostic information about flare development, but interpretation may be complicated by optical-depth effects. A flare produces heating and ionization, and hence highly variable microwave opacity, and so the lower microwave frequencies do not “see” the inner regions of the plasma. The increasing branch of the peaked curve seen in Fig. 9.5 comes from the outer layers of the source, analogous in 3D to the manner in which an optically thick chromospheric line (such as Ha) reveals different layers in the line wings. The centimeter-wave observations are extremely rich, exhibiting impulsive bursts via gyrosynchrotron emission closely aligned with the hard x-ray signatures of the impulsive phase. In fact, Neupert (1968) correlated the time integral of the microwave impulsive phase with the soft x-rays to establish the well-known relationship now usually described in terms of the hard x-rays and the derivative of the soft x-ray time series.

3.3.3 Millimeter Waves At the shortest radio wavelengths, down to the last useful atmospheric “submillimeter” window around 350 mm, the s ¼ 1 layer enters the chromosphere or even the upper photosphere under quiet conditions (Fig. 9.6). The image in the right panel (White et al., 2017) comes from Atacama Large Millimeter/Submillimeter Array (ALMA) single-dish observations, i.e., without the high resolution obtainable with the full array operated as an interferometer.

(A)

(B)

FIGURE 9.6 (A) Contribution functions at millimeter wavelengths for a quiet-Sun model at disk center (Wedemeyer et al., 2016). (B) A noninterferometric (single-dish scan) Atacama Large Millimeter/Submillimeter Array map at 230 GHz (1.3 mm) made under solar quiet conditions (White et al., 2017). Reproduced by permission of the American Astronomical Society.

3. Overview of Observations

Flares can be detected throughout the millimeter range (Krucker et al., 2013), although systematic observations with ALMA capabilities have not yet been obtained. The gyrosynchrotron spectrum can extend to high frequencies, implying the presence of highly relativistic electrons (and possibly positrons). Unexpectedly, the flare spectra can have positive slopes (Kaufmann et al., 2004). This remains unexplained, but the time profiles of the few events detected this way match the impulsive phase and thus implicate nonthermal processes. As an interesting footnote to this, observations in the mid-infrared have also become possible (Kaufmann et al., 2013; Penn et al., 2016) in a morphology strongly suggesting that of white-light flare continuum.

3.4 GAMMA-RAY OBSERVATIONS We arbitrarily take photon energies 10 and 300 keV to be the boundaries between the soft x-ray, hard x-ray, and g-ray spectral domains. The g-ray range (reaching now into the GeV range, thanks to Fermi observations) contains lines and continua, the latter dominant above about 8 MeV. Whereas all flares that produce g-rays also produce hard x-rays, the converse is not true (Shih et al., 2009); we have clearly distinct classes of events that otherwise look morphologically similar. We should understand that photons in the g-ray range cannot readily be focused, that therefore detector backgrounds limit sensitivity, and finally fluxes are small in the sense of photon numbers such that large, heavy, and expensive instruments are necessary. Solar gray observations have three main branches: prompt flare emissions, long-duration events, and cosmic-ray effects, as described subsequently. Here, “prompt” means on timescales as short as a few seconds for the acceleration of relativistic electrons and ions producing nuclear g-ray lines (Forrest and Chupp, 1983).

3.4.1 Prompt g-Ray Emission Above thresholds of a few megaelectron volts, primary high-energy ions can collide with ambient nuclei, producing secondaries that can lead to a variety of detectable radiations. The emission products include discrete g-ray lines, plus g-ray continua and other products such as neutrons. These mechanisms are well-understood except for the crucial information about the source of the high-energy primary particles and their transport. For details, see the compendious work of Reuven Ramaty and his collaborators, especially Lingenfelter et al. (1965), Ramaty et al. (1979), Murphy et al. (1987), Kozlovsky et al. (2002), and Share and Murphy (1997). From this body of theoretical work, we have the tools with which to explore the important topics of energetics, acceleration mechanisms, and abundances. Of the discrete line features, the 2.223-MeV neutron-capture line is the easiest to detect. This was first reported observationally by Chupp et al. (1973). Its presence (and that of the 0.511-MeV positron-annihilation line) require complicated processes, as described in the references just given. The appreciable (tens of seconds) delay of the 2.223- and 0.511-MeV emissions reflects the time needed for slowing down sufficiently for capture and radiation; we include them here because the

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primary particles appear promptly. A breakthrough in observations of prompt g-ray emission came with the RHESSI imaging observations reported by Hurford et al. (2006). These clearly identified the sources of the 2.223-MeV line emission to lie at conjugate footpoints of coronal magnetic structures in one case, and significantly displaced from the hard x-ray footpoint sources. In reference to the flare drawings of Fig. 9.1, either would be consistent with this observation, but an association with SEPs on open field lines would be problematic (see Fig. 9.7A). Other prompt g-ray lines include the positron annihilation line at 511 keV; the formation of this line has complexities and a serious mystery remains about its observed line width: how can it be as narrow as the RHESSI spectroscopy (Share et al., 2004) shows? More straightforward in interpretation, simple inelasticscattering lines such as 14 C (4.44 MeV) and 16 O (6.13 MeV), among others, provide a nuclear-physics method for determining elemental abundances. This view, although limited by sensitivity, is independent of the traditional views provided by atomic spectroscopy and by in situ sampling. Taken together, the whole g-ray spectrum can also constrain the primary particle distribution via forward modeling.

3.4.2 “Sustained” g-Ray Events In a process apparently separate from that responsible for the prompt g-ray emission, some flares show protracted emission well beyond the impulsive phase of the flare (Chupp et al., 1985) showed that such emissions could extend for hours and reach photon energies above 1 GeV (Kanbach et al., 1993). The Fermi/Large Area Telescope (LAT) observations have greatly improved our knowledge of these

FIGURE 9.7 (A) Reuven Ramaty High Energy Solar Spectroscopic Imager observations of SOL2003-1028 (>300 keV and 2.223 MeV) (Hurford et al., 2006). (B) Fermi/Large Area Telescope centroid determination (outermost contour at 68% confidence) of >100 MeV g-rays, interpreted as originating in p0 meson decay, for SOL2012-03-07 (Ackermann et al., 2017). Reproduced by permission of the American Astronomical Society.

3. Overview of Observations

“sustained g-ray events” (SGREs) (Ackermann et al., 2014, Share et al., 2017), so denoted because their light curves suggest that they should be distinguished from the impulsive-phase emissions (which may persist for up to a few minutes). This radiation probably results from decay of secondary pions, which needs proton energies of at least 300 MeV. This telescope can establish centroid positions for the sources of high-energy g-rays, as illustrated in Fig. 9.7B, but although this instrument has unprecedented sensitivity, it does not have sufficient angular resolution to help in understanding the detailed geometrical structures of the sources. In the case shown and in other cases, the Fermi/LAT image centroids put the high-energy g-ray sources near the surface of the Sun and near the flaring active region. But the corona is implicated for either particle transport or actual emission by the detection of three occulted events (Ackermann et al., 2017). These sustained (SGRE)sources constitute a major puzzle. The bright g-ray emission must involve a large coronal volume, and yet we have no identification of the emitting regions based on the comprehensive EUV observations from the Atmospheric Imaging Assembly or chromospheric observations. Adding to the confusion is the clear correlation with CME and SEP events (Ackermann et al., 2017). The SEPs would be ideal for g-ray production if they could enter a thick target, but the long duration of the SGREs suggests that most of the SEPs will be remote from the Sun when the g-rays appear. In addition, at least three SGREs have been identified with flares behind the limb of the Sun (Pesce-Rollins et al., 2015), which suggests complicated magnetic geometry and propagation to enable SEPs to linger near the Sun and undergo nuclear reactions. The event SOL2012-09-01 (shown in Figs. 12.3 and 12.4) strongly suggests an identification of the SGRE scenario with the hard x-ray and microwave observations of Frost and Dennis (1971) coronal hard x-ray events. This identification extends to the meter-wave radio signatures (type II, type IV, and flare continuum) that also start and extend beyond the impulsive phase (Hudson, 2017), and link the phenomena in some way with the powerful acceleration of SEPs produced by major flareeCME events. This is important for future stellar observations because these late-phase events do not fit naturally into the standard flare models and thus may require a different conceptual framework.

3.4.3 Galactic Cosmic-Ray Effects Fermi and ground-based observatories for TeV astronomy have also revealed several features of the high-energy Sun; these are not associated with any flare or CME paradigm but just relate to the coronal environment itself. This novel view may well develop into a means for studying the global coronal magnetic field on large scales. The primary signature is the “Forbush decrease” sometimes observed in neutronmonitor data after a major SEP event (Forbush, 1946), interpreted as an anomalous modulation of galactic cosmic radiation by the magnetized solar plasma launched as a CME. These modulations can even be detected remotely, such as when produced by a “backside” flareeCME event with no contemporaneous particle increase seen near Earth (Cane et al., 1993; Thomas et al., 2015).

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Specifically, Fermi/LAT routinely detects two components of solar g-ray emission outside flare times (Giglietto et al., 2012): first, a compact source at the solar disk attributed to cosmic-ray secondary processes; and second, a larger-scale halo source attributed to relativistic cosmic-ray electrons that Compton-scatter on solar photons. Both of these sources show solar-cycle modulation (Ng et al., 2016) and, with increasing effect as data accumulate, inform us about the solar environment for cosmic rays at high energies. Mechanisms for interactions of galactic cosmic rays with the body of the Sun were first discussed systematically by Seckel et al. (1991), who described the basic (p, p) interactions and modeled the outward g-ray albedo in terms of mirroring motions within funnel-shaped magnetic flux tubes. In this mirroring trajectory, the particle momentum can point at the Earth. A primary interaction at this moment will produce the equivalent of an extensive air shower (as observed at Earth’s surface), and a highly multiplied cascade of hard photons, resulting from exchanges between pair production and bremsstrahlung, will ensue. This effect will produce a narrow but bright annulus of hard x-ray and g-ray emission around the limb of the Sun (Zhou et al., 2017). Between two extreme frameworks (diffuse secondary production in the solar interior and Earth-directed shower products from mirroring trajectories), we have little knowledge owing to our ignorance of the detailed structure of the chromospheric magnetic field and of the particle dynamics. In addition to these detections of g rays, at TeV primary energies, both the Sun and the moon produce shadows on the Earth, impressed on the nearly isotropic cosmic-ray primary flux. The solar shadow comes not just from the absorption by the body of the Sun itself but also from the deflection of the cosmic rays by the coronal magnetic field. Amenomori et al. (2013) show that this detection process has enough sensitivity to detect the interplanetary sector structure, by favoring models with a heliospheric current sheet, and also that this structure follows the solar cycle. Future developments of TeV cosmic-ray observation may make this an extremely useful tool for studying long-term variations of the coronal magnetic field. Note that this analysis has required Amenomori et al. to approximate the Størmer problem for cosmic-ray transport in the low corona, via an extension of the PFSS modeling that includes the highly relevant sector structure of the solar wind (Zhao and Hoeksema, 1994).

3.5 NEUTRAL AND RELATIVISTIC PARTICLES Finally, other messengers from the high-energy Sun include neutrons, neutrinos, and energetic neutral atoms (ENAs). The neutrinos have no confirmed association with any form of magnetic activity and fall outside the scope of this review. Otherwise, neutral particles provide direct views of the Sun, just as photons do, with propagation not impeded by the magnetic field. In this section, we also mention highly relativistic ions because of the potential directness of their view of flare acceleration processes.

3. Overview of Observations

Chupp (1984) could optimistically review the “solar neutral radiations” available at that remote time, and we report some progress. Chupp’s article dealt mainly with neutron (Biermann et al., 1951) and g-ray (Morrison, 1958) observations, ignoring radiations in the rest of the electromagnetic spectrum. Energetic free neutrons can survive radioactive decay sufficiently to be detectable at the distance of the Earth, for a high enough energy. They can also penetrate the Earth’s atmosphere for direct counting by ground-based systems such as the neutron-monitor array (Shea et al., 1991) detected the canonical event SOL1991-05-24 this way (Debrunner et al., 1997). In an important development, a relatively simple instrument on the International Space Station has recorded several energetic solar neutron events, obtaining excellent significance by image modulation and thus suppressing background contributions (Muraki et al., 2012; Koga et al., 2017). ENAs also provide a clear channel to phenomena in the corona, because they propagate directly through the magnetic field without deflection. Although this potential remains to be realized, we now actually have a flare ENA detection from SOL2006-12-06, as reported by Mewaldt et al., (2009). The observations showed 2- to 5-MeV hydrogen atoms, with a clear interpretation in terms of chargeexchange reactions from SEPs near the threshold for “stripping” reionization interactions in the lower corona. This mechanism probably precludes any link between such ENAs and the flare-associated particles producing g-radiation, which must have interacted in collisionally thick regions.

3.6 SUMMARY Table 9.2 lists some observationally distinct populations of energetic particles observed near the Sun. These mainly involve flares and CMEs and reflect the presence of significant amounts of energy in the particles themselves. To some extent, this reveals observational bias, especially against the remote sensing of energetic ions. At energies of a few hundred kiloelectron volts to a few megaelectron volts, observational technique is relatively weak because of the lack of focusing optics and the difficulty of screening against background counts. The result is a high threshold of detection relative to other wavelength domains. In the radio domain (below a few gigahertz), one can have highly efficient emission mechanisms involving the coherent emission of multiple electrons, such that small-scale acceleration processes may become detectable. Table 9.2 does not attempt to list all of them, mentioning only the type II and III mechanisms closely associated with flares and CMEs. Another huge observational bias should be borne in mind. We must allow for the presence of widespread plasma instabilities leading to shock formation, turbulent cascades, and much other microphysics involving magnetism. These processes might not be detectable individually because of sensitivity limits, although there might be indirect signatures (as with “nanoflares”) of their presence (see Section 5.2).

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Table 9.2 Distinct Solar Nonthermal Particle Populations Population

Particle

Energies

Reference

1. Coronal fast-drift 2. Coronal slow-drift 3. Impulsive phase 4. Impulsive phase 5. Prompt 6. Gradual

Electrons Electrons

Metric radio Metric radio

Wild et al. (1963) Wild et al. (1963)

Electrons Ions (at Sun) Ions (heliosphere) Electrons

10e100 keV >10 MeV GeV >200 keV

7. Gradual

2e5 MeV

10. Sustained

Energetic neutral atoms Neutrons Solar energetic particles Ions

Neupert (1968) Chupp et al., 1973 Forbush (1946) Frost and Dennis, 1971 Mewaldt et al. (2009) Chupp (1984) Forbush (1946)

>20 MeV

11. Cosmic rays 12. Cosmic rays

Electrons Ions

Relativistic Relativistic

8. Gradual 9. Gradual

>10 MeV

Forrest & Chupp (1983) Abdo et al. (2011) Abdo et al. (2011)

4. SOLAR AND STELLAR “SUPERFLARES” Flares occur on stars of many classes and often must be more energetic than solar flares to become detectable (see elsewhere in this volume for further detail). In binary systems (or solitary objects with accretion disks), interacting magnetic fields may allow the orbital motions to drive magnetic storage and release (Simon et al., 1980); this concept would apply not only to the RS Canum Venaticorum (RS CVn) stars for which it was initially proposed, but also more solar-type objects such as the Alpha Centauri system. Even without this extra energy reservoir, though, solitary stars even of solar type can apparently produce “superflares” (Schaefer et al., 2000). The Kepler observatory has systematically studied the photometric variability of many objects, finding flares on a huge variety of stellar types, including cool stars of solar type (Walkowicz et al., 2011; Maehara et al., 2012). These observations have a single fixed wavelength passband, 400e900 nm, which covers much of the quiescent radiation of a solar-temperature star, but which does not extend far enough into the UV to detect the bulk of the impulsive-phase emission on the solar paradigm (Kowalski et al., 2012). Fig. 9.8 shows an example. It is especially noteworthy that this “solar-type” star does not exhibit “solar-type” background variability, as displayed in time-series observations of total solar irradiance (Willson et al., 1981). This star, KIC 11610797, exhibits relatively large (percents) smooth variability, with obvious flare peaks; the solar data exhibit more random fluctuations, no obvious flares, and instead well-defined “dips” when sunspots traverse the solar disk.

4. Solar and Stellar “Superflares”

FIGURE 9.8 Representative stellar superflare, as reported from Kepler data at 1-min cadence by Maehara et al. (2015). (A) On a timescale of many stellar rotations; (B) at high time resolution. Note the regular variations owing to persistent stellar active regions (“starspots”) modulated by stellar rotation. The time axes are in days. Much of the Kepler data have coarse time resolution (about 30 min), as indicated by the dotted histogram in (b). This particular flare was observed at about 1 min cadence. Reproduced with the author’s permission.

We can envision the Kepler superflares as white-light flares on the solar paradigm, and as such we would expect that particle acceleration would dominate the flare process; unfortunately, the Kepler stars are generally too faint for hard x-ray or g-ray detection. Osten et al. (2007) suggest the detection of nonthermal hard x-ray emission from a very energetic flare event on II Peg, a seventh-magnitude binary RS CVn star. Generally, there is a great deal of x-ray literature on stellar flares, but the interpretation of thermal versus nonthermal character seems to be different. Within the parameter range accessible to solar observations, we observe the clear variation of soft x-ray properties illustrated in Fig. 9.9. Bearing in mind the presence of systematic errors, we see a clear tendency for a weak relationship of the form 2=7 EMfTe , with uncertainty in the exponent (Aschwanden et al., 2008). Extrapolated into the domain of stellar flares, indeed this relationship appears to continue (as described in terms of magnetic properties), and also downward into the domain of solar microflares (Fisher et al., 1998). The significance of this relationship for the detection of hard x-ray sources in stellar flares lies in the weakness of the hard x-ray spectral tail of the flare bremsstrahlung emission. The more powerful the flare, the hotter its thermal source, and the more difficult the detection of this tail against the brighter thermal background fluxes. Interestingly, the detection of flare processes may become easier at higher g-ray energies. This is because of the physical difficulty of rejecting counting-rate background effects in a broad spectral band roughly centered at 1 MeV. Here, focusing optics do not work well; above this energy, one can use Compton scattering and track detection to localize photon sources spatially and thus reduce unwanted

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FIGURE 9.9 (A) Correlation between spectroscopically determined temperatures and Geostationary Operational Environmental Satellite (GOES) emission measures for isothermal fits to soft x-ray spectra of solar flares, from Feldman et al. (1996). Here, the different symbols refer to separate studies of flares in different ranges of GOES class, as identified in the original. (B) Representative count spectrum for the superflare on the RS Canum Venaticorum star II Peg reported by Osten et al. (2007), combining soft x-ray and hard x-ray observations. This joint spectral fit strongly suggests the presence of a nonthermal hard x-ray tail present above about 45 keV, where the dash-dot component (a power law) exceeds the dotted component (a high-temperature thermal spectrum). Please see the cited papers for full explanations. Figures reproduced by permission of the American Astronomical Society.

background counts. Against this advantage, of course, one has the disadvantage of the need for larger and larger collecting areas, a major difficulty for space instrumentation. The solar success of Fermi in this respect (Abdo et al., 2011; Ackermann et al., 2014) is noteworthy. At still higher photon energies, large-area Cherenkov detectors based on Earth become possible, detecting extensive air showers produced by primary g rays; Ohm and Hoischen (2018) argue that new developments in this area may lead to the detection of stellar superflares at TeV energies.

5. ADDITIONAL TOPICS 5.1 CORONAL MASS EJECTIONS, SOLAR ENERGETIC PARTICLES, AND FLARES Generally, a powerful solar flare will be accompanied by a CME on a large scale in the heliosphere and with huge fluxes of “solar cosmic ray” particles (SEPs). The SEPs themselves consist of electrons and ions at all energies, including the relativistic particles that can be detected directly by cosmic-ray detectors on Earth (Forbush, 1946) as “ground-level events.” In such cases, the radiated energy of

5. Additional Topics

the process (the flare) and the mechanical energy of the ejecta (the CME) may be of the same order of magnitude (Emslie et al., 2012). Somewhat surprisingly, the SEP fluxes can contain a substantial fraction of the total event energy, implicating particle acceleration (not simply “heating”) as the mechanism of dissipation inherent in the directly observed CME shock front, and likely other mechanisms within the solar corona as well. Emslie et al. (2012) estimate the SEPs’ total energy at “a few percent,” but we note that this estimation has many systematic limitations. They also note that the high-energy particles of the impulsive phase, either electrons or ions, may contain enough total energy to drive the whole process, something that had already aroused comment at the dawn of x-ray and g-ray astronomy (Peterson and Winckler, 1959). The relationship between flares and CMEs has had a contentious relationship since the earliest coronagraphic imagery from space. Do flares and CMEs involve common physics? For good reasons, these fields of research seemed distinct in the early days; for one thing, the observing domains do not overlap directly, because white-light coronagraphs do not see the lower solar atmosphere. Now, of course, we have EUV coronal imaging and can make clearer links. Second, events far beyond the limb can readily produce spectacular CME signatures, and yet before STEREO we had no clear way to recognize flaring on the far side. Most data point to a close relationship in the physics of the two sets of phenomena, especially for the major events. The most intense flare energy release, the impulsive phase, corresponds closely to the time of acceleration of the ejecta (Zhang et al., 2004; Temmer et al., 2008). We often see the ejecta originating in the deep atmosphere, rather than high in the corona (Dere et al., 1997; Hudson et al., 2003). Within an unresolvable timescale of a minute or so, this also coincides with the appearance of the x-ray and EUV “dimming” interpreted as the origination of the CME (e.g., Zarro et al., 1999). These relationships cannot be established precisely simply because our modern data remain insufficient even to describe the basic geometrical configuration. Fig. 9.10 illustrates the dimming phenomenon, probably first recognized as “coronal depletions” by Hansen et al. (1974), via use of the SDO/EUV Variability Experiment sun-as-a-star spectroscopy. The plot shows individual 10-s data accumulations for the Fe X line at 174.53 AA, which has an impulsive-phase peak nominally associated with footpoint excitation and chromospheric ablation, and a long-lasting dimming signature marking the ejection of the CME. As noted by Harra et al. (2016), this kind of observation suggests that a suitable instrument could detect CMEs rather directly on other stars.

5.2 FLARES, MICROFLARES, AND NANOFLARES The high temperature of the solar corona often is mentioned as a fundamental problem in solar physics, and it is natural to implicate the pervasive magnetization of the coronal plasma as a part of the mechanism. Solar magnetic activity expresses itself in obviously discrete events (flares and CMEs, for example) in which the magnetic field undergoes an unstable restructuring resulting in sudden stress

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0.5

0.0

Fe X 174.53 A

0.10

0.05 Irradiance

SOL2012-03-07 1.0 Normalized flux excess

326

0.00

-0.5 -0.05 -1.0

-1.5 23:30 00:00 00:30 01:00 01:30 02:00 02:30 UT 6-7 March 2012

-0.10 17.35

17.40

17.45 17.50 Wavelength, nm

17.55

FIGURE 9.10 Left: Time series of EUV Variability Experiment (EVE) irradiance excess at Fe X 174.53 A˚ observations of SOL2012-03-07 (X5.4), with the GOES 1e8 A˚ data overlaid and displaced for clarity. Right: Individual 10-s EVE spectra for the 174.53 A˚ line during flare peak (00:17:03 UT) and late (03:10:23 UT) in the flare, again as excesses relative to a preflare spectrum. The dimming amounts to about 10% of the preflare flux in this line.

release. These stress releases lead to copious particle acceleration and grossly distorted particle distribution functions, as discussed earlier. At the same time, we can identify persistently hot structures with an apparently gradual energy conversion, more akin to heating than to particle acceleration as such, to explain the counterintuitively high temperature of the solar corona and also perhaps the expansion of the solar wind. “Velocity filtration” (Scudder, 1992) would be an example of such a gradual process involving nonthermal particles in the tail of the distribution function in the dense atmosphere, but not necessarily flare physics as understood on a large scale. Parker (1988) proposed that the appearance of steady heating could simply represent the sum of small-scale energy releases, which he interpreted in terms of spontaneous current-sheet formations in the stressed plasma leading to “nanoflares.” These heating events would occur so frequently that their individual properties could not readily be observed; almost by definition, an individual nanoflare would not be detectable (Hudson, 1991). Although Parker based his discussion on MHD theory, Sturrock (1989), could also describe episodic heating in terms of plasma instabilities, possibly implying gross distortions of the particle distribution functions (Parker and Tidman, 1958). The observational situation is mixed, and ambiguities remain because we do not have a clear picture for the microphysics involved in such a process. Indeed, some hot loops appear not to vary appreciably with time (Brooks and Warren, 2009), implying poor detectability for any constituent nanoflares. On the positive side,

6. Conclusions

we note that Viall and Klimchuk (2012) found a basic universal tendency for signatures of plasma cooling in the corona, as though impulsive heating were occurring ubiquitously. This finding is consistent with the “smoking gun” signature of nanoflares: namely, that their time development requires some part of the coronal plasma to be at a higher temperature than the (apparently steady) average temperature, and evidence for this is often cited (Brosius et al., 2014). On the negative side, Del Zanna (2013) describes EUV spectroscopic evidence of steeply decreasing emissionmeasure distributions in active-region loops, implying an absence of hightemperature plasma in quiescent active regions. The idea of episodic heating remains attractive; for example, one could link it to the physics of magnetic reconnection, in which extreme particle distribution functions may necessarily appear (Hesse et al., 2011) in a rapidly evolving structure. Accordingly, different mechanisms may be at work in different domains (Klimchuk, 2006). Nanoflare theory includes parameters that can be adjusted to fit almost any data set; such a theory can be sufficiently descriptive, but not necessarily proven in a systematic manner.

6. CONCLUSIONS The high-energy aspects of solar activity pervade the solar atmosphere and extended corona. Most prominently, these domains host transient events of many types, universally recognized to be episodic transfers of energy stored in the magnetic field into accelerated particles. Where MHD theory suggests that the restructuring produces heating and flows, the microphysics requires particles, and observationally we see many populations of highly nonthermal particles that underlie the dissipation of magnetic energy. We can summarize the significance of high-energy solar physics, as seen from the current perspective, as the following: 1. The major phenomena, flares and CMEs, each require significant particle acceleration for energy dissipation and transport. 2. Flare and CME processes largely convert magnetic energy into particle acceleration, but the acceleration mechanisms are not yet comprehensively understood. 3. Nonthermal signatures appear not only in the corona but also in the relatively dense chromosphere and photosphere. 4. Episodic energy releases occur across a wide range of scales in the corona, evidently involving similar physical processes although it is not yet evident whether this self-similarity extends all the way to the hypothesized nanoflares. 5. Major particle populations (such as those underlying the sustained g-ray events) remain unidentified in solar imagery.

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6. So far as we can tell, stellar flaring follows the main solar paradigm (i.e., not the “sustained” pattern), but on a different scale of energy release. 7. The interpretation of flares and CMEs definitely requires a “multi-messenger” approach, in which we must understand not just the x-ray and g-ray radiation signatures but also the charged particles (SEPs), energetic neutrons, energetic neutral atoms, plasma ejecta, and even the “sunquake” acoustic waves in the solar interior (Kosovichev and Zharkova, 1998).

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10 Noe´ Lugaz

Space Science Center and Department of Physics, University of New Hampshire, Durham, NH, United States

CHAPTER OUTLINE 1. Space Weather: A Short Historical Perspective .................................................... 335 2. Solar Eruptions and Propagation ......................................................................... 337 2.1 Coronal Observations ........................................................................... 337 2.2 Observations of Coronal Mass Ejection Heliospheric Propagation ............. 339 2.3 Numerical Simulations ........................................................................ 339 2.4 Acceleration and Transport of Energetic Particles................................... 340 3. Impact at Earth .................................................................................................. 340 3.1 Geomagnetic Storms and Substorms..................................................... 341 3.2 Magnetopause and Radiation Belts ....................................................... 344 3.3 Effects of Solar Energetic Particles ....................................................... 345 4. Impact at Other Planets and Throughout the Solar System .................................... 345 4.1 Mercury.............................................................................................. 346 4.2 Venus and Mars .................................................................................. 347 4.3 Jupiter and Saturn............................................................................... 347 4.4 Outer Heliosphere ............................................................................... 347 5. StarePlanet Interaction: Space Weather in Exoplanetary Systems and Conclusions................................................................................................. 348 Acknowledgments ................................................................................................... 350 References ............................................................................................................. 350

1. SPACE WEATHER: A SHORT HISTORICAL PERSPECTIVE Solar eruptions are one of the most energetic manifestations of solar magnetism and variability. Fast and energetic eruptions often drive fast magnetosonic shock waves and are associated with particle acceleration and other high-energy phenomena. Coronal mass ejections (CMEs) and energetic particles affect Earth and the other planets in our solar system, resulting in “space weather.” During their daylong propagation between the solar corona and Earth, CMEs interact with the solar wind as well as other CMEs. This modifies their structures in a way that may affect their potential to affect Earth. Upon impacting Earth, the shock wave and CME plow The Sun as a Guide to Stellar Physics. https://doi.org/10.1016/B978-0-12-814334-6.00012-1 Copyright © 2019 Elsevier Inc. All rights reserved.

