Exoplanets 9780816529452, 0816529450

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THE UNIVERSITY OF ARIZONA SPACE SCIENCE SERIES Richard P. Binzel, General Editor Exoplanets S. Seager, editor, 2010, 526 pages Europa Robert T. Pappalardo, William B. McKinnon, and Krishan K. Khurana, editors, 2009, 727 pages The Solar System Beyond Neptune M. Antonietta Barucci, Hermann Boehnhardt, Dale P. Cruikshank, and Alessandro Morbidelli, editors, 2008, 592 pages Protostars and Planets V Bo Reipurth, David Jewitt, and Klaus Keil, editors, 2007, 951 pages Meteorites and the Early Solar System II D. S. Lauretta and H. Y. McSween, editors, 2006, 943 pages Comets II M. C. Festou, H. U. Keller, and H. A. Weaver, editors, 2004, 745 pages Asteroids III William F. Bottke Jr., Alberto Cellino, Paolo Paolicchi, and Richard P. Binzel, editors, 2002, 785 pages Tom Gehrels, General Editor Origin of the Earth and Moon R. M. Canup and K. Righter, editors, 2000, 555 pages Protostars and Planets IV Vincent Mannings, Alan P. Boss, and Sara S. Russell, editors, 2000, 1422 pages Pluto and Charon S. Alan Stern and David J. Tholen, editors, 1997, 728 pages Venus II—Geology, Geophysics, Atmosphere, and Solar Wind Environment S. W. Bougher, D. M. Hunten, and R. J. Phillips, editors, 1997, 1376 pages Cosmic Winds and the Heliosphere J. R. Jokipii, C. P. Sonett, and M. S. Giampapa, editors, 1997, 1013 pages Neptune and Triton Dale P. Cruikshank, editor, 1995, 1249 pages

Hazards Due to Comets and Asteroids Tom Gehrels, editor, 1994, 1300 pages Resources of Near-Earth Space John S. Lewis, Mildred S. Matthews, and Mary L. Guerrieri, editors, 1993, 977 pages Protostars and Planets III Eugene H. Levy and Jonathan I. Lunine, editors, 1993, 1596 pages Mars Hugh H. Kieffer, Bruce M. Jakosky, Conway W. Snyder, and Mildred S. Matthews, editors, 1992, 1498 pages Solar Interior and Atmosphere A. N. Cox, W. C. Livingston, and M. S. Matthews, editors, 1991, 1416 pages The Sun in Time C. P. Sonett, M. S. Giampapa, and M. S. Matthews, editors, 1991, 990 pages Uranus Jay T. Bergstralh, Ellis D. Miner, and Mildred S. Matthews, editors, 1991, 1076 pages Asteroids II Richard P. Binzel, Tom Gehrels, and Mildred S. Matthews, editors, 1989, 1258 pages Origin and Evolution of Planetary and Satellite Atmospheres S. K. Atreya, J. B. Pollack, and Mildred S. Matthews, editors, 1989, 1269 pages Mercury Faith Vilas, Clark R. Chapman, and Mildred S. Matthews, editors, 1988, 794 pages Meteorites and the Early Solar System John F. Kerridge and Mildred S. Matthews, editors, 1988, 1269 pages The Galaxy and the Solar System Roman Smoluchowski, John N. Bahcall, and Mildred S. Matthews, editors, 1986, 483 pages Satellites Joseph A. Burns and Mildred S. Matthews, editors, 1986, 1021 pages Protostars and Planets II David C. Black and Mildred S. Matthews, editors, 1985, 1293 pages Planetary Rings Richard Greenberg and André Brahic, editors, 1984, 784 pages

Saturn Tom Gehrels and Mildred S. Matthews, editors, 1984, 968 pages Venus D. M. Hunten, L. Colin, T. M. Donahue, and V. I. Moroz, editors, 1983, 1143 pages Satellites of Jupiter David Morrison, editor, 1982, 972 pages Comets Laurel L. Wilkening, editor, 1982, 766 pages Asteroids Tom Gehrels, editor, 1979, 1181 pages Protostars and Planets Tom Gehrels, editor, 1978, 756 pages Planetary Satellites Joseph A. Burns, editor, 1977, 598 pages Jupiter Tom Gehrels, editor, 1976, 1254 pages Planets, Stars and Nebulae, Studied with Photopolarimetry Tom Gehrels, editor, 1974, 1133 pages

Exoplanets

Exoplanets Edited by

S. Seager With the assistance of

Renée Dotson With 34 collaborating authors

THE UNIVERSITY OF ARIZONA PRESS Tucson in collaboration with LUNAR AND PLANETARY INSTITUTE Houston

About the front cover: An imaginary planetary system orbiting a cool M-dwarf star. In the foreground, a giant planet has an Earth-sized satellite, a water-rich body. The giant planet and satellite are orbiting in the star’s habitable zone. In the mid-distance is another giant planet (Saturn-sized) close to inferior conjunction with the foreground planet, and a third giant planet is in the far distance, transiting the star. The artist’s viewpoint captures the three planets at the time of a fortuitous alignment. Painting by William K. Hartmann, Planetary Science Institute, Tucson, Arizona. About the back cover: Image of the HR 8799 planetary system produced by combining the J-, H-, and Ks-band images obtained at the Keck telescope in July (H) and September (J and Ks) 2008. The inner part of the H-band image has been rotated by 1° to compensate for the orbital motion of d between July and September. The planets are listed by lower-case letter in order of chronological discovery. At center is the speckle image of the star. From Marois et al. (2008) Science, 322, 1348–1352. Image courtesy of NRC-HIA, C. Marois, and Keck Observatory.

The University of Arizona Press in collaboration with the Lunar and Planetary Institute © 2010 The Arizona Board of Regents All rights reserved ∞ This book is printed on acid-free, archival-quality paper. Manufactured in the United States of America 15 14 13 12 11 10 9 6 5 4 3 2 1 Library of Congress Cataloging-in-Publication Data Exoplanets / edited by S. Seager ; with the assistance of Renée Dotson ; with 34 collaborating authors. p. cm. -- (The University of Arizona space science series) Includes bibliographical references and index. ISBN 978-0-8165-2945-2 (hardcover : alk. paper) 1. Extrasolar planets. I. Seager, Sara. II. Dotson, Renée. III. Lunar and Planetary Institute. QB820.E87 2010 523.2’4--dc22 2010042822

Dedicated to all the people on this planet who have big dreams and succeed in making them happen.

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Contents

List of Contributing Authors and Scientific Organizing Committee.......................................................................... xiii Foreword .......................................................................................................................................................................xv Preface ........................................................................................................................................................................xvii PART I: INTRODUCTION Introduction to Exoplanets S. Seager and J. J. Lissauer . .........................................................................................................................................3 Keplerian Orbits and Dynamics of Exoplanets C. D. Murray and A. C. M. Correia . ..........................................................................................................................15 PART II: EXOPLANET OBSERVING TECHNIQUES Radial Velocity Techniques for Exoplanets C. Lovis and D. Fischer ...............................................................................................................................................27 Exoplanet Transits and Occultations J. N. Winn .....................................................................................................................................................................55 Microlensing by Exoplanets B. S. Gaudi ...................................................................................................................................................................79 Direct Imaging of Exoplanets W. A. Traub and B. R. Oppenheimer .........................................................................................................................111 Astrometric Detection and Characterization of Exoplanets A. Quirrenbach ...........................................................................................................................................................157 Planets Around Pulsars and Other Evolved Stars: The Fates of Planetary Systems A. Wolszczan and M. Kuchner.....................................................................................................................................175 Statistical Distribution of Exoplanets A. Cumming ................................................................................................................................................................191 PART III: EXOPLANET DYNAMICS Non-Keplerian Dynamics of Exoplanets D. Fabrycky ................................................................................................................................................................217 Tidal Evolution of Exoplanets A. C. M. Correia and J. Laskar .................................................................................................................................239

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PART IV: EXOPLANET FORMATION AND PROTOPLANETARY DISK EVOLUTION Protoplanetary and Debris Disks A. Roberge and I. Kamp ............................................................................................................................................269 Terrestrial Planet Formation J. Chambers ................................................................................................................................................................297 Giant Planet Formation G. D’Angelo, R. H. Durisen, and J. J. Lissauer .......................................................................................................319 Planet Migration S. H. Lubow and S. Ida . ............................................................................................................................................347 PART V: EXOPLANET INTERIORS AND ATMOSPHERES Terrestrial Planet Interiors C. Sotin, J. M. Jackson, and S. Seager ......................................................................................................................375 Giant Planet Interior Structure and Thermal Evolution J. J. Fortney, I. Baraffe, and B. Militzer ....................................................................................................................397 Giant Planet Atmospheres A. Burrows and G. Orton . .........................................................................................................................................419 Terrestrial Planet Atmospheres and Biosignatures V. Meadows and S. Seager .........................................................................................................................................441 Atmospheric Circulation of Exoplanets A. P. Showman, J. Y-K. Cho, and K. Menou .............................................................................................................471 Index

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List of Contributing Authors Baraffe I. 397 Burrows A. 419 Chambers J. 297 Cho J.Y-K. 471 Correia A. C. M. 15, 239 Cumming A. 191 D’Angelo G. 319 Durisen R. H. 319 Fabrycky D. C. 217 Fischer D. 27 Fortney J. J. 397 Gaudi B. S. 79 Ida S. 347 Jackson J. M. 375 Kamp I. 269 Kuchner M. 175 Laskar J. 239

Lissauer J. J. 319, 3 Lovis C. 27 Lubow S. H. 347 Meadows V. 441 Menou K. 471 Militzer B. 397 Murray C. D. 15 Oppenheimer B. R. 111 Orton G. 419 Quirrenbach A. 157 Roberge A. 269 Seager S. 3, 375, 441 Showman A. P. 471 Sotin C. 375 Traub W. A. 111 Winn J. N. 55 Wolszczan A. 175

Scientific Organizing Committee Tristan Guillot Marc Kuchner Doug Lin Jack Lissauer Vicki Meadows Wes Traub Stephane Udry

Sara Seager, Chair Alan Boss Adam Burrows David Charbonneau Drake Deming Debra Fischer Olivier Grasset

