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High time-resolution astrophysics (HTRA) involves measuring and studying astronomical phenomena on timescales of seconds

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High Time-Resolution Astrophysics
 1107181097, 9781107181090

Table of contents :
Contents
List of Contributors
List of Participants
Preface
Acknowledgements
List of Abbreviations
1 Radiation Processes and Models
2 HTRA Instrumentation I
3 HTRA Instrumentation II
4 X-ray Emission from Black-Hole and Neutron-Star Binaries
5 Radio Observations and Theory of Pulsars and X-ray Binaries
6 Incorporating Gamma-ray Data into High-Time Resolution Astrophysics

Citation preview

H I G H T I M E - R E S O L U T I O N A S T RO P H Y S I C S High time-resolution astrophysics (HTRA) involves measuring and studying astronomical phenomena on timescales of seconds to milliseconds. Although many areas of astronomy, such as X-ray astronomy and pulsar observations, have traditionally required high timeresolution studies, HTRA techniques are now being applied to optical, infrared and gamma-ray wavelength regimes, due to the development of high-efficiency detectors and larger telescopes that can gather photons at a higher rate. With lectures from eminent scientists aimed at young researchers and postdocs in observational astronomy and astrophysics, this volume gives a practical overview of and introduction to the tools and techniques of HTRA. Just as multi-spectral observations of astrophysical phenomena are already yielding new scientific results, many astronomers are optimistic that exploring the time domain will open up an important new frontier in observational astronomy over the next decade. ta r i q s h a h b a z is Staff Scientist at the Instituto de Astrof´ısica de Canarias, where his research focuses on the study of compact objects, such as neutron stars and black holes. He has developed methods and computer models to predict the observed lightcurves and spectra from these systems, allowing the mass of the compact object to be determined. His other research interests include high time-resolution phenomena and the study of cataclysmic variables and millisecond pulsars. He is a member of the International Astronomical Union. ´ z q u e z is Professor of Astronomy at the Instituto de j o rg e c a s a r e s v e l a Astrof´ısica de Canarias, and Leverhulme Professor at the Department of Astrophysics of the University of Oxford during 2016–2017. His research focuses on the study of galactic black holes, with emphasis on the determination of their dynamical masses. He has promoted novel strategies for deriving fundamental parameters in X-ray binaries and discovering hidden populations of galactic black-hole binaries. His other research interests include gamma-ray binaries, cataclysmic variables and millisecond pulsars. He is a member of the International Astronomical Union and the Spanish Astronomical Society. ˜ o z da r i a s is Ram´ t e o d o ro m u n on y Cajal Fellow at the Instituto de Astrof´ısica de Canarias. His research is focused on the study of accretion processes in accreting black holes and neutron stars in X-ray binaries. He has made significant contributions to this field by studying the evolution of the fast variability properties of these objects, aiming at building a universal scheme for the different accretion states. He is also leading important work on the discovery and characterisation of outflowing accretion disc winds, as well as on the nature and fundamental parameters of X-ray binaries. He is a member of the International Astronomical Union and the Spanish Astronomical Society.

Canary Islands Winter School of Astrophysics Volume XXVII High Time-Resolution Astrophysics Series Editor Rafael Rebolo, Instituto de Astrof´ısica de Canarias Previous Volumes in This Series I. II. III. IV. V. VI. VII. VIII. IX. X. XI. XII. XIII. XIV. XV. XVI. XVII. XVIII. XIX. XX. XXI. XXII. XXIII. XXIV. XXV. XXVI.

Solar Physics Physical and Observational Cosmology Star Formation in Stellar Systems Infrared Astronomy The Formation of Galaxies The Structure of the Sun Instrumentation for Large Telescopes: A Course for Astronomers Stellar Astrophysics for the Local Group: A First Step to the Universe Astrophysics with Large Databases in the Internet Age Globular Clusters Galaxies at High Redshift Astrophysical Spectropolarimetry Cosmochemistry: The Melting Pot of Elements Dark Matter and Dark Energy in the Universe Payload and Mission Definition in Space Sciences Extrasolar Planets 3D Spectroscopy in Astronomy The Emission-Line Universe The Cosmic Microwave Background: From Quantum Fluctuations to the Present Universe Local Group Cosmology Accretion Processes in Astrophysics Asteroseismology Secular Evolution of Galaxies Astrophysical Applications of Gravitational Lensing Cosmic Magnetic Fields Bayesian Astrophysics

HIGH TIME-RESOLUTION A S T RO P H Y S I C S Edited by

TA R I Q S H A H B A Z Instituto de Astrof´ısica de Canarias

´ ZQUEZ J O RG E C A S A R E S V E L A Instituto de Astrof´ısica de Canarias

˜ O Z DA R I A S T E O D O RO M U N Instituto de Astrof´ısica de Canarias

University Printing House, Cambridge CB2 8BS, United Kingdom One Liberty Plaza, 20th Floor, New York, NY 10006, USA 477 Williamstown Road, Port Melbourne, VIC 3207, Australia 314-321, 3rd Floor, Plot 3, Splendor Forum, Jasola District Centre, New Delhi – 110025, India 79 Anson Road, #06–04/06, Singapore 079906 Cambridge University Press is part of the University of Cambridge. It furthers the University’s mission by disseminating knowledge in the pursuit of education, learning, and research at the highest international levels of excellence. www.cambridge.org Information on this title: www.cambridge.org/9781107181090 DOI: 10.1017/9781316831984 © Cambridge University Press 2018

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 2018 Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall A catalogue record for this publication is available from the British Library. Library of Congress Cataloging-in-Publication Data Names: Canary Islands Winter School of Astrophysics (27th : 2015 : La Laguna, Canary Islands) | Shahbaz, Tariq, 1970– editor. | Casares Velazquez, Jorge, 1964– editor. | Munoz Darias, Teodoro, 1981– editor. Title: High time-resolution astrophysics / edited by Tariq Shahbaz (Instituto de Astrofisica de Canarias), Jorge Casares Velazquez (Instituto de Astrofisica de Canarias), Teodoro Munoz Darias (Instituto de Astrofisica de Canarias). Other titles: Canary Islands Winter School of Astrophysics (Series) ; v. XXVII. Description: Cambridge, United Kingdom ; New York, NY : Cambridge University Press, 2018. | Series: Canary Islands Winter School of Astrophysics ; volume XXVII | Lectures presented at the XXVII Canary Islands Winter School of Astrophysics, held in La Laguna, Tenerife, Spain, Nov. 9-20, 2015. | Includes bibliographical references and index. Identifiers: LCCN 2017061800| ISBN 9781107181090 (hardback ; alk. paper) | ISBN 1107181097 (hardback ; alk. paper) Subjects: LCSH: Astrophysics–Congresses. | Compact objects (Astronomy)–Congresses. Classification: LCC QB460 .C356 2018 | DDC 523.01–dc23 LC record available at https://lccn.loc.gov/2017061800 ISBN 978-1-107-18109-0 Hardback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Contents List of Contributors

page viii

List of Participants

ix

Preface

xi

Acknowledgements

xiii

List of Abbreviations

xiv

1 Radiation Processes and Models

1

Julien Malzac

2 HTRA Instrumentation I

43

Philip A. Charles

3 HTRA Instrumentation II

67

Vikram S. Dhillon

4 X-ray Emission from Black-Hole and Neutron-Star Binaries

97

Tomaso M. Belloni

5 Radio Observations and Theory of Pulsars and X-ray Binaries

133

Andrea Possenti

6 Incorporating Gamma-ray Data into High-Time Resolution Astrophysics Elizabeth C. Ferrara

vii

169

Contributors Tomaso M. Belloni

Osservatorio Astronomico di Brera, Italy

Philip A. Charles

University of Southampton, UK

Vikram S. Dhillon

University of Sheffield, UK, and Instituto de Astrofisica de Canarias, Spain

Elizabeth C. Ferrara

NASA Goddard Space Flight Center, and University of Maryland, College Park, USA

Julien Malzac

University of Toulouse, France

Andrea Possenti

INAF-Osservatorio Astronomico di Cagliari, Italy

viii

Participants Belloni, Tomaso Casares, Jorge Charles, Phil Court, James Dhillon, Vik

INAF Osservatorio Astronomico di Brera (Italy) Instituto de Astrof´ısica de Canarias (Spain) University of Southampton (UK) University of Southampton (UK) University of Sheffield (UK) Instituto de Astrofisica de Canarias, Spain Ferrara, Elizabeth NASA Goddard Space Flight Center (USA) Gambino, Angelo Francesco Universit´ a degli Studi di Palermo (Italy) Gonzalez, Miguel Universidad Nacional Aut´ onoma de Mexico, Instituto de Astronom´ıa (Mexico) Green, Matthew University of Warwick (UK) Jimenez Ibarra, Felipe Instituto de Astrof´ısica de Canarias (Spain) Julien, Malzac Universit´e de Toulouse (France) Kennedy, Mark University College Cork (Ireland) Mata S´ anchez, Daniel Instituto de Astrof´ısica de Canarias (Spain) Matranga, Marco Universit´a degli Studi di Palermo (Italy) Mu˜ noz-Darias, Teodoro Instituto de Astrof´ısica de Canarias (Spain) N´ı Chonchubhair, Deirdre Centre for Astronomy, NUI Galway (Ireland) O’Connor, Eoin Centre for Astronomy, NUI Galway (Ireland) Pala, Anna Francesca University of Warwick (UK) Palomo Nevado, Sergio Universidad de La Laguna, Instituto de Astrof´ısica de Canarias (Spain) Polednikova, Jana Instituto de Astrof´ısica de Canarias (Spain) Possenti, Andrea INAF-OAC Astronomical Observatory of Cagliari (Italy) Shahbaz, Tariq Instituto de Astrof´ısica de Canarias (Spain) Shaw, Aarran University of Southampton (UK) Steeghs, Danny University of Warwick (UK) van Doesburgh, Marieke University of Amsterdam (Netherlands) V´azquez, Veronica Instituto de Astronom´ıa. UNAM (Mexico) Wang, Louise University of Warwick (UK)

ix

Preface High time-resolution astrophysics (HTRA) concerns itself with observations on short time-scales, normally defined as milliseconds to seconds. HTRA is, therefore, an important tool in understanding the fundamental physics of radiative processes from a diverse range of objects. Understanding the radiation processes allows one to extract information encoded in the observed radiation. It allows us to test fundamental theories by measuring important physical parameters, such as temperatures, velocities, magnetic field strengths, the energy of accelerated particles and their distribution. In addition, the interpretation of the observed high time-resolution variability often requires the development of physical models describing the complex dynamical processes occurring in these extreme environments. Due to their small physical size, variability on the fastest time-scales are associated with compact stellar remnants (black holes, neutron stars, and white dwarfs), which is tightly related to their relevant physical processes and emission mechanisms. Current high time-resolution observations of compact objects are providing remarkable insights into fundamental questions, such as how black hole accretion takes place, how jets/outflows operate and the nature of the extreme gravity conditions around neutron stars and the stable orbits around stellar mass black holes. Indeed, it is becoming increasingly clear that multi-wavelength high time-resolution observation is the best way to disentangle the physical origin of the complex broadband spectral emission (e.g., accretion flow and jets) observed from compact binary systems. However, HTRA is not limited to only compact objects; for instance, transit observations involving fast timing also provide vital information on the basic parameters of exoplanets. The astronomical community has put extreme environment astrophysics as one of their key ground- and space-based research areas, implying that HTRA is critical for the success of these projects. HTRA demands the use of very fast, highly efficient, large photon-counting detectors with intrinsic energy resolution across a wide spectral range. There have been significant advances towards such detectors, e.g., the superconducting tunnel junction and microwave kinetic inductance detectors, with a realistic chance that we will see such detectors in the next decade. Combined with high time-resolution, these have the potential for revolutionizing observational astronomy over a wide range of wavelengths. Recognizing the importance of this field, the Instituto de Astrof´ısica de Canarias organised the XXVIIth Winter School on the topic of ‘High Time-Resolution Astrophysics’. The aim of the School was to bring together a number of the leading scientists working in the field of HTRA with PhD students and recent postdocs. The School tackled many aspects of HTRA and was particularly designed to provide a wide-ranging and up-todate overview of the instrumental and theoretical tools, and applications to observations at different wavelengths, necessary for carrying out front-line research in the study of HTRA. The forty lectures present a comprehensive and up-to-date introduction to the major observational and theoretical topics associated with HTRA. With emphasis on the physical processes involved, this includes applications to compact stellar objects (black holes, neutron stars, and white dwarfs), jets/outflows, interaction between highly relativistic plasma and strong magnetic fields and the relevant physical processes and emission mechanisms operating on very short time-scales. Given that it is not possible to understand their associated phenomena without covering multi-wavelength HTRA, the School took a strong multi-wavelength approach, covering HTRA at radio, optical, X-ray and Gammaray wavelengths. Furthermore, the requirements of low-noise, fast-readout detectors, xi

xii

Preface

time systems to correctly compare data from different telescopes on the ground and in space, and the use of non-conventional software tools specific for time-series analysis was addressed. The lectures were given by seven experienced research scientists, who have played key roles in the advancements made in the field of HTRA.

Acknowledgements The editors would like to thank all the lecturers for their time in preparing their classes, attending the School and writing the chapters for this book. We know that it has been a major effort on their part, and we truly appreciate their dedication throughout the process. In particular, we would like to thank Professor Tomaso Belloni for his entertaining public lecture on ‘exploring black holes with a clock’ and Professor Phil Charles for organising the poster session. Last, but not least, we thank all the students who made the School enjoyable. The success of the School was without doubt due to the hard work of our secretary Lourdes Gonz´ alez, who ensured the smooth running of the School. We would also like to thank Gabriel P´erez D´ıaz, who prepared the School’s poster, the IAC’s Centro de C´ alculo for their IT assistance and Annia Dom`enech for organising the press releases. As always, the School provides a guided tour of Teide Observatory (Tenerife), Roque de los Muchachos (La Palma) and the IAC’s headquarters in La Laguna. We thank the IAC’s Research division for helping out with these visits. We are also grateful to the Cabildo de Tenerife and the Town Hall of San Crist´ obal de La Laguna, who kindly organised visits to Teide National Park and the city, respectively. Finally, we greatly acknowledge financial support from the Severo Ochoa Centre of Excellence.

xiii

Abbreviations ACD ADAF ADIOS AGN APD ALICE ARCONS ATHENA AXMP AXTAR BB BH BHB BHXRB BIPM BL CCD CCDM CCF CCO CDAF CERN CGRO CMOS CV DBB DEPFET DM DNS EDISP EM e-ELT EMCCDS EoS EPIC ESA FBO FOV FSSC FWHM GASP GBM GPS GR GUI GW HS HBO

anti-coincidence detector accretion-dominated accretion flow advection-dominated inflow-outflow solution active galactic nuclei avalanche photodiode a large ion collider experiment Array Camera for Optical and Near-IR Spectrophotometer Advanced Telescope for High Energy Astronomy accreting X-ray millisecond pulsar advanced X-ray timing array Blackbody black hole black hole binary black hole X-ray binary Bureau International des Poids et Mesures boundary layer charge-coupled device colour-colour diagram cross-correlation function central compact objects convection dominated accretion flow European Organization for Nuclear Research Compton Gamma-ray Observatory complementary metal-oxide-semiconductor cataclysmic variable disk-blackbody DEpleted P-Channel Field Effect Transistor dispersion measure double neutron star energy dispersion electron multiplication European Extremely Large Telescope electron multiplying charge-coupled device equation of state European photon imaging camera European Space Agency flaring branch oscillation field of view fermi science support center full width at half maximum the Galway Astronomical Stokes Polarimeter gamma-ray burst monitor Global Positioning System general relativity graphical user interface gravitational wave Hot Spot horizontal branch oscillation xiv

Abbreviations HESS HFQPOs HID HIMS HiPER-CAM HMBP HMXB HRD HSS HST HTRA IBIS IMBP IMXB INTEGRAL IPC IR IRF ISCO ISS JEM-X KIDspec L3 LAPC LAT LFQPO LHC LHS LLE LMBP LMXB LOFT LT MAGIC MAMA MHD MIC MJD MKID MSP NBO NGC NICER NIR NS NS-LMXB OIR OM OPTIMA

High Energy Stereoscopic System High-Frequency quasi-periodic oscillation hardness–intensity diagram hard-intermediate state High PERformance CAMera high-mass binary pulsar high-mass X-ray binary Hardness-RMS diagram high-soft State Hubble Space Telescope high time-resolution astrophysics imager on board the INTEGRAL satellite intermediate-mass binary pulsar intermediate-mass X-ray binary INTErnational Gamma-ray Astrophysics Laboratory imaging proportional counter infrared instrument response function innermost stable circular orbit International Space Station Joint European X-ray Monitor Kinetic Inductance Detector Spectrograph low light level large area proportional counters large area telescope low-frequency quasi-periodic oscillation Large Hadron Collider low-hard state large area telescope low-energy low-mass binary pulsar low-mass X-ray binary large observatory for X-ray timing Lense–Thirring precession Major Atmospheric Gamma Imaging Cherenkov multi-anode microchannel array magnetohydrodynamic MCP-Intensified CCD Modified Julian Day microwave kinetic inductance detector millisecond pulsar normal branch oscillation New General detector Controller Neutron star Interior Composition ExploreR near-infrared neutron star neutron star low-mass X-ray binary optical/infrared optical monitor Optical Pulsar TIMing Analyzer

xv

xvi PC PCA PDS PK PMT PSU PSF PTA PWN QE QPO RB RID RMS ROI RPM RRATs RXTE S/N SAS SDD SDSS SED SEP SIMS SNR SNR SPAD SPI SSB SSS STIS STJ TAI TCB TDB TES TMT ToA TOV TS UCT ULMBP UV VERITAS VLMBP WD WISE XDINS XMM-Newton

Abbreviations proportional counter proportional counter array power density spectra post-keplerian photomultiplier tube power supply unit point spread function pulsar timing array pulsar wind Nebula quantum efficiency quasi-periodic oscillation gamma-ray burst RMS-intensity diagram root mean square region of interest relativistic precession model rotating radio transients Rossi X-ray Timing Explorer signal-to-noise small astronomy satellite silicon drift detector Sloan Digital Sky Survey spectral energy distribution strong equivalence principle soft-intermediate state supernova remnant signal-to-noise ratio Single-Photon avalanche photo-diode SPectrometer on INTEGRAL solar system barycenter solid-state spectrometer Space Telescope Imaging Spectrograph superconducting tunnel junction Temps Atomique International Barycentric Coordinate Time Barycentric Dynamical Time transition edge sensor Thirty Meter Telescope Time of Arrival Tolman–Oppenheimer–Volkoff test statistic University of Cape Town ultra low mass binary pulsar ultraviolet Very Energetic Radiation Imaging Telescope Array System very low mass binary pulsar white dwarf Wide-field Infrared Survey Explorer X-ray dim isolated neutron star X-ray Multi-Mirror Mission

1. Radiation Processes and Models JULIEN MALZAC1 Abstract This is a basic introduction to the physics of compact objects in the context of high time-resolution astrophysics (HTRA). The main mechanisms of energy release and the properties of relevant radiation processes are briefly reviewed. As a specific example, the top models for the multi-wavelength variability of accreting black holes are unveiled.

1.1. Introduction Compact objects represent a prime target for HTRA. Their physics involves strong gravitational fields, matter compressed to enormous densities, very high energy particles and huge magnetic fields. Of course, such extreme conditions cannot be produced in a laboratory on Earth. In many cases, high time-resolution observations of these objects represent a unique opportunity to test fundamental theories regarding particle interactions, the properties of dense matter or gravitation. Compact objects also inform us about fundamental astrophysical processes such as accretion and ejection in their most extreme form. Classically, there are three types of compact objects that all form mostly (but not only) through the gravitational collapse of a normal star after exhaustion of its thermonuclear fuel. White dwarfs (hereafter WDs) have masses lower than the Chandrasekhar limit (1.4 M ) and radii comparable to that of Earth. The quantum degeneracy pressure of the electrons balances the gravitation forces to prevent collapse. Neutron stars (hereafter NSs) are instead supported by short-range repulsive neutron–neutron interactions mediated by the strong force and also by the quantum degeneracy pressure of neutrons. They can have masses up to the Tolman–Oppenheimer–Volkoff (TOV) limit, which is approximately between 2 and 3 M . Above the TOV limit, a star cannot support its own weight.2 It is completely collapsed and forms a black hole (hereafter BH). The size of a NS is about 10 km; the size of a BH is given by the size of the horizon of events. The latter depends on the spin of the BH but remains comparable to the gravitational radius: Rg = GMBH /c2  1.5 MBH /M km. A typical stellar-mass BH concentrates about 10 times the mass of the sun within a radius of about 15 km. This corresponds to an average density within Rg of about 1015 g cm−3 . The average density of the matter of a NS is of the same order, i.e., 1014 times denser than Planet Earth! The surface gravity on a NS is more than 1011 times that on Earth. The dynamical timescale in the environment of compact objects is very short. For example, a Keplerian orbit at six times the radius of the object corresponds, on Earth, to a geosynchronous orbit has a period of exactly one day. This timescale is reduced to a few minutes around a WD and only a few milliseconds 1 I am extremely grateful to the organisers of the Winter School – T. Shahbaz, J. Casares and T. Mu˜ noz Darias, for inviting me to the Instituto de Astrof´ısica de Canarias and for the opportunity to deliver these lectures. I also want to thank my long-time collaborator Renaud Belmont for his inputs, in particular the material he provided for the section on radiation processes. The preparation of this manuscript was supported in part by the ANR CHAOS project ANR-12-BS05-0009 (www.chaos-project.fr). 2 Note that objects supported by the quantum degeneracy pressure of quarks, and named quark stars, have also been theorised and would have properties very similar to that of neutron stars, but their existence is disputed.

1

2

Julien Malzac

around a NS or a BH! Compact objects can also release tremendous amounts of energy in their environment in the form of radiation and powerful outflows. This combination of power and very short dynamical timescales makes compact objects one of the main targets of HTRA. Section 1.2 presents the basic mechanisms of energy release around compact objects through dissipation of gravitational, magnetic and rotational power. Then, in Section 1.3, the main emission processes responsible for the conversion of, fraction of this energy into photons are described. Finally, Sections 1.4 and 1.5 focus on models for the multi-wavelength variability of accreting BHs in X-ray binary systems involving, respectively, the accretion flow and the jets.

1.2. Compact Objects Energy can be extracted from the rotation of the compact object (pulsar), its magnetic field (magnetar), from the gravitational energy of material falling onto the compact object (accretion in binary systems) or even thermonuclear fusion of outer layers of accreted materials (X-ray bursters, classical novae). In this section, I briefly introduce the different mechanisms of energy release. A much more detailed exposition of compact objects physics can be found in classical textbooks, such as those by Shapiro and Teukolsky (1983), Frank et al. (1992), Longair (1994) and Kato et al. (2008). There are also many excellent reviews covering specific recent developments of the field, for instance, Patruno and Watts (2012) on millisecond X-ray pulsars, Watts (2012) on thermonuclear burst oscillations, Gilfanov and Sunyaev (2014) on the structure of the boundary layer in accreting neutron stars, Smith (2006) on cataclysmic variables or Done et al. (2007) on accreting BHs and NSs. 1.2.1 Rotation All stars are rotating (the rotation period of the sun is about 27 d). In the case of NSs and BHs, the angular momentum is essentially conserved during the gravitational collapse. The compactification of the star then implies faster rotation. As a result, a newborn NS can have a spin period P of the order of P = 10–100 ms. On the other hand, most WDs have slow rotation due to the removal of angular momentum during the loss of the progenitor’s outer envelope in a planetary nebula. Nevertheless, a WD can later be spun up by accretion (see Section 1.2.3). The fastest spinning WD has a period of only P = 13 s (Mereghetti et al., 2011). The rotational energy of a star can be estimated simply as Erot = IΩ2 /2 = 2Iπ 2 P −2 where I is the moment of inertia, which, for a homogeneous sphere of mass M and radius R is I = 2M R2 /5. For a NS, this energy can represent a small but significant fraction of the rest mass energy of the star:  2  −2 R P Erot −4  10 . (1.1) M c2 10 km 0.01 s In absolute terms the rotation energy of a young neutron star is comparable to the energy radiated by the sun during one billion years. However, this energy is released over much smaller timescales of the order of 103 –107 yr. Indeed, the extraction of rotational energy implies spin-down. The rotational power is related to the spin-down rate as    2  −2 ˙ P˙ P R P M d IΩ erg s−1 . (1.2) E˙ rot = = 4Iπ 2 3  3 × 1050 dt 2 P M 10 km 0.01 s P The ‘braking’ time scale over which the rotation power is released can be estimated as Erot /E˙ rot ∼ P/P˙ . Pulsars are NSs emitting beams of radiation leading to the highly

Radiation Processes and Models

3

coherent modulation of the observed light curve at the spin period. In these sources, both the spin and spin-down rate can be measured accurately (see Chapter 5 by A. Possenti in this volume), and this allows one to estimate the released rotation power. In the case of the Crab Pulsar, the observed period is P  33 ms, and the spin-down rate is P˙ /P  10−11 s−1 , which leads to E˙ rot  1038 erg s−1 . This power is comparable to the accumulated radiative output of about 105 stars like the sun. This power is larger than the observed luminosity of the source but comparable to the power required to feed the surrounding pulsar wind nebula. This coincidence indicates that the pulsar wind is powered by the rotational energy of the pulsar. The rotational power is extracted via the effects of the strong magnetic field of the neutron star. Indeed, during the gravitational collapse leading to the formation of the NS, the conservation of magnetic flux across the stellar surface implies that the magnetic field is amplified by a factor ∼ R2 /R2 , where R is the initial radius of the star, and R is the radius of the compact object. For a star of the size of the sun compacted into a 10 km radius, this amplification factor is of the order of 5 × 109 ! As a result, the typical surface magnetic field of a young NS is of the order of 1011 –1012 G. For comparison, Earth’s magnetic field is ∼0.5 G; a fridge magnet is ∼50 G; the strongest continuous field yet produced in a laboratory is about 5 × 105 G. Because of the high conductivity of the NS matter, its magnetic field dissipates only very slowly, on timescales much longer than the ‘braking’ timescale of the NS. In practice, it can be considered a constant. The young NS, therefore, behaves as a huge, spinning magnet. It is known from electromagnetic theory that a magnetic dipole rotating in a vacuum radiates. The radiated energy can be estimated simply as E˙ d =

Bp2 R6 Ω4 sin2 α  1039 6c3



Bp 1012 G

2 

R 10 km

6 

P 0.01 s

−4 sin2 α

ergs−1 , (1.3)

where Bp is the magnetic field amplitude at the poles of the compact star, and α is the misalignment angle between the direction of the dipole and the spin of the neutron star. This radiated power can be matched to the observed rotational power in pulsars in order to estimate the pulsar magnetic field, its age, etc. But in reality, the magnet is not spinning in a vacuum. Huge electric fields are generated close to the surface of the NS. Those fields extract charged particles from the NS. These charged particles distribute themselves around the star to neutralise the electric field; an extended magnetosphere is formed. Streams of charged particles leave the star at a high latitude where the magnetic field lines are open, leading to the formation of a wind. The rotation energy is dissipated mostly through the interaction of the magnetic field of the NS with the magnetosphere and the wind. Nevertheless, the energy losses remain comparable to those predicted by the simple magnetic dipole radiation model. 1.2.2 Magnetic Field Dissipation Some NSs appear to have negligible rotation energy but huge magnetic fields up to 1014 – 1016 G. These strongly magnetised NSs define the class of magnetars (see, e.g., Woods and Thompson, 2006). It has been shown (Thompson and Duncan, 1993) that such magnetic fields may have built up through dynamo processes (similar to that generating Earth or the sun’s magnetic fields) during the first moments of the life of the NS. This strong dynamo amplification occurs only if the NS is formed with a spin period P < 5 ms. After 10–30 s, the core has cooled down and is not hot enough for efficient dynamo. The magnetic braking is very efficient and the rotation energy is dissipated very quickly. After a few minutes, the rotation has slowed down to a period of a few seconds.

4

Julien Malzac

But the magnetic fields remain so strong that they can push and move material around in the star’s interior and crust. This leads to large amounts of magnetic dissipation during the first 104 yr. Magnetic dissipation in the interior of the star keeps the star hot, bright and producing mostly X-ray thermal emission. Magnetic dissipation may also occur in the surrounding magnetosphere due to twisting of the magnetic field lines and reconnection (similar to the solar corona). This can lead to bursts of non-thermal X-ray and gamma-ray radiation, which are observed in sources called ‘soft gamma-ray repeaters’. 1.2.3 Accretion Accretion is the growth of a massive object by gravitationally attracting more matter. This is an ubiquitous astrophysical process leading to the formation of planets, stars, galaxies and the growth of super-massive BHs. The gravitational energy released during accretion onto super-massive BHs also appears to regulate the joint growth of the BHs and their host galaxies. Accretion onto stellar compact objects can be observed if the compact object is in a binary system and can accrete gas from its companion star. As this type of accretion occurs in bright nearby sources evolving on human timescales it is relatively easy to observe and study, and some of the knowledge gained may be extrapolated to other accreting systems such as super-massive BHs in active galactic nuclei (AGNs). As will be discussed below, accretion onto a compact star in an accreting binary system depends on the nature of the donor star (high mass versus low mass star). It depends also on the nature of the compact star. Gas accreting onto a NS or a WD will ultimately hit the hard surface of the star. This has observable effects, such as the presence of a boundary layer in which the gas is stopped or the triggering of nuclear explosions (X-ray bursts in NSs, classical novae in WDs) when enough material has been accreted onto the surface. The structure of the accretion flow may also be affected by the strong magnetic field of the star as is the case for NSs in X-ray pulsars or WDs in polars. Such complications do not occur in accreting BHs where we can observe accretion in its ‘purest’ form; as the gas crosses the event horizon without notable effects, all the observed radiation must originate from the accretion flow. Accretion Power Accreting matter falls into the potential wells of a compact object and loses gravitational energy. For accretion to occur, gravitational energy (and angular momentum) must be dissipated away, mostly in the form of radiation. If accretion occurs at a mass accretion rate M˙ onto an object of size R and mass M , the gravitational power that is dissipated is given by the gravitational potential on the surface GM = η M˙ c2 , E˙ ac = M˙ R

(1.4)

where η = GM/Rc2 is called the accretion efficiency. It represents the amount of energy released per unit of mass energy accreted. The accretion efficiency onto a WD is of the order of 10−4 while accretion onto a neutron star reaches 0.1. Due to the absence of a hard surface during accretion onto a BH, the efficiency depends on the structure of the accretion flow. For thin accretion discs (see below), the accretion efficiency is in the range 0.057–0.42, depending on the BH spin. For comparison, accretion onto Planet Earth has η ∼ 10−9 , while the sun has η ∼ 10−6 . The efficiency of thermonuclear fusion of hydrogen is 7 × 10−3 . Accretion onto a compact object is, therefore, much more efficient than fusion at releasing energy.

Radiation Processes and Models

5

Eddington Limit The accretion power is, however, limited by the amount of matter that we are able to accrete. There is a fundamental limit on the mass accretion rate that is set by the Eddington luminosity. The Eddington luminosity is the maximum luminosity for which the gravitational force on a fluid element exceeds the radiation pressure (i.e., the maximum luminosity at which matter can be accreted). Let us consider a fluid element of mass m located at distance d from the center of the compact object, which radiates isotropically at luminosity L. The amplitude of the force of gravity is Fgrav = GM m/d2 . The radiation pressure force is proportional to the local radiation flux and is directed in opposite direction. It is given by Frad = Lκm/(4π d2 c), where the opacity κ is a measure of the effectiveness of the transfer of momentum from radiation to the fluid element. Since both forces are proportional to m/d2 , the condition for equilibrium Fgrav = Frad is independent of both the mass and distance of the fluid element. This condition defines the Eddington luminosity LE =

M 4πGM c  1.46 × 1038 κ M

ergs−1 .

(1.5)

The accreting gas is usually hot and ionised, so the opacity is dominated by electron scattering. In this case, κ = κes = 0.34 cm2 g−1 for standard abundances. This is the value of κ that was assumed in the numerical estimate given in Equation 1.5. This immediately implies a maximum accretion rate M˙ E = LE /ηc2 = 2.6 × 10−9 (M/M ) η −1

M yr−1 .

(1.6)

For a compact object with a hard surface (NS or WD), the Eddington mass accretion rate depends only on its size   R 4πcR = 1.8 × 10−8 (1.7) M˙ E = M yr−1 . κ 10 km The maximum luminosity and accretion rate are estimated for a spherical accretion geometry and radiation field, and deviations from spherical are of course expected to occur and may allow to exceed somewhat the Eddington limit (which indeed appears to be violated in some sources). Nevertheless, this gives a good estimate of the maximum power that can be extracted through accretion onto a compact object. This power is huge. The Eddington luminosity of a 10 M BH is comparable to the combined luminosity of 106 stars like the sun. Mass Transfer However, there is another limitation related to the capacity of the compact object to attract and capture gas from the donor star. This is the so-called mass transfer problem. Before falling onto the compact object, the gas must escape the pull of the donor star. The effective gravitational potential in a binary system is determined by the masses of the stars and the centrifugal force arising from the motion of the two stars around one another. One may write this potential as a function of r1 and r2 , the distance to the centre of each star, and r3 , the distance to the rotation axis φ = −

GM1 GM2 Ω2 r2 − − orb 3 , r1 r2 2

(1.8)

where Ωorb is the orbital angular velocity. The equipotential surfaces form two lobes surrounding each of the stars that are called the Roche lobes. The two lobes are connected

6

Julien Malzac

through a point called the first Lagrangian point (or L1), where the sum of the centrifugal and gravitational forces vanishes. This is a saddle point in the potential that forms a pass that the gas from the donor has to climb before being able to fall into the influence of the compact object.

Roche lobe overflow. Mass transfer may occur simply if the donor fills its Roche lobe. This may result from an increase of the stellar radius during the evolution of the star. This can also be driven by changes in the orbital parameters that make the Roche lobe smaller. This can also occur through loss of angular momentum (by emission of gravitational waves, magnetic braking, tidal torques, mass loss in a stellar wind, etc.). Also, mass transfer, over time, will change the mass ratio of the two components and affect the orbital parameters. So even if the donor fills its Roche lobe, accretion may be unstable and may not be sustained. It can be shown that stable lobe overflow can occur only if the mass of the donor is smaller than the mass of the accretor. If these two conditions are fulfilled, then steady mass accretion occurs at rates of the order of M˙ ∼ 10−10 –10−9 . Such systems are called low-mass X-ray binaries (hereafter LMXBs).

Wind accretion. A massive early type companion (O or B) can lose mass in a wind at a rate 10−6 –10−5 M yr−1 and supersonic velocity comparable to the escape velocity of the star  2GM ∼ 103 km s−1 . (1.9) vw ∼ vesc = R The compact star will gravitationally capture matter from a roughly cylindrical region with an axis along the relative wind direction. This cylinder represents the volume where the wind particle kinetic energy is less than the gravitational potential (e.g., Bondi and Hoyle, 1944; Bondi, 1952; Davidson and Ostriker, 1973; Lamb et al., 1973; Frank et al., 1992). The radius of the cylinder, called the accretion radius or gravitational capture radius, is given by racc 

2GM 2 + c2 , vrel s

(1.10)

2 2 2 = vorb + vw , and vorb is the orbital velocity of where cs is the sound speed in wind, vrel the gas. The net amount of gas captured and accreted by the compact object can be obtained by combining this relationship with Kepler’s third law and the continuity equation (assuming spherically symmetric and steady mass loss) 2 M˙ = πracc ρvrel ∼ (10−5 − 10−4 )M˙ w ∼ 10−11 − 10−9 M yr−1 .

(1.11)

We can see that both wind accretion in HMXB and Roche lobe overflow in LMXB allow mass transfer at a rate that is smaller than, and yet a significant fraction of, the Eddington limit. Keplerian Accretion Discs Once the material is captured, it would orbit the compact object indefinitely unless it can get rid of its angular momentum and be accreted. It is viscosity, and the associated viscous torques between annuli in the disc, that allows angular momentum to be transferred outwards and mass to spiral inwards. Any original ring of particles will thus spread out into a disc, with the outer radius being determined by tidal torques from the

Radiation Processes and Models

7

companion. However, ordinary molecular viscosity is completely inadequate to account for the observed properties of discs, and some kind of turbulent viscosity is invoked. The best candidate as the origin of this turbulent viscosity is the magneto-rotational instability that develops in differentially rotating flows and can generate fully developed magnetohydrodynamic (MHD) turbulence that provides efficient angular momentum transport (Balbus and Hawley, 1991, 1998). The characteristic scale of the turbulence must be less than the disc thickness, H, and the characteristic turbulent speed is expected to be less than the speed of sound, cs , since there is no strong evidence for turbulent shocks. The viscosity, ν, is therefore often parametrised by writing it as ν = αcs H, placing all the uncertainties in the unknown parameter α, taken to be less than 1. This is the standard alpha-disc model of accretion discs (Shakura and Sunyaev, 1973). This model also assumes that radiation cooling in the disc is very efficient. All the locally dissipated accretion power is radiated away on the spot in the form of blackbody radiation. As a consequence, the disc is cold and geometrically thin. From these two assumptions, it is possible to determine analytically the full disc structure as a function of the distance r from the compact object. Although there is still no reliable way of calculating α, it turns out that the observable properties of a steady accretion disc are largely independent of α. In particular, the total luminosity of the disc is simply half of the gravitational power at the disc inner radius Ri , Ldisc = 2GM M˙ /Ri . Most of the luminosity comes from the inner part of the accretion flow; 80 per cent of the power is radiated within 10Ri . The temperature profile in the disc is  3/4   1/4 Ri Ri , (1.12) 1− T (R) = T0 r r where  T0 =

3GM M˙ 8πσRi3

1/4

  6 × 107

Ri RG

−3/4 

M˙ c2 LE

1/4 

M M

−1/4 K.