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into, compress, and reconnect with Earth’s magnetic field extension into geospace (referred to as the “magnetosphere”). Changes in the magnetosphere are referred to as a “geomagnetic storm,” typically lasting about 1 day. The CME onslaught may also spawn a different dynamic mode of magnetic activity at higher latitudes, an “auroral substorm” associated with powerful electric currents flowing into, out of, and through Earth’s ionized atmosphere (the “ionosphere”). The radiation belts, a region of the geospace extending at Earth from 1.5 to 7 Earth radii and populated with relativistic electrons, can be rapidly and irreversibly modified with injections or losses of particles during the passage of CMEs. “Space weather” is a relatively new term. It was first introduced at the beginning of the space age but it became prominent in the 1990s (since 2000, for example, there have been several new journals focusing specifically on the science of space weather). However, the concepts behind space weather date back to the early days of solar physics and geomagnetism research. Space weather focuses on the solarterrestrial (or more generally solar-planetary) connection. Richard Carrington is often credited for having made the connection between a flare on the Sun and widespread auroras in Europe on the following day (Carrington, 1859). This solarterrestrial event, usually referred to as the Carrington event of 1859, can therefore be considered the initial space weather event (Tsurutani et al., 2003; Cliver and Svalgaard, 2004). Solar physics, through the observation and study of sunspots, and geomagnetism, through the observation and study of aurorae, predate space weather studies by several centuries. Sunspot and aurorae observations were reported in Asia and the Middle East as far back as the 6th century BC (Wittmann and Xu, 1987; Stephenson et al., 2004; Hayakawa et al., 2016). Carrington was able to make the connection between the solar flare and the aurorae owing to the extreme nature of the event he observed: white-light flares are rare, and only such flares could be observed at that time (flares are most easily observed in extreme ultraviolet [EUV]). In addition, a typical CME propagates between the Sun and the Earth in 3 days (the so-called 80-h rule) (Brueckner et al., 1998). For a more typical event, making the connection between the flare and aurorae at Earth requires additional observations between the Sun and the ground. The Carrington event propagated in about 18 h and was associated with extreme auroral displays (Green and Boardsen, 2006). As such, it was relatively direct for an observer to make the connection between the flare and the eruption. Other intense space weather events of the 19th century are described in Boteler et al. (1998). A review of the field of space weather from its inspection to current developments can be found in Lanzerotti (2017). The field of solar physics changed in 1971 with the first remote detection of a solar eruption, with a coronagraph made by the Solwind instrument onboard the Orbiting Solar Observatory 7 (Tousey, 1973). In the following 2 decades, debates raged regarding the relationship between CMEs and flares, as well as the exact cause of geomagnetic storms (Gosling et al. 1990). The chain of events is now relatively well understood, with a CME at the Sun being associated with a magnetic ejecta near Earth, and when the orientation of the magnetic field inside the ejecta is favorable for

2. Solar Eruptions and Propagation

magnetic reconnection with Earth’s magnetosphere, a geomagnetic storm at Earth. Numerical simulations (Odstrcil and Pizzo, 1999; Wu et al., 1999; Groth et al., 2000) and remote heliospheric imaging (Jackson et al., 2004; Harrison et al., 2008) have revealed the direct link between solar eruptions observed in the corona and the disturbances measured in situ by spacecraft at 1 astronomical unit (AU). Numerical simulations as well as combined solar wind and magnetospheric measurements by satellites have also shown how geomagnetic activity is driven by corotating solar wind streams as well as CMEs and shocks affecting Earth’s magnetosphere. Next, we will review some of these discoveries.

2. SOLAR ERUPTIONS AND PROPAGATION As discussed in Chapters 3, 6, 8 and 9, solar eruptions are essential for the Sun to release magnetic energy and helicity. On average, they occur several times per day, with a frequency ranging from about twice a week in solar minimum to five to six times per day in solar maximum (Gopalswamy et al., 2009b). Solar eruptions are associated (although not necessarily) with a one-to-one correspondence to a number of solar phenomena: filaments and prominences, sigmoids, Moreton and EUV waves, dimmings, and ribbons. Determining the physical mechanism or mechanisms leading to solar eruptions and the formation or preexistence of flux ropes that become the magnetic “core” of a CME are areas of active research and are not reviewed here. As is discussed in 3, the main important characteristics of CMEs for space weather are their direction (to determine whether they will affect Earth), speed, duration, and magnetic field strength, and the orientation of their magnetic field. Hereafter, we focus on the coronal and heliospheric effects affecting these characteristics of CMEs. A more in-depth review of CME propagation, including how it affects space weather can be found in Manchester et al. (2017).

2.1 CORONAL OBSERVATIONS In the low solar corona, the plasma is magnetically dominated (b, the ratio of thermal to magnetic pressures is less than 1). As such, magnetic forces are dominant and CMEs are affected by coronal magnetic fields. Strong coronal magnetic fields can result in failed eruptions (Ji et al., 2003; To¨ro¨k and Kliem, 2005). From 1 to 20 R1 , magnetic gradients and magnetic forces can result in CMEs rotating (Yurchyshyn et al., 2009; Isavnin et al., 2013; Kay et al., 2017) or deflecting (MacQueen et al., 1986; Kilpua et al., 2009). In many cases, CMEs do not propagate radially outward but appear to be channelled through open field lines (Liewer et al., 2015; Mo¨stl et al., 2015). In addition to these changes in the CME direction and orientation owing to forces acting on the CMEs, ideal instabilities such as the kink instability, as well as reconnection may affect the CME orientation (To¨ro¨k and Kliem, 2003; Isenberg and Forbes, 2007; Gopalswamy et al., 2009a; Shiota et al., 2010). Between 1.2 and 32 R1 , CMEs are remotely observed in white light, i.e., via Thomson scattered light, as shown in Fig. 10.1. These images provide plane-ofthe-sky projections of a CME density structure. With large-angle and spectrometric

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FIGURE 10.1 Two examples of white-light images of coronal mass ejections as observed by a STEREOA/COR2 coronagraph on Jan. 13, 2013 at 14:54 UT (A) and STEREO-A/HI-1 heliospheric imager on May 24, 2010 at 08:49 UT (B).

2. Solar Eruptions and Propagation

coronagraphs (LASCO) and COR coronagraphs, CME-driven shocks can be observed (Vourlidas et al., 2003; Maloney and Gallagher, 2011). This has shown that CME-driven shocks can form at as low as 2e5 R1 , allowing for the efficient acceleration of particles to very high energies (up to hundreds of megaelectron volts).

2.2 OBSERVATIONS OF CORONAL MASS EJECTION HELIOSPHERIC PROPAGATION Since 2003, with heliospheric imaging by the Solar Mass Ejection Imager (SMEI) instrument (Eyles et al., 2003; Jackson et al., 2004), and more recently by the Heliospheric Imagers onboard STEREO (Eyles et al., 2009; Harrison et al., 2008), CMEs have been routinely observed past 0.15 AU, which sheds light on their propagation to the inner planets (see right panel of Fig. 10.1). Reviews of SMEI and heliospheric imager (HI) observations were published in Howard et al. (2013) and Harrison et al. (2017), respectively. Combining these remote observations with in situ measurements at Mercury, Venus, Earth, and Mars, it has been possible to determine many of the heliospheric characteristics of CMEs. It is now understood that fast CMEs experience most of their deceleration in the upper corona (Liu et al., 2013), but the deceleration can continue all the way to 1 AU or beyond (Temmer et al., 2011; Hess and Zhang, 2014; Winslow et al., 2015). CME deceleration results from its interaction with the background solar wind and is often modeled as a type of hydrodynamical drag (Vrsnak, 2001; Cargill, 2004). Although determining the CME position, speed, and acceleration from remote observations depends strongly on the geometrical assumptions that are made, it is possible to image CMEs as they affect Mercury (Hu et al., 2016), Venus (Rouillard et al., 2009), or the Earth (Davis et al., 2009; Wood and Howard, 2009; Mo¨stl et al., 2009), or spacecraft in the inner heliosphere (Nieves-Chinchilla et al., 2013). Adding to the complexity of relating solar and coronal observations to in situ measurements, CMEs can deform through interaction with the solar wind (Savani et al., 2010; Lugaz and Roussev, 2011) and can have significant changes to their internal magnetic field profiles by reconnecting with the interplanetary magnetic field (IMF) (Dasso et al., 2009; Ruffenach et al., 2012) or the heliospheric current sheet (Winslow et al., 2016), or as a result of their interaction with other CMEs (Shen et al., 2012; Lugaz et al., 2013).

2.3 NUMERICAL SIMULATIONS To fill the gap between the upper corona and the near-Earth environment in the absence of remote-sensing observations, global numerical modeling was developed in the 1990s. Simulations of CME propagation were performed by a number of groups in the United States, Europe, and Asia (Chane´ et al., 2006; Groth et al., 2000; Odstrcil and Pizzo, 1999; Shen et al., 2007; Shiota and Kataoka, 2016; Wu et al., 1999). Numerical simulations have been central to the study of the interaction

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between a CME and the solar wind. Using numerical simulations, Riley et al. (1997) and Manchester et al. (2004) predicted that CMEs become distorted as they propagate; such simulations have also been used to investigate the increase in CME mass as it propagates (Lugaz et al., 2005). Both phenomena have been confirmed by observations, although the extreme CME distortion as seen in numerical simulations is only rarely observed remotely (Savani et al., 2010). These simulations now routinely complement observations of CMEs by LASCO, CORs, or HIs (Lugaz et al., 2009).

2.4 ACCELERATION AND TRANSPORT OF ENERGETIC PARTICLES CME-driven shocks and compression regions associated with CMEs as well as corotating solar wind streams are able to accelerate particles from suprathermal energies to very high energies, up to 100 s MeV per nucleon for protons, a particles, and minor ions (Axford, 1981; Bell, 1978; Lee, 1983). These solar energetic particles (SEPs) can have a strong influence on planetary atmospheres, and through particle radiation, they can strongly affect technologies in space as well as life forms. Schwadron et al. (2017) provides more details about the acceleration and effects of SEPs throughout the solar system. A general discussion of the effects of solar and galactic particle radiation on atmospheres and ionospheres can be found in Bazilevskaya et al. (2008).

3. IMPACT AT EARTH Earth is shielded from the interplanetary space through its magnetosphere, with the magnetic field at its surface orders of magnitude larger than that in the interplanetary medium at Earth’s distance (w5
105 T compared with w5 nT). The principal driver of geoeffects are (1) solar radiation, mostly from EUV and X-ray, (2) particles accelerated inside Earth’s magnetosphere as a result of reconnection with the solar transients or waveeparticle interactions, and (3) high-energy particles accelerated in the corona and interplanetary space. The solar wind speed at Earth’s orbit is faster than the fast magnetosonic speed (the characteristic speed for plasma). Therefore, the interaction between Earth and the solar wind results in the formation of a bow shock that decelerates the solar wind to speed lower than the fast magnetosonic speed. Solar X-ray emissions may increase by orders of magnitude during a solar flare; the increased emission from the Sun results in an enhanced rate of photoionization in Earth’s ionosphere, creating sudden ionospheric disturbances that disturb radio communications (Mendillo et al., 1974; Ramsingh et al., 2015). The EUV emission during a flare typically heats up the ionosphere and thermosphere, which increases the density at a given altitude. These increased densities affect spacecraft through an enhanced atmospheric drag, especially spacecraft in low-Earth orbit, and contribute to the disruption of radio signals (Tsurutani et al., 2012). All of these effects occur within tens of minutes of the start of the event at the Sun. EUV emission typically

3. Impact at Earth

increases for a few hours during a large X-class flare but the effect on the thermosphere and ionosphere lasts for a few days. Atmospheric escape at Earth associated with flares is limited but it has been reported to be associated with the polar wind or upflowing ions (Yau and Andre, 1997). The Sun primarily affects man-made technologies with no direct influence on human health (except astronauts). However, because of the ever-increasing reliance on technology, space weather may have a direct effect on everyday life, from communications to airplane travel, or possibly any system relying on electricity (Solar and Space Physics: A Science for a Technological Society, 2012).

3.1 GEOMAGNETIC STORMS AND SUBSTORMS When the IMF has a component antiparallel to Earth’s magnetic field, i.e., when the IMF has a southward component in the geocentric solar magnetospheric coordinate system, reconnection can occur on the day side of Earth’s magnetosphere. The magnetic field lines undergo a series of reconnection and convection events that were first described by Dungey (Dungey, 1961). The two main loci of reconnection are the magnetopause in the day side and the magnetotail in the night side. The magnetopause is the location where the solar wind total pressure, modified by Earth’s bow shock, is balanced by the pressure inside the magnetosphere. To first order, the dynamic pressure dominates the plasma behind the bow shock in a region referred to as the magnetosheath, whereas the magnetic pressure dominates inside Earth’s magnetosphere. In the left panel of Fig. 10.2, the Sun would be at the far right (x w 23,500 Earth radii, RE ) and Earth is at the origin. On the day side (i.e., for x > 0), the three main regions are (1) the undisturbed IMF (occurring for x > 10 RE ) which has a southward component; (2) the magnetosheath, which is clearly visible in this panel as the region of high plasma pressure; and (3) the magnetosphere, the region closest to Earth with low plasma pressure. The bow shock separates the IMF from the magnetosheath and the magnetopause separates the magnetosheath from the magnetosphere. The region of the highest pressure on the day side at high latitudes is referred to as the “cusp” regions. On the night side, the long extended magnetosphere is referred to as the “magnetotail.” Fig. 10.3 shows a schematic view of these regions and boundaries for Mercury, Earth, Jupiter, and Saturn. The right panel of Fig. 10.2 shows a compressed Earth’s magnetosphere during the passage of an extreme CME. In that case, the magnetopause has been pushed very close to Earth’s surface and the extent of the polar cap has greatly increased. The IMF orientation at Earth is dominated during quiet times by the Parker spiral (Parker, 1963), which gives an orientation primarily in the ecliptic plane with an angle of about 45 degrees with respect to the radial direction. The northesouth component of the IMF is typically around 0  5 nT. Geomagnetic storms are the periods when the rapid energy release in the magnetosphere occurs, which are associated with reconnection between the IMF and the Earth’s magnetic field. The disturbed storm-time (Dst) index, or its 1-min equivalent, the Sym-H index, is used to quantify the strength of a geomagnetic storm. These two indices are

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FIGURE 10.2 Both panels show magnetic field lines (white) and plasma pressure (color contours with unit in nPa) illustrating the state of Earth’s magnetosphere during the young sun’s era before (A) and during (B) an extreme coronal mass ejection event. The Sun is on the right side of the image; spatial units are in Earth’s radius. The magnetosheath is the region of high plasma pressure and is separated from the interplanetary magnetic field by the bow shock and from the magnetosphere by the magnetopause. From Airapetian, V.S., Glocer, A., Gronoff, G., et al. Jun. 2016 Prebiotic chemistry and atmospheric warming of early Earth by an active young Sun. Nat. Geosci. 9, 452e455.

calculated from the disturbance in the horizontal component of the magnetic field measured in low-latitude regions. Dst reaching 100 nT corresponds to an intense geomagnetic storm. Studies in the 1980s revealed that long-duration (3 h or more) southward magnetic fields are necessary to drive an intense geomagnetic storm (Gonzalez and Tsurutani, 1987). Different coupling functions to determine the energy transfer between the solar wind and Earth’s magnetosphere have been proposed, the simplest of which is the dawn-to-dusk electric field, Vx Bz (Burton et al., 1975); more complex ones involve the IMF clock angle (Akasofu, 1981) the plasma b and solar wind density (Borovsky, 2008). Long-duration southward magnetic fields occur mostly during the passage at Earth of CMEs; the magnetic ejecta are the main source of geomagnetic storm, followed by the sheath preceding the ejecta (Kilpua et al., 2017). Corotating interaction regions are the source of most non-ICME intense geomagnetic storms. During solar cycle 23 (1995e2006), there were 93 intense geomagnetic storms, more than 80 of which were caused by CMEs (Zhang et al., 2007; Echer and Gonzalez, 2004). A significant portion of intense geomagnetic storms result from a sequence of several CMEs (Xie et al., 2006; Lugaz et al., 2015). Studies (Lugaz et al., 2017) have shown (1) how preconditioning of the magnetosphere and the solar wind can result in more intense geoeffects when two CMEs occur in succession without interaction (Liu et al., 2014a), (2) how two interacting CMEs may be more geoeffective

3. Impact at Earth

FIGURE 10.3 (From top left, clockwise) The magnetospheres of Mercury, Earth, Saturn, and Jupiter. These four planets have an internal magnetic field and the shape of their magnetosphere is generally similar although the scales of the magnetospheres vary by several orders of magnitude. The interaction of the solar wind with each planet results in the formation of a bow shock (which is also present in planets without an internal magnetic field, such as Mars or Venus), whereas the shocked solar wind material is separated from the planetary plasma by a magnetopause. Because of the large solar wind dynamic pressure at Mercury’s orbit and the weak planetary magnetic field, the Hermean magnetopause is close to the surface (at a distance of about 0.4 planetary radius). The opposite situation occurs for Jupiter, with the day-side magnetopause sometimes extending to 100 planetary radii. Figure from Fran Bagenal and Steve Bartlett.

together than separate, and (3) how the interaction of more than two CMEs can result in a complex ejecta without a steady magnetic field that is not able to affect Earth strongly (Burlaga et al., 2003). Earth’s position corresponds to a typical CME transit time of 80 h (Brueckner et al., 1998), with extremes between 17 and 120 h. Two effects are to be taken into consideration to determine the likelihood of CMEeCME interaction: the typical waiting time between CMEs of the same active region (Moon et al., 2003) and the deceleration of CMEs between the Sun and the Earth (Gopalswamy et al., 2001). Strong driving of the magnetosphere may be associated with strong storms but also extreme substorms (Tsurutani et al., 2015), which are releases of energy associated with internal magnetospheric processes, primarily caused by reconnection in the night side in the magnetotail.

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3.2 MAGNETOPAUSE AND RADIATION BELTS Of significant importance to understanding the near-Earth environment are the radiation belts, which were discovered in 1959 (van Allen and Frank, 1959). As a CME affects Earth, the reconnection between the CME’s magnetic field and Earth’s magnetic field creates high-energy particles in the inner magnetosphere, radiation belts, and ring current (Miyoshi and Kataoka, 2005). These particles precipitate onto open field lines into Earth’s upper atmosphere (Hardy et al., 1985). The currents associated with these energetic particles can result in geomagnetically induced currents (GICs), which may overload the grid system and corrode pipelines (Pulkkinen et al., 2005; Ngwira et al., 2013). Because they are grounded to Earth, large electric transformers are particularly vulnerable to GICs (Kappenman, 2003). A HydroQue´bec transformer failure in Mar. 1989 associated with a large geomagnetic storm is the most often-cited space weather effect in the space age; it resulted in a regional blackout. These particles also contribute to the energy input on top of the Earth’s atmosphere. Solar radiation is typically the main source of energy deposited into Earth’s atmosphere (Roble et al., 1987), but energy from the magnetosphere may become the dominant source during geomagnetic storms (Fuller-Rowell et al., 1994). Aurorae account for a small share of this deposited energy, which is mostly associated with Joule heating (Foster et al., 1983; McHarg et al., 2005). The increase in Joule heating during geomagnetic events in the high latitude regions creates gravity waves, which globally affect Earth’s atmosphere (Yiǧit and Medvedev, 2010). CMEs affect the global magnetosphere and radiation belts in a number of different ways: (1) the high dynamic pressure associated with the CME sheath compresses the day-side magnetosphere, routinely pushing the magnetopause from about w12 RE to the near-geosynchronous orbit of 6.6 RE (Shue et al., 1998). Because Earth’s outer radiation belt typically extends up to about 7 RE , this “contraction” of Earth’s magnetosphere can result in losses of energetic particles owing to magnetopause shadowing (Turner et al., 2012). In this process, initially closed field lines, in which energetic particles are trapped, connect through the magnetopause to the magnetosheath when the magnetosphere becomes compressed. This results in a loss of energetic particles in the outer radiation belt. This same phenomenon can occur as a result of other sources of high dynamic pressure, especially compression regions associated with corotating interaction regions (CIRs). In addition, if the high dynamic pressure is combined with a southward magnetic field, the magnetopause can be pushed even farther earthward owing to the additional effect of reconnection, as discussed, for example, in Lugaz et al. (2015). An extreme case is shown in the right panel of Fig. 10.2. (2) Associated with storms and substorms, high-energy particles are injected and/or accelerated into the radiation belts (Shprits et al., 2006; Thorne et al., 2013). The competition between losses and acceleration or injection makes the study of space weather effects on the radiation belts especially complicated (Reeves et al., 2003). (3) The passage of the magnetic ejecta is characterized by a solar wind with low mach numbers. For effect, it moves the bow shock sunward and results in a number of changes in the shape and behavior of the

4. Impact at Other Planets and Throughout the Solar System

magnetosphere (Lavraud and Borovsky, 2008). Under extreme circumstances, the bow shock may even disappear as the solar wind becomes sub-Alfve´nic (Chane´ et al., 2012; Lugaz et al., 2016a).

3.3 EFFECTS OF SOLAR ENERGETIC PARTICLES High-energy particles are primarily (1) SEPs accelerated in the corona by the CMEdriven shock or reconnection and transported onto magnetic field lines to Earth; and (2) energetic storm particle (ESP) events that are particles locally accelerated by the CME-driven shock when it passes over Earth. The most energetic SEPs may reach Earth only 20 min after the start of an event at the Sun. On the other hand, ESPs arrive at Earth 1e2 days after the start of an event at the Sun. This delay is the typical propagation time between the Sun and the Earth of a fast CME. Often, these locally accelerated particles lead to a peak in measured flux as the CME-driven shock passes Earth. Although potentially the most dangerous, these can be forecasted well in advance. Both of these types of energetic particles may affect communications with satellites in orbit around Earth, such as the global positioning systems, which are used by a large variety of industries and technologies from agriculture to air traffic (Lambour et al., 2003). Energetic particles may cause spacecraft anomalies and may degrade the solar panels of satellites over time. SEPs may directly affect astronauts in the space station, or in cruise during a planetary mission (to the moon or potentially to Mars or other objects in our solar system). One of the largest SEPS events occurred in Aug. 1972, about halfway between the Apollo 16 and Apollo 17 lunar missions (Lockwood and Hapgood, 2007). Had the astronauts been on cruise between the Earth and the moon at the time, the radiation dose that they would have received would have been life-threatening.

4. IMPACT AT OTHER PLANETS AND THROUGHOUT THE SOLAR SYSTEM Although most measurements related to space weather have been made for Earth, space weather effects have been studied for all planets of our solar system. Specific reviews have been written regarding space weather effects at Venus (Futaana et al., 2017), Mars (Luhmann et al., 2017), Pluto (Bagenal et al., 2016), and planets in general (Lilensten et al., 2014). Many additional studies have been published for Mercury (Slavin et al., 2014), Jupiter (Dunn et al., 2016), Saturn (Prange´ et al., 2004), and beyond. The goal of this section is not to provide an extensive review of this wide topic, but to highlight some results that may be important for stellar astrophysics and stareplanet interactions. Comparative magnetospheric studies are an active field of research (Fig. 10.3). Reviews by Jackman et al. (2014) and Mauk and Fox (2010) focus on the radiation belts. At the core of the different space weather responses at various planets is a confluence of multiple factors: (1) the presence or absence and strength of an internal

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planetary magnetic field, (2) the nature of the planetary atmosphere and the presence of other sources of internal magnetospheric ions, and (3) the variation in the solar wind and IMF with the distance from the Sun. Of primary importance regarding the solar wind parameters are the dynamic pressure and mach number of the solar wind that influence the compression of the planetary magnetosphere and the formation of a bow shock. Because the Alfve´n and sound speeds decrease with the radial distance from the Sun, whereas the solar wind speed increases slowly with radial distances up to Neptune’s orbit or beyond, the solar wind average fast magnetosonic mach number increases from about 3 at Mercury to 5.5 at Earth and to 8e9 at Jupiter and Saturn (Slavin et al., 1985). On the other hand, because the density decreases approximately as r 2 , the dynamic pressure experienced by Mercury is about 200e300 times larger than that experienced by Jupiter (Lilensten et al., 2014).

4.1 MERCURY Mercury (top left of Fig. 10.3) has a relatively weak internal magnetic field; as the closest planet to the Sun in our solar system, it experiences a large dynamic pressure from the solar wind. Because of these two factors, Mercury’s magnetosphere is small (the bow shock and magnetopause subsolar distances are typically 2 and 1.4 RM , respectively, with RM Mercury’s radial size). To understand Mercury’s space weather we need to understand some facts: Solar wind streams have not yet interacted at Mercury’s orbital distance; and CMEs may have a higher speed because they have experienced interplanetary deceleration for much lower times, but also higher magnetic field strength and density than those measured near Earth, because they have not expanded much. CMEs have been found to result in geomagnetic storms and substorms (Slavin et al., 2014; Sun et al., 2015), although reconnection between the IMF and Mercury’s magnetic field appears to be primarily controlled by the mach number and plasma b, not the angle between the IMF and Mercury’s magnetic field (Dibraccio et al., 2013). Owing to the small magnetospheric size and the lack of internal plasma source beside the relatively tenuous atmosphere, the magnetopause can reach all the way down to Mercury’s surface during instances of CMEs (Kallio and Janhunen, 2003; Zhong et al., 2015; Winslow et al., 2017). This further depletes the atmosphere. Sub-Alfve´nic conditions (i.e., periods when the solar wind speed is slower than the local Alfve´n speed) typically occur when the IMF becomes high, whereas the solar wind remains relatively slow and sparse. They result in a different mode of solar windemagnetosphere coupling. First and foremost, the bow shock disappears, yielding more direct coupling between the planetary magnetosphere and the solar wind. Measurements by MESSENGER, a mission that orbited Mercury from 2011 to 2015, revealed that sub-Alfve´nic conditions are rare but common, occurring primarily during the passage of CMEs, with an occurrence rate of about twow2 times per year (Lugaz et al., 2016b).

4. Impact at Other Planets and Throughout the Solar System

4.2 VENUS AND MARS Between Venus and Mars orbits, the IMF strength decreases by a factor of 3, the dynamic pressure increases by a factor of 4e5, and the solar wind mach number increases from 5 to 6.5 on average. Many changes affect the solar wind: (1) CIRs form, steepen, and drive shocks. Jian et al. (2008a), for example, found that only 3% of CIRs have a shock at 0.7 AU, compared with 24% at 1 AU. The percentage is expected to reach nearly 100% at 2 AU, and CIRs are often preceded by shocks at Mars’ distance (Lee et al., 2017). (2) CMEs are often measured as being in the process of interacting near Venus’ orbit (Farrugia and Berdichevsky, 2004) or Earth (Lugaz et al., 2017), whereas interaction is mostly complete farther away, forming merged interaction regions (MIRs) (Burlaga et al., 2003), as seen, for example, in Prise et al. (2015). At the same time, CME expansion results in near doubling of their size from Venus to Mars (Jian et al., 2008b). In terms of space weather, what makes both planets special is the absence of a global internal magnetic field. Mars has crustal remnant magnetic fields (Acuna et al., 1998), primarily in its southern hemisphere. As such, the interaction of these planets with the solar wind is similar to that of comets (Ma et al., 2004), with an induced magnetosphere (Zhang et al., 1991). There is also large variability in the location of the bow shock (Vignes et al., 2000; Vech et al., 2015). The most significant difference between Mars and Venus is in the presence of a thick atmosphere for Venus, whereas that of Mars is faint. Ions can escape from both ionospheres (Cloutier et al., 1999; Dubinin et al., 2011; Jarvinen et al., 2016). This is particularly important during the passage of CMEs or CIRs because the atmospheric escape may double at Venus (Edberg et al., 2011) and may increase by more than an order of magnitude at Mars (Jakosky et al., 2015). On the other hand, particle precipitation into the atmosphere has been shown to be enhanced during CME passage over Venus (Gray et al., 2014).

4.3 JUPITER AND SATURN All of the gas giants have strong internal magnetic fields and large magnetospheres (Zarka, 1998; Bhardwaj and Gladstone, 2000). For both Jupiter and Saturn, the presence of orbiting moons results in an additional source of magnetospheric ions that must be taken into account. Of all of the magnetospheres in our solar system, Jupiter’s is the one most affected by internal processes (Io plasma torus, etc.) (Cowley and Bunce, 2001; Hill, 2001). However, it is also clearly affected by CMEs and CIRs that take the form of MIRs and interplanetary shocks and that have been found to result in an increase in radio and auroral emissions (Gurnett et al., 2002; Bunce et al., 2008), compression of the day-side magnetosphere and phenomena similar to geomagnetic storms and substorms (Clarke et al., 2009; Delamere and Bagenal, 2010; Nichols et al., 2017).