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Foreword The discovery of exoplanets is arguably the greatest scientific revolution since the time of Copernicus. Simply stated, humanity now knows for the first time as scientific fact: there actually are planets around other stars. Yet as profound as this discovery is, the trumpets have not blared and no theorists have been burned at the stake. (An early proponent of other planetary worlds, Giordano Bruno, did not fare as well in 1600.) Acclamation of exoplanets’ reality has not penetrated the global consciousness as deeply as it might merit because (I think) the public already knew that planets were there. Star Trek took us to a new planet every week; E.T. had to have some place to phone home to; etc. Even though the transition from science fiction to science fact may have seemed subtle, historical perspective from decades or centuries henceforth is likely to see this moment as another major turning point in the human perception of our place in the universe. Now, 400 years after Galileo’s observations affirmed the model of Copernicus, we are in the midst of our own golden era of discovery. Just as Galileo first applied the telescope to reveal the details of our own planetary system, that same spirit of inquisitiveness and ingenuity (with the benefit of technological advances) is now revealing details of other planetary systems. Thus it is easy to fathom why so many current researchers, and most importantly new students, are drawn so strongly to this field. The attraction of pure discovery, the opportunity to share the same experience as Galileo, is simply too alluring to resist. Thus sets the stage for this volume, Exoplanets, perhaps the most important title yet produced in the Space Science Series. The goals of this book are to merge and embrace the astrophysics of exoplanets within the disciplines of planetary science, to fully expand planetary science beyond the bounds of our own solar system, to serve as a foundation for the interface of these merging and expanding fields, and most importantly, to be the gateway for new students and researchers. To accomplish this, Series authors are each challenged to convey what we know, how we know it, and where we go from here. I thank my lucky planets that my colleague, Professor Sara Seager, embraced the editorship of this book with such clear vision and conviction for how it should be accomplished, coupled with the requisite determination and stamina. One cannot understate the effort delivered by the authors whose chapters are the cornerstones upon which the field will continue to build. Less visible, but especially deserving of thanks, are Renée Dotson and co-workers of the Lunar and Planetary Institute (LPI), who brought this volume to reality, literally page by page. The support and dedication for the ongoing success of the Space Science Series by LPI Director Dr. Stephen Mackwell and the staff of the University of Arizona Press are paramount. A grant from Dr. Wesley Traub through the NASA Exoplanet Exploration Program has made this gateway accessible to all by supporting costs for this book. No outcome is more desired than that students entering here become the future leaders of the field. Richard P. Binzel Space Science Series General Editor Cambridge, Massachusetts August 2010

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Preface This is a unique time in human history — for the first time, we are on the technological brink of being able to answer questions that have been around for thousands of years: Are there other planets like Earth? Are they common? Do any have signs of life? The field of exoplanets is rapidly moving toward answering these questions with the discovery of hundreds of exoplanets now pushing toward lower and lower masses; the Kepler Space Telescope with its yield of small planets; plans to use the James Webb Space Telescope (launch date 2014) to study atmospheres of a subset of super Earths; and ongoing development for technology to directly image true Earth analogs. Theoretical studies in dynamics, planet formation, and physical characteristics provide the needed framework for prediction and interpretation. People working outside of exoplanets often ask if the field of exoplanets is like a dot.com bubble that will burst, deflating excitement and progress. In my opinion, exciting discoveries and theoretical advances will continue indefinitely in the years ahead, albeit at a slower pace than in the first decade. The reason is that observations uncover new kinds and new populations of exoplanets — and these observations rely on technological development that usually takes over a decade to mature. For example, in the early 2000s almost all exoplanets had been discovered by the radial velocity technique. At that time, many groups around the world were working on wide-field transit surveys. But it was not until recently, a decade into the twenty-first century, that the transit technique is responsible for almost one-quarter of known exoplanets. The planet discovery techniques astrometry (as yet to find a planet) and direct imaging have not yet matured; when they do, they will uncover planets within a new parameter space of planet mass and orbital characteristics. In addition, people are working hard to improve the precision for existing planet discovery techniques to detect lower-mass planets and those further from their star. All in all, technology enables slow but sure progress, and this fuels ongoing discovery. Theory, like observations, also takes time to unfold and mature. We can anticipate an “ultimate” planet formation model similar to the “millenimum simulation” for galaxy formation and evolution. In time, incorporating detailed physics as well as being able to reproduce the generic outcome of planet populations (mass, radius, and orbital characteristics, including period) will enable a deeper understanding of planet formation and migration. Similarly, the ideal exoplanet atmosphere code of the future could be a three-dimensional Monte Carlo code that includes radiative transfer with inhomogeneous cloud coverage and surface features, a code that also solves for the temperature structure and combines with a hydrodynamical simulation to calculate the three-dimensional temperature and wind structure. Classical orbital mechanics, already reinvigorated by interesting exoplanet systems (e.g., planets in resonant orbits, hot Jupiter exoplanets that orbit in the direction opposite to the stellar rotation), also has a role to play in explaining fundamental mechanisms of how planetary system configurations came to be. Orbital dynamics modeling is driving the search for moons and other unseen planet companions by their perturbations on transiting planet signatures. Exoplanets is a unique science because it involves so many disciplines within and beyond planetary science and astrophysics. The other disciplines include geophysics, high-pressure mineral physics, quantum mechanics, chemistry, and even microbiology. While exoplanet observations clearly belong under the branch of astronomy, for many years the whole discipline of exoplanets lacked a true home. Physics departments have said “Exoplanets: It’s interesting, but is it physics?” xvii

Planetary and Earth science departments used to collecting real data in their hands from Earth and in situ measurements from solar system planets were, in the early 2000s, reluctant to believe there would ever be enough high-quality data to take the research field of exoplanets seriously. With hundreds of known exoplanets and many fascinating observational and theoretical discoveries, the whole world is embracing exoplanets, a field that can now find its home both in and spanning across planetary science, astronomy and astrophysics, and astrobiology. We use the term “exoplanets” and not “extrasolar planets” in this book. Exoplanets is the most direct derivative from the Greek language, from which the word “planet” originates. (Note that the word “extrasolar” is derived from Latin.) It is most fitting that the field be named as traditionally as possible, and there is none more traditional than Greek. The goal of this volume is to cover the range of topics in exoplanet observation and theory at the graduate student level. Each chapter of this volume aims to cover the fundamentals and recent discoveries in such a way as to be a cross between a textbook and a review article. Because the field of exoplanets moves so rapidly, we have tried to arrange each chapter so that the first two sections (the introduction and and an explanation of the basic concepts/fundamental equations) will not go out of date. The next two sections (covering recent highlights and future progress) will go out of date eventually, but serve to capture and review what is going on in a subfield. Some chapters deviate from the standard structure and, by nature of construction, the level of technical detail and writing style will vary in an edited volume with different authors for each chapter. We intend that this book will serve to inspire professional researchers and engineers and students to build a foundation for the next generation of exoplanet observational and theoretical discoveries. Exoplanets is the 37th book in the University of Arizona Press Space Science Series, the first dedicated to planets beyond our solar system. As editor of this volume, I foremost thank General Editor Richard Binzel for believing in my vision for this book and for his advice and fast turnaround on practical matters. I am extremely grateful to Renée Dotson of the Lunar and Planetary Institute for her patience and careful efforts in compiling and production of the book. The book would not have been possible without the contribution and dedication of each of the chapter authors, often going far beyond what the authors had initially anticipated. The chapter reviewers played an invaluable role in improving the quality of this book. Appreciation to William Hartmann for his original painting that graces the book’s cover. Last but not least, I am grateful to Wes Traub of NASA Jet Propulsion Laboratory for his financial support, which has enabled the book to be available for a practical purchase price. Close to two decades since the first exoplanets were discovered, the field of exoplanets has grown without bound. So many surprising discoveries have been made that by now we know to expect the unexpected. I always like to say that, for exoplanets, anything is possible within the laws of physics and chemistry. And, when asked about which planet or discovery is my favorite one, I like to reply that it is the next one, because in exoplanets, the best is yet to come. I hope you, the reader, will use this book to learn about whichever subdiscipline interests you and to build your knowledge to play a role in the future of exoplanet research. Sara Seager September 2010 Cambridge, Massachusetts xviii

Part I: Introduction

Introduction to Exoplanets Sara Seager Massachusetts Institute of Technology

Jack J. Lissauer NASA Ames Research Center

The discovery of planets around other stars, which we call exoplanets, has emerged over the past two decades as a new, vibrant, fruitful field that spans the disciplines of astrophysics, planetary science, and even parts of biology. The study of exoplanets is now an important part of the curriculum for many graduate students, and this volume is intended to bridge the gap between single-article summaries and specialized reviews. This chapter serves as an introduction both to the study of exoplanets and to this book, providing a starting point to new students and researchers entering the field.

1. A BRIEF HISTORY OF DISCOVERY

The contemporary search for exoplanets began with astrometry in the mid-nineteenth century, when a dark companion was suspected to orbit the binary star system 70 Ophiuchi (Jacob, 1855; See, 1896). Although the 70 Ophiuchi “planet” was soon discredited (Moulton, 1899), it likely represents the first published claim of a planet beyond the solar system. A century later, reports of massive planets in the 1940s were controversial and after decades of work were finally discarded as spurious signals. In the 1960s, the detection of the now infamous 24-year-period Jupiter-mass planet around Barnard’s star was announced (van de Kamp, 1963). This and a companion planet turned out to be instrument systematics. The checkered history of exoplanet detection was born [see Jayawardhana (2010) and the chapter by Quirrenbach]. A prescient two-page paper by Struve (1952) proposed that Jupiter-like planets might exist in orbits as small as 0.02 AU. Struve pointed out that high-precision radial-velocity measurements could discover such short-period planets and transits of such planets could be found from photometric observations. At the time, and for the following decades, almost all astronomers thought that Jupiter-mass planets should copy our solar system and reside in Jupiter-like orbits at 5 AU from the host star. The key in making radial-velocity observations precise enough to search for Jupiter-mass planet companions in Jupiter-like orbits was to change the reference frame from telluric spectral lines in the Earth’s atmosphere to an instrument gas cell. Campbell and Walker (1979) were the first to make this advancement, enabling an order-of-magnitude increase in radial-velocity precision, and they embarked on a 12-year search for Jupiter-like planets orbiting 21 bright Sun-like stars. In the late 1980s a substellar object orbiting HD 114762 was detected by Latham et al. (1989). Latham et al. referred to this object as a brown dwarf, but if it is very close to the minimum mass obtained by radial-velocity measurements of