(1.13)

The temperature has a maximum around 3Ri and then decreases with a distance like r−3/4 . This maximum temperature can reach a few keVs in NSs and BHs. The innermost and most luminous part of the accretion disc will therefore produce mostly X-ray radiation. The spectral energy distribution (SED) of the whole accretion disc is constituted of the sum of all the black bodies emitted at different radii in the disc with a different temperatures. Longer wavelengths allow one to probe more distant regions of the accretion flow. We note the standard Shakura-Sunyaev disc model assumes pure Newtonian physics, the general relativistic version of the thin disc model was introduced by Novikov and Thorne (1973). Hot Accretion Flows The thin accretion disc model assumes full energy thermalisation, and this implies a low temperature, a high density and a small disc scale height of H/R ∼ 10−3 – 10−4 . If, instead, the disc is hot, then the gas pressure makes the scale height larger,  0.1–0.5, and this reduces the density. Consequently, the radiation cooling is less efficient and a high temperature can be maintained. Indeed, the density becomes so small that the collision time-scales between electrons and ions of the plasma can be long compared to the accretion timescale. Then the protons acquire most of the gravitational energy through viscous heating, but this energy can only be radiated efficiently by the electrons. Since electrons and protons are decoupled, the

8

Julien Malzac

two populations end up having very different temperatures. The protons can reach ∼1012 K while the electrons are much colder (and yet very hot), ∼109 K in the innermost part of the accretion flow. In this kind of hot accretion flow, the accretion energy is not radiated away locally; it can be carried inward with the flow until it is swallowed by the BH or hits the surface of the compact star. These advectiondominated accretion flows (Ichimaru, 1977; Narayan and Yi, 1994) produce mostly non-thermal Comptonised radiation (see Section 1.3.4). Hot accretion flows have been extensively studied. Analytic solutions have been found also taking into account the effects of convection; convection-dominated accretion flows (CDAF, Narayan et al., 2000) and outflows; advection-dominated inflow–outflow solution (ADIOS, Blandford and Begelman, 1999). Hot accretion flow have also been found in numerical simulations (Stone et al., 1999). Jet Launching from Accretion Flows Accreting NSs and BHs (an perhaps also WDs) can launch highly collimated jets of magnetised plasma that travel at near light speed, carrying away a significant fraction of the accretion power (Mirabel and Rodr´ıguez, 1994; Fender and Gallo, 2014). These jets propagate over large distances and can have an enormous impact on the surrounding medium over distance scales that are far out of the gravitational reach of the BH itself. The details of the formation of those jets are unclear; none of the models and simulations take into account all the physics. But all models require magnetic fields. The mechanism proposed by Blandford and Payne (1982) involves magnetic field lines threading the accretion disc. Provided that the magnetic field lines are sufficiently inclined with respect to the disc, the centrifugal force can throw away some of the accreting material, which remains tied to the rotating magnetic field lines like a bead on a wire. Then the jet needs to somewhat collimate, and this requires large-scale coherent magnetic field structures. Another mechanism involves electromagnetic extraction of energy from a Kerr BH. Indeed, Penrose (1969) and Christodoulou (1970) have shown that up to 30 per cent of the mass energy of a maximally spinning BH can be theoretically extracted. Then, Blandford and Znajek (1977) have shown, that an accretion disc can allow this energy to be extracted and drive powerful jets. The magnetic field carried by the accreted gas remains threaded through the BH horizon. The frame of the field lines is dragged along with the rotation of the BH. These rotating field lines induce an electromagnetic force that accelerates charged plasma at relativistic speeds along the axis of rotation. Due to the radial component of the field, the particles spiral as they leave. Effects of the Magnetic Field of a Compact Star on the Accretion Flow Magnetic fields become dynamically important close to the compact object at a distance called the Alfv´en radius, Ra . At Ra , the magnetic energy is comparable to the kinetic energy of the infalling gas: ρv 2 /2  B 2 /8π. For the purpose of simple estimates, we can assume that  the accreting gas is in free 2fall in a spherical geom2GM/Ra and ρ = M˙ /(4πRa vff ). Assuming a dipole etry, so that v  vff = magnetic field around a WD or NS of radius R : B(Ra ) ∼ Bp R3 /Ra3 , we then obtain 

Bp Ra  30 km 109 G

 47 

M˙ −8 2 × 10 M yr−1

− 27 

M M

− 17 

R 10 km

 12 7 .

(1.14)

Radiation Processes and Models

9

The effects of the magnetic field are important only if Ra > R , which translates into the following condition for the surface magnetic field: − 54   12   14  M˙ M R 8 Bp > 1.5 × 10 G. (1.15) 2 × 10−8 M yr−1 M 10 km As discussed in Sections 1.2.1 and 1.2.2, in the case of NSs, this condition will be easily verified, and the magnetic fields of the star will interfere with the accretion process. At Ra , the magnetic field can force accreting material in corotation with the compact star. If the spin period of the compact star is longer than the orbital period at Ra , the centrifugal forces cannot balance gravity anymore, and the material flows along magnetic field lines onto the magnetic poles of the compact star. This forms an X-ray pulsar if the compact star is a NS and a polar in the case of a WD. The magnetic field transfers angular momentum from the accretion flow to the compact star, exerting an effective spin-up torque on the star. If the spin period of the compact star is shorter than the orbital period at Ra , the corotation implies that the centrifugal forces overcome gravity; accretion is stopped. This is the so-called propeller regime in which angular momentum is transported from the compact star to the ‘accretion’ flow. In this case, the magnetic field exerts a spin-down torque on the star. The spin of the compact star therefore evolves through propeller and accreting regimes to reach an equilibrium in which the spin period of the star is equal to the orbital period at Ra ,  67   18  − 37  − 57  7 M˙ Bp M R Peq  3 ms . (1.16) 109 G 2 × 10−8 M yr−1 M 10 km This equation shows that accretion will spin up a NS up to spins of a few ms. Such sources are observed; these are the millisecond X-ray pulsars (Wijnands and van der Klis, 1998; Harding, 2013; Tauris, 2015). Compact Stars with a Weak Magnetic Field: Boundary Layer Many compact stars appear to have a magnetic field below the threshold defined by Equation 1.15. Indeed, accretion seems to cause dissipation of the magnetic field, which may be considerably reduced in the course of the history of the accreting compact star. In this case, the accretion disc can extend undisturbed very close to the surface of the star. There is, however, a region between the accretion flow and the star where the accreting gas has to decelerate from the orbital velocity to the rotation velocity of the star, spinning up the compact star in the process. This region is called the boundary layer (BL). Since most of the kinetic energy of the gas is dissipated in the BL, the luminosity of the BL is comparable to the 2 /2 = GM M˙ /(2R ) ∼ Ldisc . total luminosity of the accretion disc: LBL ∼ M˙ vK Matter has a significant latitude velocity component in the boundary layer, spreading above the compact star surface and decelerating due to friction like a wind above the sea (Inogamov and Sunyaev, 1999). For sources with luminosity greater than 5 per cent Eddington, the local radiation flux is Eddington. In this regime, the spreading layer temperature is independent of luminosity. Emission models predict a temperature of the order of 2.5 keV, which was confirmed by observations. The size of the belt must increase with accretion rate/luminosity. For L ∼ LE , the spreading layer covers the whole surface of the compact star (Gilfanov and Sunyaev, 2014).

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Julien Malzac

1.3. Radiation Processes From radio waves to gamma-rays, compact objects radiate over the whole electromagnetic spectrum. This radiation can be emitted through thermal radiation (i.e. blackbody emission related to the temperature of the object) or non-thermal processes like synchrotron or inverse Compton. Synchrotron radiation is produced by very energetic charged particles spiralling around magnetic field lines. Relativistic particles accelerated in shocks in the jets produce synchrotron emission mostly in the radio, sub-mm and infrared (IR) bands. A similar process, called curvature radiation, is related to the propagation of particles along very curved field lines. It can be important in the magnetosphere of neutron stars. An accretion flow can be very hot and contains energetic electrons of temperatures up to a billion K. It also contains many low-energy photons produced by synchrotron radiation or coming from the outer regions of the accretion flow. These photons collide with the hot electrons and gain energy at each interaction. This process is called inverse Compton scattering. If the photon can make several interactions before escaping the hot gas, the process is called Comptonisation. It leads to the production of hard X-ray radiation. Inverse Compton from relativistic particles in the jets may also produce gamma-rays. Understanding the radiation processes allows one to extract the information that is encoded in the radiation received from these objects. It enables to measure many physical parameters of the system, such as temperatures, velocities, magnetic fields, the energy of accelerated particles and their distribution. It also helps to determine the size and geometry of the emitting regions and test the predictions of theoretical models. By simultaneously observing in different bands, we learn about the joint evolution of the different components of the systems (e.g. accretion flow and jets). In this section, I summarise the main properties of the cyclo-synchrotron, curvature radiation, bremsstrahlung, inverse Compton and photon–photon pair production. A more detailed discussion of these radiation processes can be found in classical text books such as those by Rybicki and Lightman (1986); Longair (1992), and Jackson (1999). 1.3.1 Cyclo-Synchrotron Cyclo-synchrotron radiation is the radiation produced by a charged particle accelerated by the magnetic field. In the following, I consider an e− which gyrates around the magnetic field lines with a velocity v (see Figure 1.1). The component of its velocity that is parallel to the magnetic field is a constant. The transverse component of the velocity has a constant modulus and a direction that is rotating uniformly around the magnetic field line. The constant angle α between the velocity and the magnetic field is called the pitch −1/2  is the Lorentz angle. The rotation frequency νB = νL /γ, where γ = 1 − v 2 /c2 factor of the particle, and νL = qB/2πme c = 2.8 × 106 B Hz is the Larmor frequency. The magnetic field B is expressed in G. The radius of the ‘orbit’ around the magnetic field is given by the Larmor radius: rL = p⊥ c/(2πνL ) = 1.7 × 103 p⊥ /B cm, where p⊥ = p sin α = γv sin α/c. Because the particle has a transverse acceleration, it radiates.

Figure 1.1. Emission of an electron spiralling in a magnetic field.

Radiation Processes and Models

11

The cyclo-synchrotron emission is calculated under a number of assumptions. First, the magnetic field must be uniform at the Larmor scale. This implies, for instance, that there is no turbulence at that scale. The magnetic field lines must not be too curved, as this can be the case in the magnetosphere of pulsars, for example. Moreover, as the particle radiates, its kinetic energy and Larmor radius both decrease. The energy loss must be small on the gyration timescale, in other words, we require that the cooling timescale the particle tcool  1/νB . Finally, we will present the results obtained in the classical limit in which the magnetic field is smaller than the critical limit Bc = me c3 /q = 4.4.×1013 G. Above this limit, the particle energies and orbits around the magnetic field become quantised. Power The radiated power of an accelerated particle is given by the Larmor formula, 2q 2 4  2 (1.17) γ a⊥ + γ 2 a , 3 3c where a⊥ and a represent the component of the acceleration that are perpendicular and parallel to the velocity, respectively. In the case of particles girating in the magnetic field a = 0 and a⊥ = νB v⊥ /2π. The radiated power can be written as P = 2cσT UB p2⊥ , where UB = B 2 /8π is the magnetic field energy density, and σT is the Thomson crosssection (see Section 1.3.4). Usually the distribution of pitch angles is isotropic. This is either because the distribution of radiating particles is isotropic, or the magnetic field is isotropically tangled, or both. Then the average over the pitch angles gives P =

P =

4 cσT UB p2 . 3

(1.18)

Therefore, the requirement that the cooling time of a particle tcool  γme c2 /P is longer than the giration timescale, 1/νB , implies a limit on the product of the particle energy and magnetic field γ 2 B < 2q/r02 .

(1.19)

Optically Thin Synchrotron Spectrum If the charged particle moves very slowly around the magnetic field lines, the amplitude of the electric field measured at large distance is proportional to the sine of the angle between the acceleration vector and the direction to the observer. This angle changes with time t as the particle ‘orbits’ the magnetic field and the observer observes a sinusoidal modulation of the electric field: E(t) = sin 2πνB t. The emitted spectrum is proportional to the Fourier power spectrum of the fluctuations of the electric field, which, in this case, is a Dirac delta function. In other words, the observer sees a line at frequency νB . If the particle is moving faster, the emission pattern is modified by Doppler beaming, and harmonics appear in the spectrum. For a relativistic particle, the emission is beamed within an angle 1/γ around the direction of the velocity. When this emission cone crosses the direction to the observer, she measures a briefly increased electric field. The duration of this pulse is δt  1/(νB γ 3 ).3 This pulsed modulation of the electric field at frequency νB is made of multiple lines at frequencies that are multiple of νB and extending up to a frequency νc that is comparable to the timescale of the pulse: νc =

3 3 γ νB sin α. 2

(1.20)

The line of sight remains in the emission cone during 1/(νB γ), the additional γ −2 factor is caused by photon travel time effects. 3

12

Julien Malzac

In the ultra-relativistic case (γ  1), νB vanishes and the spectrum tends toward a continuum. This limit is called synchrotron emission, while the non-relativistic case is called cyclotron emission. In fact, once one considers not only the emission from a unique particle but that of a set of particles with different directions of propagation and pitch angles, the harmonic lines are considerably broadened, even in the mildly relativistic case (see Marcowith and Malzac, 2003, for a general calculation of this case). Equation 1.18 shows that the emission increases quadratically with the momentum of the particle; the radiation induced by the magnetic fields will be more likely to be important in the synchrotron regime, which is therefore the most relevant for the observations. In this limit, the shape of the spectrum radiated by a relativistic particle is given by Rybicki and Lightman (1986) √ 3   ν 3q B dP (α, p, ν) = sin αF , (1.21) dν me c 2 νc where





F (x) = x

K5/3 (z)dz,

(1.22)

x

and Kn (z) is the modified Bessel function of second kind. Then the pitch angle averaged spectrum can be approximated as √   ν 12 3σT cUB dP (γ, ν) = sin αG , (1.23) dν νL 2νc∗ where



 3x 2 2 K4/3 (x) − K1/3 (x) . G(x) = x2 K4/3 (x)K1/3 (x) − 5

(1.24)

1/3 up to the critical In both cases, the spectrum increases with frequency dP dν ∝ ν 3 ∗ 2 frequency νc , (or νc = 2 νL γ in the angle averaged case), where there is a turnover and an exponential cutoff. The emission is therefore peaked around the critical frequency, and both νc and νc∗ scale like Bγ 2 . The condition of slow particle energy losses, given by Equation 1.19, then implies a maximum frequency at which synchrotron emission can be produced. The particles cannot radiate synchrotron emission above this energy because they cool before closing a full orbit. This maximum frequency depends only on fundamental constants and is independent of the magnetic field or particle energy: hνmax = 3 mc2 /αf = 70 MeV. Any emission above 70 MeV cannot be due to synchrotron. Of course in real situations the radiating particles are not monoenergetic but have a distribution  in energy. One common case is a relativistic Maxwellian distribution: N (γ) ∝ γ γ 2 − 1e−γ/θ , where θ is the reduced temperature: θ = kT /me c2 . In this case, the spectrum peaks at hν = 1.5 hνL θ2 , at lower frequencies, the emitted spectrum 1/3 is dP and an exponential cut-off at higher energies. Another interesting case dν ∝ ν corresponds to a power-law particle energy distribution, N (γ) ∝ γ −s , in the range 1/3 2 up to ν = 1.5 νL γmin , where γmin < γ < γmax . Then the emitted spectrum is dP dν ∝ ν dP −α the photon spectral slopes changes to α = (s − 1)/2, ( dν ∝ ν ) until it reaches a high 2 . frequency cutoff at ν = 1.5 νL γmax

Synchrotron Self-Absorption From the point of view of the electron, the emission of cyclo-synchrotron emission constitutes a spontaneous transition between energy levels. As in the case of atomic transitions, the spontaneous emission mechanisms coexist with the opposite transition,

Radiation Processes and Models

13

which happens through absorption of radiation. The transition probabilities are described by the Einstein coefficients and their relations. On can show (see, e.g., Ghisellini and Svensson, 1991) that the absorption coefficient (in cm−1 ) can be expressed as a function  ∞ (γ) dP −1 Hz−1 cm−3 ster−1 ): of the emissivity jν = 1 N4π dν dγ (erg s αν =

1 dγpjν 1 . 2me ν 2 pγ dγ

(1.25)

where αν gives the probability for a photon travelling in the magnetised plasma to be absorbed per unit of distance crossed. The specific intensity Iν (erg s−1 Hz−1 cm−2 ster−1 ) of the synchrotron radiation escaping a homogeneous medium of size L is given by the resolution of the radiative transfert equation: jν  1 − e−τsa , (1.26) Iν = αν where τsa = αν L is the synchrotron self-absorption optical depth. There are two main asymptotic regimes depending on the value of τsa . For τsa  1, the medium is optically thin, Iν  jν L, while for τsa  1, the medium is optically thick and the synchrotron emission is self-absorbed, Iν = jν /αν . As τsa decreases with photon frequency, the lowfrequency part of the spectrum is usually self-absorbed, while at higher frequency, the emission is optically thin. The transition between optically thin and thick emission occurs at frequency νt , such that by τsa (νt )  1. The value of νt increases with electron density (or Thomson depth τT = ne σT L). The exact value of the turnover frequency, as well as the shape of the spectrum in the self-absorbed regime, depends on the shape of the electron energy distribution. For a thermal distribution, the low-frequency self-absorbed part of the spectrum is a black body in the Rayleigh-Jeans regime, i.e., Iν ∝ ν 2 . For a powerlaw energy distribution, the turnover frequency can be estimated analytically as νt ∝ s+2

2

B s+4 τTs+4 , and the low-frequency self-absorbed spectrum is strongly inverted: Iν ∝ ν 5/2 . 1.3.2 Curvature Radiation Curvature radiation is emitted by relativistic particles moving along strongly curved magnetic field lines. This process is important in the magnetosphere of pulsars, where the field lines are curved and the magnetic field is so strong that the transverse component of the velocity is radiated very quickly, and then the particles just follow very closely the magnetic field lines. In this case, the acceleration is perpendicular to the fields lines and the emission is beamed in the direction of the trajectory. The observer sees a pulse every time the line of sight intercepts the beam of radiation travelling with a nearly circular motion along the magnetic field lines. This situation is in fact equivalent to synchrotron emission. The radiated power and spectrum for one particle is, therefore, given by the same formulae as for synchrotron with the Larmor frequency replaced by the rotation frequency of the relativistic particles, νL = c/2π R, where R is the curvature radius of the field lines. The monoenergetic spectrum of curvature radiation has, therefore, the same form as that of synchrotron radiation. It varies like the cube root of frequency at low frequencies and falls exponentially above the peak frequency, ν  1.5 γ 2 c/2π R. Then the full observed spectrum depends on the energy and spatial distribution of the particles and magnetic fields in the pulsar magnetosphere. However, for likely values of the parameters, it turns out that the intensity produced by the incoherent sum of a single-particle curvature radiation is not enough to explain the very high brightness temperatures of the radio emission of pulsars. It is necessary to assume that there are charged bunches containing N ∼ 1015 electrons travelling together at the same velocity and concentrated in a same volume of dimension smaller than the emitted wavelength. Then these electrons behave as a unique

14

Julien Malzac

particle of charge qN . Since, according to the Larmor formula, the radiated power scales like the square of the charge of the particle (Equation 1.17), the emission is amplified by a factor, N , with respect to the sum of the individual emission of the N particle. This amplification is due to the fact that the N electrons radiate in coherence. Such coherence effects may be responsible for the observed emission from the magnetosphere of pulsars. 1.3.3 Bremsstrahlung Bremsstrahlung is a radiation associated to the acceleration (or, in this case, braking) of a free electron interacting with the electric field of an ion. The ion is at rest, and the electron, which travels with a velocity v, is deflected. This deflection is associated with a transverse acceleration and a distant observer sees a pulse of electric field. The duration of the pulse is of the order of the duration of the interaction Δtint  b/v, where b is the impact factor (i.e., the shortest distance at which the electron approaches the ion). The photon spectrum is given by the power spectrum of the pulse, which, in the case of random incoherent pulses, is flat up to the cutoff frequency νcut  (2Δtint )−1 and exponentially cut off above that. The amplitude of the flat part of the spectrum for one particle is Iν =

8Z 2 e6 . 3πc3 m2e v 2 b2

(1.27)

In a realistic situation, we have to integrate this result not only on the energy distribution of the electrons but also on the distribution of their impact factors, since intensity and maximum energy of the spectrum for one particle both diverge when b vanishes, an artificial minimum impact factor has to be assumed. This minimum impact factor can be estimated based on physical arguments, but the full answer requires a quantum physics treatment, which introduces correction factors known as Gaunt factors. The most common situation is the case of Maxwellian electron energy distribution of temperature T . Assuming a fully ionised hydrogen plasma, the bremsstrahlung emissivity is jν ∝ ne np T −1/2 exp (−hν/kT ). The emitted spectrum is, therefore, flat up to a frequency corresponding to a photon energy comparable to the thermal energy of the electrons. The total power represents a cooling term for the plasma, √ J(T )  2.4 × 10−27 g¯f f (T )ne np T erg s−1 cm−3 , (1.28) where g¯f f (T ) is a Gaunt factor of the order of unity. Since the radiated power is proportional to square of the plasma number density, bremsstrahlung will be a very efficient cooling mechanism for dense plasma and much less efficient for diluted plasmas. In a way that is very similar to cyclo-synchrotron radiation, bremsstrahlung emission can be self-absorbed. For a thermal plasma, the emission is Rayleigh-Jeans (Iν ∝ ν 2 ) below the turnover frequency. 1.3.4 Compton Compton scattering plays an important role in the formation of the high-energy emission of compact objects. In most elementary physics textbooks, it is described as the scattering of a photon by an electron at rest. This results for the photon in a change of direction of propagation and loss of energy to the electron. In the context of high-energy astrophysics, the electron can be energetic and moving at relativistic speeds. In this case, the electron can transfer a significant amount of energy to the photon that may be up-scattered into the X-ray or even gamma-ray domain. This process is then called inverse Compton. It is, however, exactly the same process as the usual Compton diffusion but considered in a

Radiation Processes and Models

15

different reference frame. The conservation of momentum and energy during the collision gives the general formula for the change of energy for the photon: 1 − μβ ν = ,  ν 1 − μ β + (hν/γme c2 ) (1 − cos α)

(1.29)

where β = v/c is the velocity of the electron as a fraction of the speed of light, μ and μ are the cosine of the angle between the directions of the electron and photon momenta respectively before and after the interaction, ν and ν  are the photon frequencies respectively before and after the collision and α is the angle between the photon directions before and after interaction. If the photon is scattered by an electron at rest, this formula simplifies to the classic formula for Compton exchange: 1 ν = . ν 1 + (hν/me c2 ) (1 − cos α)

(1.30)

The photon loses energy in the process (electron recoil effect). If the initial photon energy is small (hν  me c2 ), hν Δν = − (1 − cos α). (1.31) ν mec2 If, on the contrary, the electron is moving fast, the change in photon energy can be seen as a Doppler effect caused by the change of reference frame. In the electron rest frame, the frequency of the incident photon is νo = γν(1 − μβ), and if hνo  me c , the change of photon energy in this frame is negligible Then going back to the lab frame, one gets 2

ν =

νo 1 − μβ νo = = ν .   γ(1 − μ β) γ(1 − μ β) 1 − μ β

(1.32) νo

= νo .

(1.33)

The maximum energy gain is obtained for a frontal collision with back scattering of the photon. Cross-Section The number of scattering events per unit time and unit volume in the lab frame for a monoenergetic beam of electrons interacting with a monoenergetic beam of photon is dn = ne nν σKN c (1 − βμ) , (1.34) dt where ne and nν are the respective electron and photon densities and, β it the reduced speed of the electrons and μ is the cosine of the angle between the two beams, as above. σKN is the Klein–Nishina cross-section    8 1 2πre2 1 8 4 + σKN = 1 − − 2 ln (1 + x) + − , (1.35) x x x x 2(1 + x)2 2 where x x =

2hν γ(1 − μβ), me c 2

(1.36)

and re = e2 /me c2 is the classical electron radius. In the non-relativistic limit (x  1), the Klein–Nishina cross-section reduces to the Thomson cross-section: σKN  8π re2 /3 = σT . This non-relativistic limit corresponds to the Thomson regime. Around x ∼ 1 the

16

Julien Malzac

cross-section starts to decrease with increasing x, in the ultra-relativistic regime: x  1, the cross-section decreases as σKN 

3 σT [1 + 2 ln(1 + x)] . 8 x

(1.37)

This strong reduction makes inverse Compton highly inefficient in the Klein–Nishina regime. Finally, the Compton cross-section scales like m−2 e . The scattering cross-section on charged particles other than an electron or positron are, therefore, smaller by a factor of at least (me /mp )2 ∼ 10−7 . These processes are, therefore, completely negligible. Radiated Power and Amplification In the Thomson regime (γhν  me c2 ), and neglecting the recoil effect (p2  hν/me c2 ), the total power radiated by one electron in the field of soft photons of energy density Uph is: P = f rac43σT cp2 Uph ,

(1.38)

where p = γβ is the reduced momentum of the electron, and Uph is the energy density of the target photon field (Rybicki and Lightman, 1986). It is interesting to note that the radiated power in this limit is independent of the actual energy of the soft photons (which do not have to be monoenergetic), or their spectrum. From Equations 1.38 and 1.34, the average photon energy amplification in the course of one interaction is given by 4 ν  = 1 + p2 . ν 3

(1.39)

Spectrum from Single Scattering of a Power-Law Energy Distribution of Relativistic Electrons Let us consider an isotropic distribution of photons of typical energy ν0 , scattering off a power-law energy distribution of electrons (i.e., N (γ) = N0 γ −s in the range γmin < γ < γmax ), in the Thomson regime. After interaction with an electron of momentum, p, the photons have an average energy of ν = ν0 (1 + 4p2 /3)  ν0 4 γ 2 /3. As a first approximation, we can neglect the dispersion of the scattered photon energies and assume that all the power radiated by the electrons of energy γ is emitted at the average scattered photon energy ν. The emitted spectrum (erg s−1 Hz −1 cm−3 ) is then 2 N0 4 dγ jν (ν) = σT cp2 Uph N (γ)  σT cUph 3 dν 3 ν0



3ν 4ν0

−( s−1 2 ) (1.40)

2 2 /3 and νmax  4ν0 γmax /3. The result is, for ν, comprised between νmin  4ν0 γmin −α therefore, a power law (jν ∝ ν ) extending up to νmax with a slope α = (s−1)/2, which is identical to that obtained in the case of synchrotron radiation. If instead, the electrons at γmax emit in the Klein–Nishina regime, the maximum energy of the Compton radiation is limited by the energy of the electrons hνmax = γmax me c2 . However, due to the sharp decrease of the cross-section in the KN regime the spectrum deviates from a power law and is strongly suppressed at photon frequencies that are above νKN  m2e c4 /(h2 ν0 ).

Multiple Compton Scattering: Thermal Comptonisation In realistic situations, the photons can undergo zero, one, or several successive scattering, before escaping from the energetic electron cloud. In the Thomson regime,

Radiation Processes and Models

17

the probability of interaction is determined by the Thomson optical depth: τT = ne σT R, where ne is the electron number density and R is the typical dimension of the medium. The probability for a photon to cross the medium without scattering any electron is exp(−τT ). In the optically thin case (τT < 1), and τT represents the average number of scatterings before escape (or, in other words its probability of interaction). For τT > 1, the photons interact at least once, and the average number of scattering events before escape is ∼ τT2 . Multiple Compton scattering implies non-negligible optical depth, and since the emitting region in compact objects is necessarily small, this also implies a significant density. If the density is large, most of the particles are thermalised. So the most relevant case in which multiple Compton scattering is important is the case of soft photons up-scattered in a hot thermal plasma of electrons (i.e., the electrons have a Maxwellian energy distribution). This process is called thermal Comptonisation. The Comptonised emission depends not only on the Thomson depth but also on the temperature, Te , of the electrons. As long as the photon energy hν is smaller than the typical energy of the electrons ∼ kTe , the photons gain energy at each interaction. In the Thomson limit, the average fractional energy gain can be approximated as Δν = 4θe + 16θe2 , ν

(1.41)

where θ = kTe /me c2 (Rybicki and Lightman, 1986). Usually one defines the Compton y parameter as the product of this average fractional energy exchange times the average number of interaction:   (1.42) y  4θ + 16θ2 τT + τT2 . Then one can show that the energy of the incident photon is amplified by a factor: −1  y y hν0 2 −1 4θ + 16θ . (1.43) A = e 1+e me c 2 In the limit of small y, the amplification factor reduces to A  1 + y. While in the limit of large y, the Comptonisation process is said to be saturated, and most of the radiation is emitted at an energy ∼ (4θ + 16θ2 )me c2 , which is comparable to average energy of the electrons. The escaping spectrum is constituted of the sum of contributions from the photons that have escaped without interaction, plus those that have interacted once before escaping, plus those that have interacted twice, trice, etc. Each of these spectral components, which are called Compton orders, forms a broad hump. The average photon energy of each hump increases with the number of interactions until it becomes comparable to the thermal energy of the electrons. These humps are, however, apparent in the spectrum only in the optically thin regime τT  1. In practice, the observed τT are often of the order of unity or larger. In this case, the Compton orders are too close and too blended to be individually distinguishable. They combine to form the power-law spectrum, extending from the typical energy of the seed photons up to the energy of the thermal electrons (Sunyaev and Titarchuk, 1980). At higher energies, the spectrum drops off exponentially. The photon index Γ (defined as Fν ∝ ν 1−Γ ) can be approximated as: Γ  C(A − 1)δ ,

(1.44)

with C  2.33 and δ  1/6 depending weakly on the temperature of the electrons, the average energy of the seed photons and the exact geometry of the comptonising plasma (Beloborodov, 1999). For instance, for electron temperatures in the range of 50–200 keV,

18

Julien Malzac

as mostly observed in X-ray binaries, and for blackbody seed photons of temperature kTbb = 0.15 keV, numerical simulations give C = 2.19, δ = 2/15 (Malzac et al., 2001). By modelling observed Comptonised spectra, it is possible to infer the temperature of the electrons and the Thomson depth. Note that the X-ray spectral slope Γ depends on both parameters (mostly via y). There is, therefore, a degeneracy that can be broken only if one observe simultaneously at higher energies and measure the Comptonisation cutoff, which is usually found around 100 keV (see Section 1.4).

1.4. Model for the Accretion Flow Accreting BHs in X-ray binary systems (hereafter BHBs) produce strongly variable radiation over the whole electromagnetic spectrum. From radio to IR bands, non-thermal emission is often detected and usually associated to synchrotron emission from very high energy particles accelerated in jets (see, e.g., Chaty et al., 2003, hereafter C03; Gandhi et al., 2011, hereafter G11). The same particles can occasionally produce detectable emission in the GeV range (Fermi LAT Collaboration et al., 2009; Tavani et al., 2009; Malyshev et al., 2013). The thermal emission from the accretion disc can be studied in the optical to soft X-ray bands. In the hard X-ray domain (above a few keV), the emission is dominated by a non-thermal component, the nature and origin of which is strongly debated, and which is associated with the emission of a hot and tenuous plasma located in the direct environment of the BH. This plasma is generically called the ‘corona’, drawing analogy from the solar corona, although the formation of the BH corona, its power feeding and emission processes are most likely very different from that of the sun. BHBs also show strong aperiodic and quasi-periodic time variability down to the millisecond scale in X-rays, optical and IR bands (see, e.g., Gandhi et al., 2010, 2008; herafter C10 and G08, respectively). They constitute prime targets for HTRA. Most BHBs are transient X-ray sources detected during outbursts lasting from a few months to a few years in duration. During outbursts, their luminosity can increase by many orders of magnitude to reach values close to the Eddington limit (LE  1039 erg/s for a 10 M BH) before going down, back to quiescence. These sources, therefore, constitute a unique laboratory to investigate how the physics of the accretion and ejection depends on the mass accretion rate onto the BH. During these outbursts, not only the luminosity, but also the broadband spectral shape changes drastically as a source evolves through a succession of X-ray spectral states, showing very different spectral and timing properties. Nevertheless, those X-ray spectral states can all be described in terms of the same spectral components arising from the thermal accretion disc, X-ray corona and jets. Their rich timing phenomenology (see Chapter 4 by T. Belloni in this volume) is poorly understood. In this section, I will focus on a model that can explain qualitatively most of the timing and spectral features arising from the accretion flow. 1.4.1 Spectral States The spectral evolution during outbursts is usually studied using hardness-intensity diagrams (HID). One of such diagrams is shown in Figure 1.2. There are two main stable spectral states, namely the soft and the hard state, corresponding respectively to the left- and right-hand-side vertical branches of the HID (see Figure 1.2). The other spectral states are mostly short-lived intermediate states associated with transitions between the two main spectral states. Some X-ray spectra of Cyg X–1 in its soft, hard and intermediate between the soft and hard states are shown in Figure 1.3. The soft spectral state is observed at luminosity levels ranging from approximately 10−2 to a few 0.1LE . In this state, the high-energy emission is dominated by the soft

Radiation Processes and Models

19

Figure 1.2. Typical track followed by the BHB GX 339–4 in the HID diagram during an outburst. The various sketches illustrate a possible scenario for the evolution of the geometry of the accretion flow in the context of a truncated disc model in which the central hot accretion flow takes the form of a jet emitting disc (Courtesy: P.O. Petrucci).

100

(g–1) dt/dg

10–2 10–4 10–6 10–8 10–10 10–1

100

101

102 –1

103

104

105

106

2

(g ) m,e (keV)

Figure 1.3. Left: Joint INTEGRAL/JEM-X, IBIS and SPI energy spectra of Cyg X–1 during four different spectral states fitted with the magnetised hybrid thermal-non-thermal Comptonisation model BELM. Right: Energy distribution of the Comptonising electrons obtained in the best fit models of the left-hand-side panel. These fits set an upper limit on the amplitude of the magnetic field in the X-ray corona at about 105 G in the harder states and 107 G in softer states (Del Santo et al., 2013).

thermal multi-blackbody disc emission peaking around 1 keV; for this reason, this state is also called ‘thermal dominant’ by some authors (Remillard and McClintock, 2006). The intense thermal radiation and hot disc temperature (∼ 1 keV) are consistent with those of a standard geometrically thin, optically thick accretion disc (Shakura and Sunyaev, 1973) extending down very close to the BH. The coronal emission is usually very weak, forming a non-thermal power-law tail above a few keV. The geometry of the corona is unconstrained in this state, but it is generally assumed to be constituted of small-scale magnetically active regions located above and below the accretion disc (Galeev et al., 1979). In these regions, the energetic electrons of the plasma up-scatter the soft X-ray photons coming

20

Julien Malzac

from the disc into the hard X-ray domain. Due to the weakness of the non-thermal features, the soft state is perfect to test accretion disc models and measure the parameters of the inner accretion disc. In particular, the detailed model fitting of the thermal emission of the accretion disc indicates that the inner radius of the disc is a constant and is independent of the luminosity of the system (Gierli´ nski and Done, 2004). This constant inner radius is believed to be located at the innermost stable circular orbit (ISCO) that is predicted by the theory of general relativity and below which the accreting material must fall very quickly across the event horizon. As the size of the ISCO is very sensitive to the spin of the BH, this offers an opportunity to constrain the spin of BHs in X-ray binaries. This may also be used to validate the spin measurements made using the relativistic iron Kα line profile, which remain the only method that can be used in AGN. This is a difficult task because measuring the disc inner radius accurately requires very detailed disc emission models taking into account the general relativistic effects as well as non-blackbody effects through detailed disc atmosphere models (Davis et al., 2005, 2006). This method also requires the knowledge of the distance and inclination of the system. Nevertheless, the most recent BHB spin estimates obtained from disc continuum and line fitting are converging (see Middleton, 2016, for a recent review of these issues). Although the soft state is associated to strong Fe absorption lines indicative of a disc wind (Ponti et al., 2014); there is no evidence so far of any relativistic jet component in soft state (see Russell et al., 2011). The hard state is observed at all luminosities up to a few 0.1 LE . In this state, the emission from the accretion disc is much weaker and barely detected; the inferred temperature of the inner disc is also lower (Tin ∼ 0.1 keV) than in soft state (Cabanac et al., 2009; Dunn et al., 2011; Plant et al., 2015). The X-ray emission is dominated by a hard power law with photon index Γ in the range, 1.5–2.1, and a high-energy cutoff around 50–200 keV. This kind of spectra is very well represented by thermal Comptonisation models with Thomson depth 1–3 and electron temperatures in the range of 20–200 keV. In addition, there is evidence for a Compton reflection component: the illumination of the thin accretion disc by the Comptonised radiation leads to the formation of a broad hump peaking around 30 keV in the high-energy spectrum and a prominent line around 6.4 keV caused by iron fluorescence (Fe Kα ) line. The amplitude R of the reflection component is defined relatively to the reflection produced by an isotropic X-ray source above an infinite slab. In the hard state, R is usually small, R < 0.3. Another characteristic of the hard state is the presence of radio emission with a flat of weakly inverted spectrum that is the signature of steady compact jets. Such compact jets are probably the most common form of jets in X-ray binaries; they appear to be present in all BH and NS binaries when in the hard X-ray spectral state. Their SED extends from the radio to the mid-IR (e.g., Fender et al., 2000; Corbel and Fender, 2002, hereafter C03; Migliari et al., 2010). The measurements of the hard state X-ray spectrum put interesting contraints on the geometry of the corona in the hard state. For instance, an isotropic corona made of active regions located above the disc (as that is envisioned for the soft state) would produce significantly stronger reflection features (R ∼ 1), which may also be enhanced by general relativistic light bending (Miniutti and Fabian, 2004). In addition, most of the illuminating radiation would be absorbed in the disc (typical disc X-ray albedo is ∼ 0.1) and re-emitted at lower energy in the form of nearly thermal radiation. This geometry would thus imply a strong thermal component with an observed flux that would be at least comparable to that of the corona. If the corona has a significant Thomson depth, as inferred by the observations (τT ≥ 1) and is radially extended above the disc, the reflection and reprocessing features might be smeared out by Compton scattering (Petrucci et al., 2001). However, detailed Monte Carlo simulations have shown that, in this case, the strong reprocessed emission from the disc illuminating the corona

Radiation Processes and Models

21

would then cool down its electrons (via inverse Compton) to temperatures that are much lower than observed (Haardt and Maraschi, 1993; Haardt et al., 1994; Stern et al., 1995; Malzac et al., 2001) and then much steeper X-ray spectra would be observed. In fact, the high electron temperature and weakness of the reflection and disc features suggest that the corona and the disc face each other with a small solid angle. In this context, a geometry that is favoured is that of the truncated disc model, where the accretion disc does not extend down to the ISCO but is truncated at some larger distance from the BH. As the disc does not extend deeply into the gravitational potential of the BH, its temperature is lower. As a consequence, the blackbody emission is much weaker than in soft state. The corona is constituted of a hot, geometrically thick accretion flow that fills the inner hole of the accretion disc (Esin et al., 1997; Poutanen et al., 1997). This hot flow Comptonises the soft photons from the disc and/or internally generated synchrotron photons to produce the hard X-ray continuum. The outer disc receives little illumination from the corona and produces only weak reflection and reprocessing features. This scenario explains qualitatively many of the spectral and timing properties of BHBs in the hard state, such as the correlation between X-ray spectral slope and reflection amplitude (Zdziarski et al., 1999) or quasi-periodic oscillations (QPO) frequencies simply by assuming that the disc inner radius changes for instance with luminosity (Done et al., 2007) The location of the disc truncation radius could be determined by a disc evaporation/condensation equilibrium (Meyer et al., 2000; Qiao and Liu, 2012). Figure 1.2 shows a sketch of the possible evolution of the geometry of the accretion flow during an outburst. Observationally, however, due to the weak disc features, the actual transition radius is very difficult to measure accurately. The most recent estimates suggest that in the bright hard state, the inner disc is actually at a few gravitational radii at most (Miller et al., 2015; Parker et al., 2015; Fabian et al., 2015). These results lead one to question whether the disc is actually truncated at all. But there is evidence that the disc recedes at low luminosity, and the truncation radius could be located much farther out (see Plant et al., 2015). At least at high luminosities, the hot flow of the truncated disc model cannot be a standard advection-dominated accretion flow (ADAF; Narayan et al. 1997). Indeed, ADAF theory does not predict hot flows that are as bright and as optically thick as observed (τT > 1) in the bright hard state. The large Thomson depth implies a large density, which causes too much cooling for a hot solution to exist at τT > 1. A possible fix to the model was proposed by Oda et al. (2010, 2012). It consists in assuming the presence of strong magnetic fields in the hot flow. The magnetic field is strong enough to support the hot flow with magnetic pressure rather than the usual thermal pressure. This increases the scale height of the flow and decreases its density, allowing for solutions with larger τT . In this context, the Thomson depth, electron temperature and coronal luminosity observed in a typical hard state of Cyg X–1 require the magnetic pressure to be a few times larger than thermal pressure (Malzac and Belmont, 2009).