4.4 OUTER HELIOSPHERE Although this is not space weather proper, MIRs composed of dozen of CMEs and CIRs have been measured in the outer heliosphere by the Voyager spacecraft

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(Burlaga et al., 1997; Richardson et al., 2005). The Voyager 1 and 2 spacecraft, launched in 1977, have crossed the first boundary between the heliosphere and the local interstellar medium (LISM): the termination shock, where the solar wind becomes subsonic. The region beyond this boundary is referred as the heliosheath. In 2012, Voyager 1 crossed the heliopause, the boundary separating the heliosheath and the LISM. MIRs have been measured past the termination shock in the heliosheath (Burlaga et al., 2011) and potentially in the LISM (Richardson et al., 2017). These long-duration periods (up to several months) of higher velocity and magnetic field strength can be traced back to specific periods when the Sun was in high activity, such as Mar.eApr. 2001 and Oct.eNov. 2003 or the solar maximum of 2012 (Liu et al., 2014b). The measured asymmetry of the termination shock by w10 AU can be partially explained by changes in dynamic and magnetic pressures during these intense periods (Stone et al., 2008). IBEX, a mission that maps the interaction of the heliosphere with the LISM, also revealed the time variability of the interaction of the global heliosphere with the LISM (Zirnstein et al., 2015; McComas et al., 2017), which will also be a topic at the core of the IMAP mission planned for the middle of the 2020s (Schwadron et al., 2016).

5. STARePLANET INTERACTION: SPACE WEATHER IN EXOPLANETARY SYSTEMS AND CONCLUSIONS The field of exoplanetary space weather dates back almost to the first low-mass exoplanet detection in the mid-2000s, with studies of the effects of stellar eruptions on the habitability of extrasolar planets (Khodachenko et al., 2007; Lammer et al., 2007). With these studies, there has been renewed interest in the long-term evolution of planetary atmospheres confronted by the more active young Sun (Lammer et al., 2008; Benz and Gu¨del, 2010), but also in the frequency and magnitude of flares and eruptions in Sun-like stars (Maehara et al., 2012; Shibayama et al., 2013). Using data from Kepler, a mission to detect exoplanets, these studies revealed that G-type stars may have flares with orders of magnitude more energy than the largest solar flares ever measured. This has led to the development of a new subfield of astrophysics focusing on the impact of stellar winds and stellar eruptions on planets and their magnetosphere and atmosphere, including for hot Jupiter (Cohen et al., 2011; Lecavelier des Etangs et al., 2012) and terrestrial-like planets (Drake et al., 2013; Vidotto et al., 2013; Cohen et al., 2014; See et al., 2014; Airapetian et al., 2017). Most of these studies are based on numerical models (Fig. 10.4), but observations of temporal variations in atmospheric losses of hot Jupiters (Lecavelier des Etangs et al., 2012) or warm Neptunes (Bourrier et al., 2016) have been reported and stellar ejections have been searched for (Leitzinger et al., 2014). Space weather is a relatively young field, but the study of the interactions among the solar wind, solar eruptions, and different planets from our solar system is a

5. StarePlanet Interaction: Space Weather in Exoplanetary Systems

FIGURE 10.4 The left column shows a star-planet interaction in which the planet is in the sub-Alfve´nic zone of the stellar wind; whereas the right column shows the interaction in the superAlfve´nic regime. The top line shows a Venus-like planet at 0.15 AU from an M-dwarf star; the bottom line displays a Venus-like planet at a Mercury distance of 0.39 AU from a Gtype star. The star is on the right-hand side of the images. Magnetic field lines are shown in white. Color contours are the ratio of oxygen ions (from planetary origin) to hydrogen ions (from stellar wind origin). Depending on the exact dynamics of the interaction between the planet and its star, the ion escape from the planet is modified and the shape and extent of the ion-induced planetary “tail” can vary significantly. Figure from Cohen, O., Ma, Y., Drake, J.J., et al. Jun. 2015 The interaction of Venus-like, M-dwarf planets with the stellar wind of their host star. Astrophys. J. 806, 41. doi:10.1088/0004-637X/806/1/41.

mature field because of numerous dedicated missions. The current behavior of the heliosphere offers a glimpse into the long-term evolution from a young and active Sun to its current state (Barnard et al., 2011). For example, it is unclear whether there are long-term cycles beyond what we can learn from historical records (Lockwood et al., 2017) and ice cores (Traversi et al., 2012; Usoskin et al., 2016). It is

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particularly interesting to determine whether periods of weak magnetic fields and weak solar activity are common. Such Maunder minima may strongly affect the possibility of interplanetary human travel owing to enhanced galactic radiation (Smith et al., 2014; Miyake et al., 2017). As such, studying other G-type stars and their planetary systems may allow us to understand our own solar wind and space weather much better, helping to comprehend the natural cycle of the Sun and its variability. Unique to our solar system, we have a combination of remote observations and in situ measurements that will soon extend from the solar corona, with the Parker Solar Probe, a mission to be launched in 2018, to the LISM with Voyager. These in situ measurements provide direct information that will not be available in the near future for exoplanetary systems, and a way to validate remote observations directly. Many of the tools and methods developed to study the solar wind, the global heliosphere, and planetary magnetospheres can be used to study exoplanetary systems and the physics of stareplanet interactions.

ACKNOWLEDGMENTS N. L. would like to thank Dr. R. Winslow for comments on a draft of this chapter. He was supported by National Aeronautics and Space Administration Grant NNX15AB87G and National Science Foundation Grants AGS1435785, AGS1433213, and AGS1460179.

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Reeves, G.D., McAdams, K.L., Friedel, R.H.W., et al., May 2003. Acceleration and loss of relativistic electrons during geomagnetic storms. Geophys. Res. Lett. 30, 1529. https:// doi.org/10.1029/2002GL016513. Richardson, J.D., Wang, C., Kasper, J.C., et al., Jan. 2005. Propagation of the October/ November 2003 CMEs through the heliosphere. Geophys. Res. Lett. 32, 3eþ. https:// doi.org/10.1029/2004GL020679. Richardson, J.D., Wang, C., Liu, Y.D., et al., Jan. 2017. Pressure pulses at Voyager 2: drivers of interstellar transients? Astrophys. J. 834, 190. https://doi.org/10.3847/1538-4357/834/ 2/190. Riley, P., Gosling, J.T., Pizzo, V.J., Jul. 1997. A two-dimensional simulation of the radial and latitudinal evolution of a solar wind disturbance driven by a fast, high-pressure coronal mass ejection. J. Geophys. Res. 102, 14677e14686. https://doi.org/10.1029/ 97JA01131. Roble, R.G., Ridley, E.C., Dickinson, R.E., Aug. 1987. On the global mean structure of the thermosphere. J. Geophys. Res. 92, 8745e8758. https://doi.org/10.1029/ JA092iA08p08745. Rouillard, A.P., Davies, J.A., Forsyth, R.J., et al., Jul. 2009. A solar storm observed from the Sun to Venus using the STEREO, Venus Express, and MESSENGER spacecraft. J. Geophys. Res. 114, A07106. https://doi.org/10.1029/2008JA014034. Ruffenach, A., Lavraud, B., Owens, M.J., et al., Sep. 2012. Multispacecraft observation of magnetic cloud erosion by magnetic reconnection during propagation. J. Geophys. Res. 117, A09101. https://doi.org/10.1029/2012JA017624. Savani, N.P., Owens, M.J., Rouillard, A.P., et al., May 2010. Observational evidence of a coronal mass ejection distortion directly attributable to a structured solar wind. Astrophys. J. Lett. 714, L128eL132. https://doi.org/10.1088/2041-8205/714/1/L128. Schwadron, N.A., Opher, M., Kasper, J., et al., Nov. 2016. Interstellar mapping and acceleration Probe (IMAP). J. Phys. Conf. 767, 012025. Schwadron, N.A., Cooper, J.F., Desai, M., et al., Nov. 2017. Particle radiation sources, propagation and interactions in deep space, at Earth, the moon, Mars, and beyond: examples of radiation interactions and effects. Space Sci. Rev. 212, 1069e1106. https://doi.org/ 10.1007/s11214-017-0381-5. See, V., Jardine, M., Vidotto, A.A., et al., Oct. 2014. The effects of stellar winds on the magnetospheres and potential habitability of exoplanets. Astron. Astrophys. 570, A99. https:// doi.org/10.1051/0004-6361/201424323. Shen, F., Feng, X., Wu, S.T., et al., Jun. 2007. Three-dimensional MHD simulation of CMEs in three-dimensional background solar wind with the self-consistent structure on the source surface as input: numerical simulation of the January 1997 Sun-Earth connection event. J. Geophys. Res. 112, 6109eþ. https://doi.org/10.1029/2006JA012164. Shen, C., Wang, Y., Wang, S., et al., Dec. 2012. Super-elastic collision of large-scale magnetized plasmoids in the heliosphere. Nat. Phys. 8, 923e928. https://doi.org/10.1038/ nphys2440. Shibayama, T., Maehara, H., Notsu, S., et al., Nov. 2013. Superflares on solar-type stars observed with Kepler. I. Statistical properties of superflares. Astrophys. J. Suppl. 209, 5. https://doi.org/10.1088/0067-0049/209/1/5. Shiota, D., Kataoka, R., Feb. 2016. Magnetohydrodynamic simulation of interplanetary propagation of multiple coronal mass ejections with internal magnetic flux rope (SUSANOOCME). Space Weather 14, 56e75. https://doi.org/10.1002/2015SW001308.

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The SolareStellar Connection

11 Gibor Basri

University of California, Berkeley, CA, United States

CHAPTER OUTLINE 1. Introduction .......................................................................................................363 2. Photometry and Starspots ...................................................................................364 3. Activity Cycles ...................................................................................................368 4. Chromospheric and Transition Region Diagnostics................................................369 5. Coronal Diagnostics ...........................................................................................371 6. Flares and Mass Loss .........................................................................................373 7. RotationeActivityeAge Relations ........................................................................376 8. Final Thoughts ...................................................................................................382 References .............................................................................................................383

1. INTRODUCTION The Sun is often characterized as an “average” star. This is largely a misimpression. The “average” star (in terms of number counts) is a little over a third the mass of the Sun, not much more than half its temperature, a few hundred times fainter, and lives more than 10 times longer. The Sun is younger and located further from the Galactic Center than most stars. It has a higher content of elements heavier than helium than average. But of course, it is “our” star. That means that we know much more about it than any other star, and it thus serves as a calibrator, anchor point, or launching point for studies of other stars. That is true for studies of stellar interiors and evolution as well as for studies of stellar atmospheres and magnetic activity. There is a rich set of connections to the Sun in the rapidly burgeoning field of asteroseismology, but in this chapter, I will concentrate on the exterior properties of the Sun compared with other main sequence stars. In particular, I will concentrate on magnetic phenomena (without which the exterior of the Sun would be boring). Because other chapters give copious detail about the magnetic Sun and the accompanying theory, this chapter concentrates on observations of other stars (with the Sun as a constant reference point). I have made no attempt to provide a comprehensive list of references; many of the works cited were chosen partly because they cite a good deal of relevant previous work. The Sun as a Guide to Stellar Physics. https://doi.org/10.1016/B978-0-12-814334-6.00013-3 Copyright © 2019 Elsevier Inc. All rights reserved.

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2. PHOTOMETRY AND STARSPOTS It might seem that the simplest sort of measurement one can imagine making about a star (other than position) is its broadband visual brightness. Indeed, catalogs of stellar positions and brightness have been around for thousands of years. Measurements get much harder if one wants to know the intrinsic brightness (luminosity in the visible band) of the star, because that requires knowing its distance and possibly the amount of interstellar extinction. They get even harder if one wants to understand stellar atmospheric variability and magnetic effects on solar-type stars, because the amplitudes of these are fractions of a percent of the luminosity. It is for these reasons that before the current century the Sun was essentially the only solar-type star for which this information was available with high enough precision and time coverage to glean much physical interest. The advent of precision photometric space telescopes (particularly Convection, Rotation and Planetary Transits [COROT] and Kepler) increased the numbers of stars with such data to hundreds of thousands. Some of these stars are much less variable than the Sun and some are very much more variable (Basri et al., 2010, Fig. 11.1). Before the 21st century, the rotation periods of a few hundred stars had been determined through the brightness variations owing to starspots. These stars are generally more active than the Sun (and younger and/or lower mass). Now there are tens of thousands of rotation periods known, and this number will continue to increase rapidly. In Section 7, I discuss some of the interesting discoveries this has made possible. It is observed that as the Sun becomes more active (the surface magnetic flux and the sunspot number increases) the total luminosity of the Sun actually becomes a bit larger, too. This is counterintuitive at first, because sunspots are dark and decrease the intensity of the part of the Sun on which they reside. However, they are accompanied by faculae, which occupy a larger area around the spots in active regions and are brighter than the quiet Sun. Things are more complicated because spots will have their maximal effect near the disk center (where they have the largest projected area to an observer) whereas faculae are brightest near the limb (for geometrical reasons having to do with the mechanism that produces them). In general, sunspots tend to produce discrete dips in the total intensity whereas faculae produce a more general brightening because they are more spatially extended than spots. It can be hard to disentangle these effects when a lot of both types of features are present. One must also remember that the Sun is seen nearly equator-on whereas stars are seen at all inclinations; the viewing angle affects how spots and faculae will influence the net brightness. A good discussion of these effects can be found in Shapiro et al. (2017) and in their chapter in this book. Regarding other stars, two distinct types of behavior are discussed by Lockwood et al. (2007) and Hall et al. (2009). Many stars follow the behavior of the Sun in displaying a net brightening when spots are more prevalent. However, sufficiently chromospherically active stars (with an Rhk of w4.8 or larger) show the opposite: the overall brightness of the star is lower when starspots are more prevalent.

2. Photometry and Starspots

4

Quiet Sun

1 0 −1 9834255

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FIGURE 11.1 Comparison of light curves from the Sun (red lines, SOHO data) and sample Kepler stars (KIC numbers label each black curve). Notice that each panel has a different vertical scale. The top 2 panels show the quiet and active Sun with comparable Kepler stars (of similar temperature and rotation period). The bottom two panels show Kepler stars that are more active than the Sun ever gets (the red line in each is the same as in the upper right). These stars have shorter rotation periods. The most active panel shows a single feature per rotation, while the others have more complicated light curves. From Basri, G., Walkowicz, L.M., Batalha, N., et al., 2010. Photometric variability in Kepler target stars: the sun among starsda first look. Astrophys. J. 713, L155.

This means that the presence of large spots overwhelms the presence of faculae. A detailed discussion of how this might work was given by Shapiro et al. (2014). The difference serves as a warning that extending the solar analogy to more active stars becomes increasingly problematic as the activity level increases. Kepler light curves are usually only differential; absolute brightness variations cannot be inferred from them. They show dips characteristic of starspots in shape and width, usually only one or two per rotation. This is because the inherent spatial resolution is poor; essentially one is really sampling a hemispheric starspot-covering fraction. A study (Montet et al., 2017) also provided absolute photometry for a few hundred Kepler stars (Fig. 11.2), with a similar outcome as before in terms of the dominance of spots increasing as the activity level increases. A basic question concerning the solarestellar connection has been phrased, “Is the Sun a typical solar-type star?” There have been persistent suggestions that it might be more inactive than the average star of its temperature and age. Some of

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FIGURE 11.2 Absolute photometry of 12 Kepler stars. The photometry is derived by measuring the behavior of many selected stars near the target star in full-frame images (collected once a month or so) by Kepler and assuming they are constant on average. The colors indicate each yearly set of measurements (because the Kepler spacecraft had four seasonal positions that placed stars on different pixels, then repeated). These stars were selected as showing potential activity cycles. From Montet, B.T., Guadalupe, T., Foreman-Mackey, D., 2017. Long-term photometric variability in Kepler fullframe images: magnetic cycles of sunelike stars. Astrophys. J. 851, 116.

these come from the photometric analysis described just previously, which generally finds that the amplitude of variability of the Sun is less than the typical comparable stars in those samples. This idea of the relatively inactive Sun continues to resurface, usually when the sample is relatively small (at most a couple hundred stars). One should worry about sample bias in those cases, because generally speaking, very quiet stars are more affected by detection limits and noise, and often show no signal, so the samples tend to favor more active stars. Indeed, the Sun itself would simply be “inactive” in many studies. More recently, there was a claim that the Sun is unusually quiet, based on a much larger sample from Kepler (Gilliland et al., 2011). This was refuted by Basri et al. (2013), who found that the variability measure in the previous work is unsuitable for this purpose and that the Sun fits clearly into the Kepler solar-type sample with respect to its photometric variability. This is true whether talking about differential variability on timescales of a day or a few years (although it does not address

2. Photometry and Starspots

the absolute variability including faculae). Approximately a quarter of the Kepler sample is more active than the Sun ever gets. It is also true, however, that many solar-type stars are more photometrically quiet than the Sun; presumably they are also older. Another clear result is that photometric variability becomes larger (in a differential sense) as one moves to cooler stars. One possible reason for this is that the contrast between magnetic and nonmagnetic regions might be larger in cool stars because there is a greater sensitivity to temperature differences in the visible (given the behavior of the Planck function). These stars also tend to show larger magnetic surface fluxes. The size of convection cells on cool stars will be smaller given their higher surface gravity, but the stars themselves are also smaller. A cogent explanation of this dependence of variability on stellar parameters has not yet been given. Other indications that the solar analogy has its limits come from the inferred locations of starspots. It is possible to map starspot locations from either Doppler imaging (Strassmeier, 2009) or Zeeman Doppler imaging (ZDI) (Vidotto et al., 2014). One general trend is that very active stars tend to have starspots concentrated near their poles. Solar-type stars will have mid- to low-latitude spots if they have dynamo action like the Sun. Rapid rotation can cause fields to drift poleward, and deep convection zones likewise will produce more polar fields (Schussler et al., 1996). More active stars also appear to have longer spot lifetimes (particularly in units of rotation period), which makes it easier to infer their rotation periods from synoptic light curves. Indeed, the techniques generally used to find stellar rotation periods struggle during many more active segments of the solar light curve because it is not periodic enough owing to ever-changing spot groups over a rotation or two. Examining the bulk of Kepler light curves for solar-type stars, one finds that there is an increasing tendency for a single rise and fall of intensity per period for the more active stars as the rotation periods decrease below w15 days, and these features can maintain their general phase and appearance for up to tens of rotations. Most of the more slowly rotating stars show two (or occasionally more) features per rotation period and change morphology more frequently. The other physical information that should in principle be discoverable from photometric light curves is a measurement of differential rotation on stellar surfaces. The Sun is a differential rotator with a period of about 25 days at the equator and 29 days at the pole (and a sine law with latitude in between). Although this can easily be tracked over the solar cycle because spots appear at higher latitudes near the beginning of a cycle and at increasingly low latitudes as the cycle progresses, it is difficult to determine the differential rotation from solar photometry that encompasses only a fraction of the cycle. On the other hand, if starspots appear all over a star and are stable over a number of rotations, the light curve and resulting change in the light curve morphology should reflect their drifts with respect to each other. Drifting spot longitudes in two-spot models have long been interpreted this way for RS Canum Venaticorum (RS CVn stars), for example. There have also been a number of claims of differential rotation measured by Doppler imaging (Strassmeier, 2009 and later references).

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There is a fundamental question, of course, whether a spot is actually drifting owing to surface shear or whether we are seeing a change in the “active longitude” where spots appear (especially if the time series is very sparse or spot lifetimes are only a rotation or two). With the advent of long and complete time series from COROT and Kepler, the possibility of really measuring stellar differential rotation has become much more promising. There have been many claims of such measurements on individual stars using a number of different spot modeling methods (Nagel et al., 2016). A large sample was analyzed using multiple closely space peaks in periodograms (Reinhold et al., 2013). Aigrain et al. (2015) showed that for solar-type differential rotation, none of the methods is particularly reliable (although this may be less of a problem for more active stars). My investigations (incorporating a much larger portion of the potential parameter space) show that disentangling the effects of differential rotation and evolution is complex and will require new methodologies. Still, the appearance of many of the Kepler light curves is strongly suggestive of differential rotation, and I believe that we will be able to extract some of this information more reliably.

3. ACTIVITY CYCLES The most well-known characteristic of sunspots is their cyclic behavior. Over an 11-year timescale, the Sun changes from having few spots per rotation to having a much larger number, and back again. The activity cycle actually is twice as long when magnetic polarity is taken into account (it switches sign every sunspot cycle). The level of spot coverage at solar maximum is different each cycle (there may be other timescales associated with these changes in cycle amplitude), and we also know that the Sun occasionally goes through several cycles during which it produces almost no spots (e.g., the Maunder minimum). One question of great interest is whether other stars also show such cycles, what their periods and amplitudes are, and how these things vary with stellar parameters such as mass, rotation, and age. The longest running study relevant to these questions has actually concentrated on Ca II emission (from the chromosphere) rather than on starspots themselves. There is every reason to believe that these two diagnostics are closely related (through the strength of the total surface magnetic field). As mentioned in the last section, the overall brightness of the star might be higher or lower during spot maximum, depending on the importance of faculae, but we expect there to be a positive correlation between a higher presence of spots and the strength of chromospheric activity (it is certainly true on the Sun). This is in the process of being tested more generally now that there are many more starspot records from Kepler and COROT, and Ca II spectra are often collected during studies of stars with transiting planets as the studies try to pin down the stellar parameters with high-resolution spectra. The longest strings of observations of Ca II emission strengths come from the Mount Wilson survey and its descendants. Pevtsov et al. (2016) have shown that

4. Chromospheric and Transition Region Diagnostics

on the Sun, the magnetic fields in the active region that cause the Ca II emission are correlated closely enough to the presence and polarity of sunspot magnetic fields to allow construction of “pseudomagnetograms” just from Ca II. The Mount Wilson study was started by Olin Wilson in 1966 (Wilson, 1978). Its results on stellar cycles were summarized by Baliunas et al. (1995). They classified stars into more active rapid rotators and less active slower rotators; this paradigm will come up several more times in this chapter. The more active stars have a tendency to be more variable, but the variability often does not show a clear cycle (looks chaotic over a few years). Clear cycles were detected more often in the less active sample, particularly among G and K stars (not many F stars show easily measurable Ca II emission). A few cycle periods of 5 years or less were found; they tend to be shorter for the more active stars. A number of G stars with cycle periods similar to those of the Sun were identified. Some stars were assigned two cycle periods on the basis of significant power at two peaks in their periodogram. The Mount Wilson sample was later found to contain a number of subgiants that look relatively inactive and were incorrectly thought to be stars undergoing Maunder minimum events. The Lowell Observatory has continued the long-term observation of Ca II emission from solar-type stars (Hall et al., 2007). We can likely expect another large study on stellar cycles from this sometime soon. Cycles are also being searched for in photometric data from the ground (Lehtinen et al., 2017) or space with Kepler (Montet et al., 2017) (Fig. 11.2). The advent of Gaia parallaxes will allow many solar-type main sequence stars to be placed more accurately onto evolutionary isochrones, which will clarify the behavior of solar-mass stars in the latter parts of their main sequence lifetimes (unfortunately, this will not work for lower mass stars because of their very long evolutionary timescales).

4. CHROMOSPHERIC AND TRANSITION REGION DIAGNOSTICS The solar chromosphere is actually difficult to see in visible or longer wavelengths; one is much better off with UV or more energetic wavelengths because the plasma ranges from 104 to 107 K above the photosphere. Images of the solar chromosphere are often taken with an Ha filter, which shows chromospheric structures nicely, because optical depth unity occurs in the chromosphere in the core of that line. Unfortunately, the Ha spectral line is not a particularly good diagnostic for stellar chromospheres, because no spatial resolution is possible. Instead, one typically sees just a deep absorption line (as one does on the integrated Sun), and because of its nonelocal thermodynamic equilibrium nature the line shape is not even a particularly useful diagnostic of physical atmospheric conditions. Differences between active and quiet stars on the scale of the solar cycle are subtle; only for very young G stars is the activity is so strong that Ha goes into emission. As one moves to cooler stars, the meaning of the depth of the Ha line becomes increasingly

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ambiguous (because hydrogen in the photosphere is increasingly in the ground state, whereas the lower level of Ha is the first excited state). This becomes confusing to the point that in early M stars it has been suggested that inactive stars may show no Ha feature, the feature grows in absorption strength as activity increases, but then it becomes shallower again (on the way to true emission) as activity increases further (Cram and Giampapa, 1987). A better visible spectral diagnostic that has strong enough opacity to place core optical depth unity in the chromosphere and is closer to local thermodynamic equilibrium and thus more sensitive to the thermal structure is the Ca II H and K resonance doublet at 393.4 and 396.9 nm. Even these lines are barely sensitive enough to register the integrated solar chromosphere, which appears as small emission bumps at the bottom of deep and broad absorption lines. However, the strength of the emission is sensitive to the amount of activity enough to serve as a measure of it. Spatially resolved solar studies show that the amount of excess H and K emission is also correlated with the underlying surface magnetic field (Pevtsov et al., 2016), so the strength of the excess emission in stars has often served as a proxy for both magnetic field and magnetic activity. Unfortunately, the Ca II resonance lines are nearly in the UV and it is difficult to get good signal-to-noise in spectra of cool stars compared with solar-type stars with the same visual magnitude. The study of stellar chromospheres in the UV received a great boost with the advent of the International UV Explorer satellite in the 1980s. The analogous resonance lines Mg II H and K are far more sensitive and easy for measuring activity changes but they are well into the space UV at 280 nm. The strongest chromospheric line is Lya, but this is even further in the UV (122 nm) and further experiences substantial interstellar absorption, so it is not an easily used chromospheric diagnostic. Above the chromosphere is the transition region (104e105.5 K), which also exhibits a number of strong UV spectral emission lines. The strongest of these include Si II, IV, C II, IV, He II, and N V in the spectral range at or longward of Lya (120e200 nm) (Fig. 11.3). It appears to be a general characteristic of both the Sun and solar-type stars that all of these diagnostics are correlated with each other (and with chromospheric and coronal diagnostics). Their flux ratios tend to follow power laws over several orders of magnitude (Fig. 11.4), and the power laws tend to be steeper for hotter lines compared with cooler lines (Oranje, 1986; Ayres et al., 1995). This leads to the convenience of not having to observe a star in all the diagnostics; any one of them observed well will give a good indication of what the activity level of the star is at the time of observation. It is also true for the Sun and stars, however, that these levels are time variable. Because of the power law relations between them, it is also true that the variability is greater for the hotter diagnostics. In a general sense, the fluxeflux relations between UV lines in stars of different activity levels are similar to those of solar regions overlying active regions of different strengths. That is, if one uses the line ratios to place solar points on the stellar relations, the Sun and stars seem to have similar structures. This requires some conversion of solar intensities to fluxes, and Judge et al. (2017) pointed out that

5. Coronal Diagnostics

FIGURE 11.3 An example of the UV spectrum of the Sun, with the major emission lines labeled. The resolution is comparable to low-resolution International UV Explorer spectra. From Mount, G.H., Rottman, G.J., 1981. The solar spectral irradiance 1200-3184 A near solar maximum - July 15, 1980. J. Geophys. Res. 86, 9193.

this is not a well-understood process, given the different center-to-limb behaviors of the lines and the unknown distribution of active regions on the stars. Their work provides an illustrative comparison of solar and solar-type stellar activity from the chromosphere to the corona, as well as references to a number of previous such articles. One other purpose to which the UV line fluxes have been put is the calculation of differential emission measures, which give a more physical idea of the state of the plasma at different temperatures in the transition region (in general, more active stars have denser and more voluminous transition regions and coronae). A good example of this type of analysis that compares solar-type stars with the Sun can be found in Jordan et al. (1987).