The search for our place in the cosmos has fascinated human beings for thousands of years. For the first time in human history we have technological capabilities that put us on the verge of answering a hierarchy of pressing questions: “Do other Earths exist?,” “Are they common?,” and “Do they have signs of life?” The study of exoplanets seeks to understand how planetary systems formed and evolved, and to understand the diversity of planetary system architectures. The concept that there may be worlds beyond Earth goes back more than 2000  years, with Epicurus (ca. 300  BCE) asserting, “There are infinite worlds both like and unlike this world of ours . . . We must believe that in all worlds there are living creatures and plants and other things we see in this world.” The thirteenth-century scholar and philosopher Albertus Magnus stated, “Do there exist many worlds, or is there but a single world? This is one of the most noble and exalted questions in the study of Nature.” Italy’s Giordano Bruno asserted that “There are countless suns and countless Earths all rotating around their suns in exactly the same way as the seven planets of our system . . . The countless worlds in the universe are no worse and no less inhabited than our Earth” (Bruno, 1584). By the time of Newton, entire books were being written on the topic, with Christopher Huygens asserting in his Cosmotheoros: Or, Conjectures Concerning the Planetary Worlds (Huygens, 1968) that “the Earth may justly liken’d to the planets  .  .  . [which have] gravity  .  .  . and animals not to be imagin’d too unlike ours . . . and even Men . . . [which] chiefly differ from Beasts in the study of Nature . . . [and who] have Astronomy and its subservient Arts: Geometry, Arithmetick [sic], Writing, Opticks [sic].” These thoughts, however modern sounding, remained speculative (C. Beichman, in preparation). 3

Exoplanets

HD 11742, then it might be just below the upper mass cutoff for planets. For the history and details of the radial-velocity technique, see the chapter by Lovis and Fischer. In addition to the radial-velocity planet search method, a technique very different from astrometry and radial velocity was emerging from radio observations of pulsar timing to take advantage of the ultraprecise radiation beaming from millisecond pulsars. Pulsars are neutron stars, remnants of massive stars with >8  M . Pulsar timing aims to detect changes as the host star orbits the planet-star common center of mass. An early, tentative claim of planetary companions to pulsars came in 1970 (Hills, 1970). The well-publicized announcement of a pulsar planet by Bailes et al. (1991) was retracted after the changes in pulsar timing were ascribed to Earth’s own motion about the Sun (Lyne and Bailes, 1992). Immediately following the retraction of this claim at a scientific conference, the first two bona fide exoplanets were announced to orbit PSR  1257+12 (Wolszczan and Frail, 1992). Dubbed “dead worlds” because of the deadly radiation from the host pulsar, pulsar planets are often ignored in favor of planets orbiting main-sequence stars. The chapter by Wolszczan and Kuchner describes the past, present, and future of pulsar planet searches. Returning to main-sequence stars, Walker et al. (1995) published results of their 12-year search for Jupiter-mass companions to nearby stars. In addition to the conclusion of their own study  — that none of the 21 Sun-like stars showed reflex motion corresponding to planets with masses less than 1–3 MJup (Jupiter masses) for orbital periods less than 15  years  — they used other evidence to state that no Jupiter-mass planets in short-period circular orbits had been detected around about 45 Sun-like stars. Later the same year, an exciting, incredible announcement was made of a 0.5  MJup planet in a 4.2-day period planet orbiting the Sun-like star 51 Peg (Mayor and Queloz, 1995). This first planet to orbit a Sun-like star — at about seven times closer to its star than Mercury is to the Sun — shattered the paradigm of our solar system as the model for planetary architecture. 51 Peg and about 100 other so-called hot Jupiters have changed the foundational concepts of planet formation and evolution. Giant planets were not expected to be found so excruciatingly close to the host star, and must have migrated inward after formation. This 51-Peg discovery is perhaps the most significant discovery in exoplanets because it marked the success of the method used to discover hundreds of exoplanets to date. Had jovian-mass planets orbiting within a few AU of their stars been rare, the field of exoplanets would have gone nowhere until surveys had operated long enough uncover the population of true Jupiters in 12-year-period orbits. The existence of short-period exoplanets allowed for a burst of exoplanet discoveries (following within a couple of years), forming the foundation for the now explosive field of exoplanets. At the present time, five different exoplanet detection techniques have been used to discover exoplanets (Fig.  1). One of the most surprising aspects of the hundreds of known exoplanets is their broad diversity: a seemingly continuous

range of masses and orbital parameters. Planet formation gives birth to planets of a wide range of masses in a wide variety of locations in a protoplanetary disk. Planetary migration allows planets to end up very close to the parent star. It is fair to say that the blank parts of the mass vs. semimajor axis diagram (Fig.  1) are unpopulated because no exoplanet discovery technique can yet reach low-mass planets with modest to large planet-star separations. In particular, any technique has difficulty discovering Earth analogs. Ultimately, in the future of exoplanets, we would like an image of an Earth twin as beautiful as the Apollo images of Earth (Fig. 2). For our generation, we are instead limited to imaging exoplanets as spatially unresolved, i.e., as point sources. The Voyager  1 spacecraft viewed Earth in such a way from a distance of more than four billion miles (Fig. 2). Earth’s features are hidden in a pale blue dot’s tiny speck of light. Even to obtain an image of a pale blue dot, the light from the host star, ~1010 times as bright as the planet, must be blocked out. This is one of the biggest challenges facing the current generation of astronomers. This book, Exoplanets, aims to introduce principal aspects of exoplanet observation and theory at the graduate student level. The book is organized into five sections. Part I is the introduction, and includes this chapter and a chapter on

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Semimajor Axis (AU) Fig. 1. Known exoplanets as of July 2010. Exoplanets surprisingly are found at a nearly continuous range of masses and semimajor axes. Many different techniques are successful at discovering exoplanets, as indicated by the different symbols. The solar system planets are denoted by the first one or two letters of their name. The horizontal line is the conventional upper limit to a planet mass, 13  MJup (Jupiter masses). The sloped, lower boundary to the collection of gray squares is due to a selection effect in the radial velocity technique. Data taken from http://exoplanet.eu/.

Seager and Lissauer:  Introduction to Exoplanets   

the fundamentals of Keplerian orbits and dynamics. Part II contains exoplanet observing techniques and findings to date, with one chapter per topic and a concluding chapter on the current statistical distribution of exoplanets. The two chapters in Part III discuss orbital dynamics in more detail. Part IV is about exoplanet formation and migration. Part V is about planet interiors and atmospheres. The goal for each chapter is to start with a conceptual introduction followed by a detailed explanation of the foundational equations and methodology. The chapters continue with current research highlights, and conclude with future prospects. The goal of this chapter is to give an introductory overview of the topics covered in Exoplanets. 2. WHAT IS A PLANET? Requirements for membership in the “planet club” received considerable attention in the press when the International Astronomical Union (IAU) “demoted” Pluto during the summer of 2006. But the IAU’s still-controversial 2006 definition only covers the four giant planets, the four terrestrial planets, and smaller objects in orbit about our Sun. What about the hundreds of (and rapidly increasing in number) extrasolar “planets” that are the subject of this book? As larger and more massive objects tend to be easier to detect than smaller ones, the first exoplanets to be found were large, and in many cases more massive than Jupiter. In our solar system, there is a gap of over a factor of 103 in mass and more than 108 in luminosity between the Sun and its most massive planet. Just as the discoveries of numerous additional small bodies orbiting the Sun have forced astronomers to decide how small an object can be and still be worthy of being classified as a planet, detections of substellar objects orbiting other stars have raised the question of an upper size limit to planethood. Stars can be defined as objects for which self-sustaining fusion provides sufficient energy for thermal pressure to bal-

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Fig. 2. Earth as viewed from space. (a)  Image from NASA’s Apollo 17 spacecraft in 1972. (b) Image from Voyager 1 at a distance of more than four billion miles. Earth lies in the center of a band caused by scattered sunlight in the camera optics.

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ance gravity. For solar composition, this yields a lower-mass limit of >0.075 M  ≈ 80 MJup. The smallest-known stars are a bit less than one-tenth as massive as our Sun. One could simply apply the word “planet” to all sizable substellar objects, but the term “brown dwarf” is generally used to denote bodies that are very similar to stars yet too small to reach internal temperatures necessary for thermonuclear fusion of significant quantities of ordinary hydrogen. Various delineations of the upper boundary of planethood incorporate formation and/or location requirements on the object in question in addition to limits on its physical properties. A few years before the Pluto planethood debate heated up, the IAU’s Working Group on Extrasolar Planets decided upon a provisional definition of the attributes required for an object outside of our solar system to be considered to be a planet. This working definition reads as follows: 1. Objects with true masses below the limiting mass for thermonuclear fusion of deuterium (currently calculated to be 13 MJup for objects of solar metallicity) that orbit stars or stellar remnants are “planets” (no matter how they formed). The minimum mass/size required for an extrasolar object to be considered a planet should be the same as that used in our solar system. 2. Substellar objects with true masses above the limiting mass for thermonuclear fusion of deuterium are “brown dwarfs,” no matter how they formed nor where they are located. 3. Free-floating objects in young star clusters with masses below the limiting mass for thermonuclear fusion of deuterium are not “planets,” but are “subbrown dwarfs” (or whatever name is most appropriate). Various hesitant qualifying statements surround the above listing (see http://www.dtm.ciw.edu/boss/IAU/div3/wgesp/ for details), lest anyone think that the definition wasn’t drawn up by a committee. A few comments are in order: This definition incorporates physical properties and location, but not formation; the theorists on the panel generally objected to the inclusion of mode of formation because of modeling uncertainties, whereas some of the observers argued for including formation-based criteria. 1, 2: Deuterium, which contains one proton and one neutron, is almost five orders of magnitude less abundant than ordinary hydrogen (one proton, no neutrons). Deuterium fuses at a substantially lower temperature than does ordinary hydrogen, but even the fusion of the entire inventory of an object’s deuterium does not provide a very large amount of energy because of deuterium’s low abundance. There is no fine dividing line dividing no-deuterium fusion from alldeuterium fusion, but the mass at which 50% of deuterium eventually fuses is now estimated at 12  MJup, down from 13 MJup estimated several years ago when the IAU exoplanet definition was written. 1: The comment regarding the lowermass limit shows that the panel was aware of the brewing Pluto controversy, but didn’t want to take a position on this matter. Note that the inner “planet” of PSR B1257+12 is less massive than any of the eight planets in our solar system, but more massive than Pluto, and it also has (at least mostly) cleared its orbital zone of debris.

