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QED and X-ray polarization from neutron stars and black holes Caiazzo, Ilaria 2019

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QED and X-ray Polarization from Neutron Stars andBlack HolesbyIlaria CaiazzoA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFDoctor of PhilosophyinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University of British Columbia(Vancouver)December 2019c© Ilaria Caiazzo, 2019The following individuals certify that they have read, and recommend to the Fac-ulty of Graduate and Postdoctoral Studies for acceptance, the thesis entitled:QED and X-ray Polarization from Neutron Stars and Black Holessubmitted by Ilaria Caiazzo in partial fulfillment of the requirements for the de-gree of Doctor of Philosophy in Physics.Examining Committee:Jeremy Heyl, Physics and AstronomySupervisorHarvey Richer, Physics and AstronomySupervisory Committee MemberIngrid Stairs, Physics and AstronomySupervisory Committee MemberMarcel Franz, Physics and AstronomySupervisory Committee MemberGary Hinshaw, Physics and AstronomyUniversity ExaminerStephen Gustafson, MathematicsUniversity ExaminerAndrew Melatos, University of MelbourneExternal ExamineriiAbstractThe emission from accreting black holes and neutron stars, as well as from thehighly magnetized neutron stars called magnetars, is dominated by X-rays. Forthis reason, spectral and timing studies in the X-rays have been extremely success-ful in broadening our understanding of compact objects in the past few decades.Soon, a new observational window will open on compact objects: X-ray polarime-try. In this work, I explore how polarized light is generated in black-hole accretiondisks, magnetar atmospheres and magnetospheres and in the accretion region ofX-ray pulsars. In the different chapters, I show how the polarization signal is sen-sitive to several unknowns in our theoretical models: the geometry of accretion inX-ray pulsars, the strength and structure of the magnetic field threading accretiondisks around black holes, the process of the non-thermal emission in magnetars.For this reason, the future X-ray polarimetry missions will be extremely helpful inconstraining our theoretical models. Furthermore, the polarization emission willprovide, for the first time, a test of one of the first theoretical predictions of quan-tum electrodynamics: vacuum birefringence. In this work, I show how this effect,previously considered only for neutron stars, plays a crucial role for black holes aswell.iiiLay SummaryNeutron stars and black holes share the same origin: they are born in the spectac-ular event that we call a supernova. Such a dramatic beginning results in extremeproperties, that place them among the most fascinating objects in the Universe. Inthe 50 years since their discoveries, observations over the entire electromagneticspectrum and in gravitational waves have brought us closer to understand theseobjects. Soon, a new type of X-ray telescope, able to detect the polarization oflight, will be in space, opening a new window on neutron stars and black holes.In this work, I show how X-ray polarization can give us answers to the questions:what is the structure and strength of magnetic fields surrounding black holes? Howcan a neutron star steal matter from an orbiting companion star? What is caus-ing the strange emission that we see in the ultra-magnetized neutron stars calledmagnetars?ivPrefaceAll the work presented in this thesis is a result of a collaboration between theauthor, Ilaria Caiazzo (I.C.), and her advisor Jeremy Heyl (J.H.).• Chapter 1. Some introductory text was adapted from the white papers [39]and [88], submitted to the Bulletin of the AAS, from [37], published in thejournal Galaxies and from [40], the book chapter “Polarimetry of Magne-tars and Isolated Neutron Stars,” of the book Astronomical Polarimetry fromthe Infrared to the Gamma-rays, currently in press by Springer. The whitepapers [39] and [88] where written by the Colibrı´ collaboration, for whichJ.H. is PI and I.C. is project scientist; I.C. wrote the bulk of [39], and partsof [88], which J.H. wrote the bulk of. All the other authors wrote part of thetext and gave useful feedback. I.C. and J.H. wrote the majority of [40], whileRoberto Turolla edited and integrated the final draft. J.H. and I.C. conceivedthe calculations of [37], while I.C. performed the calculations and wrote thebulk of the text.• Chapter 3 was adapted from [40] (see above).• Chapter 4 was adapted from the paper [87], published by the journal Galax-ies. J.H. wrote the bulk of [87], while I.C. wrote part of the text; the calcula-tions were done in part by J.H. and in part by I.C.• Chapter 5 is partly original, partly adapted from [87], published by the jour-nal Galaxies and from [40] (see above).• Chapter 6 was adapted from [38], published by the journal Physical ReviewD and from [37], published in Galaxies. J.H. and I.C. conceived the calcula-vtions of [38], while I.C. performed the calculations and wrote the bulk of thetext.• All the calculations in Chapters 7 and 8 where conceived by I.C. and J.H.,and performed by I.C. A figure in Chapter 7 was previously published in[87], by the journal Galaxies (see above).• A figure in Chapter 9 was previously published in [37], by the journal Galax-ies (see above).viTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Neutron stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.1.1 A bit of history . . . . . . . . . . . . . . . . . . . . . . . 51.1.2 Neutron stars in the X-rays: X-ray pulsars . . . . . . . . . 61.1.3 Neutron stars in the X-rays: Magnetars . . . . . . . . . . 91.2 Black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.2.1 The Kerr black hole . . . . . . . . . . . . . . . . . . . . 131.2.2 The accretion disk . . . . . . . . . . . . . . . . . . . . . 161.2.3 Black holes in the X-rays . . . . . . . . . . . . . . . . . . 171.2.4 The role of the magnetic field . . . . . . . . . . . . . . . 18vii1.3 This Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 Polarized Radiation and Its Propagation . . . . . . . . . . . . . . . . 202.1 Polarization of light and the Stokes parameters . . . . . . . . . . 212.1.1 The Poincare´ sphere . . . . . . . . . . . . . . . . . . . . 242.2 Thomson Scattering . . . . . . . . . . . . . . . . . . . . . . . . . 252.2.1 Thomson scattering in a strong magnetic field . . . . . . . 272.3 Propagation of light through an inhomogeneous and anisotropicmedium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.3.1 Propagation of light in a birefringent medium . . . . . . . 363 The Origin of Polarized Radiation in Black Holes and Neutron Stars 383.1 Black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.1.1 Polarization of an electron-scattering atmosphere . . . . . 393.1.2 The relativistic effects . . . . . . . . . . . . . . . . . . . 413.2 Neutron Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.2.1 Neutron-Star Atmospheres . . . . . . . . . . . . . . . . . 433.2.2 Condensed Neutron-Star Surfaces . . . . . . . . . . . . . 493.2.3 Neutron-Star Magnetospheres . . . . . . . . . . . . . . . 533.2.4 X-ray Pulsars . . . . . . . . . . . . . . . . . . . . . . . . 554 The QED Effect of Vacuum Birefringence . . . . . . . . . . . . . . . 574.1 Effective Action: Formal Derivation . . . . . . . . . . . . . . . . 584.1.1 The functional method . . . . . . . . . . . . . . . . . . . 584.1.2 Functional Integration . . . . . . . . . . . . . . . . . . . 624.1.3 Effective Action : Proper-time Integration . . . . . . . . . 664.2 Results for a Uniform Field . . . . . . . . . . . . . . . . . . . . . 674.2.1 Effective Lagrangian in a Constant Field . . . . . . . . . . 694.3 Index of refraction . . . . . . . . . . . . . . . . . . . . . . . . . 704.4 Propagation through the Birefringent Vacuum . . . . . . . . . . . 715 The Effect of Birefringence on Neutron-Star Emission . . . . . . . . 735.1 Vacuum Birefringence . . . . . . . . . . . . . . . . . . . . . . . 745.1.1 The Quasi-tangential effect . . . . . . . . . . . . . . . . . 78viii5.2 Plasma Birefringence . . . . . . . . . . . . . . . . . . . . . . . . 835.2.1 The vacuum resonance in the neutron-star atmosphere . . 876 QED and Polarization from Accreting Black holes . . . . . . . . . . 926.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.2 Accretion disk model . . . . . . . . . . . . . . . . . . . . . . . . 936.3 Vacuum birefringence . . . . . . . . . . . . . . . . . . . . . . . . 986.3.1 Competition with the plasma birefringence . . . . . . . . 1006.4 Depolarization in the disk plane . . . . . . . . . . . . . . . . . . 1016.4.1 Zero angular momentum photons . . . . . . . . . . . . . 1046.4.2 Maximum prograde and retrograde angular momentum pho-tons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1066.4.4 A Simulation for GRS 1915+105 . . . . . . . . . . . . . 1086.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107 The Polarization of X-ray Pulsars . . . . . . . . . . . . . . . . . . . 1127.1 Previous models: Me´sza´ros and Nagel and Kii . . . . . . . . . . . 1147.1.1 Description of the method . . . . . . . . . . . . . . . . . 1157.1.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1167.2 Polarization in the Becker and Wolff model . . . . . . . . . . . . 1187.2.1 The spectral formation model . . . . . . . . . . . . . . . 1187.2.2 Polarization at the source . . . . . . . . . . . . . . . . . . 1237.2.3 Polarization at the observer . . . . . . . . . . . . . . . . . 1317.2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 1408 Polarization of Magnetars Soft Emission . . . . . . . . . . . . . . . 1448.1 Thermal emission: Lloyd’s atmospheres . . . . . . . . . . . . . . 1458.1.1 Thermal structure and the angular dependence of the effec-tive temperature . . . . . . . . . . . . . . . . . . . . . . . 1468.2 Non-thermal emission: a twisted magnetosphere . . . . . . . . . . 1498.2.1 Non-thermal models: the 5-10 keV range . . . . . . . . . 1518.2.2 Non-thermal models: the 20-100 keV range . . . . . . . . 1558.3 Modeling the spectrum and polarization of 4U 0142+61 . . . . . . 156ix8.3.1 The hot spot . . . . . . . . . . . . . . . . . . . . . . . . 1578.3.2 The saturated Comptonization . . . . . . . . . . . . . . . 1618.3.3 The resonant Compton scattering . . . . . . . . . . . . . 1659 Conclusions and Future Perspectives . . . . . . . . . . . . . . . . . . 1719.1 The developement of X-ray polarimetry . . . . . . . . . . . . . . 1719.2 Neutron stars and black holes polarization studies with the IXPEand eXTP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1739.2.1 Black holes . . . . . . . . . . . . . . . . . . . . . . . . . 1739.2.2 X-ray pulsars . . . . . . . . . . . . . . . . . . . . . . . . 1759.2.3 Magnetars . . . . . . . . . . . . . . . . . . . . . . . . . . 1769.3 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180xList of TablesTable 5.1 Polarization-limiting radii for various sources . . . . . . . . . 78xiList of FiguresFigure 1.1 Neutron star interior. . . . . . . . . . . . . . . . . . . . . . . 4Figure 2.1 The Poincare´ sphere. . . . . . . . . . . . . . . . . . . . . . . 25Figure 3.1 Degree of polarization of an electron scattering, plane-parallelatmosphere. . . . . . . . . . . . . . . . . . . . . . . . . . . . 41Figure 3.2 Location of the parallel and perpendicular photospheres in aneutron star. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47Figure 3.3 Polarization fraction for a neutron star with Lloyd atmospheremodel. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 3.4 Polarized intensity, Lloyd atmosphere . . . . . . . . . . . . . 50Figure 3.5 Polarization Radiation from a Condensed Neutron-Star Surface 52Figure 5.1 Polarization direction at the polarization limiting radius . . . . 75Figure 5.2 The extent of the polarization averaged over the stellar surface 77Figure 5.3 The polarized emission map of a neutron star overlaid on theapparent image of the NS . . . . . . . . . . . . . . . . . . . . 79Figure 5.4 The QT effect from [237] . . . . . . . . . . . . . . . . . . . . 81Figure 5.5 Location of the parallel and perpendicular photospheres in aneutron star - strong field case . . . . . . . . . . . . . . . . . . 90Figure 6.1 ISCO and polarization limiting radius for a spinning black hole 99Figure 6.2 Monte-Carlo simulation of the depolarization of radiation froma black hole . . . . . . . . . . . . . . . . . . . . . . . . . . . 107xiiFigure 6.3 Polarization fraction at the observer for spinning a black hole,seen edge on . . . . . . . . . . . . . . . . . . . . . . . . . . 109Figure 6.4 Simulated polarization spectrum for GRS 1915+105 . . . . . 110Figure 7.1 The two emission models for X-ray Pulsars . . . . . . . . . . 113Figure 7.2 Polarization of Her X-1 as function of photon energy using theemission models of Kii [111] and Me´sza´ros and Nagel [150],averaged over the rotation of the pulsar . . . . . . . . . . . . 117Figure 7.3 Becker and Wolff accretion model. . . . . . . . . . . . . . . . 120Figure 7.4 Polarization inside and outside the accretion column . . . . . 125Figure 7.5 Cross sections for the different polarization modes. . . . . . . 127Figure 7.6 The effect of beaming on flux . . . . . . . . . . . . . . . . . 129Figure 7.7 Average Stokes parameters after beaming . . . . . . . . . . . 130Figure 7.8 Phase pattern for intensity and polarization fractions for 2 columnswithout light bending. . . . . . . . . . . . . . . . . . . . . . 132Figure 7.9 Average Stokes parameters from 2 columns without light bending133Figure 7.10 Light bending in the neutron star gravitational field. . . . . . . 134Figure 7.11 Phase pattern for intensity and polarization fractions for 2 ac-cretion columns with light bending for photon energies 1 keVand 29 keV. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135Figure 7.12 Average Stokes parameters for 2 columns with light bending . 136Figure 7.13 Phase pattern for intensity and polarization fractions for 1 ac-cretion column with different heights. . . . . . . . . . . . . . 137Figure 7.14 Average Stokes parameters for 1 column and different heights 138Figure 7.15 Dependence of the QT effect from z, φ and energy . . . . . . 139Figure 7.16 The effect of the QT effect on the average polarization. . . . . 140Figure 8.1 The spectrum of the magnetar 4U 0142+61. . . . . . . . . . . 145Figure 8.2 Intensity map for the thermal emission. . . . . . . . . . . . . 148Figure 8.3 X- and O-mode cross section in the X-rays for a magnetar mag-netic field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154Figure 8.4 Spectral shape and polarization for the hot spot model. . . . . 158Figure 8.5 Polarization map for a magnetar with and without QED. . . . 159xiiiFigure 8.6 Polarization and intensity map for the hot spot model. . . . . . 161Figure 8.7 Intensity map for the saturated Comptonization model. . . . . 163Figure 8.8 Spectral shape and polarization for the saturated Comptoniza-tion model. . . . . . . . . . . . . . . . . . . . . . . . . . . . 164Figure 8.9 Mean number of scatterings in the RCS model as function ofenergy and resulting depolarization. . . . . . . . . . . . . . . 168Figure 8.10 Spectral shape and polarization for the RCS model. . . . . . . 169Figure 9.1 Polarization degree from a black-hole accretion disk. Simula-tion for eXTP and IXPE . . . . . . . . . . . . . . . . . . . . 174Figure 9.2 Polarization degree for Her X-1 and different models. Simula-tion for eXTP and IXPE . . . . . . . . . . . . . . . . . . . . 176Figure 9.3 Different models for the polarization of magnetars . . . . . . 177xivGlossaryX-ray pulsar Accreting neutron star that shows a pulsating emission in the X-rays.Magnetar Highly magnetized neutron star, with magnetic field exceeding 1014Gauss.ISCO Innermost Stable Circular Orbit, for a black hole accretion disk.X-mode or ⊥-mode, polarization mode in which the electric field of the photon isperpendicular to the local magnetic field.O-mode or ‖-mode, polarization mode in which the electric field of the photon isparallel to the local magnetic field.Birefringent medium anisotropic medium in which the index of refraction de-pends on the polarization direction of light.xvAcknowledgmentsI would like to thank Jeremy Heyl, my advisor, for being the best advisor anyonecould ask for. Thanks to you I have learned so much and we have embarked in somany projects together, but most importantly, doing research was always fun andexciting. Thank you for being such a good friend.I would like to thank all the collaborators and friends with whom I had stimu-lating conversations and glasses of wine. In particular, I would like to thank all thepeople in our research group at UBC, headed by Jeremy Heyl and Harvey Richer.And thank you Harvey for all the good times and all the great projects that we havedone together.I would like to thank all the friends that supported me in these past years. Thankyou for being there for me when I needed a place to stay or a beer or a hug. Youhave done more for me than you can ever imagine.Finally, to Carlo. Thank you for everything.xviChapter 1IntroductionWhen a massive star runs out of fuel in its core, at the end of its life, the sourceof energy that used to balance the star’s self-gravitation, nuclear energy, is ex-tinguished. The collapse that ensues gives birth to the spectacular and extremelyenergetic event called a supernova. A supernova, however, does not destroy the starcompletely; it leaves over what remains of the star’s compact core: either a blackhole or a newly born neutron star.Being born in such a dramatic event, neutron stars and black holes (also calledcompact objects) present peculiar characteristics that place them among the mostfascinating and puzzling objects in the Universe. They uniquely provide an envi-ronment to test the laws of physics at their extremes, as density in a neutron starreaches values several times higher than nuclear density, magnetic fields are bil-lions of times higher than the Sun’s, and gravity around black holes is so strong asto trap light itself. Compact objects, however, do not like to reveal their secrets allat once. Fifty years after their discovery, we still do not know what neutron starsare made of, and the question of how black holes modify space and time aroundthem is still open.As always in the history of astronomy, the opening of a new observational win-dow on an astronomical object brings the promise of a much deeper understandingof the object itself, together with a wealth of unexpected discoveries. We are now atthe door of such an exciting time, with a new window opening on compact objects:X-ray polarimetry. Several observatories with an X-ray polarimeter on board are1now at different stages of development: in the 1–10 keV range, the NASA SMEXmission IXPE [241], scheduled to fly in 2021, and the Chinese–European eXTP[253]; in the medium range, 5-30 keV, the Indian POLIX, scheduled for launchin 2020 [175, 235]; in the hard-X-ray range, 15–150 keV, the balloon-borne X-Calibur [22] and PoGO+ [43]; and, in the sub-keV range, the narrow band (250eV) LAMP [208] and the broad band (0.2–0.8 keV) rocket-based REDSox [64].Being able to measure the polarization of X-ray photons will provide two newobservables, polarization degree and angle, which are extremely sensitive to thegeometry of the emission regions and to the structure of magnetic fields. In par-ticular, the focus of this work is on how X-ray polarization will help us probe thegeometry of the emission from accreting neutron stars (X-ray pulsars), the structureand strength of magnetic fields surrounding accreting black holes and the emissionprocesses in ultramagnetized neutron stars (magnetars). I will present models forthe polarization of the X-ray emission from compact objects based on the mostrealistic assumptions and physical models to date, and show that the observationof polarization will provide a powerful tool to understand the physical processes inaction. Specifically, I show the effects of vacuum birefringence on the polarizationfrom accreting black holes, and how the polarization signal can be used to probethe magnetic field threading the accretion disk (Chapter 6); I find the polarizationsignal of X-ray pulsars by including QED in an old model and presenting a brandnew model that fits the spectral data very well (Chapter 7); and I model the po-larization in the soft X-rays from magnetars in the context of different emissionmodels (Chapter 8).1.1 Neutron starsNeutron stars are the most compact stars in the Universe. The typical radius ofa neutron star is about 10 km and the currently measured masses of neutron starsrange between 1 and 2 solar masses. This extreme compactness leads to a veryhigh density in their cores, that can reach a few times the saturation density forterrestrial atomic nuclei. Furthermore, neutron stars rotate with periods that canbe as low as a few milliseconds and they possess very strong magnetic fields, thatrange from the 108 G of millisecond pulsars, to 1011−1012 G for radio and X-ray2pulsars, up to 1014− 1015 G for magnetars. These extreme conditions cannot befound anywhere else in the Universe and therefore, neutron stars represent uniquelaboratories where we can test our understanding in many fields of fundamentalphysics.In the fifty years since their discovery, neutron stars have never stopped puz-zling and amazing astronomers. First discovered as radio pulsars, neutron starshave revealed themselves in different fashions, over the entire electromagneticspectrum and via gravitational waves (and neutrinos). From the almost 3,000 radiopulsars detected up to now, to the radio-quiet, thermally emitting isolated neutronstars (XDINs), from the young and active magnetars with extreme magnetic fieldsand slow rotation periods, to the old and rapidly rotating millisecond pulsars, fromaccreting to merging binaries; neutron-star phenomenology is rich and we havelearned a lot from it. Yet, many puzzles remain, including the key question: Whatare neutron stars made of? This question has profound implications for the physicsof dense matter. The density reached in a neutron star’s core, several times higherthan nuclear density, is not reached anywhere else in the universe at cold tempera-tures, let alone in our terrestrial physics labs, and therefore neutron stars representthe only laboratory available to look for the equation of state for cold, dense matter.What is currently known about the neutron star internal structure is shown inFig. 1.1. Neutron stars possess a very thin atmosphere, most likely made of lightelements, either hydrogen or helium, with a thickness that can vary from someten centimeters in a hot neutron star, to a few millimeters in a cold one [2]. Un-derneath, the envelope of the star that goes from the surface to a density of about4×1011 g cm−3, called the outer crust, is made of a mostly solid lattice of neutronrich nuclei and a gas of degenerate electrons. At a density of ∼ 4× 1011 g cm−3,the neutron drip density, neutrons start to spill out of nuclei, and the inner crust istherefore made by a lattice of nuclei even richer in neutrons that coexist with a de-generate, superfluid gas of dripped, unbound neutrons and a gas of ultra-relativisticelectrons. At the inner edge of the crust, at densities of the order of 1013−1014 gcm−3, spherical shapes are no longer energetically favorable for nuclei. Competi-tion between strong interactions at short distance and Coulomb repulsion at longdistance, called frustration, leads to the formation of complex nuclear structureswith different shapes, called nuclear pasta. In the core, at densities higher than3Figure 1.1: Diagram of the internal structure of a neutron star. Credit:NASA/NICER.the nuclear density, the nuclei finally melt and create a uniform, beta-equilibratednuclear plasma, consisting mainly of superfluid neutrons, with a mixture of super-conducting protons, electrons and muons. At higher densities in the core, a phasetransition is theoretically possible, with the appearance of hyperons, pion or kaoncondensates, or even deconfined quark matter.The holy grail of neutron star observations, the mass-radius relation, if mea-sured for several neutron stars, could put stringent constraints on the equation ofstate [212]. Mass measurements of massive neutron stars exclude a number ofequations of state that predict a relatively soft dependence of pressure on density.The record holder as of December 2019 is the millisecond pulsar J0740+6620,with a measured mass of 2.14± 0.10 M [48]; the previous mass measurementsaround 2 solar masses, of the millisecond pulsar J0348+0432 at 2.01±0.04 M [6]and of the pulsar J1614-2230 at 1.97±0.04 M [53], already excluded many softequations of state. Although a number of masses of neutron stars have been mea-sured with high precision, especially for compact binaries, radius measurements4are much harder to achieve with the precision of less than a kilometer required toput stringent constraints on the equation of state.1.1.1 A bit of historyAt the 1973 Solvay Conference, Le´on Rosenfeld reminesced about a conversationbetween Niels Bohr, Lev Landau, and himself that took place in Copenhagen inFebruary 1932, right after the discovery of the neutron, in which Landau impro-vised the conception of neutron stars [1]. This is often taken as the first time theidea of neutron stars was ever proposed. In reality, Landau submitted a paper to thePhysikalische Zeitschrift der Sowjetunion before the discovery of the neutron, inJanuary 1932, in which he mentioned the possible existence of dense stars that looklike one giant nucleus [126, 247]. After this first theoretical prediction, a second,prescient hypothesis was advanced only two years later by Baade and Zwicky [7]:a neutron-rich compact object could be the remnant of the gravitational collapse ofthe core of a star after a supernova.Neutron stars were expected to be cold and faint and hard to detect; nonethe-less, some scientists kept studying the properties that such objects would have, ifthey existed. The first model of the neutron star structure in general relativity, nowcalled the TOV model, was proposed in 1939 by Oppenheimer and Volkoff [171]and separately by Tolman [230]. In early 1967, Pacini suggested that supernovaremnants such as the Crab could be powered by the radiation emitted by a rotating,strongly magnetized neutron star.The serendipitous discovery took place the same year, in the summer of 1967,by a doctoral student at Cambridge, Jocelyn Bell. Bell and her advisor, AntonyHewish, were building the Interplanetary Scintillation Array, a radio telescope, tostudy quasars, when she noticed a regular pulsation in the radio signal. In Novem-ber, they managed to record the regular pulses of the celestial object, presentlyknown as PSR B1919+21, with a period of 1.337 seconds. In the discovery pa-per, published in February, 1968, they also proposed that the source of this rapidpulsation may be a compact object, such as a white dwarf or a neutron star [86].In the first year following the discovery of the first pulsar, many theories wereproposed. Many scientists leaned toward explaining the source of the radio signal5as a binary system or a white dwarf, because both concepts where more familiarback then [133]. A few months later, however, pulsars showing much shorter pe-riods were discovered, like the Vela pulsar (89 ms) [127] and the Crab pulsar (33ms) [211]. Only an object as compact as a neutron star could rotate or vibrateat these newly observed frequencies. Moreover, these two pulsars were found in-side supernova remnants, providing a confirmation of Baade-Zwicky’s prediction.Thomas Gold, professor at Cornell University, in a paper in Nature, suggested thatthe source of the radio signal could be identified with a rotating neutron star. Heproposed the lighthouse mechanism, for which the pulsating signal is caused by abeam of radiation swept across the observer [72].For several years after the first pulsar discovery, it was widely accepted thatneutron stars could only be observed as pulsars within radio wavelengths. How-ever, in the last four decades, many different types of neutron stars, other thanrotation powered pulsars (radio pulsars), have been discovered: isolated, thermallyemitting neutron stars (XDINs), silent in radio; compact central objects (CCOs),found in supernova remnants; accreting X-ray and γ-ray emitters; magnetars, i.e.young pulsars with huge magnetic fields (up to 1015 Gauss), and rotating radiotransients (RRATs), i.e. rapidly changing objects that act as pulsars but only for afew seconds per day.1.1.2 Neutron stars in the X-rays: X-ray pulsarsEven though the first identified neutron star was a radio pulsar, the first observa-tion of emission coming from a neutron star coincided with the first detection ofX-rays from outside the solar system: the discovery of Scorpius X-1 by the Aer-obee rocket, in 1962 [68]. The correct identification of Scorpius X-1 as a binarysystem containing an accreting neutron star came only after other systems werediscovered in the 1970s by the satellite UHURU that contained a pulsating neutronstar: Centaurus X-3 and Hercules X-1 [69, 218].Accreting X-ray pulsars (to be concise I will use the term X-ray pulsars here-after) are highly magnetized neutron stars that live in a binary and accrete materialfrom a companion star. The material, mostly ionized hydrogen, gets unbound fromthe companion (either by exceeding the Roche lobe, or because of strong winds)6and becomes gravitationally bound to the neutron star. As the material gets closerto the compact object, it forms an accretion disk, and when it reaches the surface ofthe compact object, the kinetic energy of the accretion flow is converted into X-rayemission. The accretion luminosity generated in this process is given byLX ∼ GM∗M˙R∗ (1.1)where G is the gravitational constant, M∗ and R∗ are the mass and radius of the neu-tron star, respectively, and M˙ is the mass accretion rate. The presence of a strongmagnetic field on the neutron star (B∼ 1012−1013 G) disrupts the plasma flow inthe accretion disk at the magnetospheric radius, and the ionized gas is funneledalong the magnetic field lines to the magnetic poles of the neutron star, possiblyforming accretion columns above the poles. The polar caps are heated by the in-falling material, and the kinetic energy is converted into X-ray emission, whichappears to be pulsating due to the rotation of the neutron star. The position of themagnetospheric radius can be estimated as the distance from the neutron star wherethe magnetic pressure becomes equal to the ram pressure, and it is usually quite farfrom the star, rm∼ 109 cm∼ 1,000R∗ [125, 185]. The observational appearance ofX-ray pulsars