5. CORONAL DIAGNOSTICS The outer part of a solar-type star’s atmosphere is the corona, with temperatures measured in millions of kelvins. Although the corona makes a spectacular sight in total solar eclipses, one is looking at relatively faint emission caused by electron scattering of solar photospheric light. Coronae can only be studied on other stars in x-rays, where the actual plasma emission occurs. The window on stellar coronae was really opened by the Einstein satellite in the 1980s. As mentioned earlier, total coronal emission has a power-law correlation to other diagnostics of stellar activity, and the x-ray emission from stars is generally more variable than for lower temperature diagnostics (particularly below the transition region), as is true for the Sun. Thus, a snapshot survey of stellar x-ray brightness will give one only an idea of

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FIGURE 11.4 The relation between the logarithmic fluxes in the chromospheric Si II line compared with the sum of the transition region lines Si IV, C IV, and NV, as observed by International UV Explorer. The circles are main sequence stars and downward triangles are supergiants. The emptier the symbol, the hotter the star. The “e” symbols denote dMe stars and the labeled stars at the upper right are active binaries. Notice that they all lie on essentially the same linear relation. From Oranje, B.J., 1986. Magnetic structure in cool stars. IX - ultraviolet emission lines from chromospheres and transition regions. Astron. Astrophys. 154, 185.

the average emission from a given star to within a factor of a few. An excellent review of the general comparison between solar and stellar coronae was given by Rosner et al. (1985), although many more observations have been made since. For a modern compilation, see Wright et al. (2011). The x-ray emission from the Sun arises almost entirely from closed magnetic loop structures. During solar maximum, these emanate from strong active regions and can be large; they cover (in a projected sense) a substantial part of the solar disc. During solar minimum, the x-ray emission comes from much smaller (more

6. Flares and Mass Loss

widely dispersed) closed loops associated with bright point ephemeral active regions, and much of the solar disc is covered by coronal holes (which contribute much less x-ray emission). One conceptual model for this behavior over the cycle is that the Sun has two types of magnetic dynamos operating: a localized smallerscale convective turbulent dynamo that is always present and widely dispersed; and the other dynamo that arises predominantly at the base of the convection zone, produces larger scale structures, and is cyclic. In both cases, however, the corona arises from closed loops and the total coronal emission depends on both the total volume in these loops and the plasma densities within them. In the solar case, the active Sun is made brighter more by the rise in typical density than by the increased volume of the loops. When talking about stellar coronae (which are not spatially resolved and poorly temporally sampled), one has to be cautious in light of the complexity of the solar case. Clearly, there can be several types of coronal structures present at the same time in different mixes, the mix will itself change with time, the types and strengths of dynamos operating on stars can be different from the solar case, and there may be additional structures that have no solar analog. Cases that clearly fall partly outside the solar analogy include RS CVn stars that may have loops connecting two stars, M flare stars that appear to have loops that are nearly the size of the whole star, and T Tauri stars that have an accretion disk that interacts with the stellar magnetic field. The more active stars (those that show more activity than the Sun ever does) have an increasingly important coronal component with temperatures that are tens of millions of kelvins (Mewe, 1996). In particular, solar-type stars that are younger than about 1 billion years old will generally be in this active class. The source of coronal heating is still not fully understood; one theory that has become more prevalent is that “microflaring” is important, and it may be that this is increasingly the case as activity levels become stronger. The basic approach to modeling stellar coronae has centered on a simple analytic loop model proposed by Rosner et al. (1978). This assumes that all coronal emission arises from hydrostatic closed loops, which can be described by the loop length, constant pressure, and temperatures set by energy balance, with the maximum temperature at the top of the loop (simple heating functions are assumed). An example of the application of this method to stellar data can be found in Schrijver et al. (1984), in which Sun-like stars have loop structures similar to the Sun whereas more active stars must actually have more compact loops with much higher pressures (and thus heating rates). The most active stars (and lower-gravity stars) may also have loops with lengths comparable to the stellar radius (Collier Cameron, 1988). Similar lengths are implied in analyzing strong flares on M dwarfs.

6. FLARES AND MASS LOSS The largest variations of hot material and high energy emission result from flares. The cumulative energy distribution of flares follows a power law; stronger flares

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are less common (Van Doorsselaere et al., 2017), and more or less the same power law extends to even higher energies when one looks beyond the Sun to more active stars. Solar flares can be seen in diagnostics ranging from Ha to x-rays and g-rays, and the same is true for stars. The timescales for flare rise and decay can also be either similar to the Sun or substantially longer (several hours of decay) in the case of more energetic stellar flares. Flare frequencies can be very much greater than solar, and also follow a power law (stronger flares are less frequent). The Sun at its most active will generate several flares in a day but averaged over time the rate of significant flares is far lower. The most active M dwarf flare stars are essentially flaring continuously at solar energies and produce much more powerful flares at times (Fig. 11.5). As one goes increasingly younger in age, similar statements can be made for stars even up to a solar mass. A comprehensive review of stellar flares was provided by Haisch et al. (1991). There are some differences between the more energetic stellar flares and the most energetic flares observed from the Sun. In addition to dissipating total energy that can exceed solar flares by up to three orders of magnitude, the inferred geometry of these flares (especially from M dwarfs) implies that the flaring region is up to a stellar radius in dimension. Even the increase in just optical luminosity can be nearly two orders of magnitude for the strongest of the flare stars. They produce a much larger density of very high temperature material. For both the Sun and stars, there is sometimes evidence that a big flare can trigger sympathetic flares (in related magnetic structures soon afterward). This can be seen directly in solar imaging but is inferred for stars by the timing of flares after a large one; they sometimes occur much sooner than would be expected from the general flare frequency.

FIGURE 11.5 Kepler light curve for the flare star GJ 1243 (M4). This is a short cadence observation with a time resolution of about 1 min. The colors indicate a confidence level in the reality of the flare (red is best). In addition to large multiple flares and many smaller ones, the overall slow change in the amplitude of the light curve results from starspots rotating in and out of view. BJD, barycentric Julian Date. From Davenport, J.R.A., Hawley, S.L., Hebb, L., et al., 2014. Kepler flares. I. Active and inactive M dwarfs. Astrophys. J. 797, 121.

6. Flares and Mass Loss

The optical signature of all but the largest solar flares is not observable in integrated optical light even by an instrument as sensitive as Kepler. Optical flares are easier to see on cool stars because the plasma emitting the optical flare has temperatures of roughly 10,000K (heated by downward particle beams impinging on the upper photosphere), which has a much higher surface brightness than photospheres of 3000e4500K. It was therefore a surprise to find that optical flares on solar-type stars and even hotter stars were detected by Kepler with some frequency (although certainly less than on cooler stars). The energies of these events (1033e36 erg), dubbed “superflares” (Shibayama et al., 2013), can greatly exceed the largest solar flares, such as the “Carrington event” in 1859 (estimated to have an energy of about 1033 erg). Stars exhibiting superflares tend to have rotation periods of less than 10 days and show flare frequencies that are 100e1000 times greater, but the phenomenon extends to stars with the solar rotation period. It was even more surprising to find that some of the Sun-like stars hosting strong flares do not show other signs of youth. It is not clear how often superflares can arise on the Sun (at most once every few thousand years, if at all, for the stronger ones), but this question generated some concern (it would be very bad for our electronic civilization). The issue is still unresolved. A survey of the main Kepler sample (Van Doorsselaere et al., 2017) shows that F stars have more powerful flares with slower rise and decay times (but lower frequency) than G stars, whereas K and M stars show greater frequency, similar distribution of power (but extending to higher power with a shallower slope), and similar to slightly longer timescales. The distributions of flare frequency per energy (dN/dE) for solar-type stars are power laws that seem to extend across the full activity spectrum and the Sun sits in the middle of them (Shibayama et al., 2013). It appears that in a broad sense, the flaring phenomenon is similar throughout solar-type stars and the main difference between more and less active stars is the amount of magnetic flux on their surface (and its consequences). From the Sun, one can directly observe high-energy particles generated by flares; these undoubtedly are also present in stellar flares. Flares likely also produce coronal mass ejections (CMEs). There is indirect evidence for these in some stellar flares, from either radio emission (Osten et al., 2005) or sometimes blue-shifted components in the Balmer lines (Houdebine et al., 1990). The precursors of solar CMEs are eruptive prominences; those are hard to observe directly on other stars. There is evidence of large prominence-like structures seen in Ha on some active stars, such as AR Lac (Frasca et al., 2000) or AB Dor and other young stars (Collier Cameron and Woods, 1992). The amount of mass the Sun loses this way is negligible compared with its mass even over billions of years, but it has been suggested (Mullan et al., 1992) that the total output of CMEs from M dwarfs might be a significant contributor to the interstellar medium, or able to affect the stellar evolution of low-mass stars (Houdebine et al., 1990). Direct measurement of the analogs of the solar wind on other stars is difficult. They produce neither emission nor absorption that is easily measured against the stellar signal at any wavelength. Attempts have been made in the radio, but they

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cannot detect winds as weak as the Sun’s. The best method so far has been to try to detect the pileup of neutral hydrogen at the interface between the star and the interstellar medium (ISM), using the Lya line. This can only be done for nearby stars because of absorption from the ISM itself; a good summary of results is in Wood et al. (2002). They find that mass loss scales with x-ray surface flux, which implies that young solar-type stars have far greater winds than the current Sun. This is also likely the case for lower-mass stars that have longer and/or stronger active phases. Early planetary systems would be subjected to these winds along with the increased ionizing photon fluxes; these are known to be involved in water and atmospheric losses on Mars and Venus. There is a growing field of study that focuses on how planets in habitable zones (where water could in principle be a liquid on the surface) will be affected by the strong UV and high-energy fluxes and much more frequent and powerful flares coming from young stars. This issue is particularly acute for planets in the habitable zones of M dwarfs, which are both the most numerous stars and apparently more likely to host Earth-sized planets. The surface strength of magnetic activity is similar or higher on these stars compared with Sunlike stars, but the bolometric luminosities are much lower. This pushes the habitable zone much closer to the star, so planets there are subjected to greatly (up to thousands of times) higher fluxes of high-energy photons and particles. These can dissociate molecules in the upper atmosphere, ionize atoms, produce strong extended exospheres, and thus remove hydrogen even if it is originally bound up in molecules (Linsky, 2014). The slower contraction of lower mass stars off the pre-main sequence means that a planet that will be habitable on the main sequence has its water as steam for much longer. Combined with the slower decay of generally stronger magnetic activity on the main sequence for mid- to late-type M dwarfs (West et al., 2008), there is a real question as to whether these planets will be able to retain water at all (as Venus has not) even though they are in the habitable zone. This is an active and fertile area of research.

7. ROTATIONeACTIVITYeAGE RELATIONS Wilson (1963) first recognized a relation between age and the strength of Ca II emission among the Pleiades, Hyades, other clusters, and the Sun: the younger the stars, the stronger the chromosphere. Kraft (1967) noted from the same clusters and field stars that the rotation of solar-type stars decreases as a function of age; the researcher surmised correctly that this is caused by the loss of angular momentum owing to magnetized stellar winds. These trends were combined in a classic two-page paper by Skumanich (1972), who proposed that both Ca II emission and rotation decrease like the inverse of the square root of age. Implicit in these correlations is a rotatione activityeage connection: stellar magnetic activity depends on stellar rotation (faster rotation brings higher activity) and both activity and rotation decrease with age. These relations were only empirically established out to the age of the Sun; beyond that, both the Ca II emission and rotation become difficult to measure.

7. RotationeActivityeAge Relations

The theoretical basis of this fundamental relationship is twofold. First, the connection between rotation and activity is presumed to be the result of the production of magnetic fields by stellar dynamos that have a rotational dependence to their operation. This is not as clearcut as it might first appear. The cyclic solar dynamo arises at the interface between the radiative and convective layers, so shear has an important role along with rotation. The distributed solar dynamo is less obviously dependent on rotation (it could be a turbulent dynamo). These facts led to questions about how cool stars that are fully convective behave that were only resolved in this century (Browning, 2008). Second is the presumption that magnetic activity will cause a thermal magnetized wind, which will in turn apply a negative torque to the star and cause its angular momentum to decrease. As described in a previous section, we have limited knowledge about the thermal winds from stars other than the Sun. The solar wind arises predominantly from coronal holes, but the prevalence and geometry of these or analogous structures on other stars are generally unknown. Indeed, the large-scale geometry of the magnetic field is just beginning to be understood across stellar parameters using ZDI (See et al., 2017); this work also considers the possible effects of these observed field geometries on mass loss. A physical footing for the rotationeactivity connection was proposed by Noyes et al. (1984). They assumed that the relation between the local surface magnetic field and the local Ca II emission is not in itself dependent on rotation. This is consistent in principle with the related observation that the local Ca II emission is strongly correlated with the local magnetic field on the Sun (Schrijver et al., 1989; Pevtsov et al., 2016). Dynamo theory suggests in a general way (at least for a U dynamos) that the production of the field should depend on rotation through the Rossby number, defined as the ratio of the rotation period to a relevant convective overturn time. Noyes et al. (1984) showed that the rotationeactivity connection could be presented with substantially smaller scatter if one first defines a chromospheric Ca II emission variable. This is usually expressed as Rhk, which is corrected for excess emission over the expected behavior of the photospheric contribution with spectral type. This quantity is then compared with the Rossby number in a logelog plot. There are some subtleties in both the conversion of observations of the Ca II lines to Rhk and in the determination of the Rossby number. For the latter, it is not obvious at what depth the (variable) convective turnover time should be computed, and if one uses mixing length theory (which they do), what mixing length to use (they choose 2 because it minimizes the scatter). This use of a parameter to adjust the Rossby number to minimize the observed scatter continues to the present; authors tend to refer to the “empirical Rossby number” to reflect this adjustment (which is different, depending on the context and data set). Things can be clarified by using x-rays instead of Ca II; this eliminates the need for a photospheric correction. In this case, the relevant activity variable is usually taken as Lx/Lbol (the ratio between the luminosity of the star in x-rays to the bolometric or total stellar luminosity). A study by Reiners et al. (2014) shows the state of this type of analysis (Fig. 11.6). There is a “saturated” regime for Rossby numbers

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FIGURE 11.6 Presentation of the rotationeactivity connection for x-rays. The top plot shows a canonical plot with a luminosity ratio on the ordinate and Rossby number on the abscissa. The colors reflect stellar age, with blue representing very young stars ( 100 km/s) 1920 A regions of downflow with typical diameters of 4000 km (w6 arcsec). From the sample images of the solar surface. it was inferred that the disk was covered with about 4000 “explosive events” that were characterized a mean velocity of 80 km/s and a mean size of 1500 km. Versions of the HRTS had flown on rockets previous to Spacelab 2 and advanced versions were flown on a number of subsequent rocket flights. CHASE was designed to measure the solar helium abundance. However, scattered light in the instrument did not allow sufficiently accurate measurements to establish the helium abundance. SOUP got only 1 day of observations because of technical problems. However, it was possible to collect several 25-min-long sequences of white light images of the solar surface. The 30-cm-diameter SOUP telescope contained an active secondary mirror that was controlled by split photodiodes in the image plane that sampled the solar limb in two orthogonal directions. A closed-loop servo system between the limb sensors and the secondary stabilized the solar image to 0.03-arcsec root mean square. The diffraction-limited images with a 0.5-arcsec spatial resolution

5. 1980 to 2010: The Era of High-Resolution Imaging

images were perfectly aligned from frame to frame. The set of SOUP white-light images could be considered a repeat of the Stratoscope data, but with perfect pointing, uniform quality images, and time sequences of images. SOUP movies revealed apparent random distortions in the solar images. Initially, these seemed similar to those caused by seeing in observations made on the ground. However, it was soon realized the SOUP distortions were introduced by the global modes of the 5-min oscillations. 2D Fourier filtering removed them from the set of images. For the first time, it was possible to measure the properties of the granulation from images free from the distortions introduced by the 5-min oscillations. Once the global modes were removed, it was possible to generate a statistical picture of the spatial and temporal properties of the granulation. The statistical analysis, coupled with knowledge that the 5-min oscillations were present in all of the previous data, ended much of the confusion that existed from previous analysis of granulation observations. The durations of the SOUP movies were sufficiently long that, after removing the distortions introduced by the 5-min oscillations, the techniques of local correlation tracking could be used to generate an evolving surface flow-field. Test particles (“corks” inserted into the flow) provided a simulation of how small magnetic flux concentrations could be transported by the flows generated by the evolution of the surface convection. With an understanding of the role of the 5-min oscillation, it has been possible to generate cork movies from data taken on the ground (Fig. 14.2AeC). This has provided insights into how the magnetic field is dispersed as new magnetic fields emerge and decay in active regions.

FIGURE 14.2 Simulation of magnetic flux dispersal obtained from a horizontal velocity field on the surface created from local correlation tracking on the quiet Sun granulation. The panels show (A) the field covered with test particles, (B) the location of the test particles after 30 min, and (C) after 60 min. There data were not made from images made in space, but rather from images taken at the Swedish Solar Telescope (SST) on La Palma, Canary Islands, Spain, using frame selection and adaptive optics before applying three-dimensional Fourier transform to remove the 5-min oscillations. A.M. Title, Lockheed Martin Advance Technology Center/SST.

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The Spacelab 2 mission had major problems. The first was a main engine that shut down early in the flight, causing the Shuttle to abort its original orbit. This required the development of a new mission plan. The IPS controls were crossed, but this was quickly discovered on the first attempt to point at the Sun. Fortunately, there was a NASA payload specialist, Karl Henize, and two scientist astronauts, Loren Acton and John David Bartoe, on the mission. Without these three instrument experts onboard, it would have been impossible to get much science data, if any, during the 8-day shuttle mission. Launched on Aug. 30, 1991, the Yohkoh (Sun Beam) mission was designed study the x-ray Sun, and in particular, flares. It carried a glancing incidence soft x-ray telescope (SXT) with a c-axial optical telescope for imaging the solar surface in white light, a hard x-ray telescope (HXT), soft and hard x-ray spectrometers, and a gamma-ray spectrometer. The SXT had a charge-coupled device (CCD) sensor to provide 5-arcsec spatial resolution (2.45 arcsec pixels) images over a 42  42-arcmin FOV. The HXT (15e100 keV) was a Fourier transform imager that produced 4-arcsec spatial resolution over the full Sun with 0.5-s temporal resolution. In operation, SXT usually took 42  42-arcmin images of the full Sun with 5-arcsec spatial and 10-min temporal resolution. When SXT internally generated a flare signal, its FOV automatically reduced to 2  2 arcmin centered on the flare location and its temporal sampling increased to 2 s. The co-pointed HXT changed to the same 2  2-arcmin FOV with a 0.5-s cadence. The precise timing of the two imagers confirmed the coexistence of the sources of hard x-rays at loop tops and the sources of soft x-rays at the foot points of the loops. The Yohkoh data were recognized as a major resource for the entire science community. However, the Japanese government formally took the position that, having funded the mission, the data were under Japanese management. In practice, almost any request for data from the Yohkoh science team was granted, but the government’s policy served to discourage some scientists. This had the effect of limiting community involvement, especially in the first few years of the mission. Yohkoh operated until Dec. 14, 2001, when a software error caused the spacecraft to lose lock on the Sun, but for nearly a decade it provided a record of the x-ray evolution of the Sun from the decaying phase of cycle 22 to the peak of cycle 23. Yohkoh data, combined with simultaneous observations with Solar and Heliospheric Observatory (SOHO) and Transition Region and Coronal Explorer (TRACE), allowed further insights into both flares processes and upper atmospheric heating. Launched on Dec. 2, 1995, SOHO is an ESA/NASA mission and a cornerstone of ESA’s Horizon 2000 plan. It was a nice early Christmas present for the SuneEarth connection science communities. The SOHO spacecraft developed and integrated by ESA carried a complement of instruments provided by European National Space Agencies and NASA. It was launched by a NASA Atlas II rocket into an orbit around Lagrange point L1, the balance point on the SuneEarth line, of the gravitational potential of the Sun and Earth system. From the L1 position, instruments can

5. 1980 to 2010: The Era of High-Resolution Imaging

view the Sun continuously. However, because L1 is about 1 million km from the Earth, to capture SOHO’s telemetry requires multiple large antennae at locations distributed around the Earth. The NASA Deep Space Network provides the SOHO communications. In its fully operational phase, SOHO transmitted a 200kbit/s data stream. SOHO is still operational. In one package, SOHO contains instruments that surpassed nearly all previous instruments in temporal, spatial, and spectral specifications. The Solar Ultraviolet Measurements of Emitted Radiation (SUMER) collects stigmatic spectra from ˚ and the Coronal Diagnostic Spectrometer (CDS) collects stigmatic 330 to 1500 A ˚ . Both SUMER and CDS have long slits and square spectra from 150 to 800 A apertures; by using internal optics, they can scan regions of the Sun to make images from any wavelength in their spectral ranges. Both instruments can produce spectra with nearly 2-arcsec spatial resolution in the slit direction. Their cadence of the observations depends on the spatial size and the spectral resolutions that can be accommodated by their telemetry allocations. The Ultraviolet Coronagraph Spectrometer is designed to observe the corona in the spectral region of ˚ with an FOV of 1.5e10 solar radii. 500e1300 A Because of its location at L1, SOHO is in an excellent position to measure the contents of the solar wind, which will subsequently affect the magnetosphere. To accomplish that task, it includes a set of field and particle-measuring instruments. The Comprehensive SupraThermal and Energetic Particle analyzer measures electron composition, the Solar Wind Anisotropies measures the mass flux, whereas the Charge Element and Isotope Analysis System studies the ion composition. SOHO carries two surface imaging systems: the Extreme UV Imaging Telescope (EIT) and the Michelson Doppler Imager (MDI). EIT uses mirror coatings that reflect EUV at normal incidence in narrow spectral bands. Although this was demonstrated on rocket flights, a normal incident EUV telescope that could view the Sun continuously was a revolutionary step forward. Optically, EIT is a conventional RitcheyeChre´tien system whose entrance pupil is divided into four quadrants by the coatings on the primary and secondary mirrors. Images are taken in four spectral lines formed at 80,000 to 2,000,000 K. The 1024  1024 CCD detector limits the spatial resolution to 5.2 arcsec (2.6-arcsec pixels) and spans an FOV out to 1.5 solar radii. Telemetry limited the EIT’s cadence. Originally, the mission plan was for EIT to take only one image sequence a day. The value of EIT movies was soon discovered and appreciated, and it was allocated a larger fraction of the telemetry budget. The most interesting, surprising, and new observations were the “EIT waves” revealed by movies. These showed large sections of the transition region and corona dimming after the passage of a wave front expanding from a flare site. The cause of the dimming is still controversial. MDI was designed to measure the line-of-sight velocity structure of the photosphere once every 30 s with a spatial resolution of 4 arcsec (2-arcsec pixels) over a 34  34-arcsec FOV. MDI also has a high-resolution mode that has 1.25-arcsec spatial resolution over a 14  14-arcsec FOV. At L1, MDI can capture long

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uninterrupted time sequences of well-calibrated velocity maps of the solar surface. Its data have revealed both local and global properties of the solar interior. MDI’s data began a program to collect a long enough record of the Sun’s modal properties to study how the solar interior changes over solar cycles. Beside MDI, SOHO carries two other helioseismology instruments: Global Oscillations at Low Frequencies (GOLF) and Variability of Solar Radiation and Gravity Oscillations (VIRGO). The primary objective of GOLF is to detect internal gravity waves (g-modes), an experiment that requires very high photometric precision over a long observing period. GOLF data are still being collected, analyzed, and interpreted, but the jury is still out on whether g-waves have been detected. VIRGO is a precision radiance monitor that make measurements in 12 segments of the solar disk. The VIRGO science program is focused on low p-modes and g-modes. MDI’s magnetic mode produced a line-of-sight magnetogram every 90 min. Because of the freedom from systematic errors generated by seeing in the Earth’s atmosphere, MDI was able to detect and follow the motions of small magnetic feature that were observed all over the solar surface: the “magnetic carpet.” The magnetic mode of MDI is still operational and is occasionally used for crosscalibration with the Helioseismic and Magnetic Imager (HMI) on the Solar Dynamics Observatory (SDO) and magnetographs on Earth. A key component of SOHO is the Large Angle and Spectrometric Coronagraph (LASCO), which makes image images from 1.1 to 30 solar radii using a set of three coronagraphs: C1 (1.1e3 solar radii), C2 (1.5e6 solar radii), and C3 (3.5e30 solar radii). LASCO, which is still in regular operation, has created an exceptional record of how the corona has evolved over two solar cycles. Coupled with the Heliosphere Imagers (HI) and coronagraphs on the two Solar Terrestrial Relations Observatory (STEREO) spacecraft, it has been possible to track CMEs from the Sun to the Earth. Together, SOHO and STEREO have demonstrated that it is essential to understand how a CME propagates through the heliosphere, to understand the interactions of CMEs with the Earth’s magnetosphere. On Jun. 24, 1998, software errors in the commands for a series of spacecraft maneuvers caused SOHO to lose lock on the Sun and it began spinning. An intense and extremely clever effort by a recovery team regained control of the spacecraft and was able to point it toward the Sun. With the batteries recharged, the instruments were returned to operating temperature and were gradually returned to operation. By Oct. 24, 1998, SOHO was again in nearly full operation despite a very deep cold soak. Only the inner coronagraph, CI, of LASCO did not return to full operational status. The extreme cold temperatures that occurred when the spacecraft was not pointing at the Sun damaged the spacecraft’s gyros. Only one gyro was functional when SOHO was returned to solar pointing. Nevertheless, ESA developed a gyro-less mode for attitude control and the science mission continued. SOHO had a requirement for radiometric calibration to allow intercomparison of data from various EUV spectroscopic solar telescopes and spectrometers. This entailed laboratory calibration with traceable radiometric standards

5. 1980 to 2010: The Era of High-Resolution Imaging

(Pauluhn, 2002). Because of the unavoidable aging of detectors in flight, monitoring of the calibration in flight was part of the initial mission plan. Before launch, there was a comprehensive cleanliness program for all the instruments and the spacecraft. For the first 6 years of operation, resident scientists from the PI teams managed the science planning from NASA facilities at Goddard Spaceflight Center. Initially, the PIs did not have an open data policy. An exception was the MDI PI, who made the magnetograph data available in nearereal time, as well as the results from its long-duration helioseismology runs when they completed. The open data policy of TRACE that was launched 2 years after SOHO caused a conversion of most of the SOHO PIs to open data policies. A SOHO open data policy was strongly supported by both the NASA and ESA project scientists. Launched on Apr. 2, 1998, the TRACE was a NASA Small Explorer mission designed to investigate the effects on the upper solar atmosphere by the thenewell known continuously emerging “magnetic carpet.” TRACE was operated as another SOHO instrument and was included in many of the Joint Observing Programs of the SOHO mission. TRACE carried a 30-cm EUV imaging telescope that duplicated the spectral bands in SOHO’s EIT telescope, but it had a spatial resolution of 1 arcsec (0.5-arcsec pixels), an 8.5  8.5-arcmin2 FOV, and a temporal cadence that usually was 25 s, but it could take images as fast as every 10 s. TRACE was designed to exploit the full Sun measurements being collected by Yohkoh and SOHO to provide a context for its smaller FOV and higherespatial resolution imagery. TRACE’s internal image stabilization was a few hundredths of an arcsec root mean squared. Its 600-km orbit was aligned along the Earth’s dawnedusk line, so, like the instruments on SOHO, it could observe the Sun 24/7, but it could provide continuous imagery for only for 7 months of the year. In a near-Earth orbit, data could be transmitted to polar ground stations in Norway and Alaska at a rate of 2.25 Mbit/s. With one 8-min pass per 90-min orbit, it had an effective mean data rate of 200 kbit/s or about the same as that of SOHO. Occasionally, TRACE could get as many three passes per orbit. With the SOHO full-disk EUV images and MDI magnetograms always being available, TRACE could be pointed at regions that appeared to be scientifically interesting. Its high temporal cadence and arcsec resolution immediately revealed a continuously evolving chromosphere, transition region, and corona. The loop structures seen in both the transition region and corona were shown to be constantly evolving in response to the evolution of the magnetic fields at their bases. Loops associated with active regions that had been seen as slowly changing structures in the lower-resolution SOHO images were observed to be constantly evolving on timescales of minutes and sometimes down to the fastest cadence of the TRACE movies. This provided evidence that mass was continuously being inserted into loops and that they were heated low in the atmosphere.

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FIGURE 14.3 Transition Region and Coronal Explorer Images in Fe IV 171 A˚ (w750,000K). Panel (A) illustrates the fine structure of the loops and the maintenance of their width above the solar surface. The distance from the solar surface to the top of the loops is about 5 scale-heights. Panel (B) illustrates the complex connectivity of loops that can occur in active regions. A.M. Title, Lockheed Martin Advanced Technology Center/National Aeronautics and Space Administration.

Loops had a fine structure when observed with the TRACE spatial resolution. Furthermore, loop substructures did not appear to change significantly in width or intensity with the distance above the solar surface (Fig. 14.3A). TRACE movies demonstrated that the connectivity between active regions was highly dynamic (Fig. 14.3B). Because TRACE also made white light and Lya images, it was possible to align and cross-correlate the TRACE images with higher-resolution images and magnetic maps made on the ground. The TRACE team had a close collaboration with the Royal Swedish Observatory on La Palma, Canary Islands, and dedicated joint observations were scheduled in periods when exceptional seeing was expected in La Palma. The TRACE observing plans were published online; as anticipated, this led to coordinated observations by observatories around the world. The high spatial resolution and high cadence of TRACE revealed the amazing small-scale dynamics of flares, prominences, and filaments. These observations forced the realization that the flare was part of a complex set of physical phenomena that were not always associated with active regions. The existence of TRACE data stimulated an in-depth examination of how magnetic energy was built up in the corona, how the release of the energy was triggered, and how the released energy affected the observed dynamics. The TRACE observations also evoked renewed interest in regular vector magnetograms and growing interest in both magnetic field topology and MHD modeling. Most of all, TRACE data showed the solar community that combining high temporal and spatial resolution is essential to understanding both the quiet and active Sun.