1.4.2 Non-Thermal Particles In the soft state, there are indications (at least in the prototypical source Cyg X–1) that the hard X-ray emission extends as a power law at least up to a few MeV (McConnell et al., 2002). The absence of cutoff below 1 MeV indicates that the Comptonising electron distribution of the soft state corona cannot be purely thermal. Producing such gamma-ray emission through inverse Compton requires a power law like distribution of electrons extending at least up to energies of order of 10–100 times the electron rest mass energy. In the hard state, an excess with respect to a pure Comptonisation model

22

Julien Malzac

is detected in all bright sources, (e.g., Joinet et al., 2007; Droulans et al., 2010; Jourdain et al., 2012a) which is also interpreted as the signature of a population of non-thermal electrons in the corona. These findings triggered the development of hybrid thermal/non-thermal Comptonisation models. In these models, the Comptonising electrons have a similar energy distribution in all spectral state, i.e., a Maxwellian with the addition of a high energy power-law tail (Poutanen and Coppi, 1998; Coppi, 1999). These models have been extremely successful at fitting the broadband high-energy spectra of BHBs. Figure 1.3 shows examples of INTEGRAL spectra fit with an hybrid model. The transition from mostly thermal (in hard state) to mostly non-thermal (in soft state) emitting electrons is understood in terms of the radiation cooling. As a source evolves towards the soft state, the corona intercepts a much larger flux of soft photons from the disc. In this more intense soft radiation field the Compton emission of the hot electrons of the plasma is stronger. They radiate their energy faster. This makes the electron temperature significantly lower in softer states. As a consequence, the Compton emissivity of the thermal particles is strongly reduced in the soft state, and the emission becomes dominated by the higherenergy non-thermal particles. In addition, the Thomson depth of thermal electrons is found to be smaller in soft state, possibly because most of the material in the corona has condensed into the disc or is ejected during state transition. This further decreases the luminosity of the Maxwellian component of the plasma, making it barely detectable in soft state. Attempts to model the spectral evolution during spectral state transitions have confirmed that the huge change in the flux of soft cooling photon from the disc illuminating the corona drives the spectral changes of the corona (Del Santo et al., 2008). The most recent version of the model also includes the radiative effects of magnetic field on the lepton energy distribution (Belmont et al., 2008; Vurm and Poutanen, 2009). Internally generated synchrotron photons (typically in the optical/ultra-violet (UV) range) constitute a source of seed photons for the Comptonisation process. In addition, the process of synchrotron self-absorption provides an efficient coupling between leptons, which can quickly exchange energy by rapid emission and absorption of synchrotron photons, leading to very fast thermalisation of the lepton distribution on timescales comparable to the light-crossing time. In fact, it is not necessary to assume the presence of thermal Comptonising electrons in the first place. The heating mechanism could be purely non-thermal, e.g., accelerating all electrons into a power-law energy distribution. The thermalising effects of the so-called synchrotron boiler (in addition to Coulomb collisions) naturally leads to an hybrid thermal/non-thermal particle energy (Malzac and Belmont, 2009; Poutanen and Vurm, 2009). One important effect of the presence of non-thermal electrons in the corona in presence of magnetic field is the production of a strong synchrotron radiation component. The spectral break associated with the transition from optically thick to optically thin synchrotron emission is expected to be located in the optical/IR range. Thermal electrons also produce synchrotron radiation but at a much lower level. Wardzi´ nski and Zdziarski (2001) have shown that the addition of a small fraction of non-thermal electrons in the corona (carrying a fraction < 10 per cent of the total internal energy of the distribution) can increase the optical synchrotron flux by several orders of magnitude. In the framework of the truncated accretion flow model, the hybrid thermal non-thermal distribution of electrons may dominate the optical and IR flux (in hard state) through synchrotron emission, while at the same time explain the hard X-ray through Comptonisation (Veledina et al., 2013a). 1.4.3 X-ray Timing X-ray binaries harbour a strong rapid X-ray variability that presents a very complex and richly documented phenomenology (see Chapter 4 by T. Belloni these proceedings).

Radiation Processes and Models

23

On the other hand, observations show that most of the X-ray variability occurs on timescales ranging from 0.1 to 10 s. Strong variability is also observed on longer timescales. In persistent sources, it was possible to construct long-term X-ray Fourier power-density spectra (PDS) showing a nearly 1/f noise extending up to timescales of years (Gilfanov, 2010). In comparison, there is virtually no variability on timescales shorter than 0.01 s. The main problem that any model must overcome is a timescale problem. Indeed, most of the high-energy photons (and variability) must originate deep in the potential from a region of size R < 100RG ∼ 1500 km. The timescales in this X-ray emitting region are controlled by dynamical timescales of the accretion flow such as the Keplerian timescale    M R tK = 0.3 s, (1.45) 10M 50RG or the viscous timescale (comparable to the accretion timescale)  −2 H tK tvis = , R 2πα

(1.46)

where α is the usual viscosity parameter α ∼ 0.1 (see Section 1.2.3). In the case of a thin gas pressure dominated accretion disc: H/R ∼ 10−2 and tvis ∼ 104 tK ∼ 103 s, while in the case of a hot flow: H/R ∼ 0.3 an tvis ∼ 10 tK ∼ 1 − 10 s. This means that high Fourier frequencies can easily be produced in this region. But the longest observed timescales are too long to be produced in the region of main energy release. They must be generated in the outer parts of the accretion flow. A possibility could be that fluctuations are generated a large distance and propagate inward. For instance, Lyubarskii (1997) postulates fluctuations in viscosity on the local inflow timescale (itself viscous) over a wide range of distances R from the BH and computes the resulting fluctuations of the mass accretion rate at the inner disc radius. For fluctuation amplitudes that are independent of r, this gives a power spectrum p(f ) ∝ 1/f , i.e., slow variations of large amplitude. In this model, the amplitude of the fluctuations generated at a given radius is modulated by the longer fluctuations propagating from the outer region of the disc. For this reason, a linear correlation is expected between the the average flux measured within a given time interval and the root mean square (RMS) amplitude of the variability during the same time-interval. Such RMS-flux correlations are ubiquitous among accreting BH sources and difficult to produce by any other model (see discussion in Uttley et al., 2005). Another intriguing feature of the rapid variability of BH binaries is the existence of Fourier frequency–dependent delays between X-ray energy bands (the hard photons lagging behind the softer photons). Kotov et al. (2001) improved upon Lyubarskii’s model and showed that assuming a harder spectrum close to the BH produces a logarithmic dependence of time lags on energy, as observed. The shape and amplitude of the power spectrum changes drastically with spectral state (again, see Chapter 4 by T. Belloni in this volume). In the soft state, the RMS amplitude of X-ray variability is at most a small percent, while in the hard state it can reach 30 per cent. In the hard state, the PDS shows band-limited noise. An example of observed X-ray PDS in the hard state is displayed in Figure 1.4. It can be approximately represented as flat, up to a break frequency νb ∼ 0.01 Hz, at which the slope changes. Above νb , the power decreases approximatively as 1/f up to a second break at a frequency νh ∼ a few Hz. Above νh , the PDS now decreased as f −2 (or steeper). The observed band-limited noise PDS are usually well described in terms of the sum of four to five broad Lorentzians (e.g., Nowak, 2000; van der Klis, 2006). In the truncated disc scenario, the band-limited noise variability constitutes the variability that is generated due to propagating fluctuations in the hot flow. νb is naturally associated with the viscous frequency at the disc

24

Julien Malzac

Figure 1.4. Left: the observed X-ray PDS of GX 339–4 during the observations presented in G11. Right: the SED measured by G11 compared to a simulated jet SED obtained assuming that the jet Lorentz factor fluctuations have exactly the PDS as the X-ray flux (Drappeau et al., 2015).

inner radius and νh is the viscous timescale at the inner radius of the hot flow, which may differ from the ISCO (Done et al., 2007; Ingram and Done, 2011; Ingram et al., 2015). In addition, a low-frequency QPO (LFQPO) is often observed at some intermediate frequency between νb and νl . This LFQPO is widely believed to be caused by Lense– Thirring (LT) precession of the hot flow (Ingram, Done & Fragile, 2009). LT precession is a frame-dragging effect associated with the misalignment of the angular momentum of an orbiting particle and the BH spin, leading to precession of the orbit. Numerical simulations have shown that in the case of a hot, geometrically thick accretion flow, this effect can lead to global precession of the hot flow (Fragile et al., 2007). The hot flow precesses like a solid body, and the precession frequency is given by a weighted average of the LT precession frequency between inner and outer radii of the flow. Due to much longer viscous scales, the outer thin disc is not expected to be affected by global LT precession. The emission of the precessing hot flow is then naturally modulated due a mixture of relativistic Doppler beaming, light bending and Compton anisotropy. This model predicts the right range of observed LFQPO frequencies. The amplitude of the LFQPO depends on the details of the geometry and inclination of the viewing angle. The RMS is usually larger at high inclinations and can reach 10 per cent. (See also Ingram et al., 2015, for predictions of modulation of the polarisation of observed X-ray radiation.) In the truncated disc model, the evolution of the disc inner radius drives both the evolution of the photon spectrum and the timing features. As the disc inner radius decreases, νb and the LFQPO frequencies are observed to move up, as expected in the model. At the same time, varying the truncation radius change, the number of seed soft photons seen by the hot flow and the X-ray spectrum softens as the hot flow becomes gradually more efficiently cooled. Such softening of the spectrum is also observed (see, e.g., Done et al., 2007). 1.4.4 Optical Timing Besides fast X-ray variability, several accreting BH sources show a strong variability in optical presenting comparable amplitude and PDS (e.g., see Gandhi, 2009; Gandhi et al., 2010). Reprocessing Model Part of this variability could be associated to the reprocessing of the X-ray illumination in the outer disc and at the surface of the companion star. If so, the optical reprocessed flux O(t) is expected to respond linearly to the X-ray flux X(t):

Radiation Processes and Models

t O(t) = X(t )r(t − t )dt

25 (1.47)

−∞

The response function r can be calculated for given geometry. Such calculations are presented in O’Brien et al. (2002). The reprocessing model predicts that the optical should be correlated with the X-ray flux and lagging behind the X-ray light curve by a few seconds (light travel time). Moreover, the optical is expected to vary on longer timescales than the X-rays, because the sub-second timescale variability is strongly damped. Reprocessing is a mechanism that is difficult to avoid, and yet these predictions are not always verified. Fast Optical Variability from the Accretion Flow In several sources, the optical fluctuations are observed on comparable or even faster timescales than the X-ray. The cross-correlation function (CCF) of the X-ray and optical light curves show complex features involving also some level of anti-correlation (see Figure 1.5). The optical time lags can be as short as 100 ms. This suggests an additional variability mechanism in optical. Such a mechanism could be the variable contribution of synchrotron emission by non-thermal coronal particles (Veledina et al., 2011; Poutanen and Veledina, 2014). This emission could be almost as fast as the X-ray fluctuations. In the hybrid thermal-non-thermal hot flow model, moderate fluctuations of the mass accretion rate m ˙ lead to a simultaneous increase of the plasma Thomson depth and radiation power. Numerical simulations have shown this involves an anti-correlation between X-ray and optical fluxes, which is observed in several sources. Veledina and Poutanen (2015) show that the X-ray and optical variability properties of the source Swift J1753-0127 (shown in Figure 1.5) are well reproduced by a simple model assuming that the optical variability is a mixture of X-ray reprocessing plus a component that is exactly anti-correlated to the X-rays (as expected in the hot flow synchrotron model).

Figure 1.5. Various observed OIR versus X-ray CCFs. Panels a, b and d show observed optical versus X-ray CCFs of GX 339–4 (Gandhi et al., 2008), XTE J1118+480 (Kanbach et al., 2001) and Swift J1753-0127 (Veledina et al., 2011), respectively. The CCF of Swift J1753-0127 is modelled with the reprocessing plus hot flow synchrotron model. Panel c displays the IR/X-ray CCF observed in GX 339–4 by Casella et al. (2010). ©AAS. Reproduced with permission.

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LF QPOs are observed in several hard state sources in the optical band with RMS amplitude ranging from 3 per cent to 30 per cent. The optical QPOs are sometimes associated to the X-ray LF QPO but not always. These QPOs could originate from the same LT precession mechanisms leading to the formation of the X-ray QPOs. The dominant contribution to the optical synchrotron flux comes from partially self-absorbed synchrotron. The effective synchrotron self-absorption depth depends on viewing angle. Therefore, the specific synchrotron intensity produced by the hot flow is modulated by precession Is ∝ 1 − exp [−τ / cos θ(t)]

(1.48)

Depending on the geometric parameters, this can lead to QPO amplitudes comparable to those observed (see Veledina et al., 2013b). In principle, it is also possible to predict the phase lags between the X-ray and optical QPOs. Assuming a simple prescription for the angular dependence of the hot flow Comptonised radiation, Veledina et al. (2013b) predict that the X-ray and optical QPO should be either in phase or in opposition (phase 0 or π). Veledina et al. (2015) show that in Swift J1753.5-0127, the optical and X-ray QPOs are in phase, as predicted by the model. In this source, at least in the X-ray/optical CCF, the optical and X-ray PDS, their phase lags and coherence spectra are well accounted for by the synchrotron emitting hot flow model. Another possible origin for the optical LFQPO could be disc reprocessing. Indeed, the varying disc illumination caused by the LT precession of the hot flow naturally leads to the modulation of the reprocessed radiation (Veledina & Poutanen 2015). In this case, a QPO is expected only if the hot flow precession period is longer than light-crossing time of the disc, which is itself a function of the orbital separation of the binary system −1  −2/3  M c 2 Porb  Hz. (1.49) νQP Omax  Rdisc 3 1hr 10M The amplitude of the QPO, its pulse profile and the expected optical versus X-ray QPO phase lags are expected to depend on the QPO frequency. So far, none of the observed optical QPOs matches these properties and are more likely to be caused by synchrotron. 1.4.5 Caveats In the previous section, the current ‘standard’ model for the emission of accretion flows around BHs was described. It involves a truncated disc with variable inner radius, a hot inner flow harbouring non-thermal electrons, fluctuations of the mass accretion rate propagating radially inward and precession of the hot accretion flow. This complex model produces the main features of the emission. In particular, the observed simultaneous evolution of the SEDs and X-ray PDS can be understood as driven by changes in the inner disc radius. The observed optical variability and the correlations between X-ray and optical bands are qualitatively understood. There are, however, a few issues with this model that should be mentioned. First, the detailed comparison of the hybrid magnetised model and observations has brought interesting constraints on the magnetic field. In the hard state in particular, the conclusion is that either the magnetic field is strongly sub-equipartition (which would be in contradiction with theoretical models involving a magnetically dominated accretion flow, or accretion disc corona atop the disc), or the MeV excess is produced in a region that is spatially distinct from that producing the bulk of the hard X-ray radiation (Del Santo et al., 2013). In the latter case, the accretion flow could, for instance, be constituted of a truncated accretion disc surrounding a central magnetically dominated hot accretion flow, responsible for the thermal Comptonisation component, while active coronal regions above and below the outer disc may produce the mostly non-thermal Compton emission (i.e., the

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hard state MeV excess; Malzac, 2012). In the soft state, the disc inner radius moves inwards until it reaches the ISCO and the hard X-ray emission becomes gradually dominated by the non-thermal corona. But if the thermal and non-thermal particles are not in the same location, the observed correlations between the X-ray variability and optical synchrotron variability cannot be explained by simultaneous changes in Thomson depth and luminosity. A different interpretation would be required. Another interesting possibility is suggested by INTEGRAL measurements showing that the hard state MeV tail of Cyg X–1 is strongly polarised (at a level of 70 ± 30 per cent), while the thermal Comptonisation emission at lower energy is not (Laurent et al., 2011; Jourdain et al., 2012b). The only plausible mechanisms for the very high level of polarisation seems to be synchrotron emission in a highly coherent magnetic field. A natural explanation would be that the MeV excess is, in fact, a contribution from the jet (Romero et al., 2014). Conventional jet models can produce such MeV components but require some fine-tuning and quite extreme acceleration parameters (see Zdziarski et al., 2014). If the non-thermal MeV excess is caused by jet synchrotron emission, this strongly reduces the possible number of non-thermal electrons in the corona. If so, the coronal optical synchrotron component would be drastically reduced, and again, the fast optical variability becomes difficult to explain. As will be shown in the Section 1.5, a possible solution to these problems is to consider the contribution of the jets to the optical and IR emission.

1.5. Model for the Jets In this section, we will focus on the emission of BHB jets. The contribution of jets to the X-ray emission of BHBs is a controversial issue, but there is evidence that in the most documented sources, the jet X-ray emission is weak compared to the emission from the hot flow/corona (see discussion in Malzac, 2015, and reference therein). Therefore, in this section, we will adopt the conventional view that the X-rays are dominated by the accretion flow. We will focus on the contribution of jets to the observed fast variability in IR and optical bands (hereafter OIR). Indeed, in OIR, the combination of a reprocessed and a hot flow synchrotron component that is anti-correlated to the X-ray reproduces remarkably the shape of the optical/X-ray CCFs of sources like Swift J1753.5-0127 that show a strong opt/X-ray anti-correlation and a very weak reprocessing peak in their CCFs. However, several sources, such as GX 339–4 or XTE J1118+480, have an optical X-ray CCF that is very difficult to model because the main peak of the correlation is too narrow to be due to reprocessing, and the lag is too small. (See Figure 1.5 for a comparison.) This suggests some ingredient is missing in the model. Moreover, timing observations of GX 339–4 in IR also show similarly complicated CCFs with sometimes very sharp IR response lagging behind by approximatively 100 ms. Even if the hot flow contains non-thermal electrons, their synchrotron emission is unlikely to be strong at IR wavelengths. The IR variability features have been naturally ascribed to the jets, which are indeed expected to produce IR synchrotron radiation. The 100 ms lag was associated to the travel time for a fluctuation to travel from the disc to the jet IR emitting region (C10, Casella et al., 2010). In the following, it will be shown that, in fact, the jet could be responsible for most of the observed OIR variability. The interesting consequence is that the observed OIR/X-ray correlation may tell us something about the dynamics of accretion and ejection processes. Such jet models for the OIR variability were initially developed in the early 2000s in the context of the multi-wavelength observations of a very exciting BHB that had recently been discovered: XTE J1118+480 (Remillard et al., 2000). Its relatively close distance ( 1.8 kpc and the exceptionally low interstellar extinction towards the source (Garcia et al., 2000) allowed a multi-wavelength monitoring of the source during its outburst (C03 and reference therein). During the whole outburst duration, the X-ray

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Figure 1.6. The optical/X-ray correlations of XTE J1118+480 in the Fourier domain. (a) X-ray and optical power spectra. The counting noise was subtracted (see Section 3.1). (b) X-ray/optical coherence. (c) phase lags as function of Fourier frequency. A positive lag implies that the optical is delayed with respect to the X-rays (Malzac et al., 2003). Reproduced with permission ©ESO.

properties of the source, as well as the presence of strong radio emission, were typical of BH binaries in the hard state. In the radio to optical bands, a strong non-thermal component was associated with synchrotron emission from a powerful jet or outflow (Fender et al., 2001). Interestingly, fast optical and UV photometry allowed by the weak extinction revealed a rapid optical/UV flickering presenting complex correlations with the X-ray variability (Kanbach et al., 2001; Hynes et al., 2003, hereafter K01 and H03, respectively). This correlated variability cannot be caused by reprocessing of the X-rays in the external parts of the disc. Indeed, the optical flickering occurs on average on shorter timescales than the X-rays (K01, see Figure 1.6a), and reprocessing models fail to fit the complicated shape of the X-ray/optical cross-correlation function (H03, see Figure 1.5b). Spectrally, the jet emission seems to extend at least up to the optical band (McClintock et al., 2001, hereafter C03), although the external parts of the disc may provide an important contribution to the observed flux at such wavelengths. The jet activity is thus the most likely explanation for the rapid observed optical flickering. For this reason, the properties of the optical/X-ray correlation in XTE J1118+480 might be of primary importance for the understanding of the jet-corona coupling and the ejection process.

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The observations are very challenging for any accretion model. The most puzzling pieces of evidence are the following: (a) the optical/X-ray CCF shows the optical band lagging the X-ray by 0.5 s, but with a dip two to five seconds in advance of the X-rays (K01); (b) the correlation between X-ray and optical light curves appears to have timescale invariant properties: the X-ray/optical CCF maintains a similar, but rescaled, shape on timescales ranging at least from 0.1 s to few tens of seconds (Malzac et al., 2003, hereafter M03); (c) the correlation does not appear to be triggered by a single type of event (dip or flare) in the light curves; instead, as was shown by M03, optical and X-ray fluctuations of very different shapes, amplitudes and timescales are correlated in a similar way, such that the optical light curve is related to the time derivative of the X-ray light curve. Indeed, in the range of timescales where the coherence is maximum, the optical/X-ray phase lag are close to π/2 (see Figure 1.6). This indicates the two light curves are related through a differential relation. Namely, if the optical variability is representative of fluctuations in the jet power output Pj , the data suggest that the jet x power scales are roughly Pj ∝ − dP dt , where Px is the X-ray power. 1.5.1 The Energy Reservoir Model Malzac et al. (2004, hereafter MMF04) have shown that the complex X-ray/optical correlations could be understood in terms of an energy reservoir model. In this model, it is assumed that large amounts of accretion power are stored in the accretion flow before being channelled either into the jet (responsible for the variable optical emission) or into particle acceleration/ heating in the Comptonising region responsible for the X-rays. MMF04 have developed a time-dependent model that is complicated in operation and behaviour. However, its essence can be understood using a simple analogue: Consider a tall water tank with an input pipe and two output pipes, one of which much smaller than the other. The larger output pipe has a tap on it. The flow in the input pipe represents the power injected in the reservoir Pi , the flow in the small output pipe represents the X-ray power Px , and the flow in the large output pipe represents the jet power Pj . If the system is left alone, the water level rises until the pressure causes Pi = Pj + Px . Now consider what happens when the tap is opened more, causing Pj to rise. The water level and pressure (proportional to E) drop, causing Px to reduce. If the tap is then partly closed, the water level rises, Pj decreases and Px increases. The rate Px depends upon the past history, or integral, of Pj . Identifying the optical flux as a marker of Pj and the X-ray flux as a marker of Px , we obtain the basic behaviour seen in XTE J1118+480. In the real situation, MMF04 envisage that the variations in the tap are stochastically controlled by a shot noise process. There are also stochastically controlled taps on the input and other output pipes as well. The overall behaviour is therefore complex. The model shows, however, that the observed complex behaviour of XTE J1118+480 can be explained by a relatively simple model involving several energy flows and an energy reservoir. This simple model is largely independent of the physical nature of the energy reservoir. In a real accretion flow, the reservoir could take the form of electromagnetic energy stored in the X-ray emitting region, thermal (hot protons) or turbulent motions. The material in the disc could also constitute a reservoir of gravitational or rotational energy behaving as described above. In a stationary flow, the extracted power Pj + Px would be perfectly balanced by the power injected, which is, in the most general case, given by the difference between the accretion power and the power advected into the hole and/or stored in convective motions: Pi  M˙ c2 − Padv,conv . However, observations of strong variability on short timescale clearly indicate that the heating and cooling of the X-ray (and optical) emitting plasma are highly transient phenomena, and the corona

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is unlikely to be in complete energy balance on short timescales. MMF04, therefore, introduced a time-dependent equation governing the evolution of its total energy E: E˙ = Pi − Pj − Px ,

(1.50)

and we assume that all the three terms on the right-hand side are time dependent. The optical variability is produced mainly from synchrotron emission in the inner part of the jet at distances of a few thousands gravitational radii from the hole. We assume that at any time the optical flux Opt (resp. X-ray flux) scales like the jet power Pj (plasma heating power Px ). MMF04 introduced the instantaneous dissipation rates Kj and Kx Pj (t) = Kj (t)E(t),

Px (t) = Kx (t)E(t),

(1.51)

For a specific set of parameters, MMF04 generate random independent fluctuations (time series) for Kx , Kj and Pi , solve the time evolution of the energy reservoir E and then use the solution to derive the the resulting optical and X-ray light curves. (See MMF04 for details.) Combining Equations 1.50 and 1.51, we obtain the following relation for the total instantaneous jet power:   K˙ x (1.52) Pj = Pi − 1 + 2 Px − P˙ x /Kx . Kx We can see from this equation that the differential scaling Pj ∝ −P˙ x , observed in XTE J1118+480, will be rigorously reproduced provided that: (1) Kx is a constant; (2) Pi −Px is a constant. It is physically unlikely that those conditions will be exactly verified. In particular, Px is observed to have a large RMS amplitude of variability of about 30 per cent. However, the observed differential relation holds only roughly and only for fluctuations within a relatively narrow range of timescales 1 − 10 s. Therefore, the above conditions need only to be fulfilled approximatively and for low-frequency fluctuations (> 1 s). In practice, the following requirements will be enough to make sure that the low-frequency fluctuations of the right-hand side of Equation 1.52 are dominated by P˙ x : • Px  Pi , implying that the jet power, on average, dominates over the X-ray luminosity; • The amplitude of variability of Kx and Pi in the 1–10 s range is low compared to that of Pj . In other words, the 1–10 s fluctuations of the system are mainly driven by the jet activity, implying that the mechanisms for dissipation in the jet and the corona occur on quite different timescales. Figure 1.7 shows the results of a simulation matching the main timing properties of XTE J1118+480. In this simulation, jet power was set to be 10 times larger than the X-ray power. As can be seen, the resulting PDSs, CCFs, phase lag and coherence spectra are very similar to those observed in XTE J1118+480. 1.5.2 Internal Shock Model The puzzling optical/X-ray correlations of XTE J1118+480 can be understood in terms of a common energy reservoir for both the jet and the Comptonising electrons. However, the model outlined above remains a toy model, and more realistic jet physics must be taken into account. In particular, we have to explain how jet radiation is produced. The flat SED of compact jets is usually ascribed to self-absorbed synchrotron emission in a conical jet geometry (Blandford and K¨ onigl, 1979). The model postulates the presence

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Figure 1.7. Sample input time series (Panel a) and power spectra (Panel b) of Pi , Kj , Kx , resulting X-ray and optical fluxes light curves (Panel c), X-ray/optical autocorrelation and cross-correlation functions (Panel d), power spectra, coherence and phase lags (Panele; from Malzac et al., 2004).

of a standing shock located close to the base of the jet (at distances ranging between 103 to 104 RG from the BH). In this shock, electrons are accelerated with a power-law energy distribution extending up to ultra-relativistic energies and then propagate along the jet with the flow while radiating synchrotron radiation. As the electrons travel in this conical geometry, their density decreases and the peak of synchrotron emission (which corresponds to the turnover frequency between optically thin and optically thick synchrotron, see Section 1.3.1) moves toward longer wavelengths. The jet SED is made of the sum of the emitting electrons distributed all along the the jet. The radio emission is produced at a large distance while the IR is produced close to the initial acceleration region (see Figure 1.8). A flat spectrum is produced under the assumption of continuous energy replenishment of the adiabatic expansion losses. Adiabatic expansion losses are simply the internal energy losses of the gas (and tangled magnetic field) due to pressure working against the external medium as it expands in the conical geometry. The compensation of these energy losses is crucial for maintaining this specific spectral shape (Kaiser, 2006). Some dissipation mechanism is required to compensate for these losses; otherwise the radio emission is strongly suppressed, and the jet SED is too inverted. Internal shocks provide a possible mechanism to compensate for the adiabatic expansion losses by dissipating energy and accelerating particles at large distances from

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Figure 1.8. The jet SED is made of the contribution of synchrotron emitting regions distributed over the jet length (Fermi LAT Collaboration et al., 2010).

the BH. Internal shocks caused by fluctuations of the outflow velocity are indeed widely believed to power the multi-wavelength emission of jetted sources, such as gamma-ray bursts (Rees and Meszaros, 1994; Daigne and Mochkovitch, 1998), active galactic nuclei (Rees, 1978; Spada et al., 2001) or microquasars (Kaiser et al., 2000; Jamil et al., 2010). Internal shock models usually assume that the jet can be discretised into homogeneous ejectas. Those ejectas are injected at the base of the jet with variable velocities and then propagate along the jet. At some point, the fastest fluctuations start catching up and merging with slower ones. This leads to shocks in which a fraction of the bulk kinetic velocity of the shells is converted into internal energy. Part of the dissipated energy goes into particles acceleration, leading to synchrotron and also, possibly, inverse Compton emission. The energy dissipation profile of the internal shocks is very sensitive to the shape of the PDS of the velocity fluctuations. Indeed, let us consider a fluctuation of the jet velocity of amplitude Δv occuring on a timescale Δt. This leads to the formation of a shock at a downstream distance zs ∝ Δt/Δv. In this shock, the fraction of the kinetic energy converted into internal energy will be larger for larger Δv. From these simple considerations, we see that the distribution of the velocity fluctuation amplitudes over their timescale (i.e., the PDS) is going to determine where and in what amount the energy of the internal shocks is deposited. Malzac (2012) used Monte Carlo simulations to study this dependence and found that independently of the details of the model flat radio-IR SEDs are obtained for a flicker noise PDS of the fluctuations of the jet Lorentz factor. This result is illustrated by Figure 1.9, which compares the SEDs obtained for PDS of the Lorentz factor of the jet with a power-law shape with varying index α: P (f ) ∝ f −α . For larger α, the fluctuations of the Lorentz factor have, on average, longer timescales and, therefore, more dissipation occurs at larger distances from the BH. One can see from Figure 1.9 that the SED is very sensitive to the value of; for α = 1 (i.e., flicker noise), the dissipation profile scales like z −1 , and the specific energy profile is flat. In other words, the internal shocks compensate exactly for the adiabatic losses. As a result, the SED is flat over a wide range of photon frequencies. In fact, this result can

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Figure 1.9. Simulation of the internal shock model with a power-law PDS of the Lorentz factor fluctuations (P (f ) ∝ f −α ). The top left panel shows the shape of the injected PDSs, for the indicated values of the α index. The right panel shows the jet SED calculated for an inclination angle of 40 degrees and a distance to the source of 2 Kpc. (See Malzac, 2014; Malzac and Drappeau, 2015, for details.)

Figure 1.10. Synthetic light curves (left, rescaled) and power spectra at various indicated frequencies resulting from the simulation with α = 0. The injected fluctuations of the Lorentz factor are also shown. (Malzac, 2014).

also be obtained analytically (Malzac 2013). The case of flicker noise fluctuations of the jet Lorentz factor may therefore be relevant to the observations of compact jets. An interesting feature of the internal shock model is that it naturally predicts strong variability of the jet emission. Figure 1.10 shows sample light curves and power spectra obtained from the simulation with α = 1. The jet behaves like a low-pass filter. As the shells of plasma travel down the jet, colliding and merging with each other, the highest frequency velocity fluctuations are gradually damped and the size of the emitting region increases. The jet is strongly variable in the optical and IR bands originating primarily from the base of the emitting region and becomes less and less variable at longer frequencies produced at larger distances from the BH. Note that strong, coherent periodic fluctuations of of the jet Lorentz factor may lead to the formation of QPO features in the jet variability. However, in the context of the jet model, a much more likely explanation for the observed optical and IR QPOs could be the precession of the jets. Indeed, if the X-ray LFQPOs are caused by global LT precession of the hot flow, and if the jet is launched from the accretion flow, one may expect the jet to

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precesses with the flow. The (mostly) optically thin synchrotron radiation observed in IR and optical would then be modulated at the precession frequency due to the modulation of Doppler beaming effects toward the observer (Kalamkar et al., 2016). The model may be used to understand the observed complex timing correlation between the X-ray and IR bands. For example, C10 measured the CCF of the X-ray and IR light curves and found significant correlation between the two bands, with the IR photons lagging behind the X-rays by about 100 ms (see Figure 1.5c). C10 interpreted this time lag as the propagation time of the ejected shells from the accretion flow to the IR emitting region in the jet. In the framework of the internal shock model, this observation suggests that the fluctuations of the jet Lorentz factor are related to the X-ray variability of the source. Malzac, 2014 shows that the internal shock model predicts very similar IR versus X-ray CCF and lags, provided that the fluctuation of the bulk jet Lorentz factor are inversely proportional to the X-ray flux. 1.5.3 Are the Jet Lorentz Factor Fluctuations Related to the X-ray Variability? In fact, if the jet is launched from the accretion disc, the variability of the jet Lorentz factor must be related to that of the accretion disc. And we know both from theory (see, e.g., Lyubarskii, 1997) and from observations (see, e.g., Gilfanov and Arefiev, 2005), accretion discs tend to generate flicker noise variability. Therefore, flicker noise fluctuations of the jet Lorentz factor are not unexpected. In X-ray binaries, the variability of the accretion flow is best traced directly by the X-ray light curves. The left panel of Figure 1.4 shows an actual X-ray PDS observed in GX 339–4 in the hard state, which differs significantly from flicker noise both at high and low frequencies. This variability could be a good proxy for the assumed fluctuations of the jet. The right panel of Figure 1.4 shows the resulting time-averaged SED obtained if one assumes that the Lorentz factor fluctuations have a PDS that is exactly what is observed in X-rays. This synthetic SED is compared to multi-wavelength observation that are nearly simultaneous with the X-ray timing data (see Drappeau et al., 2015, for details). The model appears to reproduce pretty well the radio to IR data. This agreement is striking because the shape of the SED depends almost exclusively on the assumed shape of the PDS of the fluctuations. Although the model has a number of free parameters (jet power, inclination angle, time-averaged jet Lorentz factor, etc.) that could be tuned to fit the data, those parameters only affect the flux normalisation or shift in the photon frequency direction, but they have very little effect on the overall shape of the SED. The four mid-IR flux measurements at 1.36×1013 , 2.50×1013 , 6.52×1013 and 8.82×1013 Hz that are shown on Figure 1.4 were obtained with the Wide-field Infrared Survey Explorer (WISE; Wright et al., 2010). They represent an average over 13 epochs, sampled at multiples of the satellite orbital period of 95 minutes and with a shortest sampling interval of 11 s, when WISE caught the source on two consecutive scans. These data have revealed a strong variability of the mid-IR emission (see G11). The light curves of these observations are shown in Figure 1.11 (left panel) and are compared to light curves obtained from the same simulation, which gives a good fit to the observed radio-IR SED (right panel). The model appears to predict a variability of similar amplitude to that observed by WISE.