can vary because of several factors, including the nature of the donorstar and the parameters of the binary system, but also the geometry and physicalconditions of the emission region.The spectra of accretion-powered X-ray pulsars are usually well fitted by apower law component in the 5-20 keV range, plus a blackbody component at atemperature of about 106−107 K and a quasi-exponential cut-off at about 20−30keV [e.g. 46, 242]. Also, close to the cyclotron region, resonant scattering of pho-tons off electrons can generate absorption-like features, like cyclotron resonancescattering features or simply cyclotron lines. The cyclotron absorption featuresand the pulse shape in the X-rays are both dramatically affected by the geometricalconfiguration of the emission region, which in turn can depend on the accretionrate. A major change can happen close to the Eddington luminosity:LEdd =4piGM∗cκ= 1.26×1028(κTκ)(M∗M)ergs−1 (1.2)7where c is the speed of light, κ is the opacity and κT is the opacity due to Thomsonscattering. If the accretion rate is high, so is the X-ray luminosity (see eq.1.1),and when the luminosity approaches the Eddington luminosity, the pressure fromthe outgoing radiation becomes important in stopping the infalling gas. As firstproposed by [15], at low luminosity, the gas can freefall all the way to the neutronstar surface, and the kinetic energy of the accretion flow is only released upon theimpact with the neutron star surface, where the ionized gas is stopped mainly bynucleon-nucleon collisions. The heat is released deeply in the atmosphere. generat-ing hot-spots at the magnetic poles. The opacity of a strongly magnetized plasma islower along the field lines, and therefore the Comptonized X-rays escape predom-inantly upwards, and form a so-called “pencil-beam” pattern. As the luminosityincreases, the stopping power of radiation becomes more important and if the lu-minosity is higher than the critical luminosity Lc ∼ 4× 1036 erg s−1 ∼ 0.03LEdd[16, 21, 161], a radiation dominated shock rises above the neutron star surface,forming an extended accretion column [16, 20, 34]. In this case, photons can onlyescape through the walls of the column, and a “fan” emission pattern is expected.The higher the accretion rate, the higher the luminosity, and consequently the ac-cretion column, until an asymptotic luminosity is reached, that depends stronglyon the accretion geometry [16]. For a solid axisymmetric column it corresponds toa quarter of the Eddington luminosity, but it can exceed the Eddington luminosityby several times in the case of a hollow accretion column in which the material isconfined to a narrow wall of magnetic funnel.The emission from X-ray pulsars is hard to model, because the picture is com-plicated by the presence of a strong magnetic field, by the importance of radiationpressure in the description of the accretion flow and by the fact that the emittinggas is flowing with a high bulk velocity, up to half of the speed of light. Severalattempts have been made to calculate the spectral formation based on theoreticalmodels [112, 149, 150, 162, 246] but the results do not agree very well with theobserved profiles. On the other hand, the procedure of fitting the spectra withmulticomponent functions of energy as power laws, blackbodies and exponentialcut-offs is not easy to relate to physical properties of the source.The situation improved with the development by Becker and Wolff of a newmodel for the spectral formation that includes the effect of “thermal” and “bulk”8Comptonization of the photons by the converging flow of electrons [18–20, 244].This new model, which I will introduce in detail in Chapter 7, predicts a spectrumthat fits very well the observed profiles and returns estimates of the properties of theaccretion flow, as for example the optical thickness of the column, the temperatureof the electrons and the size of the column itself. Even though several simplifyingassumptions are made to make the treatment analytic, the Becker and Wolff modelis the current theoretical model that best fits observations, and it is the basis of mytreatment of the polarized emission from X-ray pulsars (Chapter 7).1.1.3 Neutron stars in the X-rays: MagnetarsThe history of magnetars is more recent, as the first detection of “Unusual γ-ray bursts [...] from a flaring X-ray pulsar in the constellation Dorado” dates to1979, by the space probes Venera 11 and 12 [137, 138]. These bursts were ini-tially thought to be of the same origin as gamma-ray bursts [139], but their spectrawere softer and, contrary to gamma-ray bursts, they were observed to repeat. Theidentification with a neutron star was immediate thanks to the 8-second pulsationseen in the tail of the burst [138] and to the association with the supernova rem-nant N49 [45]. The period of 8 seconds, however, was much longer compared topreviously detected neutron stars as the Crab pulsar (33 ms), and the object wasinterpreted initially as an accreting neutron star. It was only when a total of threeobjects showing the same behaviour was found that these neutron stars started tobe considered as a separate class, and they were called Soft Gamma-ray Repeaters,or SGRs [116, 128].The first magnetar models were proposed by Duncan and Thompson [56] in1992, and at about the same time by Paczynski [173]: a strong magnetic field is thecause of both the SGR activity and of the very long periods observed. Given the lo-cation of the 8-second period SGR 0526-66 in the center of the supernova remnantN49, a magnetic field of the order of 1014−1015 G is required to brake the pulsarfrom a birth period of milliseconds in the typical lifetime of a supernova remnant(about 10,000 years). Moreover, the high magnetic field, especially if it is evenhigher in the interior of the star, can function as a reservoir for the energy neededto explain the SGR activity. Thompson and Duncan [226, 227] also demonstrated9that the gamma ray bursts can be explained by large-scale reconnection events, andthat the decay of the strong magnetic field can power the quiescent emission (seealso [92]).In the meantime, a new class of pulsars was discovered that had persistent softX-ray emission and long pulsation periods but no sign of a binary companion [58,78, 84, 101, 204]. These sources, called Anomalous X-ray Pulsars or AXPs, wereinterpreted as very-low-mass X-ray binaries [146, 234]. The first identification ofAXPs as SGRs in quiescence was advanced by Thompson and Duncan [227], and itwas confirmed less than ten years later when two AXPs exhibited SGR-like bursts[66, 109].Nowadays, SGRs and AXPs are considered to be the same class of sources:magnetars. Confirmation on the strength of the magnetic field has come from themeasurement of the spin down rates of a few magnetars [117, 118], of which bothmagnitude and sign showed a very good agreement with the predictions from themodels. Even if the origin is still debated, absorption features have been detectedin magnetar spectra that have been mostly interpreted as proton cyclotron fea-tures from a magnetar-strength magnetic field, confirming in many cases the highmagnetic field value inferred from spin-down measurements (e.g., 5 keV absorp-tion line from SGR1806-20 [100]; 8.1 keV absorption line from 1RXS J170849-4009104 [187]; 4 keV and 8 keV emission lines from 4U 0142+62 [67]).I will not present a detailed review of the phenomenology of magnetars, thereader is invited to read the exhaustive review by Kaspi and Beloborodov [107].However, the main observational characteristics of magnetars can be summarizedas [232]:• long pulsation periods, in the range 2-12 s;• large spin-down rates: P˙ ∼ 10−13−10−11 s s−1, which convert to magneticfields of the order 1014−1015 G if interpreted as due to magnetic braking;• a persistent X-ray luminosity of the order 1033− 1036 erg s−1 in the soft(0.5-10) and in the hard (20-100 keV) X-ray range;• many exhibit bursting activities, comprising of short bursts of about 0.1-1 s,the most common, with peak luminosity of∼ 1039−1041 erg s−1 and thermal10spectra; intermediate bursts of ∼ 1-40 s, with peak luminosity of ∼ 1041−1043 and also thermal spectra; and the exceptionally rare giant flares, withan energy output of ∼ 1044− 1047 erg s−1. Giant flares were only detectedthree times, and all three events started with an initial spike of ∼ 0.1−0.2 s,followed by a long pulsating tail (lasting a few hundred seconds) modulatedat the neutron star spin period.In Chapter 8, I will model the polarized persistent emission of magnetars inquiescence. Spectra of magnetars are best studied in the soft X-rays (0.3− 10keV), thanks to decades of observations from instruments such as XMM-Newton,Chandra and Swift [169]. In this range, spectra are well parametrized by an ab-sorbed blackbody component (kT ∼ 0.3−0.5) and a steep power law, with photonindex between −2 and −4 (see Fig 8.1). In some sources, good fits of the ob-served spectra are obtained with a double blackbody as well [79]. These are onlyphenomenological parametrizations; however, the thermal component is thought tocome from the hot surface of the neutron star, while the steep power law is thoughtto be caused by a combination of atmospheric and magnetospheric effects.Hard X-ray observations with INTEGRAL and RXTE, later confirmed by NuS-TAR, have shown an inversion in the spectrum at about 20 keV in a few persistentsources: a “hard tail” was detected, with a positive slope, that extends to hundredsof keV. This means that a non-thermal process is causing the bulk of the magnetar’semission. The origin of this hard emission is still debated, and proposed mecha-nisms range from thermal bremsstrahlung in the surface layers of the star, heatedby a downward beam of charges, to synchrotron emission from pairs created in themagnetosphere [224], to resonant Compton scattering (RCS) of seed photons on apopulation of highly relativistic electrons [13].1.2 Black holesThe definition of a black hole is quite simple: a black hole is an object so compactthat not even light can escape from it. The fact that light is trapped in a black hole,causally separates the region where the black hole lives from the rest of the uni-verse: there is no possible communication between the two regions of spacetime.The boundary of the isolated region of spacetime is called event horizon.11Shortly after the Newtonian theory of gravity was developed, in the 17th cen-tury, John Michell and Pierre-Simon Laplace discussed the possibility of an objectso compact that not even light, that at that time was thought to be particle-like andwith a characteristic velocity, could escape from it [11]. The first formal solutionfor a black hole in full general relativity was developed only a year after Einsteinproposed the theory in 1915, by Karl Schwarzschild in 1916 [57, 201]. The firstrigorous calculation for the formation of a black hole from gravitational collapsewas performed by Oppenheimer and Snyder [170] in 1939, but the full understand-ing of the crucial properties of a black hole did not come until later. For example,the first to realize the existence of an event horizon was David Finkelstein [63] in1958.The first suggestion of a connection between black holes and astrophysicalobjects was advanced by Zeldovich and by Salpeter in 1964 [196, 252], who sepa-rately proposed the idea that supermassive black holes are the engines of quasars.The first strong observational evidence for the existence of black holes only camein the next decade, thanks to X-ray and optical observations of the X-ray binaryCygnus X-1 in 1972, [32, 238]. Today, more than 20 binary systems containinga stellar-mass black hole have been found [141, 190], together with tens of su-permassive black holes at the center of galaxies [114]. In addition, starting fromSeptember 2015, the date of the first detection of merging black holes by LIGO[3], the gravitational waves emitted from the coalescence of two black holes havebeen detected from many systems [164, 222].Black holes are divided in two main categories: stellar-mass black holes, formedby the collapse of massive stars, that can have masses in the range of 3−100 M,and supermassive black holes, found at the center of galaxies, with a wide rangeof masses, between 105 and 1010 solar masses. The existence of a third class withmasses in between, called intermediate-mass black holes, is still debated. The fo-cus of my work is on accreting stellar-mass black holes in X-ray binaries, but mostof my results can be extended to supermassive black holes in active galactic nuclei(or AGNs), as they are independent of the black hole mass (see Chapter 6).121.2.1 The Kerr black holeOne of the most subtle consequences of general relativity is the no-hair theorem,for which black holes can be fully characterized by a small number of parameters(they have no “hair”) [159]. In known black-hole solutions of the Einstein equa-tions, such parameters are mass, angular momentum and charge. Since we expectno charge on astrophysical black holes, the spacetime that surrounds a black holecan be nearly exactly described just by two parameters: mass (M) and angular mo-mentum (J), and the solution to the Einstein’s equation is called the Kerr metric,found in 1963 by Roy Kerr [110]. The Schwarzschild metric is the special casewith J = 0.The specific angular momentum or spin of the black hole is identified by theparameter a = J/cM, where c is the speed of light. It is often convenient to ex-press the spin value in terms of a dimensionless spin parameter, a? = a/Rg, whereRg = GM/c2 is the gravitational radius. The value of a? lies between 0 for aSchwarzschild hole and 1 for a Kerr hole rotating at critical velocity. While themass gives the scale of the system, the spin parameters modify the geometry of thespacetime. In general relativity, the choice of the coordinate system is arbitrary. Inthe case of a Kerr black hole, the Boyer-Lindquist coordinate system is a conve-nient choice. In natural units (G= c= 1), the spacetime interval in the Kerr metricis expressed in the form:ds2 = gttdt2+2gtφdtdφ +grrdr2+gθθdθ 2+gφφdφ 2=−(1− 2MrΣ)dt2− 4aMr sin2 θΣdtdφ +Σ∆dr2+Σdθ 2+(r2+a2+2Mra2 sin2 θΣ)sin2 θdφ 2 (1.3)whereΣ≡ r2+a2 cos2 θ (1.4)∆≡ r2+a2−2Mr (1.5)The metric is stationary (independent of t) and axi-symmetric about the polar axis13(independent of φ ). In this coordinate system, the radial coordinate of the eventhorizon is given by ∆= 0RH = Rg(1+√1−a2?)(1.6)and it ranges from 2Rg for a Schwarzschild hole, to Rg for a hole rotating at criticalvelocity (a? =±1).It is useful to consider a stationary observer, which is an observer at fixedcoordinates r and θ , but that is rotating at a constant angular velocityΩ=dφdt=uφut(1.7)where u is the four-velocity of the observer. Since the observer has to follow atime-like worldline, the following condition applies−1 = gµνuµuν = (ut)2[gtt +2Ωgtφ +Ω2gφφ ] . (1.8)Imposing the quantity in the square brackets to be negative returns the conditionΩ− <Ω<Ω+ (1.9)whereΩ± =−gtφ ±√g2tφ −gttgφφgφφ. (1.10)Ω− vanishes when gtt = 0; this occurs atR0 = Rg(1+√1−a2? cos2 θ). (1.11)This means that observers between RH and R0 cannot be static, and they must orbitthe black hole with Ω > 0. The surface r = R0(θ) is called the boundary of theergosphere.It is insightful to look at the geodesics in the equatorial plane. By symmetry,a geodesic that starts tangent to the equatorial plane will remain in the equatorial14plane. At θ = pi/2, the Lagrangian is given by [206]2L =−(1− 2Mr)t˙2− 4aMrt˙φ˙ +r2∆r˙2+(r2+a2+2Mra2r)φ˙ 2. (1.12)From it, we can obtain two integrals of motionpt ≡ ∂L∂ t˙ =−E (1.13)pφ ≡ ∂L∂ φ˙ = L . (1.14)A third integral of motion can be obtained by setting gµν pµ pν = −m2, which isthe same as imposingL =−m2/2. After some algebra, one can obtainr3r˙2 =V (E,L,r) (1.15)= E2(r3+a2r+2Ma2)−4aMEL− (e−2M)L2−m2rwhere V can be regarded as the effective potential for radial motion in the equatorialplane. Circular orbits correspond to geodesics with r˙ = 0; which requires V = 0and ∂V/∂ r = 0. This yieldsEcirc =r2−2Mr±a√Mrr(r2−3Mr±2a√Mr)1/2 (1.16)Lcirc =√Mr(r2∓2a√Mr+a2)r(r2−3Mr±2a√Mr)1/2 (1.17)where the upper sign refers to corotating or prograde orbits, i.e. orbits with angularmomentum parallel to the black hole spin, while the lower sign corresponds tocounterrotating or retrograde orbits. Circular orbits exist for all radii greater thanthe limiting orbit, when the denominator is equal to zero, which is the photoncircular orbit:Rph = 2Rg{1+ cos[23cos−1(∓a?)]}. (1.18)At this radii, photons with zero angular momentum orbit the black hole in a circularorbit. For a Schwarzschild black hole, a? = 0, the photons can orbit the hole in15either direction and the circular orbit is at Rph = 3Rg, while for a Kerr black holethe prograde and retrograde orbits are at different radii: for a? = 1, rph = Rg forprograde and Rph = 4Rg for retrograde orbits.If we consider the motion of a test-particle around a massive body in Newto-nian gravity, equatorial circular orbits are always stable. In the case of the Kerrmetric, for r > Rph, circular orbits exist, but not all orbits are stable. Orbits withE/m > 1 are unbound, which means that, given an infinitesimal outward perturba-tion, a particle in such an orbit would escape. Moreover, even if a circular orbit isbound, it can still be unstable. Stability requires ∂ 2V/∂ 2r ≤ 0. The limiting case(∂ 2V/∂ 2r = 0), yields the radius for the Innermost Stable Circular Orbit or ISCO:rI = Rg{3+Z2− [(3−Z1)(3+Z1+2Z2)]1/2} (1.19)Z1 ≡ 1+(1−a2?)1/3[(1+a?)1/3+(1−a?)1/3]Z2 ≡ (3a2?+Z21)1/2For a? = 0, rI = 6Rg, while for a? = 1, rI = Rg for prograde orbits and RI = 9Rgfor retrograde orbits.1.2.2 The accretion diskAs light cannot escape from a black hole, when a black hole is detected in the X-rays the observed light is usually coming from an accretion disk. Accretion disksin black-hole binaries are formed because the gas transferred from the stellar com-panion has to lose its angular momentum before it can accrete onto the black hole.In the case of supermassive black holes, the accretion disk material comes fromthe surrounding interstellar medium. Depending on their origin, accretion diskscan have different characteristic and properties. An accretion disk is consideredgeometrically thin (thick) if its semi-thickness h at a distance r from the black holeis much less than r (is about r). The disk is considered optically thin (thick) ifh λ (h λ ), where λ is the photon mean free path in the disk.In Chapter 6 I will employ the accretion disk model of Novikov and Thorne[168, N&T]. The N&T accretion disk model is the general relativistic generaliza-tion of the Shakura-Sunyaev model [205], set in the Kerr spacetime. It assumes a16geometrically thin, optically thick disk, radiation dominated, where the motion ofthe gas is determined by the gravitational field of the black hole (the impact of thegas pressure is ignored). Also, the direction of the angular momentum of the diskis assumed to be aligned with the spin of the hole.1.2.3 Black holes in the X-raysX-ray binaries are grouped in two classes depending on the mass of the companionstar: low-mass X-ray binaries (LMXB), where the donor star has a mass . 3 M,and high-mass X-ray binaries, with companion masses & 10 M. In HMXB con-taining a black hole, like Cygnus X-1, accretion is typically due to the strong windsfrom the companion star, which is a continuous process, and the sources are usuallypersistent in the X-rays. LMXB, on the other hand, are usually transient sources,with the notable exception of GRS 1915+105, which has been continuously brightsince 1992. The peculiarity of GRS 1915+105 is in the very large accretion disk,which does not easily get depleted.In black-hole X-ray binaries and AGNs, accretion to the central black holetakes place via a geometrically thin, optically thick accretion disk. The spectralshape of the disk emission can be well fitted by a multi-temperature blackbody,where the temperature at each radius depends on the accretion rate and the blackhole mass. The peak temperature is reached close to the ISCO and it is in thesoft X-rays (0.1–1 kev) for black hole binaries and in the optical and UV bands(1–10 eV) for AGNs [168, 205]. The photons emitted by the disk are thoughtto be Compton up-scattered in an optically thin corona, which produces a power-law spectrum in the hard X-rays [217, 229]. The geometry of the corona is stillunknown, but it is believed to be a quite compact cloud of optically thin plasmalaying above and below the central object. Some of the up-scattered photons in thecorona are reflected back into the line of sight by the disk. This reflection emissionpresents particular features, that include an iron Kα fluorescence line at 6.4 keVand a reflection hump that peaks at ∼30 keV, formed via inelastic scattering fromfree electrons [65, 193].The same black-hole binary can be found in different spectral states, whichare thought to be related to different accretion rates. For a detailed description the17reader is directed to the reviews by McClintock and Remillard [141], Remillardand McClintock [190]. The different spectral components become more or lesspredominant depending on the state of the source. In the soft state, the X-rayluminosity is high and the thermal component from the disk is the predominantemission. In Chapter 6, I will estimate the polarization degree of the emissionfrom the disk, and therefore I will focus on the soft state.1.2.4 The role of the magnetic fieldAccretion disks have to transfer angular momentum outward in order for matterto radially fall inward toward the central object. Black-hole accretion disks, asmost astrophysical accretion disks, are rarefied, and angular momentum transferdue to molecular viscosity is inefficient and cannot lead to accretion [184]. Con-ventional accretion disk models invoke viscous and magnetic torques to transportangular momentum outwards in the disc [8, 9, 83, 205]. In § 6.2 I will calculatethe minimum magnetic field strength needed for accretion to occur in the α-model[168, 205], which assumes the magnetic field and the turbulence in the flow to bethe source of shear stresses. Another possible mechanism for angular momentumtransfer is given by winds: angular momentum flows along open magnetic fieldlines that leave from the accretion disk surface, and it is eventually expelled in aoutgoing wind [30].Information on the strength and structure of magnetic fields around black holesis hard to obtain by direct observations. From the analysis of the spectra of twoGalactic stellar-mass black holes, Miller et al. [154, 155, 157] showed that a wind isgenerated from the disk as close as 850 GM/c2 to the hole. In the paper, Miller andhis collaborators obtained an estimate of the strength of the magnetic field whendifferent magnetic process are assumed to be driving the wind [157]. The onlyindication that we have on the magnetic field structure closer to the central enginecomes from interferometry observations of the radio polarization from Sagittar-ius A*, the supermassive black hole at the center of the Milky Way, which showsevidence for a partially ordered magnetic field on scales of 12 GM/c2 [105]. InChapter 6, I describe how X-ray polarization measurements from black-hole accre-tion disks could provide a way to probe, for the first time, the strength and structure18of the magnetic field close to the event horizon.1.3 This ThesisThe main goal of this thesis is to study the polarization of light in the X-rays foraccreting black holes, X-ray pulsars and magnetars, starting from realistic physicalmodels and including the effects of vacuum birefringence and general relativity. InChapter 2, I introduce the concept of polarization, how it is described in the Stokesparameters formalism, and how it can change because of scattering or when lightpropagates in a birefringent medium. In Chapter 3, I focus on neutron stars andblack holes and describe how polarized radiation is generated in black-hole accre-tion disks and neutron star atmospheres. In Chapter 4, I start from the Lagrangianof quantum electrodynamics to derive from first principles the prediction of vac-uum birefringence. In Chapter 5, I describe how birefringence, both in plasmaand in the vacuum, can affect the polarization of neutron stars. In Chapter 6, Iderive the polarization signal of black-hole accretion disks in Kerr metric and in-cluding the QED effect of vacuum birefringence. In Chapter 7, I calculate thepolarization signal of bright X-ray pulsars, including general relativity and QED,for existing models and I present my new model, based on a physically realisticaccretion scheme. In Chapter 8, I employ realistic atmosphere models for the ther-mal emission of magnetars and different models for the non-thermal emission andI calculate the polarization signal in the context of the different models. Finally,Chapter 9 summarizes my results and reinterprets them in the context of upcomingpolarimetry missions.19Chapter 2Polarized Radiation and ItsPropagationThe polarization of light indicates the direction in which a photon’s electric fieldoscillates. The direction may remain constant, as in the case of linear polariza-tion, or change with time, as for circular or elliptical polarization. In this lattercase, the direction of the oscillating field draws a circle (or an ellipse) in the planeperpendicular to the propagation direction of the photon.In order for radiation to be polarized at emission, the emitting medium has tohave some sort of anisotropy: the electric field of the photons must have a preferredaxis of oscillation. In Chapter 3 I will show that the presence of a strong magneticfield in neutron stars has the role of breaking the symmetry of the atmosphere anddetermining the preferred axis, and it is at the origin of the polarized emissionfrom the compact objects. But even when radiation is unpolarized at emission,polarization can be built through the scattering of photons off charged particles, asI will show in the following sections. This effect is enhanced if in the scatteringmedium a strong magnetic field is present, as I will show in § 2.2.1.Usually, as light travels without being scattered or absorbed, its polarizationparameters remain unchanged. However, if light travels through a birefringentmedium, in which the index of refraction depends on the polarization direction(and I will show in Chapter 4 that even vacuum can be birefringent), the anisotropyof the medium will affect the polarization of light and change its direction. In § 2.320I will introduce a formalism to describe the change in polarization as light traversesa birefringent medium.2.1 Polarization of light and the Stokes parametersBy superposing two plane waves with orthogonal polarization directions, one candescribe the most general state of polarization of a wave. We can focus on theelectric field only, as the magnetic field will follow at 90◦. The generic expressionfor a monochromatic, linearly polarized wave can be written asE = E 0ei(k·r−ωt) (2.1)where E 0 is a real vector and its direction determines the direction of polarization. Inow take k to be in the zˆ direction and focus on an arbitrary point in space, let us sayr = 0. A wave with a generic polarization state can be written as (see Chapter 15of [42] or Chapter 2 of [195])E = E 0e−iωt = (E1xˆ+E2yˆ)e−iωt (2.2)where now E 0 is a complex vector with componentsE1 = E1eiφ1 , E2 = E2eiφ2 (2.3)For fully polarized light, the Jones calculus can be used to describe the polar-ization state of the wave. In theJones calculus, the polarization state is described bya two-component vector, the Jones vector, and, in the case of Cartesian coordinates,the two components correspond to E1 and E2, the x and y complex amplitudes ofthe electric field, while the effects of an optically active material on the polariza-tion state are expressed by 2×2 matrices. The advantage of Jones calculus is thatit includes a description of the absolute phase of the wave. However, it assumesthe light to be fully polarized, and therefore cannot be used to represent scatteringor partially polarized beams. I will now introduce the Stokes parameter formal-ism and the Mueller calculus, which is the formalism I will employ in this work.The Mueller calculus does not keep track of the phase of the wave as it consid-21ers only the time-averaged intensity of light, and it can describe partially-polarizedradiation.Taking the real part of eq. 2.2, I can write the physical components of theelectric field:Ex = E1 cos(ωt−φ1), Ey = E2 cos(ωt−φ2) (2.4)These equations describe the movement of the tip of the electric field in the x− yplane and they trace an ellipse: a generic polarization state is an elliptical polar-ization state. If the ellipse’s axes are parallel to the coordinate axes, the x and ycomponents can be written asEx = E0 cosβ cosωt, Ey =−E0 sinβ sinωt (2.5)where −pi/2 < β < pi/2. For β > 0 the ellipse is traced clockwise and the polar-ization state is called right-handed, while for β < 0 the ellipse is traced in a coun-terclockwise sense and the polarization state is called left-handed. For β = ±pi/4we find the special case of circular polarization, while for β = 0 or β =±pi/2 thewave is linearly polarized.In the general case in which the ellipse’s axes are rotated with respect to thecoordinate axes of an arbitrary angle χ , eq. 2.5 becomes:Ex = E0(cosβ cosχ cosωt+ sinβ sinχ sinωt) (2.6a)Ey = E0(cosβ sinχ cosωt− sinβ cosχ sinωt) (2.6b)This generic expression is equal to the expression in eq. 2.4 if we takeE1 cosφ1 = E0 cosβ cosχ, (2.7a)E1 sinφ1 = E0 sinβ sinχ, (2.7b)E2 cosφ2 = E0 cosβ sinχ, (2.7c)E2 sinφ2 =−E0 sinβ cosχ, (2.7d)The state of polarization can always be described by the quantities E0, β and χ . Analternative set of parameters that is often used to describe the polarization state is22given by the Stokes parameters:I = E 21 +E22 = E20 (2.8a)Q = E 21 −E 22 = E 20 cos2β cos2χ (2.8b)U = 2E1E2 cos(φ1−φ2) = E 20 cos2β sin2χ (2.8c)V = 2E1E2 sin(φ1−φ2) = E 20 sin2β (2.8d)Since the polarization state can be derived from three parameters, the Stokes pa-rameters are not independent and they are bound by the relationI2 = Q2+U2+V 2. (2.9)The Stokes parameters are useful because they relate to physical properties ofradiation: I is proportional to the intensity of the ray, and the constant of propor-tionality is usually set equal to 1, as done in eq.s 2.8; V measures the ratio of theprincipal axes of the polarization ellipse and therefore gives a measure of the “cir-cularity” of the wave (V = 0 means linearly polarized light); Q or U is the remain-ing independent parameter and it measures the orientation of the ellipse relative tothe x-axis (Q = I means vertical linear polarization and Q = −I means