5. 1980 to 2010: The Era of High-Resolution Imaging

The combination of Yohkoh, SOHO, GOES, and TRACE provided a powerful set of measurements for a solid base to test theories, models, and numerical simulations. TRACE had an “open data policy.” Data were available to all members of the science community as soon as they were available and no access preference was given to the PI team. All of the SOHO science teams except GOLF adopted an open data policy, having seen the success of the TRACE example. One consequence of the success that the TRACE open data policy brought was that the NASA Heliophysics division required an open data policy for all of its future missions. Launched on Feb. 5, 2002, the Reuven Ramaty High-Energy Solar Spectroscopic Imager (RHESSI) uses a fixed collimator on a rotating spacecraft to create a rotation modulation collimator. Reconstruction techniques are capable of making x-ray images with 2-arcsec spatial resolution from 4 to 100 keV and 7-arcsec spatial resolution images to 400 keV. It also can produce 36 arcsec spatial resolution images in gamma-ray lines. Its cryogenically cooled germanium crystal detectors provide spectral resolution. Before RHESSI, there had been major questions regarding how electrons were accelerated and how the accelerated electrons could penetrate deep into the transition region, chromosphere, and upper chromosphere. Comparisons of TRACE white light images and RHESSI maps and spectra established that white light photospheric enhancements are cotemporaneous with hard x-ray emission. Another crucial result of the RHESSI mission is the small size and rapid rate of change of the hard x-ray enhancements. Although the RHESSI data have not established the mechanisms of electron acceleration or superheated plasma at loop tops, or how particles penetrate deep into the lower atmosphere, they have placed strong limits on these high-energy release processes. Launched Sep. 22, 2006, Hinode (Sunrise) is a JAXA/ISAS mission. A dawne dusk 680-km orbit allows the instrument complement to view the Sun 24/7 during 7 months of the year. Hinode carries a 50-cm solar optical telescope (SOT) that consists of an optical telescope assembly (OTA) and focal plane package (FPP), an x-ray imaging telescope (XRT), and an EUV imaging spectrograph (EIS). These instruments were designed to operate as a system to investigate how the surface magnetic field affects the upper solar atmosphere. Normally obtaining 15 passes a day, Hinode has an effective data rate of about 300 kbit/s and a maximum uncompressed data rate of 1.8 Mbit/s. The OTA provides 0.2- to 0.3-arcsec spatial resolution images to the science instruments in the FPP. The OTA does not make full Sun images. For fine pointing control, it uses a correlation tracker (CT) located in the FPP to generate a fine pointing signal that controls a folding mirror in the OTA. In operation, the image stabilization system achieves a pointing stability of 0.029 arcsec (3s) for periods of several hundred seconds. Because the CT tracks a small image of solar granulation, there is a random slow drift of the image delivered to the FPP. The small drift has no effect on image quality but it needs to be considered when a precision alignment of

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sequences of images is required. Included in the OTA is a continuously rotating polarization modulator that is synchronized with the exposure cycles of the cameras in the FPP. The OTA was designed and built in Japan and the FPP was designed and built in the United States. The specifications of the SOT and the interfaces between the OTA and FFP systems were jointly developed. Although many properties of both the OTA and FPP could be tested and verified independently, it was necessary to assemble the SOT from the OTA and the FPP into a system for final testing; this was carried out over a period of months at the National Astronomical Observatory of Japan in Tokyo. NASA management was concerned about this method of instrument development because of the difficulty of assigning responsibility in the case of problems. It was not a problem for the ISAS management because their only concern was the proper performance of the complete SOT. Once the FPP was selected, there was only one SOT team. There are two imagers in the FPP: the broadband filter imager (BFI) with ˚ -wide spectral bands in the visible 0.2-arcsec spatial resolution in a set of 2- to 5-A range and the narrowband filter imager (NFI), which can produce 0.35-arcsec ˚ spectral resspatial resolution images in a set of spectral lines by using a 90-mA olution, tunable birefringent filter combined with a set of prefilters. The BFI and NFI share a 2048  1024 single CCD camera. The BFI has 0.054-arcsec pixels, a 218  109-arcsec FOV, and an exposure range between 0.03 and 0.8 s. The BFI images are oversampled to achieve maximum image quality. The NFI has 0.168-arcsec pixels, a 328  164-arcsec FOV, and a normal exposure range of 0.12e1.6 s. The FPP includes a spectropolarimeter (SP), which has a spectral resolution of ˚ spectral sampling, which are determined by a 12-mm slit and the 25 and 21.55 mA 12-mm pixels of its CCD detector, respectively. The slit length is 162 arcsec and the spatial resolution along the slit is 0.159 arcsec, which critical samples of the spatial resolution provided by the slit. The minimum step for rastering is 0.148 arcsec. The polarization modulator in the OTA rotates at 10 Hz and the camera readouts are synchronized to the rotation rate of the modulator. In operation, the system SP achieves a polarization sensitivity of 10 3. There had been SPs on the ground for nearly 20 years before the SP on Hinode. Some of the ground-based systems had significantly higher sensitivity to polarization than the Hinode SP. Nevertheless, the initial SP maps of vertical and horizontal magnetic field were truly revolutionary. There was virtually no region of the solar surface that was free of small magnetic features and nearly the entire quiet Sun was covered by regions of low-lying, very weak horizontal fields. Many of the features revealed by SP had either never been seen before or only been hinted at in observations made on the ground. The great difference between the data produced by SP and ground-based instruments is a combination of seeing variations that blur together small-scale, opposite-polarity regions, seeing-introduced image quality differences that occur during the modulation cycle, and unappreciated systematic effects caused by seeing.

5. 1980 to 2010: The Era of High-Resolution Imaging

In parallel with the development of the SP, advances occurred in analysis techniques. Faster computers enabled much fuller use of all of the information encoded in the spectral line shapes. The ability to deconvolve, pixel by pixel, the line shapes of both the portion of the pixel containing the polarization signal and the portion that was nonmagnetic was essential for determining the magnetic field strength and the direction the vector field. Accurate values for the field strength and vector direction are especially critical in Sunspots. It has long been hoped that the surface vector field would be the key to estimating the energy stored in active regions. Such estimates are not yet possible. Notable problems remain that are caused by unmeasured currents in the upper photosphere and chromosphere that generate additional magnetic fields; the MHD models that generate an estimate of the stored energy depend on the choice of boundary conditions and the 180-degree ambiguity in the direction of the vector field that is inherent in interpreting the polarizations signals. Nevertheless, the time sequences of vector magnetograms produced by the SP are a major step forward in estimating both the energy storage and rates of energy release during a flare. Both the SP and the NFI produced high-resolution maps of the magnetic field. However, because the NFI was an imager, although it had lower signal-to-noise than the SP, its magnetograms could be obtained at a sufficiently high cadence to follow the evolution of the smallest magnetic field features. A clear picture of how the emerging, fragmenting, canceling, and disappearing inter-network fields serve to replace and maintain the network field has emerged from the NFI data. There is a quantitative estimate of the amount of flux that emerges in the internetwork per unit time. The BFI and NFI stopped operating on Feb. 25, 2016, after 9.4 years of operation in space, because of a short in the CCD camera’s power supply resulting from a component failure. The SP is still in full operation. The XRT telescope is similar in overall design to the SXT on Yohkoh, and it has a significantly higher spatial resolution. The combination of telemetry limitations coupled with the fact that the spacecraft pointed all of the instruments at the targeted region of SOT and EIS has limited the XRT to a few full Sun images each day. XRT revealed that the pervasive x-ray bright points both were dynamic and had a small-scale internal structure. XRT and EIS observations added data necessary for developing an understanding of loop structure and evolution. With the loss of the BFI and NFI camera, the telemetry allocations for the working instruments were increased. Hinode science operations are planned and uploaded to the spacecraft from a science center at ISAS. Hinode has an open data policy and the science community can submit proposals for observations that use any and all of the instruments. Proposals are normally reviewed monthly by a team designated by the science working group (SWG). Normally, proposals are accepted. The SWG operates without a time allocation review process managed by ISAS, NASA, or ESA. Early in the mission, the Hinode’s transmitted X-band began to degrade; by the end of the first year in orbit, the S-band transmitter became the telemetry channel. To

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limit the data loss, the number of ground station passes was increased and careful mission planning has allowed Hinode to achieve its design goals. Hinode has an open data policy. Launched Oct. 26, 2006, STEREO is a pair of identical spacecraft that are in orbits just inside and just outside that of the Earth’s. STEREO A orbits in 347 days and moves ahead of the Earth’s position by 21.635 degrees per year whereas STEREO B orbits in 387 days and lags behind the Earth’s position by 21.99 degrees per year. On Feb. 6, 2011, the spacing between A and B was 180 degrees, and for the first time the entire Sun could be imaged by merging data from both spacecraft. Each STEREO payload contains two sets of instruments: the SuneEarth Connection Coronal and Heliospheric Imager (SECCHI) for imaging the Sun and the In-Situ Measurements of Particles and CME Transients, which includes instruments to measure energetic particles, the 3D distributions of the solar wind electrons, and the interplanetary magnetic field. SECCHI consists of an EUV imager (EUVI), a pair of coronagraphs, COR 1 (FOV 1.4e4 solar radii) and COR 2 (FOV 2.5e15 solar radii), and the HI. HI can map the brightness of the heliosphere with an FOV that extends out in two cones with opening angles of 20 and 70 degrees centered at 14 and 53.7 degrees, respectively, off the spacecrafteSun line. COR 2 has an 8-degree Sunecentered FOV. Together, HI and the coronagraphs can follow a CME from the vicinity of the Sun to beyond the Earth. These three instruments demonstrated how CMEs become the interplanetary CMEs that travel through the heliosphere. It is clear from the imagery produced by SECCHI that point measurements in the heliosphere are insufficient to characterize the CME as it travels through the heliosphere. EUVI is a 9.8 cmeaperture normal incidence telescope with spectral bands that are nearly identical to those on SOHO EIT. EUVI has 3.2-arcsec spatial resolution and an FOV of 1.7 solar diameters (2048  2048 CCD with 1.6-arcsec pixels). In Mar. 2011, the Atmospheric Imaging Assembly (AIA) on the EartheSun line added its EUV imagers. This allowed the entire Sun to be imaged in the EUV until 2015, when the STEREO spacecraft rotated to be on the opposite side of the Sun from the Earth. One of the great contributions of SECCHI has been detecting active regions that emerge on the solar surface in areas that are invisible to instruments on the EartheSun line. The full Sun maps have provided insights into instabilities that would not be understandable with a view on only half of the Sun. Launched on Feb. 11, 2010, the SDO is in a geosynchronous orbit that allows it to observe the Sun 24/7 and transmit continuously at a data rate of 130 Mb/s to a pair of dedicated ground stations located at White Sands, New Mexico. The SDO data rate is more than two orders of magnitude greater than that of any previous solar satellite. In 1 day, it delivers as much data as TRACE transmitted in 5 years. SDO carries three instruments: the AIA, the HMI, and the EUV Variability Experiment. Operating together, they produce images of the full Sun formed from 6000K to 20 MK with high-resolution Dopplergrams, line-of-sight and vector

5. 1980 to 2010: The Era of High-Resolution Imaging

magnetograms, and measurements of solar variability in the EUV and the x-ray region of the spectrum. Every day, more than 2 TB of compressed data are collected and archived. AIA consists of four 20-cm, dual-channel, normal incidence telescopes, each of which has a 41-arcmin FOV. Each telescope contains an active secondary that is controlled by a dedicated guide telescope. The secondaries reduce the effect of spacecraft jitter and serve to align the optic axes of telescopes internally to the center of the Sun. Any one of the four guide telescopes can be selected to provide a slowly varying correction signal to the spacecraft for fine pointing. The spatial resolution of AIA is pixel limited to 1.2 arcsec (0.60-arcsec pixels). The telescopes obtain images in the EUV spectral bands used in EUVI and EIT and an additional three EUV wavelengths that have not previously been used in solar satellites. There ˚ ; the temperature are also narrow band filters for images of the photosphere, 4500 A ˚ ˚ minimum, 1600 A; and a 1700-A filter to separate CIV from the temperature minimum. Every 12 s, eight 4096  4096 14-bit images are captured, nearly losslessly compressed, and transmitted directly to the ground. No onboard storage or additional processing is performed on the spacecraft. The data go from White Sands to Goddard Spacecraft Center for reformatting and then directly to the Joint Science Operations Center in Palo Alto where currently more than 250 million images are archived. AIA generates over 30 million images per year. Except to observe a very few exceptional events, SDO is always pointed at Sun center. AIA has an automatic exposure control that is normally on to manage the large intensity changes that can occur in flares. However, exposure adjustments are applied only to alternate 12 s cycles. This ensures proper exposures of nonflaring regions even during flares. Consequently, during flares, the cadence for properly exposed nonflaring and flaring areas may drop to 24 s. The purpose of this strict operating protocol is to understand the conditions that occur on the Sun previous to an unusual event, as well as what is happening over the entire solar surface during an event and as its consequence. It is hoped that by operating for many years, AIA and its predecessors will produce a historical record long enough to be able to gain more understanding of how the outer atmosphere of the Sun responds to an evolving sequence of solar cycles. Another advantage of unchanging operations is that other observatories on the ground or in space can plan on the availability of AIA data. A quick look AIA data is produced within minutes after being taken; the data are available for planners and forecasters in near real time. Before opening the front doors of the AIA telescopes, it was anticipated that with 1.2-arcsec spatial resolution and 12-s temporal sampling, virtually every structure on the Sun would be observed to be in motion. This was expected because TRACE with its 10-s cadence showed continuous evolution in loops and the quiet Sun. However, TRACE, with its limited FOV, only occasionally showed large-scale loop oscillations whereas AIA almost always detects such oscillations during and after flares and CMEs. AIA movies demonstrated the presence of fast waves associated with flares and filament ejections, whose wave fronts can travel across the solar surface at rates as high as 3000 km/s. Waves resulting

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from M- and X-class flares often transverse a solar diameter, causing solar features in their path to move like blades of grass in the wind. These fast waves had not been recognized previously because the earlier EUV instruments did not have the temporal resolution to sample their evolution. The SDO fast waves are distinct from the EIT waves. There is some evidence that the SDO fast waves are part of the same phenomena as the MoretoneRamsey waves occasionally seen in Ha observations. A primary design goal of AIA was to detect the thermal histories associated with feature evolution. TRACE saw structures changing, but its cadence through a range of images made at different temperatures was insufficient to separate motions from temperature changes. Because AIA takes images in eight temperature bands in 12 s, it is straightforward to discriminate among motion, heating, and cooling effects. Although the AIA cycle is 12 s, the images are exposed simultaneously in groups of four, so half of the observations have a time separation of approximately 1 s. The readout time for all four cameras is about 2 s. HMI has 1-arcsec spatial resolution (0.5-arcsec pixels) and an image cadence of ˚ in circularly 3.75 s that allows a sequence of wavelength steps through Fe 6173 A polarized light to be obtained in 45 s. Each set of measurements produces full Sun maps of the Doppler velocity, line-of-sight magnetic field, the continuum intensity, and the spectral line shapes. The data generate velocity maps of the entire solar surface that are used to understand the interior of the Sun employing the techniques of global and local helioseismology. The Dopplergrams and continuum images are also useful for developing maps of the surface flows that can have a role in creating gradients in the surface magnetic field. Vector magnetograms are made every 45 min. HMI has a second channel dedicated to full Stokes measurements. HMI is the first instrument to make vector magnetic field maps over the entire solar surface without interruption (24/7). The long duration of the observations of the global and local solar modes have arrived at sufficient precision to map the meridional flow in the solar interior. It is well-known that a surface meridional flow exists from equator to pole, but the nature and location of the return flow had been uncertain. The newest interior maps revealed a return flow in the mid-convection zone. This result is not in agreement with any current dynamo model and has important consequences for understanding the solar cycle. The combined archives of the MDI and HMI modal data represent the history of two magnetic solar cycles and contain the data for a test of any numerical simulation of the solar dynamo. However, because of the number of scale heights from the bottom of the convection zone to the solar surface, a full numerical simulation of the solar dynamo is not yet possible owing to the limitations of the largest current computers. At the present rate of computer development, it will be some time before simulations will be able to indicate how magnetic fields developed in the interior provide observables on the visible surface. Shown in Fig. 14.4 are series of magnetograms and the corresponding EUV images made from the beginning of cycle 23 to very near the end of cycle 25. The nearly full cycle of AIA images and HMI line of sight and vector magnetograms

5. 1980 to 2010: The Era of High-Resolution Imaging

MIN Cycle 22

MAX Cycle 23 mdi

MDI

MIN Cycle 23 mdi md

MDI

MAX Cycle 24

MIN Cycle 24

HMI

HMI

mdi m

MDI

EIT

EIT

EIT

XRT

XRT

1 May 1996

1 June 2000

1 Feb 2009

1 Feb 2014

1 Mar 2018

FIGURE 14.4 Two solar cycles of magnetograms and extreme UV images obtained from the Michelson Doppler Imager (MDI) and Extreme UV Imaging Telescope (EIT) on the Solar and Heliospheric Observatory (SOHO) and magnetograms and x-ray images obtained from the HMI and XRT on the Solar Dynamics Observatory (SDO). These data were extracted from the public archives of the European Space Agency/National Aeronautics and Space Administration (NASA) SOHO and the NASA SDO missions. Public magnetogram data are available from 1967 from the Mount Wilson and Kit Peak Observatory archives. In addition, x-ray images from mid-1992 are available from the Yohkoh archive. The purpose of this figure is to show that the long life and high quality of solar images from space are building a database for understanding the solar cycle. HMI, Helioseismic and Magnetic Imager; MAX, maximum; MIN, minimum; XRT, x-ray imaging telescope. A.M. Title, Lockheed Martin Advance Technology Center/SST.

together with various feature-finding algorithms have allowed successful experiments with various artificial intelligence (AI) techniques. These appear to be able to predict large flares significantly better than the National Oceanic and Atmospheric Administration prediction center operators. The AI experiments are at an early stage but they already show considerable promise. If their neural networks can be transferred to data processors on future space missions, a greatly increased effective data rate will be achieved. All of the SDO data are publically available as soon they goes through incoming processing. The delay is usually less than 30 min. This has to occur because the production of the released data cannot fall behind the high incoming data rate. In addition to the actual data, there are a variety of software tools for analyzing the data and a structure, the Heliophysics Knowledge Base, that catalogs observations made by the scientific community on ground-based and space-based instruments. An online tool, the JHelioviewer (Mu¨ller et al., 2017), can make images and movies using AIA and MDI data.

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Launched Jun. 27, 2013, the Interface Region Imaging Spectrograph (IRIS) is a NASA Small Explorer Mission designed to investigate conditions in the chromosphere, transition region, and corona. IRIS is in a dawnedusk orbit and for 7 months of the year it can observe the Sun 24/7. IRIS contains two spectrographs whose wavelength regions are in the near UV (NUV) and the far UV (FUV). The spectrographs are fed by a 19-cm UV telescope with a spatial resolution of 0.33 (FUV) to 0.4 (NUV) arcsec. The exposure cycle can be as fast as 2 s. The 175-arcsec-long and 10-mm-wide spectrograph slit is in a UV reflecting mirror. This allows images to be captured simultaneously with the spectra in a selection of wavelength bands that include the Mg H and K lines and several silicon lines. An active secondary mirror that is controlled by a signal generated by a guide telescope stabilizes the solar image. The secondary can be commanded in discrete steps to generate 3D data cubes of x-y position and wavelength. The mass and energy to support the corona and solar wind must be transported from the photosphere through the chromosphere and transition region (interface region), which is a complex, dynamic, and continuously evolving volume with a mixture of neutral and highly ionized gas threaded by magnetic fields. State-ofthe-art numerical simulations guided the selection of the temporal, spatial, and spectral requirements of the IRIS mission (Martı´nez-Sykora, 2017). Although the simulations cannot yet contain all of the physics necessary to capture the interface region fully, it was hoped that they would contain enough physics to provide the focus for instrument design process. IRIS is now in an extended mission phase has been in operation for some time. During this period, an international team of scientists has executed a continuous comparison of the observations with the features shown in a series of simulations. This has been an iterative process that has led to introductions of more physics into the models, which have resulted in a closer agreement in both the shapes of the spectral profiles and the details of their temporal evolution. Comparison of the observed spectra and the spectra generated in the simulations revealed the importance of ioneneutral interactions. Ioneneutral processes have been shown to have a key role in the observed heating of the chromosphere above the magnetic fields in the network. This has led to a good understanding of the processes that create the spicule phenomenon. Computer power has increased to the level that simulations of the 3D response of the chromosphere, transition region, and corona can be created with higher spatial and temporal resolution than is currently observable. However, that simulations are more detailed than can be currently observed does not mean that the simulations are correctly emulating the Sun. Each improvement in the simulations, whether by increased spatial or temporal resolution of more sophisticated physical models, must be validated by real observations. The detailed observations of SOHO, Hinode, SDO, RHESSI, and IRIS have been providing a database that covers two solar cycles and can be used to validate theories or models of physical process occurring in the photosphere, chromosphere, transition region, and corona. Fig. 14.4 illustrates the time span of two solar cycles

6. What Is the Status of the Questions of 1896 and 1962?

with line-of-sight magnetic field and EUV images. The simulation/observation process is an international effort that has been generously supported with computer time provided by ESA, NASA, and many universities. As part of the AIA open data policy, some simulation data volumes have been released.

6. WHAT IS THE STATUS OF THE QUESTIONS OF 1896 AND 1962? Of Charles Young’s questions, the first, the peculiar rotation of the Sun’s surface, has been answered by helioseismology coupled with numerical simulations, which have demonstrated how a very small temperature difference between pole and equator is sufficient to drive the differential rotation seen in the interior and on the surface. Second, the periodicity of the spots and their distribution, remains open and will remain so until a new generation of numerical simulations can cover the span from the convection zone to the visible surface and reproduce a reasonable approximation of Sunspot evolution over a solar cycle. Third, the variations in the amount of solar radiation at different times exist from the comprehensive observations over the past two cycles. Fourth, the problem of the corona and prominences, is still a work in progress. The challenging tasks of the Space Studies Board are really nests of questions. The first, Evolution has seen a great deal of progress that has resulted from comprehensive measurements of the nuclear reaction rates, a new understanding of the conditions required to drive the nuclear reactions, and the discovery of stellar events that create the environments that produce elements. Astroseismology, the Kepler and Gaia missions, and advances in astrophysics have produced an outline of the generation and evolution of cool stars similar to the Sun. The second, Photosphere is well-understood from a combination of observations and numerical simulations that those observations have validated. In the case of the third, Chromosphere, observations have produced an excellent composite picture of the structures that constitute the chromosphere. High-resolution spectra coupled with numerical simulations have explained how some but not all of the structures observed in the chromosphere are heated. There are also partial answers to how the regions of the chromosphere transport heat energy into the corona. For the fourth, Solar Activity, there is general agreement that the source of the energy for flares, filaments, and CMEs, are nonpotential magnetic fields created by convection, surface flows, and flux emergence. MHD models can reproduce many of the features seen during periods of solar activity, but the models have not been able to predict flares reliably based on observations of the measured magnetic fields and surface flows. Numerical simulations have produced amazing simulations of Sunspots based on the temporal evolution of the top of the convection zone and a large magnetic flux tube inserted into the bottom of the layer. However, how such a flux tube would get near the top of the convection zone is not known.

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In the case of the fifth, Corona, there is a good model of the global form of the corona based on potential magnetic field calculations whose boundary conditions are a line-of-sight magnetic map of the entire Sun and the assumption that the field is normal to an imaginary surface at 2.5 solar radii. However, these calculations do not capture the evolution of the corona over emerging active regions or the dynamical changes in the corona that occur during flares and filament excitations and in CMEs. Unfortunately, any field extrapolation depends on knowledge of the magnetic field over the entire solar surface, which cannot be obtained from a single viewpoint. Dynamic changes in the corona require knowledge of the magnetic fields that have their origin above the photosphere, a region where there are few measurements.

7. COMING SOON Planned to be launched about 2019, the Solar Orbiter is an ESA mission launched on a NASA Atlas II. Its goal is to explore and sample the inner heliosphere and to make high-resolution images, magnetograms, and spectral data. The closest approach to the Sun will be 0.28 AU or about 60 solar radii. At closest approach, the thermal load on the Solar Orbiter will be 13 solar constants. Helios A (1975) and Helios B (1976) had perihelions of 0.31 AU and 0.29 AU, respectively, and both collected good data. Although the Orbiter’s thermal design is significantly more difficult than that of Helios because windows in the thermal shield are required for its optical instruments, the Helios missions have provided a pathfinder. With each 150-day orbit, the Solar Orbiter’s orbital inclination to the ecliptic plane will increase; in the later parts of its mission, it will enable the best ever views of the Sun’s polar regions to be obtained. Launched Aug. 12, 2018, the Parker Solar Probe is a NASA mission designed to get much closer to the Sun than any previous mission. Perihelion is 0.046 AU, so at closest approach it will be about 9 solar radii from the solar surface. The Solar Probe includes an array of instruments for in situ measurement of particles and one sidelooking optical instrument for examining the corona. The probe spacecraft’s thermal load will be about 45 times higher than that which will be experienced by the Orbiter, or about 585 solar constants (w800 kW/m2).

REFERENCES Alexander, J.K., 2017. Science Advice to NASA: Conflict, Consensus, Partnership, Leadership, Monographs in Aerospace History No. 57, NASA SP-2017-4557. Bethe, H.A., 1939. Energy production in stars. Rev. 55, 434. Burbidge, E.M., Burbidge, G.R., Fowler, W.A., et al., 1957. Synthesis of the Elements in Stars. Rev. Mod. Phys 29, 547.

References

Friedman, H., Lichtman, S.W., Byram, E.T., 1951. Photon counter measurements of solar x-rays and extreme ultraviolet light. Phys. Rev. 83, 1025. Giacconi, R., Gursky, H., Paolini, F.R., et al., 1962. Evidence for x rays from sources outside the solar system. Phys. Rev. Lett. 9, 439. Leighton, R.B., Noyes, R.W., Simon, G.W., 1962. Velocity Fields in the Solar Atmosphere. I. Preliminary Report. Astrophys. J. 135, 474. Logsdon, J.M., 1995. Exploring the Unknown, NASA, SP-4407. Martı´nez-Sykora, J., De Pontieu, B., et al., 2017. On the generation of solar spicules and Alfve´nic waves. Science 356, 1269. Mu¨ller, D., Nicola, B., et al., 2017. JHelioviewer. Time-dependent 3D visualization of solar and heliospheric data. Adv. Astrophys. 606, A10. Neugebauer, M., Snyder, W., 1962. Solar plasma experiment. Science 138, 1095. Newell Jr., H.E., 1980. Beyond the Atmosphere: Early Years of Space Science. NASA SP-4211, Reprint of the NASA History Series, Washington, D.C. Parker, E.N., 1958. Dynamics of the interplanetary gas and magnetic fields. Astrophys. J. 128, 664. Pauluhn, A., Huber, M.C.B., von Steiger, R. (Eds.), 2002. The Radiometric Calibration of SOHO, ISSI Scientific Report (ISBN 1608e280X). Schwarzschild, M., 1959. Photographs of the solar granulation taken from the stratosphere. Astrophys. J. 130, 345. Space Research-Direction for the Future. Publication 1403 NAS_NRC, 1966, 182e187. Vernazza, J., Avrett, E.H., Loeser, R., 1973. Structure of the solar chromosphere. Basic computations and summary of the results. Astrophys. J. 164, 605. Young, C.A., 1892. The Sun. Kegan Paul, Trench, Trubner & Co, Ltd.

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12.2 Oddbjørn Engvold1, Jack B. Zirker2

Rosseland Centre for Solar Physics, Institute of Theoretical Astrophysics, University of Oslo, Oslo, Norway1; National Solar Observatory, Sunspot, NM, United States2

CHAPTER OUTLINE 1. Introduction .......................................................................................................420 2. Strategies and Conditions ...................................................................................421 2.1 Preferred Types of Observatory Sites ..................................................... 421 2.2 Optical and Technical Solutions ........................................................... 422 2.3 Seeing Correction Techniques .............................................................. 423 2.3.1 Adaptive Optics .......................................................................... 423 2.3.2 Deformable Mirrors ..................................................................... 425 2.3.3 Multiconjugate Adaptive Optics..................................................... 425 2.3.4 Multiobject Multiframe Blind Deconvolution.................................... 426 3. Observations With Modern Solar Facilities...........................................................427 3.1 Swedish 1-Meter Solar Telescope ......................................................... 428 3.2 Phillip R. Goode Solar Telescope .......................................................... 430 3.3 The German Vacuum Tower Telescope and GREGOR at Teide Observatory in the Canary Islands ......................................................... 430 3.4 New Vacuum Solar Telescope............................................................... 432 3.5 French THEMIS Telescope at Teide Observatory..................................... 432 3.6 Interferometric Atacama Large-Millimeter/Submillimeter Array................ 432 4. Outlook for the Future.........................................................................................433 4.1 Daniel K. Inouye Solar Telescope.......................................................... 434 4.2 European Solar Telescope .................................................................... 436 4.3 Plans for 8-Meter Telescopes ............................................................... 437 4.4 Concluding Remarks............................................................................ 437 5. Summary ...........................................................................................................438 Acknowledgments ...................................................................................................438 References .............................................................................................................439

The Sun as a Guide to Stellar Physics. https://doi.org/10.1016/B978-0-12-814334-6.00015-7 Copyright © 2019 Elsevier Inc. All rights reserved.