6   

Exoplanets

3. PLANETARY DYNAMICS By analyzing Tycho Brahe’s careful observations of the orbits of the planets, Johannes Kepler deduced the following three laws of planetary motion: 1. All planets move along elliptical paths with the Sun at one focus. 2. A line connecting a planet and the Sun sweeps out equal areas DA in equal periods of time Dt



∆A = constant ∆t

(1)

Note that the value of this constant differs from one planet to the next. 3. The square of a planet’s orbital period about the Sun (in years) is equal to the cube of its semimajor axis. Although Kepler’s laws were originally found from careful observation of planetary motion, they were subsequently shown to be derivable from Newton’s laws of motion together with his universal law of gravity. Newton’s analysis corrected some small errors in Kepler’s laws, and generalized them to apply to objects not orbiting the Sun. Two mutually gravitating bodies of masses m1 and m2 travel on elliptical paths about their mutual center of mass, with an orbital period T given by

T2 =

4π2 a3 G (m1 + m 2 )

(2)

where a refers to the semimajor axis of relative orbit of the two bodies and G is the universal gravitational constant. The majority of known exoplanets have been found by tracking the small variations in their star’s velocity as they moves in response to the planet’s gravitational influence (see chapter by Lovis and Fischer). Newton’s laws imply that all massive bodies have influence on one another, so that in systems with more than two bodies, orbits are not perfect ellipses. The planet Neptune was discovered by studying deviations of the path of Uranus from an ellipse that could not be accounted for by the perturbations of the then-known planets. Analogous deviations have been used to precisely specify the masses and orbital inclinations of the two largest planets known to orbit the pulsar PSR  B1257+12 (see chapter by Wolszczan and Kuchner), as well as the two resonant giant planets orbiting the small main-sequence star GJ 876 (Correia et al., 2010). Newton demonstrated that the gravitational force exerted by a spherically symmetric body is equivalent to that of a point particle of the same mass. However, rotation and other processes can cause deviations from spherical symmetry, and these asymmetries affect orbits, especially for bodies whose separations are not substantially larger than their sizes. General relativity implies differences from elliptical trajectories that are most profound for orbits close to massive bodies. Astrophysical bodies are neither perfectly rigid nor perfectly

fluid. Tidal forces can deform a body, and since the strength of the tidal force increases with proximity, so does the amount of deformation. These variations produce flexing in a body traveling on an eccentric orbit, which can dissipate energy as heat within the body at the expense of damping the orbital eccentricity. Moreover, a tidally deformed body exerts a different gravitational force as a result of its nonspherical distribution of mass. For example, the Moon raises tides on Earth, producing a tidal bulge. This bulge points almost along the Earth-Moon line, but as Earth cannot respond instantaneously to the pull of the Moon, Earth’s rotation carries the bulge slightly ahead of the Earth-Moon line. The resulting skewed mass distribution within Earth exerts a torque on the Moon, causing its orbit to expand (at the rate of a few centimeters per year) and Earth’s rotation to slow. The basic dynamics of the two-body problem are described in the chapter by Murray and Correia. The chapter by Fabrycky describes planetary perturbations, the principal effects of stellar oblateness, general relativity, and tidal forces as they apply to exoplanet research. The chapter by Correia goes into more detail on tidal evolution. 4. OBSERVATIONAL TECHNIQUES Exoplanets are being discovered with several different detection techniques (Fig.  1). Each technique has its own selection effects in mass (or radius) and semimajor axis parameter space. Here we introduce and compare the different planet-finding techniques. Radial velocity: In the presence of a planet, a star orbits the planet-star common center of mass. The star’s motion can be described by three components: one along the observer’s line-of-sight to the star, and two components on the plane of the sky. The radial-velocity technique measures the star’s lineof-sight motion with a selection effect toward more-massive planets close to the star. Radial velocity is the most mature of the planet-finding techniques, giving it the advantage of being able to push to relatively low planetary masses around bright stars. The major disadvantage of the radial-velocity technique is that only an exoplanet’s minimum mass (given by Mp sin i, where Mp is the planet mass and i is the orbital inclination) is measured because the orbital inclination of the planet orbit is undetermined. The radial-velocity technique is described in the chapter by Lovis and Fischer. Astrometry: Astrometry measures the position of stars by the two components of motion in the plane of the sky. The main advantage of astrometric measurements is that they can provide the planetary mass (without the sin i factor inherent in radial-velocity searches) and all planetary orbital parameters. Although the astrometry technique has a long history, with the first claimed exoplanet detections in the mid-nineteenth century, astrometry has yet to discover an exoplanet (as of 2010). For details, see the chapter by Quirrenbach. Timing: Timing techniques for exoplanet detection include both pulsar timing (Wolszczan and Frail, 1992) and time perturbations of stars with stable oscillation periods (e.g., Silvotti et al., 2007) (see the chapter by Wolszczan and

Seager and Lissauer:  Introduction to Exoplanets   

Kuchner for details). Because the planet detection techniques radial velocity, astrometry, and timing infer the presence of the planet from the star’s orbital motion, only the planet’s mass and orbital elements can be derived. Gravitational microlensing: Gravitational microlensing oc‑ curs when a foreground star happens to pass very close to the observer’s line of sight to a more distant background star. The foreground star acts as a lens, magnifying the background source star, as a function of time, by an amount that depends on the angular separation between the lens star and source star. If a planet is orbiting the lens star, the lightcurve may be further perturbed, resulting in a characteristic, short-lived signature of an exoplanet and yielding the planet mass and planet-star physical separation. The major weakness of the microlensing planet-finding technique is that the lensing occurence is a one-time event that will never repeat. The advantages of the current state of microlensing over other planet-finding techniques (except for pulsar timing) is the sensitivity to low planetary masses (potentially to Earth’s mass) at relatively large planet-star separation, aiding in understanding the statistical distribution of exoplanets (see Fig.  1). The theoretical and observational foundation for microlensing is laid out in the chapter by Gaudi. We now turn to the two planet-finding techniques that can also be used to observe physical characteristics beyond planetary mass. Transits: Exoplanet transits (or primary eclipses) occur when a planet passes in front of its star as seen from Earth. When the planet goes behind the star, the event is called an occultation or secondary eclipse. (The term “transit” is used for small bodies going in front of larger bodies, “occultation” for larger bodies going in front of smaller bodies, and “eclipse” for bodies of about the same size moving in front of each other.) Most transiting planets have been discovered from transit surveys searching large numbers of stars for the characteristic drop in brightness that is indicative of a planet transit. Transiting exoplanets are also detected via follow-up photometric observations to known radial-velocity planets and these tend to be orbiting bright stars, favorable for additional observations. Transits are described in the chapter by Winn. For an exoplanet to show transits, the exoplanet-star orbit must be aligned with the observer. For a random orientation of stellar inclinations (for zero eccentricity and Rp  1) planetary caustic is approximately ∝ q1/2d−2, and the lengthto-width ratio is ~1  +  d−2. The shape is independent of q in this approximation. The overall size of the close (d < 1) planetary caustics are approximately ∝  q1/2d3, and their shape is also independent of q. The vertical separation between the two planetary caustics in the close topology is ∝ q1/2d−1. For the resonant case (dc ≤ d ≤ dw) there exists a single, relatively large caustic with six cusps. For fixed q, the resonant caustic is larger than either the central or planetary caustics. The large size of these caustics results in a large cross-section and thus an enhanced detection probability. Indeed, in the first planet detected by microlensing, the source crossed a resonant caustic. The large size also means that the lightcurve deviation can last a significant fraction of the duration of the event. However, resonant caustics are also “weak” in the sense that for a large fraction of the area interior to or immediately outside the caustic, the excess magnification relative to a single lens is small (see Fig. 7). The exceptions to this are source positions in the vicinity of the cusp located on the planet/star axis pointing toward

Gaudi:  Microlensing by Exoplanets   

the planet, and source positions near the “back end” of the caustic near the position of the primary, which are characterized by large negative deviations relative to the single-lens magnification. The precise shape of the resonant caustic depends sensitively on d, and thus small changes in the value of d lead to large changes in the caustic morphology, as can be seen in Fig. 5. As a result, the effects of orbital motion, which result in a change in d over the course of the event, are expected to be more important for resonant caustic perturbations. The size of resonant caustics scales as q1/3, in contrast to planetary caustics, which scale as q1/2, and central caustics, which scale as q.

3. PRACTICE OF MICROLENSING 3.1. Lightcurves and Fitting The apparent relative motion between the lens and the source gives rise to a time-variable magnification of the source: a microlensing event. Is is often (but not always; see section 3.2) a good approximation that the source, lens, and observer are in uniform, rectilinear motion, in which case the angular separation between the lens and source as a function of time can be written as

(a)



10

1

–2

–1

0

1

2

80 60

(b)

d = 1.25

(

u (t) = τ 2 + u20

)

12



(28)

where τ  ≡ (t − t 0) / tE, t0 is the time of closest alignment, which is also the time of maximum magnification, u0 is the impact parameter of the event, and tE is the Einstein ring crossing time

Central Caustic Perturbations

100

91

40

tE ≡

θE µ rel

(29)

where μrel is the relative lens-source proper motion. Figure 3 shows the magnification as a function of time for a microlensing event due to a single lens, with impact parameters of u0 = 0.01, 0.1, 0.2, . . . , 0.9, 1.0, which serve to illustrate the variety of lightcurve shapes. Of course, what is observed is not the magnification, but the flux of a photometered source as a function of time, which is given by

F (t) = Fs A (t) + Fb

(30)

Here Fs is the flux of the microlensing star, and Fb is the flux of any unresolved light (or “blended light”) that is not

20 80 60

(c)

d = 1.25

40

20 80 60

d = 0.80

(d)

d = 0.80

(e)

40

20

Magnification

80 60 40

20 –0.04

–0.02

0

(t–t0)/tE

0.02

0.04

Fig. 8. Example lightcurves of planetary perturbations arising from the source passing close to the central caustic in a highmagnification event, for a planet/star mass ratio of q  = 0.001. Panel (a) shows the overall lightcurve. The impact parameter of the event with respect to the primary lens is u0 = 0.02, corresponding to a peak magnification of Amax ~u −1 0  = 50. Panels (b)–(e) show zooms of the lightcurve peak. Two different cases are shown, one case of the wide planetary companion with d = 1.25 [(b),(c)], and a close planetary companion with d = 0.8 [(d),(e)]. These two cases satisfy d ↔ d−1 and demonstrate the close/wide degeneracy. The source passes close to the central caustic; two example trajectories are shown in Fig.  6 and the resulting lightcurves including the planetary perturbations are shown in panels  (b)–(e). The dotted line shows the magnification with no planet, whereas the solid lines show the planetary perturbations with source sizes of ρ* = 0, 0.003, and 0.01 (lightest to darkest). In panel  (e), the lightcurve for ρ* = 0.03 is also shown. In this case, the primary lens transits the source, resulting in a “smoothed” peak. Although the planetary deviation is largely washed out, it is still detectable with sufficiently precise photometry.