horizontallinear polarization, while U = I means polarized light at 45◦ with respect to thevertical).Until now, I have treated light that is 100% polarized, which means that thepolarization state of all photons in the beam adds up to a certain polarization state.However, the polarization state of a beam can be varying stochastically with time,or the polarization states of the photons can cancel each other when summing overthe beam, and in this case light is said to be unpolarized. For unpolarized lightQ =U = V = 0. In general, a light beam can be partially polarized, and it can beregarded as the superposition of a beam of polarized light and a beam of unpolar-ized light (see § 15.2 and § 15.3 of [42]). In this case:I2 ≥ Q2+U2+V 2 (2.10)23and the total degree of polarization is given byΠ=√Q2+U2+V 2I≤ 1. (2.11)Other useful quantities are the linear degree of polarization or linear polarizationfractionΠl =√Q2+U2I(2.12)and the circular degree of polarization or circular polarization fractionΠc =|V |I. (2.13)In Mueller calculus, the polarization state of a beam is described by a 4-dimensional vector, S = (I,Q,U,V ), where the 4 components are the 4 Stokesparameters, and the effect of the element of an optical system on the polarizationstate of a beam is described by the Mueller matrix:S′ = MS (2.14)where M , the Mueller matrix, is a 4×4 matrix.2.1.1 The Poincare´ sphereThe Poincare´ sphere is a useful graphical tool to depict the polarization state of abeam of light. The radius of the sphere is usually set equal to 1 or to the intensity ofthe polarized fraction (√Q2+U2+V 2). If we consider the case of the unit sphere,the polarization vector is defined ass =1S0S1S2S3 , where (S0,S1,S2,S3) = (I,Q,U,V) (2.15)The polarization states of fully polarized radiation are mapped onto the surface ofthe sphere. Linearly polarized states are positioned on the equator of the sphere,while purely circularly polarized states correspond to the north (right-handed) and24Figure 2.1: The Poincare´ sphere. Polarization states are mapped onto thesurface of the sphere, identified by the vector s. Linearly polarizedstates are positioned on the equator of the sphere (in green). Circu-larly polarized states correspond to the north (right-handed) and south(left-handed) poles of the sphere.south (left-handed) poles of the sphere. On the equatorial plane, the s1 axis spansthe states from vertically to horizontally polarized light, while s2 represents ±45◦polarized light. Any point between the equatorial plane and the poles represents anelliptical polarization state.Only fully polarized light is represented at the surface of the sphere, whilepartially polarized light will be located at a radius equal to its polarization degree(eq. 2.11), with completely unpolarized light being mapped to the origin of thesphere.2.2 Thomson ScatteringThomson scattering is the process of photons scattering off free electrons. Theelectron oscillates in response to the incoming electromagnetic wave and re-emitsa photon in a new direction. I will first address the case of linearly polarized in-25coming radiation in a non-magnetized medium and I will extend the results tounpolarized light. I will describe the case of scattering in a strongly magnetizedmedium in § 2.2.1.If the charge oscillates at velocities that are not relativistic, the magnetic fieldof the photon can be ignored and the force acting on the electron is just the LorentzforceF = mer¨ = eεE sinωt (2.16)where ε is the polarization vector and ω is the frequency of the incoming radiation.If I indicate with d = er the dipole moment of the electron, I can writed¨ =e2Emeε sinωt (2.17)Using the Larmor’s Formula, I can obtain the time-averaged power of the emittedradiation as [195]dPdΩ=sin2Θ4pic3〈d¨〉2 = e4E28pim2ec3sin2Θ (2.18)P =e4E23m2ec3(2.19)whereΘ is the angle between the incident polarization vector ε and the propagationdirection of scattering. I can find the cross section by dividing the power emittedby the incident flux (which is simply cE2/8pi)(dσdΩ)polarized=e4m2ec4sin2Θ= r2e sin2Θ (2.20)σT =8pie43m2ec4=8pi3r20 (2.21)where re is the classical radius of the electron, and σT is the Thomson cross sec-tion. The scattered radiation remains linearly polarized in the plane of the incidentpolarization ε and the direction of the outgoing radiation.I will now consider the incoming radiation as unpolarized. Unpolarized radia-tion can be described as the independent superposition of two beams of light lin-26early polarized in orthogonal directions. Without loss of generality, I can chooseone beam to be polarized in the same plane as the outgoing radiation direction n,with polarization vector ε 1 . The angle Θ now indicates the angle between ε 1 andn, while the angle between ε 2 (the orthogonal beam’s polarization vector) and n ispi/2. The differential cross section for the unpolarized radiation will be the averageof the cross sections for scattering of the two beams(dσdΩ)unpol=12[(dσdΩ(Θ))pol+(dσdΩ(pi/2))pol]=12r20(1+ sin2Θ)=12r20(1+ cos2 θ) (2.22)where θ = pi/2−Θ is the angle between the incident radiation and the scatteredone. I can easily derive the degree of polarization of the scattered radiation as thetwo polarized intensities in the plane and perpendicular to the plane of scatteringare in the ratio cos2 θ :Π=Πl =1− cos2 θ1+ cos2 θ(2.23)Even when the incident radiation is totally unpolarized, some fraction of thescattered light is linearly polarized in the plane of scattering, and the polarizationfraction increases with θ : if we look in the direction of the incident radiation, atθ = 0, as we expect we see no net polarization since, by symmetry, all directionsin the plane are equivalent. On the other hand, since the electron motion is con-fined in the plane perpendicular to the incident wave, if we look in the directionperpendicular to the incident wave we see 100% polarized light.2.2.1 Thomson scattering in a strong magnetic fieldStrong magnetic fields affect the motion of electrons by forcing them to movemainly along the field lines. This tendency strongly alters the interaction betweenphotons and electrons: in Chapter 3 I will show how strong magnetic fields alteropacities in neutron star atmospheres and in Chapter 4 I will describe how plasmaand vacuum can become birefringent in the presence of a strong magnetic field.27Thomson scattering is no exception. Following the calculations performed by Chou[44], I will develop a formalism to analyze the change in the Stokes parametersafter Thomson scattering in the presence of a strong magnetic field.In contrast with the previous section, now the presence of a strong magneticfield alters the equation of motion of the electronsmev˙ =−eE (t)− ecv×B (2.24)whereE (t) = (Exxˆ+Eyyˆ+Ezzˆ)eiωt = (E‖ cosα xˆ+E⊥yˆ−E‖ sinα zˆ)eiωt (2.25)is the photon’s electric field and B = Bzˆ is the uniform, static magnetic field. Also,I indicate with α the angle between the incident radiation and the magnetic field,and with E‖ the component of E (t) in the plane of B and k, the wavevector of theincident wave, and with E⊥ the component perpendicular to the plane.If I write the induced electron acceleration as v˙(t) = (v˙xxˆ+ v˙yyˆ+ v˙zzˆ)eiωt , theneq. 2.24 yieldsv˙x =emeω2E‖ cosα+ iωωcE⊥ω2c −ω2(2.26)v˙y =emeω2E⊥− iωωcE‖ cosαω2c −ω2(2.27)v˙z =emeE‖ sinα (2.28)The dipole radiation field emitted by the electron is given by [103]E e(x, t) =−eDc2[rˆ× (rˆ× v˙)] (2.29)where rˆ = D/D is the unit vector directed from the position of the electron to theobserver, and D is the distance between the emission region and the observer.It is now convenient to transition to spherical coordinates, where (r,θ ,φ) in-dicates the direction of the scattered radiation. The velocity of the electron in this28system is given byv˙r = v˙x sinθ cosφ + v˙y sinθ sinφ + v˙z cosθ (2.30)v˙θ = v˙x cosθ cosφ + v˙y cosθ sinφ − v˙z sinθ (2.31)v˙φ =−v˙x sinφ + v˙y cosφ (2.32)where v˙x, v˙y and v˙z are given in eq. 2.28. Eq. 2.29 therefore yieldsE e = Eeθ θˆ +Eeφ φˆ=reD{[ζ cosθ(u1+ i xu2)−u3]θˆ +ζ (u2− i xu1)φˆ } (2.33)where re = e2/mec2 is the classical radius of the electron andx =ωcω, ζ =1x2−1 (2.34)u1 = E‖ cosα cosφ +E⊥ sinφ , (2.35)u2 =−E‖ cosα sinφ +E⊥ cosφ , (2.36)u3 = E‖ sinα sinθ (2.37)The Stokes parameters for the incident radiation can be written in terms of theparallel and perpendicular components of the electric fieldI = S0 = E‖E∗‖ +E⊥E∗⊥Q = S1 = E‖E∗‖ −E⊥E∗⊥U = S2 = E‖E∗⊥+E⊥E∗‖V = S3 = i(E‖E∗⊥−E⊥E∗‖ ) (2.38)while the Stokes parameters of the scattered radiation can be expressed in terms of29the emitted electric field components derived in eq. 2.33I′ = S′0 = EeθEe∗θ +EeφEe∗φQ′ = S′1 = EeθEe∗θ −EeφEe∗φU ′ = S′2 = EeθEe∗φ +EeφEe∗θV ′ = S′3 = i(EeθEe∗φ −EeφEe∗θ ) (2.39)From eq.s 2.33, 2.38 and 2.39 we can express (I′,Q′,U ′,V ′) as a function of theincident Stokes parameters (I,Q,U,V ), of the direction of the incident radiationwith respect to the magnetic field, expressed by the angle α , of the direction of theoutgoing radiation, expressed through the polar and azimuthal angles θ and φ , ofthe energy of the photon, encoded in ω , and of the strength of the magnetic field,in ωc, or more precisely by the ratio of the two, encoded in x. In Mueller calculus,the relations can be written in a matrix formI′Q′U ′V ′= r2e2D2M11 M12 0 M14M21 M22 0 M240 0 M33 M34M41 M42 0 M44IQUV (2.40)For the full expressions of the matrix elements Mi j see [44]. In the cases that Iwill consider in this work, azimuthal symmetry is always present, and therefore I30can average over φ , which yieldsM11 =ζ 22(1+ x2)(cos2α+1)(cos2 θ +1)+ sin2α sin2 θ (2.41a)M12 =ζ 22(1+ x2)(cos2α−1)(cos2 θ +1)+ sin2α sin2 θ (2.41b)M14 = −2ζ 2xcosα(1+ cos2 θ) (2.41c)M21 =ζ 22(1+ x2)(cos2α+1)(cos2 θ −1)+ sin2α sin2 θ (2.41d)M22 =ζ 22(1+ x2)(cos2α−1)(cos2 θ −1)+ sin2α sin2 θ (2.41e)M24 =2ζ 2xcosα sin2 θ (2.41f)M33 =0 (2.41g)M41 = −2ζ 2x(1+ cos2α)cosθ (2.41h)M42 =2ζ 2xsin2α cosθ (2.41i)M44 =2ζ 2(1+ x2)cosα cosθ (2.41j)After taking the average over φ , I find that all the matrix elements that involve Uare equal to zero. I can therefore reduce the matrix to a 3×3 matrix where the thirdelement correspond to the circular polarization parameter V : I′Q′V ′= r2e2D2M11 M12 M13M21 M22 M23M31 M32 M33 IQV (2.42)and where M13 = M14 of eq. 2.41c and so forth.The angular dependence of the incoming and outgoing radiation can be ex-panded in a series of orthonormal functions in α and θ . Since the matrix elementsare only functions of cosα , cos2α , sin2α and the same for θ , the only important31functions for the expansion are given byf1(α) =√154sin2α; (2.43)f2(α) =√62cosα; (2.44)f3(α) =5√34(cos2α− 15)(2.45)and the same for θ . I can write the Stokes parameter in this new basis:I = l1× f1(α)+ l2× f2(α)+ l3× f3(α)Q = l4× f1(α)+ l5× f2(α)+ l6× f3(α)V = l7× f1(α)+ l8× f2(α)+ l9× f3(α)I′ = l′1× f1(θ)+ l′2× f2(θ)+ l′3× f3(θ)...and so forth, where li do not depend on angles. In this way, I can rewrite thescattering matrix in eq. 2.42 as a 9x9 matrix in this new basisl′1l′2...l′9= re2D2a1,1 a1,2 · · · a1,9a2,1 a2,2 · · · a2,9....... . ....a9,1 a9,2 · · · a9,9l1l2...l9 (2.46)where the matrix elements are just functions of x and ζ , and the angle dependenceis conveyed by the f functions. In this way, I can efficiently compute the effects ofthe external magnetic field on the scattered radiation.I will employ this formalism in Chapter 7 and in Chapter 8 to analyze the ef-fect on the X-ray polarization of Compton scattering in strong magnetic fields. Therelation expressed in eq. 2.46 is valid in the instantaneous rest frame of the elec-trons; if the motion of the electrons is relativistic, I will have to consider beamingeffects. Also, I have not considered any energy transfer between the electron andthe photon, which I will have to include in the case of Compton scattering.322.3 Propagation of light through an inhomogeneous andanisotropic mediumIn this section I will follow the work of Kubo and Nagata [119], who analyzedthe change in the polarization state as light travels through an inhomogeneous andanisotropic medium employing the Stokes parameters formalism. Maxwell’s equa-tions in an inhomogeneous medium can be written as∇×H = 1c∂D∂ t, ∇×E =−1c∂H∂ t(2.47)∇ ·D = 0 , ∇ ·B = 0 (2.48)D = [ε]E , B = H (2.49)where E , D, B, and H are the electric vector, the electric induction, the magneticinduction, and the magnetic vector, respectively. The tensor [ε] is a non-Hermitianand asymmetrical dielectric tensor with general complex elements, which repre-sents the various types of birefringence and absorption in the medium.I take the coordinate system to be (x1,x2,x3) and light to be propagating alongthe x3 direction. Without loss of generality, the tensor [ε] can be written as [ε] =[ε ′]s + i[ε ′]a + i[ε ′′]s− [ε ′′]a, where [ε ′]s and [ε ′′]s are symmetric real tensors and[ε ′]a and [ε ′′]a are antisymmetric real tensors. The tensor [ε] with respect to prop-agation in the direction of the x3 axis can be written as the two-dimensional tensor[ε¯] =[ε11 ε12ε21 ε22](2.50)=[ε ′s11+ iε′′s11 ε′s12+ iε′a12+ iε′′s12− ε′′a12ε ′s21+ iε′a21+ iε′′s21− ε′′a21 ε′s22+ iε′′s22](2.51)I can obtain the E vector equation from eq.s 2.47 and 2.49:1c2∂ 2[ε]E∂ 2t=−∇×∇×E , (2.52)33and I can express the waves propagating in the x3 direction asE j = G j exp[i(k0Φ−ωt)], (2.53)Φ=∫ [12(ε′s11+ ε′s22)]1/2dx3 (2.54)where E j are the components of the electric field, G j are the complex electric fieldamplitudes of E j (without the rapidly varying phase), ω is the frequency of light invacuum, and k0 = ω/c.For a weakly inhomogeneous, anisotropic and optically active medium, eq. 2.52yields∂∂x3[G1G2]=i2[Ω1+ iT0+ iT1 Ω2+ iΩ3+ iT2−T3Ω2− iΩ3+ iT2+T3 −Ω1+ iT0− iT1][G1G2](2.55)where Ω1 =C(ε′s11−ε′s22), Ω2 = 2Cε′s12, Ω3 = 2Cε′a12, T0 =C(ε′′s11+ε′′s22), T2 = 2Cε′′s12,T3 = 2Cε′′a12 and C = k0/(ε′s11+ ε′s22)1/2.The expression in eq. 2.55 represents the evolution of the polarization stateas described by the Jones formalism, in terms of the complex amplitudes of theelectric field. I want now to rewrite the expression in eq. 2.55 in terms of theStokes parameters and the Mueller calculus. This translation can be performedwith the help of Wolf’s coherency matrix, J . Wolf’s matrix is a 2×2 matrix whosecomponents are given by Ji j = E∗i E j, where the Ei are the complex amplitudesof the electric field. The relation between the coherency matrix and the Stokesparameters (S0,S1,S2,S3) = (I,Q,U,V ) is given by [74]J =123∑i=0σ S (2.56)where S = [S0,S1,S2,S3], σ t = {σˆ0, σˆ1, σˆ2, σˆ3}, σ t is the transpose of σ andσˆ0 =[1 00 1], σˆ1 =[1 00 −1], σˆ2 =[0 11 0], σˆ3 =[0 i−i 0](2.57)34In other words, the Stokes parameters are the components of the coherency matrixof the radiation when expanded in the basis of {σˆ0, σˆ1, σˆ2, σˆ3}, where σˆ0 is the unitmatrix and σˆi are the Pauli matrices. Therefore we can writeS = E †σE = G†σG (2.58)where E † is the adjoint of E . The representation of eq. 2.55 with the matrix σbecomes∂G∂x3=i2[iT0σˆ0+(Ω1+ iT1)σˆ1+(Ω2+ iT2)σˆ2+(Ω3+ iT3)σˆ3]G=i2(Ω+ iT )G (2.59)where Ω = Ω1σˆ1 +Ω2σˆ2 +Ω3σˆ3 = {0,Ω1,Ω2,Ω3} and T = T0σˆ0T1σˆ1 +T2σˆ2 +T3σˆ3 = {T0,T1,T2,T3}. From the derivatives of S with respect to x3 and eq. 2.59, Ican derive the change in the Stokes parameters as light travels in the x3 direction∂S∂x3=i2G†[(Ω+ iT )†σ −σ (Ω+ iT )]G (2.60)Using the commutation relations σˆ jσˆk = σˆ0δ jk− iε jkmσˆm ( j,k= 1,2,3) in eq. 2.60,I can derive∂S∂x3= [ω]S = {[ω]s+[ω]a}S (2.61)where[ω]s =T0 T1 T2 T3T1 T0 0 0T2 0 T0 0T3 0 0 T0 , [ω]a =0 0 0 00 0 −Ω3 Ω20 Ω3 0 −Ω10 −Ω2 Ω1 0 (2.62)and [ω]s and [ω]a are the symmetric and antisymmetric parts of [ω] respectively.Eq. 2.61 can be rewritten in a simple vectorial manner if the normalized Stokesvector is considered s = (S1/S0,S2/S0,S3/S0)∂ s∂x3= Ωˆ× s+(Tˆ × s)× s (2.63)35where Ωˆ = (Ω1,Ω2,Ω3) is the birefringence vector and Tˆ = (T1,T2,T3) is thedichroic vector. The component T0 drops out since only the relation among thecomponents S1, S2 and S3 is considered.In the following chapters, I will not consider any dichroic effect (for the photonenergies that I am interested in, in the X-rays, there is no pair production), andtherefore Tˆ = 0.2.3.1 Propagation of light in a birefringent mediumA birefringent medium is an anisotropic medium in which the index of refractiondepends on the polarization direction of light. The simplest birefringence is calleduniaxial, which means that the optical anisotropy is driven by a single axis, and themedium is still symmetric for rotation around this special axis. The birefringencethat I will consider in the following chapters is given by the presence of a strongmagnetic field, and the direction of the magnetic field sets the special axis.As shown in § 2.1, light can always be considered as a superposition of waveslinearly polarized in orthogonal directions. In the case of uniaxial birefringence,as light travels through the birefringent medium, the component polarized parallelto the special axis can propagate faster (or slower) than the orthogonal component,and the difference in velocity is given by ∆v = c∣∣∣ 1n‖ − 1n⊥ ∣∣∣, where n‖,⊥ are the in-dices of refraction in the parallel and the perpendicular mode. In this case, thebirefringent vector amplitude is given by |Ωˆ| = |k0∆n| where ∆n = n‖− n⊥ andk0 is the wavenumber of the radiation in the vacuum. Therefore the change in theStokes parameters is given by (eq. 2.63)dsdλ= Ωˆ× s (2.64)where λ measures the length of the photon path in the medium. The directionof Ωˆ points toward the polarization of the faster mode on the Poincare´ sphere ofpolarization states.This equation may seem more familiar if one considers the Faraday rotation ofpolarized light passing through a weakly magnetized plasma. In this case, Ωˆ pointstoward the S3-direction, corresponding to the circular polarization, so the polariza-tion direction of linearly polarized light will rotate. In general, if the direction of Ωˆ36is constant, the vector s will circle the direction of Ωˆ. If |Ωˆ| is sufficiently large, thevector s will circle the direction of Ωˆ even in the case in which Ωˆ changes directionand magnitude, if it does so sufficiently gradually. In particular, if the polarizationstate is initially parallel (or perpendicular) to Ωˆ, that is, the initial polarization isparallel (or perpendicular) to the special axis, the polarization state will remainnearly parallel (or perpendicular) to Ωˆ as long as [93]∣∣∣∣∣Ωˆ(dln |Ωˆ|dλ)−1∣∣∣∣∣≥ 0.5. (2.65)If this condition holds, the polarization states evolve adiabatically, and the polar-ization direction will follow the direction of the birefringence.37Chapter 3The Origin of PolarizedRadiation in Black Holes andNeutron StarsIn the previous chapter I showed how polarization is described in the Stokes for-malism and how it can change when propagating toward the observer. In this chap-ter, I focus on compact objects and describe how polarized radiation is generatedin the atmospheres of neutron stars and in black-hole accretion disks.3.1 Black holesThe simplest and best understood spectral state of accreting black holes is the ther-mal state, which is characterized by the predominance of the thermal emission bythe disk and thus is well fitted by a multi-temperature blackbody peaking in thesoft X-rays (see § 1.2.3). In this section, I will only consider the thermal polarizedemission from the accretion disk itself; for polarized emission from the corona see[199] and references therein. The physical model usually employed is that of a geo-metrically thin, optically thick disk [168, 205], and the polarized radiation from theinner disk can be well described by an electron-scattering dominated atmosphere.383.1.1 Polarization of an electron-scattering atmosphereThe polarization degree of a plane-parallel, electron-scattering atmosphere, was de-rived by Chandrasekhar [42] in 1960. In his book, instead of using the usual Stokesvector S =(I,Q,U,V ), Chandrasekhar uses the polarization vector I =(Il, Ir,U,V ),whereIl =12(I+Q) and Ir =12(I−Q) (3.1)are the intensities in two directions at right angles to each other, and U and V are thesame Stokes parameters as in the original set. In the case of Thomson scattering,we can take Il = I‖ as the intensity in the plane of scattering and Ir = I⊥ as theintensity in the perpendicular direction. In this basis, if we indicate with Θ theangle between incident and scattered light, from eq. 2.22 the scattered intensity inthe direction Θ can be written as(σTdΩ′4pi)RIdΩ (3.2)where I and dΩ are the polarization vector and solid angle of the incident beam oflight, dΩ′ is the solid angle of the scattered light in the Θ direction andR =32cos2Θ 0 0 00 1 0 00 0 cosΘ 00 0 0 cosΘ . (3.3)I now characterize the radiation field at each point by the intensity vectorI(θ ,φ) = [Il(θ ,φ), Ir(θ ,φ),U(θ ,φ),V (θ ,φ)] (3.4)where θ and φ are the polar angles of an appropriate coordinate system. If I writethe scattering opacity asκ =σTρn =σTmp1+X2(3.5)where n is the number of scattering centers per unit volume, ρ is the mass density,mp is the proton mass and X is the hydrogen mass fraction, I can write the equation39of radiative transfer as− dI(θ ,φ)κρds= I(θ ,φ)−S(θ ,φ) (3.6)where S(θ ,φ) is the vector source function for I(θ ,φ).The contribution dS(θ ,φ ;θ ′,φ ′) to the source function arising from the scat-tering of a beam of radiation of solid angle dΩ′ in the direction (θ ′,φ ′) is givenbyRIdΩ′4pi(3.7)if I(θ ′,φ ′) is referred to the directions parallel and perpendicular to the plane ofscattering.After a considerable gymnastics with angles, one can integrate the contribu-tions of dS(θ ,φ ;θ ′,φ ′) and find the total source function asS(θ ,φ) = 14pi∫ pi0∫ 2pi0P(θ ,φ ;θ ′,φ ′)I(θ ′,φ ′)sinθ ′dθ ′dφ ′ (3.8)where the matrix P(θ ,φ ;θ ′,φ ′) is a mixture of R and rotation matrices and its fullexpression is given in § 17.1 of [42]. If I now define µ = cosθ and µ ′ = cosθ ′, Ican write the transfer equation in the plane-parallel atmosphere asµdI(θ ,φ)dτ= I(τ,µ,φ)− 14pi∫ +1−1∫ 2pi0P(µ,φ ;µ ′,φ ′)I(τ,µ ′,φ ′)dµ ′dφ ′ (3.9)In the case of black hole accretion disks, the axial symmetry requires the polar-ization vector of the emitted photon to be in the direction parallel or perpendicularto the plane of the disk and perpendicular to the propagation direction of the pho-ton. For this reason, U =V = 0 and I can rewrite eq. 3.9 asµddτ(Il(τ,µ)Ir(τ,µ))(3.10)=(Il(τ,µ)Ir(τ,µ))− 38∫ +1−1(2(1−µ2)(1−µ ′2)+µ2µ ′2 µ2µ ′2 1)(Il(τ,µ)Ir(τ,µ))dµ ′.Because I am not considering any illumination of the disk, the boundary conditions400.0 0.2 0.4 0.6 0.8 1.00.000.020.040.060.080.100.12Degree of PolarizationFigure 3.1: Degree of polarization of an electron scattering, plane-parallelatmosphere as a function of µ as tabulated in Table XXIV of [42].to find a solution areIl(0,−µ) = Ir(0,−µ) = 0 (3.11)andIl(τ,µ)< eτ and Ir(0,−µ)< eτ for τ → ∞ (3.12)for convergence.The exact solutions of eq. 3.10 that satisfy the boundary conditions were cal-culated in § 68 of [42]. The intensity at the surface for a direction parallel (Il(0,µ))and perpendicular (Ir(0,µ)) to the disk plane are tabulated in Table XXIV of [42]together with the degree of polarization. For light traveling in the vertical direction(µ = 1) the two intensities are equal and therefore the degree of polarization iszero, while at the limb (µ = 0), the ratio between the two is about 25 percent andtherefore the mean polarization fraction is about 11.7 percent (see also Fig. 3.1).3.1.2 The relativistic effectsThe Chandrasekhar [42] result represents the seed radiation emitted at the surfaceof the accretion disk, in the fluid frame. The observed radiation will be much41distorted by relativistic effects. Since these effects are stronger closer to the blackhole, and since the temperature of the disk is higher closer to the hole, the changesin the polarization due to relativity are stronger for higher energy photons, whichare emitted on average closer to the black hole.Another effect is caused by returning radiation: photons emitted very close tothe black hole are strongly deflected by gravitational lensing, and can bend overthe black hole and intersect the accretion disk a second time. The disk is expectedto be highly ionized (T ∼ 1 keV), and therefore the returning radiation can scatteroff the disk at large angles and reach te observer, which naturally leads to highpolarization.Schnittman and Krolik [198] calculated the polarization of the emission fromthe disk including all these effects. For the direct radiation, they found that thegreatest changes in polarization are due to gravitational lensing and relativisticbeaming. The former causes the far side of the disk to appear warped up toward theobserver and thus that part of the disk has a smaller effective inclination, reducingthe polarization seen by the observer. The latter, relativistic beaming, reduces theeffective inclination and thus the degree of polarization of photons emitted alongthe direction of orbital motion (blue-shifted photons), while it increases effectiveinclination and degree of polarization for photons emitted against the direction oforbital motion (redshifted photons). Additionally, light bending rotates the po-larization direction, decreasing the polarization degree of the spatially integratedemission. These effects are most important close to the black hole, where the gasis hottest and photons are emitted with higher energies. They find that for lowerenergies, the direct radiation follows the Chandrasekhar result, and depends onlyon the inclination angle. Higher energies probe the inner disk regions and the rel-ativistic effects reduce the polarization degree, to a factor that depends on the spinof the black hole.The direct radiation is polarized in a direction parallel to the plane, with a po-larization degree of a few percent, while the scattered returning radiation is highlypolarized in a direction perpendicular to the plane, especially for observers at highinclination angles. Therefore, when Schnittman and Krolik [198] take into ac-count the returning radiation as well, they find that at low energies, where photonsare coming mainly from the outer parts of the disk, the emission is dominated by42direct radiation and the polarization is of the order of a few degrees parallel tothe disk. At intermediate energies (above 1 keV), the returning radiation starts todominate and the polarization degree reaches a minimum as the two contributionscancel each other. At high energies, the polarization degree goes up again, but thistime the direction is perpendicular to the disk. The energy range at which the tran-sition happens decreases with higher black hole spins (the polarization minimumis reached above 10 keV for a? = 0 and at around 2 keV for a? = 0.9).An additional effect that can change the polarization of photons as they travelin the magnetosphere of the disk is the QED effect of vacuum birefringence. Noneof the previous calculations of the polarization of accreting black holes in the X-rays has taken into account this effect. In Chapter 4, I explain in detail the originof the birefringence, and in Chapter 6 I calculate the effect of QED for photonstraveling parallel to the disk plane, showing that QED has to be taken into accountif we want to understand future polarimetric observations.3.2 Neutron StarsThe polarized emission of isolated neutron stars comes from their surface, eitherfrom an atmosphere (§ 3.2.1) or from a condensed surface (§ 3.2.2), and thus I willfocus on this region first. Afterwards (§ 3.2.3), I will describe how scattering in themagnetosphere can diminish the extent of polarization.3.2.1 Neutron-Star AtmospheresThe emission processes in a neutron star’s atmosphere are strongly influenced bythe magnetic field. Isolated neutron stars have magnetic fields that range from 1011to 1015 Gauss; such strong magnetic fields can constrain the motion of particles inthe atmosphere and the geometry of emission.In the atmosphere of a typical neutron star, the temperature is much less thanthe electron cyclotron energy,kT = 0.086 keVT106 K h¯ωc = h¯ eBmec = 11.6 keVB1012 G, (3.13)and much higher than the proton cyclotron energy (∼ 6 eV at 1012 G). This means43that the typical photon energy is not sufficient to excite motion across the magneticfield lines and the scattering and absorption cross-sections depend strongly on thepolarization state of the photon and its direction of motion [41]. Furthermore,as the typical electron energy is also much smaller than the cyclotron energy, theelectrons are found in the ground Landau level and are restricted to move along thefield lines.The cyclotron energy is also much larger than the typical energy of electrons inatoms and the strong magnetic field squeezes the electron clouds around the nuclei,increasing the binding energies; therefore, the structure and binding energies ofatoms, if atoms indeed exist at the surfaces of neutron stars, are expected to bedramatically different [194, 223], so even small atoms such as hydrogen may have asignificant neutral fraction in the high temperatures of the neutron star atmosphere.The composition of the surface of isolated neutron stars is uncertain, and thereforecurrent atmospheres models span a wide range of compositions: hydrogen [214],helium [145], carbon [216], mid-Z elements [160] and iron [186]. For simplicity, Iwill consider fully ionized hydrogen atmospheres in the discussion that follows, butthe general polarization properties of emerging radiation depend on the geometryof the polarization states and on how they interact with free and bound electrons,so the results for hydrogen are illustrative of other compositions.In the atmosphere, if a photon propagates in a direction that is perpendicular tothe field, and its energy is far from the cyclotron energies, its polarization modeswill remain nearly linear within the atmosphere, so that the transverse componentof the electric field of the wave is either within the plane containing the magneticfield direction and the wave vector (parallel or ordinary mode) or perpendicularto that plane (perpendicular or extraordinary mode). Also, if the radiation is inthe extraordinary mode, a small longitudinal component (EL) is present along thedirection of propagation: the ratio to the transverse field (ET ) is given by [148]:∣∣∣∣ELET∣∣∣∣=∣∣∣∣∣ ω2pωcω (ω2−ω2c −ω2p)∣∣∣∣∣≈ ω2pωωc for ω  ωc (3.14)where ωp = (4pinee2/me)1/2 is the plasma frequency, and ne is the number densityof electrons. This longitudinal electric field is typically smaller by a large factor44relative to the transverse field. On the other hand, if the photon is propagatingalong the field, there is no longitudinal component of the photon’s the electric field,and the transverse electric field in both modes is perpendicular to the magneticfield, and cannot accelerate the electrons unless the energy is close to the cyclotronenergy.If we focus on photons traveling nearly perpendicular to the magnetic fielddirection (the angle between the direction of propagation of the photon and themagnetic field θ is about pi/2), the non-relativistic scattering cross sections forthe two polarization modes, ordinary (1) and extraordinary (2), become [41, 80–82, 85, 106, 115, 213]:σ1 ≈ σT sin2 θ (3.15)σ2 ≈ σT(ω2(ωc−ω)2+ cos2 θ). (3.16)where in the ordinary mode, the cross section tends to the Thomson cross section(σT ), while in the extraordinary mode the transverse electric field can only ex-cite the electrons close to the cyclotron resonance. For radiation that propagatesapproximately along the magnetic field (θ about zero), the cross section for bothmodes is reduced dramatically, as in both cases the electric field is mostly perpen-dicular to the magnetic field:σ1 ≈ σT(ω2(ωc+ω)2+12sin2 θ)(3.17)σ2 ≈ σT(ω2(ωc−ω)2+12sin2 θ). (3.18)A similar result holds for intermediate angles. As ω gets closer to ωc, the extraor-dinary mode’s cross section increases, until it becomes larger than the ordinarymode’s. Very close to ωc however, the energy transfer from photons heats up theelectrons and equations 3.15, 3.16, 3.17 and 3.18 are no longer valid, as dampingeffects become important [148].Because the reduction in the cross section results from the restriction of theelectron motion along the field lines and the geometry of the polarization modes,45the cross sections for other processes such as free-free, bound-free and atomictransitions also depend on the polarization state and the direction of propagation[36, 50, 129, 163, 177, 182]. The properties of the emission from the atmosphereof neutron stars will depend sensitively on the strength of the magnetic field and itsdirection relative to the vertical.To illustrate the various effects on the generation of polarization, I will considera simple plane-parallel atmosphere consisting of magnetized, fully ionized hydro-gen from Lloyd [130]. The neutron star atmosphere is incredibly thin comparedto the radius of the star (centimeters vs. kilometers), so the plane-parallel approx-imation is appropriate; however, across the surface of the star, the magnetic fieldwill vary in magnitude and direction, so the flux emergent through the surface willalso depend on the location. Potekhin [181] present a comprehensive review ofneutron-star atmospheres in general. To calculate the emission from the entire sur-face, a set of neutron-star atmosphere models must be computed accounting alsofor the surface temperature distribution. I will employ the same atmosphere modelsin Chapter 8 and I will present a prescription on how to add the contribution fromdifferent latitudes of the neutron star surface.I will now focus on the situation where the magnetic field is perpendicular tothe surface. Fig. 3.2 illustrates the various processes at play for a magnetic fieldstrength of B = 1012 and three values for the surface temperature, from top to bot-tom T = 0.4, 1.0, 2.5×106 K. I will discuss the trends in Fig. 3.2 in conjunctionwith the polarization fraction depicted in Fig. 3.3, because the polarization of theemergent radiation clearly reflects the relative location of the two photospheres. Inboth these figures, the photospheres and the polarization fractions where calculatedfrom the radiative transfer code developed by Lloyd [130]. The photosphere hereis intended as the layer in the atmosphere where the total optical depth from thesurface is unity. The ordinary mode (indicated as ‖, in orange) is less strongly af-fected by the magnetic field, and therefore I will start by addressing its behaviour.For small photon energies, the opacity is dominated by free-free absorption, whichdecreases with photon energy as E−2, so higher energy photons decouple deeperwithin the atmosphere, as long as their energy is still below the limit at whichelectron-scattering opacity starts to dominate over free-free opacity. Above thislimit (at about 100.5 keV in Fig. 3.2), the constant electron scattering opacity dom-463.