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1. INTRODUCTION A new generation of ground-based solar telescopes and their associated instruments is gradually replacing established observatories. The main driver for the development of today’s solar observatories and the planned next-generation 4-m class telescopes is the quest for small-scale resolution combined with high polarimetric sensitivities. One particular challenge is to ensure a signalenoise polarimetric accuracy better than the current 10
3 for increasingly higher spatial resolution. This becomes especially important for observations of the chromosphere and corona with magnetic fields that are notably weaker than in the photosphere. To resolve the structure and dynamics of chromospheric magnetic fields, observers also need higher spatial and temporal resolution over larger areas and longer times. Tracking the development of chromospheric flares in three dimensions and at high temporal rate creates the demand for larger photon fluxes and therefore larger mirrors. Fortunately, meter-class, temperature-insensitive telescope mirrors with low scattered light and residual polarization are available and are being gradually adopted. However, the atmosphere will blur the images produced by such large mirrors unless remedies are found. In the pursuit for higher angular and temporal resolution, several strategies were adopted in the past. These included locating sites with the best natural seeing (2.1); avoiding ground turbulence with high towers, evacuating the telescope, redesigning the dome, and controlling internal temperatures (2.2). These measures remain useful despite the notable development of adaptive optics (AO) that correct wavefront distortions at kilohertz rates (2.3). In fact, the development and deployment of AO systems has revolutionized solar observing. We briefly describe the simplest versions and their extensions, such as multiconjugate systems (2.3). In combination with a site that boasts moderately good seeing for hours, this equipment can deliver near-diffraction resolution performance. Postobservation restoration of a single unique image, using many weakly distorted short-exposure images, is still important. With more light at the focal plane, beam splitters can direct different wavelength bands to different instruments for simultaneous observations. The combined observations yield a more fully detailed description of a solar phenomenon than was possible even a few years ago. Ongoing technical progress and science at several major observatories today are discussed in the third subsection. We focus largely on current strategies and solutions resulting from the demands for multiple-wavelength and highresolution observations and pay less attention to the recent past. Then we survey one 4-m facility being built (the Daniel K. Inouye Solar Telescope [DKIST]) and one (the European Solar Telescope [EST]) in the design and planning stage and finally touch on ideas and plans for extremely large future (8-m) telescopes.

2. Strategies and Conditions

2. STRATEGIES AND CONDITIONS 2.1 PREFERRED TYPES OF OBSERVATORY SITES The siting of an observatory is fundamental to its ultimate optical performance (Dunn, 1985; von der Lu¨he, 2009). An accumulated body of experience from operating observatories, supported by test evidence, shows that mountains in dry desert regions, which lie in coastal regions, benefit from the thermal inertia of coastal waters. Better still for solar observations are mountain sites on small islands that are surrounded by large oceanic bodies and located in trade wind zones. The jet stream’s position varies with the seasons. In the northern hemisphere, they tend to move further south during winter and spring months. Solar telescopes located in large lakes (Goode et al., 2012; Goode and Cao, 2013) also benefit from the thermal stabilizing effects of the surrounding water. A combined high-altitude lake site in Tibet is being considered for a planned 8-m Chinese solar telescope (Beckers et al., 2013). High mountain sites are most often situated above the cloud-forming atmospheric inversion layer. Current sites around 2400 m above sea level in the Canarias archipelago notably benefit from this advantage. Excellent daytime seeing on the La Palma site appears preferentially on days with wind coming from the northeast to northwest direction, which is due to local mountain topology. The stable air from the ocean remains undisturbed over the steepest part of the island, whereas air from the other direction will be disturbed by the usually warmer caldera. The sites for daytime as well as nighttime telescope observatories are nearly always based on detailed measurements and studies of the atmospheric conditions and properties. The influence on image quality from atmospheric turbulence representing fluctuation in temperature can be represented by two parameters called, respectively, Fried’s parameter r 0 (Fried, 1966) and t0, the Greenwood time (Greenwood, 1977), corresponding to the timescale over which the changes in the turbulence become significant. Fried’s parameter has the dimension of length and defines an important length scale over which phase errors in a wavefront are on the order of 1 radian. Both parameters are essential for construction and operation of AO systems (see Section 2.3), where r 0 determines the spacing of the actuators needed and t0 determines the required correction speed. Air turbulence is a manifestation of wind shear and atmospheric airflows are nearly always turbulent air bubbles owing to ground heating. Telescope apertures are therefore usually placed at the top of a tower around 20e40 m high to be above the most serious disturbance of the ground layer. Modern seeing monitors are also able to map the height structure of the atmospheric seeing (Beckers and Liu, 2002). This information is also central in application of multiconjugate AO (MCAO) correction of seeing (see Section 2.3).

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2.2 OPTICAL AND TECHNICAL SOLUTIONS The optical and technical solution and structure of a solar telescope are to a large degree determined according to how they solve the heat load problem. The need for a tall supporting tower construction to bring the entrance aperture above the disturbing ground-layer turbulence influences also the optical-mechanical solution. Concern for polarimetric use and performance also has a role. The common constructions of today’s actively operating solar telescopes are presented and discussed in the following subchapter. Solar telescopes are subjected to significant solar flux heat loads, which are not negligible (w1 kw/m2). Heat from tube walls, reflecting optical surfaces and other components, causes internal telescope seeing as a result of warm air bubbles rising the length of the telescope tube. However, the main contributor to internal seeing appears to be the “boundary layer” of warm air directly in front of the primary mirror. The internal seeing problem was first successfully solved with the National Solar Observatory (NSO) Richard B. Dunn Solar Telescope (DST), in which the internal light path is in vacuum, made possible by a sealing entrance window (Dunn, 1964). Other well-functioning vacuum solar telescopes followed. Several of these are discussed in detail in Section 3. An aperture entrance window of a vacuum telescope must sustain the pressure load of around 1 bar, which requires a diameterethickness ratio of about 10. That inevitably limits the aperture diameter of vacuum telescopes to a maximum of 1 m. An idea to replace the evacuated volume by helium gas, which would allow for larger apertures and a thin entrance window, was tested and showed that a heliumfilled telescope light path was nearly as good as a vacuum (Engvold et al., 1983). The low refractive index and high thermal conductivity of helium effectively subdue the internal seeing. The French 90-cm THEMIS solar telescope at the Teide Observatory at the island of Tenerife, was the first to use a helium-filled telescope tube. The 50cm aperture Synoptic Optical Long-term Investigations of the Sun Vector SpectroMagnetograph was also designed to be filled with helium at slightly above ambient pressure (Harvey, 2014). After a number of years, helium was replaced by nitrogen gas owing to an acute shortage of helium, which led to slightly degraded image quality. A 45-cm aperture open telescope by the Dutch engineer Robert Hammerschlag (Hammerschlag and Bettonvil, 1998; Rutten, 1999) provided the first convincing demonstration of an open solution. Later, it was adopted for the 1.5-m German GREGOR telescope at Teide on Tenerife, Canary Island and the 1.6-m aperture GST at Big Bear Lake, California. The open solution is also accepted for the planned future 4-m telescopes, DKIST and EST, and for even larger future facilities. The internal telescope seeing should be notably reduced by the free flow of air that continuously removes warm air bubbles.

2. Strategies and Conditions

2.3 SEEING CORRECTION TECHNIQUES Over the past decades, the restoration of spatially degraded data has become an indispensable tool in exploring the full potential of solar ground-based observational data (van Noort, 2017). The Earth’s atmosphere contains turbulent eddies that possess different temperatures and densities and that are moving horizontally as high-altitude winds. Such eddies have different indexes of refraction, so that light from a distant astronomical source is refracted into different paths. As a result, a wave arrives at Earth with modified phases across the wavefront, so-called “aberrations.” Modern ground-based solar telescopes correct wavefront distortions over a limited field of view by use of real-time AO (Rimmele, 2000, 2004). Higher-order multiconjugate systems of AO can triple the corrected area (Denker et al., 2005). High image quality is becoming attainable over the full field of view by use of post facto restoration techniques, which will be briefly discussed next.

2.3.1 Adaptive Optics The purpose of an AO system is real-time correction of wavefront phase errors that the atmosphere introduces. The errors change in milliseconds and vary across the entrance aperture of the telescope. Hence, the AO system must be capable of responding at such rates. For discussion of the current state-of-the-art of solar AO systems, the reader is referred to Rimmele and Marino (2011), Rimmele et al. (2013), Schmidt et al. (2016), and Denker et al. (2018). A basic tutorial is posted on the Web by the staff of the Cerra Telolo Inter-American Observatory: (http:// www.ctio.noao.edu/watokovin/tutorial/intro.html). In operation (Fig. 12.2.1 A), a small fraction of the light from the telescope primary mirror is directed to the deformable mirror (DM), where wavefront corrections (optical pathlengths) are applied. The corrected image is sent through a beam splitter to a camera. The reflection from the splitter is sent to the wavefront sensor, whose output (phase errors) the processor uses to calculate new piston displacements. The wavefront sensor is the key element in the AO system. The Shacke Hartmann sensor (Fig. 12.2.1 B) is a popular choice. It consists of an array of small identical lenslets that sample the wavefront at the telescope exit pupil at a large number of positions. The more positions, the better the overall correction, but at a higher cost. The lenslets have the same focal length and are focused on a charge-coupled device (CCD) at the common image plane. Each lens captures a small part of the pupil, called a subpupil, and forms an image of the source, say a sunspot, on the CCD detector at the image plane. When an incoming wavefront is plane, all the images form a regular grid that is defined by the lenslet array geometry. When the wavefront is distorted, the images are displaced from their nominal positions. A bend in the wavefront that falls on a lenslet (a local phase error) displaces the position on the CCD of the focus of the lenslet; this displacement is recorded in two orthogonal directions, x,y. The displacement is proportional to the local wavefront slope over the lenslet. In this way, a

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FIGURE 12.2.1 The principle of adaptive optics. (A) The main adaptive optics components are the deformable mirror, the wavefront sensor, and a control system that includes a wavefront reconstructor. A beam splitter sends a small fraction of the light to the wavefront sensor whereas most of the light is distributed to the science instrument(s). (B) Schematic of a ShackeHartmann wavefront sensor (Cerro Tololo Inter-American Observatory Web tutorial). From Claire Max, Center for Adaptive Optics, UC Santa Cruz.

2. Strategies and Conditions

ShackeHartmann measures the local wavefront slopes. The wavefront itself can be reconstructed from the arrays of measured slopes, up to a constant that is of no importance for imaging. The wavefront reconstructor accepts the set of measured slopes and calculates the path differences that must be applied by the DM to cancel the phase errors. The calculation can be carried out with matrix multiplication, at least in principle.

2.3.2 Deformable Mirrors Early DMs consisted of discrete segments, each controlled by three piezoelectric actuators. Nowadays, a common technology is to bond a thin faceplate to an array of piezoelectric actuators. The actuators are not produced individually, but rather a multilayer wafer of piezo-ceramic is separated into individual actuators. This type of DM is used in the GEMINI AO systems.

2.3.3 Multiconjugate Adaptive Optics Classical AO systems apply the same wavefront modification equally to all directions in the field-of-view (FOV) of the telescope. Optical aberrations resulting from turbulent airflows at various heights above the observatory site vary with view angle. The classical AO wavefront correction is valid over a limited viewing angle, usually called the isoplanatic angle, the Fried parameter r0 (Fried, 1966). Typical values for Fried parameter at modern observatory sites are r0 ¼ 10e20 cm at 500 nm. To reach resolutions less than 1 arcsec, which are essential in current high-resolution solar research, the FOV is commonly restricted to about 10 arcsec. That is too small to cover even a small active region. The current solution is to apply MCAO. MCAO is a further development of the original AO concept (Beckers, 2000; Denker et al., 2005). It consists of correcting for the turbulence located at various heights in the Earth’s atmosphere with more than one DM. Each DM is associated with a certain distance from the telescope. The benefit of MCAO is a noticeable increase of the compensated FOV, size as demonstrated in Fig. 12.2.2. Experiments to develop MCAO for the Sun were carried out at the GREGOR telescope and reported in a conference in 2013 (Schmidt et al., 2014). The system employed three DMs conjugated to 0, 8, and 25 km line of sight distance. Two correlating ShackeHartmann wavefront sensors were incorporated: an on-axis wavefront sensor with 1.0-cm subapertures and 10 arcsec FOV, and a loworder multidirection wavefront sensor with 19 5-cm subapertures distributed over a 1-arcminute FOV. Three observatories are collaborating to develop a reliable MCAO system: the Goode Solar Observatory, the US National Solar Observatory, and the Kiepenheuer Institute for Solar Research. Their current system (CLEAR) employs three DMs to correct for turbulence at three altitudes in the atmosphere and one wavefront sensor. Tests in 2016 and 2017 at the 1.6-m Goode telescope showed high resolution over an FOV approximately three times the size (30 arcsec) a single AO would cover. Post facto image corrections via speckle masking provide further reductions of remaining wavefront errors (Denker et al., 2007).

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FIGURE 12.2.2 Images of solar granulation and sunspot recorded with classical adaptive optics (AO) system (left) and multiconjugate adaptive optics (MCAO) of the Phillip R. Goode Solar Telescope at Big Bear Observatory (right). The MCAO-corrected observations show a clearly visible widened corrected field-of-view compared with quasisimultaneous observations with classical AO. From Dr. D. Schmidt, MCAO project scientist, Big Bear Solar Observatory.

2.3.4 Multiobject Multiframe Blind Deconvolution To achieve high image quality over the full FOV, advanced image restoration techniques such as speckle masking or multiobject multiframe blind deconvolution (MOMFBD) remain mandatory (Noort et al., 2006). These techniques use a large number of images of an object within a short interval such that the object stays the same (the solar scene) while the wavefront aberrations vary. Tens to hundreds of exposures are typically used for restorations.

3. Observations With Modern Solar Facilities

Simultaneous recording of the same solar area in different passbands with one or possibly two spectrometers allows the further reduction of residual wavefront aberrations in areas of the FOV that is not fully corrected by the AO. The method is referred to as MOMFBD (van Noort et al., 2005; Rouppe van der Voort et al., 2006). The two passbands may represent greatly varying solar scenes such as the narrowband Ha line core and continuum images. This method achieves nearperfect alignment between images from the different passbands, which is particularly advantageous for sequentially recorded images from tunable filter instruments because it allows for the integrity of the spectral profiles (Fig. 12.2.3). The MOMFBD method can also be applied to spectrographic data to obtain pristine spectra (van Noort, 2017). High-resolution imaging spectropolarimetry involving multidimensional data sets requires versatile data-processing pipelines (de la Cruz Rodrigues et al., 2015).

3. OBSERVATIONS WITH MODERN SOLAR FACILITIES Better understanding of the complex dynamic structures of the inner solar atmosphere over the whole range from the large to the smallest is stimulating modern ground-based observatories toward observations of higher spatial and temporal resolution. Coordinated use of various types of postfocal instruments is making simultaneous high cadence two-dimensional (2D) recordings of small-scale dynamic and magnetic structure possible over a range of heights from the deep photosphere to the chromosphere. In addition to spatial resolution, the light-collecting power of a large telescope permits observations of high sensitivity. Coordinated observing with space instruments has also become common.

FIGURE 12.2.3 A small sunspot and nearby chromosphere and photosphere observed with the CHROMIS spectrometer of the Swedish 1-m Solar Telescope, in the Ca II K line center, the two opposite line wings, and the continuum. The spatial resolution is 0.1 arcsec. From Professor Luc Rouppe van der Voort (University of Oslo).

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The 1.6-m reflecting McMathePierce solar telescope at Kitt Peak National Observatory in Arizona was inaugurated in 1962 (Pierce, 1964; Plymate, 2001). Until a few years ago it was the largest aperture solar telescope in operation. The telescope uses a heliostat at the top of its main tower to direct the Sun’s light down a long shaft to its image-forming mirrors. For a long time, it was a premier telescope for infrared observations and for Fourier transform spectroscopy. Subarcsec resolution became available after the introduction of AOs. The Kitt Peak mountain site observatory at 2096-m altitude remains favored for both daytime and nighttime observations. The DSTof the NSO at Sacramento Peak, New Mexico was inaugurated in 1969 and named after its creator in 1998. The telescope revolutionized the capabilities of solar telescopes and became a trailblazer for all modern high-resolution solar observing facilities. With its 76-cm entrance aperture located about 40 m above ground and with the evacuated light path, it became the first truly successful high-resolution solar telescope. The first testing of an AO system that was developed by Lockheed Associates (Acton and Smithson, 1992) was also done at DST and daily operations became successively strengthened by being upgraded with high-order AO systems (Rimmele, 2000). Polarimetry remains a challenging, important tool for high-resolution solar observations (van Noort and Rouppe van der Voort, 2008). Very low polarimetric noise levels were commonly achieved by sacrificing spatial and temporal resolutions (Gandorfer et al., 2004). Successfully combining AO and speckle reconstruction or MOMFBD has resulted in sensitivity better than 10
3 for close to diffractionlimited imaging (Collados et al., 2007; Bello Gonza´lez and Kneer, 2008). The following presentations of some modern ground-based observatories, covering visual and infrared to radio wavelengths, illustrate how the improvements of facilities have progressed.

3.1 SWEDISH 1-METER SOLAR TELESCOPE The Swedish Solar Telescope (SST) is located on the rim of an extinct volcanic Caldera de Taburiente and the observatory site at Roque de los Muchachos on the island of La Palma, Spain. The original Swedish Vacuum Solar telescope was a refractor with a lens 47-cm in diameter. The telescope was evacuated to eliminate internal seeing. The site was (and remains) extraordinary; the isoplanatic patch external to the telescope approached the full diameter of the aperture in size for extended periods, leading to diffraction-limited resolution (Fig. 12.2.4 A). Around 2001, the SST was replaced by a 98-cm aperture evacuated refractor (Scharmer et al., 2003). AO was integrated from the beginning (2002). In 2013, the AO system was upgraded to a higher-order system (an 85-electrode monomorph DM from CILAS). This combination produced the highest spatially resolved images of solar granulation ever recorded, at a resolution of 0.15 arcsec. Two modes of observing the AO-corrected images are provided: a spectrograph mode and a polarimeter mode. A TRI-Port Polarimetric Echelle-Littrow instrument is a Littrow spectrograph operating at a wavelength range about 380e1100 nm. It has a moderate resolution for a solar telescope, with R being approximately

(A) Swedish Solar Telescope. (B) German GREGORY solar telescope. (C) German Vacuum Tower Telescope. From (DeG) French THEMIS telescope. Chinese Fuxian Lake, New Vacuum Solar Telescope; Big Bear, Goode Solar Telescope; and Dutch Open Telescope.

429

From Owners of the various observatories.

3. Observations With Modern Solar Facilities

FIGURE 12.2.4

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200,000. The CRisp Imaging SpectroPolarimeter (CRISP) Fabry-Pe´rot spectrometer, which has been in use since 2008, operates from 510 to 860 nm and is able to measure polarization by using liquid crystal modulation combined with a polarizing beam splitter (Ortiz and Rouppe van der Voort, 2010). A second spectrometer, also type Fabry-Pe´rot, CHROMospheric Imaging Spectrometer (CHROMIS) is designed for use at wavelengths in the range of 380e500 nm. This instrument is most frequently centered on the chromospheric H and K lines of Ca II and the Hb Balmer line. A synchronized set of wide-band images secured via a prefilter enable near-perfect alignment between sequentially recorded images from CRISP and CHROMIS, which are corrected for atmospheric seeing with the noted MOMFBD technique. Observing runs with these two spectrometers are frequently coordinated with Atmospheric Imaging Assembly (AIA) and Helioseismic and Magnetic Imager (HMI) onboard the Solar Dynamics Observatory, IRIS and Hinode in space, and thereby enhance the scientific value of the combined data. As an example of the science that subarcsecond observations can reveal, we mention a study of the interlaced flows and magnetic fields in a sunspot penumbra (Scharmer et al., 2013). In 1909, Evershed discovered radially outward flows in the penumbrae. Observations from Hinode at arcsecond resolution (Franz and Schlichenmaier, 2013) suggested the presence of horizontal oppositely directed magnetic fields but only in the outer penumbra. Relying on a resolution of 0.15 arcsec, the Swedish group was able to trace the field into the innermost penumbra and to derive the 3D temperature, velocity, and field.

3.2 PHILLIP R. GOODE SOLAR TELESCOPE The Big Bear Solar Telescope, with a 0.5-m primary mirror, was built by Cal Tech around 1969 in the middle of Big Bear Lake, California, at an altitude of 2 km. The lake water suppressed the ground-level turbulent convection that normally produces bad seeing and blurry images. For 3 decades, Big Bear earned a reputation for long sequences of highly resolved images of flares and active regions. In 1997 the observatory passed to the New Jersey Institute of Technology. The optics were replaced or upgraded and the telescope was renamed the Phillip R. Goode Solar Telescope (GST) (Cao et al., 2010). The original primary mirror was replaced by a 1.6-m mirror in 2009. GST is an open telescope structure with an equatorial mount. The telescope is protected by a dome that employs a wind gate and an exhaust system to control the air flow. Three generations of AO, including a multiconjugate version, were installed to triple the former corrected FOV (Schmidt et al., 2017). Focal plane instruments include a visible and a near-infrared spectropolarimeter (Fig. 12.2.4 B).

3.3 THE GERMAN VACUUM TOWER TELESCOPE AND GREGOR AT TEIDE OBSERVATORY IN THE CANARY ISLANDS Three major solar telescopes comprise the Teide Observatory in the Canary Islands: the Vacuum Tower Telescope (Kiepenheuer Institute of Solar Physics [KIS]); the

3. Observations With Modern Solar Facilities

Gregor Telescope (run by a German consortium); and the THEMIS Telescope (run by a French and Italian consortium) (Fig. 12.2.4 C). The Vacuum Tower saw first light in 1989. The primary mirror has a diameter of 70 cm and a focal length of 46 m. Two coelostat mirrors at the top of the tower direct sunlight through the vacuum entrance window at the top. KIS developed one of the first AO systems in 1998. The main components are a ShackeHartmann wavefront sensor with 36 subapertures, a DM, and a high-speed digital camera. A corrected image with resolution of 0.2 arcsec is available to all focal plane instruments. Since 2003, the AO system has been installed and available to all focal plane instruments. These include a visible light spectropolarimeter, an infrared (1.5-mm) polarimeter, and an echelle spectrograph. A German consortium inaugurated in 2012 the 1.5-m Gregorian telescope named GREGOR, the third largest in the world (Schmidt et al., 2012). It replaces the previous Gregory Coude´ Telescope. It has an open-air design that depends on AO to obtain high angular resolution. Its high-order AO system has a 256-actuator DM and a 156 subaperture wavefront sensor. In 0.6 arcsec seeing, the telescope resolution comes close to a diffraction limit of 0.1 arcsec. Postfocal instruments include a long-slit spectrograph (Gregor infrared spectrograph) and a 2D spectrometer with two FabryePerot interferometers in a collimated beam (Figure 12.2.5). Both can be used as full vector polarimeters. Coordinated observations with AIA and IRIS are mutually stimulating and important.

FIGURE 12.2.5 High-resolution spectrum and image of a sunspot recorded with the GREGOR’s infrared spectrograph. The five subpanels, from left to right are: (1) the intensity spectrum of the iron line at 1564.8 nm along the slit; (2) the scanned intensity map, assembled from all slit positions; and the panels (3), (4), and (5) displaying maps of the linearly polarized light in two directions and the circularly polarized light. From Dr. Rolf Schlichenmaier and Kiepenheuer-Institut fu¨r Sonnenphysik.

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The combination of high spatial resolution and polarimetric sensitivity of the GREGOR telescope have led to new insights in small-scale magnetic fields in solar photosphere and chromosphere that are presented in a special issue of Astronomy and Astrophysics (cf. Peter et al., 2016).

3.4 NEW VACUUM SOLAR TELESCOPE The 1-m New Vacuum Solar Telescope of the Fuxian Solar Observatory is located at altitude 1720 m above sea level at the Fuxian Lake in Yunnan, China (Rao et al., 2014, 2016). The telescope is an alt-azimuth Gregory-Coude´ system (Liu et al., 2014) and feeds postfocal instruments for spectrometry and polarimetry in visible and near-infrared bands (Liu and Xu, 2011). Its high-order wavefront correction loop consists of a DM with 151 actuators and a correlating ShackeHartmann wavefront sensor with 102 subapertures.

3.5 FRENCH THEMIS TELESCOPE AT TEIDE OBSERVATORY THEMIS is a 90-cm aperture alt-azimuth mounted RitcheyeChre´tien telescope specially built for spectropolarimetry over a broad spectral range by a consortium of French and Italian institutions in 1990. THEMIS uses a thin entrance window and is filled with helium for low-stress birefringence. The spectrograph allows simultaneous observations of up to 10 wavelengths from 400 to about 1300 nm, which provide opportunities to perform 3D inversion of the magnetic field structure in the solar atmosphere (Paletou and Molodij, 2001; Bommier, 2011). The overall instrumental performance has gradually been strengthened and a classical AO system was put into operation. The telescope has been open to the international community since 1999 (Bommier and Sahal-Bre´chot, 2007) (Fig. 12.2.4 D-G).

3.6 INTERFEROMETRIC ATACAMA LARGE-MILLIMETER/ SUBMILLIMETER ARRAY Radio wavelength observatories were less favored for studies for which very high spatial resolution was essential until the interferometric Atacama Large Millimeter/Submillimeter Array (ALMA) became available in 2014 for recording and imaging solar radiation at millimeter wavelength (White et al., 2017). The ALMA radio telescope is located in the Chilean Andes at an altitude of 5000 m and is composed of 66 high-precision antennas operating on wavelengths from 0.32 to 3.6 mm. Its main array has 50 antennas, each with 12-m diameters, which act together as an interferometer. This arrangement is complemented by a compact array of four antennas with 12-m diameters and 12 antennas with 7m diameters. The antennas can be configured with spacing of distances from 150 m to 16 km, and thereby ensure subarcsec spatial resolution. Its modern design enables the antennas to work together as if they were a single giant telescope (Fig. 12.2.6).

4. Outlook for the Future

FIGURE 12.2.6 (A) The Atacama Large Millimeter/Submillimeter Array (ALMA) in the Atacama Desert (Chile), photographed from above with a camera installed on a hexacopter. (B) A microwave-length (1.25-mm) image observed with ALMA of an area containing a sunspot, as shown on the full disk of the Sun. From the National Radio Astronomy Observatory and Associated Universities, Inc.; computer graphics by ESO.

ALMA has demonstrated its impressive capabilities by observing a large variety of targets ranging from stellar flare eruptions and protoplanetary disks to galactic nuclei. The antennae were designed specifically so that the Sun’s strong radiation would not affect its instruments and thereby the Sun could be observed directly. Regular observations of the Sun started in 2016 (Shimojo et al., 2017). ALMA represents a notable leap forward in terms of spatial resolution at millimeter wavelengths and thereby became an unprecedented new diagnostic potential for studying the Sun (Wedemeyer, 2016).

4. OUTLOOK FOR THE FUTURE The Coronal Solar Magnetism Observatory (COSMO) is a proposed ground-based facility with unique capabilities for magnetic field measurements in the solar atmosphere and corona to increase our understanding of solar physics and space weather (Oakley et al., 2016). The facility consists of three instruments: (1) a meter-class aperture coronal magnetometer devoted to obtaining the highest quality polarimetric data; (2) a chromosphere and prominence magnetometer; and (3) a white-light polarized-brightness coronagraph. The aim of COSMO is to enhance the scientific output of existing observatories that are ground based and in space, and new 4-m observatories. Plans for and the construction of new, larger aperture solar telescopes are encouraged by the noted urge for high spatial and temporal resolution and polarimetric sensitivity in observations of the Sun. A new Chinese large solar telescope with a 1.8-m aperture is under construction (Rao et al., 2016). It will be a classic Gregorian configuration telescope with an open structure, alt-azimuth mount, retractable dome, and large mechanical d-rotator. Also, a 2-m class Indian national large solar

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telescope is proposed for high-resolution solar studies (Hasan et al., 2010). It will be an on-axis alt-azimuth Gregorian multipurpose open telescope and located at the Pongong Lake site in the Himalayan Mountains. Two near-future large aperture solar telescopes, the American DKIST and the European EST, both 4-m class telescopes, are expected to offer unparalleled spatial resolution and simultaneous wavelength coverage that is not yet provided by any other instrument, ground-based or space-borne (Marino and Rimmele, 2017).