1.6. Conclusion Compact objects can release huge amounts of energy on very short time-scales through extraction of their own rotational energy, dissipation of their powerful magnetic fields and the gravitational energy of accreted material, which can be also gravitationally

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Figure 1.11. Left: the observed mid-IR variability as observed by WISE in 4 bands. Right: sample synthetic light curves obtained from the same simulation shown in Figure 1.4. (Malzac and Drappeau, 2015).

compressed to the point of thermonuclear fusion. A fraction of this energy is converted into radiation, which is emitted non-thermally through synchrotron, curvature radiation inverse Compton and bremsstrahlung. Based on the properties of these radiation processes and the energy dissipation mechanisms, time-dependent emission models are developed, which, through comparison to data, allow to constrain the physical conditions in the emitting regions. Such comparisons may also shed new light on the physics of compact objects requiring, in turn, to revise the models. In the the particular case of accreting BHBs, the truncated disc model including a hot inner flow with non-thermal electrons, fluctuations of the mass accretion rate propagating radially inward, and precession of the hot accretion flow produces the main features of the multi-wavelength spectra and variability. In particular, in some sources, the observed optical variability and the correlations between X-ray and optical bands are qualitatively understood. However, in some other sources, a strong contribution of the jet to the IR and perhaps even optical variability seem to be required. Fast jet synchrotron variability may result from the disc variability through a tight coupling between jet and accretion flow on short time-scales, possibly through a common energy reservoir. The IR jet radiation is likely to be produced through synchrotron emission of particules accelerated in colliding shells of gas (internal shocks). This internal shock model naturally predicts the formation of the observed SEDs of compact jets and also predict a strong, wavelength-dependent variability that resembles the observed one. The model also suggests a strong connection between the observable properties of the jet in the radio to IR bands and the variability of the accretion flow as observed in X-rays. If the model is correct, this offers a unique possibility to probe the dynamics of the coupled accretion and ejection processes leading to the formation of compact jets. Future multi-frequency HTRA observations combined with further modelling may reveal this fast dynamic coupling.

REFERENCES Balbus, S. A. and Hawley, J. F. 1991. A Powerful Local Shear Instability in Weakly Magnetized Disks. I–Linear Analysis. II–Nonlinear Evolution. Astrophys J, 376(July), 214–33. Balbus, S. A. and Hawley, J. F. 1998. Instability, Turbulence, and Enhanced Transport in Accretion Disks. Rev Mod Phys, 70(Jan.), 1–53. Belmont, R., Malzac, J. and Marcowith, A. 2008. Simulating Radiation and Kinetic Processes in Relativistic Plasmas. Astron Astrophys, 491(Nov.), 617–31. Beloborodov, A. M. 1999. Accretion Disk Models. Page 295 of: Poutanen, J., and Svensson, R. (eds), High Energy Processes in Accreting Black Holes. Astro Soc P, 161, 295.

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2. HTRA Instrumentation I PHILIP A. CHARLES1 Abstract The range in wavelength and time resolution of current instrumentation for carrying out astrophysical studies has increased dramatically over the last five decades. Here I will give a brief historical review of time-domain astronomy, followed by a summary of the facilities available now from X-ray to near-infrared wavelengths. I will then give a glimpse of various remarkable technologies under development for the next generation of groundand space-based observatories that will take such studies to unprecedented levels.

2.1. Introductory Remarks First, what exactly is ‘high time-resolution’ for astronomers? Even today, there are those for whom the answer will be 10 minutes, while for others it is 10 microseconds! It has certainly been moving to ever shorter times over the last 50 years, and it is set very much by both the technology in use and the wavelength range in which it is applied. However, even when the field has a paucity of photons, there is a growing expectation of timing the arrival of those photons with the microsecond capability we all now effectively take for granted with the global GPS network. Achieving that is often a different matter! This chapter is based on a series of lectures at the XXVII Canary Islands Winter School of Astrophysics on the physical instrumentation with which we undertake research in HTRA. I begin with a brief historical overview of the subject, which is almost entirely post–Second World War and driven significantly by developments in space astronomy, particularly those in high-energy astrophysics. Then I examine the range of HTRA instrumentation available today across a very broad spectrum, from both ground- and space-based facilities. Finally, I look at the latest technological advances and what they will hopefully mean for future HTRA. This chapter is structured to provide a quick reference for the relevant literature on HTRA instrumentation and current developments. It loosely follows the lectures that I presented at the Winter School.

2.2. History of HTRA from the 1960s to the 1990s Even by the end of the 1950s, observational astronomy was limited to optical and radio wavelengths and to timescales of a second or longer (at the very best). But this was to change dramatically in the 1960s and 1970s as space technology opened up windows to the universe all the way from γ-rays to the far-IR (FIR), regions from which we had been excluded on the ground as a result of atmospheric absorption. This is shown very nicely in figure 2 of Harwit (2003), which reviews the accessible wavelength regions and observable time resolutions as a function of decade. Indeed, what is remarkable is how these new windows all immediately had time resolutions of ∼ 0.01 s available, and by 1

I am grateful to the organisers of the Winter School, Tariq Shahbaz and his Local Organising Committee, for the opportunity to deliver these lectures. I am also grateful to the Instituto de Astrof´ısica de Canarias and its Director, Professor Rafael Rebolo, for their hospitality at that time and subsequently during the preparation of this manuscript.

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Philip A. Charles Table 2.1. HTRA Science Drivers and Typical Timescales Objects

Physical Process

AGN

Flares Light-crossing time ISCO∗ Disc flares V∼16   V∼22 Opt/X-ray correlations Eclipse timing Disc flickering Correlations/QPOs Magnetospheric Thermal X-ray bursts Occultations

BH Binaries

LMXBs/CVs Pulsars Solar System ∗

Timescales (now)

Timescales (2020s)

mins–hours

secs

ms ∼20 ms 5–10 s 1μs 10 ms 1 ms 50 ms

∼ns 1μ s ∼1 ms

Innermost stable circular orbit

the turn of the millenium, this had improved to approaching 1 μs across almost the entire electromagnetic spectrum. The time domain for astronomical research has opened literally within a single human lifetime. Harwit’s review goes on to point out how various revolutionary astronomical phenomena are only revealed once these windows are opened with appropriate time resolution. Excellent examples include the discovery of radio pulsars, GRBs (γ-ray bursters), X-ray bursters and millisecond X-ray pulsars, and Table 2.1 is an update of that in Shearer (2008) for HTRA science drivers, the typical timescales achievable with current technology together with that under development for the next decade. 2.2.1 Fast Optical Photometry in the 1950s and 1960s Even in the pre-computer era, it was possible to undertake fast (∼ seconds) photometry with non-imaging, photoelectric photometers, albeit of relatively bright objects, such as cataclysmic variables (CVs). Indeed, using only a chart recorder as the output device for his photomultiplier tube (PMT), Walker (1956) was able to see coherent oscillations at a period of 71 s in the traces produced by the CV, DQ Her. This was an early glimpse of ‘high-energy’ processes near a compact object – as it is now recognised to be a rapidly spinning magnetic WD. Of course, using data on chart recorders was the only way of examining radio data, and that is exactly what Jocelyn Bell Burnell was doing when she discovered the first radio pulsar in 1967. The fastest of the early pulsars found was the Crab (at 33 ms), and even with the rudimentary technology of the time, its optical counterpart (a 15th magnitude star that had been seen for years on photographic images of the Crab Nebula) was found to be pulsating with the same period. It was the arrival of the first mini-computers in the early 1970s that combined with PMTs to really open up these high-speed studies, providing a dramatic improvement on the many-minutes, limiting time resolution of photographic plates. This technology was superbly exploited by Warner and Nather with their Texas/UCT photometer in uncovering many surprising phenomena in their studies of close, interacting binaries. Indeed, it contributed significantly in building up our current picture of such systems, obtained from time-resolved eclipse studies, dwarf-nova oscillations, disc flickering, etc.

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2.2.2 HTRA in Space It was during this same time period that regular access to space began, starting with rocket flights in the 1950s and 1960s. These rocket flights discovered the first cosmic X-ray source in 1961 (Sco X-1), which subsequently led to the first X-ray satellite in 1971 (Uhuru). However, identifying these first X-ray sources was a challenging problem as those early X-ray experiments had very poor angular resolution (typically degrees). But the nature of X-ray detectors on spacecraft meant that they had very good time resolution, so various techniques were developed that translated temporal resolution into spatial information. The simplest way was to wait until the moon was passing across regions containing X-ray sources of interest, at which point rockets were then launched in order to time the lunar occultation. The moon’s position in the sky was very precisely catalogued, and so the time would tell you exactly where the X-ray source was located. The early identification of the Crab Pulsar immediately led to a rush to look for similar optical counterparts to other rapid radio pulsars, such as that in the Vela remnant, spinning only slightly slower with a period of 89 ms. However, these proved fruitless, and it took more than a decade before its 23rd magnitude counterpart was discovered. Indeed, even today, there are only five other optical pulsar counterparts known, and all, apart from the Crab, are fainter than Vela (the faintest, Geminga, is at 26th magnitude!). 2.2.3 X-ray Detector Technology The first two decades of X-ray astronomy were dominated by usage of X-ray proportional counters as the detector, shown schematically in Figure 2.1. The principle is simply an extension of the Geiger counter. The operating voltage of the anode is set such that an X-ray photon entering through the thin plastic detector window ionises the gas inside, creating a cloud of photoelectrons that drift rapidly towards the anode. The causes further ionisations through internal collisions, thereby amplifying the number of electrons and producing a final pulse at the anode, the size of which is proportional to the energy of the incident photon. Whilst this clearly has no imaging capability, the pointing direction of the detector is controlled through a simple honeycomb collimator mounted directly above (and helping physically support) the window, restricting the field of view to be a fraction of a degree or so. Some positional information could be added to this detector by

Figure 2.1. (Left) Schematic of one of the RXTE PCA’s five X-ray proportional counters (Bradt et al., 1993). X-rays are collimated into the sealed counter through a thin window where they ionise the enclosed mixture of xenon and methane, producing a cloud of photoelectrons that are then attracted to the ∼ 2 kV anodes. Note the surrounding anti-coincidence chambers, which are blind to cosmic X-rays but are sensitive to cosmic rays and other energetic particles, thereby greatly reducing the background count rate. Reproduced with permission ©ESO. (Right) Early demonstration of the power of LAPCs, a rocket-borne observation of Cyg X-1 (Rothschild et al., 1974) displaying extremely rapid erratic variability and flickering, now considered the hallmark of BHXRBs. ©AAS. Reproduced with permission.

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making the anode resistive and timing the X-ray pulse arrival time at each end. A 2-D version of this (using a resistive disc as the anode with read-outs at the four corners) was the Imaging Proportional Counter (IPC) at the heart of the first X-ray imaging satellite (the Einstein X-ray Observatory). Given the importance of proportional counter technology over the last 50 years, it is worth looking at its properties in more detail. If the detector gas has an ionisation energy of weV, then an incoming X-ray of energy, E, initially will produce N (= E/w) photoelectrons, initially, which are subsequently amplified via collisions within the chamber en route to the anode (w is 26.2 eV for Ar, 21.5 eV for Xe). This process induces fluctuations 2 = F.N , where F that are lower than expected from Poisson statistics, with a variance σN is the Fano factor, whose value is 0.17 for Ar and Xe. The resulting energy resolution is  σN wF ΔE = 2.35 = 2.35 , (2.1) E N E which is obviously poor (approaching 100 per cent) at low energies, and limited their spectroscopic capabilities in early soft X-ray surveys. Typically, ΔE/E = 0.4E −1/2 , and the energy range of the detector is governed by (a) the window material and thickness at low E, and (b) the gas mixture and its pressure at high E. In practice, early rocket and satellite X-ray survey experiments were mostly limited to the 0.2–20 keV range. The sensitivity of proportional counters is set by the background count rate against which X-ray sources are observed. This background is made up of two components: (i) a local (solar wind-generated) charged particle background, B1 , which will scale with detector volume; and (ii) a diffuse, uniform (across the sky) X-ray background, B2 , which is entirely astronomical in origin (at low energies, it is dominated by the ∼106 K gas of the local interstellar medium (ISM), itself due to old supernova remnant (SNR) shocks, whereas at higher energies it is due to myriad unresolved, distant AGN). This means that the background counts detected in a t seconds observation will be (B1 + ΩAB2 )t, where Ω is the solid angle of the detector of effective area A. Consequently, the S/N ratio, σSN R obtained from a source of strength S, is σSN R = 

SAt (B1 + ΩAB2 )t

(2.2)

and hence will only increase as A1/2 , a feature of non-imaging systems, and ultimately limited by source confusion to sources brighter than ∼ mCrab. Moving to imaging systems drastically reduces the detector volume and hence the background count rate, so sensitivity can be improved by orders of magnitude. But X-ray imaging telescopes have nothing like the collecting area of simple proportional counters, and so, for HTRA work, the latter are still by far the best technology to use. Indeed, the pulse detection in a proportional counter takes ∼1 μs (limited only by the ion mobility), and because the background is not varying, the S/N ratio for variability work will increase as A. The power of this simple technique is nicely demonstrated in a mere 25 s of data obtained during a rocket-borne proportional counter observation of Cyg X-1 (see Figure 2.1, from Rothschild et al., 1974), which revealed extremely rapid, erratic variability and flickering – now accepted as the hallmark of a BH X-ray binary. And from the practical point of view of limiting telemetry to data associated with real X-ray photons only, the onboard detector processing can use various techniques (guard counters and pulse-rise-time discrimination) to discard background events. Figure 2.2 shows the AstroSat Large Area X-ray Proportional Counters (LAXPCs), the most recent mission to incorporate these large-area proportional counters for HTRA X-ray astronomy work.

HTRA Instrumentation I

47 36

0

Collimator TXT Flow

9

300

0 10

47 155

Detector Housing

19 4

0 119

Figure 2.2. AstroSat LAXPC detector (left) during testing prior to its launch in 2015, and schematic (right) showing the mechanical collimator in place above the detector. (From AstroSat Handbook, courtesy of Space Science Programme Office of ISRO.)

In terms of obtaining accurate (∼ arcmins) locations of the bright X-ray sources, an earlier technique that was used to convert temporal information into spatial locations was to modulate the collimator of the detector (e.g., by rotating it) around a known pointing position. This provided a modulation to a bright X-ray source’s signal by an amount that depended on where it was relative to the pointing position and was the method by which Sco X-1 and others were first located and identified. But the major step forward in X-ray astronomy came with using standard proportional counter technology on a satellite mission, and that was Uhuru in 1971. The greatly extended observing times combined with excellent (sub-second) time resolution immediately led to the discovery of X-ray pulsars, in particular Her X-1 and Cen X-3, with periods of a few seconds. But the extended observing time revealed that both were also eclipsing systems with obvious orbital periods of a few days. This was followed by a number of highly successful X-ray astronomy missions in the 1970s (OAO-Copernicus, OSO-8, SAS-3, ANS, Ariel-V, HEAO-1), culminating with the first true imaging mission, NASA’s Einstein Observatory. The first European Space Agency (ESA) mission of the 1980s combined both conventional proportional counters and X-ray imagers in their EXOSAT observatory, but it had one radically different approach compared to any previous astronomical satellite, and that was a spacecraft orbit that was inspired by the early lunar occultation experiments of the 1960s. ESA launched EXOSAT into an extremely high (two-thirds of the way to the moon) and long (∼ 4 d), elliptical orbit that would enable it to eventually be able to observe lunar occultations of almost every bright X-ray source.2 Whilst that feature was never actually exploited, it was the long satellite orbit that would make EXOSAT a revolutionary tool for HTRA, and that was because it allowed for long, uninterrupted observations of X-ray sources. This is in contrast to all conventional low-Earth spacecraft, whose ∼ 100 min orbits are punctuated by Earth blockage and restricted operations in charged-particle regions, leading to typical observing efficiencies of only ∼ 40 per cent, making studies of short-period X-ray binaries much more difficult. It was the success of EXOSAT’s orbit that led to it being adopted by both the major X-ray observatories of the current era. However, while this extended orbit does lead to a much higher overall 2 Because of the long gestation time of EXOSAT, by the time it was launched in 1983, the advent of X-ray imaging telescopes meant that the occultations were superfluous, and not a single one was observed.

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observing efficiency, it does leave the spacecraft well outside Earth’s magnetosphere and hence unprotected from the intense particle storms that can occur during solar flares. When these happen, the instrumentation has to be shut down.

2.3. Current Major Facilities for HTRA from X-ray to IR Wavelengths Large-area proportional counters (LAPCs) have been in use for >50 years, and they are still in use today. That is because for studying the brightest X-ray sources at the highest time resolutions (∼μs), you must have a very large number of photons, and that simply means you must have a large collecting area. There is still no easier and cheaper way of obtaining this than through the construction of LAPCs. Beginning with NASA’s HEAO-1 in 1977 and ESA’s EXOSAT in 1983, there has been an almost unbroken run of missions that had such detectors as their main components, and these are summarised in Table 2.2. Perhaps the most successful of them all so far has been the Rossi X-ray Timing Explorer (RXTE), which had a 15-year operational lifetime, leaving an enormous archive of high-quality data that the recently launched Indian mission, AstroSat, hopes to build upon. As a demonstration of what such instruments can achieve, Figure 2.3 shows the extraordinary light curve of the X-ray transient, V404 Cyg, on 1989 May 30 obtained by the Ginga LAPC (Tanaka, 1989) during the peak of that outburst. Note the logarithmic scale of this plot. Even at ∼ 200,000 ct/s the detector is not saturating (it can easily handle these levels), but something within the source itself is preventing it from going above this level (possibly it is Eddington limited). And it is varying by orders of magnitude within minutes. No other source, before or since, has shown such remarkable variability. And V404 Cyg underwent a second huge, but shorter (∼ 2 weeks), X-ray outburst in June 2015, but sadly this was just a few months prior to the launch of AstroSat, which in its LAXPC had what would have been the perfect instrument with which to study this outburst. However, for studying fainter sources, LAPCs suffer from their very high particle and diffuse X-ray background, making it much harder to follow their short-term variations. It becomes essential to move to imaging systems, which allow the detectors to be much smaller in volume, and the focussing of the source X-rays enhances them with respect to the diffuse X-ray background, thereby providing enormous gains in sensitivity. To achieve

Table 2.2. Large-Area X-ray Proportional Counter Missions Mission

Dates

Uhuru Ariel-V HEAO-1 A-1 HEAO-1 A-2 LED HEAO-1 A-2 MED HEAO-1 A-2 HED EXOSAT ME Ginga LAC RXTE PCA RXTE HEXTE AstroSat LAXPC

70–73 74–80 77–79



N = no. of individual PCs

  

83–86 87–91 95–12 

15–

Total Area (N∗ ) (cm2 ) 840 580 10,500 800 800 2,400 1600 4,000 6,500 1,600 10,800

(2) (2) (7) (2) (1) (3) (8) (8) (5) (2) (3)

E range (keV) 2–20 1.5–20 0.25–25 0.15–3 1.5–20 2.5–60 1–50 1.5–37 2–60 15–250 3–80

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Figure 2.3. The LAPC on Ginga observed this extraordinary light curve (left) from the BHXRB V404 Cyg during its 1989 outburst, with associated dramatic spectral variability (right). This included large (local) changes in the X-ray absorbing column (up to ∼ 5×1023 cm−2 ). From Tanaka (1989). Credit ESA Special Publication.

X-ray

~3 kv

~108 electrons

e– Electron cascade in resistive glass tube

Lx(1038 erg s–1)

0.8

Channel electron multiplier

XTE/PCA 2–20keV 0.6 EXO 0748-676

0.4 0.2 0.0 30

HST/STIS 1400Å

Chevron pair of microchannel plates

e–

X-ray image

Fl (10–15 erg cm–2 s–1 Å–1)

25 20 15 10 5 0

Gemini–S 5500Å

0.6 0.4 0.2 0.0

position-sensitive read-out

0

50

100

150

200

Elapsed Time (s)

Figure 2.4. (Left) Microchannel plates are formed from thousands of tiny glass channel electron multipliers, each of which provides a very high gain of electron clouds produced by incoming X-rays. Normally operated as an angled (‘chevron’) pair so that radiation cannot pass straight through (Seward and Charles, 2010). (Right) X-ray burst from EXO 0748-676 observed simultaneously at X-ray (RXTE), UV (HST) and optical (Gemini-S) wavelengths, demonstrating an optical lag of 4 s with respect to the X-rays as they illuminate this LMXB’s accretion disc (Hynes et al., 2006). ©AAS. Reproduced with permission.

the imaging, however, requires the use of grazing incidence X-ray mirrors, the first of which were Einstein and then EXOSAT. But this gain in sensitivity comes at a cost of collecting far fewer X-ray source photons and increased difficulty in achieving so very short time variability. And while position sensitivity is possible with proportional counters, as employed on both Einstein and ROSAT, there are other technologies that can achieve better spatial resolution. An example of such technology, microchannel plates (MCPs), as shown schematically in Figure 2.4, amplify the photoelectrons created by the

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incoming high-energy photon as a result of the high voltage across the plate. The resulting charge cloud is then read out and centroided in order to provide the location of the initial photon. This technique works from X-rays all the way to UV photons and formed the basis of one of the first-generation detectors, MAMA (the multi-anode microchannel array), on Hubble Space Telescope (HST), and then as both near- and far-UV detectors on GALEX. Unfortunately, one of the main limitations of such detectors is their maximum total count rate of ∼ 30,000 ct/s, and that is from the entire detector, not just from one source (and hence explains why observations even now with the XMM-Newton OM are restricted according to the brightest star within the target field of view). Furthermore, the photocathode used to turn the incoming UV photons into photoelectrons has a maximum efficiency of ∼ 10 per cent, whereas modern charge-coupled devices (CCDs) are now approaching 100 per cent efficiency, so why should we continue using MCPs? The answer is that the MCPs provide photon-counting detection, which has zero noise associated with it, whereas the read-out of a CCD image has an unavoidable noise. And in the UV part of the spectrum, the sky is remarkably dark compared to the optical, and so it is possible for the lower-efficiency MCPs to provide better UV images than would be possible using CCD detectors (see, e.g., Morrissey and KCWI Team, 2011). However, there are now ‘low light level’ (or L3) CCDs becoming available that use on-chip amplification prior to read-out to reduce this effect (these are also known as EMCCDs, electron multiplication charge-coupled devices, see Chapter 3 by V. Dhillon in this volume for full details). A very nice example of the use of these technologies for multi-wavelength HTRA is in Figure 2.4, where RXTE, HST and Gemini-S obtained simultaneous observations of the LMXB X0748-676 (Hynes et al., 2006) during which an X-ray burst occurred and was detected at all wavelengths. Furthermore, the fast timing allowed the delay of the UV and then (later still) the optical bursts with respect to the X-rays to be clearly measured. HST was using the Space Telescope Imaging Spectrograph (STIS) MAMA detector for this work, whereas Gemini-S was using its Acquisition Camera, a conventional CCD but one where binning and windowing allowed for the 1 s exposures to be read out with only ∼ 0.3 s dead time between exposures. It is also important to note that these HST data do not have absolute timing information of the same accuracy as provided by RXTE and Gemini-S. 2.3.1 Solid-State X-ray Detectors To improve at low energies on the poor spectral resolution of proportional counters (PCs), it is necessary to move to materials that have much lower work functions in the detection of X-ray photons, and that means moving to solid-state technologies. As shown in Table 2.3 the effective energy resolution of materials such as Si (used by the first such detector, the solid-state spectrometer, or SSS, on the Einstein Observatory) is both far better than PCs and constant with energy. A schematic of the SSS is shown in Figure 2.5

Table 2.3. X-ray Energy Resolution Material

w (eV)

Fano Factor

Ar Xe Si Ge CdTe

26.2 21.5 3.6 3.0 4.4

0.17 0.17 0.12 0.13 0.11

R (=E/ΔE) (@ 6 keV) 5–10 

25–50 55 3–50

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Figure 2.5. (Left) Schematic of the Einstein SSS in which incoming X-rays produce ion pairs in the Si-doped reverse-biased junction, which is maintained at 80 K so as to reduce thermal noise (Seward and Charles, 2010). (Right) Schematic of a microcalorimeter operation, as used on Hitomi. Incoming X-ray photons deposit energy in a very cold absorber (at 50 mK), producing a thermal pulse that is then measured (Mitsuda et al., 2010). © 2010 Society of Photo Optical Instrumentation Engineers.

(left), where a reverse-biased junction creates a depletion layer that acts as the X-ray detecting volume. Incoming X-ray photons create ion pairs, but there is no avalanche as in PCs. Consequently, to reduce noise, the detector must be cooled to 80 K and employ low-noise pre-amps. It is small in volume and is located in the focal plane of the X-ray telescope. The Einstein SSS was a great success but only operated for about nine months, at which point it had run out of cryogen. Over the last 20 years, an extension of this technology to even better spectral resolution has been achieved through the development of micro-calorimeters, shown in Figure 2.5 (right). By maintaining the detector at ultra-low temperatures (≤ 100 mK) the total energy of the incoming X-ray photon can be measured through the thermal pulse it induces, leading to an energy resolution, ΔE, which depends on the detector’s temperature and thermal capacity. The device flown recently on Astro-H (launched February 2017) used an array of 32 absorbers operated at 50 mK, each of which was less than 1 mm square and only 8 μ thick, giving an energy resolution of 7 eV over the range 0.3–12 keV (Mitsuda et al., 2010). This provided the highest spectral resolution ever obtained for cosmic X-ray spectroscopy and was also combined with high time-resolution capability. Sadly, Astro-H (or Hitomi, as it was renamed in orbit) only worked for barely a month before an attitude control failure led to the loss of the spacecraft. Nevertheless, the power of this technology was clearly demonstrated (see the 2016 July 6 edition of Nature) as it provides high detection efficiency, high spectral resolution and fast timing capacity, essentially the ideal X-ray astronomy detector. Since the turn of the millenium, the dominant X-ray astronomy missions have been Chandra and XMM-Newton, with the largest collecting areas (of grazing incidence mirrors) yet assembled. XMM-Newton is the largest, and Chandra has the finest angular resolution (∼ 1 arcsec, comparable to ground-based imaging, neither of which are likely to be superseded for at least a decade. Both use arrays of CCDs as their X-ray detectors, but they are used in photon-counting mode rather than as integrating devices as in the optical. That is because the X-ray photon flux is much smaller than in the optical (and the telescopes’ effective areas do not compare with conventional telescopes) and each X-ray photon deposits far more energy in the CCD, thereby producing many more photelectrons. Indeed, as for the SSS, the energy resolution of CCDs is far higher than PCs, and so the full range of attributes (location, time, energy) can be measured for each individual photon. It is these data that are transmitted to ground, rather than full images, as the data rate is then far lower. The CCDs employed are 1024 × 1024 devices.

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Philip A. Charles Table 2.4. X-ray CCD Missions

Mission

Dates

E Range (keV)

Effective Area (cm2 @1 keV)

ΔE (@6 keV)

ASCA Chandra XMM-Newton SWIFT – XRT Suzaku – XIS AstroSat – SXT

93–01 99– 99– 04– 05–15 15–

0.4–10 0.2–10 0.1–15 0.2–10 0.2–12 0.3–8

1300 340 ∼ 1000 125 ∼ 1000 125

2% 1–5 % 2–5 % 2% 2% 2.5 %

Table 2.5. XMM-Newton EPIC Timing Capabilities for Point Sources MOS Central CCD (pixels)∗ 600×600 300×300 100×100 100×600 (pixels)† 376×384 198×384 63×64 64×200 64×180 ∗ †

Full frame Large window Small window Timing (1-D)

Time Resolution (millisecs.)

Live Time ( %)

2600 900 300 1.75

100 99.5 97.5 100

73.4 47.7 5.7 0.03 0.007

99.9 94.9 71.0 99.5 3.0

Max. Count Rate (s−1 ) [mCrab] 0.50 1.5 4.5 100

[0.17] [0.49] [1.53] [35]

2 3 25 800 60,000

[0.23] [0.35] [3.25] [85] [6,300]

pn CCDs Full frame Large window Small window Timing (1-D) Burst (1-D)

1 pixel = 1.1 arcsec 1 pixel = 4.1 arcsec

This means that with 1 s binning and 16-bit encoding, full images would require a 16 Mb/s data stream, which is not manageable. Instead, event lists of [x, y, t, E] are compiled on-board, and this makes a much lower data rate to be transmitted for subsequent processing on the ground. Thus, this allows observers to subsequently generate X-ray images in whatever time or energy bin they require for the nature of the observation, a very powerful feature. This technology was first used in the Japanese ASCA mission, equipped with ∼ 5 times the Einstein collecting area, but with CCDs providing far better spectral resolution than the IPC. Such detectors are also part of the ASCA, SWIFT and Suzaku missions (see the summary in Table 2.4). But these devices have significant limitations for high time-resolution work: the readout demands of CCDs provide serious constraints on their use for HTRA, as the onboard processing of individual photons to determine their 2-D position limits the time resolution to a few seconds. However, there are approaches available to improve the time resolution to the order of milliseconds, by using a windowed timing mode, in which the image is reduced to 1-D only, but the full spectroscopic capability is still available. Of course, this only works for bright sources, as there will be an increased background level. This range of options is nicely demonstrated in Table 2.5, which lists the fast timing options that are available from XMM-Newton’s European photon imaging camera (EPIC)-pn cameras. In normal operation, time resolution is limited to ∼ 0.2 s, but this can be extended down to ∼ 5 ms with the windowing technique. However, all of these

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53

are restricted to relatively faint sources of a few mCrab (up to 25 mCrab with the small window), otherwise pile-up (multiple photons arriving during the CCD observing interval) will occur. There are special non-imaging timing modes that can reach 30 μs (for sources of < 800 mCrab), or even 7 μs for the very brightest sources (up to 60 Crab), but the efficiency of operation of this latter option is extremely low (3 per cent). This is thus a very different operational region from that available to LAPCs. Similar restrictions apply with Chandra. However, both XMM-Newton and Chandra are in high, ∼ 4 d, orbits that permit extended, uninterrupted viewing of sources for many hours at a time, sometimes even days. This is a particularly valuable tool to use on short-period (∼ hours) X-ray binaries that show variability on orbital and super-orbital timescales (e.g., Smale et al., 1988). The SWIFT X-ray telescope provides an observing capability that complements all those described so far. Its name is not an acronym but an indication of its principle operational mode, namely the ability to respond swiftly to transient events, thereby providing a ‘high time-resolution’ mode of a completely different nature than what had been available hitherto. Targeted of course at GRBs, it can slew and begin observing with its X-ray telescope and optical monitor (OM) within minutes of the detection and approximate localisation of a GRB, a feature that has been critical to its huge success in this field (and very dramatically better than the ∼days response possible with earlier high-energy missions). However, such a capability is also perfect for studying new X-ray transients or following state changes in a wide range of X-ray binaries and monitoring (via many short observations) over transient outbursts or through periods of unusual source behaviour, and SWIFT has been extremely productive in these ways over its lifetime, as recently reviewed by Gehrels and Cannizzo (2015), Kennea (2015) and Levan (2015). Extending X-ray imaging capability to high-energy X-rays (significantly above 10 keV) has been recently achieved through the NuSTAR mission (Harrison, 2015). Since 2000, developments in multi-layer coating of grazing incidence optics have given NuSTAR an operational energy range of 3–79 keV, combined with an angular resolution of about an arcminute. Given the lower photon fluxes and much higher particle background at these energies, such an imaging capability has drastically improved sensitivity compared to earlier missions.

2.3.2 Optical/IR Detector Developments CCDs are very much the workhorse detectors for a wide swathe of astronomical studies. But their (relatively) extended read-out times and associated read-out noise provide limitations that require workarounds for HTRA, as discussed above for X-ray applications. Similar issues apply at ground-based observatories, and windowed read-out (to reduce read-out times from minutes down to a few seconds) has been used since the early days of CCDs. A significant improvement came about 20 years ago through frame-transfer devices, whereby the CCD can be divided into two electronic halves, one to act as the live imager, which shifts the image at the end of each exposure very rapidly (∼μs) into the read-out half. The image can then be read out while the ‘live’ half continues in parallel with the next exposure (thereby eliminating the dead time associated with each read-out). Two extremely successful instruments, the UCT/CCD (O’Donoghue, 1995) and ULTRACAM (Dhillon et al., 2007) have operated in this way since the mid-1990s onwards, and full details of this and the latest CCD developments are dealt with in Chapter 3 by V. Dhillon in this volume. Since 2000, there have been developments in optical photon-counting detectors, and we will now look at two of these developments.

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Figure 2.6. (Left) Schematic of one tantalum element (which is 25 μ square) of an STJ device, as used in S-Cam2, and which is operated at 0.32 K. Incoming photons deposit energy in this element, thereby breaking Cooper pairs in the superconductor and releasing free electrons in proportion to the energy of the incident photon. Credit ESA. (Right) S-Cam2 light curve of the eclipsing dwarf nova IY UMa, obtained with the William Herschel Telescope (WHT). Times of eclipse of the WD and hot spot (HS) are marked, as are the emergence from eclipse of the disc and HS (Steeghs et al., 2003).

1. STJs Developed at ESTEC, S-Cam is based on the superconducting tunnel-junction (STJ) detectors used from the mm to far-IR bands both on the ground and in space. As part of a research program to exploit this technology at X-ray energies, it was recognised by Perryman et al. (1993) and demonstrated by Peacock et al. (1996) that it could be extended to work at optical wavelengths as well (indeed, providing the first non-filtered measurement of the energy of an optical photon). Shown schematically in Figure 2.6, the incident photons break Cooper pairs in the superconductor. Furthermore, the detected optical photon would produce a sufficient number of electrons in this process to provide some (R∼ 10–50) spectral information as well. Given that they work well over the entire visible spectrum with close to 100 per cent quantum efficiency (QE), provides ∼ 1 μs timing accuracy and nondispersive, albeit crude, spectroscopy, they have the potential to become the ‘ideal’ astronomical detector. Their main limitation is the practical difficulties associated with (a) producing arrays of these devices (current versions have been small, typically 107 >106 >106 >106

ns ns 0.1 ns ns ns ns

80 % 60 % 60 % >90 % >90 % >90 %

5 >20 >50

∼5 1 4 10 keV.

X-ray Emission from Black-Hole and Neutron-Star Binaries

115

Figure 4.13. Top panels: detections of single HFQPO peaks in GRO J1655-40, peaking at ∼ 300 Hz and ∼ 450 Hz. Bottom panels: corresponding phase-lag spectra (from M´endez et al., 2013).

Figure 4.14. Left panel: double HFQPO in GRS 1915+105 (from Belloni and Altamirano, 2013a). Right panel: distribution of the detection of single HFQPOs in GRS 1915+105 (from Belloni and Altamirano, 2013b).

• When two peaks are observed simultaneously, in GRO J1655-40 and GRS 1915+105, the lower peak has soft lags and the upper peak has hard lags (see Figure 4.13; M´endez et al., 2013). For XTE J1550-564, as mentioned, the 180 Hz and 280 Hz peaks have similar lags, which suggests they might be the same peak at different frequency. • In the case of GRO J1655-40 and XTE J1550-564, the frequencies of the two peaks are around a 3:2 ratio (see Abramowicz and Klu´zniak, 2001). • The case of GRS 1915+105, as in many other respects, needs to be treated separately. The original detection was of a peak at 65–67 Hz (Morgan et al., 1997). Since the observations where the oscillation was found showed very strong variability on timescales of 10–20 s, a flux-selected analysis led to the discovery of another peak at 27 Hz, not simultaneous to the 67 Hz one (Belloni et al., 2001). Shortly afterwards, a 41 Hz peak simultaneous to the 67 Hz one was discovered from a subset of RXTE observations (Strohmayerb, 2001b). A semi-automatic study of all RXTE observations for a total exposure time of 5 Ms led to the detection of 51 QPO peaks in as many observations. Most of the centroid frequencies were in the 63–71 Hz range (see Figure 4.14, left panel), indicating that ∼ 67 Hz (the average value) must be a fundamental frequency in the system (Belloni and Altamirano, 2013b). A time-resolved analysis indicates that the detections are confined to a restricted region of the HID of GRS 1915+105 (which has a peculiar and complex shape), again corresponding to the highest count rates (Belloni and Altamirano,

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Tomaso M. Belloni 2013b). An additional peak at 34 Hz (see 4.14, right panel), simultaneous with the 67 Hz one and consistent with half of its frequency, was found in data corresponding to a subregion in the HID, just as the 41 Hz one corresponded to a separate subregion (Belloni and Altamirano, 2013a). This system therefore showed multiple centroid frequencies: 67 Hz, 27 Hz, a pair 41–67 Hz and a pair 34–67 Hz. The sequence 27:41:67 corresponds to 2:3:5, while 34:67 is 1:2. Independent of the model, 67 Hz is too low to be the Keplerian frequency at the innermost stable circular orbit for GRS 1915+105, being too slow even for the highest mass allowed for the BH in the system (Reid et al., 2014) and zero spin.