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0log10(E/keV)32101log 10(/g cm3 )T=0.4, 1, 2.5×106 K  B=1012 G photosphere photosphere= p= vFigure 3.2: An illustration of the locations of the parallel and perpendicularmodel photospheres for B = 1012 G and T = 0.4, 1, 2.5×106 K usingthe models of Lloyd [130]. The radiation in higher temperature modelsdecouples deeper within the star at larger densities.inates, and the density of decoupling approaches a constant value.For the extraordinary mode (⊥, in blue) the trends are somewhat more compli-cated. For energies below the proton cyclotron resonance, the opacity is so smallthat the photosphere lies at the plasma frequency, quite deep in the star compared tothe ordinary mode’s photospere, as the collective oscillations of the plasma domi-nate the generation of the extraordinary photons. The polarization fraction at theseenergies is quite high in the X direction. At the proton cyclotron resonance, thecross-sections for scattering and absorption increase, drawing the photosphere toshallower depths and we can see it both in Fig. 3.2 and in Fig. 3.3 as a dip at about6 eV. The ordinary mode does not interact with the cyclotron resonances, so it isnot affected at this energy. Above the proton cyclotron resonance, the photospherefollows the plasma frequency until free-free absorption takes over. Because theelectric-field geometry of the extraordinary mode depends on the photon energy,473.0 2.5 2.0 1.5 1.0 0.5 0.0 0.5 1.0log10(E/keV)1.000.750.500.250.000.250.500.751.00(FF)/(F+F)B=1012 G4×105K106K2.5×106KFigure 3.3: The polarization fraction as a function of energy for B = 1012 Gand T = 0.5,1,2.5,×106 K using the models of Lloyd [130]. The dip at6 eV corresponds to the proton cyclotron line (see text).the dependence of the free-free opacity with energy is shallower in this mode, sothe depth of the photosphere does not increase as quickly as for the ordinary mode.The two photospheres approach each other. This reduces the extent of polariza-tion in the total emission. Finally, as the photon energy approaches the electroncyclotron resonance at 11.6 keV, the opacity for the extraordinary mode increasesdramatically because of the resonant cross section, and the photosphere of the ex-traordinary mode lies above that of the ordinary mode and the direction of thenet polarization switches to the ordinary mode (the fraction becomes negative inFig. 3.3).I will now consider the general case, where the direction of the magnetic fieldcan vary. We can see from the angular dependence of the scattering cross sectionsin Eq. 3.15 through 3.18 that the cross section is dramatically decreased when thephoton is traveling along the direction of the magnetic field. If we examine ra-diation traveling along the magnetic field direction it should decouple at a larger48depth and at a higher density than radiation traveling in other directions. Further-more, away from the cyclotron line, the contribution by the two polarization statesof photons traveling along the magnetic field should be nearly equal as their crosssections are also nearly equal. Fig. 3.4 depicts the specific intensity for radiationnear the peak of the spectrum as a function of the angle between the propagationdirection and the vertical for the case in which a magnetic field of 1012 G is directedat 30 degrees away from the vertical. The radiation is nearly fully polarized in theextraordinary mode direction and approximately isotropic except for very shallowangles where the intensity is diminished (limb-darkening) and within about ten de-grees of the direction of the magnetic field, where the intensity is much larger andthe radiation is not polarized. In this general case, the intensity depends not onlyon the zenith angle but also on the azimuthal angle relative to the local magneticfield direction. This dramatically increases the numerical effort in calculating aspectral model both relative to the unmagnetized case and to the situation wherethe magnetic field points in the vertical direction. It is this latter, more restrictivesituation that is most often treated in the literature [76, 99, 214, 215], even if manyworks, including the model employed in this section, consider a varying inclinationof B [75, 130, 179, 248].3.2.2 Condensed Neutron-Star SurfacesThe properties of matter that form the surface regions of neutron stars are stronglyaffected by the strong magnetic fields [194]. Neutron stars may have a solid sur-face [33, 35, 122, 124, 143, 144, 181, 231, 233]; in this case, the emission willessentially depend on the reflectivity (Rv) of the metallic surface. If one focuseson the interface between the vacuum outside and the condensed surface, one canargue by detailed balance that the intensity emerging from the surface is given byIν ,X/O(θk) =(1−Rν ,X/O(θk))Bν(T ) (3.19)where Bν(T ) is the intensity of blackbody radiation at the temperature of the sur-face and θk is the angle with respect to the surface normal. The typical densityof the condensed surface is ≈ 102–103 g/cm3, significantly larger than terrestrialmetals, and the plasma frequency within the surface is about 1 keV, so even at X-4975 50 25 0 25 50 75Angle with Respect to Vertical [degrees]0123456IntensityI + IIIFigure 3.4: The polarized intensity at E = 0.32 keV, near the spectral peak,along a slice through sky containing the magnetic field direction. Themagnetic field is directed at thirty degrees from the vertical. The atmo-sphere is calculated for B = 1012 G and T = 106 K using the models ofLloyd [130]. The units of intensity are 1019 erg/cm2/s/sr/keV.ray energies we would expect the surface to be highly reflective and the emissivity(1−R) to be small. However, there are some additional complications as the metalis highly magnetized and the ions can damp the radiation within the metal.To address these complications we can relate the reflectivity to the electromag-netic modes within the condensed surface which is essentially a magnetized plasma(see § 5.2). Using the Fresnel equations, which establish the boundary conditionsacross the surface, we find the reflectance of the p-polarization (with the electricfield along the plane of incidence) to beRp =∣∣∣∣cosθt −ncosθkcosθt +ncosθk∣∣∣∣2 , (3.20)50and for the s-polarization (with the electric field normal to the plane of incidence)Rs =∣∣∣∣cosθk−ncosθtcosθk +ncosθt∣∣∣∣2 , (3.21)where nsinθt = sinθ (Snell’s Law) and n is the index of refraction within the con-densed material that forms the surface. We have neglected the index of refractionabove the surface, the magnetic permeability of the material and the small longi-tudinal component of the X-mode. The Fresnel equations are defined in terms ofthe polarization states relative to the interface. The labels s and p refer to whetherthe radiation is polarized with its electric field in the plane containing the incomingray and the normal to the surface (p) or perpendicular to it (s). The s-polarizationis parallel to the interface itself. Both above the surface and within the condensedmaterial, the propagation modes are determined relative to the magnetic field direc-tion. Both regions are birefringent, so at the interface there are two reflected wavesand two transmitted waves; thus, the complete picture is composed of a reflectioncoefficient for the X-mode to reflect into the X-mode, for the X-mode to reflectinto the O and the other possibilities as well as the corresponding transmission co-efficients that can be obtained by expanding the propagation modes in terms of themodes defined at the surface.The left panel of Fig. 3.5 depicts the reflectivities and the emissivity for acondensed iron surface with a thin hydrogen atmosphere above it from Potekhinet al. [183]. The feature in the reflectivities at 0.25 keV results from the proton-cyclotron line that affects the polarization states in the region above the condensedsurface. The increase in the reflectivities at about 0.125 keV corresponds to theion-cyclotron frequency within the surface. The feature at 0.4 keV is given byEC = Ec,i+E2p,eEc,e(3.22)within the condensed surface (where Ec,i is the ion-cyclotron energy, Ep,e is theelectron-plasma energy and Ec,e is the electron-cyclotron energy). These two en-ergies (0.25 and 0.4 keV) feature strongly in the observed polarization signaturefrom the surface as local extrema in the polarization fraction. The right panel de-picts the results also from Potekhin et al. [183] for a condensed iron surface without51A&A 546, A121 (2012)Fig. 8. Emergent spectra (top panel) and temperature structures (bottompanel) for the fiducial model atmosphere (solid curve) and for model at-mospheres that are calculated using the fixed-ions approximation for thereflectivity calculations (dashed curves), and the inner boundary condi-tion from Paper II (dotted curves). In the top panel the diluted blackbodyspectrum that fits the high-energy part of the fiducial model spectrum isalso shown (dash-dotted curve).methods. In our case, the deepest atmosphere point is the up-per point of the condensed surface. The temperature correctionat this point is obtained as follows: the total flux at the bound-ary between the atmosphere and the condensed surface is fixedand, therefore, the following energy balance condition has to besatisfied:H0 =σSBT 4eff4π=12∫ ∞0dE∫ 1−1(IXE (µ) + IOE (µ))µ dµ= Btot kRL + JR + H−. (31)Here, σSB is the Stefan-Boltzmann constant, µ = cos θk, andBtot =∫ ∞0BE dE ,kRL =12 Btot∫ ∞0BE dE∫ 10(1−R) µ dµ,JR =12∫ ∞0dE×∫ 10(IXE (µ)(RXX + ROX) + IOE (µ)(RXO + ROO))µ dµ,H− =12∫ ∞0dE∫ 0−1(IXE (µ) + IOE (µ))µ dµ. (32)Fig. 9. Top panel: dimensionless emissivities for coefficients of reflec-tion RXX (dashed curve), RXO (dotted curve), ROX (dash-dot-dottedcurve), and ROO (dash-dotted curve). The quantities are calculated atthe bottom of the fiducial model atmosphere for the angle between theradiation propagation and magnetic field, θk = 10◦, together with the to-tal dimensionless emissivity (solid curve). Bottom panel: dimensionlessoutward specific intensities (inner boundary condition) at the bottomof the fiducial model atmosphere for the X-mode (solid curve) and O-mode (dashed curve). For comparison, the dotted curve shows the samefor the X-mode, calculated using the inner boundary condition fromPaper II (in this case the dimensionless specific intensity of the O-modeequals 0.5).Generally, the condition (31) is not fulfilled at a given temper-ature iteration. Therefore, we perform a linear expansion of theintegrated blackbody intensity:H0 = (Btot + ∆Btot)kRL + JR + H−, (33)and find a corresponding temperature correction∆T =π4σSBT 3(1kRL(H0 −Btot kRL −JR−H−)) . (34)This last-point correction procedure is stable and has a conver-gence rate similar to the Unsöld-Lucy procedure at other depths.We also changed the depth grid for a better description ofthe temperature structure in thin-atmosphere models. In semi-infinite model atmospheres that do not have a condensed surfaceas a lower boundary, a logarithmically equidistant set of depths isused. However, in thin-atmosphere models, such a set yields in-sufficient accuracy at the boundary between the atmosphere andcondensed surface. To improve the description of the boundary,we divide the model atmosphere into two parts with equal thick-nesses and use logarithmically equidistant depth grids for eachA121, page 8 of 14A. Y. Potekhin et al.: Spectra of neutron stars with metallic surfacesFig. 7. Degree of linear polarization Plin (Eq. (28)) as a function of pho-ton energy E for condensed Fe surface. The values of B, directions ofthe field and the wave vector, and line types are same as in Fig. 6.In the fixed-ion case, it is sufficient to set Eci → 0 and to replaceEq. (26) byJB1 =J1( ˜EC)0.1 + 0.9 ( ˜EC/E)0.4· (27)For the second mode, no additional fitting is needed, becauseR2 = 2R−R1 and J2 = 2J −J1.Figure 6 compares the use of Eqs. (23)–(26) to numerical re-sults. The upper panel shows the case where the field lines areperpendicular to the surface. In this case the line at EL disappearsfrom mode 1, so the line in R seen in Fig. 3 for θk , 0 is entirelydue to mode 2. As soon as the field is inclined, the line is redis-tributed between the two modes (the lower panel of Fig. 6). Inthe latter case the numerical results show a more complex func-tional dependence R1(E) in the range Eci < E < ˜EC, which isnot fully reproduced by our fit, for the reasons discussed above.The azimuthal angle ϕ enters the fit only through α. As aconsequence, the fit is symmetric with respect to a change insign of ϕ. This property may seem natural at first glance; how-ever, we note that the numerical results do not strictly obey thissymmetry, which holds for the nonpolarized beam, but not foreach of the polarization modes separately. We have checked thatthis is not a numerical artifact: because the magnetic field vec-tor B is axial, there is no strict symmetry with respect to the (x, z)plane. A reflection about this plane would require simultaneousinversion of the B direction in order to restore the original re-sults. However, as long as the electromagnetic waves are nearlytransverse (i.e., Kz in (4) and (A.24) are small), the asymmetryis weak, allowing us to ignore it and thus keep the fit relativelysimple.The analytic approximations in Eqs. (11) and (23) allowone to evaluate the degree of linear polarization of the emittedradiationPlin = (J1 −J2)/2J = (R2 −R1)/(2−2R). (28)For example, the two panels of Fig. 7 show Plin for the samedirections of the magnetic field and the photon beam as in therespective panels of Fig. 6. We see that the analytic formulae,originally devised to reproduce the normalized emissivities, alsoreproduce the basic features of Plin(E). Although the feature atE ∼Epe is absent in the top panel of Fig. 6, it reappears in thetop panel of Fig. 7 due to the contribution of R2 in Eq. (28).3. X-ray spectra of thin atmospheres3.1. Inner boundary conditionsPropagation of radiation in an atmosphere is described by twonormal modes (see Sect. 2.2.1). At the inner boundary of a thinatmosphere, an incident X-mode beam of intensity IXE gives riseto reflected beams in both modes, whose intensities are propor-tional to IXE , and analogously for an incident O-mode. Therefore,the inner boundary conditions for radiation transfer in an atmo-sphere of a finite thickness above the condensed surface can bewritten asIXE (θk, ϕ) = 12 JX(θk, ϕ)BE(T ) + RXX(θk, ϕ) IXE (π−θk, ϕ)+RXO(θk, ϕ) IOE (π−θk, ϕ), (29)IOE (θk, ϕ) = 12 JO(θk, ϕ)BE(T ) + ROO(θk, ϕ) IOE (π−θk, ϕ)+ROX(θk, ϕ) IXE (π−θk, ϕ), (30)where IME (M = X, O) are the specific intensities of the X- andO-modes in the atmosphere at ρ = ρs, RMM′ are coefficientsof reflection with allowance for transformation of the incidentmode M′ into the reflected mode M, and JM are the normalizedemissivities. The latter can be written by analogy with J1,2 asJX = 1 −RX and JO = 1 −RO, where RX = RXX + RXO andRO = ROO + ROX (cf. Paper I).Ho et al. (2007) retained only the emission terms 12 JMBE onthe right-hand sides of Eqs. (29), (30). The reflection was takeninto account in Paper II, but calculations were performed ne-glecting ROO, ROX, and RXO, under the assumption that RXX isequal to R and does not depend on ϕ. Here we use a more re-alistic, albeit still approximate, model for RMM′ , described inAppendix B.3.2. ResultsHere, we illustrate the importance of the correct description ofthe reflection for computations of thin model atmospheres abovea condensed surface. To this end, we have calculated a few modelatmospheres with normal magnetic field (therefore, θB = ϕ = 0,and αr = αi = θk), taking the model with B = 4 × 1013 G,effective temperature Teff = 1.2 × 106 K, and surface density Σ =10 g cm−2 as a fiducial model. In the fiducial model the free-ionsassumption for condensed-surface reflectivity is used.For these computations we use the numerical code describedin Suleimanov et al. (2009), with a modified iterative procedurefor temperature corrections. We evaluate these corrections usingthe Unsöld-Lucy method (e.g., Mihalas 1978), which gives a bet-ter convergence for thin-atmosphere models than other standardA121, page 7 of 14Figure 3.5: Left (Fig. 9 of [183]): The upper panel shows the four emissivi-ties for coefficients of reflection as a function of energy RXX (dashedcurve), RXO (dotted curve), ROX (dash-dot-dotted curve), and ROO(dash-dotted curve), together with the total dimensionless emissivity(solid curve). The lower panel shows the resulting dimensionless out-ward specific intensities for the X-mode (solid curve) a d O-mode(dashed curve). Right: Degr e of linear polarization as a function ofphoton energy E for condensed Fe surface from Fig. 7 of [183]. Theupper panel depicts the case where the magnetic field is normal to thesurface θB = 0 and several directions of the photon relative to the n rmaldirection θk: 0◦ (red), 30◦ (magenta), 45◦ (green) and 60◦ (blue). T elower panel holds the magnetic field and photo dir ction at 45◦ relativeto the normal and examines the emission as a function of the azimuthaldirection: 0◦ (red), 45◦ (gre n) and 90◦ (blue). The solid lines depictthe numerical res lts, and the dashed lines show a fit.a atmosphere abov it. W see that t e extent of polarization is typically muchsmaller than for an atmosphere. Furthermore the direction of the polarization de-pends on the energy of the photon, and the photon energy where the polarizationswitches from the ordinary to the extraordinary mode depends on the strength ofthe magnetic field through the electron and ion cyclotron frequencies and on the52density of the surface layers through the plasma frequency [233]. Although thetotal polarization is somewhat lower than for the case of the neutron-star atmo-sphere, the condensed surface leaves many exciting signatures on the polarization,most importantly that the surface is indeed condensed.3.2.3 Neutron-Star MagnetospheresAlthough the typical density of the plasma in the magnetosphere is too low formany photons to be produced there, the cross section for scattering of photonsfrom the surface may be large within the cyclotron resonance. The observed spec-tra of isolated neutron stars are characterized by one or two thermal components(below about one keV) [35, 197] and possibly a power-law component decliningtoward higher energies (above one keV) especially in magnetars [108, 136] (seeFig. 8.1). The spectra of magnetars often have an additional power-law componentthat becomes important from 10 keV to 100 keV [28, 77, 121, 147]. The sourceof the low-energy power-law component is often interpreted to be resonant inverseCompton scattering (RCS) onto mildly relativistic electrons/positrons flowing inthe twisted magnetosphere [134, 228]. The origin of the harder power-law compo-nent is less clear, although it might still be related to RCS, possibly onto a differentcharge population(s) [12, 14, 26, 62, 188, 189, 236].RCS in MagnetarsIn magnetars the expected particle density due to charges flowing along the mag-netic field lines, required to sustain the non-potential field, might be too low to builda sizable Thomson scattering depth [228]. On the other hand, this could be easilyachieved in the cyclotron resonance [12, 14, 26, 62, 165, 166, 188, 189, 228, 236,250, 251]. Given the typical energy of the photons from the surface, the cyclotronenergy at scattering should be about 1 keV, much less than the rest-mass energyof the electron, so the scattering region must lie at several stellar radii, where themagnetic field is about 1011−1012 G.The resonant scattering occurs when the photon frequency in the rest frame ofthe electron equals the cyclotron frequency; for an electron moving with velocity53v = βc and Lorentz factor γω =ωcγ(1−βµ) ≡ ωD (3.23)where ω is the photon frequency in the stellar frame andµ = kˆ · Bˆ (3.24)is the cosine of the angle between the propagation of the photon and the magneticfield, also in the stellar frame. In the frame of the electron, the scattering is non-relativistic so we can use the cross sections from Eq. 3.17 and 3.18 to understandhow the Compton scattering affects the two polarization states. In particular, forphotons traveling along and across the field we see that only the extraordinaryor perpendicular mode is resonantly scattered. Therefore, the incoming radiationfrom the atmosphere, which is mostly polarized perpendicular to the magnetic fielddirection, remains polarized perpendicular to the field direction after the resonantscattering, and then we expect the high-energy power-law component to be stronglypolarized perpendicular to the local magnetic field direction as well. From a moredetailed treatment [219], which includes geometric considerations, one finds thatthe resonant scattering can switch the polarization states. In fact, the cross sectionsare related in the following way [85, 166]:σ1−1 =13σ1−2 =pi2rec2δ (ω−ωD)cos2α, (3.25)σ2−2 = 3σ2−1 =3pi2rec2δ (ω−ωD) (3.26)where re is the classical radius of the electron and α is the angle between theincident photon direction and the magnetic field as measured in the rest frame of theelectron (cosα = (µ−β )/(1−βµ)). Again I use the convention that (1) indicatesthe ordinary or parallel mode and (2) indicates the extraordinary or perpendicularmode,. The resulting emission is polarized but less than fully polarized. If oneassumes that the initial photons are completely in the extraordinary mode, after asingle scattering the polarization fraction is reduced by 50% and it decreases withsubsequent scatterings [62]. One can conclude that the resonant scattering process54typically destroys the polarization.3.2.4 X-ray PulsarsThe studies of polarization are more mature for magnetars and thermally emittingneutron stars, and much less developed for X-ray pulsars. The models developedby Me´sza´ros and Nagel in the 1980s [M&N, 149–151], are still the most used inthe field. Their calculations assumed a static, homogeneous atmosphere (with con-stant density, temperature and magnetic field) and two possible geometries: a slab,with the magnetic field perpendicular to the surface, and a column, with the fieldparallel to the walls. In order to calculate the spectrum of the outgoing polariza-tion, they solved the approximate radiative transfer equations separately for the twopolarization modes, following the so-called Feautrier method [153], including vac-uum, thermal and incoherent scattering effects. In their model, photons are mainlyproduced by thermal bremsstrahlung, and the polarization of the X-ray signal isdriven by the difference in opacities between the two polarization modes. An al-ternative model, which however ignores the effect of vacuum birefringence and thecontribution from Comptonization, was calculated by Kii [111].M&N predict the smallest linear polarization degree being coincident with themaximum flux for the “pencil beam”, i.e. when photons propagate along the field(see § 1.1.2), and viceversa, a peak in polarization degree when the flux is at max-imum for the “fan beam”. Therefore, phase resolved measurements of the linearpolarization could help distinguishing between the two scenarios.However, the M&N models do not include relativistic effects and are basedon quite crude assumptions on the physics of the emission region; for example,they assume a static atmosphere even if the ionized plasma is expected to reachthe surface of the neutron star at a considerable fraction (up to ∼ 0.5) of the speedof light. Moreover, the spectral shape obtained in [149] fails to describe the morerecent observations of luminous X-ray pulsars [e.g. 244], expecially the flatten-ing at low energies. For this reason, also in view of the upcoming polarimeters,a new, upgraded model is needed to predict the polarization parameters that wewill observe from X-ray pulsars in the near future. In Chapter 7, I calculate thepolarization degree of X-ray pulsars in the context of the M&N model, including55relativistic effects and the QED effect of vacuum birefringence (see Chapter 4). Inaddition, I present a new model for the polarization parameters of X-ray pulsarsbased on the accretion model by Becker and Wolff [20]. As already mentionedin Chapter 1, Becker and Wolff analytically modeled the channeled steady-stateaccretion flow at the surface of the neutron star as a radiating plasma heated bya radiation-dominated shock above the neutron star surface and obtained a goodspectral fit for luminous X-ray pulsars as Her X-1.56Chapter 4The QED Effect of VacuumBirefringenceQuantum electrodynamics or QED is the relativistic quantum field theory of elec-trodynamics. It is usually thought to apply only to the realm of the very small.However, its effects can be important on macroscopic scales in extreme environ-ments, like the ones attained inside and around astrophysical compact objects, suchas neutron stars and black holes.In classical electrodynamics, photons do not interact with other electromag-netic fields as Maxwell equations are linear in the fields. In QED, the presence ofa Dirac current in the vacuum results in an addition to the usual action integral ofthe electromagnetic field that is more than quadratic in the fields. This implies thatthe interaction between the fields is not linear as photons can interact with virtualelectron-positron pairs as they travel through a magnetized vacuum. As a result,the speed at which light travels through the vacuum depends on its polarization andon the strength of the field. In other words, in the presence of a magnetic field thevacuum becomes birefringent, i.e., it acquires an index of refraction that is differ-ent depending on the angle between the direction of the photon’s polarization andthe magnetic field.In § 4.1, I derive the effective Lagrangian of QED from the classical Lagrangianof the electromagnetic field coupled to a Dirac field using modern functional tech-niques. In § 4.3, I calculate the index of refraction in the case of a uniform magnetic57field. In § 4.4 I use the formalism introduced in § 2.3 to describe how birefringenceaffects the propagation of polarization radiation.4.1 Effective Action: Formal DerivationThe Lagrangian of QED isL = ψ¯(iγµ∂∂xµ+ eγµAµ −m)ψ− 14FµνFµν (4.1)where the interaction between the fermionic fields and the external field is givenby the Feynman rule− ieγµ A˜0µ(q) . (4.2)This rule must be taken into account in all fermion propagators, including inter-nal lines such as in the vacuum polarization and in photon splitting processes.The symbols γµ are the Dirac matrices that span the spinorial components of thefermion fields.If the external field is sufficiently weak, the interactions with the field may betreated as perturbations, as a series of