4.1 DANIEL K. INOUYE SOLAR TELESCOPE DKIST, the first in a generation of 4-m telescopes, is under construction on the Hawaiian Island of Maui. It will be the largest solar telescope in the world, with a primary mirror 4 m in diameter (Tritschler et al., 2016). The National Solar Observatory and the University of Hawaii will operate the facility and share observing time with 22 international partners. First light is projected for 2019. Funds for the telescope were provided by the US National Science Foundation. Several potential sites were tested (Hill et al., 2006) and the telescope will finally be located on the perimeter of an extinct volcano, Haleakala, at an altitude of 3 km above the sea. It will replace the former Mees Solar Observatory that has been operating since the 1960s. The site faces into the dry gentle trade winds over the caldera and has been proven to have good seeing for at least several hours per day. The atmosphere above the site is clean and the sky is a deep blue, which allows for observations of the corona. The telescope has an off-axis Gregorian optical configuration open to the air and free of any obstruction in the optical path, such as a secondary mirror and its spider. An alt-azimuth mount, with an independently rotating coude´ platform, has been built (Fig. 12.2.7). The telescope opens to the sky from a dome at the top of a tower 42 m tall to avoid ground-level turbulence. Superhigh-quality AO will be necessary to reduce or eliminate aberrations introduced by the atmosphere, even by layers at 5 or 10 km altitude. The goal is to reach an angular resolution of 0.03 arcsec in visible light, with AO and under excellent seeing conditions. This resolution would correspond to a size of 20 km on the Sun. The Goode Solar telescope on Big Bear Lake has served as a test bed for the DKIST Gregorian model, and with an advanced AO system it has achieved nearly the diffraction limit of the 1.6-m primary. The jump to a 4-m primary may require further development of AO technology. Simulations with state-of-the-art AO (Morino and Schmidt, 2016) have yielded promising results for DKIST, which were reported at a meeting in 2016. One of the many challenges in meeting the goals of the DKIST is dumping the kilowatts of heat collected by the primary and maintaining ambient temperatures for the optics. One of the virtues of the off-axis Gregorian is the accessibility of the prime focus, for which heat rejection mirrors and cooling equipment can be employed.

4. Outlook for the Future

FIGURE 12.2.7 Optical layout of Daniel K. Inouye Solar Telescope (DKIST). The following two abbreviations “Alt” and “Az” stand for, respectively, “Altitude” and “Azimuth”. From Ruth Kneale of the DKIST Team.

The DKIST project has listed on its homepage the instrumentation suite that will be available at first light if all goes well. All instruments are served by the AO system and include: A 2D near-infrared spectropolarimeter, working with AO A rapid broadband filter graph, working in visible light A slit-based dual-beam spectropolarimeter, using visible light A visible tunable FabryePerot filter

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Finally, a novel instrument, the cryogenic near-infrared spectropolarimeter (Cryo-NIRSP) offers the possibility of measuring coronal magnetic fields outside the limb of the disk. The site has a proven record of low sky brightness and the primary mirror had specified a very low scattered light level. Thus, the conditions are favorable to make unique observations with the Cryo-NIRSP. The individual instruments can be grouped to make parallel, simultaneous observations of chosen events. Only the Cryo-NIRSP will need exclusive use of the beam. The DKIST will open a new era in solar observations, with the opportunity to answer several critical issues in solar physics.

4.2 EUROPEAN SOLAR TELESCOPE The EST is a next-generation large-aperture solar telescope. It is a pan-European project involving 21 research institutions in 15 countries across Europe. EST will be optimized for studies of the magnetic coupling between the deep photosphere and upper chromosphere. This will require diagnostics of the thermal, dynamic, and magnetic properties of the plasma over many scale heights by using multiple wavelength imaging, spectroscopy, and spectropolarimetry. EST is specifically optimized for high precision and sensitivity polarimetry. To achieve its ambitious goals, EST will be a 4-m aperture telescope specializing in high spatial and temporal resolution and using various instruments simultaneously that can efficiently produce 2D spectral information (Fig. 12.2.8). EST will be provided with powerful AO, which will be further strengthened by an MCAO system to correct for contributions to seeing from certain heights in the atmosphere (Soltau et al., 2010). That is to ensure a fully corrected FOV of 1 arcmin  1 arcmin.

FIGURE 12.2.8 Artist’s impressions of the future European Solar Telescope. From the EUROPEAN SOLAR TELESCOPE.

4. Outlook for the Future

The optical configuration of the EST will be Gregorian with a primary on-axis mirror with a central obscuration. The primary mirror and associated mirror cell will both be mounted above the elevation axis and will be directly exposed to the wind. As an integrated part of the AO system, the surface of the primary mirror will be actively controlled by 96 axial actuators in the back of the plate cell. The EST optical configuration will provide good polarimetric performance and a diffraction-limited Coude´ focus covering a spectral range from UV to infrared, i.e., wavelengths from 390 nm to 2.3 mm. The two Canary Islands, La Palma and Tenerife, are the candidate sites to host the EST. The EST commissioning and operation phase are planned for the late 2020s.

4.3 PLANS FOR 8-METER TELESCOPES The pioneering closed-loop experiments with MCAOs for solar observations are successful and are being applied for nighttime telescopes as well (Beckers et al., 2013; Rigaut et al., 2014). The prospect of providing diffraction-limited imaging with gradually larger aperture telescopes raises the possibility of building solar telescopes with apertures beyond 4 m. The science objectives of an 8-m Chinese Giant Solar Telescope (CGST) are similar to those of the 4-m DKIST and the EST. The CGST is a joint effort from the Yunnan Astronomical Observatory, Chinese Academy of Sciences (CAS); the National Astronomical Observatories, CAS; the Purple Mountain Observatory, CAS; the Nanjing University; the Nanjing Institute of Astronomical Optics; and the Beijing Normal University. Various technical solutions are under consideration. One of these is a Gregorian-type telescope with a continuous ring made out of segments or a multiple-aperture ring consisting of six off-axis telescopes. The CGST will have an infrared MCAO for studies in near-infrared wavelengths with higher magnetic field accuracy compared with visible wavelengths. The Trari Namtso at an altitude of 4600 m in Tibet is likely to be selected for the CGST.

4.4 CONCLUDING REMARKS From documented studies provided by modern solar telescopes in operation and documents preparing for DKIST and EST, one may conclude that ground-based instruments provide opportunities that do not currently exist in space or will in the near future. These include large-aperture multistage AO telescopes that will achieve 0.0025-arcsec spatial resolution and broad spectral range using very high-spectral, spatial resolution spectrometers. Multiple instruments will be operating simultaneously at data rates that are far in excess of what is now technically possible from space. In addition, space instruments are inevitably partly out of date before launch, so ground-based instruments and new techniques can be executed on a somewhat faster time scale.

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5. SUMMARY Observations and computations of the Sun’s activity over the past few decades have convinced many solar astronomers that the key processes connecting plasma and magnetic field occur at spatial scales smaller than existing telescopes can resolve. To reveal the key interactions, angular resolution less than 0.1 arcsec will probably be necessary, along with other requirements. Observers have therefore undertaken replacing current telescopes and instruments with new technology. Meter-class primary mirrors for telescopes in an open-air Gregorian configuration are proving their worth, but only after air turbulence in the telescope and above it are dealt with. Evacuation of the air remains an effective solution for mirrors smaller than 1-m diameters. Larger mirrors require active real-time correction of the image with so-called AO. The independent, parallel development of AO systems at several institutions has been a rousing success story, although it has taken almost a decade to mature. All of the major observatories now operate some version of an AO system that is coupled with a meter-class telescope, up to 1.6 m. Focal plane instrumentation is also undergoing a transformation to take advantage of larger photon fluxes and lower scattered light. More extensive combinations of instruments, working in parallel, are producing more complete observations of rapidly changing events. For example, observers are beginning to report the behavior of individual magnetic fibrils in such objects as arch filament systems, white light flares, and sunspot penumbras. At the same time, advanced MCAO systems are being developed by a consortium of institutions. Early versions at Big Bear Observatory are capable of delivering diffraction-limited resolution over a distance three times larger (30 arcsec) than was previously possible. This is an advantage in observing a large active region, which may span 60 arcsec or more. The solar community is looking forward to the completion and inauguration of the first 4-m solar telescope (DKIST) and the beginning of the second (EST). The CGST (8 m) is in the planning stage and preliminary designs for its AO system are progressing. Whether the project is feasible and fundable will depend on the experience with the 4-m telescopes.

ACKNOWLEDGMENTS We are grateful for comments and suggestions from Jack Harvey, Luc Rouppe van der Voort, Jean-Claude Vial, and Sven Wedemeyer during preparation of this chapter.

References

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Solar Data and Simulations

13 Neal Hurlburt

Lockheed Martin Advanced Technology Center, Palo Alto, CA, United States

CHAPTER OUTLINE 1. Introduction .......................................................................................................443 2. Solar Data .........................................................................................................444 3. Transformation and Provenance ..........................................................................446 4. Search and Discovery.........................................................................................447 5. Data Fusion and Analysis ....................................................................................449 6. Solar Simulations...............................................................................................450 7. Early Simulations ...............................................................................................451 8. Shift Toward Solar Similitude..............................................................................453 9. Future of Data and Simulation .............................................................................455 References .............................................................................................................458

1. INTRODUCTION Our only means of studying the Sun are through remote telescopic observation and in situ measures of ejecta spewing away from it. Therefore, we must rely on a detailed understanding of fundamental physics to interpret what we find. The most basic properties of the Sun were derived from this combination, evolving from astrometric data combined first with geometric theory and then Newtonian theories of motion and gravity, and then added to surface temperatures derived using spectral observations and theories of blackbody radiation. Our first direct measures of the Sun’s magnetic properties came from using the Earth itself as an in situ probe, with the resulting geomagnetic storms, and then through the direct comparison of observations of spectral lines with advances in atomic physics. Today we have an abundance of data collected from a fleet of spacecraft and ground-based instruments that generate a range of data that are open to the worldwide research community and complex simulations based on our current understanding of the physical processes involved in interpreting existing observations and informing designs of future instrumentation.

The Sun as a Guide to Stellar Physics. https://doi.org/10.1016/B978-0-12-814334-6.00016-9 Copyright © 2019 Elsevier Inc. All rights reserved.

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In this chapter, we first review the state of available data and the data environment surrounding them. This is followed by a similar review of simulations, which is then followed by a brief extrapolation into the future based on these reviews.

2. SOLAR DATA There is a natural disconnect between what we can observe and what we would like to know about the physical state of the Sun. Although we desire to know physical properties such as pressure, temperature, and composition at all locations and times, observing methods limit us to a few point-wise direct measures of these far from the Sun and sets of solar images and spectra with limited spatial resolution, temporal sampling, and fields of view. Thus, much of what we learn about the Sun comes from inversions and inferences using these data combined with models that represent our understanding of the physical processes that relate what we measure to what know. The path that transforms data to knowledge can be broken down into defined processes including instrument design and operations; data capture methods, data calibration, transformation to physical observables, and data provenance; data storage and transport; data discovery; data analysis and data fusion; and finally, comparison with models and simulations. Each of these steps relies on a known and reasonably accurate understanding of the artifacts and limitations of the previous steps. The usefulness of data collected from an instrument ultimately depends on its physical and operational properties, as well as the level of knowledge about those properties (Fig. 13.1). There are always trade-offs in instrument design that lead to optical artifacts such as vignetting, distortion, point spread function, and scattered light. In addition, ground-based observations must deal with atmospheric effects of visibility and seeing. If properly characterized, these artifacts can be further minimized during the calibration process. Solar physics has historically concentrated on pushing the limits to uncover the morphologies and dynamics of solar phenomena, partly owing to the difficulty of calibrating extended objects such as the Sun. As a result, absolute calibration knowledge has typically been less essential than other fields. Details regarding the state of the instrument, the quality of the resulting data, and features or events found during the observation or after reviewing the data are available at various stages of data capture and processing. These can be captured by properly designed processing pipelines as annotations to the final data set. Other information is critical, particularly pointing accuracy, because stationary reference features often are unavailable in the instrument’s field of view. Observing logs and postobservation notes are also valuable for archival studies, which can be captured as annotations to the data collections. Currently, the most common means of capturing image data is with arrays of charge-coupled devices. These arrays introduce artifacts caused by variable gain

2. Solar Data

FIGURE 13.1 Data collected by an instrument inherit properties both of the instrument and regarding how it is operated. This includes observatory properties, such as where it is located (ground or space) and what observational constraints exist (dayenight cycle, seeing conditions, etc.). It also contains details about how the instrument is used and properties of the instrument itself, such as its field of view (where it is pointing and the spatial extent of the observation) and the wavelength or band of wavelengths observed.

across the array and dark currents that shift the offset. Readout of these detectors also introduces noise. The resulting images often must be compressed, which may introduce additional artifacts. Once the raw data have been collected, they can be processed to remove as much of the instrument and detector artifacts as feasible. This depends on the availability of both calibration data during testing and operation and the appropriate algorithms and computing power. It is common to characterize the processing using terms of raw, levels 0, 1, and higher. Raw typically refers to data as collected directly from the instrument. Level 0 image data are a reconstruction of the detector array. Level 1 has detector effects removed, which typically includes identifying bad pixels, correcting for variations in gain (i.e., flat-fielding), and removal of dark currents. It may also include permutations such as transposes or flips to give a rough orientation with solar north. Most solar image and spectral data from level 0 upward are stored in the Flexible Image Transport System (FITS) format. Since the Solar and Heliospheric

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Observatory (SOHO) launched in 1996 (Domingo et al., 1995), the solar community has used a consistent set of key words describing the images. This has provided the foundation for the development of common tools for analysis and easy file exchange for missions and researchers.

3. TRANSFORMATION AND PROVENANCE The most computationally expense portion of solar data processing is the steps beyond level 1, where the data are transformed to physically meaningful quantities. This is where corrections for optical artifacts, distortion, seeing, and point spread

FIGURE 13.2 Concept map describing the steps involved in generating higher-level data products from the Helioseismic and Magnetic Imager (HMI) velocity maps, which are derived from sets of HMI level 1 data. The yellow boxes are intermediate data products whereas the green boxes are archived data. The intervening processes shown in pink transform one product to the next with the associated purple boxes indicating the prelaunch development status.

4. Search and Discovery

functions come into play. The data may also be inverted to extract velocities, densities, and temperatures from the corrected intensity images. An example of the products derived from the Helioseismic and Magnetic Imager (HMI) (Scherrer et al., 2012) images, part of the Solar Dynamics Observatory (SDO) (Pesnell et al., 2012), is shown in Fig. 13.2. Appropriate use of the resulting data products in downstream applications requires an understanding of the processing heritage, or data provenance. Provenance is usually captured in the FITS header as comments or key words. One element of data provenance is documentation of the data pipelines used to create the data product. Concept maps such as that seen in Fig. 13.2 provide a visual means for documenting these pipelines. More robust tracking can be provided by using selfdefining workflow systems such as Pegasus, used by the German SDO data center1 or such as Kepler, used in processing data collected by the Solar Optical Telescope on Hinode (Kosugi et al., 2007). Annotations of the data sets shown in Fig. 13.1 provide another means to capture these details. Different computational models have been employed for higher-level processing by different solar communities. The choice depends on a trade-off among the computational, data transport, and provenance requirements. Studies of the solar atmosphere have traditionally worked directly with level 1 data, with perhaps the additional steps of image registration and rotation. The computing model in this case has a central data center for level 0/1 data that provides routines to the user to process data downloaded from that center locally. Alternatively, helioseismology involves several computationally demanding processing steps involving long time series of image data. In this case, a centralized, high-performance computing facility is a better fit to manage the large data volume while controlling the overall provenance of the resulting products.

4. SEARCH AND DISCOVERY Solar data are distributed worldwide; most host institutions provide open data access through websites and Web services. Several tools have been developed by the solar community to assist in finding solar data that are relevant to particular investigations. Fig. 13.3 displays a concept map of the metadata describing solar observations. Given the community standard for image metadata since SOHO, all solar data contain instrument-level details such as the observable wavelength range, time of exposure, and resolution for each file. These parameters form the foundation of the Virtual Solar Observatory2 (Hill et al., 2009) developed by a consortium of solar institutions and supported by the National Aeronautics and Space Administration

1

http://www2.mps.mpg.de/projects/seismo/GDC-SDO/. http://sdac.virtualsolar.org.

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FIGURE 13.3 Concept map of searchable metadata for solar observations.

Solar Data Analysis Center. Most solar image data available online are discoverable through this interface. Richer descriptions of the data sets take into account details of the coverage of any particular instrument, including which region of the Sun is observed, what the instrument operating mode is at the time, how long a particular observing sequence lasts, and at what cadence data are collected. Host sites typically possess this information but it may be inaccessible or difficult to obtain online. The Lockheed Martin Solar and Astrophysics Laboratory’s data center, which hosts the Interface Region Imaging Spectrograph (IRIS), Hinode, and Atmospheric Imaging Assembly data, is an example of making these metadata available for enhanced search. Ultimately, researchers want to find data that have something interesting contained within them. One means of helping in the search and discovery phase is to identify such data either through comparison with externally provided lists of features or events or by actively mining the data using specific algorithms or more general machine learning methods to classify the data. The results of these

5. Data Fusion and Analysis

can then be added to the search services. The Heliophysics Event Knowledgebase3 (HEK) (Hurlburt et al., 2010) is an example of attempting to do both. A faceted search in which one can start the search on any of the central nodes (Feature, Coverage, Instrument, etc.) provides users with a flexible tool for search and discovery. The VSTO (Fox et al., 2009) and HELIOS (Bentley et al., 2011) systems were early efforts in the ability to search all solar data. The heksearch4 tool, based on the HEK system, is the most recent interface of this type.

5. DATA FUSION AND ANALYSIS With the ability to obtain images efficiently from multiple instruments, the next step in the chain is data fusion, the act of combining data from multiple sources, and analysis. The Helioviewer5 project (Mu¨ller et al., 2009) and Festival6 (Auche`re et al., 2008) take advantage of common metadata to provide multiple-instrument visualizations, such as are displayed in Fig. 13.4. It also integrates a subset of the features and events contained within the HEK and provides tools for movie generation and downloads and sharing via social media.

FIGURE 13.4 Data from multiple missions can easily be fused using tools such as Helioviewer.

3

http://www.lmsal.com/hek. http://www.lmsal.com/heksearch. 5 http://helioviewer.org. 6 http://www.ias-u-psud.fr/stereo/festival. 4

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The solar physics community established a common data processing and analysis environment, known as SolarSoft (Freeland and Handy, 1998), based largely on the Interactive Data Language in the 1990s. It has evolved and matured over the following decades by promoting standard FITS metadata and access methods and by integrating new instruments and missions via multilingual software trees. The project grew out of the Yohkoh mission (Ogawara et al., 1991) expanded for multiple instruments by the SOHO mission and for new viewpoints by the STEREO mission (Thompson, 2006). SolarSoft currently supports virtually all solar instruments, with many software packages that are distributed worldwide, including the Festival package described previously. Studies using these software tools dig deep into the various data products to explore and understand the detailed morphologies and dynamics of solar phenomena. The rise of the Python language in scientific computing has inspired the development of SunPy,7 which mirrors many of the functions of SolarSoft but with access to a large library of routines including those for numerical methods (NumPy), scientific computing (SciPy), and astrophysics (Astropy). The move toward such tools is partly motivated by the availability of advanced methods in data analytics and machine learning in the broader Python community and by the need for higher-performance computing architectures such as graphics processing units and big-data environments for scalable analysis to support these methods. Many of the methods have been available for many years: it is the concurrence of large uniform data sets, high-performance computing tools, and researchers who know how to use them that has moved the methods into the mainstream and defined the new field of data science. Examples of these applications include investigations into flare prediction (Bobra and Couvidat, 2015) using support vector machines, differential emission measure inversion using compressed sensing (Cheung et al, 2015), and velocity estimationetrained deep neural networks (DNNs) (Asensio Ramo et al., 2017). Modern data science approaches may eventually supplant the entire processing chain that we have described in this section. One could imagine feeding the raw level 0 data and associated metadata into a DNN and training it to produce high-quality inverted data and from then letting the DNN figure out how to deal with all of the image artifacts.

6. SOLAR SIMULATIONS Like other fields in astrophysics, there is little opportunity for solar physicists to conduct controlled laboratory experiments. Instead they must turn to models and simulations to explain their observations and validate theoretical arguments. Unlike

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http://sunpy.org.

7. Early Simulations

other astrophysical systems, solar physicists have an (over)abundance of data that require interpretation. It is only within the past few decades that computing power and computational methods have been able to begin to bring these together. The general approach to simulating solar phenomena has been first to identify the essential physics that must be included to address the specific scientific question under investigation and then to attempt to exclude or parameterize nonessential processes. The type of physical processes that make up modeling the Sun (or other stars) include coping with highly stratified, compressible fluids and their interaction with magnetic fields; atomic physics and radiative transfer; and the effects of chemical composition and rotation. The relative importance of these processes varies with the temperature and density of the solar plasma, which naturally leads to the standard classification of solar layers. In the solar interior, radiative transfer is well-described as a diffusive process and the composition and atomic physics is encapsulated as variations of diffusivity, heat capacities, and latent heat with temperature and pressure. The explicit dynamics are dominated by rotation, convection, and, to some extent, magnetic fields. The situation in the low corona is reversed with the dynamics controlled by radiative cooling of a wide variety of atomic ions and structured by the dominant magnetic field. The effect of compressibility has an important role from beneath the photosphere through the transition region; the chromosphere presents the full complexity of all of these effects aside from rotation. For the most part, simulations in solar physics are based on the magnetohydrodynamics (see Eqs. (7.11)e(7.18) presented in Chapter 7), but with various approximations and assumptions applied, depending on the region and processes being studied. The phenomena observed on the Sun are highly dynamic, variable, and closely coupled. The details of the solar dynamo reveal themselves in the disruption of the quiet Sun’s granulation patterns forming sunspots at the surface, leading to explosive events in the corona leading, which in turn lead to disruption of large portions of the solar atmosphere and solar wind. Simulations have two roles when confronted with these interacting systems: focus on idealized systems and parameter surveys to reveal the range of possible behaviors; or focus on a particular region or phenomenon and attempt to mimic the observed behavior. The earliest simulations focused on the former approach; as we have built up an understanding of the possibilities, they have transitioned increasingly to the latter approach.

7. EARLY SIMULATIONS The study of magnetoconvection was one of the earliest problems to be explored numerically. Weiss (1966) conducted simulations of nonlinear, two-dimensional magnetoconvection in the Boussinesq limit (Chapter 7 section 2.5, p. 10). He demonstrated that magnetic flux is expelled from the center of the convective cell. Later studies revealed a rich range of possible dynamics in both two and three dimensions, which are reviewed in Proctor and Weiss (1982) and

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Weiss and Proctor (2014). These included patterns and dynamics that mimicked those seen within sunspots; their surroundings depend primarily on field strength and inclination (Hurlburt et al., 2000; Hurlburt and Rucklidge, 2000). One-dimensional hydrodynamic investigations from the early 1970s to the late 1990s explored the behavior of acoustic oscillations in the solar photosphere and chromosphere. Stein and Schwartz (1972) developed a finite difference code to examine the ability of acoustic modes, such as the 5-min oscillations to heat the chromosphere in the presence of ionization and related effects. This was followed by studies of the nonlinear dynamics of the oscillations (Leibacher et al., 1982) and eventually to detailed comparisons between time-averaged, shock-driven heating found in the simulations and semiempirical models derived from spectral observations (Carlsson and Stein, 1997). Graham (1975, 1977) carried out early nonlinear simulations of fully compressible convection in two dimensions, finding a clear asymmetry between the rising and descending flows; Hurlburt et al. (1984, 1986) determined that nonlinear pressure effects decelerated the former and accelerated the latter. Subsequent simulations revealed that the downflow regions were also sites of increased vorticity (Brummell et al., 1996) and that the surrounding inflows could become supersonic (Cattaneo et al., 1990). The dynamics of convection in deep, rotating shells similar to the solar convection zone were initiated by Gilman (1983). Subsequent simulations are reviewed in detail in Chapter 4, Section 4.31, to which we refer the reader. Efforts to understand flux emergence were initially studied using the thin flux tube approximation, in which a single magnetic tube is assumed to rise buoyantly across the convection zone under the influence of magnetic buoyancy, rotation, and turbulent drag (Moreno-Insertis, 1983; Calagari et al., 1995; Fan, 2009; Moreno-Insertis and Emonet, 1996). These demonstrated that magnetic flux could indeed rise to the solar surface in patterns similar to the observed Hale and Joy laws (the ordering of polarities for leading and following sunspots and the tendency for leading spots to be closer to the equator) within certain parameter ranges. Soon after compressible and anelastic simulations in two and three dimensions, Longcope et al. (1996) and Fan et al. (1998a,b) demonstrated that the survival of such flux tubes was susceptible to shredding by the vigorous surface convection and required a significant twist in the field to survive. In the solar corona, where beta is expected to be much less than unity, simulations initially concentrated solely on the structure and evolution of the magnetic fields. Understanding the mechanisms associated with flares and coronal mass ejections (CMEs) was one focus of these studies. Forbes and Priest (1983, 1984) and Forbes (1990) studied magnetic reconnection in the corona and its consequences using a two-dimensional, flux-corrected transport (FCT) code. These studies explored a catastrophic model for CME release. These were soon followed by the three-dimensional simulations of Mikic and Linker (1994a) using a semiexplicit finite difference method exploring the consequences of sheared photospheric fields and the breakout model of Antiochos et al. (1999), also based on FCT methods.

8. Shift Toward Solar Similitude

An alternative approach to studying coronal fields is to focus on the hydrodynamics along specified field lines or flux tubes with the assumption that flows are limited to move along those fields. Such one-dimensional simulations have been used to investigate the radiative stability of flows along the loops starting with Vesecky et al. (1979) and to permit detailed studies of the sources of coronal heating (Aschwanden et al., 2000) and prominences (Karpen et al., 2006).

8. SHIFT TOWARD SOLAR SIMILITUDE As computing power increased, simulations transitioned to three dimensions and began incorporating more complete sets of physical processes. This led to the shift from using simulations to uncover basic solar behaviors and guiding physical intuition to using them to replicate phenomena and be directly compared against observations. Nordlund (1985) pioneered this approach with his studies of photospheric convection that combined aspects of radiative transfer and atomic physics into an anelastic convection code. For the first time, observers could directly recognize the characteristic quiet-sun granulation patterns in his simulations. This encouraged further development of “realistic” simulations in the research community. The increasing complexity of the resulting solar convection codes also led to the coalescence of individual researchers into collaborations working with common code bases. Studies of photospheric convection continued to be refined and expanded by Nordlund et al. (Stein and Nordlund, 2012) and were joined by those using the MURaM (Vogler et al., 2005) and CO5BOLD (Wedemeyer et al., 2004) codes. These codes included improved and expanded approaches to simulating the physical processes, such as a greater number of atomic species and improved treatments of radiative transfer. They also incorporated higher-order numerical schemes with lower numerical diffusivities. Meanwhile, coronal simulations were gaining increasingly realistic treatments of the energy equations. The Magnetohydrodynamics on a Sphere (MAS) (Mikic and Linker, 1994b; Lionello et al., 2001) and Bats-R-Us (Toth et al., 2012) codes captured the highly structured corona, with its sharp, anisotropic gradients in the magnetic field, using automatic mesh refinement methods, in which the density of the mesh is concentrated in high-gradient regions. Sunspots pose one of the most difficult challenges in simulation owing to the combination of highly stratified convection, intense magnetic fields with plasma-b near unity, and complex radiative transfer including molecular chemistry. Schussler and Vogler (2006) made the first realistic simulation of magnetoconvection within a sunspot umbra. This was soon followed by a series of improvements and expansions of the MURaM code until full sunspots were simulated by Rempel et al. (2009) and Rempel (2015). These captured almost all visible elements of sunspots, including umbral and penumbral magnetoconvection surrounded by a vigorous network of granular convection. These series of papers confirmed the general behavior of solar

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magnetoconvection that had been laid out in simpler studies: the distinctive patterns and behaviors displayed in sunspots could be explained as the convective response to the strength and inclination of the sunspot’s magnetic field. The question of why the field has a particular field structure remains to be understood. In part, this is because of a lack of understanding of the large-scale source of sunspot magnetic fields and to its interactions with surrounding supergranular scale flows. Cheung et al. (2007, 2010) used the MURaM code to model the final stages of flux emergence as it breaks through the photosphere and interacts with the intense convective motions there. Their results show the formation of proto-sunspots with hints of penumbra. However, fully formed penumbra still require the imposition of unrealistic upper- or lower-boundary conditions to force sufficient inclination in the field. This may result from the use of periodic boundary conditions on the sides (Rempel and Schlichenmaier, 2011) or it may be a call for better treatment of the small-scale flux pumping (Brummell et al., 2008). The coronal and subphotospheric simulations remained distinct at the end of the 20th century, each assuming idealized boundary conditions at their interface and largely ignoring the intervening chromosphere. By the turn of the 21st century, this was becoming untenable. Attempts to provide a simple coupling were confronted with dealing with the full range of physical processes that are present in the chromosphere and transition region. The Bifrost code (Gudiksen et al., 2011) was the first to address these complexities fully. It enabled simulations that could capture granular magnetoconvection and its interactions with magnetic loops extending into the low corona. The ability to simulate the interface region between the photosphere and corona enabled missions such as IRIS (De Pontieu, 2014), which is in orbit. The Bifrost code provides the critical link for interpreting the otherwise indecipherable spectral profiles observed in that region. Its ability to model a particular physical scenario and then generate synthetic spectra that can be compared directly with observation opens a new way to do solar physics. Similar advances are being made with other simulations. The MAS eclipse predictions, generated from synoptic maps of the photospheric field before the past several solar eclipses, provide guidance in preparing for observing campaigns. A rendering of Aug. 21, 2017 eclipse is displayed in Fig. 13.5. Eventually, solar simulations will become fully integrated with observational data as they have in terrestrial weather simulations, using data assimilation techniques developed in that field. In the solar case, this requires driving simulations from observed photospheric measurements of magnetic field and velocities. Fisher et al. (2015), for example, successfully used a magnetofrictional code to generate a simulated eruption driven by the evolving vector magnetic fields in the photosphere as observed by SDO/HMI. At this point, solar physics has a deep and rich set of simulation tools that span a wide range of physical parameters and processes present in the Sun. With the increase in computing power and advances in methods, each code-based consortium is expanding and may merge as they cross domains: Global-scale simulations of

9. Future of Data and Simulation

FIGURE 13.5 (Left): A synthetic solar eclipse based on a Magnetohydrodynamics on a Sphere simulation can be directly compared with the Aug. 21, 2017 eclipse. Here, the plasma density within the simulation is used to generate the polarization brightness as viewed from Earth owing to scattering by coronal electrons. (Right): The eclipse as observed (Adler, 2017). The simulation captures the overall structure, including the two helmet streamers in the lower hemisphere. Courtesy of Predictive Sciences, http://www.predsci.com/corona/aug2017eclipse/home.php.

the dynamo and magnetic flux emergence (Chen et al., 2017) displayed in Fig. 13.6 are becoming fully compressible and nearly resolving granular scales while the radiative hydrodynamics models of photospheric convection are reaching broader and deeper. They are also reaching higher to merge with transition region (Martinez-Sykora, 2017) and coronal models (Rempel, 2017), leading to the first self-consistent simulations of flares (Cheung et al., 2018), as seen in Fig. 13.7.