4.4. Comparison between NS and BH Systems Comparison When a new galactic transient X-ray source is observed in the sky – even in absence of detected pulsations or X-ray bursts, unambiguous telltales of the presence of a NS – in most cases, the X-ray properties are sufficient for the identification of the nature of the compact object, although not a conclusive one as no definitive criterion for the presence of a BH in the system has yet been found. Below, I will outline the main differences and similarities in X-ray properties between classes of sources. 4.4.1 Quiescent Emission BH transients in quiescence have a very low X-ray emission that contrasts with the relative optical brightness of the accretion disc, a discrepancy that can be solved with the presence of an advection-dominated accretion flow (ADAF), where the flow is radiatively inefficient (see Narayan et al., 1996). However, in the case of a NS, the advection energy stored in heat in the ADAF is not lost through the event horizon and must be released on the star’s surface. Therefore, the quiescent luminosity of a NS transient is expected to be lower than in the BH case, given the same accretion rate. Accretion rate depends on the binary orbit, with tighter systems being fainter (see Menou et al., 1999) but is not expected to be different between different classes of systems. Deep observations with current instruments confirm both statements (see Figure 4.15). The theoretical situation is, however, more complicated, and the claim that this results in an indication of the existence of event horizons has been critically discussed by Abramowicz et al. (2002). 4.4.2 X-ray Bursts Thermonuclear X-ray bursts are X-emissions originating on the surface of a NS and as such a direct evidence of the presence of said surface. It follows that, by definition, none of the currently known BHBs have shown an X-ray burst. However, it has been suggested that the absence of X-ray bursts can be taken as an indication of the absence of a surface and therefore as an indication of the BH nature of a source (Narayan and Heyl, 2002). A statistical analysis of RXTE data in relation to burst models numerically confirmed this statement (Remillard et al., 2006b). Also in this case, Abramowicz et al. (2002) have criticised this approach as a detection of the presence of an event horizon. 4.4.3 Energy Spectra As mentioned in the introduction, several spectral ‘signatures’ have been proposed in the past to identify the presence of a BH in an X-ray binary. The very soft thermal component observed from the accretion disc in soft states cannot be considered as a signature, as low-B neutron-star binaries can have accretion discs extending to roughly the same inner

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Figure 4.15. Quiescent luminosity for X-ray transients as a function of orbital period. Only the lowest observed value for each system is shown. Filled symbols represent BH binaries, empty symbols neutron-star binaries. (From Garcia et al., 2001). ©AAS. Reproduced with permission.

radius as BHs. Measurements of inner disc radii through high-resolution spectroscopy are also unlikely to yield results that would allow a strong identification. A good discussion of this comparison throughout the full spectrum can be found in Lin et al. (2007), but no clear dividing line between classes of systems is found. The hard component (often referred to as ‘hard tail’), which in the past was compared simply in terms of flux or power-law index, can now be also analysed in terms of more complex physical models. Recently, Wijnands et al. (2015) reported the results of the spectral analysis of a sample of NS LMXBs showing that at lower luminosities, in the range 1034 – 1035 erg/s, BH systems have systematically harder (lower power-law photon index) spectra than NS (see 4.16). The separation between classes of sources appears to be marked, and additional data will be able to test this method. Another recent proposal for a clear distinction between classes of sources was presented by Burke et al. (2017). Fits with Comptonisation models to RXTE data of systems in the hard state show that BH and NS systems display different values of parameters. BHs correspond to higher values of the Comptonisation parameter y and higher amplification ratio A (hard flux divided by seed flux). While the y−A relation appears to be common, there is a clear segregation (Burke et al., 2017). Comparing the evolution of both transient and persistent systems in the HID, one can see that there is no major difference, indicating that the underlying physics must be, as expected, the same (see Mu˜ noz-Darias et al., 2014). At the same time, results like those of Burke et al. (2017) and Lin et al. (2007) indicate that the same models can be applied, with significant differences in physical parameters. 4.4.4 Time Variability The comparison of fast-timing properties in BH and NS sources is complex but very promising. The deconvolution of PDS into a combination of Lorentzian components (Olive et al., 1998; Belloni et al., 2002) has allowed a homogeneous treatment of all systems,

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Figure 4.16. Plot of photon index - luminosity for a sample of NS and BH X-ray binaries (from Wijnands et al., 2015). ©AAS. Reproduced with permission.

allowing to link different classes of sources across different states (see Figures 4.3 and 4.11). By comparing the two figures, one can see that in the hard states, the power distribution appears similar: a more peaked (QPO) component, a broad component at lower frequencies (Lb ) and two broad components at higher frequencies (L and Lu ). The main difference between the NS and BH case is that the relative power of L and, in particular, Lu are higher in the NS case. Indeed, the difference in power above 10–50 Hz has been proposed as an empirical method to distinguish BH from NS binaries (Sunyaev and Revnivtsev, 2000). In the hard states, no counter-example has been found. Sunyaev and Revnivtsev (2000) interpret it as an effect of the presence of a boundary layer in NS binaries, although the presence of the same number of components in the PDS seems not to be consistent with this idea. The identification of similar components in both classes of systems (see Belloni et al., 2002) has allowed a detailed comparison. Identifying the most common QPO in BHBs (type-C) with the HBO in NS systems (see the following paragraph) and comparing its frequency with that of the Lb component has led to a rather tight correlation (called WK correlation, see Figure 4.17 and Wijnands and van der Klis, 1999). Z sources appear to deviate from the correlation, which otherwise extends over three orders of magnitude. (See Bu et al., 2015, for an updated version.) Belloni et al. (2002) added data to the correlation and noticed that there is a parallel correlation following a 1:1 relation when characteristic frequencies are considered. However, the noise component related to these points was found to be an additional one (Lh ). The WK correlation connects two homogeneous observables through different classes of systems and indicates that the underlying phenomenon does not depend on the nature of the compact object. There is the indication of a segregation of BHB points to lower frequencies, but this is expected on the basis of mass scaling.

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Figure 4.17. Frequency of type-C QPO and HBO versus break frequency for a sample of NS and BH systems. Left panel: BHB in black, atoll sources in red and accreting millisecond pulsars in cyan. Panel: all points from the left panel in black, Z sources in red and sources with more than one QPO is observed in cyan (from Wijnands and van der Klis, 1999).

Figure 4.18. PB correlation as published by Belloni et al. (2002), with HFQPO points removed as many of them were later found not to be significant. Two points were added (five-point stars in the upper right), corresponding to the two simultaneous detections of type-C and HFQPO analysed in Motta et al. (2014a,b).

There is another correlation that links different classes of systems over three orders of magnitude, the so-called PBK correlation (Psaltis et al., 1999a; Belloni et al., 2002). Unlike the WK correlation, here non-homogeneous quantities are included. The version of the correlation by Belloni et al. (2002) is shown in Figure 4.18, modified to remove incorrect points and to add the two points from Motta et al. (2014a,b) (as discussed later in this paragraph). For NS binaries with kHz QPO, what is plotted in Figure 4.18 is the frequency of the HBO versus that of the lower kHz QPO. For hard-state sources,

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both containing BH and NS, the plotted quantities are the type-C QPO for BH or the low-frequency QPO for NS versus the characteristic frequency of the lower of the highfrequency broad component L . Added to the plot are the two only RXTE detections of a HFQPO and a simultaneous type-C QPO for BHBs from Motta et al. (2014a,b). Therefore, at x < 20 Hz, the y axis includes a broad component, and at x > 100 Hz, it includes the lower kHz QPO. The 20–100 Hz interval is covered by points from the anomalous Z source Cir X-1, which bridges the gap in the plot. Although there seems to be a parallel correlation at very high frequencies, the correlation is very well defined and links not only BH and NS sources but also broad and narrow components, suggesting that the broad features observed in the hard state mark the same physical frequencies as kHz QPOs (and HFQPOs) in softer states. Interestingly, an extension of this correlation by three more orders of magnitude at low frequencies has been presented when adding the frequencies of dwarf-nova oscillations and quasi-periodic oscillations observed in the optical band from cataclysmic variables (Warner et al., 2003). The extension of the correlation looks impressive, although it is not clear how to relate the signals observed in a different band from white dwarf systems, where the inner region of the accretion flow cannot exist due to the physical size of the compact star (more in the next section). As presented earlier, at low frequencies NS binaries feature three ‘flavors’ of QPOs: HBOs, NBOs, and FBOs and three are the ‘flavors’ of BH QPOs: type A,B and C. We have seen compelling evidence that links HBOs and type-Cs. Casella et al. (2005) compared in detail the QPO types and suggested a one-to-one correspondence between them, with NBOs corresponding to type-B and HBOs to type-C. More work is needed to build on these similarities. Motta et al. (2011), on the basis of the analysis of a large sample of observations of GX 339-4, suggested the possibility that type-A QPOs are high-frequency extensions of type-C QPOs. Of course, it is natural to explore the possibility that the two kHz QPOs and the two HFQPOs are the same signal observed in the vicinity of neutron stars and BHs respectively. The comparison is not simple, as very few HFQPO have been detected until now. There are clear phenomenological differences: kHz QPOs are observed to span a rather broad range of centroid frequencies, while the frequencies of HFQPOs appear to vary only slightly if they vary at all. Moreover, the fact that kHz QPOs have been observed frequently and HFQPOs have not implies that the typical fractional RMS of the latter must be lower. Until new observations with more sensitive instruments are available, it is difficult to make a full comparison.

4.4.5 Mass Measurements from X-rays Spectral analysis, whether from the continuum emission or from fluorescence lines, does not yield a mass for the compact object, which must be assumed on the basis of optical observations. Adding timing information can deliver this information, but, of course, a model must be assumed. The problem with past and current observatories is that they were optimised for either timing or high-resolution spectral analysis, which means that co-ordinated observations are needed, introducing further complication. A comparison was attempted on the NS binary 4U 1636-53 with simultaneous observations with RXTE and XMM-Newton (Sanna et al., 2014). The inner disc radius obtained by the fitting of a relativistically distorted profile to the iron emission line was combined with the information from the upper kHz QPO under the assumption (see next paragraph) that it represents the Keplerian frequency at the same radius. Six separate couples of simultaneous observations yielded inconsistent results on the mass, indicating that either one of the models is not correct or the two signatures do not originate from the same radius.

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Fast time variability offers a potentially unique possibility for the study of effects of general relativity in the strong-field regime and to discriminate the nature of the compact object. The information from timing signals like QPO centroid frequencies is essentially model independent and must originate in the innermost regions of the accretion flow. In the recent years, several models have been proposed for the interpretation of these signals, and almost all of them include the presence of frequencies from general relativity. In particular, in almost all cases, the highest detected feature corresponds to the Keplerian frequency at a certain radius, which can be the ISCO or larger (see previous paragraph). Ideally, a successful model must be able to apply to both the NS and BH cases and explain the main observable facts outlined in the previous sections. One particular model (called relativistic precession model, hereafter RPM) has received particular attention. The model identifies the three observed frequencies (type-C and HFQPOs for BH binaries, HBO and kHz QPOs for neutron star binaries) to a combination of fundamental frequencies set by general relativity. The low-frequency peaks are identified with the nodal precession frequency (Lense-Thirring), while the high-frequency peaks would be the periastron precession frequency and the Keplerian frequency, all corresponding to the same orbit, which identifies a ‘special’ radius in the accretion flow (see Stella and Vietri, 1998, 1999; Stella et al., 1999). This model has the advantage of being extremely simple, relying on basic relativistic frequencies, and at same time with the disadvantage of being extremely simple, as it associates the frequencies to a special radius but does not address the issues of how the oscillation is produced and why that (variable) radius is special (it cannot be the ISCO, as the two highest frequencies are identical at ISCO). Originally applied to a sample of pairs of kHz QPOs, this model did not provide a precise fit but provided frequencies in the correct range and with roughly the correct dependence (see the left panel in Figure 4.19 and Stella and Vietri, 1999; Boutloukos et al., 2006). Applied to Figure 4.19 (where accreting millisecond pulsars are not included), the model makes three predictions; all of which observed that the difference Δν between the centroids of kHz QPOs must increase at low frequencies, decrease at high frequencies and have a maximum around 350–400 Hz. The increase part is covered only by data from Cir X-1, a rather anomalous Z source (Boutloukos et al., 2006). The decrease part is more scattered, as expected if not all NS have the same mass, although the source with the

Figure 4.19. Left panel: plot of the difference Δν between the centroid frequencies of kHz QPO pairs as a function to the centroid of the upper peak (after Stella and Vietri, 1999; Boutloukos et al., 2006) for all RXTE detections in the literature from NS LMXBs (excluding accreting millisecond pulsars). The lines are predictions from the RPM for three NS masses (indicated in solar masses). Right panel: distribution of all Δν values on the Y axis of the plot in the left panel (after M´endez and Belloni, 2007). The line is a Gaussian fit with centroid ∼ 305 Hz.

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highest statistics, Sco X-1, shows a decrease steeper than expected (see Stella and Vietri, 1999). Notice that a more direct way to examine the relation between the frequencies of the two kHz QPO peaks is plotting one versus the other (see Figure 4.5, where the RPM prediction for 2 M is shown). In this plot, Cir X-1 and Sco X-1 are marked separately, as are accreting millisecond X-ray pulsars, which are systematically below the correlation and can be brought back to it only by multiplying both frequencies by a factor of 1.5 (see Section 4.2.3). The RPM was found also to interpret naturally the PBK correlation, which also contains very broad features (Stella et al., 1999), although it would not be applicable to the white-dwarf case because of the size of the compact star being too large. Within the RPM, there is no direct dependence of kHz QPOs from the spin of the NS (although the model lines would be slightly modified by rotation), and there is, indeed, evidence that the spin does not have influence (see Section 4.2.3): Figure 4.19 (extended from that in M´endez and Belloni, 2007) shows the distribution of all published Δν values (again without millisecond accreting pulsars) with a simple Gaussian fit, indicating that the typical Δν value is around 300 Hz. Why accreting millisecond pulsars, the only systems where we directly observe the pulsation, are shifted by a factor of 1.5 is unclear, although also considering correlations involving the HBO suggest that both kHz peaks are shifted, but not the HBO (see van Straaten et al., 2005; Linares et al., 2005). To test the validity of the model and the relation to the spin frequency, one needs an extreme source, either very slow or very fast. In 2010, an accreting accretion-powered pulsar with a rotation frequency of 11 Hz was discovered in outburst (Bordas et al., 2010). Unfortunately, only one kHz QPO was observed, and it was not possible to discriminate between the dependence on spin (Δν was expected to be ∼ 11 Hz or ∼ 300 Hz in the two scenarios). However, a low-frequency QPO was observed in six observations, moving in the range 35–50 Hz, which was identified as a HBO because of its intrinsic properties and positioning in the correlation diagrams (Altamirano et al., 2012). Together with the observed kHz QPO, this results to be too fast to be associated to with Lense-Thirring frequency at the same radius. More observations will be needed to firmly confirm the lack of association. The curves in the left panel of Figure 4.19 can be calculated also for BHs (i.e., higher masses) and obviously would lie to lower frequencies, merging with the others at their low end. For a 10 M BH, the curve would peak around 120 Hz. Of course, for low-frequency broad features, the model does not allow one to discriminate the compact object, but notice that Figure 4.19 indicates that the RPM identifies all sources excluding Cir X-1 as NSs. As previously mentioned, there are only a few detections of HFQPOs and fewer detections of pairs of them. Moreover, the presence of HFQPOs seems to be mutually exclusive with that of type-C QPOs. However, there is a single observation of GRO J1655-40, the only RXTE observation in twelve years of operation in which all three frequencies (one type-C and two HFQPOs) have been observed (see Strohmayer, 2001a, and Figure 4.20). From these three values, measured with considerable precision, it has been possible to apply the RPM relations and obtain three physical parameters: the radius R of the orbit to which they are related, the mass M and the spin a of the compact object (Motta et al., 2014a). The derived parameters are R = 5.677 ± 0.035 rg , M = 5.307 ± 0.066M , and a = 0.286 ± 0.003. Although with a small error bar, the derived mass is compatible with the precise dynamical measurement of M = 5.4 ± 0.4M (Beer and Podsiadlowski, 2002). Being based only on one detection, this result will have to be corroborated by more observations, but in principle, this would constitute the first direct evidence of the presence of a black hole in an X-ray binary, as well as a precise measurement of its spin. In addition, since a and M define fully the system, it is possible to measure the radius of the

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Figure 4.20. Power density spectrum of the only RXTE observation of a BHB with a triplet of QPOs. The main panel shows the 18 Hz type-C QPO; the inset shows the high-frequency part in two separate energy bands to highlight the two HFQPOs (from Motta et al., 2014a).

Figure 4.21. Frequency correlations in the full sample of RXTE observations of GRO J165540. The red lower points represent type-C versus type-C, therefore lie on a 1:1 line. The three blue upper points are the frequencies of the triplet of QPOs used to determine mass and spin of the BH. The black points are the ν and νu frequencies as a function of type-C QPO (ν versus type-C in the PBK correlation).The lines are the dependencies predicted by the RPM for Keplerian, periastron precession and nodal precession (from top to bottom) for the best fit parameters from the blue upper points. The red vertical band indicates the maximum nodal frequency set by the presence of an ISCO in the system (from Motta et al., 2014a).

ISCO and therefore the maximum values that the frequencies of the three peaks can reach. While the few detections of HFQPOs are all around the same values, a large number of type-C frequencies were measured throughout the two outbursts of this system, and their frequencies are distributed between 0.1 Hz and 28 Hz, the latter only marginally above the maximum value allowed by the model (red lower points in Figure 4.21, see Motta et al., 2014a). If confirmed by more data, this constitutes a direct measurement of the presence of the ISCO. Finally, adding all detections of broad components in the PDS, identified as ν and νu (black points in Figure 4.21, the PBK correlation), they line along the prediction of the model. The same analysis can be done on the only observation of XTE J1550-564 that shows a type-C and a single HFQPO (Motta et al., 2014b). Here the third parameter needed to

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solve the system of GR equations cannot be the second HFQPO, so the mass measured from optical observations was used. The same analysis led to the measurement of a spin of a = 0.34 ± 0.01, with all type-C frequencies being below the maximum ISCO values and the broad components fitting the model. The identification of the physical origin of the observed frequencies is not sufficient for a solid model, and a modulation process is necessary. A promising model has been proposed for the BH case, based on a thermal disc with an inner truncation radius, below which the accretion flow is geometrically thick and consistent with spectral models (see, e.g., Poutanen et al., 1997). This model interprets type-C QPO as arising from the frame-dragging precession of the inner flow, where the flux is modulated by effects like selfoccultation, relativistic effects and changes in projected area. Although each radius has a different precession frequency, the overall result is an oscillation at a frequency intermediate between those at the outer and inner boundaries of the geometrically thick precessing region (Ingram et al., 2009), with the broadband noise arising from propagation effects (see earlier in this chapter and Ingram and Done, 2010, 2011; Ingram and van der Klis, 2013). This model does not address (yet) the high-frequency oscillations nor the case of NSs, but it is being investigated further through the search and detection of modulation of the effects of the inner-flow precession on the properties of the radiation reflected by the geometrically thin accretion disc. A modulation at the QPO period of the iron line due to reflection has been detected, strengthening the applicability of the model (Ingram et al., 2016). Finally, an alternative model for low-frequency QPOs, the transition layer model, interprets type-C QPOs as viscous magneto-acoustic oscillations of a spherical transition layer between the Keplerian flow and a sub-Keplerian region near the BH (Titarchuk and Fiorito, 2004). This model predicts a relationship between the photon index of the hard spectral component and the frequency of the type-C QPO, which is used to measure the mass of the BH (see Gliozzi et al., 2011).

4.5. Conclusions In this chapter, I have presented the current standpoint in a selected number of topics regarding X-ray emission from BH and NS binaries. The past two decades have seen a dramatic increase in our knowledge of the process of accretion onto compact objects. We now know that the presence of collimated relativistic jets and powerful wind outflows from X-ray binaries, which I did not touch here, is impossible to ignore if we want to understand the physics of accretion. At the same time, high-resolution spectral information, longterm coverage of transient systems and high-sensitivity timing analysis have proven to be essential, in particular when combined. With current instruments, we now have the tools to start disentangling the effects of accretion that are in common between NS and BH sources and those connected to the nature of the compact object. This will allow us to use these powerful sources to study both astrophysics and basic physics, making the best use of these cosmic laboratories. REFERENCES Abbott, B. P., Abbott, R., Abbott, T. D. et al. 2016. Observation of gravitational waves from a binary black hole merger. Phys Rev Lett, 116(6), 061102. Abramowicz, M. A. and Klu´zniak, W. 2001. A precise determination of black hole spin in GRO J1655-40. Astron Astrophys, 374(Aug.), L19–L20. Abramowicz, M. A., Klu´zniak, W. and Lasota, J.-P. 2002. No observational proof of the blackhole event-horizon. Astron Astrophys, 396(Dec.), L31–L34.

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5. Radio Observations and Theory of Pulsars and X-ray Binaries ANDREA POSSENTI1 Abstract This chapter introduces the basics of pulsar phenomenology and then reviews the link between various classes of binary pulsars with their X-ray emitting binary progenitors. The bulk of the chapter is devoted to describing the methodology – called ‘pulsar timing’ – with which pulsar-clocks can be exploited as tools for setting up experiments of fundamental physics. Some applications are also presented.

5.1. Pulsars for Newcomers Most neutron stars are the leftover from the Type II (i.e., core-collapse) supernova explosion, which is experienced by stars initially more massive than about 8–10 M at the end of their nuclear burning life. The exact threshold value for the initial mass of the star is still matter of scientific discussion, as well as the upper bound (currently believed to be at about 20–25 M ), above which the star undergoes the gravitational collapse to a black hole (Fryer 2013). This threshold holds for a star that evolves in isolation. Evolution in binary or multiple systems (see Section 5.2) can significantly modify the last stages of a star’s life, leading to the formation of neutron stars from progenitors that largely violate the range of initial masses reported at the beginning of this paragraph (Podsiadlowski 2007). Correspondingly, the characteristics of the associated supernova event can be very different with respect to the typical core collapse (van den Heuvel 2010; Tauris et al. 2015). Radio pulsars are a subgroup of the larger family of neutron stars. In particular, their very rapid rotation (from ∼0.1 Hz to ∼700 Hz) and ultra high surface magnetic field (from ∼107 G up to ∼1014 G) support a coherent process of highly anisotropic emission of radio waves. Since, in general, the neutron star’s rotational axis is misaligned with respect to the beamed radio emission, this translates in the observation of a repetition of pulses, often described as the lighthouse effect. In this introductory chapter, I will briefly report on a series of (mostly) phenomenological facts about pulsars. A thorough review and analysis can be found in a number of very good resources (Lyne and Graham-Smith 2012; Ghosh 2007; Lorimer and Kramer 2004; Manchester and Taylor 1977). 5.1.1 Energetics Radio pulsars belong to the category of the ‘rotationally powered’ neutron stars, because their energetics is dominated by the progressive depletion of their rotational kinetic energy. A simple relation can be written, linking the total spin-down losses E˙ rot to two observable quantities P and P˙ , i.e., the rotational period and its time derivative: E˙ rot = 4π 2 IN S P˙ P −3 ∼ 3.9 × 1032 IN S,45 P˙ −14 P −3 erg/s, 1

(5.1)

AP acknowledges the precious help of Alessandro Ridolfi, Caterina Tiburzi and Marta Burgay in preparing some of the figures, as well as the contribution of all the collaborators at INAF-Osservatorio Astronomico di Cagliari and of the pulsar researchers all around the world, who are constantly making this field of research intriguing and amazing.

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where the rotational parameters are related to the spin frequency ν and its derivative 2 ν, ˙ according to P = 1/ν, P˙ = −(ν)/(ν ˙ ), where the period is reported in seconds and the period derivative is expressed in units of 10−14 . The moment of inertia IN S is dependent on the equation of state (EoS) describing the neutron star matter, the mass of the specific neutron star and also on its rotational period. For a large class of EoSs, for typical neutron star masses around 1.4 M and for a not too fast spin rate, the moment of inertia is IN S ∼ 1045 g/cm2 . It can be experimentally shown that only a small fraction of the spin-down power estimated earlier is emitted in the radio band. A figure of merit is given by the ‘pseudoluminosity’ at a radio frequency f Lf = 4π d2 Δf Sf

(5.2)

where d is the pulsar distance, Δf the bandwidth of the observation centred around the frequency f, and Sf is the flux density of the source at frequency f . By adopting reference values for the quantities, one typically obtains Lf  E˙ rot , and an efficiency ˙ ∼10−4 − 10−8 is common. It is worth noting that Lf is calculated as if the of Lf /Erot radio emission were isotropic, which is certainly a wrong assumption, hence the use of the more appropriate name, pseudo-luminosity. A much larger fraction of the pulsar spin-down power goes into electromagnetic emission at high energy, in the X-ray band (with typical efficiency of ∼ 10−5 − 10−3 , Shibata et al., 2016) and in the γ-ray band (with efficiency up to few tenths; Hui et al., 2017). The remaining portion of E˙ rot is expected to be converted in accelerated particles. 5.1.2 Notional Pulsar Quantities Related to P and P˙ A useful set of physical quantities can be associated to a pulsar once the values of P and P˙ have been experimentally determined. The underlying assumption is that the total pulsar spin-down power equals the power released in vacuum by a rotating magneto-dipole with 3 18 3 3 magnetic moment μ ∼ BN S RN S = 10 BN S RN S,6 G cm , where RN S , RN S,6 and BN S are, respectively, the neutron star radius, the same quantity in units of 106 cm (the latter being the expected order of magnitude for a neutron star’s radius) and the surface (assumed dipole) magnetic field at the neutron star’s equator in Gauss E˙ rot =

2 2 |¨ μ|2 = 3 μ2 (2π/P )4 , 3c3 3c

(5.3)

where the rightmost equality has been derived assuming that the magnetic inclination (i.e., the angle between the magnetic and spin axes of the neutron star) is 90◦ . Rearranging equations 5.1 and 5.3 for E˙ rot , it is easy to derive a formula for the equatorial surface magnetic field of the pulsar:      3c3 12 6 6 ˙ BN S = IN S /RN S P P = 3.2 × 10 IN S,45 /RN S,6 P P˙ −14 G (5.4) 8π 2 while integrating the associated differential equation in P (t) over the time t, it is possible to obtain the time evolution of the spin period of the pulsar and, in particular, its age:   P P P P02 Tage = yr, (5.5) 1− 2 ∼ = τc = 1.6 × 106 ˙ ˙ ˙ P 2P 2P 2P−14 where P0 is the spin period of the pulsar at birth. The approximate value τc is usually termed as ‘characteristic age’ or, equivalently, as ‘spin-down age’ of the pulsar. Because that is obtained under the assumption that P0 is negligible with respect to the present

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106 yr), the latter representing the vast majority of the whole population. (for τc ∼ This evidence, as well as the observed slow-down of the pulsar’s rotation, are in good agreement with an evolutionary path starting somewhere in the upper-left corner of the diagram at the time of the supernova explosion. The newborn pulsar then rapidly proceeds towards the bulk of the ordinary pulsar population and finally spends most its active life in the middle-right part of the diagram, close to the death line, until the radio emission completely switches off. Even after that, it is very likely that the magneto-dipole emission keeps on braking the pulsar’s rotation and the now radio-quiet neutron star enters deeper into the pulsar’s graveyard, reaching a spin period of tens or hundreds of seconds.

Most pulsars located in the globular clusters do not appear in the diagram, since their P˙ is usually deeply affected by the gravitational pull of the associated stellar system. 2

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Figure 5.1. The P − P˙ diagram. The radio pulsars in the Galactic field are displayed with grey dots, whereas an empty circle highlights pulsars located in a binary or multiple system. Rotating radio transients (RRAT, e.g., Keane and McLaughlin 2011) are indicated by purple diamonds, whereas red triangles are associated to magnetars (Mereghetti et al., 2015). Pulsars associated to supernova remnants are also indicated with yellow pentagons. Dashed lines refer to the labelled values of the dipolar magnetic field, while dash-dotted lines are associated with equal spin-down age and equal spin-down luminosity. The data were collected querying the ATNF pulsar catalog at www.atnf.csiro.au/research/pulsar/psrcat/ (Manchester et al. 2005) and the magnetar catalog at www.physics.mcgill.ca/ pulsar/magnetar/main.html. Courtesy A. Ridolfi.

A second concentration of pulsars is found in the lower-left corner of the P − P˙ diagram. They are characterized by a spin period in the range 1.4 → 100 ms and P˙ in the range 10−21 → 10−16 sec/sec. Since the time of the discovery of the first object of the latter category in 1982 (PSR B1937+21; Backer et al. 1982), the pulsar community wondered about the origin of the dichotomy between the two populations of pulsars. A key factor for addressing this issue can be clearly seen upon inspecting the P − P˙ diagram. Among the second population, the pulsars hosted in binaries dominate the isolated ones (despite an observational bias that negatively affects the discovery of the former), and their percentage, about 70 per cent, is almost two orders of magnitude larger than the percentage of binaries in the ordinary pulsar population. This strongly suggests that a binary system membership must play a significant role in the formation of the second class of objects. The currently most supported model was presented in its original form in 1982 (Alpar et al. 1982) and goes under the name of the ‘recycling scenario’. The pulsars belonging

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to the second group of objects are globally called ‘recycled pulsars’. According to this model, the recycled pulsars are very old neutron stars that formed in a binary system and were then re-accelerated by the accretion of mass and angular momentum from the companion star. During this phase, the underlying compact object copiously emits X-rays and belongs to the group of the accretion-powered neutron stars. Despite the accretion process, which leads to a significant decrease of the surface magnetic field – and hence of P˙ (e.g., Patruno 2012; Istomin and Semerikov 2016; Colpi et al. 2001 for an historical perspective) – the larger gain in the spin rate compensates it, allowing the neutron star to cross again the death valley and, when the accretion of matter is over, to restart, shining as a radio pulsar. The details of the process of recycling strongly depend on the binary parameters of the system and, most notably, on the mass of the companion star and on the orbital separation after the supernova explosion that produced the neutron star. In the following sections, various general evolutionary pathways are described and matched to some of the observed systems, aiming to give a representation of the major classes of the recycled pulsars and of the associated binary progenitors. A thorough analysis of the binary pulsar zoo is given in many excellent papers and reviews, like, e.g., Verbunt (1993), Rappaport et al. (2001) or Tauris et al. (2012). We finally note that an alternative hypothesis for explaining the second cluster of pulsars in the P − P˙ diagram is to assume that they were directly formed with values of the spin rate and of the surface magnetic field close to those currently observed. Most of these theories invoke the collapse of a white dwarf into a neutron star caused by accretion (Tauris et al. 2013; Freire and Tauris 2014). 5.2.1 Binary Pulsar Families Two main physical quantities shape the distribution of the population of the recycled pulsars in the P −P˙ diagram: the intensity of the surface magnetic field of the neutron star BN S and the mass of the companion star Mc,ini at the beginning the recycling process. In fact, the accretion process onto the neutron star is regulated by the balance between the kinetic pressure exerted by the infalling plasma and the magnetic pressure associated with BN S . Various authors have investigated the details of this interplay (see, e.g., Frank et al. 2002 for a complete discussion) showing that a magnetic centrifugal barrier is produced at a radial distance rmag from the neutron star; this is called the ‘magnetospheric radius’ 1/7

−2/7

−2/7

−4/7

rmag ∼ 3000 km MN S RN S,6 Lacc,37 μ30

,

(5.7)

where MN S is the mass of the neutron star in solar masses, RN S,6 the radius of the neutron star in units of 106 cm, Lacc,37 is the luminosity due to the accretion of plasma 3 onto the neutron star surface in units of 1037 erg/s and μ30 = 0.5 BN S,12 RN S,6 is the 30 3 magnetic moment of the neutron star in units of 10 G cm (in the latter formula, the surface magnetic field is expressed in units of 1012 G). Accretion onto the neutron star surface can occur only if the peripheral velocity of the lines of force of the magnetic field at rmag (lines of force that are rotating solidly with and at the velocity of the neutron star) is smaller than the Keplerian velocity of the plasma at rmag . If this condition does not hold true, the magnetic field propels the infalling plasma away from the neutron star, preventing accretion. Hence the name ‘centrifugal barrier’, which acts as a major inhibitor for both the transfer of mass and of angular momentum to the neutron star, thus effectively halting the re-acceleration of the compact object when it reaches a short enough spin period. Given the small power index of most of the variables in the

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magnetospheric radius formula (as well as the small range of values that can be attributed to them in the typical astrophysical context), the value of the limiting period at which the barrier appears depends mainly on BN S , which can span more than three orders of magnitude. In view of the assumed relation between BN S and P˙ (see the beginning of this section), an additional line can be readily drawn in the P − P˙ diagram, called the 6/7 ‘spin-up line’ (see Fig. 5.2). It scales as P ∝ BN S (translating into P˙ ∝ P 4/3 ), with the −3/7 exact location mainly fixed by Lacc . The occurrence of the spin-up barrier naturally explains why neutron stars cannot be recycled down to very short periods of order of milliseconds and why they are distributed in the elongated area between the spin-up line and the death line. The maximum achievable spin rate depends on the value of BN S when the accretion halts, and that, in turn, depends on the past history of the neutron star (which sets the value of BN S at the beginning of the accretion process), as well as on the nature and duration of the accretion process. Although the details are still a matter of debate, it is generally assumed that the surface magnetic field progressively decays while the neutron star undergoes accretion and/or is spun up. That directly calls in play the second basic parameter mentioned at the beginning of Section 5.2, namely Mc,ini , which regulates the duration of the accretion process and the total mass transferred during this period. Fully Recycled Pulsars The recycled pulsars with very fast spin rates (usually called ‘fully recycled pulsars’) result from the evolution of a neutron star low-mass X-ray binary (NS-LMXB), in which the original companion was a relatively light main sequence star with mass 1−2 M . After the first supernova took place, and provided that the binary remains bound in this catastrophic event (which, in turn, requires that the supernova is not completely symmetric and a so-called ‘natal kick’ is imparted to the newborn neutron star; e.g., Ng and Romani 2007; Rankin 2015), the further evolution of the system proceeds at the slow rate set by the nuclear timescale (i.e., a few billion years) of the companion star. When the latter finally evolves off the main sequence, its expansion can lead – in suitably close systems – to overfill its own Roche lobe, and matter starts flowing into the Roche lobe of the neutron star. Here the matter settles into an accretion disc, slowly converging towards the magnetospheric radius of the neutron star, due to dissipative (viscous) effects that perturb the dynamics of the otherwise Keplerian disc. In view of the slow evolution of the companion star, the transfer of matter and angular momentum across rmag can continue up to a few tens of seconds or even hundreds of millions of years, allowing for a very significant decay of BN S and providing the amount of transferred mass (at least 0.1 M ; e.g., Burderi et al. 1999) which is necessary for a neutron star in the pulsar graveyard to reach a period of a few millseconds. During the Robe lobe overflow phase, the system shines as a bright neutron star-LMXB, while the strong tidal forces circularise the system and the companion progressively loses all its external layers. The typical endpoint of this evolution is a so-called low-mass binary pulsar (LMBP), hosting a rapidly spinning (often a period of less than 10 ms) pulsar in a circular orbit of ∼ 0.5 d → months with a white dwarf (most likely a helium white dwarf) companion having a mass typically ∼0.1 − 0.3 M . It is worth noting that this evolutionary picture predicts both a well-defined relation between the He-white dwarf mass and the orbital period (e.g., Tauris and Savonije 1999; Shao and Li 2012), and another tight relation between the orbital eccentricity (e, predicted in the range ∼10−6 → 10−3 ) and the orbital period (e.g., Phinney and Kulkarni 1994). These theoretical relations are compatible with the most precise data accumulated so far (e.g., Corongiu et al. 2012; Burgay et al. 2013), thus providing a good test for the recycling model.