discrete interactions. On the other hand, ata field strength of BQED = m2c3/(eh¯) = 4.4× 1013 G, the gyration energy of anelectron or, equivalently, the potential energy drop across its Compton wavelength,is equal to its rest mass. For this reason, when the field exceeds a critical value ofapproximately BQED/2, this series fails to converge. Essentially, each term in thesum of diagrams is equally large in this limit.In the following sections, I derive the effective action of a general field config-uration to one-loop order from the QED Lagrangian (Eq. 4.1) using the methodof functional integration. The key results for X-ray polarization are the indexof refraction (Section 4.3) and how polarization changes as radiation propagatesthrough an inhomogeneous birefringent medium (§ 4.4).4.1.1 The functional methodThe connections between the theory of quantum and statistical fields are manifold.The following derivation of the effective action and Lagrangian will exploit theseconnections. The final results that I present here are well known in the special-58ized literature for quantum field theory in strong fields (e.g., [54, 55]) and olderintroductory texts (e.g., [29, 102]), but they are typically absent from recent intro-ductory texts (e.g., [135, 180]). I present a derivation of the effective action usingfunctional techniques familiar from modern treatments of quantum field theory(e.g., [180]) and statistical mechanics. The analysis in this section draws on § 11.3and § 11.4 of [180].In the functional method of quantum field theory, correlation functions can bederived from the functional derivative of a generating functional. For example, fora field theory governed by the lagrangianL , the generating functional is given by:Z[J] =∫Dφ exp[i∫d4x(L + Jφ)], (4.3)where J is an external current and φ is the field. The integral∫Dφ indicates anintegration over all field configurations. In this case, correlation functions can bederived as:〈0|Tφ(x1)φ(x2)|0〉= Z[J]−1(−i δδJ(x1))(−i δδJ(x2))Z[J]∣∣∣∣J=0. (4.4)where T is the time-ordering operator. The generating functional has many simi-larities with the partition function of statistical mechanics. In particular, it consistsof an integral over the quantum phases (or statistical weights) of each possible stateof the system, and the source J(x) plays the role of an external field. In the case ofQED, the generating functional takes the following form:Z[Jµ , η¯ ,η ] = exp(− ih¯E[Jµ , η¯ ,η ])=∫DAµDψDψ¯ expih¯∫d4x(L + JµAµ + η¯ψ+ ψ¯η). (4.5)where Jµ and Aµ are the electromagnetic current and vector potential respec-tively, η and η¯ are the fermionic currents and ψ and ψ¯ are the fermionic fields.E[Jµ , η¯ ,η ] is an energy functional that corresponds to the vacuum energy. Brack-ets are used to denote functionals (integrals of the fields over the entire spacetime)while parentheses indicate functions. The variables η , η¯ ,ψ and ψ¯ are Grassmann59numbers. Grassmann numbers are introduced to implement the anti-commutingnature of the fermion wave function: they anti-commute, which means that ηψ =−ψη and η2 = 0. Grassmann numbers behave differently from commuting num-bers under integration and differentiation as well. For a more complete introductionto the properties of Grassmann numbers, I redirect the reader to § 9.5 of [180].Following the example of [180], it is useful to compare the generating func-tional to the partition function of a specific system, in their case a magnetic system:Z(B) = e−βF(B) =∫Dsexp[−β∫dx(H [s]−Bs(x))], (4.6)where β = 1/kT , B is the external magnetic field, s(x) is the local spin field,H [s]is the spin energy density and again∫Ds is the integral over all spin configurations.In eq. (4.5), each field configuration receives a phase proportional to the integralof the Lagrangian over spacetime, i.e., the action. The constant of proportional-ity is i/h¯; in statistical physics, the constant of proportionality is −β = −1/kT ,and the states are weighted by energy and not action. Drawing the analogy fur-ther, the functional E[Jµ , η¯ ,η ], would correspond to the Helmholtz free energyF(T,V,N) in statistical mechanics; it is the vacuum energy as a function of theexternal sources Jµ , η¯ and η . In the case of a magnetic system, I can find themagnetization at a certain temperature by differentiating F(B):−dFdB∣∣∣∣β fixed=1βddBlogZ=1Z∫dx∫Dss(x)exp[−β∫dx(H [s]−Bs(x))](4.7)=∫dx〈s(x)〉 ≡M .At zero temperature, the ground state is the state of lowest energy, while at T 6= 0the preferred state is the state that minimize the Gibbs free energy:G = F +MB . (4.8)60Therefore:dGdM∣∣∣∣β fixed=dFdM∣∣∣∣β fixed+MdBdM∣∣∣∣β fixed+B=dBdMdFdB∣∣∣∣β fixed+MdBdM∣∣∣∣β fixed+B = B . (4.9)G is extremal at B = 0 and the corresponding value of M. In general, the moststable state corresponds to the minimum of G(M): G(M) represent the preferredstate at a temperature greater than zero that includes all the thermal fluctuations.For the quantum field, the result proceeds similarly: the functional derivativeof E[ ] with respect to one of the currents yields the classical field, i.e., the vacuumexpectation value of the corresponding field, which I denote as A0µ(x)δE[Jµ , η¯ ,η ]δJµ(x)= ih¯δδJµ(x)lnZ=−∫DAµDψDψ¯Aµ(x)exp ih¯∫d4x(L + JµAµ + η¯ψ+ ψ¯η)∫DAµDψDψ¯ exp ih¯∫d4x(L + JµAµ + η¯ψ+ ψ¯η)=−〈Ω|Aµ(x)|Ω〉≡−A0µ(x). (4.10)where I use the symbol δ to denote a functional derivative and where |Ω〉 denotesthe vacuum state. This correspond to a weighted average over all possible quantumfluctuations.Generally, when one considers the properties of the magnetized vacuum, it isthe fields that are specified, not the currents. The effective action is related to E[ ]through a Legendre transformation, just as the Gibbs free energy G is related to F :Γ[A0µ , ψ¯0,ψ0] =−E[Jµ , η¯ ,η ]−∫d4y(Jµ(y)A0µ(y)+ η¯(y)ψ0(y)+ ψ¯0(y)η(y)).(4.11)The functional derivative of the effective action, Γ[ ], with respect to one ofthe classical fields yields the distribution of the corresponding current. Using theanalogy with thermodynamics, the effective action is the vacuum energy with thedistribution of the fields fixed.614.1.2 Functional IntegrationComputing the effective action begins with the expression for Z[ ], the partitionfunction, specifically by expanding the classical action with currents about the val-ues of the classical fields,∫d4x(L + JµAµ + η¯ψ+ ψ¯η)=∫d4x(L [A0µ , ψ¯0,ψ0]+ JµA0µ + η¯ψ0+ ψ¯0η)+∫d4x[∆Aµ(x)(δLδAµ− Jµ)+∆ψ¯(x)(δLδψ¯− η¯)+(δLδψ−η)∆ψ(x)]+12∫d4xd4y[(∆Aµ(x))(∆Aν(y))δ 2LδAµ(x)δAν(y)+(∆ψ¯(x))δ 2Lδψ¯(x)δψ(y)(∆ψ(y))+(∆Aµ(x))δ 2LδAµ(x)δψ(y)(∆ψ(y))+(∆ψ¯(x))δ 2Lδψ¯(x)δAµ(y)(∆Aµ(y))]+Higher Order Terms(4.12)where ∆Aµ(x) = Aµ(x)−A0µ(x), the difference between the electromagnetic fieldincluding the quantum fluctuations and the classical electromagnetic field, and sim-ilarly for the other fields. Since the functional derivatives will be evaluated atψ0(y)= ψ¯0(y)= 0, the last two terms vanish. Furthermore, second derivatives withrespect to the same Grassmann field also vanish because of the anti-commutativenature of the fields. Let us now evaluate for an example the termδLδAµ− Jµ = δδAµ(ψ¯(iγµ∂∂xµ+ eγµAµ −m)ψ− 14FµνFµν)− Jµ= ψ¯eγµψ− Jµ − 14δδAµ(Aµ,νAµ,ν −Aν ,µAµ,ν −Aµ,νAν ,µ +Aν ,µAν ,µ)= ψ¯eγµψ− Jµ − 14δδAµ(2Aµ,νAµ,ν −2Aµ,νAν ,µ)= ψ¯eγµψ− Jµ −←∂∂xνAµ,ν +←∂∂xνAν ,µ . (4.13)62The←∂ notation indicates that the resulting derivative is an operator that differen-tiates something to the left. As the derivative lives within a integral over all ofspacetime, one can use integration by parts to simplify the result further if oneassumes that the boundary terms vanish:δLδAµ−Jµ = ψ¯eγµψ−Jµ+ ∂∂xν(Aµ,ν −Aν ,µ) = ψ¯eγµψ−Jµ+ ∂∂xνFµν (4.14)The first-order derivatives vanish when the fields satisfy the field equations.Although there is not an explicit relationship that connects the currents to theclassical fields that they generate, I will impose that the currents Jµ , η¯ and η alongwith the classical fields A0µ , ψ¯0 and ψ0 satisfy the field equations and evaluate allof the functional derivatives at the values of the classical fields. Thereby, I focus onthe quantum fluctuations about a classical field configuration. In this case the firstorder terms in the expansion vanish and the vacuum energy E[ ] to lowest order isa Gaussian functional integral,E[Jµ , η¯ ,η ] =−∫d4x(−14F0µνF0,µν + JµA0µ)+ih¯ ln∫DAµDψ¯Dψ expi2h¯∫d4xd4y[(∆Aµ(x))(∆Aν(y))δ 2LδAµ(x)δAν(y)+(∆ψ¯(x))δ 2Lδψ¯(x)δψ(y)(∆ψ(y))]=−∫d4x(−14F0µνF0,µν + JµA0µ)−ih¯2lnDet[δ 2LδAµ(x)δAν(y)]+ ih¯ lnDet[δ 2Lδψ¯(x)δψ(y)]+Constant Terms(4.15)I integrated over all of the possible field configurations by assuming that aparticular field configuration is a point in an infinite dimensional space and bylooking at the functional derivatives as infinite dimensional matrices. Let us look63at the following term in detail to see how this works:EAµ = ih¯ ln∫DAµ expi2h¯∫d4xd4y[(∆Aµ(x))(∆Aν(y))δ 2LδAµ(x)δAν(y)]= ih¯ ln∫∏(dAµ(xi))expi2h¯∑i, j(∆Aµ(xi))(∆Aν(y j))[δ 2LδAµ(xi)δAν(y j)]= ih¯ ln∫∏(dAµ(xi))exp[−∑i(∆Aµ(xi))2(− i2h¯λi)](4.16)where λi are the eigenvalues of the matrix[δ 2L /(δAµ(xi)δAν(y j))], and I havechosen the eigenvectors as a basis for performing the integral over the field con-figurations,∫∏(dAµ(xi)). Now I will perform the integration over each of thedAµ(xi) to yieldEAµ = ih¯ ln∏i(− i2h¯λi)−1/2=− ih¯2ln∏i(λi)− ih¯2 ∏i(− i2h¯)=− ih¯2lnDet[δ 2LδAµ(x)δAν(y)]+Constant Terms. (4.17)The constant terms absorb the constant prefactor in front of the integral,−i/(2h¯),as well as some divergent terms. The symbol Det denotes the functional determi-nant over both the spacetime and the spin space in the case of the Dirac fields. Onesubtlety is the plus sign in front of the functional derivative involving the Grass-mann fields ψ¯(x) and ψ(y). Simply, the integral of∫dψ¯dψ exp(−ψ¯aψ) =∫dψ¯dψ(1− ψ¯aψ)=∫dψ¯dψ(1+aψ¯ψ)=∫dψ¯(aψ¯) = a and not2pia, (4.18)where the first equality is not an approximation because the higher terms of theexpansion vanish. This unexpected result comes from the anti-commuting natureof the fields. Performing the Legendre transformation yields an expression for the64effective action to lowest order (one loop),Γ[A0µ ] =∫d4x(−14F0µνF0,µν)+ih¯2lnDet[δ 2LδAµ(x)δAν(y)]− ih¯ lnDet[δ 2Lδψ¯(x)δψ(y)](4.19)where I have dropped the constant terms from the expression. So far, I have notbeen concerned with renormalizing the effective action, but the functional deter-minants are probably divergent. The effective action vanishes as the classical fieldvanishes, so I have to subtract two terms corresponding to the functional determi-nants in the absence of an external field. This renormalizes the zero-point energyand yields,Γ[A0µ ] =∫d4x(−14F0µνF0,µν)− ih¯ lnDet[δ 2Lδψ¯(x)δψ(y)]∣∣∣∣Aµ=A0µ+ ih¯ lnDet[δ 2Lδψ¯(x)δψ(y)]∣∣∣∣Aµ=0=∫d4x(−14F0µνF0,µν)− ih¯ lnDet[/Π−m/p−m](4.20)where/Π= γµΠµ = iγµ∂∂xµ+ eγµA0µ = γµ pµ + eγµA0µ . (4.21)The functional derivative of the Lagrangian with respect to the vector potentialis same for all values of the classical vector potential as along as the fermionicclassical field vanishes. The effective action contains the classical Maxwell actionof electrodynamics and an additional term that quantifies the effects of the vacuumfluctuations of the Dirac (here electron-positron) fields.I will use the linear algebra result, lnDetA = TrlnA, to simplify the expressionfor the effective action further. I use the convention that Det and Tr span bothcoordinate and spin space, while tr and det just cover the spinorial components65(the analysis in this subsection builds upon § 4.3.3 and § 4.3.4 of [102]),Γ[A0µ ] =∫d4xLeff =∫d4x(−14F0µνF0,µν)− ih¯Trln[/Π−m/p−m](4.22)I would like to put the logarithm in a more manageable form.Tr ln[/Π−m/p−m]= Trln[/Π+m/p+m]=12Tr ln[/Π2−m2/p2−m2]. (4.23)The first equality holds since the charge conjugation matrix C satisfies CγµC−1 =−γTµ , so C /ΠC−1 =−/ΠT (similarly for /p), and the trace of an operator is invariantunder transposition. The second equality results from summing the first two ex-pressions.4.1.3 Effective Action : Proper-time IntegrationI use the identitylnab= limε→0∫ ∞0dss(exp is(b+ iε)− exp is(a+ iε)) (4.24)to expand the logarithmTrln[/Π−m/p−m]= (4.25)− 12∫d4x∫ ∞0dsse−ism2e−εstr(〈x|exp(is/Π2)|x〉−〈x|exp(is/p2)|x〉)and obtain the proper-time expression for the effective Lagrangian density [203],Leff =−14F0µνF0,µν +ih¯2∫ ∞0dsse−ism2e−εstr(〈x|U(s)|x〉−〈x|U0(s)|x〉) (4.26)where U(s) is the time-evolution operator governed by the Hamiltonian,H =−/Π2 =ΠµΠµ − 12eσµνF0µν . (4.27)66where σµν = i2 [γµ ,γν ]. U0(s) is the analogous operator for vanishing externalfields. Please notice that, for convenience, I dropped the limit for ε→ 0 in eq. 4.25,but we have to keep in mind that I will have to take ε → 0 at the end of the cal-culation. Equation 4.26 forms the basis of the worldline numerics technique thatfacilitates the calculation of the effective action for arbitrary field configurations[70, 140].4.2 Results for a Uniform FieldI will select a particular frame and gauge to calculate the trace and obtain an ex-pression for the effective Lagrangian from a uniform electromagnetic field. I beginwithtr(〈x|exp(is/Π2)|x〉)= tr(〈x|exp(isΠµΠµ)|x〉)tr(〈x|exp(i2esσ µνF0µν)|x〉)(4.28)since σ µνF0µν commutes with ΠµΠµ for constant fields.In a general frame, the eigenvalues of i2 esσµνF0µν are ±es(a± ib), where(a+ ib)2 = (E+ iB)2 = |E|2−|B|2+2iE ·B , (4.29)E and B are the classical electric and magnetic fields respectively, and a and b areLorentz invariants of the field (see [202]). I can therefore rewrite the second traceastr(〈x|exp(i2esσ µνF0µν)|x〉)= 4cosh(eas)cos(ebs) . (4.30)The evaluation of the first trace is more complicated. I now choose a framesuch that E‖B, and a ≡ |E| and b ≡ |B|. With no loss of generality, I can assumethat the magnetic and electric fields point in the z-direction and select a gauge withA3 =−at and A1 =−by; this yields,ΠµΠµ = (P0)2− (P2)2− (P1+ ebX2)2− (P3+ eaX0)2 (4.31)67Using the commutation relation [x, px] =−i, I can define the shift operatore−ipxc f (x)eipxc = f (x+ c) (4.32)which allows me to writeΠµΠµ = exp(−iP2P1eb− iP0P3ea)[(P0)2− (P2)2− (ebX2)2− (eaX0)2]×exp(iP2P1eb+ iP0P3ea)(4.33)To evaluate the trace itself I use the momentum representationtr(〈x|exp(isΠµΠµ)|x〉)=∫ dp3dp1(2pi)4dp0dp′0dp2dp′2 exp[i(p′0− p0)(t+p3ea)]×exp[i(p′2− p2)(y+p1eb)]〈p0|exp[is(P20 − e2a2(X0)2)]|p′0〉×〈p2|exp[−is(P22 + e2b2(X2)2)]|p′2〉=e2ab(2pi)2∫ ∞−∞dp0〈p0|exp[is(P20 − e2a2(X0)2)]|p0〉×∫ ∞−∞dp2〈p2|exp[−is(P22 + e2b2(X2)2)]|p2〉(4.34)Let us examine the last of the two integrals in detail.∫ ∞−∞dp2〈p2|exp[−is(P22 + e2b2(X2)2)]|p2〉= Trexp[−is(p2+ e2b2x2)]= Trexp[−2isH ] (4.35)where H is the Hamiltonian of a harmonic oscillator with unit mass and springconstant k = e2b2. Using the known eigenvalues of the system yields an expressionfor the integral,Trexp[−2iH s] =∞∑n=0exp[−2iebs(n+12)]=12isin(ebs). (4.36)The result for the first integral is similar except here k = −e2a2. Therefore, the68complete expression for eq. 4.28 istr(〈x|exp(is/Π2)|x〉)=−i e2ab(2pi)2coth(eas)cot(ebs). (4.37)Taking the limit of this expression as a and b vanish yieldstr(〈x|exp(is/p2)|x〉)=− i(2pi)21s2. (4.38)4.2.1 Effective Lagrangian in a Constant FieldSubstituting this result into 4.26 yields an expression for the effective Lagrangian,Leff =−14F0µνF0,µν +h¯2(2pi)2∫ ∞0dsse−ism2e−εs[e2abcoth(eas)cot(ebs)− 1s2](4.39)For small values of s, the integrand diverges as e2(a2−b2)/(3s). Since this is pro-portional to the classical Lagrangian, it can be absorbed through a renormalization,or a scale change, of all fields and a corresponding scale change of charge. I iden-tify the quantities thus far employed with a zero subscript, and introduce new unitsof field strength and charge according to [202](a+ ib)2 = (1+Ce20)(a0+ ib0)2 (4.40a)e2 =e201+Ce20(4.40b)C =112pi2∫ ∞0dssexp(−m2s) (4.40c)This yields the renormalized expression for the effective Lagrangian,Leff =− 14F0µνF0,µν+h¯8pi2∫ ∞0dsse−ism2e−εs[e2abcoth(eas)cot(ebs)− 1s2− 13e2(a2−b2)](4.41)69Performing a Wick rotation and substituting ζ = sm2 yieldsLeff =a2−b22+α8pi2B2QED∫ ∞0dζζe−ζ[abB2QEDcot(ζaBQED)coth(ζbBQED)+1ζ 2− 13a2−b2B2QED](4.42)where α = e2/(h¯c) and I have taken ε → 0.4.3 Index of refractionFrom the effective Lagrangian, I can derive the index of refraction for low-energyphotons by defining the macroscopic fields as the generalized momenta conjugateto the fields [29],D =∂L∂E= E+P, H =−∂L∂B= B−M, (4.43)and linearizing these relations about the background field [4]. For an external mag-netic field this yields [89]n‖ = 1−α4piX1(1ξ)sin2 θ +O[( α2pi)2](4.44)n⊥ = 1+α4pi[X (2)0(1ξ)ξ−2−X (1)0(1ξ)ξ−1]sin2 θ +O[( α2pi)2].(4.45)where ξ = B/BQED, n⊥ and n‖ are the index if refraction for the perpendicular andparallel modes respectively (see below),X1(1ξ)=23ξ − 13+8[lnA−∫ 1/(2ξ )+11lnΓ(v)dv]+23Ψ(12ξ)+1ξ[2lnΓ(12ξ)−3lnξ + ln(pi4)−2]− 12ξ 2,(4.46)70X (2)0(1ξ)ξ−2−X (1)0(1ξ)ξ−1 =23+1ξ[−2lnΓ(12ξ)+ lnξ + ln4pi+1]+1ξ 2[Ψ(12ξ)−1](4.47)and lnA= 112−ζ (1)(−1)≈ 0.248754477. The functions X0(1/ξ ) and X1(1/ξ ) arerelated to the effective action and its derivative with respect to a in the limit ofa→ 0 [90].My naming convention is the following: if εµναβFµνFαβ = 0 or ~E ·~B = 0, thephoton is in the perpendicular mode, otherwise it is in the parallel mode, where Fµνis the sum of field tensors of the wave and external field. In the weak field limit,n−1 ∝ ξ 2; while in the strong field limit n‖−1 ∝ ξ and n⊥ approaches a constant[89]. In particular, in the weak field I getn‖ = 1+α4pi1445ξ 2 sin2 θ n⊥ = 1+α4pi845ξ 2 sin2 θ (4.48)and a birefringence ofn‖−n⊥ =α4pi215ξ 2 sin2 θ . (4.49)4.4 Propagation through the Birefringent VacuumAs polarized radiation propagates through a birefringent medium, the direction ofpolarization changes. In particular, the evolution of the normalized Stokes vectors = (S1,S2,S3)/S0 is given by (eq. 2.64, in § 2.3)dsdλ= Ωˆ× sIn the case of the magnetized vacuum, the birefringence vector is given by|Ωˆ|= |k0∆n|= α15νc(B⊥BQED)2(4.50)71where ν is the frequency of the radiation. The value of ∆n is the difference in theindex of refraction for the two polarization states (eq. 4.49), and the equality holdsin the weak-field limit of QED. The direction of Ωˆ points toward the polarizationof the perpendicular mode on the Poincare´ sphere of polarization states.If the magnitude of Ωˆ is high, the polarization modes decouple, the evolution isadiabatic and the polarization direction follows the direction of the birefringence.The condition for adiabatic evolution is given in eq. 2.65:∣∣∣∣∣Ωˆ(dln |Ωˆ|dλ)−1∣∣∣∣∣≥ 0.5.In Chapter 5 I will show how this effect can dramatically change the polariza-tion of neutron stars, while in Chapter 6 I will analyze how the effect of vacuumbirefringence changes the polarization of X-ray photons as they travel in the mag-netosphere of accreting black holes.72Chapter 5The Effect of Birefringence onNeutron-Star EmissionIn Chapter 3, I describe how polarized radiation is generated in the atmospheres ofneutron stars and in black hole accretion disks. However, as light travels throughthe magnetosphere of these objects, its polarization state can still change because ofthe effect of birefringence, and I need to account for it to understand the observedpolarization. In Chapter 6, I will analyze the effect of vacuum birefringence on thepolarization from black-hole accretion disks. Here, I will focus on neutron starsinstead.As described in § 2.3, the presence of a strong magnetic field can make amedium birefringent: the index of refraction in the medium depends on the an-gle between the polarization of the photon and the magnetic field. In the case ofthe magnetized vacuum, the birefringence is caused by the interaction of photonswith virtual electron-positron pairs: it is easier to excite virtual electrons along thedirection parallel to the magnetic field than perpendicular to it, and thus photons inthe ordinary mode travel slower than photons in the extraordinary mode (see Chap-ter 4). If real electrons and positrons are present, they can also interact with photonsand the presence of a strong magnetic field causes the plasma to be birefringent. Inthis chapter, I analyze the effect on polarization of vacuum birefringence (§ 5.1),of plasma birefringence (§ 5.2), and the interplay between the two effects (§ 5.2.1).735.1 Vacuum BirefringenceIn § 2.3 I showed that in a birefringent medium, in which the anisotropy is set bythe magnetic field, the two polarization modes, parallel and perpendicular to themagnetic field, are decoupled if (eq. 2.65)∣∣∣∣∣Ωˆ(dln |Ωˆ|dλ)−1∣∣∣∣∣≥ 0.5 .where λ measures the length of the photon path in the medium and Ωˆ is the bire-fringence vector, given by (eq. 4.50)|Ωˆ|= |k0∆n|= α15νc(B⊥BQED)2.In this case, the evolution is called adiabatic, and the photon polarization followsthe direction of the local field lines.In the case of neutron stars, for which the magnetic field is dipolar (B≈ µr−3,where µ is the magnetic dipole moment of the star and r is the distance from thecenter of the star) the adiabatic condition of eq. 2.65 translates into∣∣∣∣∣ α15 νc µ2 sin2βr6B2QED r6∣∣∣∣∣≥ 0.5 (5.1)where β is the angle between the dipole axis and the line of sight. If I define thepolarization-limiting radius (rPL) to be the distance at which the equality holds, Ifind that the polarization will follow the direction of magnetic field out torPL =( α45νc)1/5( µBQEDsinβ)2/5≈ 1.2×107(µ1030 G cm3)2/5( ν1017 Hz)1/5(sinβ )2/5 cm. (5.2)Figure 5.1 illustrates the propagation of radiation away from the surface of theneutron star toward a distant observer. For X-ray photons coming from near thesurface of a neutron star with a surface field of 1012 G, the polarization-limiting74radius is much larger than the star, according to eq. 5.2, so the observed polarizationof the photons will reflect the direction of the magnetic field at a large distancefrom the star and not at the surface. For a much more weakly magnetized star,the polarization-limiting radius will be comparable to the radius of the star, so theobserved polarization will reflect the field structure close to the star.Figure 5.1: Radiation leaving the surface of a neutron star follows geodesicsso that the bundle of rays that reaches the distant observer is approxi-mately cylindrical. The three-dimensional coordinates (x,y,z) are givenin terms of the radius of the neutron star (R). If the polarization-limitingradius is small, the final polarization will reflect the magnetic field struc-ture near to the star where the bundle covers a large fraction of hemi-sphere so the field structure varies a lot over the bundle at this point, andthe final polarization will also vary a lot over the image. On the otherhand, if the polarization-limiting radius is large, the field structure overthe ray bundle is simpler, and the polarization direction will not varymuch over the image. Since the magnetic field is assumed to be that ofa dipole (aligned with the z-axis), it has axial symmetry and differentimages will be distinguishable by the observer’s magnetic inclinationangle i. Adapted from [207].75Fig. 5.2 depicts the broadband polarization from the entire visible surface of aneutron star with a mass of 1.4 M, and temperature and field strength at the mag-netic pole of 106.5 K and 2×1012 G respectively, using the fully ionized hydrogenatmospheres discussed in § 3.2.1. The proton cyclotron line can be seen as a dip atabout 10 eV, and the electron cyclotron line is at the right end, at about 22 keV. Thethermal structure of neutron stars is affected by the presence of the strong magneticfield, and the thermal flux through the surface varies as B0.4 cos2ψ , where ψ is theangle of the local magnetic field with respect to the normal [91, see also § 8.1.1].The polarized fraction is plotted in terms of the total flux polarized perpendicu-lar and parallel to the projection of the magnetic moment of the star onto the sky.A value of 1 indicates radiation fully polarized perpendicular to the moment, and−1 indicates radiation fully polarized parallel to the moment. In the upper panel,the magnetic moment makes an angle of 30 degrees with respect to the line ofsight, and in the lower panel the angle is 60 degrees. In each panel, the upper setof curves traces the result including vacuum birefringence in the magnetosphereand the lower curves neglect it. Vacuum birefringence dramatically increases theexpected polarization fraction after integrating over the stellar surface. With vac-uum birefringence, the expected polarized fraction is larger for smaller neutronstars and larger at higher energies until one approaches the cyclotron resonance.The trend with stellar radius is simply due to the fact that the bundle of rays forsmaller stars is smaller so it subtends a smaller fraction of the magnetosphere at thepolarization-limiting radius; furthermore, as the energy of the photons increases,the polarization-limiting radius also increases, increasing the expected polarizedfraction. Without vacuum birefringence, both of these trends are reversed. Further-more, the polarized fraction is larger when the magnetic field makes a larger anglewith the line of sight. If one assumes that the flux is largest when the magnetic fieldis closest to the line of sight, one would expect the polarized fraction and the fluxto be somewhat anti-correlated. In particular, the off-pulse radiation is more po-larized than on pulse, so polarized emission off-pulse may come from the neutronstar itself rather than the background.Table 5.1 lists several classes of possible sources and their polarization limit-ing radii. For the magnetars and XDINS, the polarization-limiting radius is muchlarger than the radius of the star. As the emission in these sources is expected763 2 1 0 1log10(E/keV)0.00.20.40.60.8(FF)/(F+F)30 degrees8 km10 km12 km3 2 1 0 1log10(E/keV)0.00.20.40.60.8(FF)/(F+F)60 degrees8 km10 km12 kmFigure 5.2: The extent of the polarization averaged over the stellar surfaceas a function of energy, angle and stellar radius using the fully ionizedhydrogen atmospheres discussed in § 3.2.1. Here, the perpendiculardirection is defined to be perpendicular to the projection of the magneticmoment of the star into the sky. In the upper panel, the magnetic polemakes an angle of 30◦ with the line of sight. In the lower panel, theangle is 60◦. The lower set of curves trace the results without vacuumpolarization, and the results for the upper curves include it.77to come from a larger region of the stellar surface or magnetosphere (in the caseof the non-thermal emission from magnetars [219]), a large increase in the ob-served polarization fraction due to QED is also expected. Although the ratio of thepolarization-limiting radius to the stellar radius is also large for the X-ray pulsars(XRP), as we shall see in Chapter 7, the effect for these objects is more subtle.The QED effects for more weakly magnetized stars such as millisecond XRPs (msXRPs) and strongly magnetized white dwarfs have not yet been explored. Chap-ter 6 focuses on the effect of QED on accreting black holes [see also 37, 38].Table 5.1: The expected polarization-limiting radii for various sources; thetypical observing times are for eXTP at 2–8 keV for the magnetars(4U 0142+61) and XRPs (Her X-1) from the text and for RedSOX [64]at 0.2–0.8 keV for RX J1856.5-3754 to make a four-sigma detection.R [cm] B [G] µ [G cm−3] rpl at 4 keV[cm] rpl/R t obsMagnetar 106 1015 1033 3.0×108 300 10 ksXDINS 106 1013 1031 4.7×107 50 * 1 ksXRP 106 1012 1030 1.9×107 20 100 ksms XRP 106 109 1027 1.2×106 1.2AM Her 109 108 1035 1.9×109 1.9Black Hole 106+ ? N/A See Chapter 6* XDINS have little emission at 4 keV, and therefore will be difficult to observewith eXTP and IXPE. The value of rpl/R at 0.4 keV is 32, so vacuum birefringenceis important for observations with soft-X-ray polarimeters.5.1.1 The Quasi-tangential effectFigure 5.3 depicts the final polarization states across the image of the neutron starsurface assuming that the radiation is initially in the extraordinary mode, that is, theelectric field is perpendicular to the local magnetic field. The left panel shows thecase where the vacuum birefringence is neglected, and the right panel show the casewhere the surface field is about 1012 G and the frequency is 1017 Hz or an energy ofabout 0.4 keV. This is appropriate for a thermally emitting neutron star such as oneof the X-ray dim neutron stars (XDINS). The effect of the vacuum polarization isto comb the polarization direction to be aligned with the direction of the magnetic78Figure 5.3: The polarized emission map of a neutron star overlaid on the ap-parent image of the NS. The left panel depicts the observed map ofpolarization directions if one assumes that the surface emits only inthe extraordinary mode (perpendicular to the local field direction) andneglects the vacuum birefringence induced by QED. The right panelshows the polarization map including birefringence for a frequency ofν = (µ/(1030G cm3))−21017 Hz. The ellipses and short lines describethe polarization of a light ray originating from the surface element be-neath them. The lines and the major axes of the ellipses point towardsthe direction of the linear component of the polarization direction. Theminor to major axis ratio provides the amount of circular polarization(s3). The observer’s line of sight makes an angle of 30◦ with the dipoleaxis. For comparison, if one assumes that the entire surface is emittingfully polarized radiation, the net linear polarization on the left sums upto about 13%, while it is 70% on the right.axis of the star and dramatically increase the observed total polarization from about13% to about 70%. For more strongly magnetized neutron stars the effect is moredramatic. Linearly polarized radiation can also be converted to circularly polarizedradiation, if the radiation happens to pass through the polarization-limiting radiuswhen it is propagating approximately tangential to the field (in Fig. 5.3 this effectis shown by the ellipses near the polar cap in the right panel). This is called theQuasi-Tangential effect.In their 2009 paper, Wang and Lai [237] showed that the polarization of X-ray79photons can change significantly when they cross the quasi-tangential (QT) point,where the photon momentum is nearly aligned with the magnetic field, and that thenet effect, when averaged over a finite emission area, is to decrease the fraction oflinear polarization.Not all light rays go through a real tangential point, where the photon wavevec-tor k is perfectly aligned with the local magnetic field direction; however, there isalways a point in the photon path, called QT point, where the angle between kand B, θB, reaches a minimum. The magnetic field around the QT point can beexpressed, without loss of generality, asBX =BRs, BY = εB (5.3)in the fixed XY Z frame where Zˆ ‖ kˆ. HereR is the curvature radius of the projectedmagnetic field line in the X−Z plane and s measures the distance from the QT pointalong the Z− axis. At the QT point, s= 0 and ε = sinθB. Depending on the strengthof the vacuum birefringence at the QT point, the outcome for the polarization ofthe photon crossing the point can be different.In § 2.3 I have shown that whenever the vacuum birefringence dominates thephoton polarization modes are decoupled and evolve independently following thelocal magnetic field lines (eq. 2.65). Wang and Lai [237] introduced an equivalentcondition to eq. 2.65, which states that the photon modes are decoupled if Γad 1(eq. 2.12 in [237]), where they call Γad the adiabaticity parameter. The value of Γadat the QT point is given by (eq. 3.22 in [237])Γt ' 1.0×108E1B213ε3R1 (5.4)where E1 = Ep/(1 keV), B13 = B/(1013 G) and R1 = R/(10 km). Wang andLai [237] show that in both limiting cases of adiabatic (Γt  1) and non-adiabatic(Γt  1) propagation, the polarization direction is unchanged when the photontraverses the QT point. The only interesting effect is for the intermediate case,Γt ∼ 1. In this latter case, even if a photon is in a pure mode prior to QT crossing,it will come out of the QT region in a mixture of the two modes.In the same paper, Wang and Lai [237] give a prescription to account for the8010 2 10 1 100 101 102Wt/Wem0.40.50.60.70.80.91.0F Q/FQ0.0 0.5 1.0 1.5 2.0 2.5 3.00.000.250.500.751.001.251.501.75f()Figure 5.4: Left panel: the depolarization effect of QT propagation on linearpolarization; Wt is the width of the QT effective region, Wem is the widthof the emission region and FQ (F¯Q) is the polarized radiation flux before(after) traversing the QT region. Same as Figure 11 in [237]. The ver-tical beige lines highlight the region where the effect is stronger, andthe red vertical line pinpoints the peak of the effect, at Wt/Wem ∼ 1.82.Right panel: the function f (ψ), which in [237] is called f (θµi).QT effect in case of a dipolar field. In particular, they calculate the effect on theemission coming from the polar cap. They find that the region in which the QTeffect is important is the region where Γt . 