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Exoplanets

being lensed. The latter can include light from a companion to the source, light from unrelated nearby stars, light from a companion to the lens, and (most interestingly) light from the lens itself. Microlensing experiments are typically carried out toward crowded fields in order to maximize the event rate, and therefore one often finds unrelated stars blended with the microlensed source for typical groundbased resolutions of ~1′′. Even in the most crowded bulge fields, most unrelated background stars are resolved at the resolution of the Hubble Space Telescope (HST). Figure 10 shows the fields of two events as observed from the ground with typical seeing, with HST, and with groundbased adaptive optics (AO).

Planetary Caustic Perturbations 6

(a)

d = 1.25 4

2

1

–2

–1

0

1

2

6

(b) 4

2

–0.45

–0.4

–0.35

–0.3

–0.25

–0.2

6

(c)

d = 0.80 4

2

1

–2

–1

0

1

2

6

Magnification

(d) 4

2

0.2

0.25

0.3

(t–t0)/tE

0.35

The observed flux as a function of time for a microlensing event due to a single lens can be fit by five parameters: t0, u0, tE, Fs, and Fb. It is important to note that several of these parameters tend to be highly degenerate. There are only four gross observable properties of a single-lens curve: t0, the overall timescale of the event (i.e., tFWHM), and the peak and baseline fluxes. Thus u0, tE, Fs, and Fb tend to be highly correlated, and are only differentiated by relatively subtle differences between lightcurves with the same values of the gross observables but different values of u0, tE, Fs, and Fb (Wozniak and Paczynski, 1997; Han, 1999; Dominik, 2009). As a result of these degeneracies, when fitting to data it is often useful to employ an alternate parameterization of the single-lens model that is more directly tied to these gross observables, in order to avoid strong covariances between the model parameters. In practice, several different observatories using several different filter bandpasses typically contribute data to any given observed microlensing event. Since the flux of the source and blend will vary depending the specific bandpass, and furthermore different observatories may have different resolutions and thus different amounts of blended light, one must allow for a different source and blend flux for each filter/observatory combination. Thus the total number of parameters for a generic model fit to an observed dataset is Nnl  +2  × NO, where Nnl is the number of (nonlinear) parameters required to specify the magnification as a function of time, and NO is the total number of independent datasets. Since the observed flux is a linear function of Fs and Fb, for a given set of Nnl parameters that specify A(t), the set of source and blend fluxes can be found trivially using a linear least-squares fit. Thus in searching for the best-fit model, one typically uses a hybrid method in which the best-fit nonlinear parameters (e.g., t0, u0, tE for a single lens) are varied using, e.g., a downhill-simplex, Markov Chain Monte Carlo (MCMC), or grid-search method, and the specific best-fit Fs and Fb values for each trial set of nonlinear parameters are determined via linear least squares. The addition of a second lens component increases the number of model parameters by (at least) three, and enormously increases the complexity. As discussed in section  2.3, the magnification pattern (magnification as a

0.4

0.45

Fig. 9. Example lightcurves of planetary perturbations arising from the source passing close to the planetary caustic for a planet/star mass ratio of q = 0.001. Panels (a) and (c) show the overall lightcurves, whereas panels (b) and (d) show zooms of the planetary deviation. Two cases are shown, one case of the wide planetary companion with d = 1.25 [(a),(b)], and a close planetary companion with d  = 0.8 [(c),(d)]. In both cases, the impact parameter of the event with respect to the primary lens is u0 = 0.3. The trajectories for the lightcurves displayed are shown in Fig. 5. The dotted line shows the magnification with no planet, whereas the solid lines show the planetary perturbations with source sizes of ρ* = 0, 0.003, 0.01, and 0.03 (lightest to darkest).

Gaudi:  Microlensing by Exoplanets   

function of source position) for a binary lens is described two parameters: d, the separation of lens components in units of the Einstein ring of the lens, and q, the mass ratio of the lens. Four additional parameters specify the trajectory of the source through the magnification patterns as a function of time: t0, u0, tE, and a. The first three are analogous to the single lens case: u0 is the impact parameter of the trajectory from the origin of the lens in units of θE, t0 is the time when u = u0, and tE is the Einstein ring crossing time. Finally, a specifies the angle of the trajectory relative to the binary-lens axis. Thus 6 + 2 × NO parameters are required to specific the lightcurve arising from a generic, static binary lens. There are many possibles choices for the origin of the lens, as well as the mass used for the normalization of θE. The optimal choice depends on the particular properties of the lens being considered, i.e., wide stellar binary, close stellar binary, or planetary system (Dominik, 1999b). For planetary systems, one typically chooses the location of the primary for the origin and normalizes θE the total mass of the system. Lightcurves arising from binary lenses exhibit an astonishingly diverse and complex phenomenology. While this

(a)

1˝

(b)

1˝

(c)

1˝

(d)

1˝

Fig. 10. (a) A 15" × 15" I-band image of the field of planetary microlensing event OGLE-2005-BLG-071 obtained with the OGLE 1.3-m Warsaw telescope at Las Campanas Observatory in Chile. (b)  Same field as the top panel, but taken with the Advanced Camera for Surveys instrument on the Hubble Space Telescope in the F814W filter. (c)  22"  × 22" H-band image of the field of planetary microlensing event MOA-2008-BLG-310 obtained with the CTIO/SMARTS2 1.3-m telescope at Cerro Tololo InterAmerican Observatory in Chile. (d) Same field and filter as the panel above, but taken with the NACO instrument on VLT. In all panels, the arrow indicates the microlensing target.

93

diversity makes for a rich field of study, it complicates the interpretation of observed lightcurves mightily, for several reasons. First, other than a few important exceptions (i.e., for planetary caustic perturbations; see section  2.1), the salient features of binary and planetary lightcurves have no direct relationship to the canonical parameters of the underlying model. Thus it is often difficult to choose initial guesses for the fit parameters, and even if a trial solution is found, it is difficult to be sure that all possible minima have been located. Second, small changes in the values of the canonical parameters can lead to dramatic changes in the resulting lightcurve. In particular, the sharp changes in the magnification that occur when the source passes close to or crosses a caustic can make any goodness-of-fit statistic such as χ2 very sensitive to small changes in the underlying parameters. This, combined with the shear size of parameter space, makes brute-force searches difficult and time-consuming. The complicated and highly corrugated shape of the χ2 surface also causes many of the usual minimization routines (i.e., downhill simplex) to fail to find the global or even local minimum. These difficulties are compounded by the fact that the magnification of a binary lens is nonanalytic and time-consuming to calculate when finite source effects are important. Aside from their diverse phenomenology, binary lens lightcurves also have the important property that they can be highly degenerate, in the sense that two very different underlying lens models can produce very similar lightcurves. These degeneracies can be accidental, in the sense that with relatively poor quality data or incomplete coverage of the diagnostic lightcurve features, otherwise distinguishable lightcurves can provide equally good statistical fits to a given dataset (Dominik and Hirshfeld, 1996; Dominik, 1999a; Albrow et al., 1999b). This is particularly problematic for caustic-crossing binary-lens lightcurves in which only one (i.e., the second) caustic crossing is observed. In this instance, it is typically the case that very different models can fit a given dataset. [An important corollary is that, given an observed first caustic crossing, it is very difficult to predict the time of the second caustic crossing well in advance. For further discussion, see Albrow et al. (1999b) and Jaroszyn´ski and Mao (2001).] The more insidious degeneracies, however, are those that arise from mathematical symmetries in the lens equation itself (Dominik, 1999b). For example, in the limit of a very widely separated (dw >> 1) or very close (dc  1 is ~(θE,p/θE)d−1  = q1/2d−1 (Di Stefano and Scalzo, 1999b). Indeed, since microlensing is directly sensitive to the planet mass, planets can be detected even without a primary microlensing event (Di Stefano and Scalzo, 1999a). Even free-floating planets that are not bound to any host star are detectable in this way (Han et al., 2005). Microlensing is the only method that can detect old, free-floating planets. A significant population of free-floating planets planets is a generic prediction of most planet formation models, particular those that invoke strong dynamical interactions to explain the observed eccentricity distribution of planets (Goldreich et al., 2004; Juric and Tremaine, 2008; Ford and Rasio, 2008).

to planets orbiting stars all along the main sequence, from brown dwarfs to the main-sequence turn-off, as well as planets orbiting white dwarfs, neutron stars, and black holes. 4.5. Sensitivity to Planets Throughout the Galaxy Because microlensing does not rely on light from the planet or host star, planets can be detecting orbiting stars with distances of several kiloparsecs. The microlensing event rate depends on the intrinsic lensing probability, which peaks for lens distances about halfway to the typical sources in the galactic bulge, and the number density distribution of lenses along the line-of-sight toward the bulge, which peaks at the bulge. The event rate remains substantial for lens distances in the range Dl ~ 1−8 kpc. Roughly 40% and 60% of microlensing events toward the bulge are expected to be due to lenses in the disk and bulge, respectively (Kiraga and Paczynski, 1994). Specialized surveys may be sensitive to planets with Dl < 1 kpc (Di Stefano, 2008; Gaudi et al., 2008b), as well as planets in M31 (Covone et al., 2000; Chung et al., 2006; Ingrosso et al., 2009). 4.6. Sensitivity to Multiple-Planet Systems For low-magnification events, multiple planets in the same system can be detected only if the source crosses the

200

Total 150

dΓ/dlog(M/M )

100   

100

NS 50

M

D S+ B

4.4. Sensitivity to Planets Orbiting a Wide Range of Host Stars The hosts probed by microlensing are simply representative of the population of massive objects along the line of sight to the bulge sources, weighted by the lensing probability. Figure 11 shows a model by Gould (2000) for the microlensing event rate toward the galactic bulge as a function of the host star mass for all lenses, and also broken down by the type of host, i.e., star, brown dwarf, or remnant. The sensitivity of microlensing is weakly dependent on the host star mass, and has essentially no dependence on the host star luminosity. Thus microlensing is about equally sensitive

WD 0

0.01

0.1

BH 1

10

log(M/M )

Fig. 11. The rate of microlensing events toward the galactic bulge as a function of the mass of the lens, for main-sequence (MS) stars and brown dwarfs (BD) (0.03 M   < M < 1 M  ) (bold dashed curve) and white dwarfs (WD), neutron stars (NS), and black hole (BH) remnants (solid curves). The total is shown by a bold solid curve. From Gould (2000).