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This evolutionary picture obtained an additional strong support with the discovery (SAX J1808.4-3658 the first recognized system, Wijnands and van der Klis 1998) of the so-called accreting X-ray millisecond pulsars (AXMPs, counting 16 objects as of June 2017.3 They are accreting binaries showing coherent X-ray pulsations with millisecond periods arising from a neutron star orbited by a low-mass companion. However, the key test for the recycling theory has been provided in the most recent years, with the discovery in the globular cluster M28 of IGR J18245-2452 (aka PSR J1824-2452I; Papitto et al. 2013) and the subsequent monitoring of that and of two other systems (Archibald et al. 2009; Bassa et al. 2014) plus one candidate (Bogdanov 2016), the members of the new class of the ‘transitional pulsar’s known so far: i.e., J1023+0038, J1227-4859 and the candidate 1RXS J154439.4-112820. They are neutron star binaries swinging from a phase of mass accretion in which they show coherent X-ray millisecond pulsations to a phase of radio emission during which they appear as fully recycled radio pulsars with a spin period compatible with that seen during the X-ray phase. They represent the long-sought link between the population of LMXBs (and, most notably, the AXMPs) and the population (or at least a part of the population) of LMBPs. Although the general picture is understood, there are still various open problems in the evolution leading to LMBPs. For instance, (a) a persistent discrepancy between the spinperiod distribution of AXMPs and LMBPs (the former tending to have shorter periods than the latter; Papitto et al. 2014); (b) the nature of the evolutionary link between AXMPs and the recently proposed subclasses of very-low-mass binary pulsars (VLMBPs) and of ultra-low-mass binary pulsars (ULMBPs), having companion masses of the order of 0.01 M and 0.001 M , respectively (Possenti 2013); (c) the still undetermined prevailing mechanism in the formation of the isolated fully recycled pulsars (Phinney and Kulkarni 1994). All these issues might be better addressed in the near future with a careful study of the so-called ‘eclipsing pulsars’, i.e., binary pulsars displaying regular or irregular eclipses and/or other distortion of the radio signal along their orbit. These effects are attributed to plasma poured into the system by the companion and not accreted onto the neutron star. The eclipsing pulsars represent a majority among the VLMBPs and ULMBPs and have been long suspected to be progenitors of the isolated pulsars. Interestingly enough, their numbers have significantly grown (as of 2015, 30 are catalogued, four times more than 10 years ago), mostly thanks to the observations of the Fermi satellite, which unveiled a large series of pulsar candidates, some of which have later shown to be fully recycled eclipsing pulsars. Mildly Recycled Pulsars Pulsars with a surface magnetic field of the order of 109 − 1010 G and a spin period ranging from ∼10 ms up to few 100 s ms are called ‘mildly recycled pulsars’. Their slower spin rate with respect to the case of the fully recycled pulsars is a consequence of the larger mass Mc,ini of the companion star at the onset of the interaction between the two stars, which makes the evolutionary phases faster. The reduced duration of the mass transfer impacts the amount of angular momentum accreted onto the neutron star and allows for a smaller decrease in the surface magnetic field. > 2 M but below ∼8 M , the most common evolutionary In particular for Mc,ini ∼ path takes the binary to experience a stage of intermediate-mass X-ray binary (IMXB), ending up with the formation of a binary pulsar in a circular orbit with a relatively massive (∼ 0.5 M → 1.3 M ) compact companion, typically a carbon-oxygen white 3

https://apatruno.wordpress.com/about/millisecond-pulsar-catalogue

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Figure 5.2. Evolution of the spin of a neutron star binary in the period versus surface magnetic field diagram (obtained from the P − P˙ diagram using the relation between P, P˙ , and BN S ).

dwarf or a oxygen-neon-magnesium white dwarf. In general, the endpoints of this kind of evolution populate the class of intermediate-mass binary pulsars (IMBPs). > 8 M , the expansion of the mass-donor star initially triggers a phase For Mc,ini ∼ during which the neutron star accretes matter from the strong stellar wind released by the companion star. At this stage, the system can appear as a high-mass X-ray binary (HMXB). However, for a suitable orbital separation, the rapid evolution of the outer parts of the donor can create the conditions for initiating a ‘common envelope’ phase, during which the neutron star spirals in towards the core of the companion while ejecting the companion star’s outer layers. At the end of the common envelope phase, the binary transforms into a neutron star orbited by a He-burning star, which is the leftover of the inner parts of the companion star. If the subsequent supernova explosion of the He-star does not disrupt the binary, we are left with a system comprised of two neutron stars in an eccentric orbit. These systems are usually known as double neutron star (DNS) binaries (although they can be alternatively categorized as high-mass binary pulsars, HMBPs). Given the short timescale of the duration of the mass exchange, the neutron star born first can have only accumulated a small amount of angular momentum, and the magnetic field experiences only a modest decay. Thus, in average, the original neutron stars in these systems are distributed towards longer spin periods than the IMBP (typically above ∼ 20 ms) and correspondingly higher surface magnetic fields. The neutron star born second has the features, and undergoes the expected evolution, of an ordinary, not recycled pulsar. Given its short lifetime (a few tens of millions of years) with respect to the mildly recycled pulsar companion, one is not very likely to observe the binary at the initial stage, when both the neutron stars are active as a radio pulsar. In fact, only one double pulsar has been discovered so far (PSR J0737-3039A/B; Burgay et al. 2003; Lyne et al. 2004) out of a population of a dozen DNSs. Additional Binary Pulsars Zoology The splitting of the recycled binary pulsars into two major families (fully recycled and mildly recycled) only captures a small portion of the large phenomenology observed in

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the couple hundred objects discovered so far. Not only can each family be subdivided again into a series of subgroups (e.g., the already reported case of the DNSs and of the IMBPs for the mildly recycled pulsars and the case of the VLMBPs and of the ULMBPs for the fully recycled objects), but there are binary pulsars with mixed characteristics: for example, a DNS system (J1906+0746; Lorimer et al. 2006) in which the only visible pulsar is an apparently ordinary pulsar, a fully recycled binary pulsar with a companion star in a high eccentric orbit (PSR J1903+0327; Champion et al. 2008) or a non-recycled pulsar younger than its massive white dwarf companion (PSR J1141-6545; Kaspi et al. 2000), also in an eccentric orbit. Assessing the nature of the companion star and the evolution of such exotic systems can only be done on a case-by-case basis. The last very intriguing addition to the variegated zoology of pulsars is demonstrated by PSR J0337+1715, the first example of a fully recycled pulsar in a hierarchical triple system composed of compact objects only: i.e., a neutron star and two white dwarfs (Ransom et al., 2014).

5.3. The Art of Pulsar Timing 5.3.1 The Case of Isolated Pulsars The vast majority of the physical and astrophysical information resulting from the observation of the radio pulsars is obtained via pulsar timing procedures. At the time of the discovery, only four numbers are associated with the new pulsar. They represent the rough estimates of the rotational period P of the source, its celestial co-ordinate (most often RA and DEC referred to the 2000.0 epoch) and the dispersion measure, DM (see the following paragraphs). For the typical observing set-ups of a pulsar search < 10μs−100μs and the celestial experiment, P is initially known to an accuracy of ∼ co-ordinate is known to an accuracy of 1−10, arcmin precision. The DM reflects the integrated column density of free electrons along the line of sight to the pulsar, and usually it is initially known with uncertainty of the order of 0.1 − 10 pc cm−3 . The timing procedure comprises two main parts: observational and interpretational. The first consists in repeated observations of the times of occurrence (often referred as times of arrival, ToAs) of a given feature in the pulsar profile with respect to a selected system of time reference. The second part involves searching the aforementioned ToAs for systematic trends on as many timescales as possible, ranging from minutes to decades. This requires the introduction of a timing model, i.e., a formula capable to reproduce all the already observed ToAs and to predict the following. This formula is built aiming to use the minimal series of parameters. If no suitable formula is found for a set of parameters, the timing model is extended (adding parameters) or rejected in favour of another model. When a model eventually accurately describes the collected ToAs, the values of the associated parameters represent the derived properties of the pulsar. A very good discussion about the pulsar timing procedures can be found, e.g., in chapters 7 and 8 of Lorimer and Kramer (2004), as well as in chapters 4 and 5 of Lyne and Graham-Smith (2012). The Collection of the ToAs There are two main reasons why a single pulse emitted by a pulsar is not suitable for associating it to a ToA. On one hand, it is due to the faintness of most pulsars at radio frequencies typically used for performing the timing, i.e., from ∼100 MHz to a few GHz: the single radio pulses are weaker and typically buried in the noise, resulting from the combined contributions of the sky and the detector system. On the other hand, any sequence of single pulses from a given pulsar shows varying shapes between one pulse to the

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Figure 5.3. A pictorial representation of the procedure of folding. In particular, the pulsar data are usually folded in sub-integrations, i.e., subsections of the whole duration of an observation, during which the rotational period of the pulsar is considered constant. Courtesy Alessandro Ridolfi.

other, as well as remarkable scatter in the distribution of the rotational phases at which the pulses occur. These two effects prevent a meaningful determination of the needed reference feature (as previously described in this section) in the profile of a single pulse. To overcome these two difficulties, a relatively large series of pulses (usually from a few hundreds to a few thousands) are summed in rotational phase, assuming a constant rotational period corresponding to the expected apparent spin period of the source at the time of the observation (see Figure 5.3). This procedure, also known as folding, improves the signal-to-noise (S/N) ratio of the profile (by the square root of the number of collected pulses) and stabilizes the shape of the profile. In order to preserve the spectral information, many folded pulse profiles over a series of Nch sub-bands are required. At the end of the day, a timing observation usually results in an array of Nsub × Nch integrated (i.e., folded) pulse profiles, associated with Nsub consecutive intervals, called sub-integrations, in which the total observation time is split. The dispersive effects of the ionized component of the interstellar medium produce a spread in the times of arrival of the pulses across each sub-band: in particular, the pulses arrive earlier at higher frequency, with the relative delay among the ToAs of the same pulse at two frequencies, fup and fdown , which follows a inverse square law of the observing frequency −2 −2 − fup )DM, t(DM ; fup , fdown ) = D(fdown

(5.8)

2

e where D = 2πm = 4.148808(3) × 103 MHz2 pc−1 cm3 s is the dispersion constant, e ec the electron charge, me the electron mass and c the speed of the light in vacuum. DM is defined as the path-integral

d DM = ne dl (5.9) 0

If not accounted for, the dispersion would cause the pulses to be completely cancelled within a large enough sub-band (see Figure 5.4). In view of that, a procedure called dedispersion is applied to the data, either by entirely removing the effect in each sub-band separately (coherent de-dispersion) or by keeping the width of the sub-bands narrow enough so that the smearing of the pulse is smaller than the time resolution of the data. Of course, the procedure of de-dispersion requires the knowledge of the DM of the given pulsar.

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Figure 5.4. When travelling across the interstellar medium (ISM), the pulses emitted from a pulsar are spread in time, according to the observed radio frequency. The plot on the left shows the typical parabolic relation between the observing frequency (across the band reported in the vertical axis) and the delays due to the dispersion in the ISM, expressed in units of the pulsar phase on the horizontal axis. The target of this observation was a pulsar (B1933+16) rotating at 359 ms period and the DM of which is 158 pc cm−3 . For these parameters, the total delay across the selected band equals the duration of an entire rotation of the underlying neutron star. Therefore, no pulse would emerge from the data if the radio signal were detected in one single frequency channel covering the whole band (see lower panel). Splitting the data acquisition into multiple channels provides the possibility to correct for the dispersive effects of the ISM and recover the pulses (see lower panel of the picture on the right). Courtesy A. Ridolfi.

As previously mentioned, each of the Nsub × Nch observed pulse profiles have to be time-tagged. That is usually accomplished by exploiting the time information produced during the observations by an accurate clock that is available at the radio observatory. That is, in turn, regularly monitored and adjusted by using the time which is distributed via the Global Positioning System (GPS). The last step in the determination of a ToA requires the comparison of each of the Nsub × Nch observed profiles (and the associated time-tags) with a ‘template profile’. The latter is a very high S/N profile, which (i) results from the coherent sum of the profiles of many observations of the given pulsar at a given frequency, or (ii) it can be a simulated noiseless profile reproducing the observed shape of the pulse at a given frequency. In a heuristic way, a ToA is finally obtained by first measuring the shift in time among the observed pulse profile and the template profile (see Figure 5.5) and then adding that shift to the reference time tagged to that pulse profile. As a rule of thumb, the characteristic uncertainty in the determination of a ToA scales linearly with the width of the pulse (in seconds) and inversely with the S/N of the pulse profile. The ToAs measured with this procedure are called topocentric ToAs, since they reflect the times of arrival of the pulses as collected at the focal plane of the radio telescope. The cadence and the duration of a timing campaign are strictly related to the objectives of the specific experiment. For ordinary and not very interesting pulsars, the campaign usually lasts about one year with a cadence of one observation per month. For high-precision timing of pulsars that are part of a pulsar timing array (PTA), the observations extend for many decades with a cadence of up to twice a month. A much

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Figure 5.5. A pictorial view of the process for extracting a topocentric ToAs. After having chosen a reference (also known fiducial) point in the profile of a pulsar, a ToA is obtained by determining the shift among a template profile and the profile of a sub-integration. Courtesy Alessandro Ridolfi.

more intense coverage (reaching one or more observations a day within a limited time span) is sometimes needed for unveiling relativistic effects in a binary pulsar system and/or is required at the beginning of the timing procedure for close binary systems. The Modelling of the ToAs A timing model is considered satisfactorily when the associated timing formula can exactly account for all the neutron star rotations occuring within the time span covered by the available observations. The validity of a given timing model is then supported (or falsified) by its predictive capabilities: i.e., if the model is able to predict the time of arrival of the successive pulses. Once it is assumed that the ToA of a pulse coincides with reference epoch tep (usually chosen within the total time span of the available observations), the number of rotations, N (t), of the neutron star occurred from tep to a generic time, t, can be described by a power series for the time difference between the generic time, t, and tep , namely 1 1 N (t) = νep × (t − tep ) + ν˙ ep × (t − tep )2 + ν¨ep × (t − tep )3 + ... (5.10) 2 6 Here, the parameters νep , ν˙ ep , ν¨ep , ... represent the neutron star spin frequency and the first, second, and higher-order derivatives of νep at the epoch tep . In order to accomplish the aim of exactly counting all the neutron star rotations, one has to determine the set of parameters νep , ν˙ ep , ν¨ep , ... for which all N (ti ) as well as N (tfut ) are close to an integer number, where ti is the i-th observed ToAs and tfut represents the ToA of any future pulse. The quality of a timing model can be expressed by introducing the so-called timing residuals R(ti ), which are defined as R(ti ) = N (ti ) − n(ti )

(5.11)

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Figure 5.6. An example of the timing residuals resulting from what could be deemed a good timing ‘solution’ of an ordinary pulsar. No trend is evident in the residuals along the ∼ 2 yr data-span. The residuals are consistent with zero.

where n(ti ) is the nearest integer to N (ti ). An acceptable rotational model (often referred also as a coherent timing solution) requires R(ti )  1 for all the observed ToAs in the time interval ranging from tep to tep + Δtspan , where Δtspan is the total duration of the time span covered by the performed observations (see Figure 5.6). On a practical side, the introduction of the time residuals also suggests an obvious way for approaching a satisfactorily timing model, i.e., varying the parameters νep , ν˙ ep , ν¨ep , ... in order to bring all the R(ti ) as close as possible to zero. This is typically done by minimizing the expression  2 R(ti ; νep , ν˙ ep , ν¨ep , ...) 2 , (5.12) χ = Σi i where i is the uncertainty on the i-th ToA in units of the pulsar spin period, and i runs over all the available ToAs in the given time span. The formula above can be generalized, assuming that the timing model depends on p parameters α1 , α2 , ..., αp . In this case, one has to apply a least squares fit with p free parameters, in order to minimize the expression  2 R(ti ; α1 , α2 , ..., αp ) . (5.13) χ2 = Σi i Both a numerical and a visual test are applied to qualify the goodness of a timing solution resulting from the aforementioned multi-parameter fit. On one hand, one requires that the RMS of all the residuals for the set of the best fit parameters to satisfy RMS/P −3 < . On the other hand, a plot of the residuals with respect to the times ti should not ∼ 10 show any recognizable trend, and the residuals should be randomly distributed about the null mean value.

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Barycentring the ToAs In order to properly combine the ToAs for a given source taken at various times and at various telescope sites, it is necessary to report all of them using a common reference time, a unique timescale and a common reference system. An obvious choice is to transfer all the topocentric ToAs to the solar system barycentre (SSB), which is a suitable approximation for an inertial frame. However, it is necessary to compensate for the effects of the different frequencies used for the observations: this suggests referring all the ToAs to a unique frequency, which is nominally chosen to be infinite. All of that implies that one needs to apply the following time-transfer formula: ti,bary = ti + tclock −

D + ΔR + ΔE + ΔS , f2

(5.14)

where ti are the topocentric ToAs and ti,bary are called the barycentric ToAs. The various terms in the conversion formula, Equation 5.14, are examined in the following paragraphs. The tclock term goes under the name of clock correction. Its aim is to account for the drifts and/or irregularities experienced by the clock (most often a maser) located at each radio telescope site and thus imprinted in the topocentric ToAs. In order to do that, a series of corrections are applied, eventually leading one to link the time observed at the radio telescope to the Terrestrial Time realization of the Temps Atomique International (TAI) [i.e., TT(TAI)]. Since 1971, the latter differs from the TAI by only a constant offset. The conversion from the time of the local clock to TT(TAI) is the most accurate that can be done in real time, often using the time of the GPS system as an intermediate step. When performing high-precision timing, it is possible to further retroactively improve the quality of the adopted corrections by referring the times to the TT(BIPMyy) time. This is the best available reference time, the relation of which to TT(TAI) is tabulated for each year ‘yy’ by the Bureau International des Poids et Mesures (BIPM). The second term in Equation 5.14 accounts for the already mentioned effect (see Equation 5.8) due to the dispersion of the radio pulses in the interstellar medium. In particular, the factor D(ti ) is given by D(ti ) = D × DM(ti ),

(5.15)

where f is the Doppler-corrected observing frequency (the correction accounting for the relative movement of the radio telescope with respect to the SSB). By comparing Equations 5.8, 5.15 and 5.14, it is evident that the reference frequency is implicitly assumed to be infinite in the ToA conversion formula, Equation 5.14. The other three terms in Equation 5.14 are known as the Roemer delay, Einstein delay and Shapiro delay and can be respectively written as: ΔR =

r · n (r · n)2 − |r|2 + c 2cd

 Gmk v2 dΔE = + ⊕2 − constant 2 dt c dk,⊕ 2c

(5.16)

(5.17)

k

2GM ln (1 + cos θ). (5.18) c3 In equation 5.16, n represents the unit vector associated with the segment joining the SSB with the pulsar, whereas r is the vector associated with the segment joining the SSB with Earth. The Roemer delay is associated with the relative motion of the telescope ΔS = −

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Figure 5.7. The motion of Earth around the solar system barycentre produces periodic variations of 1 yr in the length of the optical path followed by the radio waves emitted by a pulsar and reaching Earth. If the celestial coordinates of the pulsars are poorly known, the errors in the assumed right ascension (Δα) and declination (Δδ) of the source reflect in annual variations in the time of arrival of the pulses, as seen from the solar system barycentre. By fitting for the amplitude and the phase of the resulting curve, it is usually possible to determine Δα and Δδ and thus to infer the celestial coordinates with high astrometric accuracy. The level of achievable precision depends on the ecliptic coordinates of the pulsar.

with respect to the SSB (see Figure 5.7) and in almost all cases produces the largest contribution to Equation 5.14. In particular, annual variations of up to about 16 cos bE minutes are expected in the ToAs from a given source at ecliptic latitude bE because the radius of Earth’s orbit is about 8 light-minutes. The ToAs from an ordinary pulsar can be typically determined with millisecond precision, whereas for recycled pulsars, microsecond precision is obtained. These values correspond to a fraction ∼10−6 (or ∼10−9 for recycled pulsars) of the aforementioned annual variations (but only for pulsars that are very close to the ecliptic pole). As a consequence, once these annual variations are properly accounted for in the timing model, the celestial position of a radio pulsar can be roughly determined with ∼10−6 (or ∼10−9 for recycled pulsars) precision. That translates into a sub-arcsec (or sub milli-arcsec for the recycled pulsars) precision in the celestial coordinates of the pulsar. Thus, the timing of a pulsar over ∼yearly timescales provide a powerful way of obtaining an astrometric determination of the position of a source. It is worth noticing that this is in striking contrast with the usually very poor angular resolution of each observation producing a ToA, reflected in the already mentioned large uncertainty in the pulsar coordinates at the time of its discovery. We also note that the rightmost term in Equation 5.16 results from the consideration of the curvature of the wave-front carrying the electromagnetic pulses. In fact, for pulsars located at large d, this term becomes completely negligible with respect to the other in the right hand side of the equation, reflecting the fact that the wave-fronts tend to become progressively flatter for pulsars positioned at increasing distances. The variations in the residuals due to the rightmost term have a typical period of six months and can be measured when the pulsar is close (typically less than 1–2 kpc), bright and rapidly rotating, thus ensuring very narrow error bars on each observed ToAs. In agreement with this description, a successful measurement of that term directly provides constraints on the distance of the pulsar, via the measurement of the pulsar parallax, as depicted in Figure 5.8.

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Figure 5.8. The geometrical scheme underlying the measurement of a pulsar’s distance d via the determination of the trigonometric parallax, namely the angle π ˆ . Provided that the pulsar is not too far and the precision achievable in the determination of its ToAs is good, the pulsar timing procedure reveals the apparent variations in the celestial coordinates of the pulsar due to the Earth’s orbital motion. The maximum effect will repeat every six months, corresponding to the largest change in the relative position of Earth with respect to the pulsar.

Of course, a meaningful application of the Roemer delay term must be supported by a very precise knowledge of the orbit of Earth around the sun. It also requires knowledge of the location and the motion of the major bodies in the solar system at the time of the observations, as well as the time evolution of the non-uniform Earth rotation. This information is embedded in Earth and in the other solar system bodies ephemeris, which are routinely updated and announced either by the Jet Propulsion Laboratory 4 and/or by the bulletin regularly published by the International Earth Rotation Service.5 The Einstein’s delay ΔE is a superposition of the relativistic time delay and of the gravitational redshift caused by the motion of the Earth and by the mass of the other solar system bodies. In fact, the velocity of Earth with respect to the SSB (v⊕ ∼ 10−5 c) is not constant, as a consequence of the non-circular Earth motion. Thus, variations in ΔE are expected, and its time derivative can be expressed like Equation 5.17, where G is the gravitational constant, mk are the masses of the other solar system bodies and dk,⊕ are the distances from these bodies to Earth. The last equation of the group of three is associated with the Shapiro delay ΔS . The effect was first theoretically reported in Shapiro (1964) and is due to the extra time (with respect to a flat space) needed for a radio wave to follow the geodesic in the curved gravitational field of a celestial body. The formula in Equation 5.18 refers to the case of the Shapiro delay imposed by the sun. A similar formula holds true for all solar system bodies, while θ, in the context of a generalized definition, is the angle resulting from the vector indicating the direction from the pulsar to the telescope with the other vector pointing from the telescope to the source responsible for the Shapiro delay – Sun and Jupiter in most cases. ‘Solving’ an isolated pulsar means obtaining a reasonably low RMS for the entire set of residuals and not seeing any trends in the distribution of the residuals over the data

4 5

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Figure 5.9. Trends that appear in the time residuals when one parameter at a time in the adopted timing model is not set at its optimum fit value. In real life, the trend shown by the residuals of an unsolved pulsar will be a superposition of these trends displayed. Courtesy A. Ridolfi.

span of the observation. Getting there is usually possible using an iterative process. Since the discovery parameters are never precise enough, they will leave a clear trend in the residuals. As shown in Figure 5.9, an erroneous value of νep will impart a linear trend to the residuals. An imprecise value of the pulsar spin frequency derivative ν˙ ep will leave a parabolic shape in the timing residuals distribution. Furthermore, if the celestial coordinates are not precise enough, a sinusoidal trend with a 1 yr periodicity can be easily recognized in the residuals, whereas a proper motion of the pulsar is associated to a 1 yr sinusoid with a linearly varying amplitude. Finally, a sinusoidal trend with the periodicity of a semester is an indication of the occurrence in the ToAs due to parallax effect because of the orbital motion of Earth around the Sun. When this effect is visible, a direct geometrical determination of the distance to the pulsar can be obtained. Similarly, a wrong determination of the DM is shown as a parabolic trend when plotting the residuals from a given observation of a pulsar at various radio frequencies. In practice, one tries to remove one trend at a time, typically starting with the most evident one, resulting from a non-optimal νep . This can be done on a short data span of few weeks. However, if the data span is too short, a strong covariance occurs between ν˙ ep and the celestial coordinates of the pulsar. This is due to the difficulty of recognizing and disentangling the parabolic shape in the residuals (associated to the required corrections to the value of ν˙ ep ) with respect to an arc ascribed to wrong celestial coordinates. Hence, for an isolated pulsar, one has to wait longer than a semester, and typically almost a full year, before one can break the covariance and obtain a significant determination of both the position and spin-period derivative. When a new set of parameters – ‘pulsar ephemeris’ – is available, it replaces the older set in the process of folding and de-dispersing the raw pulsar data. Of course, this applies to the new observations but, whenever possible, also to the previous data. This, in turn, often leads to an improved determination of the old ToAs, from which a new fitting process produces an even better ephemeris, i.e., a set of parameters with a smaller intrinsic uncertainty than the previous set. The iterations continue until two consecutive loops in the process do not produce statistically significant differences in the values of the total RMS of the residuals or in the uncertainties associated with the derived parameters. At that point, one can say that one has produced the most refined timing solution, given the available dataset.

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If the pulsar observations are not interrupted, the aforementioned best solution will have to undergo additional changes in the subsequent years. On one hand, the trend due to the proper motion of the pulsar (and, more rarely, that due to the parallax) may become evident, and thus two (or more) additional parameters would need to be entered in the model, in turn requiring to refit all the ToAs, producing a new ephemeris. On the other hand, the pulsar may undergo timing irregularities (described later in the chapter) that prevent one to extend the temporary best solution to a longer data span. In the latter case, there are several approaches. The first option is to produce a set of ‘local timing solutions’, each of them reporting the interval of time, usually expressed in Modified Julian Day (MJD). A second option is to try to join the local solutions by introducing additional parameters in the timing model that have poor or no physical meaning, but which allow the observer to track the rotation phase of the star over the whole available data span. The adoption of one or the other option depends on the necessities of the specific kind of investigation. For instance, the second option is the most sensible if the aim is to exploit the ephemeris to properly associating a rotational phase to a list of time-tagged photons collected by a high-energy detector aboard a satellite. Generally, the first option is adopted if the aim is to study the physical processes underlying the timing irregularities experienced by the pulsar. When the operation of barycentring the ToAs is performed, the Terrestrial Time has to be translated into a suitable time measured at the SSB. In this respect, it is worth noting that the ephemeris are always reported as barycentric values; i.e., they are valid as measured at the SSB. Moreover, in more recent radio pulsars studies, the parameters that depend on the time are reported adopting Barycentric Coordinate Time (TCB), which is the default choice for tempo2 (www.atnf.csiro.au/research/pulsar/tempo2, (Hobbs et al. 2006; Edwards et al. 2006), the most popular code for deriving timing solutions (see Figure 5.10). In physical terms, TCB represents the proper time associated with

New par

F1

F2

New tim

Restart

Undo

DM

PMRA

PMDEC

PX

0531+21 (nms = 50.983 ms) pre–fit

10

–4

2×10

–4

y

0

x

–10

pre–fit post–fit date orbital phase serial day of year frequency TOA error year elevation rounded MJD

F0

Prefit Residual (sec)

RE–FIT

DECJ

–4

RAJ

–100

0

100

MJD–54212.1 Quit

Clear

Measure

List

Print

Highlight

Help

plk v.3.D (G.Hobbe)

Figure 5.10. A typical screenshot of tempo2 when investigating an isolated pulsar. In the upper part, one can select the parameters of the adopted timing model to be fitted.

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an observer located at the SSB, provided the gravity effects of all the solar system bodies (including the Sun and Earth) are absent. That is equal to saying that all the effects of general relativity (GR) on the time are accounted for, and hence removed when translating the ToAs to the SSB. That task is implicitly done by the application using the correction factors ΔR , ΔE and ΔS that appear in Equation 5.14. It is useful to highlight that the adoption of TCB is a relatively recent choice made by most of the radio pulsar community and that this choice is not usually implemented in most of the high-energy telescopes aboard the satellites orbiting Earth. In fact, the latter (as well as most of the old radio pulsar ephemeris) use Barycentric Dynamical Time (TDB). In order to avoid confusion, it is then very important to check if the adopted timescales are identical for the radio and X-ray observations. If not, software is available for converting the data to a common time-reference scale. 5.3.2 The Case of the Binary Pulsars For a pulsar in a binary system, additional terms will have to be introduced in the timetransfer function of Equation 5.14. In particular, one needs to convert the pulsarcentric ToAs (i.e., the ToAs referred to the pulsar proper time measured near the pulsar’s surface, where the radio pulses are assumed to be emitted) into the ToA that would be measured if the pulses were emitted at the barycentre of the pulsar binary system, if the local space-time were flat and if all the relativistic effects related to the presence of a pulsar companion (see later) were accounted for. In all practical cases, the required corrections are embedded in four additional terms appearing in Equation 5.14, which eventually reads tBin i,bary = ti,bary + ΔRBin + ΔEBin + ΔSBin + ΔABin .

(5.19)

The four rightmost terms in Equation 5.19 account for the geometrical delays and the Doppler effect appearing in a purely classical physics treatment of the orbital motion, but they also include the corrections to the predictions of the Newtonian physics that take place in strong gravitational fields and/or when the stars move with high relative speed. The nature of these effects, and the information which can be extracted from them, are the focus of Section 5.4. As to the last addendum, namely ΔABin , it is associated with the variations of the aberration along the orbital path of the pulsar, which is indeed an effect entirely described by classical physics. Nevertheless, the two related parameters result to be almost degenerate with the relativistic parameters, and, therefore, aberration is often dealt with in the context of relativistic effects. Not surprisingly, the largest contribution is due to the Roemer term ΔRBin . In agreement with the original Roemer observation of delays and/or anticipations in the time of occurrence of phenomena involving Jupiter satellites, the pulses from a binary pulsar arrive earlier when the pulsar is in the near side of its orbit (with respect to the SSB) around the companion and later when the pulsar is on the far side of the orbit. The procedure for ‘solving a binary pulsar’ is more involved and less easy to describe with respect to the case of an isolated pulsar (discussed earlier). Indeed, it first depends on the duration of the binary period. For a short enough orbital period (typically hours to a few days), the immediate objective is to approximately derive the Keplerian parameters via a series of shortly spaced, repeated observations of the target. In the most difficult cases, this requires one to plan observations to measure for acceleration and velocity shown of the pulsars and then apply suitable codes with the aim of fitting these quantities with a model including the Keplerian parameters. In fact, experience shows that one can reconstruct an orbit from the ToAs, only if a preliminary solution

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that is close to the correct values of Keplerian parameters is fed to the ToA fitting code, most often tempo6 or tempo2.7 Once the Keplerian orbital parameters are determined (typically over a data span not longer than a few orbital periods), one can start searching for the trends already discussed in the case of the isolated pulsars, which, as anticipated, requires data covering about one year. For the majority of the relativistic binary pulsars (the orbital period of which is typically short), the relativistic effects on the orbit start becoming evident and measurable only after a few years of observations; thus their inclusion in the model takes place well after the Keplerian, rotational, positional and kinematical parameters of the pulsar have been properly assessed. There might be exceptions though. For example, in the case of the double pulsar the relativistic effects, most notably the advance of the periastron, manifested only after a few days of observation (Burgay et al. 2003), and three relativistic parameters were measurable within a few months. When the orbital period of the binary pulsar is comparable to or longer than the annual motion of Earth, the aforementioned approach cannot be applied, and solving the system requires one to simultaneously account for all (orbital and non-orbital) parameters. This might create additional complications to the process. As a matter of fact, the way one treats every binary pulsar is different, and solving these systems is a scientific art. Measuring the Pulsar and the Companion Star Masses In the context of classical physics (namely, by solving the two-body problem in Newtonian gravity), it is easy to show that from the modulation of the ToAs along the orbit, it is possible to extract five of the Keplerian parameters that describe the binary system (see Figure 5.11). In particular, the timing of a binary pulsar gives the possibility to determine the orbital period Pb , the longitude of the periastron ωp (typically measured with respect to the ascending node of the orbit), the epoch of the passage at the periastron T0 , the eccentricity e and the projection of the semi-major axis along the line-of-sight x = ap sin i, where i is the orbital inclination (equal to 0◦ for face-on orbits and 90◦ for edge-on orbits). An additional sixth Keplerian parameter, the position angle on the sky of the ascending node (van Straten et al. 2001) can also be determined in favourable cases. A useful quantity resulting from a suitable combination of some Keplerian parameters is the mass function of the pulsar, given by f (Mp ) =

(Mc sin i)

3

(Mp + Mc )

2

3

=

4π 2 (ap sin i) , GPb2

(5.20)

where Mp and Mc are the masses of the pulsar and the companion star, respectively. For any adopted value of Mp , a lower limit for Mc results from the assumption of an edge-on orbit. For an assumed value of Mp , a complete determination of Mc is possible only if the mass function of the companion star can also be determined, e.g., via the observation of the radial velocity curve of the companion using optical spectroscopy. If the mass function f (Mc ) of the companion star is measurable, then by looking Equation 5.20, it can be seen that the ratio of the two mass functions gives the ratio of the masses:  3 Mp f (Mc ) = (5.21) f (Mp ) Mc 6 7

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Figure 5.11. Graphical representation of a pulsars orbit. Courtesy A. Ridolfi.

The occurrence of eclipses in the system might in principle provide an opportunity for an independent determination of the orbital inclination i and Mc (again for an assumed value of Mp ). However, in practice, this approach, commonly used for the eclipsing spectroscopic binary systems observed in the optical band, cannot applied to the case of the binary pulsars. On one hand, the tiny size of the neutron star makes it very unlikely to detect any effect on the observed optical light curve of the companion star when the pulsar transits in front of it. On the other hand, although the eclipse of the radio emission of a pulsar when it passes behind the companion is not an uncommon phenomenon, the eclipse itself is, in general, caused not only by the spherical body of the companion, but also by the surrounding plasma, the geometry of which is usually irregular and difficult to determine. Another way to independently estimate Mc is to perform both accurate photometry and high-resolution spectroscopy of the companion star in the optical band and then find the theoretical stellar structure model that best fits the observed data – the latter typically being the effective temperature of the star surface, its total emitted luminosity (these two quantities give the radius of the star) and the surface gravity, which (coupled with the constraints on the radius) leads to constraints on Mc . Since the availability of high-resolution spectroscopy provides a radial velocity curve for the companion star, the mass ratio can also be derived by combining the optical data with the radio timing observations of the pulsar. At the end of the day, both Mc and Mp can be separately obtained for this favourable case. However, this method relies on very precise

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high-resolution spectroscopy, which is hard to perform for the vast majority of the pulsar companions, even with 8 m class telescopes. Moreover, this approach is based on a set of a priori assumptions about the stellar structure of the companion star. Although stellar evolutionary models have reached a very high level of reliability, their predictions might still be questionable when the star results from the complex evolution in a binary system. As described in Section 5.4, a simultaneous determination of both Mp and Mc in a binary pulsar system can be obtained by the observations of relativistic effects. The optical observations described earlier take into account the fact that only a single relativistic effect is enough to obtain constraints on both masses. However, usually a more accurate determination of Mc and Mp is possible when two relativistic effects are measured from the analysis of radio timing data. In contrast with the previously described case, this measurement does not depend on any detailed assumptions about the stellar structure of the companion star but only on the fact that the companion is a spherical star with a size much smaller than the orbital separation. Good and Bad Timers As anticipated in the preceding text, only a fraction of the ∼2,650 radio pulsars (as of May 2018) catalogued so far behave well as timekeepers. Various kinds of timing irregularities affect the vast majority of the pulsars in the form of glitches (i.e., sudden increases in the spin frequency), anti-glitches and, in most other cases, the so-called timing noise. For a long time (e.g., Anderson and Itoh 1975; Shabanova 2007), the glitching behaviour has been thought to be due to a sudden reconfiguration of the matter in the neutron star’s interior, but the scientific discussion about this hypothesis is still open. Recently, progress has been made in the interpretation of the timing noise, which has been suggested to be a phenomenon related with instabilities involving the magnetosphere of the pulsars (Lyne et al. 2010). Whatever its origin, the two effects are mostly observed among the youngest and ordinary pulsars (Hobbs et al. 2010). The recycled pulsars, at least within the limit of the level of precision of the currently available ToA measurements, (e.g., Shannon and Cordes 2010), seem much less affected (or influenced only at a low level, e.g., Cognard and Backer 2004) by these effects. In summary, the recycled pulsars (and, in particular, those with the highest timing stability, i.e., not showing a significant signature of timing noise) are better timekeepers than the ordinary pulsars. On top of that, there is the fact that the precision in the determination of the ToAs scales approximately with the width of the pulses and hence, in turn, with the spin rate. Therefore, the rapid rotation of the recycled pulsars leads one to measure their ToAs with a much higher accuracy than for the ordinary pulsars. On the other hand, being a recycled pulsar with a good timing stability is not enough to be a good clock for high-precision timing. There are at least two other features of a recycled pulsar that significantly contribute to qualifying it as a potential high-quality clock: namely, the shape of its pulse and its flux density.

5.4. Some Examples of Pulsar Uses: Gravity Theories Tests For some binary pulsar systems, the accuracy of the collected ToA is so high that by using only the description of the system (obtained in the context of the classical physics), one cannot derive any acceptable timing solution. Clearly, additional physics is needed, in order to interpret the behavior of these systems, but . . . what specific physics? Some of the most intriguing applications in the study of pulsars result from the possibility of addressing that question and sometimes deriving unique constraints on the fundamental physical laws that govern the universal phenomenon called gravity.

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In fact, shortly after the publication of the theory of GR in 1915, a flurry of alternate theories of gravity started appearing. Among theories is a very large group called ‘metric theories of gravity’, the only ones that had some chance to be viable (Will 2006). Three assumptions characterize the metric theories: (i) a symmetric metric must exist; (ii) the world lines of all test bodies are geodesics of the metric; (iii) when tested and applied in a local reference frame in free fall, all the non-gravitational laws of physics are those written in the framework of special relativity. The combination of these three assumptions implies that, within any metric theory, the phenomenology of what is generically called gravity can be intimately associated with the occurrence of ‘curved space-time’. A comprehensive summary of most gravity theories proposed so far is available – e.g., in the textbook by Will (1993) and in a series of updates regularly published by the same author (Will 2006, 2010, 2014). Of course, one can in general imagine the presence of various kinds of fields, associated to additional tensorial, vectorial and scalar contributions. Hence, a large family of tensor/scalar theories, tensor/vector theories or bi-tensor theories appeared. In contrast to the special case of GR (which does not have any tunable parameters), the other metric theories incorporate a few free parameters, which, in principle, could be set by suitable observations. On the other hand, the presence of these unknown parameters makes these theories intrinsically less predictive with respect to the theory of GR. A very useful framework for constraining a very large class of metric theories of gravity, and for understanding the consequences of their application, is given by the parametrized post-Newtonian (PPN) formalism. It investigates all the metric theories of gravity in the context of the so-called ‘weak-field’ conditions, in practice, considering perturbations of the order of 1/c2 with respect to Newtonian physics. The deviations from the Newtonian physics are expressed by the values assumed in a list of 10 PPN parameters, which have been introduced to represent a list of specific physical effects (Will 2006). In order to better quantify the meaning of the aforementioned weak-field conditions, it is useful to introduce the dimensionless ratio  = Egrav /Erest , where Egrav represents the classical gravitational potential energy Egrav ∼ GM 2 /R associated with a body of mass M and radius R, whilst Erest is the rest mass energy of the same body, Erest = M c2 (where c is the speed of light in vacuum, and G, the gravitational constant). For our planet,  ∼ 10−10 , and for the Sun,  is only ∼ 10−6 . Therefore, all the experiments performed in our planetary system since the very famous eclipse-based experiment set-up by Eddington in 1919 have been performed in the so-called weak-field limit of gravitational physics. A good review of some of the past and future experiments of this class can be found in Peron (2014) and Ciocci et al. (2016), respectively. Given the fact that GR has no adjustable parameters, one single discrepancy between observations and predictions would imply the falsification of the theory, at least in the range of the performed observations. As a matter of fact, GR has survived all these tests (e.g., Will 2014 for a summary) and today it is more alive and trusted than ever. The continuum space-time and the deterministic valence of GR still do not combine well with the discrete components and the probabilistic approach of quantum theories. It is well known that the problem of finding a unified picture is important, mostly in being able to describe the events that took place during the very early phase of the evolution of the universe. It is possible that in such extreme conditions, GR (and/or quantum theories) fails and other gravity or particle physics theories replace – or better yet, incorporate – the existent theories. Descriptions of some of the different flavors of these putative theories can be found in Angus et al. (2013), Stecker et al. (2015) or Antoniadis and Cotsakis (2017).