9. FUTURE OF DATA AND SIMULATION As these simulations approach or exceed the fidelity of observational techniques, the strategy for advancing the subject is evolving. Simulations are no longer being used only to uncover physical processes that might be observed but are instead generating synthetic data that can directly confront observation. Because the latter will always be constrained by physical limits such as mean-free paths, imaging system resolution, and coverage, simulations will become more of a means to extend observations to what cannot be observed and to use observations to predict future events, including space weather forecasts. Other emerging trends are reshaping data processing. The increase in both the number of sensors and their ability to collect at data rates higher than can be

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FIGURE 13.6 Active region emerging through the simulated solar photosphere after being generated from emerging flux of a dynamo simulation. Courtesy of Chen, F., Rempel, M., Fan, Y., 2017. Emergence of magnetic flux generated in a solar convective dynamo. I. The formation of sunspots and active regions, and the Origin of their Asymmetries. Astrophys. J. 846, 149.

transmitted to a processing center encourages future instrument designs to include onboard data processing. This was foreshadowed by the experiences of the Michelson Doppler Imager on SOHO (Scherrer et al., 1995), which had custom electronics to compute magnetograms and Dopplergrams onboard to accommodate the bandwidth limitations imposed by its deep-space location. One consequence of this approach was that the imperfections in the instrument could not be fully characterized on-orbit. This introduced errors in the processed images that could not be corrected after the fact on the ground. The availability of flexible, modern computing hardware and software makes it feasible to move the full data-processing pipeline onboard, thereby permitting tuning of the pipeline on the fly. Simulation may also offer a means to increase the scientific return of deep-space observations. Space missions orbiting the Earth are already limited by their ability to bring data to the ground, and as their capacity grows or the distance increases, the problem becomes worse. Similar problems are emerging in ground-based systems such as the Daniel K. Inouye Solar Telescope (Berukoff et al., 2016), for which image selection on the mountaintop is already incorporated into the data processing plan to reduce data sent down the mountain to 10 GB/s. A data-driven model of reasonable fidelity could be used to compress data collected at remote sites, say at

9. Future of Data and Simulation

FIGURE 13.7 A self-consistent solar flare generated by MURaM. Although not driven directly by observed photospheric fields and flows, this simulation was designed to mimic the observed emerging flux seen in National Oceanic and Atmospheric Administration Active Region 12,017. Courtesy of Cheung, M., Rempel, M., Chintzoglou, G., Chen, F., Testa, P., Martinez-Sykora, J., Sainz Dalda, A., DeRosa, M., Malanushenko, A., Hansteen, V., De Pontieu, B., Carlsson, M., Gudiksen, B., McIntosh, S., 2018. Realistic radiative MHD simulation of a solar flare. Nature Astron. (in press).

the L5 Lagrange point, 1 AU away from the Earth, returning only significant differences between the model and observation. Back on the ground, cloud computing is becoming more cost-effective. It is already being used as a backup service for some missions and it is possible that many of the instrument-based data centers that exist in the solar community may eventually merge into a unified system in the cloud, with researchers buying computer time to compute near the data rather than buying disks to store data for local analysis. At the other extreme, high-speed research networks such as the Pacific Wave project8 are displaying network transfer rates that exceed the read and write speed of local disk arrays, effectively making all data local. If these networks become widespread, it is unclear whether it matters where the data are stored or where

8

http://www.pacificwave.net.

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simulations are calculated. One thing that is certain is that future data center architects will have a wide range of options.

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Leibacher, J., Gouttebronze, P., Stein, R., 1982. Solar atmospheric dynamics. II - nonlinear models of the photospheric and chromospheric oscillations. Astrophys. J. 258, 393. Lionello, R., Linker, J.A., Mikic, Z., 2001. Including the transition region in models of the large-scale solar corona. Astrophys. J. 546, 542. Longcope, D.W., Fisher, G.H., Arendt, S., 1996. The evolution and fragmentation of rising magnetic flux tubes. Astrophys. J. 464, 999e1011. Martinez-Sykora, 2017. Astrophys. J. 847, 36. Mikic, Z., Linker, J., 1994. Disruption of coronal magnetic field arcades. Astrophys. J. 430, 898. Mikic, Z., Linker, J.A., 1994. Disruption of coronal magnetic field arcades. Astrophys. J. 430, 898. Moreno-Insertis, F., 1983. Rise times of horizontal magnetic flux tubes in the convection zone of the sun. Astron. Astrophys. 122, 241. Moreno-Insertis, F., Emonet, T., 1996. The rise of twisted magnetic tubes in a stratified medium. Astrophys. J. Lett. 472, L53eL56. Mu¨ller, Daniel, et al., 2009. JHelioviewer: visualizing large sets of solar images using JPEG 2000. Comput. Sci. Eng. 11, 38. https://doi.org/10.1109/MCSE.2009.142. Nordlund, A., 1985. Solar convection. Sol. Phys. 100, 209.  Ogawara, Y., et al., 1991. The SOLAR-A mission: an overview. In: Svestka, Z., Uchida, Y. (Eds.), The Yohkoh (Solar-A) Mission. Springer, Dordrecht. Pesnell, W.D., Thompson, B.J., Chamberlin, P.C., 2012. The Solar Dynamics Observatory (SDO). Sol. Phys. 275 (3). https://doi.org/10.1007/s11207-011-9841-3. De Pontieu, B., The IRIS Team, 2014. SoPh..289.2733D. Proctor, M.R.E., Weiss, N.O., 1982. Magnetoconvection. Rep. Prog. Phys. 45, 1317. Rempel, M., 2015. Numerical simulations of sunspot decay: on the penumbra-evershed flow-moat flow connection. Astrophys. J. 814, 125. Rempel, M., 2017. Extension of the MURaM radiative MHD code for coronal simulations. Astrophys. J. 834, 10. Rempel, M., Schlichenmaier, R., 2011. LRSP 8 3. http://www.livingreviews.org/lrsp-2011-3. Rempel, M., Schussler, M., Knolker, M., 2009. Radiative magnetohydrodynamic simulation of sunspot structure. Astrophys. J. 691, 640. Scherrer, P.H., Bogart, R.S., Bush, R.I., et al., 1995. Sol. Phys. 162, 129. Scherrer, P.H., Schou, J., Bush, R.I., et al., 2012. The Helioseismic and Magnetic Imager (HMI) Investigation for the Solar Dynamics Observatory (SDO). Sol. Phys. 275, 207. https://doi.org/10.1007/s11207-011-9834-2. Schussler, M., Vogler, A., 2006. Magnetoconvection in a sunspot Umbra. Astrophys. J. 641, 73. Stein, R., Nordlund, A., 2012. On the formation of active regions. Astrophys. J. 753, L13. Stein, R., Schwartz, R., 1972. Waves in the solar atmosphere. II. Large-amplitude acoustic pulse propagation. Astrophys. J. 177, 8070. Thompson, W.T., 2006. Astron. Astrophys. 449, 791. Toth, G., et al., 2012. Adaptive numerical algorithms in space weather modeling. J. Comp. Phys. 231, 870. Vesecky, J.F., Antiochos, S.K., Underwood, J.H., 1979. Numerical modeling of quasi-static coronal loops. I. Uniform energy input. Astrophys. J. 233, 987. Vogler, A., Shelyag, S., Schussler, M., et al., 2005. Simulations of magneto-convection in the solar photosphere. Equations, methods, and results of the MURaM code. Astron. Astrophys. 429, 335.

References

Wedemeyer, S., Freytag, B., Steffen, M., et al., 2004. Numerical simulation of the threedimensional structure and dynamics of the non-magnetic solar chromosphere. Astron. Astrophys. 414, 1121. Weiss, N., 1966. The expulsion of magnetic flux by eddies. Proc. Roy. Soc. 293, 310. Weiss, N.O., Proctor, M.R.E., 2014. Magnetoconvection. Cambridge University Press. ISBN 975-0-521-19055-8.

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14

Jean-Claude Vial1, Andrew Skumanich2 Senior Scientist Emeritus, Institut d’Astrophysique Spatiale, CNRS-Universite´ Paris-Sud, Orsay, France1; Senior Scientist Emeritus, High Altitude Observatory, National Center for Atmospheric Research, Boulder, Colorado, United States2

CHAPTER OUTLINE 1. Helioseismology...............................................................................................463 2. Solar Dynamo...................................................................................................464 3. Magnetic Activity and Monomials......................................................................464 4. Data Analysis and Modeling..............................................................................465 5. Magnetic Fields and Magnetography .................................................................467 6. Required Spatial Resolution..............................................................................467 7. Nanoflare Heating and Parker’s Model...............................................................468 8. Magnetohydrodynamic Modeling .......................................................................468 9. Flares..............................................................................................................468 10. Long-Term Activity............................................................................................469 11. Instrumentation: What Do We Need?..................................................................470 12. The Issue of Data Volume and Mining................................................................470 References .............................................................................................................472

1. HELIOSEISMOLOGY As mentioned in Chapter 1, major progress has been made in helioseismic sounding of the solar interior, an achievement predicted to be impossible by Sir A. Eddington. The very concept itself came from a combination of apparently innocuous observations (5 minevelocity photospheric oscillations) along with the design of ultrahighresolution and stable spectrometers of the “cell resonance” and later, Michelson types. The frequency resolution of the global modes thus discovered was then improved with ground-based networks and well-situated space instruments (e.g., at the L1 Lagrangian point between the Sun and Earth for the Solar and Heliospheric Observatory [SOHO]), allowing for noninterrupted observations. Actually, the first tools for analysis were borrowed from Earth seismology, although the respective physical conditions are different in the two bodies. Of course, techniques specific to the Sun have been applied and developed, leading to the impressive results shown in Chapter 4. The Sun as a Guide to Stellar Physics. https://doi.org/10.1016/B978-0-12-814334-6.00017-0 Copyright © 2019 Elsevier Inc. All rights reserved.

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The “extrapolation” of helioseismology to other stars resulted from “natural curiosity” but it was also made possible by progress in space technology (e.g., extremely accurate positioning that allowed for the detection of minute stellar orbital variations, with the GAIA Spacecraft, S/C). Opening such a window onto a universe billions of times more crowded has allowed access to different objects in which quantities (such as rotation or magnetic fields inside giants or nearby dwarfs) larger by orders of magnitude can be determined, although they still are elusive or disputed inside the Sun. Does this move toward a stellar window opening mean that helioseismology has reached some limits? Certainly not. With the extension of the SOHO mission, a firm proper detection of gravity modes should be (or is being) reached and perhaps confirms a fast internal (core) rotation. The solar tachocline where the velocity shear occurs also needs further studies (Chapter 7). The local helioseismology should internally confirm the hemispheric rule of helicity (chirality), for instance, through at least two solar cycles.

2. SOLAR DYNAMO This helicity rule explained in Chapter 6, although discovered with standard magnetography performed for a sufficient time, is of major importance and is proposed to be the “simple” result of Parker’s dynamo model (1955a,b). This raises more than a few questions: Is the same physics at work in other celestial objects (e.g., galaxies)? Is the conservation of helicity the rule from the laboratory to the Sun and stars? The issue also has an earthly dimension: If the emergence of sunspots and flux ropes finally leads to coronal mass ejections (CMEs) through the ejection of helicity, is there some possibility of predicting CMEs (and their impact on Earth) through precisely measuring the time evolution of helicity? Of course, this does not solve the difficulties in predicting the nature of plasma when arriving at Earth, including its mass (Howard and Vourlidas, 2018). Concerning similar prediction capabilities, statistical databases are being built for relating “sun-quakes” with the properties of the “incident” accelerated particles and perhaps providing some short-term clues about space weather. Of course, determining the internal magnetic field is a more problematic but nevertheless fascinating objective for understanding the solar dynamo (Chapters 6 and 7). This topic is interesting not only for stellar studies but also for space weather prediction of the magnitude and periods of cycles (or their absence).

3. MAGNETIC ACTIVITY AND MONOMIALS It is significant that algebraic monomial expressions seem to systematize the various parameters describing magnetic activity. These systemizations are well-described in part by several authors: Ayres in Chapter 2, Faurobert in Chapter 6, and Basri in Chapter 11. Next, we will attempt to perform a synthesis that leads to the conclusion that a discordance occurs that needs to be resolved.

4. Data Analysis and Modeling

Phenomena that do not have a characteristic scale, either of length or time (think “exponential” description), we classify as self-similar. Such phenomena often occur when evolving systems are in an “asymptotic” state and can be described by monomials, i.e., “power laws.” A variety of basic “independent” variables is used by different observers, depending on the nature of the observational method, which clouds intercomparisons: rotational velocities, or angular rate U, if one assumes the radius is basically a constant; magnetic field B (or its equivalent “magnetic flux”; and time (age)). Ayres presents in his Fig. 7 a log-log plot of the coronal x-ray and transition region (Si IV) luminosities* (normalized) against U for solar-like stars. In the asymptotic (straight line) region, we find that L_XR w U2.9 and L_TR w U1.9. To this we add the chromospheric Ca II emission luminosity* that L_CaIIey w U (refer to Ayres, Fig. 6). From this, one might conclude that the magnetic heating processes are different in the three regions. Next, consider Faurobert, who lists the observed magnetic field dependence of both CaII emission luminosity and coronal x-rays. According to her, there are contradictory monomials L_CaIIe w B (Skumanich et al., 1975) and also L_CaIIe w B0.5 (Schrijver et al., 1989) that have yet to be resolved. Also, we have that L_XR w B1.13 (Pevtsov et al., 2003). The Pevtsov relation, which is essentially linear, corroborates the Skumanich calibration. Now the magnetic heating processes in the corona and chromosphere are essentially the same. With the two different magnetic field calibrations for chromospheric emission, we enter two different universes. In the case of the linear B field universe, we require B w U (a linear dynamo relation). Thus, Pevtsov’s x-ray result may be interpreted as L_XR w U1.13 and not U2.9. We have a conundrum. Could the x-ray data used by Ayres be contaminated by high-energy events? In the case of the square-root B field universe, one finds B w U2 and L_XR w U1.132 so that the x-ray conundrum remains. This conundrum may be resolved by the new effort to use asteroseismology to determine (global) magnetic fields through their effect on the sound frequencies (Gonc¸alves dos Santos, 2017).

4. DATA ANALYSIS AND MODELING The subtlety and variety of the tools developed to analyze the data are amazing and relevant to various fields ranging from data mining and processing (Chapter 13) to sophisticated plasma and atomic physics (Chapter 5).

*A y

proxy for the total region luminosity. “e” refers to “emission.”

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For instance, ne´cessite´ faisant loi (necessity rules), the quality and quantity of spectroscopic measurements have required complex and efficient methods and codes of nonlocal thermodynamic equilibrium radiative transfer in which it is possible to address the two to three spatial dimensionality (along with time variations) and take into account multilevel atoms and ions. The disagreement between well-observed and computed profiles of resonance lines has led to an increase in complexity with the introduction of some partial resonance scattering in the emission processes (Chapters 5.1 and 5.2). This has had an impact on the wings of these lines, which are enhanced in the presence of velocity fields larger than a couple of Doppler widths. When stars have a strong rotation rate, this subtle effect is certainly smeared (although well present), but in other cases, might it be important for a proper diagnostic? Atomic physics has also provided a wealth of new data that are a prerequisite for any spectral analysis. Tools such as CHIANTI and ADAS are easily available for colleagues who are not necessarily educated in atomic physics. However, the analysis of solar UV spectra sometimes leads to surprising results that one can be tempted to interpret in terms of models, e.g., variations of abundances. This should be exercised only after other possibilities have been excluded (Mok et al., 2016). Actually, the solar atmosphere is increasingly viewed as a combination of timevarying, interconnected, small-scale structures and its analysis requires more and more complex tools. The concepts of detailed balance have to be thrown away in many cases of ionization. This has an impact on the popular differential emission measure (DEM) derivation and meaning; the DEM concept also relies on a unique gradient of temperature that is valid for a one-dimensional (1D) layer. Moreover, whenever strong gradients are present (e.g., temperature across magnetic field lines), ambipolar diffusion has to be taken into account (Zweibel, 2015). This plasma-neutral drift has been introduced in the basic chromospheric Vernazza, Avrett, and Loeser models (becoming Fontenla, Avrett, and Loeser models, Fontenla et al., 1990), in prominences (Fontenla et al., 1996) and in spicules (MartinezSykora et al., 2017). Such a process leads to more complexity in the issue of the neutral versus ionized plasma in those regions. One can expect that this physics will be systematically included in further modeling, whether solar or stellar. As shown by Ballester et al. (2018), the issue of partial ionization is important for many processes, whether magnetohydrodynamic (MHD), Alfven waves, heating, reconnection, conduction, etc. It is increasingly taken into account in the modeling of the chromosphere and other solar features because we know that cool material exists in the corona itself. Another process biasing the current DEM analysis consists of radiative processes (such as “external” illumination), which can add substantially to atomic-level populations (Jordan et al., 1979), possibly providing an explanation for anomalous lines ratios in the transition region (Gontikakis and Vial, 2016). Such radiative processes might also work in the corona (Jejcic et al. 2017); the combination of strong illumination from an active region and internal opacity can explain the discrepancies between the observed and modeled coronal loops.

6. Required Spatial Resolution

If magnetic reconnection is present at all scales, departures from Maxwellian distributions must be taken into account in computing excitation and ionization crosssections (Chapter 5.1). In some regions (e.g., the solar wind), modeling must take into account Hall MHD (Galtier and Buchlin, 2007). In some fast-evolving events (flares), one needs to take into account a microphysics still to be developed: for instance, the need for time-dependent radiative transfer.

5. MAGNETIC FIELDS AND MAGNETOGRAPHY Determination of the magnetic field (Chapter 5.3) has evidenced fantastic progress since the early measurements of Hale. With more and more sophisticated magnetographs, one has access to the vector field at different altitudes with some critical measurements in the chromosphere itself. However, the search for weak magnetic fields (prominence and turbulent) has required developments in atomic physics (Hanle´ effect) that seem not to have been fully achieved (Lopez-Ariste, 2015), because the magnitude of the field seems to depend on the technique (Zeeman vs. Hanle´) and, of course, the spatial resolution. The coronal magnetic field, mainly “known” from extrapolations above the photosphere, can be only episodically measured across active regions. It is on the priority list of major ground-based projects such as the Daniel K. Inouye Solar Telescope and the European Solar Telescope (Chapter 12.2). Linear polarimetry in the coronal UV through the Hanle´ effect (Bommier and Sahal-Brechot, 1982) has been proposed for some time (Vial et al., 2006) and is included in future space projects (Peter et al., 2012). However, it is not easy to say whether the Hanle´ technique will be applicable to stars (e.g., for their large prominences and their eruptions), but the magnitude of the magnetic field remains an important issue for the relationship between (habitable or not) exoplanets and their hosts.

6. REQUIRED SPATIAL RESOLUTION For most of these issues, the critical question remains the “actual” size of solar features: spicules, mottles, filament threads, umbrae, etc., The most popular fine structure consists of cylinders (threads), but there are examples of sheet-like features (Judge et al., 2012). A clear example of threads is provided by filament or loops in which the perpendicular conductivity (kperp) is neglected. However, what if there are small magnetic structures that involve many opportunities for kperp to have a roˆle? What if the plasma is not fully ionized? (See earlier discussion.) This implies that one should improve the spatial resolution by orders of magnitude, a dream with the current instrumentation and perhaps a useless one when the layers one wants to detect are much smaller than the photon mean free path.

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7. NANOFLARE HEATING AND PARKER’S MODEL This remark is still more valid with the kind of threads or fibers that nanoflaring (Parker, 1988) implies in loops (Chapters 2, 6 and 9). Of course, one is tempted to build statistics of the “flares” energy and extrapolate toward very low energies (Chapters 2 and 9). However, only the radiative output (in some lines) is measured and the derivation of the actual energy involved may be hazardous (Aletti et al., 2000). Another puzzling result concerning coronal loops is the slow replenishment cycle (a few hours) compared with the duration of the processes (conduction, evaporation, etc.) involved (Auche`re et al., 2014), which could be explained by the fine structure of these loops (Froment et al., 2018; Mok et al., 2016). The various processes of chromospheric and coronal heating are now welldocumented (Chapters 1, 3 and 11) but there is still some divergence (but not necessarily incompatibility) among the various candidates. The most popular mechanism remains magnetic reconnection, which is important down to the upper photosphere (Chapter 3). It results in current sheets in which resistivity has some role in the otherwise conductive corona and leads to Ohm heating (Chapter 6). It initiates the propagation of Alfven and MHD waves, which can heat plasmas located far away. It also accelerates particles that, following magnetic field lines, will impinge in remote dense areas and contribute to heat the local plasma. It is certainly a major challenge for solar physics to sort out the main mechanisms at work in the different solar features and to confirm the existence of the large current sheets detected by the UV coronagraph and spectrometer instrument on SOHO. Actually, one can even say that it would be a major step to “detect” and characterize the reconnection itself unambiguously, a task already difficult for geophysicists who nevertheless benefit from in situ measurements in the Earth magnetosphere.

8. MAGNETOHYDRODYNAMIC MODELING Observers will be helped by the radiative predictions issued from 3D MHD simulations, including 3D radiative energy transport. In view of the overwhelming number of free parameters in stellar physics, it is difficult to imagine how these solar MHD sophisticated techniques will be applied in stellar atmospheres. However, it is clear that the huge progress made in MHD solar physics (Chapter 7) will benefit other astrophysical fields.

9. FLARES The strongest manifestation of solar activity is the flare and its accompanying manifestations (particles acceleration, prominence eruption, and CMEs), as shown in Chapter 9. On the Sun, it is possible to combine complete remote sensing in the

10. Long-Term Activity

electromagnetic spectrum and the in situ detection of particles, velocity and magnetic fields, waves, and more. For stars, remote sensing is incomplete in terms of wavelength coverage and obviously, spatial resolution. As for stellar in situ measurements, this does not mean that unexplored capabilities do not exist for solar flares themselves. For instance, the 100-eV to 1-keV window that is so important for the formation of the terrestrial ionosphere is still not well-documented. The extreme UV (EUV) dimming (Harra et al., 2016) is reminiscent of “black” flares found by Henoux et al. (1990). Despite the many achievements in flare studies, it is amazing to realize that the precise particle acceleration process itself is still unknown. For further progress, one needs access to high-energy imagery and to have a “multimessenger” approach (Chapter 9). Stellar superflares have been detected in the visible range, a wavelength range in which solar flares produce a small addition of emission. Actually, the source of superflare emission is a matter of debate (see Heinzel and Shibata, 2018, who propose the emission of overlying loops), whose conclusion affects the diagnostics of the whole superflare process. It will be fascinating to observe these superflares over the whole electromagnetic range. In radio, for instance, will it be possible to sound the stellar coronae through the measurements of the equivalent of Type III? Will it be possible to detect super-CMEs through, e.g., Doppler dimmings in the EUV (as on the Sun) (Harra et al., 2016)? And what about the detection of gamma lines in stellar superflares?

10. LONG-TERM ACTIVITY As with all other areas in astronomy (not to mention cosmology), solar and stellar studies allow us to build the history of our star. In this respect, despite the (complex) use of geophysical proxies, the best tools rely on the models of stellar evolution and the observations of solar-like stars. In terms of solar analogues, such observations sometimes provide surprising results such as the low activity of our Sun compared with its colleagues (Chapter 8). It would be interesting to see how this activity is distributed between large and small scales. The main tools consist in the H and K lines of Ca II, ZeemaneDoppler imagery, and interferometry. There is no doubt that the two latter techniques will be improved with time (what about very longebaseline interferometry for solar studies?). One can wonder whether the first could be replaced by the observation in the Mg II h and k lines despite the drawback for these lines of not having been recorded in previous decades. On a routine mode, solar irradiance measurements (Chapter 8) should be continued and performed in increasingly large spectral windows. The reconstruction of irradiance and spectral irradiance now relies on proxies and semiempirical models. There is no obstacle for extending such reconstruction to stars. The observation of their cycles should bring a wealth of information. Can one imagine stellar measurements equivalent to the total solar irradiance?

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Going backward in time, the faint early Sun paradox seems to question the formation of liquid oceans on Earth: but is it a stellar problem or a planetary one (related, e.g., to methane formation)? A very strong wind from the young Sun also could have depleted the Mars atmosphere and its magnetic field (Chapter 10). This depletion also could have been the result of exceptional flares (Chapter 9) such as currently occuring on a large variety of stars (including solar-like ones).

11. INSTRUMENTATION: WHAT DO WE NEED? Since the early Naval Research Laboratory rocket flights at the beginning of the 1960s, space has had an ever more important role in solar physics (Chapter 12.1). This increasing role relies on the unique capabilities of space (access to the whole electromagnetic spectrum, continuity of observations, etc.) and instrumentation, which is the result of huge technological progress. As usual, the share between space and terrestrial-based instrumentation is a matter of debate. As shown in Chapter 12.2, for now, spectropolarimetry at high spatial resolution and sensitivity (very, very photon-hungry) can be performed from large-aperture (e.g. Green Bank Observatory) instrumentation, provided the involved multiconjugate adaptive optics meets the diffraction limit of the telescopes. We have to warn the taxpayers that the size of the instruments will not decrease and that solar physics needs both types of instrumentation, if possible, within increasingly larger collaborations. We mentioned the need for (3D plus time) imaging and spectroscopy, but the line-of-sight sounding that scanning line profiles allows is not enough to determine the actual geometry of the observed structures, which is often complex. As far as the Sun is concerned, observations performed with the EUV imaging telescope and STEREO spacecraft and the use of tomographic techniques (Frazin, 2000; Barbey et al., 2008) lead the path toward systematic measurements made at different angles, e.g., from L4 and L5. Of course, in the immediate future, stellar observations will not provide the wealth of details and the whole big picture extending from the star to the relevant exoplanet, as shown in Fig. 14.1.

12. THE ISSUE OF DATA VOLUME AND MINING For a given mission (whether space instrumentation or dedicated ground-based observatories), the amount of data gathered, combined with ancillary data from other instruments on the same platform or ground-based observatories, has led to the concept of data sharing under an open policy, as evidenced with SOHO, the Solar Dynamics Observatory, Hinode, etc. The very large databases thus built have to be distributed, interconnected, and easily accessible to facilitate data mining. One can say that this activity now pertains to instrumentation and must be started from the very beginning of projects. In parallel, a huge combination of effort (people,

12. The Issue of Data Volume and Mining

FIGURE 14.1 Snapshot from a movie of the combined STEREO/Sun Earth Connection Coronal and Heliospheric Investigation (SECCHI) telescope fields of view on Aug. 1, 2010. The full movie is available at the SECCHI site (https://secchi.nrl.navy.mil/movies/SECCHI_ 20100801_20.mpg). Top row: EUVI (extreme UV imager), COR1 and COR2 (coronagraphs) from STEREO-A flanked by the HI1 (red) and HI2 (blue) (heliographic imagers) on STEREO-A (left) and STEREO-B (right), respectively. The Earth is at the center of both HI2 images, marked by a few saturated columns (owing to its brightness). The galaxy is also visible on HI2-A. Bottom row: Zoomed-in composites of the EUVI 304A (red), COR1 (green), and COR2 (blue) images from STEREO-A (right) and -B (left), respectively. From the National Aeronautics and Space Administration and the STEREO/SECCHI Team.

software, and hardware) is being put forward for collaborative modeling (see Chapter 13). In the perspective of joint solar and stellar studies, let us mention the sunstardb initiative, which consists of building a relational database of stellar properties and magnetic activity proxy time series (Egeland, 2018).

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