3 and the width of this region can beexpressed as (eq. 4.32 in [237])Wt ' 2.7×10−2(B2∗13E1)−1/3 f (ψ)R∗ (5.5)where B∗ is the magnetic field at the pole and f (ψ) is a dimensionless function ofthe angle between the magnetic axis and the line of sight, which they call θµi andwhich I will be calling ψ in Chapter 7. Once the width of the QT effective regionhas been calculated, the linearly polarized radiation flux (F¯Q) can be obtained fromthe ratio between the width of the QT effective region (Wt) and the emission region(Wem); the numerical result, taken from Figure 11 of [237], is shown in the leftpanel of Figure 5.4, where FQ is the flux of linearly polarized radiation prior topassing the QT region.In [237], the authors are only interested in ψ < 90◦, while in the cases ofinterest of this work, ψ can be higher than that. For this reason, I have calculatedf (ψ) for all angles. First, I define rqt as the distance from the center of the star to81the QT point. I indicate with δ the angle between the magnetic axis and rqt, andsince the magnetic field is dipolar, the relation between δ and ψ is given bycos2(ψ−δ ) = 4cos2 δ3cos2 δ +1(5.6)The relation between the impact parameter b and rqt isrqt sin(ψ−δ ) = b = (R∗+ z)sinψ (5.7)where z is the height above the star of the part of the column that we are consider-ing. This yieldsrqt =(R∗+ z)sinψsin(ψ−δ ) (5.8)I can write the dipolar field as (eq. 4.28 of [237])B =− µr3qt+3rqtr5qt(µ · rqt) (5.9)where r is the distance from the center of the star, and since at the QT point µy = 0By =3yqtr5qtµrqt cosδ (5.10)The field strength at the QT point is related to the field strength at the pole byB = B∗(R∗rqt)3(3cos2 δ +14)1/2, and B∗ =2µr3qt(5.11)This yieldsε =ByB=3yqtrqtcosδ(3cos2 δ +1)1/2(5.12)From ε , I can find the width of the QT regionWtR∗=2yqtR∗= ε2rqt3R∗(3cos2 δ +1)1/2cosδ(5.13)82From eq. 5.4, I can determine the value of ε for which Γt . 3ε(Γt = 3) = (3×10−8)1/3 ∗ (E1B213R1)−1/3 (5.14)I still need the radius of curvatureR, which for a dipolar magnetic field readsR =rqt3(3cos2 δ +1)3/2|sinδ |(cos2 δ +1) (5.15)I finally have all the ingredients to find f (ψ) of eq. 5.5f (ψ) = 7.7×10−2(rqtR∗)8/3(10kmR∗)1/3(12|sinδ |(cos2 δ +1)cos3 δ (3cos2 δ +1))1/3(5.16)where the relation between ψ and δ is given in eq. 5.6. This result is shown in theright panel of Figure 5.4 and it reproduces the function f (θµi) shown in Figure 10of [237] for ψ = θµi < 90◦.If one integrates eq. 2.64 along the photon path, the QT effect will come natu-rally from the integration, as can be seen in Fig. 5.3. However, the numerical resultobtained in this section will be useful in § 7.2, where the only effect of the vacuumbirefringence is due to the QT effect and there is no need to integrate eq. 2.64 foreach photon path.5.2 Plasma BirefringenceAt low photon energies or high plasma densities, the plasma may play an importantrole in the propagation of polarized radiation through the atmospheres and magne-tospheres of neutron stars and black holes. The typical energy where plasma andvacuum trade off, if one assumes that the density of the plasma is approximatelythe Goldreich and Julian [73] density, is in the infrared, but it will increase withthe density of the plasma. If the magnetosphere carries substantial currents as inmagnetars [e.g. 228], the effect of the plasma will be more important. The com-bination of plasma and vacuum birefringence results in an eigenvalue equation for83the complex amplitudes of the electric field [148],[ηxx−n2 ηxyηyx ηyy−n2ρ][ExEy]= 0 (5.17)where ηi j are components of the dielectric tensor and I have assumed that thephoton propagates along the z−axis and that the magnetic field lies in the x−z−plane. The parameter ρ ,ρ = 1− 4αQED45pi(BBQED)2sin2 θ , (5.18)accounts for the magnetization of the vacuum and the tensor η characterizes thedielectric response of the plasma and vacuum. Since I am considering photonfrequencies well above the plasma frequency (ω  ωp =√4pie2n/m ) and belowthe cyclotron resonance (ω <ωc = eB/(mc)), the eigenvalues of the matrix, n21 andn22, are of the order unity. These conditions hold in the neutron star magnetospherefor the photon energies from the visual to the X-rays but do not necessarily hold inthe atmosphere.I can employ the formalism of Kubo and Nagata [120] (§ 2.3) by noting that themagnitude of Ωˆ is related to the two eigenvalues of the matrix (n21,2), in particular|Ωˆ|= k0∣∣∆n‖⊥∣∣ where∆n‖⊥ = n‖−n⊥ =n2‖−n2⊥n‖+n⊥≈n2‖−n2⊥2(5.19)≈ 12[(ηxx−ρ−1ηyy)2+4ρ−1|ηxy|2]1/2(5.20)≈ sin2 θ[αQED30pi(BBQED)2− 12ω2pω2ω2cω2c −ω2](5.21)≈ sin2 θ αQED30pi(BBQED)2[1− ω2pω21vr](5.22)wherevr =αQED15pi(BBQED)2 ω2c −ω2ω2c(5.23)84and where I have assumed n1,n2 ≈ 1.If ω2vr ≈ ω2p, the value of ∆n‖⊥ vanishes; in this regime (close to the vacuumresonance), the modes of the plasma plus vacuum are approximately circular andin the resonance, the difference in index of refraction between the two helicities isgiven by∆n±,res = [n+−n−]res =−cosθωω2pωc(ω2−ω2c,i) . (5.24)where ωc,i = ZieB/(mic) is the ion cyclotron frequency. In general, the helicity ofthe photon is an adiabatic invariant for the polarization states and(∆n±)2 = (∆n±,res)2+(∆n‖⊥)2 (5.25)The direction of Ωˆ is given by the direction of the eigenvector with the largereigenvalue on the Poincare´ sphere. The eigenvectors of eq. 5.17 yield the polariza-tion vectors~e+ = [icosθm,sinθm,0] , ~e− = [−isinθm,cosθm,0] . (5.26)where I have labelled the states by their helicity rather than parallel and perpendic-ular. The angle θm lies between 0 and pi/2 and characterizes the mixing betweenthe parallel and perpendicular polarization states. For θm = 0, the electric field inthe + state is parallel to the x−axis and for θm = pi/2, it is parallel to the y−axis. Ican determine the value of θm from the ratio of the linear to the circular portion ofthe birefringencetan2θm =∆n±,res∆n‖⊥. (5.27)85I map~e1 onto the Poincare´ sphere using the definitionsI = S0 = |~e1 · xˆ|2+ |~e1 · yˆ|2 = 1 (5.28)Q = S1 = |~e1 · xˆ|2−|~e1 · yˆ|2 = cos2θm (5.29)U = S2 =∣∣∣∣~e1 · 1√2 (xˆ+ yˆ)∣∣∣∣2− ∣∣∣∣~e1 · 1√2 (xˆ− yˆ)∣∣∣∣2 = 0 (5.30)V = S3 =∣∣∣∣~e1 · 1√2 (xˆ+ iyˆ)∣∣∣∣2− ∣∣∣∣~e1 · 1√2 (xˆ− iyˆ)∣∣∣∣2 = sin2θm, (5.31)yielding the direction of Ωˆ on the Poincare´ sphere. Finally this yieldsΩˆ = |∆n±|ωc cos2θm0sin2θm . (5.32)I find that |∆n‖⊥|  ∆n±,res is large so the modes of the plasma and vacuum arelinear except in the vicinity of the cyclotron resonance and of the two vacuumresonance frequencies, where ∆n‖⊥ ≈ 0.In the neutron star magnetosphere, we expect the net charge density of theplasma to be at least the Goldreich and Julian [73] density ρ =−~Ω ·~B/(2pic). Thetotal charge density could be some multiple ζ (the multiplicity) of this, yielding anestimate of the local plasma frequency ofωp = (2ζΩωc|cosβ |)1/2 = 14.88 GHz(ζ cosβ )1/2(P1 s)−1/2( B1012 G)1/2(5.33)where β is the angle between the spin axis and the magnetic field direction locally.The first vacuum resonance energy in the magnetosphere is typically ath¯ωv,1 = 0.068 eV(ζ cosβ )1/2(P1 s)−1/2( B1012 G)−1/2. (5.34)The vacuum resonance energy increases as r3/2 further into the magnetosphere ifthe density follows the Goldreich-Julian expresion and the magnetic field is dipolar.86Although the plasma density may be sufficiently large for this energy to reachinto the visual range (ζ ∼ 103), I do not expect the magnetospheric density to belarge enough (ζ ∼ 108) for this energy to reach the X-ray regime; in this casethe magnetosphere would become somewhat opaque even outside of the resonantenergies (see § 5.2.1 for the role of plasma birefringence in the atmosphere).Below the vacuum resonance energy, the plasma dominates the birefringenceand I can calculate the magnitude of the birefringent vector|Ωˆ|= ζ |cosβ |Ωωcωcsin2 θ . (5.35)The polarization-limiting radius for low photon energies will be determined bythe plasma birefringence. On the other hand, for radiation in the visual and blue-ward for pair multiplicities less than 103, the vacuum birefringence dominates theevolution of the polarization in the magnetosphere, and the magnitude of the bire-fringence vector can be calculated from eq. 2.64.This section considered just the weak-field limit. The strong-field limit issomewhat more complicated. For a detailed treatment of the propagation of radia-tion through the combined plasma and vacuum for arbitrary fields consult [98, 123].5.2.1 The vacuum resonance in the neutron-star atmosphereAs the density of plasma in the atmosphere is much larger than in the magneto-sphere, the energy of the vacuum resonance is also much larger and typically inthe range of X-ray energies. Therefore, it may be important for the propagation ofX-ray radiation above the photosphere. At the resonanceh¯ωv = h¯ωpv−1/2r = 2.0( n1022cm−3)1/2 1012 GBkeV. (5.36)Deep within the atmosphere, the density is large and the birefringence is dominatedby the plasma: ∆n‖⊥ < 0, |∆n‖⊥|  ∆n± and 2θm ≈ pi . We can see from eq. 5.28to 5.31 that, in this regime, the polarization modes are linear: S1 = −1 and S3 =0; also, the + mode is polarized along the y-axis (perpendicular to the magneticfield). As the radiation in the + mode propagates upward and through the vacuumresonance, it will remain in the + mode if the condition of eq. 2.65 holds (the87adiabatic criterion). At low densities the vacuum dominates so ∆n‖⊥ > 0, 2θm ≈ 0and therefore, for the + mode, S1 = 1 and S3 = 0. As the photon crosses theresonance, the polarization is transformed from perpendicular to parallel.I can calculate whether the adiabatic criterion holds as the radiation passesthrough the vacuum resonance. The index of refraction difference reaches a min-imum value of ∆n±,res at the resonance precisely. In the resonance, the change inthe value of Ωˆ is entirely in its direction at a rate of 2θ ′m = (∆n‖⊥)′/∆n± so wehave∣∣∣∣∣∣Ωˆ(1|Ωˆ|∣∣∣∣∣∂ Ωˆ∂x3∣∣∣∣∣)−1∣∣∣∣∣∣=∣∣∣∣∣ωc (∆n±,res)2(∆n‖⊥)′∣∣∣∣∣ (5.37)=ωcHρ[cosθωω2pωc(ω2−ω2ci)]2[sin2 θ αQED30pi(BBQED)2]−1=ω3cHρ cotθωc(1− ω2c,iω2)22αQED15pi(BBQED)2=(EγEad)3.This yields the abiabatic energy [98, 123]E3ad =15pi2αQED(1− ω2c,iω2)2tan2 θh¯cHρ(mc2)2 ≈ (2.6 keV)3(1− ω2c,iω2)2tan2 θ1 cmHρ(5.38)where Hρ is the density scale height along the ray (in the kˆ direction), typicallykT/(Amukˆ ·~g)≈ 8 mm for a temperature T = 106K, surface gravity kˆ ·~g= 1014 cm s−2and for hydrogen atmosphere, A = 0.5.If one considers blackbody emission, for effective temperatures greater thanabout 5× 106 K, the energy of the typical photon is greater than the adiabaticenergy, so much of the radiation will pass through the resonance adiabatically.Typically, light element atmospheres peak at higher photon energies than 3kTeff,so the effect would be even more pronounced. If the vacuum resonance occurs inthe neutron star above both the photosphere for parallel polarization and the pho-tosphere for perpendicular polarization (which I found in § 3.2.1), the main effect88is to switch the polarization coming from the surface from mostly perpendicular tothe magnetic field to mostly parallel, if the energy of the photon is greater than Ead.Below this energy, the polarization would remain perpendicular.For sufficiently strongly magnetized neutron stars (magnetars), the vacuum res-onance lies above the photosphere for radiation polarized perpendicular to the mag-netic field and below the photosphere for parallel photons. In this case, the effectivephotosphere for perpendicularly polarized photons with energies greater than theadiabatic energy will lie at the vacuum resonance itself which is at a lower tempera-ture than the photosphere for low-energy perpendicularly polarized radiation. Evenin this case, the bulk of the radiation will emerge in the perpendicular polarization,and the emission of parallel photons will be diminished for photon energies abovethe vacuum resonance energy at the photosphere. I can estimate the number densityof electrons at the photosphere for the parallel polarization to be n≈ (σT Hρ)−1 soh¯ωv,‖ ≈ 4 keV(BBQED)−1( g1014cm s−2)1/2( T106 K)−1/2. (5.39)Although this equation is not accurate for magnetic fields approaching or exceed-ing BQED, one can see that the role of the vacuum resonance in the formation ofthe spectrum will be crucial for the magnetars where the magnetic field exceedsBQED and the temperatures exceed several million degrees. The number density atthe photosphere for the perpendicular polarization is larger by a factor of ω2/ω2c ,giving the energy of photons in the vacuum resonance at this surface ofh¯ωv,⊥ ≈ 45 keV( g1014cm s−2)1/4( T106 K)−1/4(5.40)or approximately the electron cyclotron energy, whichever is smaller.Fig. 5.5 illustrates how the vacuum resonance affects the location of the pho-tosphere for the perpendicular mode photons. At low energies, the photospherelies at a density where the plasma frequency equals the frequency of the photon,then it runs at nearly constant density punctuated by a dramatic drop in the den-sity at the proton cyclotron line and then above the adiabatic energy it follows thedensity at which photon frequency equals the vacuum resonance energy (this is a892.0 1.5 1.0 0.5 0.0 0.5 1.0log10(E/keV)432101234log 10(/g cm3 )T=5×106 K  B=1014 G photosphere photosphere= p= vFigure 5.5: An illustration of the locations of the parallel and perpendicularmodel photospheres for B= 1014 G and T = 5×106 K using the modelsof Lloyd [130]. This is similar to Fig. 3.2, but for the strong field case.The proton cyclotron line has moved into the X-rays; the key new thingis that the ⊥-mode photosphere lies along the vacuum resonance line athigh energies, because the ⊥-mode photons have a significant parallelcomponent at the vacuum resonance so the opacity for them is reallyhigh there. The vacuum resonance is effectively their photosphere. Infact a significant amount of energy is deposited in this layer so it formsa local maximum in the temperature of the atmosphere.constant multiple of the plasma frequency that depends on the strength of the mag-netic field). For photons above the adiabatic energy, the structure of the atmosphereis rather complicated. At high densities, where the birefringence is plasma domi-nated, but not so high that the the plasma is opaque to photons in the perpendicularmode, the bulk of the energy flux is carried by photons in the perpendicular mode,while those in the parallel mode are trapped. As the radiation approaches the den-sity of the vacuum resonance, both the photons in the parallel mode and those inthe perpendicular mode become circularly polarized and both couple strongly to90the plasma. At the resonance density, the flux carried in the perpendicular mode isdumped back into the plasma. Just below the resonance density the modes becomemainly linear again, so the perpendicular mode is no longer well coupled to theplasma and again travels freely, and the photosphere for the perpendicular modefollows the vacuum resonance density above the adiabatic energy. For strongermagnetic fields, the proton cyclotron line can lie above the adiabatic energy, so thestructure of the line, even without polarization information, will be affected by theresonance [98].This picture, in which above the adiabatic energy the radiation behaves adia-batically and below this energy it does not, is a gross approximation. In detail, thebehaviour near the vacuum resonance will also depend on the imaginary portion ofthe index of refraction [148, 178], and the entire mode description may collapse.How to treat photons passing through the vacuum resonance within the atmosphereis still uncertain, and it is often treated as such in the calculations [76, 98, 123, 249].However, it is clear that when the radiation passes through the vacuum resonanceoutside the atmosphere, as in more weakly magnetized stars (see Fig. 3.3), the res-onance can switch the final polarization state of the radiation depending on whetherthe energy lies above or below the adiabatic energy, leaving an imprint of the localdensity scale height on the outgoing radiation, which could be a powerful diagnos-tic of the surface gravity in the polarization of the outgoing radiation.91Chapter 6QED and Polarization fromAccreting Black holes6.1 IntroductionIn the theory of accretion disks around black holes and astrophysical accretion ingeneral, magnetic fields play a crucial role. They are expected to be the mainsource of shear stresses, without which accretion cannot occur [8, 205]. More-over, magnetic fields in the inner regions of black-hole accretion disks are thoughtto lead to the formation of relativistic jets through the Penrose–Blandford–Znajekmechanism [31, 220]. As I already mentioned in § 1.2.4, however, direct measure-ments of the magnetic field strength and structure in the accretion disk of a blackhole are hard, and the only estimates to date come from the spectral analysis of thewinds from two Galactic stellar-mass black holes [154, 155, 157], and they probeonly the field quite far from the horizon.The first estimate of the structure of the magnetic field close to the event hori-zon of a black hole comes from polarimetric studies of the radio emission fromSagittarius A*, the supermassive black hole at the center of the Milky Way [105].In this Chapter, I show that X-ray polarimetry could provide an additional tool toprobe the strength and structure of the magnetic field close to the event horizon ofaccreting black holes if the effect of vacuum birefringence is properly accountedfor in the modeling of the polarization.92If only classical electrodynamics is considered, at energies higher than 1–2keV, the polarization of a photon emitted by the accretion disk is not affected bythe presence of a magnetic field. The linear polarization of X-ray photons staysthe same as they travel through the magnetosphere of the hole all the way to theobserver. At lower photon energies, the presence of a magnetized corona could de-stroy the linear polarization of X-ray photons due to the effect of plasma birefrin-gence [52, 149] (see § 6.3.1). In quantum electrodynamics (QED), the vacuum isalso expected to be birefringent in presence of a magnetic field. This effect, whichwas one of the first predictions of QED, has never been proven. Recent observa-tions of the visible polarization from a radio-quiet neutron star [152] have stronglyhinted that vacuum birefringence is indeed affecting the photons’ polarization. Ifthe vacuum is indeed birefringent, after photons are emitted from the disk, theirpolarization will change as they travel through the magnetized vacuum. A detailedderivation of the vacuum birefringence in QED is described in Chapter 4.In this Chapter, I assume the strength of the magnetic field in the accretion diskto be the minimum needed for accretion to occur if an α-model structure of thedisk is considered. I find that the effect of vacuum birefringence on the photonpolarization becomes important, depending on the angular momentum of the blackhole and that of the photon, around 10 keV, for both stellar-mass and supermassiveblack holes. A stronger (weaker) field would shift this range to lower (higher)energies. Observation of the X-ray polarization from accretion disks in the 1–30keV range, if properly modeled with QED, would both probe the strength of themagnetic field and test the currently accepted models of astrophysical accretion.6.2 Accretion disk modelBlack-hole accretion disks are rarefied; thus, angular momentum transfer due tomolecular viscosity is inefficient and cannot lead to accretion [184]. In current the-ories of astrophysical accretion disks, magnetic fields and turbulence are expectedto be the source of shear stresses. In this section, I will calculate the minimummagnetic field strength needed for accretion to occur in a α-model disk [205]. Therelation between the tangential stresses between layers in the disks and the mag-93netic field is given by [205]tφˆ rˆ = ρcsvt +B24pi= αP (6.1)where ρ is the mass density, cs is the speed of sound, vt is the turbulence velocity,P is pressure and tφˆ rˆ is the shear stress as measured in a frame of reference movingwith the gas. The last equality is the simplifying assumption of the α-model: theefficiency of the angular momentum transfer is expressed with one parameter, α .Since turbulence in the disk is generated by shear instability caused by the samemagnetic field [8], I expect the viscosity term and the magnetic field term to be ofthe same order. The minimum strength for the magnetic field to generate the shearstresses needed for accretion is then of the order B∼ (4piαP)1/2.In this work, I will model the accretion disk physics using the Novikov andThorne (N&T) model [168] for a geometrically thin, optically thick disk (see also§ 1.2.2). For simplicity, in order to split expressions into Newtonian limits timesrelativistic corrections, N&T introduced the following functions (from now on Iwill use c = G = 1), which are equal to one in the non-relativistic limit:A =1+a2?/r2?+2a2?/r3? (6.2a)B =1+a?/r3/2? (6.2b)C =1−3/r?+2a?/r3/2? (6.2c)D =1−2/r?+a2?/r2? (6.2d)E =1+4a2?/r2?−4a2?/r3?+3a4?/r4? (6.2e)F =1−2a?/r3/2? +a2?/r2? (6.2f)G =1−2/r?+a?/r3/2? (6.2g)N =1−4a?/r3/2? +3a2?/r2? (6.2h)where r? = r/M and a? = a/M. The last expression,N , is not from Novikov andThorne [168] and corresponds to the quantity called C in Riffert and Herold [191].In the N&T accretion disk model, the disk lies in the equatorial plane (θ = pi/2),94matter rotates in quasi-circular orbits with angular velocityω =dφdt=√Mr31B. (6.3)and the inner edge of the disk is assumed to be coincident with the innermoststable circular orbit of the Kerr metric (or ISCO, see eq. 1.19). Also, the angularmomentum of the disk is assumed to be aligned with the spin of the hole.In order to calculate the pressure in the disk, I have to analyze the local verticalstructure of the disk near the equatorial plane. The easiest way is to perform thecalculations in the local orbiting frame at the center of the disk (z= 0). In this iner-tial frame of reference, all that is needed are the following equations, in which theNewtonian value is multiplied by the relativistic corrections defined in eqs. (6.2). Iwill need, of course, the equation for hydrostatic equilibrium in general relativity.I use the correction to the N&T equilibrium found by Riffert and Herold [191]:dPdΣ=−ω2zB2NC(6.4)where dΣ = ρdz. Since I am interested in the mid-plane, where by symmetry Iexpect the vertical density profile to reach a local maximum, I consider ρ to beapproximately constant near the mid-plane.Next, I will need an expression for how the energy is generated inside the disk.The viscous heating generated by friction between adjacent layers is given by [168]dFdz=32ω tφˆ rˆC−1BD (6.5)where F is the energy flux. I assume the energy transport to be radiative:F =− 1κRdPraddΣ(6.6)where κR is the Rosseland mean opacity.For the equation of state, in order to calculate the vertical structure, I assumethat in the central part of the disk pressure is dominated by radiation. However, I95still leave the possibility of a z dependence in the equation of state:P =1χ(Σ)Prad . (6.7)From eqs. (6.4), (6.6), and (6.7) I get:−κRF = d(χ(Σ)P)dΣ =dχdΣP+χdPdΣ=dχdΣP+χ(−ω2z)B2NC(6.8)Thus, from eqs. (6.5) and (6.7):αP = χ2ω3κRBND− 2F3ωd lnκRdzCBD+2ωz3κRdχdzBND− 23κRωddz(dχdΣP)CBD(6.9)In the mid-plane this becomes:αPc = χc2ω3κRBND−Pc 23κRωddz(dχdΣ)∣∣∣∣z=0CBD∼ χc 2ω3κRBND−Pc 23κRωχcρch2∣∣∣∣z=0CBD(6.10)where h is the typical scale height of the disk. The second term is negative becauseχ decreases with z and Σ and it reaches its maximum at z = 0, so its derivative atz = 0 is less than 0.Rewriting κRρc = 1/λ (mean free path), I obtain:Pc23κRωχcρch2= χcPc2λ 2ωλh2(6.11)In this expression, h2/λ 2 corresponds to the number of mean free paths that aphoton needs to perform a random walk out of the disk, while λ/c is the time forone mean free path. I can then rewrite this expression in terms of the diffusiontime:χcPc2λ 2ωλh2= χcPc2ωtdiff= χcPctrotpitdiff(6.12)96where trot is the time needed by the disk to undergo a complete rotation and tdiff isthe diffusion time. Since trot  tdiff, this term is much smaller than the first one.The relativistic corrections do not affect this result because the value of C /(BD)is less than one from the ISCO to infinity and it goes to one at infinity. I can thenwrite the strength of the magnetic field in the mid-plane as:B2 ∼ 4piαPc ∼ χc 2ω3κRBND= χc8pi3κR√Mr3ND. (6.13)Since radiation dominates the pressure in the mid-plane of the disk, I can takeχc ∼ 1. Moreover, it is safe to assume that in the innermost part of the disk theopacity is dominated by electron scattering:κR = κes =8pi3mp(e2mec2)2(1+X)2(6.14)where mp and me are the proton mass and the electron mass respectively and X isthe hydrogen mass fraction. For a 10 M black hole at the ISCO, r = rI , I obtainB2 = (0.36−1.22×108 G)2(M10M)−1(1+X2)−1(6.15)where the first value is for a? = 0 and the second is for a? = 0.999 (the valuediverges for a? = 1). This is a crude estimate of the minimum magnetic fieldstrength needed to generate enough shear stresses for accretion to occur. Bothglobal magneto-hydrodynamic (MHD) simulations [200] and shearing box simu-lations [97] show that, when moving away from the mid-plane, the magnetic pres-sure decreases toward the photosphere. However, the expression in eq. (6.15),with the radial scaling of eq. (6.13), reproduces the strength of the magnetic fieldat the photosphere obtained with shearing box simulations by Hirose et al. [97]for a 6.62 M black hole at a radius of 30 GM/c2. Likewise, the expressions ineqs. (6.13) and (6.15) reproduce both the strength and the radial decrease of themagnetic field along the photosphere in Fig. 3 of Schnittman et al. [200], who per-formed a global MHD simulation for a 10 M black hole. Regarding the estimatesobtained by Miller et al. [157] for GRS 1915+105 at 850, 1,200, 3,000 and 30,00097GM/c2, eqs. (6.13) and (6.15) reproduce their minimum estimate at every radius,the one obtained by assuming MHD pressure, while it is two orders of magnitudeless than their estimates obtained by assuming a magnetocentrifugal driven windor an α-model pressure. For the purposes of this work, I will then use the analyt-ical expression found in eq. (6.13) for the minimum magnetic field strength at thephotosphere.6.3 Vacuum birefringenceFrom eq. 4.50 in Chapter 4 and eq. 6.13, I can estimate the amplitude of the bire-fringent vector in the vacuum just above the accretion disk. Reintroducing all theconstants yields the magnitude of the birefringent vectorΩˆ= k0∆n =αQED15k0(BBQED)2sin2 θ = k0h¯mp15pim2ec21(1+X)√GMr3NDsin2 θ(6.16)And the left term of eq. 2.65 (the adiabaticity condition) becomes∣∣∣∣∣∣Ωˆ(1|Ωˆ|∂ |Ωˆ|∂x3)−1∣∣∣∣∣∣' Ωˆ(r)r' k0 h¯mp15pim2ec21(1+X)√GMrN (r)D(r). (6.17)Equating this expression to 1/2, I can calculate the polarization limiting radius, i.e.the distance from the hole at which the adiabaticiy condition breaks down, to berpc2GM=(2k0h¯mp15pim2ec(1+X)N (rp)D(rp))2. (6.18)The polarization-limiting radius is a rough indication of the distance from thesource at which the polarization of light is not affected by the birefringence any-more. In Fig. 6.1, the energy of the photon at which rp is equal to rI is plottedagainst the spin of the black hole (solid red line). The dotted line represents theISCO (right y−axis). This means that, for rapidly spinning black holes, the ef-fect of QED will be important around a photon energy of 10 keV or lower, while98101214161820E[keV]23456r Ic2/GM0.0 0.2 0.4 0.6 0.8 1.0a⋆Figure 6.1: The plot shows, on the left, y axis, the energy at which rp = rI(solid red line). On the right, y axis, the ISCO for a black hole as func-tion of the spin parameter a (dashed black line).for slowly spinning black holes, QED will affect the polarization only above 10-20 keV. However, if the magnetic field strength is higher (or lower), the energy atwhich QED becomes important decreases (or increases) as the inverse square ofthe magnetic field strength. The effect of vacuum birefringence, if properly mod-eled, can therefore provide an indication on the strength of the magnetic field thatthreads the accretion disk. It is worth noticing that this result depends on the spinof the black hole but not on the mass, so it stands for both stellar-mass and super-massive black holes. The polarization-limiting radius estimate does not account forlight bending, which causes the photon’s path in the strong magnetic field regionto be longer due to the gravitational pull of the hole. For this reason, photons atenergies lower than the one plotted in Fig. 6.1 could still be affected by the vacuumbirefringence, depending on their angular momentum (see Sec. 6.4).996.3.1 