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From an observational point of view, the issue is whether the proposed alternate metric theories of gravity is potentially able to incorporate GR and work in extreme physical regimes. To answer this question, solar system experiments are not enough, and it is necessary to perform experiments for probing GR and/or the alternate gravity theories in strong-field conditions. This, in turn, implies finding cosmic laboratories for which the aforementioned dimensionless quantity  approaches unity. There are at least three astrophysical contexts in which the aforementioned conditions are satisfied and which involve tight binary systems including compact objects such as neutron stars and/or black holes. • During the final phase of the coalescence of a binary that includes neutron stars and/or black holes, the orbital velocity approaches c and the orbital separation approaches the size of the star(s). In this situation, the strong-field condition (i.e.,  → 1) is certainly met. Due to the mass of the involved stars, these events of spiral-in and/or final coalescence were long suspected to be primary targets for the ground-based gravitational wave (GW) detectors (such as aLIGO and aVIRGO) for the future space detector eLISA and for the pulsar timing arrays (PTAs). In the historical detections of GWs from the merging of two black holes (obtained by aLIGO in September 2015, Abbott et al., 2016) and from the merging of two neutron stars (observed by aLIGO and aVIRGO in August 2017, Abbott et al., 2017) are magnificient proofs of the opportunity of using the mentioned kind of binaries for testing the behavior of gravity in extreme conditions. • The processes of electromagnetic emission (often in the X-ray band) that occurs in neutron star and/or black hole binaries close to the event horizon is, in principle, another very promising laboratory. In particular, from the investigation of the spectral and timing features arising from the neighbourhood of the last stable orbit of the accretion disc that surrounds the neutron star or a black hole hosted in a binary system with a mass-losing companion star. XMM-Newton and (in the past) RossiXTE are instruments that are/were well suited for starting to explore this field. Future X-ray missions with much larger collecting areas should provide remarkable constraints on GR in the strong-field regime. • Some pulsars in compact binary systems also present superb tools for investigating theories of gravities due to the combination of two factors, namely the occurrence of  ∼ 0.2 at the pulsar’s surface8 and being stable clocks, which allows observers to precisely account for both the rotation and the orbital motion of the star by means of the procedures explained in Section 5.3. In the following, three stereotypical cases will be reported, in which pulsars have been used to constraining GR, alternate gravity theories and fundamental principles related to gravity. Illustrative Example: Testing GR with the Double Pulsar The most widely used framework for testing both GR and a very large category of gravity theories was originally proposed by Damour and Deruelle (1985, 1986) and 8 Since the orbital separation is always much larger than the radius of the involved compact star(s), the orbital motion of all known binary pulsar systems does not nominally occur in a strong-gravity regime. Nevertheless, a common feature of most gravity theories (GR being the remarkable exception) is the occurrence of significant effects imparted to the orbital motion by the ‘self-field energy’ (a strong-field effect) of the binary components, which becomes important > 0.1. This is the basis for the possibility of using some binary pulsars to for bodies with  ∼ measure the level of the deviations from the motion predicted by GR, and, in turn, for testing both GR and alternate theories in the presence of strong-field effects.

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exploits the post-Keplerian (PK) parameters. Their usefulness stems from two features: (a) the Post-Keplerian parameters are all phenomenological quantities, i.e., they can be measured following a well-defined operational prescription, which is not dependent on their physical meaning (if any) or on the chosen theory of gravity to be investigated; (b) for any selected theory of gravity (in the large group of the metric theories), one can express all the PK parameters as function of the Keplerian parameters of the binary system (which can be measured with high precision) and a function of the pulsar and companion star’s mass. Given these features (a and b), once the Keplerian parameters have been measured, any PK parameter can be written as a function of the unknown pulsar and companion star’s mass. Therefore, the observation of two PK parameters results in the possibility of separately inferring the mass of the two bodies in the binary. Of course, the values of the masses are strictly dependent on the theory of gravity used for calculating the aforementioned functional forms. Along this line of reasoning, it is obvious that the measurement of NPK > 2 parameters leads one to derive NPK − 2 independent tests of self-consistency for the chosen gravity theory. A graphical representation is often useful to better highlight the power of this methodology (see Figure 5.12). For any given gravity theory, the allowed areas in the mass-versus-mass plot is between a pair of lines (shown by a thin strip in the diagram), which are associated to the extremes of the confidence interval for the available measurement of each specific PK parameter. The existence of a common area for all the pairs of lines (strips) implies that the specific gravity theory survives the test (see left panel of Figure 5.12). If no area of overlap is found (e.g., right panel of Figure 5.12) then the theory cannot avoid rejection. For the theory of GR, the functional forms for the five most commonly measured PK parameters are listed in the following publications: Damour and Deruelle (1986); Taylor and Weisberg (1989); Damour and Taylor (1992).

Figure 5.12. The diagrams report on the two axes the masses of the components of a pulsar binary for which three PK parameters have been measured. The panels are built adopting two putative gravity theories. Each different strip shows the combined constraints on the masses of the two bodies, which results from the same measured value of a given PK parameter (including its uncertainty) for the gravity theory that is specific to that panel. The shaded areas in the bottom part of the diagrams are excluded by the the mass function of the pulsar. Left panel : An area of overlap for the three strips is present; hence this gravity theory passes the test. Right panel: No common area occurs for the three strips; this imposes the rejection of the gravitational theory.

ω˙ γ P˙ b r s

Radio Observations and Theory of Pulsars and X-ray Binaries  −5/3 Pb = 3 (T M )2/3 (1 − e2 )−1 2π  1/3 Pb 2/3 = e T M −4/3 m2 (m1 + 2m2 ) 2π −5/3    m1 m2 Pb 192π 73 2 37 4 = − 1 + e + e (1 − e2 )−7/2 5 2πT 24 96 M 1/3 = T  m2  −2/3 Pb −1/3 T M 2/3 m−1 = x 2 ≡ sin i 2π

159 (5.22) (5.23) (5.24) (5.25) (5.26)

In the above equations, m1 and m2 represent the masses of the two stars, M = m1 + m2 , x = a sin i and T ≡ GM /c3 = 4.925490947 μs. As to the physical meaning of the five PK parameters: • • •

ω˙ is representative of the relativistic advance of the periastron, γ accounts for the gravitational redshift and time dilation, P˙b is associated with the orbital shrinking. If one assumes GR, the decreases of the orbital period are caused by the emission of quadrupolar gravitational radiation, • The parameters r and s ≡ sin i are the so-called rate and shape of the Shapiro delay (Shapiro 1964), i.e., a time delay due to the deformation of the space-time surrounding the companion star. Similar sets of equations can be written for alternate gravity theories. (Some examples are reported in Will 1993, 2006.) At variance with the case of GR, they include one or a few free parameters. Double neutron star binaries in close orbits are the best targets for the application of the methodology described in the previous paragraphs. The first of these binary systems, PSR has B1913+16, was discovered at Arecibo in 1974 (Hulse and Taylor 1975). The pulsar’s spin is a period of 59 ms and the system has a highly eccentric orbit (e = 0.61) with an orbital period of about 7.8 hr. The measurement of three PK parameters is possible, namely ω, ˙ γ, and P˙b . From the first two, the masses of two pulsars were separately measured with high accuracy (Weisberg et al. 2010). By inserting these masses in Equation 5.24, a very accurate prediction for P˙ bGR was obtained, which was then compared with observations, after subtracting non-relativistic contributions. (For a complete discussion of these terms, see Damour and Taylor 1991.) The agreement between the predictions of GR and the observations was remarkable (it is currently at the 0.2 per cent level; Weisberg et al. 2010), and the study of the B1913+16 binary provided the first wondrous – although indirect – demonstration of the emission of GWs from a binary system, the results of which were awarded the Nobel Prize for Physics in 1993. PSR B1534+12 (Wolszczan 1991) was the first pulsar in a double neutron star system in which all the 5 PK parameters reported in equations 5.22→5.26 were measured, although the observed P˙b is largely contaminated by the Shklovskii contribution 2 /(cd)], which is difficult to estimate due to the uncertainties in (P˙ b /Pb )Shk = [v⊥ the value of the pulsar’s distance (Stairs et al., 2002). By far, the best laboratory for testing the limits of GR in strong field conditions is PSR J0737-3039, often also reported as the double pulsar system. The first of the two pulsars present in this system was discovered in April 2003 (Burgay et al. 2003) during a 1.4 GHz survey of new pulsars at high-galactic latitudes (Burgay et al. 2006), using the multibeam receiver of the Parkes radio telescope. The Keplerian parameters were

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constrained after three observations covering a total of 15 hr. The orbital period is only 2.4 hr, and the eccentricity is e ∼0.09. Later observations also showed that both compact objects were observable as radio pulsars (Lyne et al. 2004): PSR J0737-3039A (hereafter mentioned as psrA) is a mildly recycled pulsar rotating with a spin period of ∼22 ms, whereas PSR J0737-3039B (hereafter psrB) is an ordinary pulsar with a spin period of ∼2.7 s. The uniqueness of the double pulsar as a laboratory results from various favourable characteristics. First, the short orbital separation ranging from about 8.0 × 105 km and 1.2 × 106 km; secondly, the high relative velocity of the two neutron stars, ∼10−3 c; thirdly, the high orbital inclination, ∼89◦ ; lastly, the possibility of observing (at least for some of the time) both the stars as radio emitters, thus having two timekeepers in the binary. In particular, the first two features imply the occurrence of relevant relativistic effects, while the third characteristic improves the capability of detecting some of these effects. In fact, the relativistic advance of periastron ω˙ of psrA (Burgay et al. 2003) was measured using two observations taken only one week apart. It amounts to ∼16.9◦ yr−1 , much larger than ever measured before. Three further PK parameters were measured in less than half a year, γ and the two Shapiro delay parameters. The first direct determination of the mass ratio R (Lyne et al. 2004) in a double neutron star binary was also achieved. The orbital decay P˙b was also detected in only 1.5 yr of observations; the emission of GWs will lead the two neutron stars to coalesce in ∼85 million years. After an additional four years of observations, the angular velocity ΩB associated with the relativistic precession of the spin axis of psrB (Breton et al. 2008; Stella and Possenti 2009) was also detected and measured. In summary, in less than five years of collecting data, seven constraints (i.e., seven strips) could be drawn in the mass-versus-mass plot reported in Figure 5.13: five resulting from the PK parameters, plus the two associated to the mass-ratio R and ΩB . A tiny region of overlap of all the strips was present, providing 7 − 2 = 5 successful tests of GR in strong-field conditions (Kramer and Stairs 2008). Among the published results, the most stringent test is for the parameter s of the Shapiro effect: for that, the prediction is in agreement with the observations at the 0.05 per cent level (Kramer et al. 2006). Despite the (supposedly temporary) disappearance of the radio signal emitted from psrB (Perera et al. 2010), the continuous accumulation of ToAs is leading to large improvements in the measurement of all the PK parameters with respect to the already published results (Kramer et al. 2018, in preparation). This will open intriguing perspectives for the detection of new effects (e.g., aberration) and for setting new constraints to some fundamental laws of physics, such as the existence of preferred frames (Wex and Kramer 2007). Around the corner (i.e., with the advent of new, powerful instruments, like the Square Kilometer Array), there will be the chance to also reveal second-order relativistic effects related to the spin of psrA, such as the LenseThirring effect (Stella and Possenti 2009; Wex and Kramer 2010; Kehl et al. 2016). The determination of the spin contribution to the advance of the periastron in the double pulsar binary will provide an opportunity to directly measure the moment of inertia of psrA (Damour and Schafer 1988) and hence to significantly narrow the list of the acceptable EoS of matter at nuclear densities (Lattimer and Schutz 2005). Illustrative Example: Constraining Alternate Gravity Theories As discussed in the previous section, the PSR J0737-3039 binary is the best test bed for constraining the level of validity of the predictions of GR in strong-field conditions. However, it is not the optimal system for constraining alternate gravity theories.

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Figure 5.13. Mass-versus-mass plot for the PSR J0737-3039 system (aka the double pulsar). The Newtonian mass functions of the two pulsars lead one to exclude the shaded regions. The mass ratio R determines another constraint. There are six additional pairs of lines that enclose a region in the diagram, which is compatible with the predictions of GR for the observed values of the five PK parameters and of the relativistic precession rate ΩB of the spin axis of psrB. A tiny quadrilateral region results from the intersection of all the mentioned constraints (Breton et al. 2008).

The reason resides in the two bodies of the system – two neutron stars – being very similar in both mass and size. In most alternate theories, the deviations from the predictions of GR are strongly enhanced when the degree of compactness (i.e., the ‘self-energy’) of the two stars are significantly diverse. That certainly holds true for the binaries including a neutron star, typically having ns ∼ 0.1−0.2, and a white dwarf with wd ∼ 10−4 . An illustrative case is that of the PSR J1738+0333 system (Jacoby 2005), which contains a white dwarf with mass ∼0.18 M orbiting the pulsar in an almost circular orbit e ∼ 3×10−7 with an orbital period of ∼8.5 hr. After about 10 years of timing at Arecibo, a very good determination of the intrinsic P˙ b of the system was obtained, while the masses of the white dwarf and, in turn, of the pulsar were derived (see Section 5.3.2 for details) from accurate optical photometry and spectroscopy (Antoniadis et al. 2012). The precise < 7 per cent accurate values of the masses and of the Keplerian parameters provided a ∼ GR −15 ˙ . prediction for the orbital damping in the context of GR Pb = −27.7+1.5 −1.9 × 10 A comparison with the observed P˙ bint = −25.9 ± 3.2 × 10−15 places very significant upper limits on the emission of dipole GWs (Freire et al. 2012). In fact, the emission of

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Figure 5.14. Experimental constraints on the values for |α0 | (logarithmic scale) and β0 (linear scale). They are the parameters that appear in function a(ψ) = α0 ψ + 0.5 β0 ψ 2 , which describes the coupling of matter in the universe with an hypothetial scalar field ψ. The vertical axis in the plot represents the Brans-Dicke theories (Brans and Dicke 1961; Will and Zaglauer 1989). The lowest part of the vertical axis (the one that tends to −∞) corresponds to the case of GR. The permitted regions are those located underneath the solid lines, each related to the observations of various systems (PSR B1534+12, PSR J0737-3039, PSR B1913+16, PSR J11416545 and PSR J1738+0333), as well as the results of various experiments carried out in the weak-field limit of the solar system, amongst which are the Lunar Laser Ranging (LLR) and the Cassini mission observations. The shaded area is compatible with all the tests. The importance of the role of PSR J1738+0333 in setting some of the current limits is depicted (Freire et al. 2012).

dipole GWs (on top of the quadrupole component resulting from the tensorial gravity field) is one of the most conspicuous effects that are expected from the application of tensor/scalar gravity theories9 . Hence, the aforementioned constraints on the amount of dipole radiation translates to significant constraints in the diagram of Figure 5.14. Illustrative Example: Testing SEP in Triple System Another milestone in pulsar science was set with the recent discovery of a pulsar in a multiple system, J0337+1715, with two other compact objects, namely two white dwarfs (Ransom et al. 2014). The lightest white dwarf (∼0.197 M ) orbits the 2.7 ms pulsar in 9

The tensor/scalar gravity theories are often invoked in the framework of the grand unification theories (i.e., Kaluza-Klein, superstring etc.) and as a ‘natural’ hypothesis to account for the past (or the current) phases of accelerated expansion of the Universe (Damour and Esposito-Far`ese 1992, 1996).

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−4

about 1.63 d in an almost circular orbit (e = ∼ 10 ), whereas the heaviest white dwarf (∼ 0.410 M ) follows a much wider trajectory of ∼327 d duration with an eccentricity of e ∼ 10−2 . The accurate timing of the system (Archibald 2015) has also allowed us to infer the pulsar’s mass (∼1.438 M ), the orbital inclination of the inner orbit (∼39.2◦ ) with respect to the line of sight, as well as the small angle between the orbital plane of the inner and the outer white dwarfs. As a matter of fact, this triple compact object system represents the best cosmic benchmark for three-body dynamics. Even more interestingly, it promises to become the reference laboratory for putting tight constraints on the strong equivalence principle (SEP). In fact, the inner binary is immersed in the gravitational field generated by the outermost white dwarf, which is located at only ∼ 2 light-mins from the inner binary. Hence the possibility of carefully investigating if the two inner stars (which have significantly different compactness parameter ) are falling differently in the external gravitational field. REFERENCES Abbott, B. P., Abbott, R., Abbott, T. D. et al. 2016. Observation of gravitational waves from a binary black hole merger. Phys Rev Lett, 116(6), 061102. Abbott, B. P. et al. 2017. GW170817: Observation of gravitational waves from a binary neutron star inspiral. Phys. Rev. Lett, 119(16), 161101. Abdo, A. A., Ajello, M., Allafort, A. et al. 2013. The second fermi large area telescope catalog of gamma-ray pulsars. ApJS, 208(Oct.), 17. Acero, F., Ackermann, M., Ajello, M. et al. and Fermi-LAT Collaboration. 2015. Fermi large area telescope third source catalog. ApJS, 218(June), 23. Alpar, M. A., Cheng, A. F., Ruderman, M. A. and Shaham, J. 1982. A new class of radio pulsars. Nature, 300(Dec.), 728–730. Anderson, P. W., and Itoh, N. 1975. Pulsar glitches and restlessness as a hard superfluidity phenomenon. Nature, 256(July), 25–27. Angus, G. W., Diaferio, A., Famaey, B. and van der Heyden, K. J. 2013. Cosmological simulations in MOND: The cluster scale halo mass function with light sterile neutrinos. Mon Not R Astron Soc, 436(Nov.), 202–211. Antoniadis, I., and Cotsakis, S. 2017. Infinity in string cosmology: A review through open problems. Int J Mod Phys D, 26, 1730009–2056. Antoniadis, J., van Kerkwijk, M. H., Koester, D. et al. 2012. The relativistic pulsar-white dwarf binary PSR J1738+0333 – I: Mass determination and evolutionary history. Mon Not R Astron Soc, 423(July), 3316–3327. Archibald, A. 2015 (Apr.). Testing Einstein’s theory of gravity in a millisecond pulsar triple system. In APS Meeting Abstracts. Archibald, A. M., Stairs, I. H., Ransom, S. M. et al. 2009. A radio pulsar/X-ray binary link. Science, 324(June), 1411. Backer, D. C., Kulkarni, S. R., Heiles, C., Davis, M. M. and Goss, W. M. 1982. A millisecond pulsar. Nature, 300(Dec.), 615–618. Bassa, C. G., Patruno, A., Hessels, J. W. T. et al. 2014. A state change in the low-mass X-ray binary XSS J12270-4859. Mon Not R Astron Soc, 441(June), 1825–1830. Bogdanov, S. 2016. A NuSTAR Observation of the Gamma-ray-emitting X-ray Binary and Transitional Millisecond Pulsar Candidate 1RXS J154439.4-112820. Astrophys J, 826(July), 28. Brans, C. and Dicke, R. H. 1961. Mach’s principle and a relativistic theory of gravitation. Phys Rev, 124(Nov.), 925–935. Breton, R. P., Kaspi, V. M., Kramer, M. et al. 2008. Relativistic spin precession in the double pulsar. Science, 321(July), 104.

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Papitto, A., Torres, D. F., Rea, N. and Tauris, T. M. 2014. Spin frequency distributions of binary millisecond pulsars. Astron Astrophys, 566(June), A64. Patruno, A. 2012. Evidence of fast magnetic field evolution in an accreting millisecond pulsar. Astrophys J, 753(July), L12. Pavlov, G. G., Sanwal, D. and Teter, M. A. 2004. Central compact objects in supernova remnants. Page 239 of: Camilo, F., and Gaensler, B. M. (eds), Young Neutron Stars and Their Environments. IAU Symposium, vol. 218. Perera, B. B. P., McLaughlin, M. A., Kramer, M. et al. 2010. The evolution of PSR J0737-3039B and a model for relativistic spin precession. Astrophys J, 721(Oct.), 1193–1205. Peron, R. 2014. Testing general relativistic predictions with the LAGEOS Satellites. Advances in High Energy Physics, 2014(May). Phinney, E. S. and Kulkarni, S. R. 1994. Binary and millisecond pulsars. Ann Rev Astron Astrophys, 32, 591–639. Pierbattista, M., Harding, A. K., Gonthier, P. L. and Grenier, I. A. 2016. Young and middle age pulsar light-curve morphology: Comparison of Fermi observations with γ-ray and radio emission geometries. Astron Astrophys, 588(Apr.), A137. Podsiadlowski, P. 2007 (Mar.). Electron-capture supernovae and accretion-induced collapse of ONeMg white dwarfs. In: Paths to Exploding Stars: Accretion and Eruption. Possenti, A. 2013 (Mar.). Binary pulsar evolution: Unveiled links and new species. Pages 121–126 of: van Leeuwen, J. (ed), Neutron Stars and Pulsars: Challenges and Opportunities after 80 years. IAU Symposium, vol. 291. Rankin, J. M. 2015. Toward an empirical theory of pulsar emission. XI. Understanding the orientations of pulsar radiation and supernova kicks. Astrophys J, 804(May), 112. Ransom, S. M., Stairs, I. H., Archibald, A. M. et al. 2014. A millisecond pulsar in a stellar triple system. Nature, 505(Jan.), 520–524. Rappaport, S., Pfahl, E., Rasio, F. A. and Podsiadlowski, P. 2001. Formation of compact binaries in globular clusters. Page 409 of: Podsiadlowski, P., Rappaport, S., King, A. R., D’Antona, F., and Burderi, L. (eds), Evolution of Binary and Multiple Star Systems. Astronomical Society of the Pacific Conference Series, vol. 229. Shabanova, T. V. 2007. Slow glitches in the pulsar B1822-09. Astrophys Space Sci, 308(Apr.), 591–593. Shannon, R. M. and Cordes, J. M. 2010. Assessing the role of spin noise in the precision timing of millisecond pulsars. Astrophys J, 725(Dec.), 1607–1619. Shao, Y. and Li, X.-D. 2012. Formation of millisecond pulsars from intermediate- and low-mass X-Ray binaries. Astrophys J, 756(Sept.), 85. Shapiro, I. I. 1964. Fourth test of general relativity. Phys Rev Lett, 13(Dec.), 789–791. Shibata, S., Watanabe, E., Yatsu, Y., Enoto, T. and Bamba, A. 2016. X-ray and rotational luminosity correlation and magnetic heating of radio pulsars. Astrophys J, 833(Dec.), 59. Stairs, I. H., Thorsett, S. E., Taylor, J. H. and Wolszczan, A. 2002. Studies of the relativistic binary pulsar PSR B1534+12. I. Timing analysis. Astrophys J, 581(Dec.), 501–508. Stecker, F. W., Scully, S. T., Liberati, S. and Mattingly, D. 2015. Searching for traces of Planck-scale physics with high energy neutrinos. Phys Rev D, 91(4), 045009. Stella, L. and Possenti, A. 2009. Lense-Thirring precession in the astrophysical context. Space Sci Rev, 148(Dec.), 105–121. Tauris, T. M. and Savonije, G. J. 1999. Formation of millisecond pulsars. I. Evolution of low-mass X-ray binaries with P orb 2 days. Astron Astrophys, 350(Oct.), 928–944. Tauris, T. M., Langer, N. and Kramer, M. 2012. Formation of millisecond pulsars with CO white dwarf companions - II. Accretion, spin-up, true ages and comparison to MSPs with He white dwarf companions. Mon Not R Astron Soc, 425(Sept.), 1601–1627. Tauris, T. M., Sanyal, D., Yoon, S.-C. and Langer, N. 2013. Evolution towards and beyond accretion-induced collapse of massive white dwarfs and formation of millisecond pulsars. Astron Astrophys, 558(Oct.), A39.

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Tauris, T. M., Langer, N. and Podsiadlowski, P. 2015. Ultra-stripped supernovae: progenitors and fate. Mon Not R Astron Soc, 451(Aug.), 2123–2144. Taylor, J. H., and Weisberg, J. M. 1989. Further experimental tests of relativistic gravity using the binary pulsar PSR 1913 + 16. Astrophys J, 345(Oct.), 434–450. van den Heuvel, E. P. J. 2010. High space velocities of single radio pulsars versus low orbital eccentricities and masses of double neutron stars: Evidence for two different neutron star formation mechanisms. New A Rev, 54(Mar.), 140–144. van Straten, W., Bailes, M., Britton, M. et al. 2001. A test of general relativity from the three-dimensional orbital geometry of a binary pulsar. Nature, 412(July), 158–160. Verbunt, F. 1993. Origin and evolution of X-ray binaries and binary radio pulsars. Ann Rev Astron Astrophys, 31, 93–127. Weisberg, J. M., Nice, D. J. and Taylor, J. H. 2010. Timing measurements of the relativistic binary pulsar PSR B1913+16. Astrophys J, 722(Oct.), 1030–1034. Wex, N. and Kramer, M. 2007. A characteristic observable signature of preferred-frame effects in relativistic binary pulsars. Mon Not R Astron Soc, 380(Sept.), 455–465. Wex, N. and Kramer, M. 2010. Generic gravity tests with the double pulsar. ArXiv e-prints, Jan. Wijnands, R. and van der Klis, M. 1998. A millisecond pulsar in an X-ray binary system. Nature, 394(July), 344–346. Will, C. M. 1993. Theory and Experiment in Gravitational Physics. Cambridge University Press. Will, C. M. 2006. The confrontation between general relativity and experiment. Living Rev Relativ, 9(Mar.), 3. Will, C. M. 2010. The confrontation between general relativity and experiment. In: Ciufolini, I., and Matzner, R. A. A. (eds), General Relativity and John Archibald Wheeler. Astrophysics and Space Science Library, vol. 367. Will, C. M. 2014. The confrontation between general relativity and experiment. Living Rev Relativ, 17(June), 4. Will, C. M. and Zaglauer, H. W. 1989. Gravitational radiation, close binary systems, and the Brans-Dicke theory of gravity. Astrophys J, 346(Nov.), 366–377. Wolszczan, A. 1991. A nearby 37.9-ms radio pulsar in a relativistic binary system. Nature, 350(Apr.), 688–690. Woods, P. M. and Thompson, C. 2006. Soft gamma repeaters and anomalous X-ray pulsars: Magnetar candidates. Pages 547–586 of: Lewin, W. H. G., and van der Klis, M. (eds), Compact stellar X-ray sources. Cambridge University Press.

6. Incorporating Gamma-ray Data into High-Time Resolution Astrophysics ELIZABETH C. FERRARA1 Abstract At high energies, high time-resolution data is limited by statistics, with gamma-ray instruments like Fermi-LAT detecting fewer than a single photon per day for the average source. However, the time of arrival for each high-energy photon is known very accurately. This means that high-energy data can still be useful for sources with timing signatures, such as pulsars or galactic binaries. With its all-sky observing strategy, the LAT also provides monitoring for sources with gamma-ray signals associated with flares or state transitions. Transitional pulsars are a prime example of these sorts of systems, as transitions between their low-mass X-ray binary and rotation-powered states appear to correlate with an offset in overall gamma-ray flux. Here we discuss the Fermi mission and instruments, the wide variety of gamma-ray sources, and details of the maximum likelihood analysis method. We also describe some recommendations for using gammaray data when investigating sources with time signatures that are singificantly shorter than the time separating individual gamma-ray events.

6.1. Introduction Gamma rays are the signature of non-thermal physical processes throughout the universe, regardless of scale. These extremely energetic photons are the result of relativistic effects, Compton upscattering, nuclear decay, or possibly exotic physics. In the realm of high time-resolution astrophysics, the gamma-ray regime is necessarily poorly understood, as individual gamma-ray detections from bright sources occur on the order of minutes to hours apart. Incorporating gamma rays into high time-domain astrophysics thus usually requires adding timing information from other wavebands into the analysis.

6.2. Gamma-ray Production and Detection Gamma radiation in the universe is produced exclusively through non-thermal processes, as there are no materials that can get hot enough to generate blackbody radiation above temperatures of several KeV. Instead, gamma rays are produced by highly accelerated charged particles. These particles can be electrons, positrons, protons, or even heavier nuclei. They get accelerated in a number of ways, but once they are moving near the speed of light, gamma-ray production is possible. The primary astrophysical mechanisms to produce high-energy gamma rays are synchrotron radiation, inverse Compton scattering, bremsstrahlung radiation, and pion production and decay. Relativistic boosting occurs when photons are upscattered by electrons moving near the speed of light. This process is most commonly found in astrophysical sources that contain accretion disks (to generate the ambient photon field) and relativistic jets (which provide high-energy electrons). Shocks are most commonly seen where relativistic jets 1 This work was supported by the Fermi project office and NASA grant NNX12AO86G. The author thanks the organizers of the Winter School in Astrophysics for their hospitality during the school and also thanks collaborators Steve Fegan, Tyrel Johnson, and Liz Hays for their assistance in preparing these lectures.

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Figure 6.1. The physical process with the highest interaction cross section for gamma rays (shown here for iron) depends strongly on energy. Processes with a steep energy dependence are not ideal choices when designing a wide-band detector. (Credit: Wikimedia Commons)

meet the ambient medium, or on the leading edges of supernova remnants. In shocks, particles are accelerated as they travel back and forth across the shock boundary and are eventually boosted to cosmic ray energies. Curvature radiation is produced by relativistic electrons in strong magnetic fields. For millisecond pulsars, the neutron star is rotating so rapidly that electrons moving along magnetic field lines do not have to travel very far above the surface to get to relativistic speeds. Fundamentally, gamma rays are a by-product of massive charged particles being accelerated to relativistic speeds. While the product of the most extreme environments in the universe, particle acceleration and the resulting gamma rays have been detected from sources over a wide range of distance scales. Once produced, gamma rays are difficult to detect. Because they have such high energies, they tend to be scattered and absorbed in matter. This makes focusing gamma rays basically impossible with current technologies. Instead, astronomers use detectors tuned to detect the results of those interactions. The cross section for those interactions depends on the scattering material and on the energy of the photon (see Figure 6.1). At low energies (< 100 keV), gamma rays primarily interact with matter via coherent scattering, or by the photoelectric effect, but the cross sections depend very strongly on energy and so are not good choices for a wide-band detecter. Instead, low-energy gammaray instruments use the properties of Compton scattering to detect and localize the origin of an incoming gamma ray. Above 10 MeV, the pair production mechanism takes over, meaning that gamma-ray detection turns into a particle physics problem. By including high atomic weight material in gamma-ray detectors, the photons become more likely to pair produce into charged particles that can be easily tracked. Astrophysical gamma rays are completely scattered by Earth’s atmosphere. At very high energies, that scattering can be detected as flashes of light created via Cherenkov

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radiation. Ground-based observatories such as MAGIC, HESS, and VERITAS use arrays of optical telescopes to detect these flashes. They then model the light shower to determine the initial photon trajectory and energy. Below ∼50 GeV, these flashes are no longer detectable, and the detector must move above the atmosphere.

6.3. A Brief History of Gamma-ray Astronomy The first gamma-ray detectors in space were launched in the 1960s on the U.S. military’s Vela satellites and intended to monitor international nuclear weapons testing. However, they found that a number of bright gamma-ray flashes detected by the spacecrafts were not nuclear tests but instead had an astrophysical origin. At first, scientists theorized that these bursts might be coming from neutron stars in our galaxy. But we now know that these were the first detections of gamma-ray bursts (GRBs), bright flashes of gamma rays produced at cosmological distances. Since that time, four other gamma-ray focused missions have been flown. SAS-2 (Small Astronomy Satellite) flew from November 1972 to June 1973. It was the first mission to map the gamma-ray sky with any level of detail and the first to detect the unusual gammaray source now known as ‘Geminga.’ In 1975, the European Space Agency (ESA) launched the COS-B mission. During its 6.5-year lifetime, COS-B completed the first catalog of 25 gamma-ray sources and mapped the gamma-ray distribution along the galactic plane. The Compton Gamma-ray Observatory (CGRO) was launched by NASA in 1991 and flew for nine years. The observatory’s four detectors were designed for a wide variety of gamma-ray science, spanning six decades in energy. Science topics included detection of GRBs (BATSE), as well as low- (OSSE), medium- (COMPTEL), and high-energy (EGRET) gamma rays. In the high-energy regime, EGRET expanded the gamma-ray catalog to 271 sources.

6.4. The Fermi Gamma-ray Space Telescope Launched in 2008, NASA’s Fermi Gamma-ray Space Telescope is the successor to these missions. It hosts two instruments: the Gamma-ray Burst Monitor (GBM; 8–40 keV) with multiple large NaI and BGO scintillators observing the entire unocculted sky, and the Large Area Telescope (LAT; 20 MeV–1 TeV) that can view 20 per cent of the sky at any given time. The mission was designed to perform a long-term survey of the entire highenergy gamma-ray sky, with the ability to detect and characterize transient and variable sources. The observing profile for the spacecraft rocks to alternating hemispheres for each 95-minute orbit. While Fermi is in ‘sky survey’ mode, every point on the sky is observed by the LAT for at least 30 minutes every three hours. In addition, an autonomous repoint capability is available for bright transients (like GRBs) detected by either the GBM or the LAT. The duration of a repoint can be adjusted, but it is currently 2.5 hours. The mission is anticipated to continue at least through 2020, which will make it the longest-flying gamma-ray mission to date. The instrument team’s four-year point source catalog contained more than 3,000 sources, increasing the known gamma-ray source population by more than an order of magnitude. The team works hand in hand with the ground-based community to trigger rapid follow-up observations to GRBs, Galactic novae, and bright AGN flares by rapidly releasing GCN notices about these events. 6.4.1 Fermi’s Gamma-ray Burst Monitor The Fermi Gamma-ray Burst Monitor (Meegan et al., 2009) provides simultaneous low-energy spectral and temporal measurements for all GRBs within the LAT FOV. The combined GBM and LAT effective energy range spans more than seven energy

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decades from 10 keV to 300 GeV. The GBM extends the energy coverage from below the typical GRB spectral break at ∼ 100 keV to above the LAT’s low-energy cutoff for interinstrument calibration. Furthermore, the GBM’s sensitivity and FOV are commensurate with the LAT’s to ensure that many bursts will have simultaneous low-energy and highenergy measurements with similar statistical significance. The GBM also assists the LAT in detecting and localizing GRBs rapidly by providing prompt notification to the ground of a burst trigger. Finally, the GBM provides coarse GRB locations over a wide FOV that can be used to repoint the LAT at particularly interesting bursts (both inside and outside the LAT FOV) for gamma-ray afterglow observations. These locations are also sent to the ground to notify external follow-up observers. The GBM detects gamma-ray bursts at a rate of about 250 per year, which makes it the most prolific GRB instrument on orbit at this time. Of these, only a few percent have concurrent detections by the Large Area Telescope (LAT), Fermi’s higher-energy instrument. In addition, some Fermi-detected GRBs are also detected and observed by the SWIFT observatory. While rare, these bursts provide an opportunity to see the GRB phenomenon across a wide range of the electromagnetic spectrum.

6.4.2 Fermi’s Large Area Telescope The Large Area Telescope (Atwood et al., 2009) uses similar detector technology to that used by ground-based particle accelerators. It consists of 16 towers in a four-byfour square configuration. Each tower contains an 18-layer silicon tracker (TKR), with interleaved tungsten foils to promote pair production by incoming photons. Below the tracker lies an eight-layer hodoscopic CsI crystal calorimeter, used to measure photon energies up to (and beyond) a TeV. Surrounding these components is a segmented anticoincidence detector (ACD), used to veto any signal caused by incoming charged particles (see Figure 6.2). Unlike previous instruments, the LAT ACD is segmented into 89 individual tiles. Each plastic scintillating tile is embedded with optical fibers that feed into photomultiplier tubes (PMTs) that can detect hits by charged particles. The fibers are combined so that alternating fibers are fed into different PMTs for redundancy. That way, the loss of a single PMT does not affect the usefulness of the tile. To date, no ACD PMTs have failed during the mission. An incoming photon passes undetected through the ACD and enters the tracker. As it travels through, if it passes near a heavy tungsten nucleus, the photon may convert to a e− e+ pair. These charged particles then interact with the silicon strip detectors in alternating XY planes, spaced to allow flight time between each layer. The tracker strips are read from both ends, and the split point for that readout is completely configurable, making the tracker readout fully redundant. These hits allow the trajectories of the charged particles to be mapped, thus providing the incoming direction of the photon. Finally, the two particles deposit their energy into the cesium iodide logs. The resulting flash of light is measured to determine the energy of the original photon. The light yield is measured by PIN diodes at either end, with two channels for high and low gain. The ratio of the measurements from each end of the log provides a longitudinal position for the energy deposition. The calorimeter logs are arranged in eight layers of 12 crystals each, in alternating directions. This provides a fully three-dimensional image of the energy shower, which allows for reasonable reconstruction even when a significant portion of the energy is lost at the bottom of the instrument. Of course, this is a great simplification. The actual process of correctly reconstructing a photon’s true direction and energy requires instrument simulation as well as preflight

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Figure 6.2. Fermi’s Large Area Telescope uses a modular design for tracker, calorimeter, and anti-coincidence detector. (Credit: NASA)

Figure 6.3. The effective area response of Fermi’s Large Area Telescope as a function of energy (left) and incidence angle (right) for the Pass 8 event reconstruction. (Credit: FermiLAT Collaboration)

and on-orbit calibrations (Abdo et al., 2009). The gamma-ray emission detected by the LAT is a combination of astrophysical point sources, astrophysical diffuse emission, and instrumental background emission that changes based on the position and orientation of the spacecraft. In order to create instrument response functions, simulated events are run through the reconstruction pipeline, and the results are compared against the original photon energy and direction. Analyzing the results of the simulations allows the instrument team to design a series of cuts on the raw data that produce data sets with varying levels of non-astrophysical background events, called “event classes.” In LAT parlance, each event reconstruction

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Figure 6.4. Gamma rays are produced by a wide range of sources, from gamma-ray bursts at cosmological scales to terrestrial gamma-ray flashes here on Earth. The yellow text denotes classes of gamma-ray emitters that were not known prior to the launch of the Fermi observatory. (Credit: NASA)

software build is called a “pass”, while each set of data cuts for classification is a “revision”. A specific version of the reconstructed LAT data can be completely described with these two values. Using data cuts to remove background events also cuts real photons, resulting in lower source statistics for lower background event classes. Analysis of gamma-ray data is, therefore, an exercise in balancing signal versus background. In addition to event classes, the LAT instrument team has subdivided the full data set into quarters based on the quality of the direction reconstruction (PSF) or the quality of the energy reconstruction (EDISP).2 When statistics are not a concern for detection, these “event types” can be used to remove events with the worst reconstructions to improve either localization (using the PSF event types) or energy resolution (using EDISP event types). Since these data sets are 100 per cent overlapping, it is not possible to select on both direction and energy resolution quality. Details of both event classes and event types are given in Section 6.7.1.