Gaudi:  Microlensing by Exoplanets   

planetary caustics of both planets, or equivalently only if both planets happen to have projected positions sufficiently close to the paths of the two images created by the primary lens. The probability of this is simply the product of the individual probabilities, which is typically O (1%) or less (Han and Park, 2002). In high-magnification events, however, individual planets are detected with near-unity probability regardless of the orientation of the planet with respect to the source trajectory (Griest and Safizadeh, 1998). This immediately implies all planets sufficiently close to the Einstein ring radius will be revealed in such events (Gaudi et al., 1998). This, along with the fact that high-magnification events are potentially sensitive to very low-mass planets, makes such events excellent probes of planetary systems. 4.7. Sensitivity to Moons of Exoplanets Because microlensing is potentially sensitive to planets with mass as low as that of a few times the mass of the Moon (Bennett and Rhie, 1996), it is potentially sensitive to large moons of exoplanets. A system with a star of mass M, planet of mass mp, and moon of mass mm corresponds to a triple lens with the hierarchy mm  0.1 M and separations a  > 0.5 AU, including free-floating planets (Bennett and Rhie, 2002; Bennett et al., 2009). This range includes analogs to all the solar system planets except Mercury. If every main-sequence star has an Earth-mass planet in the range 1−2.5 AU, the survey would detect ~500 such planets within its mission lifetime. The survey would also detect a number of habitable Earth-mass planets comparable to that of the Kepler mission (Borucki et al., 2003), and so would provide an important independent measurement of η . When combined with complementary surveys (such as Kepler), a spacebased microlensing planet survey would

Saturn

Detailed simulations of such a next-generation microlensing survey have been performed by several groups (Gaudi et al., unpublished; Bennett, 2004). These simulations include models for the galactic population of lenses and sources that match all constraints (Han and Gould, 1995, 2003), and account for real-world effects such as weather, variable seeing, moon and sky background, and crowded fields. They reach similar conclusions. Such a survey would increase the planet detection rate at fixed mass by at least an order of magnitude over current surveys. Figure 13 shows the predictions of these simulations for the detection rate of planets of various masses and separations using a survey including MOA-II, OGLEIV, and a Korean telescope in South Africa. In particular, if Earth-mass planets with semimajor axes of several AU are common around main-sequence stars, a next-generation microlensing survey should detect several such planets per year. This survey would also be sensitive to free-floating planets, and would detect them at a rate of hundreds per year if every star has an ejected Jupiter-mass planet. Ultimately, however, the true potential of microlensing cannot be realized from the ground. Weather, seeing, crowded fields, and systematic errors all conspire to make the detection of planets with mass less than Earth effectively impossible from the ground (Bennett, 2004). As outlined in

105

1

10

Semimajor axis (AU)

1

1

10

100

1000

Mass (M )

Fig. 13. Expectations from the next generation of groundbased microlensing surveys, including MOA-II, OGLE-IV, and a KMTNet telescope in South Africa. These results represent the average of two independent simulations, which include very different input assumptions but differ in their predictions by only ~0.3 dex. (a) Number of planets detected per year as a function of semimajor axis for various masses, assuming every star has a planet of the given mass and semimajor axis. (b) Number of planets detected per year as a function of planet mass, normalized by the number of ~10 M found to date by microlensing (indicated by the large dot). The solid line is the prediction assuming an equal number of planets per logarithmic mass interval, and the dotted curve assumes that the number of planets per log mass scales as mp− 0.7 (Sumi et al., 2010). In both cases, planets are assumed to be distributed uniformly in log(a) between 0.4 and 20 AU. Arrows indicate the locations of the 10 published exoplanets.

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determine the demographics of both bound and free-floating planets with masses greater than that of Mars orbiting stars with masses less than that of the Sun. The basic requirements for a spacebased microlensing planet survey are relatively modest: an aperture of at least 1 m, a large FOV camera (at least ~0.5 deg2) with optical or near-IR detectors, reasonable image quality of better than ~0.25", and an orbit for which the galactic bulge is continuously visible. The Microlensing Planet Finder (MPF) is an example of a spacebased microlensing survey that can accomplish these objectives, essentially entirely with proven technology, and at a cost of ~$300 million excluding the launch vehicle (Bennett et al., 2009). Interestingly, the requirements for a spacebased microlensing planet survey are very similar to, or less stringent than, the requirements for a number of the proposed dark-energy missions, in particular those that focus on weak lensing measurements. Thus, it may be attractive to consider a combined darkenergy/planet-finding mission that could be accomplished at a substantial savings compared to doing each mission separately (Gould, 2009). Acknowledgments. I would like to thank S. Dong for preparing Fig.  10, A. Udalski for reading over the manuscript, and two anonymous referees for catching some important errors and providing comments that greatly improved the chapter.

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Sumi T., Bennett D. P., Bond I. A., Udalski A., Batista V., et al. (2010) A cold Neptune-mass planet OGLE-2007-BLG-368Lb: Cold Neptunes are common. Astrophys. J., 710, 1641–1653. Thomas C. L., Griest K., Popowski P., Cook K. H., Drake A. J., et al. (2005) Galactic bulge microlensing events from the MACHO collaboration. Astrophys. J., 631, 906–934. Tsapras Y., Street R., Horne K., Snodgrass C., Dominik M., et al. (2009) RoboNet-II: Follow-up observations of microlensing events with a robotic network of telescopes. Astron. Nachr., 330, 4–11. Udalski A. (2003) The optical gravitational lensing experiment. Real time data analysis systems in the OGLE-III survey. Acta Astron., 53, 291–305. Udalski A., Szymanski M., Kaluzny J., Kubiak M., Krzeminski W., Mateo M., Preston G. W., and Paczynski B. (1993) The optical gravitational lensing experiment. Discovery of the first candidate microlensing event in the direction of the galactic sulge. Acta Astron., 43, 289–294. Udalski A., Szymanski M., Kaluzny J., Kubiak M., Mateo M., Krzeminski W., and Paczynski B. (1994) The optical gravitational lensing experiment. The early warning system: Real time microlensing. Acta Astron., 44, 227–234. Udalski A., Zebrun K., Szymanski M., Kubiak M., Pietrzynski G., Soszynski I., and Wozniak P. (2000) The optical gravitational lensing experiment. Catalog of microlensing events in the galactic bulge. Acta Astron., 50, 1–65. Udalski A., Jaroszynski M., Paczynski B., Kubiak M., Szymanski M. K., et al. (2005) A jovian-mass planet in microlensing event OGLE-2005-BLG-071. Astrophys. J. Lett., 628, L109–L112. Uglesich R. R., Crotts A. P. S., Baltz E. A., de Jong J., Boyle R. P., and Corbally C.  J. (2004) Evidence of halo microlensing in M31. Astrophys. J., 612, 877–893.

van Belle G. T. (1999) Predicting stellar angular sizes. Publ. Astron. Soc. Pacific, 111, 1515–1523. Vermaak P. (2000) The effects of resolved sources and blending on the detection of planets via gravitational microlensing. Mon. Not. R. Astron. Soc., 319, 1011–1019. Vermaak P. (2003) Rapid analysis of binary lens gravitational microlensing light curves. Mon. Not. R. Astron. Soc., 344, 651–656. Wambsganss J. (1997) Discovering galactic planets by gravitational microlensing: Magnification patterns and light curves. Mon. Not. R. Astron. Soc., 284, 172–188. Witt H. J. (1990) Investigaion of high amplification events in light curves of gravitationally lensed quasars. Astron. Astrophys., 236, 311–322. Witt H. J. (1995) The effect of the stellar size on microlensing at the Baade window. Astrophys. J., 449, 42–46. Witt H. J. and Mao S. (1994) Can lensed stars be regarded as pointlike for microlensing by MACHOs? Astrophys. J., 430, 505–510. Witt H. J. and Mao S. (1995) On the minimum magnification between caustic crossings for microlensing by binary and multiple stars. Astrophys. J. Lett., 447, L105–L108. Wozniak P. and Paczynski B. (1997) Microlensing of blended stellar images. Astrophys. J., 487, 55–60. Yoo J., DePoy D. L., Gal-Yam A., Gaudi B. S., Gould A., et al. (2004) OGLE-2003-BLG-262: Finite-source effects from a point-mass lens. Astrophys. J., 603, 139–151. Zakharov A. F. (1995) On the magnification of gravitational lens images near cusps. Astron. Astrophys., 293, 1–4. Zakharov A. F. (1999) On the some properties of gravitational lens equation near cusps. Astron. Astrophys., 18, 17–25.

Direct Imaging of Exoplanets Wesley A. Traub Jet Propulsion Laboratory, California Institute of Technology

Ben R. Oppenheimer American Museum of Natural History

A direct image of an exoplanet system is a snapshot of the planets and disk around a central  star. We can estimate the orbit of a planet from a time series of images, and we can estimate the  size, temperature, clouds, atmospheric gases, surface properties, rotation rate, and likelihood of  life on a planet from its photometry, colors, and spectra in the visible and infrared. The exoplanets  around stars in the solar neighborhood are expected to be bright enough for us to characterize them  with direct imaging; however, they are much fainter than their parent star, and separated by very  small  angles,  so  conventional  imaging  techniques  are  totally  inadequate,  and  new  methods  are  needed. A direct-imaging instrument for exoplanets must (1) suppress the bright star’s image and  diffraction pattern, and (2) suppress the star’s scattered light from imperfections in the telescope.  This chapter shows how exoplanets can be imaged by controlling diffraction with a coronagraph  or interferometer, and controlling scattered light with deformable mirrors.