Competition with the plasma birefringenceThe inferred presence of a corona above the inner regions of the disk introduces thepossibility of a competing Faraday rotation due to the plasma birefringence. Theeffects of plasma birefringence for black hole accretion disks were studied in detailin a paper by Davis et al. [52] and comprise of a reduction of the photons linear po-larization in a range of energies that depends on the strength of the magnetic field,on the energy of the photons and on the distance to the black hole of the emissionregion. In this section, I estimate the photon energy above which the vacuum bire-fringence dominates over the plasma. If I write the two photon polarization modesas|e1〉= cosψ|a〉+ isinψ|b〉 (6.19a)|e2〉= sinψ|a〉− icosψ|b〉 (6.19b)where|a〉= −sinθ0cosθ , |b〉= 010 , (6.20)in the cold plasma limit I obtainb =1tan2ψ' ωBω[1+Vω2−ω2Bω2B]sin2 θ2cosθ(6.21)where ωB = eB/mec2 is the cyclotron frequency,V =αQED15pi(BBQED)2( ωωp)2(6.22)measures the influence of the virtual e+ e− pairs in the strong magnetic field rela-tive to the real electrons of the plasma and ωp is the plasma frequency [149].For an accretion disk in the keV range, we are in the limit for which ωωB. Ifb goes to zero, the polarization becomes circular, and without the presence of QED,the Faraday rotation induced by the plasma would destroy the linear polarization,as in that case b'ωB/ω 1. The limit for which the QED and the plasma effects100are similar is for b ∼ 1. Since ωB/ω  1, in order for b to be about 1, Vω2/ω2Bneeds to be much greater than 1, so I can neglect the first term in the brackets ofeq. (6.21) and then obtainb' αQED15pi(BBQED)2 ω3ω2pωB=eBE360pi2nem2e h¯2c6(6.23)where E is the energy of the photon and ne is the number density of electrons.If I assume the optical depth over a distance comparable to the ISCO to be low:τ = neσT rI ' 0.2 (6.24)where σT is the Thomson cross section, I obtain that b∼ 1 forE = 2.11−2.43 keV(M10M)− 16 ( τ0.2) 13(1+X2) 16(6.25)where the first value is for a? = 0 and the second value is for a? = 1. Because bscales as E3, at higher energies the plasma birefringence does not destroy the linearpolarization of the photons thanks to the predominance of QED, which rendersthe propagation modes approximately linear. The energy at which QED begins todominate scales slowly with the assumed magnetic field strength, in fact as B−1/3.6.4 Depolarization in the disk planeTo better understand how vacuum birefringence affects the polarization of pho-tons traveling through the black hole magnetosphere, in this section I will as-sume a simple structure for the magnetic field threading the accretion disk, andI will study how the polarization changes for photons traveling parallel to the diskplane. Recent observations of the radio polarization coming from the region closeto the event horizon of Sagittarius A* suggest the presence of a partially organizedfield [105]. It is reasonable to assume the magnetic field to be organized on somelength-scale that reflects the competition between the magnetic field itself, whichwould tend to be organized, and the shear of the disk, which prevents big structuresfrom forming. I therefore assume the disk to be divided into regions of constant101magnetic-field direction, which is also the structure often assumed for the mag-netic field in the plane of the disk by magnetohydrodynamics (MHD) simulations[174]. I pick two different length-scales to test how my assumption on the size ofthe magnetic loops affects the results. Since I expect the length scale to be relatedto both the distance to the hole and to the size of the hole itself, I first divide thedisk into five regions, each twice as large as the previous one: from the ISCO totwice the ISCO, to 4 times the ISCO, to 8 times the ISCO, to 16 times the ISCO,and to infinity. For simplicity, I call this configuration the 2-fold configuration.In the second configuration, the regions of constant magnetic-field direction areeach 1.5 times as large as the previous one: from the ISCO to 1.5 times the ISCO,to 2.3 times the ISCO, to 5.1 times the ISCO, to 7.6 times the ISCO, to 11 times theISCO, to 17 times the ISCO, and to infinity. For simplicity, I call this configurationthe 1.5-fold configuration.As a photon travels through a magnetized birefringent vacuum with differencein index of refraction ∆n, the polarization direction rotates around the birefringentvector Ωˆ asdΘdτ= ∆np ·uh¯c(6.26)where p is the four-momentum of the photon, u is the four-velocity of the disk thatanchors the field and τ is the proper time elapsed in the frame of the disk. I wantto calculate the final depolarization of the photon, so I integrate along the geodesic∆Θ=∫∆np ·uh¯c(dxµdr)uµdr (6.27)to determine the total rotation of the polarization of a photon across the Poincare´sphere. The polarization of an individual photon will perform a random walk acrossthe Poincare´ sphere, and the total rotation of the polarization along the path is givenby eq. 6.27, where the extremes of the integral are the ISCO and infinity. Thedirection of the individual step, is given by eq. 2.64.For simplicity, I only consider photons traveling near the plane of the disk. In102the equatorial plane, the spacetime interval in the Kerr metric (eq. 1.3) becomesds2 = gttdt2+2gtφdrdφ +gφφdφ 2+grrdr2 (6.28a)gtt =−1+2M/r (6.28b)gtφ =−2Ma/r (6.28c)gφφ = r2(1+a2/r2+2Ma2/r3) = r2A (6.28d)grr = (1−2M/r+a2/r2)−1 =D−1 (6.28e)The four-velocity of an observer rotating with the disk can be easily obtained re-membering thatuφ =dφdτ= ωut (6.29)From its definition, gµνuµuν =−1, I obtainur = 0 (6.30a)ut =√−1gtt+2gtφω+gφφω2=BC−12 (6.30b)uφ =dφdτ= ωut = ωBC−12 (6.30c)(ur = 0 because we are in the local orbiting frame), andut = (gtt+gtφω)ut =−GC− 12 (6.31a)uφ = (gφφω+gtφ )uφ =√MrFC−12 (6.31b)In order to study the path of the photon along its geodesic, it is useful to cal-culate quantities that do not change along the path. From the dot-product of thefour-momentum of the photon and two of the Killing vectors of the metric, I findtwo quantities that remain constant along the geodesics: the energy and angularmomentum of the photonE =−ξt · p =−(gtt pt+gtφ pφ ) (6.32a)L = ξφ · p = gφφ pφ +gtφ pt (6.32b)103I call the specific angular momentum L/E = l. I analyze three cases: a photoncoming from the ISCO with zero angular momentum (l = 0), a photon initiallyrotating with the disk (maximum prograde l+) and a photon initially going againstthe rotation of the disk (maximum retrograde l−).6.4.1 Zero angular momentum photonsIf l = 0, from eqs. (6.32a) and (6.32b), I obtain, for the photon,dφdt=− gtφgφφ= 2Mar3A −1 (6.33)From the null-geodesic condition ds2 = 0, I find:dtdr=√√√√ grrg2tφgφφ−gtt=A12D−1 (6.34)I can then write the second part of eq. (6.27) as(dxµdr)uµ =dφdruφ +dtdrut =−(AC )− 12B. (6.35)The first part becomes:p ·u = E(−ut+buφ ) =−Eut (6.36)Using ∆n from eq. (6.16), and changing the integration variable to a dimensionlessone (r? = r/M), eq. (6.27) becomes∆Θ= EK∫sin2 θr−32? A− 12B2N (DC )−1dr? (6.37)where K = mp/[15pim2ec2(1+X)].1046.4.2 Maximum prograde and retrograde angular momentumphotonsFrom eqs. (6.32a) and (6.32b), I obtain, for the photon,dφdt=− lgtt+gtφgφφ + lgtφ(6.38)By imposing dr2 = 0, at the point of emission (the ISCO), I obtain the values forthe maximum prograde specific angular momentum (l+) of a photon rotating withthe disk and the maximum retrograde specific angular momentum (l−) for a photonin retrograde motion:l± =gtφ ±√g2tφ −gφφgtt−gtt = r(−2a?/r2?±D1/21−2/r?)(6.39)Since l is a constant along the geodesic, I calculate l± at the ISCO. Depending onthe spin of the black hole, however, the ISCO can be inside the retrograde photonorbit (the prograde photon orbit is always inside the ISCO). In this case, I calculatel− at the retrograde photon orbit (no photons can escape in retrograde motion froma smaller orbit than the retrograde photon orbit).Employing eq. (6.38) in the null-geodesic condition, I find the path of the pho-ton:dtdr=gφφ + lgtφrD(l2gtt+2lgtφ +gφφ )1/2(6.40a)dφdr=−(lgtt+gtφ )rD(l2gtt+2lgtφ +gφφ )1/2(6.40b)Defining a dimensionless angular momentum as l? = l/M, I can then write the105second part of eq. (6.27) as(dxµdr)uµ =dφdruφ +dtdrut=l√Mr− r2BrC 1/2(l2gtt+2lgtφ +gφφ )1/2=l?/r3/2? −BC 1/2(l2?gtt/r2?−4l?a?/r3?+A )1/2(6.41)The first part becomesp ·u = E(−ut+ luφ ) = EC−1/2(l?/r3/2? −B) (6.42)I can then rewrite eq. (6.27) as∆Θ= EK∫sin2 θr−32?NDC(l?/r3/2? −B)2(l2?gtt/r2?−4l?a?/r3?+A )1/2dr? (6.43)where K = mp/[15pim2ec2(1+X)] is the same as in the previous section.6.4.3 ResultsEquations (6.37) and (6.43) allow us to calculate the path that the polarization of aphoton takes across the Poincare´ sphere in each region. To calculate the directionof the step, I rotate s around Ωˆ [eq. (2.64)]. In each region, I take the angle betweenthe magnetic field and the photon, θ , and the angle between s and Ωˆ as random.In order to visualize the depolarization effect of the partially ordered field onthe single photon, I first perform a Monte Carlo simulation for 60 photons forthe 2-fold configuration, calculating the evolution of their polarization from theISCO to infinity. Each photon is emitted with the same angular momentum andthe same energy at infinity from the ISCO of a black hole rotating with a? = 0.84(as the AGN NGC 1365 [192]). I repeat the same calculation for photons travelingwith zero, 90% of the maximum prograde and 90% of the maximum retrogradespecific angular momentums and for three different energies: 3, 5 and 7 keV (atinfinity). The results are shown in Fig. 6.2. Figure 6.2 depicts a solid Poincare´sphere, in which the dots represent the end-point of the polarization vectors. The106Figure 6.2: Monte-Carlo simulation of the depolarization of radiation from ablack hole with a = 0.84 (as NGC 1365) for three photon energies (asmeasure by a distant observer): 3 keV (left), 5 keV (middle) and 7 keV(left). Polarization is represented on the Poincare´ sphere: the dots rep-resent the end-point of the polarization vector. The initial polarizationvector is indicated by a dark blue dot. The violet dots are photons thatreceive a large blue shift (90% of l+) , the yellow dots are zero-angular-momentum photons and the copper receive a large red shift (90% of l−)on their way from the ISCO to us.dark blue dot indicates the initial polarization, which is the same for every photon.Without the QED effect, the polarization would be frozen at the emission and thefinal polarization at infinity would be the same for all photons: still the dark bluedot. The other dots indicate the final polarization of the photons, calculated withinQED. The yellow dots indicate the end-point of the polarization vector for the zeroangular momentum photons; the violet dots correspond to the photons that receivea large blue shift (l+ photons) and the copper dots represent the photons that receivea large red shift (l− photons). We can immediately see that the final polarizationis different from the initial one for all the photons, with a much bigger effect forred-shifted photons and for high-energy photons.The same Monte-Carlo simulation, this time with 6,000 photons, for both the2-fold configuration and the 1.5-fold configuration, for different energies from 1to 80 keV (at the observer), and for four values of a?: 0.5, 0.7, 0.9 and 0.99, isshown in Fig. 6.3. Both plots show the polarization fraction obtained as an averageof the final linear polarization of all the 6,000 photons against the photon energy.Results are shown for both the 2-fold configuration (solid lines) and the 1.5-fold107configuration (dashed lines). The left plot shows the final polarization fraction ofthe zero angular momentum photons (black lines), the blue-shifted photons (bluelines) and the red-shifted photons (red lines) for a black hole rotating with a? =0.9. The right plot shows the polarization fraction of red-shifted photons for fourdifferent a?: 0.5 (green lines), 0.7 (light blue lines), 0.9 (red lines) and 0.99 (purplelines).In Figure 6.3, the dashed lines show the results for the 1.5-fold configurationand the solid lines show the results for the 2-fold configuration. I find that, if themagnetic loops are smaller, the depolarization effect is reduced linearly with thesize of the loops: in this example, the dashed lines fall on top of the solid lines ifI re-scale them by 2/1.5. However, the solid lines show peaks that are not presentin the dashed lines. For example, for a hole rotating with spin a? = 0.99 in the2-fold configuration (purple solid line, right panel) the polarization fraction peaksat 7 keV and then again at 14 keV, at 21 keV and so on. These peaks are due to thefact that at those energies the integral in Equation (6.43) reaches, in the first zoneof the disk, an average value of pi , and therefore, the polarization vector remainscloser to the S1− S2 plane. In the 1.5-fold configuration, this does not happenbecause the first region is smaller and the second region has a bigger effect on thefinal polarization, washing out the peaks. Ideally, the presence of features in thepolarization spectrum such as the peaks shown for the 2-fold configuration couldprovide hints on the structure of the magnetic field in the disk.All of the aforementioned results are independent of the black hole mass.6.4.4 A Simulation for GRS 1915+105As an example, I simulated the observed polarization of the black-hole binaryGRS 1915+105. GRS 1915+105 is a bright microquasar that hosts a rapidly spin-ning black hole. Measurements of its spin, which rely on observations in bothX-rays and optical, seem to indicate a spin parameter a? & 0.98 [142, 156]. I as-sumed an inclination angle of 75◦ [60, 158], and I used the polarization spectrafrom Figure 7 of Schnittman and Krolik (2009) [198]. To calculate the effects ofthe vacuum birefringence, I assumed that the bulk of the radiation comes from nearthe ISCO and has zero angular momentum.1080.10.20.30.40.50.60.70.80.9Linearpolarizationfraction1 10Photon Energy [keV]0.10.20.30.40.50.60.70.80.9Linearpolarizationfraction1 10Photon Energy [keV]Figure 6.3: Final polarization fraction vs. photon energy calculated in the 2-fold configuration (solid lines) and in the 1.5-fold configuration (dashedlines). Left plot, left to right: maximum retrograde (90% l−) angularmomentum photons (red), zero angular momentum photons (black) andmaximum prograde (90% l+) angular momentum photons (blue), com-ing from the ISCO of a black hole with a? = 0.9. Right plot: 90% l−photons for, left to right, a? = 0.99 (purple), 0.9 (red), 0.7 (light blue)and 0.5 (green).Figure 6.4a shows the observed polarization degree for two spin parameters,a? = 0.95 and a? = 0.99, both with and without including QED. If QED were notincluded in the model, it would be easy to mistake a black hole actually spinning ata? = 0.99 (blue line) with one spinning at a? = 0.95 (green line). In the left panelof Figure 6.4, all the models were calculated assuming the minimum magnetic fieldneeded for accretion to occur in an α−model (Equation (6.13)). In Figure 6.4b, Ishow the effect of a stronger magnetic field. The red and the blue lines are thesame as in Figure 6.4a: a? = 0.99 and the minimum magnetic field, with and with-out QED, while the black line represents a model with the same parameters but amagnetic field 2.5 times stronger. We can see that the curves are very different,with the QED effect being much stronger for the stronger magnetic field, and thatthe peaks have shifted into the 2–8 keV range. Of course, the magnetic field struc-ture that I used in this work is just a toy model, but the peaks show that the QEDeffect can be sensitive to the magnetic field structure, and the upcoming polarime-109Figure 6.4: Observed polarization degree for the black-hole binary GRS1915+105. (a) Model with a? = 0.99 with QED (blue line) and with-out QED (red line); model with a? = 0.95 with QED (yellow line) andwithout QED (green line). (b) Model with a? = 0.99 with QED and theminimum magnetic field (blue line) and without QED (red line); modelwith a? = 0.99 with QED and 2.5 times the minimum magnetic field(black line).ters would be sensitive enough to detect them.I want to stress that these figures show preliminary calculations, and furtherwork is required to model the expected polarization degree. Indeed, our modelassumes the flux to be dominated by photons coming from close to the ISCO andwith nearly zero angular momentum, which could be a good assumption for high-energy photons but the contribution of photons coming from more distant regionshas to be properly included in the calculations for low-energy photons. Moreover,the structure of the magnetic field that I employed is just a simple toy model, andbetter calculations are needed to make a prediction on whether features like thepeaks in the polarization degree would be detectable and at which energies theywould be present.6.5 ConclusionsIn Figure 6.3, all photons were emitted with the same polarization. If the vacuumwere not birefringent, their final polarization would still be the same, and the finallinear polarization fraction would still average at one. I can therefore conclude thatvacuum birefringence has a big impact on the polarization of X-ray photons, espe-110cially for fast-spinning black holes and for red-shifted (retrograde) photons. Thereason the effect is stronger for higher spinning parameters is because the ISCO iscloser to the event horizon and, therefore, the magnetic field is stronger, but alsobecause photons perform more orbits around fast-spinning holes, staying longer inthe strong magnetic field region. Retrograde photons are more affected for two rea-sons: they perform more orbits around the black hole with respect to zero angularmomentum and prograde photons, and they receive a red-shift, which means thattheir energy at emission was higher.The results shown in Figure 6.3 were obtained for the minimum magnetic fieldneeded to generate enough shear stresses for accretion to occur in an α-model forthe accretion disk. The actual magnetic field threading the accretion disk could behigher, leading to a stronger effect of the vacuum birefringence on the polariza-tion. In general, a stronger (or weaker) magnetic field would shift the x−axis ofFigure 6.3 to a lower (higher) energy range, and the shifting would scale with thesquare of the magnetic field, as shown in Figure 6.4.The simulations presented for GRS 1915+105 are not intended to be predictiveas more detailed models are required for the structure of the magnetic field close tothe disk plane and for the contribution to the total emission from photons emittedat different distances to the central engine. However, they show that vacuum bire-fringence has an effect on the observed polarization of fast-spinning black holesthat can be detected.My analysis was restricted to edge-on photons, traveling close to the disk plane,where the magnetic field is expected to be partially organized on small scales. Fur-ther studies are needed to calculate the effect of vacuum birefringence for photonscoming out of the disk plane, where we expect the magnetic field to be organizedon large scales. In this case, the effect of QED could be the opposite of what hap-pens for edge-on photons: the organized magnetic field could align the polarizationof photons traveling through the magnetosphere, resulting in a larger net observedpolarization.111Chapter 7The Polarization of X-ray PulsarsX-ray pulsars are highly magnetized neutron stars that live in a binary and accreteionized gas from a stellar companion. The pulsating nature of their X-ray emissionwas interpreted quickly after their discovery as resulting from the channeling alongmagnetic field lines of accretion gas onto their magnetic poles [5, 185]. However,it was immediately clear that the high pulse fraction detected was impossible toexplain merely by the presence of isotropically emitting hot spots on the surface ofthe rotating neutron star, and that a strong beaming of the radiation was required[71]. A possible beaming mechanism is naturally given by the strong magneticfield: the cross-sections of the elementary processes of interaction between radia-tion and matter have a strong dependence on the angle between the magnetic fieldand the propagation direction of the photons, and at small angles with respect tothe direction of the magnetic moment one can see deeper in the atmosphere. Ifthe kinetic energy of the infalling material is deposited deep in the atmosphere,then the emission from the hot spots will have a characteristic “pencil” beam pat-tern [15, 49, 71]. As described in § 1.1.2, an alternative model invokes a radiativeshock above the surface of the neutron star, in which the infalling gas is sloweddown considerably by radiation before reaching the surface and an accretion col-umn is formed above the magnetic pole in which the ionized gas is slowly sinking.In this second scenario, the photons escape from the walls of the column and theemission has a “fan” beam pattern [16, 20, 34, 51]. Due to the low resistivity, thedepth to which the plasma penetrates into the dipole field is small compared to the112Figure 7.1: The two emission models for X-ray Pulsars: on the left, the gascan freefall all the way to the neutron star surface, and the kinetic energyof the accretion flow is only released upon the impact with the neutronstar surface; the Comptonized X-rays escape predominantly upwards,and form a so-called “pencil-beam” pattern. If the luminosity is higherthan the critical luminosity, a radiation dominated shock rises above theneutron star surface, forming an extended accretion column (right). Inthis case, photons can only escape through the walls of the column, anda “fan” emission pattern is expected.magnetospheric radius, and the accretion channel is more likely to look like a thinwall of funnel more than a solid, axisymmetic column.The continuum X-ray emission of accreting X-ray pulsars is often described byphenomenological models, including an absorbed power law extending up to∼ 100keV with a roll-over at ∼ 30− 50 keV or a broken power law [59, 172]. Severalattempts have been made to develop spectral models that link the X-ray emissionto the accretion physics [51, 111, 112, 149, 150, 162, 246] but the modelling iscomplicated by the fact that the accretion regions are radiation-dominated, which113means that the radiation transfer is coupled with the hydrodynamics of the flow;by the presence of a relativistic bulk motion in the infalling gas, which in turnsmakes the modeling of Compton upscattering more difficult; and by the strongmagnetic field, which changes all the cross sections for scattering and absorption.All these complications should be addressed self-consistently and the study of thepolarization parameters should be tailored to the spectral formation model. Ofthese attempts, only Me´sza´ros and Nagel [150] and Kii [111] have addressed theproblem of polarization (see § 3.2.4).In the next section, I calculate the polarization signal of X-ray pulsars in thecontext of the currently available models for polarization by Me´sza´ros and Nagel[150] and by Kii [111]. In these models, the emission is coming from a hot slab atthe magnetic pole of the neutron star, and therefore we are considering the “pencilbeam” case. Both models solve the problem of radiative transfer separately forthe two polarization modes (parallel and perpendicular to the magnetic field) andtherefore calculate at the same time the flux and the polarization degree of theemitted radiation. In § 7.1, I take the emission at surface provided by the modelsand I add the effects of gravitational lensing and of vacuum birefringence (seeChapter 4) to find the polarization at the observer.In § 7.2, I will consider a different model of spectral formation, the Beckerand Wolff [20] model (see also § 1.1.2). In this model, which analyzes the emis-sion from an accretion column, in the “fan beam” context, the directional depen-dence of electron scattering is treated in terms of mode-averaged cross sections,and therefore the problems of polarization and of radiative transfer are consideredseparately, and no information is given on the polarization of light. The model,however, predicts a spectrum that fits very well the observed profiles and providesinsights on the properties of the accretion flow. In § 7.2 I show that the polarizationparameters can be calculated independently of the radiative transfer solution, andthat a robust prediction can be made in the context of the model.7.1 Previous models: Me´sza´ros and Nagel and KiiIn this section, I calculate the polarization degree in the context of the models byMe´sza´ros and Nagel [150] and by Kii [111], in the slab geometry. I use the re-114sults of [150] to estimate the total flux from the region. However, [150] do notreport the polarization fraction as function of inclination angle, and therefore I em-ploy the results of [111] to estimate the polarized fraction and the intensity as afunction of direction from the slab. Both Me´sza´ros and Nagel and Kii solve theFeautrier equations for the radiative transfer [153] in the assumption of completeFaraday depolarization, i.e. they assume that the two polarization modes stay dis-tinct as the photons propagate through the slab and they track the 2 modes insteadof the full Stokes vector. In their model, photons are mainly produced by thermalbremsstrahlung, and the polarization of the X-ray signal is driven by the differencein opacities between the two polarization modes.In order to pick the temperature and magnetic field strength, I take as an ex-ample Her X-1, a bright X-ray pulsar with kT ∼ 8 keV [151] and cyclotron energyεc ∼ 38 keV [244]. As the emission in Her X-1 is strongly pulsed, I choose ageometry that results in a large pulsed fraction (an orthogonal rotator).7.1.1 Description of the methodIn the slab geometry, the X-ray emission comes from a slab of uniform temperatureat the polar caps of the neutron star, heated by the infalling gas; in my calculations,I assume the region of emission to comprise about 6 degrees of the stellar surface.Each element on the neutron star surface emits highly polarized radiation, andthe value of polarization at the surface is given by [150] and [111]. In particular,the total intensity from the slab as function of inclination angle with respect to thesurface and of energy is shown in Fig. 1b of [150], while Fig. 1a of the same paperonly shows the flux in the two polarization modes integrated over angles. In orderto get the differential flux for each mode, I use the polarization degree at surfaceshown in Fig. 4b of [111] as function of energy and inclination angle. In this way,I have the flux in the two polarization modes and the value of Q/I at the surface ofthe neutron star.At emission, the direction of polarization is correlated with the direction of themagnetic field. However, the magnetic field orientation varies over the surface ofthe neutron star. Summing the polarized intensities of the 6◦ polar cap slightlyreduces the net polarization. In order to calculate the polarization degree at the115observer, I use a ray-tracing code, initially ignoring the effect of vacuum birefrin-gence, and simply parallel-transporting the polarization vector along the geodesics.In this way, I obtain the “QED-off” polarization degree, which is shown in the leftpanel of Fig. 7.2.When I include QED, I have to calculate the rotation induced by vacuum bire-fringence, and for that I solve the equations of the polarization evolution (eq. 2.64)through the neutron-star magnetosphere using an adaptive Runge-Kutta methodas outlined in [94]. Additionally, I have to consider the fact that all of the pho-tons coming through the atmosphere pass through the vacuum resonance region,in which the linear contribution to the birefringence from QED cancels that of theplasma [98] (see § 5.2.1). In this region, the value of Ωˆ swings from pointingalong a particular direction in the s1− s2−plane up toward s3 and back onto thes1− s2−plane in the opposite direction. As we know from eq. 2.65, if this hap-pens slowly enough, the photon polarization will follow the direction of Ωˆ, andthis is in fact what happens for photons with energies greater than about 350 eV.The polarization state is switched from perpendicular to parallel.7.1.2 ResultsThe effect of vacuum birefringence was already shown in Fig. 5.3, in § 5.1.1, whichdepicts the final polarization states across the image of the neutron star surface, as-suming that the radiation is initially in the extraordinary mode, with and withoutQED. In this section, the effect is similar, but the initial intensities in the two po-larization modes are taken to be the ones from [150] and [111] instead of being100% in X. From Fig. 5.3, one can see that, when the emission is restricted to theregion near the magnetic pole, the effect of QED is subtle: the projected magneticfield direction is well aligned near the pole even without vacuum birefringence inthe left panel and the final polarization in the right panel is also well aligned ingeneral but substantial circular polarization can be generated near the pole, due tothe quasi-tangential effect [237] (see § 5.1.1). Because X-ray polarimeters onlydetect linear polarization, the circularly polarized radiation does not contribute tothe observed polarization fraction, and the net effect of the vacuum birefringenceis to reduce the polarization fraction integrated over the emission region and the116−0.20−0.15−0.10−0.050.000.05Q/I[PolarizationDegree]2 3 4 5 6 7 8Energy [keV]9 km10 km11 km15 km −0.25−0.20−0.15−0.10−0.050.000.050.10PolarizationDegree0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0Energy [keV]9 km10 km11 km12 km15 kmFigure 7.2: The polarization of Her X-1 as function of photon energy usingthe emission models of Kii [111] and Me´sza´ros and Nagel [150], aver-aged over the rotation of the pulsar. A positive value of the polarizationdegree indicates that the polarization direction is perpendicular to theprojection of the rotation axis onto the plane of the sky. The differentcolors represent different stellar radii, as indicated in the legend. Theleft panel shows the energy interval between 2 and 8 keV while theright panel shows the energy interval between 0.2 and 2 keV. In the leftpanel, the dashed lines give the result without vacuum birefringence (wehave reversed the polarization direction in this case for ease of compar-ison, see the text) and the solid lines include the QED effect. A positivevalue of Q/I indicates that the net polarization is perpendicular to theprojected spin axis of the stars in the QED-on case, and parallel in theQED-off case.rotational phase as shown by the left panel of Fig. 7.2.The results are