6.5. High-Energy Astrophysical Processes While the physical processes that produce gamma rays are few, they can be found from a very wide range of astrophysical sources all across the universe. These fall into a few basic categories: • Compact objects with accretion disks and relativistic jets Many gamma-ray sources fall into this category. The most common are active galactic nuclei (AGN), galaxies with supermassive black holes that are in a period 2

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of active accretion. In addition, when very massive stars collapse to form a black hole, they may briefly form an accretion disk and jet (MacFadyen and Woosley, 1999). This is one theory for the origin of “long” GRBs with durations greater than a few seconds. When two neutron stars merge to form a black hole, the same process may occur but with a much shorter duration (“short” GRBs). Stellar mass black holes in binaries may also have an accretion disk + jet structure (microquasars). Interestingly, while no known microquasar has been detected by Fermi to date, high-mass binaries containing neutron stars undergoing active accretion have been detected as gamma-ray sources. • Shock fronts Shocks are seen in locations where highly accelerated particles slam into a slower, colder, ambient medium. For long GRBs, this process has been theorized to be one possible explanation for the multiple peaks seen in time series data. As a relativistic jet is formed, it will send particles outward, slamming them into the layers of stellar envelope that have been shed by the star in its giant phase (Zhang et al., 2003). In supernova remnants (SNRs), the expanding shock accelerates particles along the leading edge, boosting their energy each time the particle crosses the shock front. This is the mechanism responsible for the acceleration of cosmic rays in our galaxy (Reynolds, 2008). In addition, as these cosmic rays escape the remnant, they may interact immediately with the ambient medium, producing a gamma-ray glow. Shocks also likely play a role in the variability seen in AGN light curves, as well as the transient gamma-ray emission seen from stellar novae (Ackermann and Fermi-LAT Collaboration, 2014). • Strong, rotating magnetic fields Compact objects like neutron stars have their magnetic fields frozen into their crust. The surface of these dense stellar remnants is a hot plasma, easily capable of ionizing the surrounding medium, which frees electrons that stream along the neutron star’s magnetic fields. As the neutron star rotates, the electrons are trapped by the magnetic field lines and follow them away from the star. At the magnetic poles, these lines are open, and the electrons produce radio emission. The radio beam is tight, and it is seen from Earth as radio pulses as the rotation moves the beam in and out of Earth’s line of sight, hence the term “pulsar.” Away from the poles, the lines are closed, and the electron must move farther and farther from the pulsar until it approaches the speed of light. The electron must emit gamma rays as a cooling mechanism, allowing it to continue to curve back toward the star’s surface. Gamma-ray emission, however, occurs over a wide area at quite a distance from the star, making the gamma rays fan out in a wide swath (Caraveo, 2014). For young pulsars, the radio emission beam from the poles will often be completely distinct from the gamma-ray fan beam. However, nearly all millisecond pulsars that are detected in gamma rays also are seen in radio, indicating that the radio beam for MSPs must be broader than for young pulsars (see, e.g., Venter and Harding, 2014). • Particle decay and annihilation A wide variety of radioactive high–atomic weight elements produce gamma rays when they decay, producing a series of gamma-ray emission lines before becoming stable. These nuclear lines all lie in the keV–MeV range and are below Fermi’s sensitivity range. In addition, it is possible that dark matter particles – especially the weakly interacting massive particles (WIMPs) that are theorized based on

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• Magnetic reconnection events When magnetic field lines reconnect, a significant amount of energy is released. As part of this process, nearby electrons will be accelerated to high velocities along the new field lines. When they hit nearby target material, they can emit gamma rays. This emission is seen during strong solar flaring events (Holman et al., 2011). In some instances the gamma-ray emission is bright enough to localize the emitting region on the sun’s disk, confirming that the gamma rays are positionally coincident with the location of an optical flare (Ajello et al., 2014). Magnetic reconnection has also been postulated to be the mechanism behind flaring events from the Crab pulsar wind nebula (Cerutti et al., 2013). These flares were first detected by Fermi and were quite a surprise, given that the Crab had been used for many years as a gamma-ray calibration source and was believed to have a very stable flux value.

6.6. Astrophysical Sources That Produce Gamma Rays The gamma-ray sky is a highly variable one. The processes mentioned above are nearly all impulsive at some time scale or another, or come with an inherent periodicity. The kinds of objects that create gamma rays are at the extreme ends of their distributions in mass, magnetism, and periods. However, because gamma-ray data are sparse, most analyses are focused on time scales ranging from months to years. These time scales would seem to be incompatible with the goal of high time-resolution astronomy. The power of time-domain analysis actually drastically improves the sensitivity of gamma-ray instruments to sources with periodic signals. Pulsars, gamma-ray binaries, and (perhaps) a supermassive black hole binary are all sources with periodicities that have been detected in the gamma-ray regime. The recent discovery of pulsars in lowmass binaries that appear to transition between accretion and rotational states has brought gamma-ray astronomy into the high time-resolution arena. All-sky gamma-ray observations allow for constant monitoring of such sources and can quickly inform the scientific community of changes in state. Overall, gamma-ray variability can be divided into three different types: 1. Impulsive events – Gamma-ray bursts and solar flares occur on very rapid time scales, with emission lasting only a few hundreds to thousands of seconds. They occur either once or very infrequently. In addition, magnetic reconnection in the Crab, stellar novae, and flaring events from long-period Be binaries extend a bit longer, typically on the scale of a week or so. Regardless, these events do not have periodicities and are not predictable. 2. Irregular variability – AGN vary in the gamma rays, sometimes on time scales as short as a few minutes. However, while the longer time-scale variations may tell us something about the structure of the emission region, the rapid variability appears to be more chaotic in nature. 3. Regular/periodic variability – Both young pulsars and recycled millisecond pulsars (MSPs) produce periodic gamma-ray emission. In addition, gamma-ray binaries in short orbits will produce a modulation in the gamma-ray flux at the orbital period

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of the system. Transitional pulsars produce more gamma rays during accretion than during their rotation-powered phase. However, there is not yet sufficient information to determine if these transitions are periodic (Stappers et al., 2014). Here we will focus on galactic gamma-ray sources, as they were the focus of the Winter School. 6.6.1 Gamma-Ray Pulsars Pulsars are the most numerous gamma-ray source in our galaxy. Pulsars are rapidly rotating stellar cores left behind after a supernova. These compact sources provide the necessary ingredients to generate high-energy gamma rays: strong magnetic fields frozen into a rotating object capable of producing charged particles at its surface. Pulsars emit beams of radiation due to particle acceleration along their magnetic field lines. Because gamma-ray emission requires that the particles move near the speed of light, only those pulsars with light cylinders close to the star will generate sufficient gamma rays to be detectable by Fermi. This explains why there is a gamma-ray “death line” that has been seen in young pulsars (Guillemot et al., 2016). Most pulsars are detected in the radio waveband as a series of radio pulses with intervals associated with their rotation rate. However, some pulsars have been seen to pulse only in gamma rays. It is possible that all MSPs are able to generate gamma rays, and the lack of a detection for some may simply be due to viewing geometry (Venter and Harding, 2014). Shortly after a pulsar is formed, its crust is relatively unstable. As the crust cools and settles, or when strong magnetic reconnection events produce energetic restructuring of the surface, the neutron star’s moment of inertia changes. This change affects the pulsar’s rotation period, resulting in a shift in the the pulse arrival times, known as a “glitch.” By the time the neutron star cools enough for pulse periods to remain very stable, the gamma-ray production mechanism has stopped. Pulsars in binaries, however, may be given a second chance at gamma-ray production through mass transfer (see Figure 6.5). This stage arises as the endpoint of a long evolution of the system. The pulsar spins down; the companion evolves into the giant phase; mass is transferred during the low-mass X-ray binary (LMXB) phase; and the pulsar is spun back up to millisecond periods, emerging as a millisecond pulsar when the accretion disk finally dissipates. Such millisecond pulsars (MPSs) have incredibly stable pulse periods and are frequently gamma-ray emitters. Studying these sources provides new insight into gamma-ray emission mechanisms, as well as informing our understanding of the fundamental properties of dense nuclear material. 6.6.2 Supernova Remnants In the MeV regime, gamma-ray emission from SNRs is produced by pion decay. While at high energies, gamma-ray emission originates at the shock front as particles traverse the shock boundary multiple times, gaining energy each times. Eventually they are energetic enough to escape the shock. At this point, they interact with the ambient medium and produce gamma rays with a characteristic spectrum (Reynolds, 2008). Supernova remnants are not variable sources in the gamma-ray regime. Pulsar Wind Nebulae Hidden beneath the emission from pulsars, gamma-ray pulsar wind nebulae (PWNe) are difficult to detect. In general, like SNRs, these sources are non-varying. However, the PWN associate with the Crab pulsar has been seen to vary significantly in the gamma rays. Long considered a constant source that could be used as a reference standard,

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Figure 6.5. Gamma rays from pulsars are produced by electrons traveling along the closed magnetic field lines. As they approach the light cylinder (where the co-rotating frame is moving at the speed of light), the electrons will emit curvature radiation, resulting in the characteristic cutoff spectrum seen by Fermi (Aliu and MAGIC Collaboration, 2008).

Figure 6.6. The evolution of a stellar binary system travels through the LMXB phase before finally producing an isolated millisecond pulsar. (Credit: Bill Saxton, NRAO)

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Fermi’s monitoring of the Crab has shown that the nebula displays long time-scale variations (Wilson-Hodge et al., 2011). In addition, flaring events lasting several days have been seen in the PWN. It is believed that these flares may be the result of magnetic reconnection events between the gaseous filaments in the PWN and the neutron star that powers it (Cerutti et al., 2013). 6.6.3 Gamma Rays from Stellar Binaries Stars in binaries can generate gamma rays in several different ways. In all instances, however, one of the stars must have completed its life, exploded as a supernova and left behind a stellar remnant: a white dwarf, neutron star, or black hole. To date, the only “normal” star that has been detected in gamma rays is our sun; and the flux, even during flares, is too small to be seen from stars outside our solar system. Stellar Novae Until recently, no one expected that stellar novae could be a source of gamma rays. However, early in the Fermi mission, routine software looking for rapid flaring events detected a possible flare along the galactic plane. At the same time, optical astronomers reported a symbiotic nova that had been detected in the same region. Detailed analysis showed that the gamma-ray emission began shortly after the peak of the optical emission, and lasted for ∼1 week (Abdo et al., 2010). At the time, the production of gamma rays was speculated to be a result of the type of system in which the nova occurred, with a red giant star as the white dwarf’s companion. In such a scenario, the shock wave would propagate off the surface of the white dwarf and into the bloated photosphere of the red giant. Much like for supernova remnants, gamma rays could be produced on the leading edge of that shock, with the emission fading as the shock slowed in the ambient medium. However, since then, a number of classical novae have also been detected in gamma rays (see Figure 6.7). These are wider binary systems where the companion star is not close enough to the white dwarf to be the target material for gamma-ray production from a shock. Astronomers now believe that the gamma-ray emission may be the result of several interacting shock fronts (Metzger et al., 2015). If proven, this mechanism would be similar to one speculated to produce gamma-ray emission in GRBs. High-Mass Binaries High-mass (X-ray) binaries (HMXBs) contain a massive accreted object and a highmass donor star, from which material is being stripped and accreted onto the compact object. Gamma rays are produced by bipolar relativistic jets. HMXBs fall into two classes, depending on whether the massive object is a neutron star (neutron-star HMXB) or a black hole (microquasar). Only five such sources have been detected in gamma rays; Cygnus X-3, LS 5039, LSI +61 303, 1FGL 1018.6-5856, and LMC P3 in the Large Magellanic Cloud. The identification of a gamma-ray binary is based on the modulation of gamma rays at the orbital period of the binary. This modulation is due to geometric effects (Fermi- LAT Collaboration and Ackermann, 2012). In LSI +61 303, an additional long-term modulation is seen in gamma rays (see Figure 6.8). This 1,667-day superorbital period has been detected in other wavebands as well, including in the TeV gamma-ray regime (Ackermann et al., 2013). Multi-wavelength correlation studies are underway to help explain this second periodicity. While gamma rays have been detected by a few neutron-star HMXBs, Cyg X-3 is the only microquasar that has been shown to be a gamma-ray source (Corbel et al., 2012). This is curious, as both types of systems are fed by accretion. In addition, gamma rays

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Figure 6.7. Gamma-ray light curves from four different classical novae show clear similarities (Ackermann and Fermi-LAT Collaboration, 2014).

are seen for accreting supermassive black holes in the large AGN population. This may imply that detection of gamma-ray emission in accreting black hole systems requires more stringent geometric constraints than in neutron-star systems. In addition, the pairing of an energetic pulsar with a main sequence star creates a system that is fundamentally different from other gamma-ray binaries. High-energy emission originates from interactions between the energetic pulsar wind and the stellar wind of its companion. In cases where the companion is massive, gamma-ray flares arise as the pulsar passes through the companion’s equatorial disk. The flare durations are typically ∼ weeks (Caliandro et al., 2015). PSR B1259−63 and PSR J2032+4127 are pulsar binaries with very different orbital periods. Transitional Pulsars Some pulsars in binary systems undergo occasional transitions, switching between a pulsation-powered (normal pulsar) state during which the pulsar is spinning down (and the pulse period is increasing), and an accretion-powered (LMXB) state during which the pulsar is spinning up (and the pulse period is decreasing), or vice versa (Stappers et al., 2014). While in the accreting/LMXB state, the persistent (non-pulsed) gamma-ray

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Figure 6.8. The average gamma-ray flux of LS I +63◦ 303 is seen to vary over orbital periods. But in addition, a superorbital modulation can be seen in the data. Here, the left column shows portions of the orbit near periastron, and the right shows orbital phases near apastron. The superorbital modulation is seen at all phases but is much more visible near apastron (Ackermann et al., 2013). ©AAS. Reproduced with permission.

flux associated with these systems is higher than that seen in the normal pulsar state (see Figure 6.9). This is unexpected, as standard LMXBs have not been seen in the high-energy gamma-ray band (Roy et al., 2015). Colliding Wind Binaries Only one colliding wind binary is seen in gamma rays, and this is Eta Carinae. It was detected early in the Fermi mission, so it is reasonably bright. Low-level periodic modulation has made the identification of the gamma-ray source certain. However, the gamma-ray emission mechanism is unclear, though it may be due to shocks that exist on the wind boundaries (Reitberger et al., 2015).

6.7. Analysis of Sparse Gamma-Ray Data In most wavebands, photons are collected into a summed unit before being read out. In the gamma-ray regime, photons are sparse, and so detectors read out a single event at a time. The brightest persistent source in Fermi’s sky is the Vela pulsar. But the spacecraft receives, on average, only one photon every 2.6 minutes from that source. During that time, the pulsar has rotated ∼ 1766 times, with each rotation providing a radio pulse. With such sparse a data set, it seems impossible to imagine that one could perform high time resolution astronomy in the gamma rays. However, the timing information for each event is known very accurately – to better than a microsecond (Atwood et al., 2009).

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Figure 6.9. The gamma-ray flux from PSRJ1023+0038 was seen to rise significantly at the same epoch that regular radio observations found that pulsations were no longer detected. Further multi-wavelength studies found that the system had begin to exhibit LMXB-like properties in the X-rays (Stappers et al., 2014). ©AAS. Reproduced with permission.

This means that gamma-ray data is well suited for sources where the timing information is already known. The analysis methods, however, will differ significantly from those used in other wavebands. 6.7.1 The Fermi-LAT Data Set The full LAT data set, as well as analysis software and instructions for performing a wide range of analyses with LAT data are available from the Fermi Science Support Center website.3 The data are provided in several forms, including a query-able database, weekly data files via FTP, and short-duration data files used for specific analyses. Since the quality of the reconstruction varies based on the energy, incidence angle, and other factors, the instrument team has provided information that allows the user to make quality cuts based on that information. These cuts can be made in two ways: • Event Class – Event classes (Table 6.1) have been defined to give users an idea of the amount of background contamination they are allowing into their data set. Before providing the data to Fermi Science Support Center (FSSC), the LAT instrument 3

http://fermi.gsfc.nasa.gov/ssc/

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Table 6.1. Event Classes Defined for Standard LAT Data Files Event Class Name

Bitmask

P8R2 TRANSIENT020

16

P8R2 TRANSIENT010

64

P8R2 SOURCE

128

P8R2 CLEAN

256

P8R2 ULTRACLEAN

512

P8R2 ULTRACLEANVETO

1024

Residual Background

Analysis purpose

Transient event class with background rate equal to two times the IGRB∗ reference spectrum Transient event class with background rate equal to one times the IGRB∗ reference spectrum Residual background rate that is comparable to P7REP SOURCE

Bright transient events

Identical to SOURCE below 3 GeV, 2–4 times lower background rate than SOURCE above 3 GeV Intermediate background rate between CLEAN and ULTRACLEANVETO Rate is 2–4 times lower than the background rate of SOURCE class between 100 MeV and 10 GeV

Faint transient events

Most analyses, good sensitivity for analysis of point sources and moderately extended sources Slightly more sensitive to hard spectrum sources at high galactic latitudes Analysis of all-sky diffuse and large-scale extended sources Use to check for CRinduced systematics. Also for studies of diffuse emission that require low levels of CR contamination



IGRB is the Isotropic Diffuse Gamma-Ray Background. The reference spectrum is given in [REF].

team makes cuts on the raw data that classify events based on the probability that they are photons. Each class has its own instrument response functions (IRF) that describe the response of the instrument given the cuts made to create the class. Most of the event classes are nested, with each “better” class reducing the level of background present in the data set. The exception to this is data associated with solar flare analysis, as these events are determined temporally and can come from any of the classes. • Event Type – Event types (Table 6.2) allow subselection within the chosen event class. There are three ways the data may be divided; by location of gamma ray–tophoton conversion (FRONT/BACK), by quality of the point spread function (PSF), or by quality of the energy dispersion (EDISP). Event-type selections may not be mixed, as they each sub-divide the same data set. In the PSF and EDISP event types, the data are divided by quartiles, with quartile 0 having the best (narrowest) quality and quartile 3 having the worst (broadest) quality. Using standard analysis tools, each selected event type can be analyzed independently and the results combined. This combined likelihood analysis provides a significant improvement over the more generalized analysis for data sets with many events.

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Event Type FRONT BACK

Bitmask 1 2

Description Events converting in the front section of the tracker Events converting in the back section of the tracker PSF Type Partition

Event Type PSF0 PSF1 PSF2 PSF3

Bitmask 4 8 16 32

Description First (worst) quartile in the quality of the reconstructed direction Second quartile in the quality of the reconstructed direction Third quartile in the quality of the reconstructed direction Fourth (best) quartile in the quality of the reconstructed direction EDISP Type Partition

Event Type EDISP0 EDISP1 EDISP2 EDISP3

Bitmask

Description

64 128 256 512

First (worst) quartile in the quality of the reconstructed energy Second quartile in the quality of the reconstructed energy Third quartile in the quality of the reconstructed energy Fourth (best) quartile in the quality of the reconstructed energy

Information about the performance of the various event types can be found at the LAT Performance webpage,4 maintained by the LAT instrument team. In addition to the standard LAT event data describe above, the LAT Low-Energy (LLE) data set can be used to investigate short–time-scale impulsive events, like gammaray bursts, that have spectra that peak below the 100 MeV low-energy cutoff for standard LAT data. Because the instrument background is large at low energies, LLE data are not recommended for anything other than transient events. Indeed, this data set is only produced for such events. 6.7.2 A Combination of Signals The all-sky gamma-ray data (see Figure 6.10) contain contributions from multiple astrophysical sources and backgrounds, as well as instrumental backgrounds. These must all be accounted for during the analysis. Handling Background Emission The LAT’s relatively large PSF causes contributions from many different sources to overlap at low energies. Since most of these contributions are not uniform, any analysis must fit the various contributions simultaneously. Several bright backgrounds exist that must be accounted for: 1. Particle events – Low Earth orbit is a radiation-rich environment. As a result, particle showers that occur near the observatory can be detected and improperly classified as gamma rays. The frequency of such events generally follows Earth’s geomagnetic cutoff. Particle events from solar energetic activity can be identified and removed due to their proximity in time to solar flares.

4

www.slac.stanford.edu/exp/glast/groups/canda/lat Performance.htm

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Figure 6.10. This gamma-ray all-sky map includes contributions from point sources, diffuse sources, and several sources of background. Since the data are sparse, all contributors must be taken into account when performing a fit to the data. (Credit: NASA)

2. Cosmic rays interacting with galactic gas/dust – The structure of this background traces the distribution of hydrogen and dust in our galaxy. This contribution must be modeled, and an all-sky galactic diffuse model mop, divided up into many energy bins, has been produced by the LAT instrument team for use in standard point source analyses. 3. Earth’s atmosphere – Cosmic rays also interact with the gas of Earth’s atmosphere, making Earth’s limb the brightest gamma-ray source seen by Fermi-LAT. Because of the wide field of view on the LAT, essentially all-sky survey-mode observations include a significant Earth limb component. To remove this, the analysis process can use the arrival direction for the event to determine if it originated from Earth. This is called making a “zenith cut.” 4. Irreducible instrument background – No detector is perfect, and the LAT data set will contain a certain level of instrumental noise. This is relatively straightforward to characterize, and is accounted for as an isotropic background with a known spectrum. The galactic diffuse model and the isotropic spectrum are characterized by the LAT instrument team each time they generate a new reconstruction and classification of the data. They generate files corresponding to these two elements, which are provided in the standard installation of the Fermi science tools. 6.7.3 Handling Gamma-Ray Point Sources The large gamma-ray point spread function causes low-energy photons from multiple point sources to overlap. For this reason, contributions from the various point sources must all be considered when fitting the data. This process is described below in (Section 6.7.4). When performing a point source analysis, it is much easier to use a catalog of gammaray point sources to use as an input into the process, rather than attempting to determine the location of all the point sources in your region of interest (ROI). The LAT instrument

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team has provided point source catalogs for a variety of different data sets. The most recent all-sky, all-energy catalog (3FGL Acero and Fermi -LAT Collaboration, 2015) uses four years of ‘Pass 7 reprocessed’ data, and detected 3,033 sources. In order to properly account for the low-energy PSF, it is important to consider point sources that lie outside, but near, the region you are analyzing. The LAT team recommends including point sources within at least five degrees from the edge of your ROI, with spectral properties fixed to the values from the catalog analysis. This may not be perfect for sources that are variable, but it is certainly better than not considering those sources at all. 6.7.4 Using Maximum Likelihood A standard technique of dealing with sparse data is to use the “maximum likelihood” method, also known simply as “Likelihood.” This method compares the data to a model and determines how likely it is that the model describes the data, as compared to a null hypothesis. When performing a likelihood fit, parameters in the model are adjusted until the overall likelihood value is maximized. The result of that fit quantifies the statistical significance of the final model. The likelihood method can only ask the question, “How well does the model as given fit the data?” Models that are missing sources or have improper spectral forms for the components will not fit the data well. However, the likelihood result will not be able to provide that feedback. As a result, the process of getting a model that well describes the data is an iterative process. Initial Source Model Creation In an analysis of LAT data, all of the various contributions to the gamma-ray signal (including backgrounds) are incorporated into an XML model, which is then fitted to the full data set. This means the model will include the isotropic and galactic background models, spectral models for all the point sources in the region, and those within an annulus around the outside of the region to account for spread from the PSF. If the region includes an extended source, the model will also provide information about the spatial signature of the source. The easiest starting point is to incorporate information from an all-sky source catalog. This provides source positions, as well as initial spectral models and parameter values that are reasonable first guesses. Additional components can then be added to the model file as needed. Since this is a relatively standardized process, the LAT user community has generated scripts that allow the model creation process to be mostly automated. These scripts are available from the FSSC. For data sets that are much shorter than the duration used by the catalog, it will be necessary to remove weak sources from the model file. These are sources that were in the catalog, but for which a short data set is not sufficiently constraining on their spectral parameters. There are several methods for doing this, but using the python version of the analysis tools provides the most flexibility. For data sets that include data outside the time frame used by the catalog analysis, it will be necessary to add sources to your model to account for new, weak sources or variable sources that have flared during that period. Excesses in the data are discovered by generating a residual map. Each excess should then be localized before adding the source to the model. A spectral analysis should not be considered complete until the residuals are essentially flat (no excesses/deficits of more than 5σ).

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Instrument Response Functions The IRFs are calculated by performing detailed Monte Carlo simulation of photons of various true energies (E) and true directions (ˆ p) interacting with a simulated detector. The instrument response function is defined as: − → R(E  , pˆ , E, pˆ, L (t)) =

− → dσ (E, pˆ, L (t)).   dE dˆ p

(6.1)

→ − where E  is the measured energy of the photon, and pˆ is the measured direction. L (t) contains all of the information about the telescope location and state. The IRF can be decomposed into three functions (or matrices): − → − → − → − → p ; E, pˆ, L (t))A(E, pˆ, L (t)). R(E  , pˆ , E, pˆ, L (t)) = D(E  ; E, pˆ, L (t))P (ˆ

(6.2)

− → Here, the effective area A(E, pˆ, L (t)) is the projected area of the detector multiplied by its efficiency (the probability that a photon will interact in the detector and pair-produce). − → The energy dispersion D(E  ; E, pˆ, L (t)) reflects how well or poorly the instrument mea− → sures the photon energy. And the point spread function P (ˆ p ; E, pˆ, L (t)) reflects how well or poorly the instrument measures the photon direction. IRFs for each event class and event type combination have been generated by the instrument team and are provided with the standard Fermi science tools installation. Performing the Likelihood Fit Once the model file and data files are prepared, run the likelihood fitter to adjust the parameters in the model to their best-fit values. This can be done in two ways: • Unbinned Analysis – An unbinned fit evaluates the data on an event-by-event basis, without the loss of position or energy information caused by binning. This process is best used on short–time-scale data sets, to keep the computation time from becoming excessive. In order to perform an unbinned analysis, the effect of the instrument response on the diffuse components must be precomputed. • Binned Analysis – Performing a binned analysis is much faster than an unbinned analysis on a data set of any given size. The trade is the loss of some information due to binning in position and energy. However, consideration of separate event types can help compensate for those losses. Additional computation time is saved because the instrument response for diffuse components is applied as part of the overall fit. At its most fundamental, the likelihood value (L) is the probability of obtaining the data given the input model (the distribution of gamma-ray source positions, intensities, and spectra). Maximizing L gives the best match of the model to the data. Given a set of data, one can bin them in multidimensional (energy, sky pixels, etc.) bins. L is the product of the probabilities (pk ) of observing the detected counts in each bin (k). When the number of counts in each bin is small, their distribution is characterized by the Poisson distribution. This means L is the product of the probabilities of observing nk counts in each bin when the number of counts predicted by the model is mk . L =



pk =

 mnk e−mk k

k

nk !

(6.3)

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L can be rewritten as: L =



e−mk

 mnk k

k

nk !

k

= e−Npred

 mnk k

k

nk !

,

(6.4)

where Npred is the number of counts predicted by the model for bin k. This is the situation when performing a binned likelihood fit. If bin sizes get infinitesimally small, then nk ≈ 1, and we are left with a product running over the number of photons i:  mi (6.5) L = e−Npred i

This is the limit in which the unbinned likelihood fit is performed. It is easier to handle the logarithm of L so we usually maximize:  ln(mi ) − Npred (6.6) ln L = i

Thus, as a general rule, the equation to be maximized in either version of likelihood is:   nk ln(mk ) − mk (6.7) ln L = k

k

The first term in Equation 6.7 increases as the model predicts counts in bins where they actually occur, and the second term demands that model counts be parsimoniously allocated. In order to quantify the significance of the fit, we must consider the model components of the source region S that contribute to the observed counts:   si (E, t)δ(ˆ p − pˆi ) + SG (E, pˆ) + Seg (E, pˆ) + Sl (E, pˆ, t). (6.8) S(E, pˆ, t) = i

l

Here E is true energy, t is time, and pˆ is true direction. The four terms in Equation 6.8 correspond respectively to the point sources in the data region, the galactic and extragalactic diffuse sources, and sources outside the data region that may contribute events to the data. This model is folded with the instrument response (R from Equation 6.2) to obtain p ,t ): the predicted number of counts in the measured quantity space (E  ,ˆ

M (E, pˆ, t) = dEdˆ pR(E  , pˆ , t; E, pˆ)S(E, pˆ, t). (6.9) SR

We perform this integral over the full source region (data plus external contributions). In the standard analysis, only steady sources are assumed, reducing S(E, pˆ, t) to S(E, pˆ). This means the function to be maximized is now:  ln L = ln M (Ej , pˆj , tj ) − Npred (6.10) j

where the sum is performed over photons in the ROI, and the predicted number of counts is:

Npred = dE  dˆ p dtM (E  , pˆ , t). (6.11) ROI

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To save CPU time, a model-independent quantity for exposure (ε) as a function of position is precomputed by using the response function:

dE  dˆ p dtR(E  , pˆ , t; E, pˆ). (6.12) ε(E, pˆ) = ROI

which gives:

Npred =

dEdˆ pS(E, pˆ)ε(E, pˆ).

(6.13)

SR

Thus, the final equation that is maximized by the fit is:

 ln M (Ej , pˆj , tj ) − dEdˆ pS(E, pˆ)ε(E, pˆ) ln L = j

(6.14)

SR

The likelihood ratio test is used to determine the significance of a specific source in the ROI. This compares the maximum likelihood value with the source included in the model, with the likelihood value that would result if the source were not present. The value that results from that comparison is the test statistic (T S):   Lmax,0 T S = −2 ln (6.15) = −2(ln Lmax,0 − ln Lmax,1 ) Lmax,1 where the final term is called the delta-log-likelihood (or Δ ln L). While the TS is not significance, Wilk’s Theorem states that “In the limit of a large number of counts, the TS for the null hypothesis is asymptotically distributed as χ2n where n is the number of parameters characterizing the additional source.” This means that if the additional source has only two degrees of freedom, then the source significance (σ) is: √ σ = TS (6.16) The likelihood ratio test can also be used to test changes made to the overall source model. Estimating Errors and Calculating Upper Limits In the regime where a source has a large number of counts, a Gaussian fit to a Poissonian distribution is sufficient for calculating errors (see Figure 6.11, top). However, gamma-ray sources often contribute only a handful of counts to the overall fit. In order to estimate the error associated with a particular parameter for such sources, a likelihood profile must be calculated in which the one parameters is moved over a grid of values near its maximum, and all other parameters are maximized. A likelihood value is calculated at each grid point, producing a likelihood profile with a distribution that should behave as χ21 . This means that the 95 percent of errors can be determined by measuring the parameter values at which the 2Δ ln L = 1. In some cases, a source will not be detected at all, and an upper limit must be calculated. The upper limit calculation is also calculated from the likelihood profiile, except that it requires calculating a one-sided interval rather than a two-sided one. For a 95 percent upper limit, the upper limit is the value at which the 2Δ ln L = 2.71 (see Figure 6.12). This is reasonable in some circumstances, but this can also provide misleading results if the fit from the parameter grid moves a value to a regime that is nonphysical (e.g., moving a flux below zero). In these cases, it is better to use a quasi-Bayesian upper limit.

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Figure 6.11. Likelihood profiles for large (top) and small (bottom) number statistics. Gaussian approximations (blue) are good enough when a source has a large number of counts. But for most gamma-ray sources, it is necessary to generate a Poissonion likelihood profile (red) in order to properly calculate the errors. (Credit: S. Fegan)

To do this, the likelihood profile is integrated over the portion of parameter space that is physically possible until it reaches the specified percentage of the total. Evaluating the Fit Results As with any analysis, the best way to check the quality of the fit is to look at the residuals. An initial look at the fitted spectrum for the whole ROI will show whether the overall residuals are flat spectrally. In addition, the fitted source model can be used to generate a map of the expected counts. This can then be combined with the original data to get a significance map of the spatial residuals. Excesses and deficits of more than 5σ indicate areas that have not been properly modeled. Use the spectral and spatial residuals to refine the output model. This can be done by adding or removing sources, by freeing or fixing spectra parameters for some sources, or by changing which spectral model is used for each source. It is important to keep track of the number of degrees of freedom that are changed in this way, as that information will be needed when calculating the significance of the change. In addition, be aware that the significance of changes made to the spectral model can only be calculated if the two

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Figure 6.12. Calculating upper limits requires calculating a one-sided interval of the likelihood profile. (Credit: S. Fegan)

Figure 6.13. Left: Significance map of residuals in the 3C 279 region. An unmodeled point source is clearly visible in the field. Right: The same data set but with the source correctly modeled. Residual maps usually make clear when a new point source should be added to the spectral model.

models are nested (i.e., in a limit, the more complex model becomes the simpler model). If the change in the likelihood value is large enough, calculating an exact significance may not be necessary. Localizing New Sources When a new source is detected, it will be necessary to localize the source in the gamma rays. This requires a fitted spectral model that includes the source at or near the expected position. The best way to find a good target position for the new source is to calculate a TS map in the immediate area of the source. This uses an input source model without

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the source included and performs a pixel-by-pixel likelihood fit adding a putative point source at that location. The resulting map should provide a maximum in the pixel where the new source should be added to the model. Next, an initial spectral fit should be performed using the model that includes the new source. This will move the spectral parameters for the new source to their best-fit values. Finally, the output spectral model can be used to localize the new source to the best-fit position. Since this final step must be performed using an unbinned method, it is advisable to use only the time period when a variable source is bright. For new, nonvarying sources, a smaller sky region can be used to reduce computation time.

6.8. Characterizing the Gamma-Ray Sky on Short Time Scales On short time scales, the number of events can become small with respect to the number of degrees of freedom in a typical spectral model. The loss of information caused by binning the data becomes untenable. For this reason, short–time-scale analyses are performed using unbinned likelihood methods. 6.8.1 Generating Time Series Time series analysis for variable sources (such as flaring AGN, stellar novae, gamma-ray binaries, or transitional pulsars) fundamentally follows the same process as the spectral analysis described in Section 6.7.4. The full data set is divided into smaller time spans (typically days to weeks), and an unbinned likelihood analysis is performed. Because the data set is small, many faint sources are fixed to their catalog values and contribute very little to the observed counts. Variable sources should be left free initially, but if found not to be flaring, their parameters may also be fixed. Once the time span for an analysis drops below a day, only the very brightest sources can provide time series information via a likelihood analysis. Analyses of impulsive events like gamma-ray bursts and solar flares utilize the higher-background transient classes and fit the background rates as a function of time. Persistent astrophysical sources contribute very few counts to an analysis that covers a period of hours to minutes, as is common for solar flare and GRB analyses. 6.8.2 Using Time Signatures with Gamma-Ray Data Because the event arrival times are known very accurately, gamma-ray sources with characteristic time signatures can provide a new method for optimizing the data set. Using the timing information, pulse phase and/or orbital phase values can be assigned to each event in the data. By cutting the data as a function of phase, a pulsar can be either enhanced, or eliminated from the data set. For bright pulsars, gamma-ray phase-resolved spectroscopy can be performed. For gamma-ray binaries, the data can be folded to search for orbital modulation. These modified data sets can then be run through a standard analysis, but using only a part of the original data. As a result, the apparent exposure is too large by whatever fraction has been eliminated from the data. This has the effect of causing likelihood to report flux values that are too small by that fraction. It is important to remember to rescale the resulting flux values from any analysis that has used phase information to down-select data. By folding gamma-ray data using pulsar timing information originating in the radio, gamma-ray pulsations have been detected in a number of young radio pulsars, as well as many millisecond pulsars – a source of gamma rays that was unconfirmed before

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the launch of the Fermi mission. Indeed, radio follow-up observations of previously unassociated Fermi sources have produced a large number of newly discovered MSPs, most of which are in binary systems (see, e.g. Camilo et al., 2015). Pulsar astronomers have also developed methods to search the gamma rays directly for pulsations. These searches have uncovered a number of new, young pulsars, as well as several isolated MSPs (Pletsch et al., 2012b). However, discovery of binary MSPs in gamma-ray data requires some orbital information to be provided from other wavebands. As a result, optical and X-ray investigations of bright, pulsar-like Fermi unassociated sources are ongoing, searching for modulation at the orbital period. These searches have already borne fruit, with at least one new MSP discovered using optically derived orbital parameters (Pletsch et al., 2012a).

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