1. INTRODUCTION The  fi rst  direct  images  of  exoplanets  were  published  in  2008, fully 12 years after exoplanets were discovered, and  after more than 300 of them had been measured indirectly  by radial velocity, transit, and microlensing techniques. This  huge time lag occurred because direct imaging of exoplanets  requires extraordinary efforts in order to overcome the barriers imposed by astrophysics (planet-star contrast), physics  (diffraction), and engineering (scattering).  The structure of this chapter is as follows. Section 1 discusses the scientifi c purpose of direct imaging of exoplanets,  and includes a glossary of terms. Section 2 discusses basic  physical concepts, including brightness, contrast, wavefronts,  diffraction,  and  photons.  Section  3  discusses  coronagraph  and  interferometer  concepts.  Section  4  addresses  speckles  and adaptive optics. Section 5 sketches recent results from  exoplanet imaging and lists current projects. Section 6 outlines future prospects for exoplanet imaging on the ground  and in space. 

fainter than its star, and separated by 12.7 arcsec (98 AU at  7.7 pc distance). It was detected at two epochs, clearly showing  common motion as well as orbital motion (see inset).  Figure 2 shows a near-infrared composite image of exoplanets HR 8799 b,c,d by Marois et al. (2008). The planets  are  at  angular  separations  of  1.7,  1.0,  and  0.6  arcsec  from  the star (68, 38, and 24 AU at 40 pc distance). The H-band  planet-star  contrasts  (ratio  of  planet  to  star  fl ux)  are  about  10–5, i.e., roughly 12 mag fainter. The planets would be much 

1.1. Exoplanet Images We illustrate with three examples of direct imaging; as it  happens, all three examples are of young, self-luminous objects. Figure 1 shows the dust ring and exoplanet Fomalhaut b  by  Kalas et al. (2008),  in  the  visible. The  central  star  was  suppressed using a combination of methods described in this  chapter:  the rectangular-mask coronagraph (section 3.13) and  angular differential imaging (section 4.12). Kalas et al. used  the Hubble Space Telescope (HST) Advanced Camera for Surveys (ACS) in coronagraph mode. The planet is about 23 mag 

Fig. 1. Visible-wavelength image, from the Hubble Space Telescope,  of  the  exoplanet  Fomalhaut  b.  The  planet  is  located  just  inside a large dust ring that surrounds the central star. Fomalhaut  has been blocked and subtracted to the maximum degree possible. 

111

112 

    Exoplanets

Fig. 2. Near-infrared  image,  from  the  Keck  telescope,  of  the  exoplanets  HR  8799b,  c,  and  d.  The  central  star  has  been  suppressed with angular differential imaging, coupled with adaptive  optics. The splatter of dots in the center of this image is simply  the small amount of leftover light from the central star that could  not be subtracted by ADI, so it is an artifact. 

fainter were it not for their youth and consequent internal heat  sources, putting their effective temperatures in the 1000 K regime (section 1.3). Marois et al. used the groundbased Gemini  and Keck telescopes for these observations. Their techniques  included minimizing diffraction using ADI, and minimizing  atmospheric speckles using adaptive optics (section 4). If this  system were instead the Sun and solar system, then Jupiter  would be buried in the inner one-fourth of the speckle fi eld,  Earth  would  be  in  the  inner  one-twentieth  radius,  and  both  would be 4 to 5 orders of magnitude fainter than the speckles. Figure 3 shows a near- and mid-infrared composite image  of the b Pictoris system, from Lagrange et al. (2010). The star  itself has been subtracted using a reference star image, and  independently with ADI. The composite shows the edge-on  dust disk plus the planet b Pic b at two epochs, 2003 (left  and above) and 2009 (right and below). The proper motion of  b Pic is north, not northwest, so the planet is co-moving, and  not a background object. The planet age is ~12 m.y., much  younger than the HR 8799 or Fomalhaut planets. The mass is  ~9 MJup, and the semimajor axis in the 8–15 AU range, with  a period as short as 17 years. The star is at 19 pc, and the  star-planet separations shown are in the 0.3–0.5 arcsec range. These images set the stage for our goal in this chapter, direct  imaging  of  planets  from  Earth-  to  Jupiter-sized  around  nearby stars. 1.2. Exoplanet Spectra The spectrum of an exoplanet tells us about its composition, clouds, thermal structure, and variability, as discussed in  the chapter by Burrows and Orton  for giant planets, and the  chapter by Meadows and Seager for terrestrial planets. A direct  image of an exoplanet permits us to obtain a spectrum, using  a conventional spectrometer or an integral fi eld spectrometer.  The resolution will be low, because the planet is faint; however,  since  many  molecular  bands  are  intrinsically  low-resolution  features, we can still learn much about an atmosphere. 

Fig. 3. Near- and mid-infrared composite image of b Pic b and the  b Pic dust disk, from the ESO 3.6-m and Very Large Telescopes,  with the star subtracted. The planet is shown at two epochs, 2003  and 2009, demonstrating co-moving position with the star as well  as orbital motion.

1.3. Hot, Young, and Mature Exoplanets Hot Jupiters have not yet been directly imaged, but their  large thermal fl ux, 10 –3 to 10 –4 times the parent star, means  that  they  will  likely  be  imaged  in  the  future. Their  extreme  closeness to the parent star requires extreme angular resolution,  so the images will come from long-baseline interferometers,  not from single-dish telescopes.  Young, self-luminous planets were the fi rst to be directly  imaged,  because  their  high  temperature  and  large  size  give  them a strong, detectable fl ux, and their large distances from  their  parent  stars  makes  them  easier  to  see  in  the  halo  of  atmospherically or instrumentally scattered star light. These  young, self-luminous planets are likely to continue to be prime  targets for detection in the near future, owing to this combination of favorable parameters. However, young planets cool  off in a few tens of millions of years, so they will be found  only around young stars, and not around nearby (older) stars.  A mature exoplanet may be defi ned here as one with an effective temperature that is roughly comparable to its star-planet  equilibrium temperature (section 2.5). These planets, like those  in  the  solar  system  or  around  mature  nearby  stars,  will  be  fainter in the infrared than young, self-luminous planets, and  therefore will require more sophisticated techniques to image  them.  In  addition,  most  of  them  probably  will  be  closer  to  their stars than the ones in HR 8799 and Fomalhaut, and will  therefore potentially be detectable by single-dish telescopes,  but will require the full power of the techniques in this chapter. 

Traub and Oppenheimer:  Direct Imaging of Exoplanets     

1.4. Radial Velocity, Transits, Lensing, and Astrometry Currently, the techniques of radial velocity (RV), transits, and gravitational lensing are more productive than direct imaging. Remarkably, these techniques, including astrometry and direct imaging, perform nearly independent roles for exoplanet science, so each of them is valuable. Radial velocity has been very successful in measuring masses and periods of planets with masses greater than several Earths and in short-period orbits. Transits have been valuable in measuring the diameters and periods of giant planets, and in combined-light mode have measured temperature distributions, spectral features, and thermal inversions in two gas giant planets. Transits will also be valuable for determining mass and orbit statistics of distant planets, but its geometric bias precludes using it for the vast majority of nearby systems. Ultimately, exoplanet science will require direct images and spectra of exoplanet systems. For this information, planets around nearby stars will be essential, because these systems will have larger apparent sizes and photon fluxes than more distant systems, and will therefore be relatively accessible to the techniques in this chapter. A combination of astrometry and imaging will provide the mass, period, orbit, and spectroscopic characterization for these planets, down to and including Earth-mass ones. 1.5. Solar System and Exoplanet Systems There is a strong connection between solar system and exoplanet science. Until exoplanets were discovered in the early 1990s, it was widely thought that exoplanet systems would resemble our solar system. But with the discovery of hot Jupiters, it is now clear that our system is but one of many possible types. There are several points of comparison. Planetary migration and chaotic episodes are now thought to be common to all systems. Self-luminous planets are also common. Dust and debris structures are common as well. Our picture of the evolution of the solar system is strongly influenced by what we are learning about exoplanet systems. In particular, our picture of the evolution of Earth, and of life itself, may well depend on what we learn about habitable-zone terrestrial exoplanets. And for these planets especially, because of their faintness and small angular separations, the techniques in this chapter will be very important. 1.6. Glossary Some of the terms used in this chapter are briefly defined here for reference: Visible: wavelength range ~0.3–1.0 µm Near-infrared: ~1.0–2.5 µm Mid-infrared: ~2.5–10 µm Far-infrared: ~10–200 µm Photometry: broadband (~20%) flux measurement Color: ratio of two broadband fluxes Spectrum: narrowband (> Tequil

113

Mature planet: Teff ≈ Tequil Terrestrial planet: 0.5 M  < Mp < 10 M Gas giant planet: 10 M  < Mp < 13 MJup Habitable zone (HZ): liquid water possible on surface, 0.7 AU < a/L1/2 s   0) is transmitted in state ZB, but nulled in state B, so chopping between these gives the zodi, times the transmission pattern, as

+ (89)

+ +



I 2 (SZB) = cos2 (πθBcos (α) λ) cos2 (πθb sin (α) λ) (90)



I 2 (ZB) = sin 2 (πθBcos (α) λ) cos2 (πθb sin (α) λ) (91) I 2 (B) = cos2 (πθBcos (α) λ) sin 2 (πθb sin (α) λ)

(92)

A sketch of the resulting fringes, i.e., the transmission pattern projected on the sky, is shown in Fig. 18. We see that the star (at q = 0) is transmitted in state SZB, but nulled in states ZB and B, so chopping between these gives the star flux as

Null Measurement

(b)

~5 Hz rate

(c)

Interferometric



 πθBcos (α)  2 πθb sin (α) I (zodi) sin 2  cos    λ λ    

The beam pattern is known, and for a zodi disk that extends over several periods of the pattern the effective transmission is about a factor of 1/2. Therefore the contrast, zodi/star, is given by twice the ratio of equations (94) to (93). References include Colavita et al. (2008) and Barry et al. (2008). 3.22. Nullers to Measure Exoplanets The thermal infrared space missions TPF-I and Darwin, in the U.S. and Europe, are designed to detect and characterize exoplanets, down to and including Earth-like ones, around nearby stars. Here we describe the preferred method of operation that these projects have in common. The TPF-I/Darwin schematic is identical to the KIN schematic. However, in TPF-I/Darwin the four collecting mirrors are mounted on four free-flying separated spacecraft, and the combining unit is another spacecraft at (x,y) = (0,0) but above the plane at z >> B. The key fundamental difference is in the phases added to each beam, which are specified in Table 8. Inserting these phases into equation  (89) and working through as before, we find the following detected intensities, again dropping a factor of 16

(d)

Chopping Null

(94)

 πθBcos (α) π 2  πθb sin (α) I 2 (left) = cos2  +  sin   (95) 4 λ λ  

Null

Peak

(a)

(93)

I 2 (ZB) − I 2 (B) =

Inserting the phases for each state, collecting terms, squaring to get intensities, and dropping a factor of 16, we find



I 2 (SZB) − I 2 (ZB) = I (star)



j=1

i2 π θ cos(α)(− B 2)+θ sin (α)(+ b 2) λ+φ A

135

~1 Hz rate

Fig. 18. Keck Nuller output is shown.

TABLE 8. TPF-I/Darwin phase states. State

φA

φB

left

0

π

0 + π/2 π + π/2

right

0

π

0–π/2

φC

φD

π–π/2

Signal null + null +

star + left zodi left planet star + right zodi right planet

136 

    Exoplanets

 πθBcos (α) π 2  πθb sin (α) I 2 (right) = sin 2  +  sin   (96) 4 λ λ    In both states the star at q = 0 is nulled by the sin2(0) = 0 term. For a planet at any value of (q, a) the signal is

I 2 (right) − I 2 (left) =

 2 πθBcos (α) 2  πθb sin (α) I (planet) sin   sin   λ λ    

(97)

So the planet will generate a signal given by its intensity times the beam pattern on the sky. As the array rotates about the line of sight, the pattern will sweep across the planet, producing a modulation pattern that is uniquely characteristic of the planet’s brightness, and also its radial (q) and azimuthal (a) position in the sky, so that a map of its position can be unambiguously reconstructed after one-half of a full rotation of the array. Importantly, any symmetric brightness component will be removed by the chopping, so if the zodi is bright and symmetric, it will drop out of the signal stream. Specifically, if I zodi(q, a) = I zodi(q, a + p), then the zodi signal is

I 2 (right) − I 2 (left) = 0

(98)

This is helpful in detecting the planet, especially if the zodi is bright. Obviously asymmetries in the zodi will be detected, but since these might be generated by planets in the first place, this will be of value to measure. Interestingly, by reprogramming the TPF-I chopping sequence we can easily measure either the symmetric part of the target signal or the asymmetric part. And in fact this could be done in a single chopping sequence if desired. Thus a full picture of the target can be built up. The fringe pattern scales with wavelength, so the output beam should be dispersed onto a detector array, the same as for KIN. As an example, if the individual mirrors are 2 m in diameter, then at 10  µm wavelength the FWHM of each diffractionlimited beam pattern on the sky is l/D = 1.0 arcsec FWHM, which just barely will accommodate a Jupiter at 10 pc (q = 0.5  arcsec radius). In each wavelength interval, all the light from the system falls on a single pixel. The modulation of that signal gives us a picture of the target at an angular resolution of l/2B = 0.010 arcsec, assuming a baseline of up to about B = 100 m, although in principle there is no limit to B. References include Beichman et al. (1999) and Cockell et al. (2009). 3.23. Golden Rule When thinking about an interferometer, and the image that could be formed in its focal plane, there is an important geometrical consideration that should be noticed: the “golden rule” of reimaging systems. This rule says that in order to have a wide field of view at a detector, the relative geometry

of the input pupil must be preserved at the output pupil, to within a constant magnification factor. This rule was originally formulated for multi-telescope arrays such as the original Multi-Mirror Telescope (MMT) with its six primary mirrors, and it applies to later systems such as TPF-I. There are three kinds of systems that do not obey this rule, and in each case the focal planes have extremely narrow fields of view. The first kind is the pupil densification system of a large array of telescopes in space, covering a baseline of several thousand kilometers, and phased up on a planet around a nearby star, for the purpose of making a true, spatially resolved image of that planet. In this concept the widely spaced collecting telescopes (the input pupil) are reimaged as a close-packed array (the output pupil), followed by an imaging lens. Because the two pupils are not related by a single scaling factor the output image has a very small field of view, which in this case is designed to be slightly larger than the diameter of the planet being imaged. The second kind is the pupil-mapping coronagraph discussed in section 3.12, in which the output image of the first two mirrors has a field of view on the order of a few times l/D, not sufficient to image a planetary system. Here the pupils are both circular, but the rays are rearranged, thereby essentially forcing a variable magnification factor between the two pupils, as a function of radial distance. However, in this case the image can be subsequently passed through a reversed set of optics, largely restoring the useful field of view. The third kind is the family of interferometers discussed in sections  3.21 and 3.22 above. Here, by superposing the output pupils on top of each other, the exit pupil (a single opening) is clearly not a scale copy of the multiple openings in the entrance pupil pattern. This extreme case has a correspondingly tiny field of view, essentially just the diffraction beamwidth of the individual telesopes (l/D). References include Traub (1986), Labeyrie (1996), and Pedretti et al. (2000). 3.24. Visible Nuller The visible nuller (Fig. 19) is a coronagraph-interferometer hybrid that can be used with segmented-mirror telescopes, such as the Thirty Meter Telescope (TMT), and is similar in plan view to the KIN and TPF-I, except that the baselines are more equal because they have to fit within a roughly circular primary mirror footprint. All the equations for KIN and TPF-I can be applied to the visible nuller, and in particular it can measure the symmetric as well as asymmetric parts of the target brightness distribution. References include Shao et al. (2008). 4. SPECKLE CONCEPTS All the discussions about methods in the preceding sections have dealt with idealized situations using perfect optics. In reality, nothing is perfect, and as work in exoplanet imaging has progressed, the effect of small variations in the photon propagation path lengths across the collected wavefronts has

Traub and Oppenheimer:  Direct Imaging of Exoplanets     

which is a pair of peaks (speckles), each similar in width to the original single peak, and separated by about twice that width. In other words, where we once had a single image of the star, we now have two adjacent images. Smaller phase jumps will produce intermediate results, i.e., a pair of speckles but with unequal intensities and smaller separation. Clearly, we could continue to subdivide the pupil into smaller segments, producing about as many speckles as there are distinctive phase patches across the pupil.

become a significant issue. In imaging, the effects of wavefront errors manifest themselves as speckles. Speckles appear as shown in Fig. 20. They are by far the dominant source of noise that must be controlled in order to image exoplanets. 4.1. Speckles from a Phase Step The basic idea of speckles can be demonstrated with a simple example. Suppose that the wavefront incident on a telescope is advanced by a phase step f/2 on one half of the pupil, and delayed by f/2 on the other half. With reference to equation (60), the net amplitude in the focal plane becomes

A 2 (θ) =

∫ ∫

+D 2

0 0

−D 2



e + iφ 2 e i2 πx1θ λ dx1 +

e

− iφ 2 i2 πx1θ λ

e

dx1

4.2. Speckles from Phase and Amplitude Ripples If we use an ideal telescope to image a star, and the wavefront from the star has been slightly distorted by possibly random phase and amplitude fluctuations, from an intervening atmosphere or from the telescope itself, then the natural result is a weakened star image surrounded by a halo of speckles. If the fluctuations are large, then the speckles will dominate and the star image will become just another speckle. If the telescope is a coronagraph, so the diffraction pattern is suppressed, then the speckles will certainly dominate. Any continuous wavefront across the pupil can be represented by a sum of sine and cosine waves. For reference, we recall the standard result from Fourier analysis

(99)

The resulting amplitude in the focal plane is



A 2 (θ) =

sin (πDθ 2λ) π D θ 2λ

cos (πDθ 2λ + φ 2) D

(100)

If the wavefront has f  = 0, i.e., no phase jump, then we recover the standard diffraction result



A 2 (θ, φ = 0) =

sin (πDθ λ) πDθ λ

A (x) =

(101)



sin 2 (πDθ 2λ)

Pupil input S

π D θ 2λ

(102)

D

Bright output

∑ (a

n

cos (2 πnx D) + b n sin (2 πnx D))

an =



2 D

∫

+D 2

−D 2

A (x) cos (2 πnx D)dx

Dispersive components for achromatic nulling

(104)

Coherent fiber-optic array spatial filter

Image plane (real image)

θ4 null in pupil overlap area

10 0 Variable delay

Variable pupil shear

Null contrast

Null output

Dispersive components for achromatic nulling

(103)

where A(x) is any real function on the interval x = (–D/2,+D/2). The cos( ) and sin( ) functions form an orthogonal basis set, and the coefficients are obtained by projecting A(x) onto this basis set and using the orthogonality. Multiplying both sides by cos(2pmx/D) or sin(–) and integrating we find

which is a single peak at the origin. However, if the total phase step is f = p, i.e., a half-wavelength, then we get A 2 (θ, φ = π) =

∞

n =0



D

137

10 –5

10 –10 0.4

0.5

0.6

0.7

0.8

0.9

1

Wavelength (µm)

Fig. 19. Visible Nuller schematic is shown. If the input pupil is a segmented mirror, then the shear s is set to a multiple of the segment spacing, in one pass through the interferometer, and set to a multiple along a different axis, in a second interferometer, and only the doubly overlapped segments are used in the fiber output stage.

138 

 

    Exoplanets

bn =

2 D

∫

+D 2

−D 2

A (x) sin (2 πnx D)dx  

(105)

This method of representing functions in terms of a basis  set can be extended to complex functions. Let A(x) be any  complex function on the interval x = (–D/2,+D/2). Then A(x)  can be expanded as  A (x) =

+∞

1 c n e i2 πnx D 2 n =−∞  

∑

(106)

where the coeffi cients are  cn =

2 D

∫

+D 2

−D 2

A (x) e − i2 πnx D dx

 

(107)

and n = 0, ±1, ±2, . . . As an example, we could write A(x) = A0eif(x) where A0  is a constant amplitude and f(x) is a spatially-varying phase,  e.g., linear for a tilted wavefront, etc. If f is complex, then the  imaginary part represents the spatial variation of amplitude,  which  could  also  be  absorbed  in  the  coeffi cient  as A(x)  =  A0e–Im(f). If f(x)