shown in Fig. 7.2 for a range of stellar radii from 9 to 15 km.The solid lines trace the extent of linear polarization including QED, and the dashedlines neglect it. Across the 2–8-keV energy range, the trend in Q/I can be explainedby looking at the opacities in the atmosphere. At low energies, the opacity forphotons in the parallel polarization is larger than for the perpendicular polarization,so the bulk of the radiation emerges in the perpendicular mode. However, as thephoton energy approaches the cyclotron energy (here taken to be ∼ 38 keV), the117opacity for the perpendicular mode increases and one gets more emission in theparallel mode.Because the vacuum resonance reverses the polarization direction for all ofthe photons above about 350 eV, I have switched the sign of the value of Q/I forQED-off case for ease of comparison. Without an independent measurement of theprojection of the spin axis of the star into the plane of the sky, measurements just inthe 2-8-keV band cannot measure this polarization flip. The key effect of the QEDbirefringence in this harder band is to slightly reduce the polarization fraction.I can apply Eq. 2.65 to determine how the conditions in the atmosphere, inparticular the density scale height, determine the critical energy above which thepolarization switches as the radiation passes through the vacuum resonance [98].In the case of Her X-1, I have taken the temperature of the atmosphere to be 8 keV[151]; with the temperature fixed, the density scale height only depends on thecomposition of the atmosphere which is that of the donor star and the surface grav-ity; therefore, the photon energy at which the polarization direction flips dependson the surface gravity and can be used to measure the radius of the star. For thecurves in Fig. 7.2 I have assumed a mass of 1.4 M for the neutron star.7.2 Polarization in the Becker and Wolff modelIn this section, I will present a new model for the polarization parameters in theX-rays, based on the accretion model developed by Becker and Wolff [20].7.2.1 The spectral formation modelIn their 2007 paper [20, B&W07], Becker and Wolff propose a new model forspectral formation in luminous X-ray pulsars that quite successfully reproducesthe phase-averaged spectrum of bright X-ray pulsars as Hercules X-1 (Her X-1).In their model, the ionized gas accreted from the companion star is funneled insidea column at the polar caps of the neutron star. The strong magnetic field keepsthe gas confined inside the column as in a “pipe”, which is however transparent forradiation. Fig. 7.3 shows the geometry of the accretion column: the ionized gas freefalls from the accretion disk along the field lines to the top of the column, wherethe speed of the flow is supersonic; inside the column, radiation pressure slows118down the gas until it comes to rest at the bottom of the column. Seed photonsin the column are produced by a combination of bremsstrahlung, cyclotron andblackbody radiation, and are scattered by electrons through Compton scattering.Blackbody photons are emitted by a thermal mound at the bottom of the column,so that the mound’s surface represent the photosphere for creation and absorptionof photons and the opacity in the rest of the column is given by electron scatteringonly. Bremsstrahlung and cyclotron photons are emitted throughout the column.The observed radiation comes from the walls of the column, in a “fan beam”.In their 2005 papers, Becker and Wolff [18, 19] considered only the effect ofbulk Comptonization, for which photons are upscattered in energy through a first-order Fermi energization, ignoring the effects of thermal Comptonization. Thedifference between “bulk” and “thermal” Comptonization is mainly in the motionof the scattering centers: in the first case, photons gain energy interacting withelectrons that are part of a converging flow, as opposed to the stochastic motion ofscattering centers in the case of thermal scattering. Depending on the temperatureof the electrons in the column and their infalling speed, both effects can be impor-tant. Neglecting thermal effects corresponds to considering a flow in which ther-mal velocities are considerably smaller than the converging bulk velocity, whichseems to be a good assumption for low-luminosity X-ray pulsars like X Persei andGX 304-1 [18, 19], but fails to describe the spectra of bright X-ray pulsars. BulkComptonization alone leads to a steep power law in the hard X-rays, and there-fore the down-scattering of high energy photons due to thermal Comptonization isneeded to explain the low spectral index observed in the 1-20 keV range and thequasi-exponential cut off observed at about 20-30 keV in bright X-ray pulsars. InB&W07 both thermal and bulk effects are included.The inclusion of thermal Comptonization, described mathematically by theKompaneets equations [113], makes the analytic treatment of the transport equa-tions more difficult. For this reason, the authors in B&W07 use an approximatevelocity profile instead of using the exact one employed in their 2005 paper. Theupstream flow above the column is composed of fully ionized hydrogen moving atsupersonic speed, reaching about half the speed of light. Inside the radiation dom-inated column, electrons slow down as they transfer energy to the radiation fieldand stop at the stellar surface. Instead of using the exact solution for the velocity119Figure 7.3: Up: artist rendition of an accreting X-ray pulsar; the accretiondisk is disrupted at the magnetospheric radius and the ionized gas fun-neled to the magnetic poles. Credits: NASA/NuSTAR. Down: the ac-cretion column in Becker and Wolff model (adapted from B&W07).profile derived by Becker [17], the authors in B&W07 use a particular form for thevelocity profile that approximates the exact solution and also makes the transportequation separable in energy and space:v(τ) =−Aτ‖ (7.1)120where A is a constant and τ‖ is the optical depth in the direction parallel to themagnetic field (and the column vertical axis). τ‖ increases vertically, and is equalto zero at the stellar surface. A is calibrated by equating the velocity at the sonicpoint to the exact velocity, which yieldsA = 0.20(M∗M)(R∗10km)−1ξ , ξ =pir0mpcM˙(σ‖σ⊥)1/2(7.2)where M∗ and R∗ are the mass and radius of the neutron star, r0 is the radius of thecolumn, mp is the mass of the proton, c is the speed of light, M˙ is the accretion rate,and σ‖ and σ⊥ are the cross sections for photons travelling parallel and perpendic-ular to the magnetic field. These quantities are the means of those introduced inChapter 3 averaged over photon energy and polarization state.Another approximation in the B&W07 model is given by the treatment of thescattering cross sections. Since radiation pressure is dominant in the column, thedynamical structure of the flow is closely tied to the spatial and energetic distri-bution of the radiation, making the coupled radiation-hydrodynamic problem ex-tremely complex. For this reason, in B&W07 the directional dependence of theelectron scattering is treated in terms of the constant, energy- and mode-averagedcross sections σ‖ and σ⊥. In particular, σ⊥ is set≈ σT , the Thomson cross section,while σ‖ is expressed in terms of the accretion rate M˙, of the radius of the accretioncolumn r0, and of the dimensionless parameter ξ , which determines the importanceof the escape of photons from the accretion column in the radiation transfer equa-tion. Both r0 and ξ are free parameters, recovered by fitting the model to thespectrum. The expression used is given in eq. 83 of B&W07:σ‖ =(pir0mpcM˙ξ)2 1σ⊥(7.3)I will not use these averaged cross sections in my calculation of the polarizationparameters, as I will calculate angle- and energy-dependent cross sections for thedifferent modes in § 7.2.2.I will base my calculations on the accretion model of B&W07, and use thefitted parameters for the model obtained for Her X-1 in [244]; specifically:121• the radius of the accretion column r0 = 107 m;• the strength of the magnetic field B = 4.25×1012 G;• the dimensionless parameter ξ = pir0mpc/[M˙(σ‖σ⊥)1/2] = 1.36;• the dimensionless constant A = 0.38, defined in eq. 7.1;• the height of the accretion column zmax = 6.6 km (see Fig. 7.3), see also nextsection.What is the height of the column?The height of the accretion column in B&W07 is found by equating the approxi-mate velocity at the top of the column to the local free-fall velocity(2GM∗R∗+ zmax)1/2= cAτmax where τmax =( σ‖σ⊥)1/4(2zmaxAξ r0)1/2(7.4)Thenzmax =R∗2[(1+C1)1/2−1] where C1 = 4GM∗r0ξAc2R2∗(σ⊥σ‖)1/2(7.5)which gives, for the fitted parameters in [244], zmax ∼ 6.6 km.Imposing the free-fall velocity at the top of the column corresponds to assum-ing that the radiative shock extends to the entire column, where the electron veloc-ity smoothly changes from free-fall to zero. Another possibility would be assuminga strong, adiabatic shock in a thin layer at the top of the column and then a radiativeflow in the rest of the column. In an adiabatic, radiation-dominated shock, the jumpin velocity is never less then 1/7 [17], and therefore in this case I would impose thevelocity at the top of the column to be 1/7 (or less) of the local free-fall velocityinstead of being equal:17(2GM∗R∗+ zmax)1/2& cAτmax (7.6)122which yields zmax . 1.4 km. The two assumptions give different results for thepolarization parameters, as I will show in § 7.2.3.7.2.2 Polarization at the sourceInside the accretion column, the seed photons for Comptonization are a mixture ofbremsstrahlung, cyclotron and blackbody photons. Blackbody photons are emittedin the thermal mound at the bottom of the column, while bremsstrahlung and cy-clotron photons are emitted throughout the entire column. The main contributionto the seed photons come from bremsstrahlung [20, 244]. Following the formalismintroduced in § 2.2.1, I can take the average polarization of bremsstrahlung photonsinside the column as my incident polarization, with Stokes parameters (I,Q,V ),and then apply the matrix in eq. 2.46 to calculate the final polarization. But first ofall, I need to know the average number of scatterings that a photon undergoes inthe column as a function of energy.Polarization inside the columnFrom B&W07, I can calculate the optical depth for photons moving horizontallyoutward of the column, called τ⊥ in the paper. First, I need to calculate the densityprofile from the accretion rate M˙, kept as a constant fixed parameter of the model.From eq. 19 of B&W07:M˙ = pir20ρ|v|= pir20ρAτ‖c (7.7)where ρ is the density of the gas and where I used eq. 7.1 to express the electronsvelocity v. I can therefore calculate the perpendicular optical depth, knowing thatthe opacity is dominated by electron scattering:τ⊥ =r0ρσ⊥mp∼ r0ρσTmp=M˙σTmppir0Aτ‖c(7.8)where I have employed the fact that the scattering cross section of photons movingperpendicular to the magnetic field is close to the Thomson cross section σT atall energies except very close to the cyclotron energy, where it is even higher (seeFigure 7.5). τ‖ increases in the vertical direction and is of the order 1 at the top123of the column. Employing the fitted parameters from [244] I find that τ⊥ ∼ 500 atthe top of the column, and therefore greater than 500 throughout the column. Thisyields an average number of scatterings per photonNsc ∼ τ2⊥ & 250,000 (7.9)For Compton scattering, the energy transfer for a single scattering is given by [195]∆ε/ε ∼ (γ2−1). 0.15 (7.10)where γ = 1/√1−β 2 is the Lorentz factor of the scattering center, the electron,and I have used β = 0.5, which is the velocity of the electrons at the top of thecolumn.From the estimates in eq.s 7.9 and 7.10, it is clear that an average photon hasto undergo many scatterings before it can escape the column and that in the finaltens of scatterings, the energy of the photon is very close to its final energy, i.e. itsenergy when it finally manages to escape from the column. Thus, during the finaltens of scatterings of each photon, the elements of the scattering matrix in eq. 2.46will remain approximately unchanged.Multiplying a vector by the same matrix many times brings the vector close tothe matrix’s eigenstate with the largest eigenvalue, unless the vector itself is in anorthogonal eigenstate. Depending on the magnitude of the ratio between the largesteigenvalue and the rest, this process takes relatively few interactions. It is easy tosee that, except for energies very close to the cyclotron line, the largest eigenvalueof the scattering matrix is orders of magnitude higher than the other eigenvalues.For this reason, I can safely assume that, independently of the initial polarizationstate of the photon, its Stokes vector will be in the matrix’s predominant eigenstatejust after a few scatterings.Therefore, the predominant eigenstate of the scattering matrix in eq. 2.46 rep-resents the polarization state of radiation inside the column, in the rest frame ofthe electrons. In the left panels of Figure 7.4 the Stokes parameters for the Comp-tonized radiation inside the column are shown a as a function of energy and angleθ , which is the angle with respect to the magnetic axis, zˆ. The results are shownfor θ < pi/2 and are specular for θ > pi/2. As expected, except for very small124100 101 102Energy [keV]0.00.10.20.30.40.50.60.7I0.00.20.40.60.81.01.21.4100 101 102Energy [keV]0.000.050.100.150.200.250.30I0.00.20.40.60.81.01.21.4100 101 102Energy [keV]1.000.750.500.250.000.250.500.751.00Q/I0.00.20.40.60.81.01.21.4100 101 102Energy [keV]0.80.60.40.20.00.20.40.60.8Q/I0.00.20.40.60.81.01.21.4100 101 102Energy [keV]1.00.80.60.40.20.0V/I0.00.20.40.60.81.01.21.4100 101 102Energy [keV]0.40.30.20.10.00.10.20.3V/I0.00.20.40.60.81.01.21.4Figure 7.4: Average Stokes parameters. The left panels depict the polariza-tion parameters of radiation inside the column, while panels on the rightrepresent the polarization after the photons have gone through the regionof last scattering and left. The calculation does not include beamingeffects. From top to bottom: intensity I (arbitrary units), linear polar-ization fraction Q/I and circular polarization fraction V/I against theenergy of the photons. The color code represents the angle with respectto the magnetic field θ .angles, photons are always nearly linearly polarized in the ordinary mode (positiveQ) for energies lower than the cyclotron line (∼ 37 keV), while photons aroundthe cyclotron line present a mixture of extraordinary and circular polarization. Atthe cyclotron line, photons propagating along the magnetic field (at small θ ) have125no linear polarization (the two linear polarization modes are equally perpendicularto the magnetic field), and have a strong circular polarization due to resonant cy-clotron scattering (electrons can be excited to the second Landau level). Photonspropagating at θ ∼ pi/2, on the other hand, can resonantly scatter only if in theX-mode, and therefore Q/I is equal to −1.This picture represents the polarization state of photons propagating inside thecolumn, but in order to find the average polarization parameters of photons leav-ing the column, I have to take into account the difference in the cross sectionsfor the different polarization modes, and therefore the difference in the volume ofthe optically thin region close to the column’s walls. Since the cross section ismuch smaller, I expect the volume of the region of last scattering for extraordinaryphotons to be much larger than for ordinary photons, reducing the extent of linearpolarization at all energies.Region of last scatteringIn order to find the polarization of light escaping from the accretion column, I haveto consider the difference in volume of the region of last scattering between thedifferent polarization modes. The volume of the optically thin region close to theexternal wall of the column is proportional to the square of the sine of the incidentangle divided by the total cross section of the polarization mode at hand, whichwill also depend on the incident angle α .I first calculate the cross sections for the different modes by applying the matrixto a polarization vector completely polarized in the mode under consideration andwith a certain distribution in incoming angle α and then by averaging over theoutgoing angle θσ = [...]∫ −11I′(α,θ)d(cosθ) . (7.11)In this way, I obtain the angle and energy dependent cross sections σ‖ for theordinary mode (1,1,0), σ⊥ for the extraordinary mode (1,-1,0), and σ+ and σ− forthe two circular polarization modes (1,0,1) and (1,0,-1) respectively (see also [85])126100 101 102Energy [keV]10 310 1101103105+/T0.00.20.40.60.81.01.21.4100 101 102Energy [keV]10 310 1101103105/T0.00.20.40.60.81.01.21.4100 101 102Energy [keV]10 310 1101103105/T,/T0.00.20.40.60.81.01.21.4Figure 7.5: These plots show the dependence of the cross sections on theincident angle α and on the energy of the photons. Top left: σ+/σT .Top right: σ−/σT . Bottom: in this plot both σ‖/σT and σ⊥/σT areshown. σ⊥ does not depend on angle and it is shown as the solid blackline, while σ‖ is color-coded with respect to the angle. When α = 0,σ‖ = σ⊥.σ‖ = σT[sin2α+ cos2αx2+1(x2−1)2](7.12a)σ⊥ = σTx2+1(x2−1)2 (7.12b)σ+ =12σT[sin2α+(x2+1)(1+ cos2α)−4xcosα(x2−1)2](7.12c)σ− =12σT[sin2α+(x2+1)(1+ cos2α)+4xcosα(x2−1)2](7.12d)where σT is the Thomson cross section. The different cross sections are shown inFigure 7.5. They all seem to diverge at the cyclotron line; however, for x very close127to 1, the energy transfer from photons heats up the electrons and damping effectsbecome important [148]. The lower panel depicts both σ⊥ (black solid line) andσ‖ (color coded with α) and we can see that they become equal when α = 0, i.e.when photons propagate along the magnetic field, as expected.If I now indicate with (I,Q,V) the average polarization state inside the columnand with (I’,Q’,V’) the polarization of the outgoing radiation I can writeQ = O−X (7.13)where O and X are the intensities of the ordinary and the extraordinary modesinside the column. The outgoing intensities will beO′ =12(Q+ I)V‖ (7.14)X ′ =12(I−Q)V⊥ (7.15)where Vi ∝ sinθ/σi. I can therefore write the Stokes parameters of the radiationcoming out of the column asI′ =12(Q+ I)V‖+12(I−Q)V⊥ (7.16)Q′ =12(Q+ I)V‖−12(I−Q)V⊥ (7.17)V ′ =12(V + I)V+− 12(I−V )V− (7.18)The right panels of Figure 7.4 show the average Stokes parameters after the radi-ation has gone through the region of last scattering. Comparing to the left panels,we can see that the linear polarization is reduced at low energy even for high anglesbecause the low value of σ⊥ favour the emission of extraordinary photons. Inten-sity is drastically lowered at high energies because all the scattering cross sectionsbecome divergent close to the cyclotron line. For the same reason, radiation at thecyclotron line is completely unpolarized.1280.0 0.5 1.0 1.5 2.0 2.5 3.0′0.0000.0010.0020.0030.0040.005Flux, arbitrary units0.0 0.5 1.0 1.5 2.0 2.5 3.0′0.000000.000020.000040.000060.00008Flux, arbitrary unitsFigure 7.6: The effect of beaming on flux for β = 0.4 and different photonenergies. Solid blue line: flux, without beaming; solid orange line: flux,with beaming. Left panel: photon energy 1 keV; right panel: photonenergy 29 keV.Relativistic beamingThe previous calculations were performed in the instantaneous rest frame of theelectrons. Electrons are flowing down the column with a velocity that goes fromabout 0.5 c at the top to zero at the bottom. For this reason, the emission from thecolumn, especially from the top, where most of the radiation is coming from, willbe beamed. If I indicate with a prime the quantities after beaming, I getθ ′ = cos−1(cosθ −β1−β cosθ)(7.19)E ′p =Epγ(1+β cosθ)(7.20)I′ = I(E ′pEp)3(7.21)where Ep is the energy of the photons and β = Aτ‖ is the speed of the electronsdivided by c.Figure 7.6 shows the effect of beaming on flux for β = 0.4 and for two photonenergies, 1 keV and 29 keV: radiation is strongly beamed toward the surface ofthe star (high θ ′). Figure 7.7 shows the average Stokes parameters after beaming,always for β = 0.4. Please notice that the angle θ ′ in the colour bar now doesnot go from 0 to pi/2, as in the previous plots, but from 0 to the angle wherethe intensity peaks, at about 2.7 radians (see also Figure 7.6). The main effect129100 101 102Energy [keV]0.80.60.40.20.00.20.40.60.8Q/I0.51.01.52.0′100 101 102Energy [keV]0.40.30.20.10.00.10.20.3V/I0.51.01.52.0′100 101 102Energy [keV]0.00.10.20.30.40.50.6I0.51.01.52.0′Figure 7.7: Average Stokes parameters after beaming, for β = 0.4. Top left:linear polarization fraction Q′/I′; top right: circular polarization frac-tion V ′/I′; bottom: intensity I′ (arbitrary units). The color code rep-resents the angle with respect to the magnetic field after beaming θ ′.Please notice that the angle θ ′ goes from 0 to the angle where the inten-sity peaks, at about 2.7 radians (see also Figure 7.6).of beaming is shifting the angle where intensity peaks toward higher angles (andtherefore toward the surface of the star) and to move the cyclotron line to higherenergies at higher angles.Emission from an orthogonal rotatorI can finally sum the contribution of the entire column and find the polarization pa-rameters at emission. In the B&W07 model, most of the flux is coming from the topof the column, where τ⊥ is smaller. Throughout the column, the relation between130flux, τ‖ and z (vertical direction coordinate along the column) is the followingL(z)Ltot=(zzmax)3/2=(ττmax)3(7.22)In order to integrate the emission from the column, I divide the column in fractionsof equal flux, where I calculate the beaming, and sum all the contributions.I consider the presence of two columns at the two magnetic poles of the neutronstar. We expect to see two columns if the magnetic field has a bipolar structure,except if the field is stronger at one pole, in which case the stronger pole wouldswipe the gas from the accretion disk at a radius outside of the reach of the weakerpole. In this section, I assume an orthogonal rotator, with the magnetic field alwayslying in the same plane with the line of sight. From the orthogonal rotator, theresults for any other rotational geometry can be inferred.I am still calculating the polarization parameters at emission, so I do not includeany light bending, and therefore I can use θ as the angle between the line of sightand the magnetic field and also as the phase (I will now abandon the prime). Thereis always an entire column visible and the other one is partially covered by the star.If zmax is the height of the column, for each θ the part of the back column that wesee is given byzmax−R∗(1sinθ−1).One column is at θ and one column is at pi−θ . The radiation pattern with rotationphase is shown in the upper plots of Figure 7.8 for two photon energies, 1 keV and29 keV. The sudden rise in intensity at about pi/5 is due to the fact the back columnstarts to be visible and, since the emission from a column is highly beamed towardthe surface of the star, the back column is the one dominating the emission. Thesame effect can be seen on the average Stokes parameters in Figure 7.9.7.2.3 Polarization at the observerNow that I have calculated the emission pattern and polarization at the source, thenext step is to analyze how polarization changes as photons propagate to the ob-server. Specifically, I have two additional effects to take into account: gravitational1310.0 0.5 1.0 1.5 2.0 2.5 3.001234I0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.20.40.60.81.01.21.4I0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.10.20.30.40.50.60.70.8Q/I, V/I0.0 0.5 1.0 1.5 2.0 2.5 3.00.60.40.20.00.20.4Q/I, V/IFigure 7.8: Phase pattern for intensity and polarization fractions for 2columns without light bending. Upper panels: intensity I (arbitraryunits). Lower panels: linear polarization fraction Q/I solid orange line,circular polarization fraction V/I solid orange line. Left panels: photonenergy 1 keV; right panels: photon energy 29 keV.lensing and the effect of vacuum birefringence (see also Chapter 4).Gravitational lensingBecause neutron stars are very compact, their strong gravitational field affects thepropagation of light around them, and general relativity needs to be included whencalculating the photons’ path. In general relativity, the path of light is bent bygravity, and therefore the image of the star results distorted at the observer. Becauseof light bending, the angle between the magnetic field and the photon momentumat emission, which I call θ , is now different from the phase, that from now on Iwill call φ , and from the angle between the vertical direction of the column (zˆ) andthe line of sight, that I will call ψ (see Fig. 7.10). The relation between θ and ψ in132100 101 102Energy [keV]0.60.40.20.00.20.40.60.8Q/I0.20.40.60.81.01.21.4100 101 102Energy [keV]0.40.30.20.10.00.10.20.3V/I0.20.40.60.81.01.21.4100 101 102Energy [keV]01234I0.20.40.60.81.01.21.4Figure 7.9: Average Stokes parameters from summing up the two columnwith energy and phase. Top left: linear polarization fraction Q/I; topright: circular polarization fraction V/I; bottom: intensity I. The colorcode represents the phase angle θ .general relativity is given by [23]sinθ =bR√1− RgR(7.23)ψ =∫ ∞R−uψurdr =∫ ∞Rdrr2[1b2− 1r2(1− Rgr)]−1/2(7.24)where Rg = 2GM∗/c2 is the gravitational radius of the neutron star, b is the impactparameter and R is the distance of the emission region from the center of the star.In [23], R is the radius of the star, while, in the case of emission from a column,R is the radius of the star plus z, the height along the column under consideration.For the column in the front of the star, I have to integrate between R = R∗+ zand infinity to get ψ . For the column in the back, I have to be more careful. For133Figure 7.10: Lensing in the neutron star gravitational field. θ is the anglebetween the vertical direction of the column (zˆ) and the photon mo-mentum at emission, ψ is the angle between zˆ and the line of sight ,and φ is the rotation phase of the star.each light ray, labelled by the impact parameter b, I have to calculate the minimumdistance from the center of the star of the light path, defined by eq.7.23 whensinθ = 1, and integrate from R = R∗+ z to the minimum distance and then fromthe minimum distance to infinity.Depending on the compactness of the star, it is possible to see both sides ofone column, from the front and from the back, because of light bending. Thus, foreach phase φ , I have to sum the contribution from the front column at ψ = φ andat ψ = 2pi−φ and from the back column at ψ = pi−φ and at ψ = pi+φ , makingsure of which part of the two columns is not blocked by the neutron star itself.Furthermore, light bending magnifies the back column, and this effect, togetherwith relativistic beaming, means that most of the emission comes from the backcolumn. The huge magnification that we can see in Figures 7.11 and 7.12 at φ ∼ 0and at φ ∼ pi hinges on the very particular geometry that I am considering: it re-quires the column to be pointing essentially directly away from us, creating anEinstein ring around the star. The result of this effect is that any pulsed fractioncan be achieved by this model, just by varying the geometry of the star. In this par-ticular geometry, the back column is magnified at φ ∼ 0, and the linear polarizationfraction at low angles is still high for low energies because it is dominated by the1340.0 0.5 1.0 1.5 2.0 2.5 3.00100200300400500600700800I0.0 0.5 1.0 1.5 2.0 2.5 3.0050100150200250I0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.10.20.30.40.50.60.70.8Q/I, V/I0.0 0.5 1.0 1.5 2.0 2.5 3.00.70.60.50.40.30.20.10.00.1Q/I, V/IFigure 7.11: Phase pattern for intensity and polarization fractions for 2 ac-cretion columns with light bending. Upper panels: intensity I. Lowerpanels: linear polarization fraction Q/I solid orange line, circular po-larization fraction V/I solid orange line. Left panels: photon energy 1keV; right panels: photon energy 29 keV.beamed emission from the back column, for which θ is high.Before I move on to address the effects of QED, I would like to stop and analyzethe effect of lowering the height of the accretion column (see § 7.2.1). In particular,I will consider zmax = 6.6 km and zmax = 1.4 km. In order to better see the effect,it is simpler to show the case with only one column instead of two.From Figure 7.13 we can immediately see that the pulse fraction is different inthe two cases: in the case with zmax = 1.4 km, the back column is blocked by thestar at φ = pi and therefore there is no huge magnification as in the zmax = 6.6 kmcase, for which the back column is always in sight. The same effect can be seenin the upper panels of Figure 7.14. Also from Figure 7.14, we can see that thereis little effect on the polarization fraction; the main effect is due to a shift in thecyclotron depolarization feature with angle, which is more pronounced in the case135100 101 102Energy [keV]0.60.40.20.00.20.40.60.8Q/I0.20.40.60.81.01.21.4100 101 102Energy [keV]0.30.20.10.00.1V/I0.20.40.60.81.01.21.4100 101 102Energy [keV]0200400600800I0.20.40.60.81.01.21.4Figure 7.12: Average Stokes parameters for 2 columns with light bending.Top left: linear polarization fraction Q/I; top right: circular polariza-tion fraction V/I; bottom: intensity I. The color code represents thephase angle φ , which goes from 0 to pi/2.of a highest column because of the stronger beaming from the fast electrons at thetop of the column.QED: the quasi-tangential effectIn Chapter 5, I introduced how the effect of vacuum birefringence can affect thepolarization of X-ray radiation from neutron stars. For the geometry that I amconsidering in this chapter, the magnetic field is always in the same plane as theemission column and of the line of sight, and therefore the effect of vacuum bire-fringence cannot affect the direction of polarization. However, it can still destroysome of the linear and circular polarization due to the so-called quasi-tangential(QT) effect. In § 5.1.1 I introduced the QT effect on the emission coming from apolar cap, which is similar to the case in this chapter, with the difference that in1360.0 0.5 1.0 1.5 2.0 2.5 3.00255075100125150175I0.0 0.5 1.0 1.5 2.0 2.5 3.001020304050I0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.10.20.30.40.50.60.70.8Q/I, V/I0.0 0.5 1.0 1.5 2.0 2.5 3.00.00.10.20.30.40.50.60.70.8Q/I, V/IFigure 7.13: Phase pattern for intensity and polarization fractions for 1 accre-tion column with different heights and photon energy at 1 keV. Upperpanels: intensity I. Lower panels: linear polarization fraction Q/Isolid orange line, circular polarization fraction V/I solid orange line.Left panels: zmax = 6.6 km; right panels: zmax = 1.4 kmthis case the radiation is coming from the walls of the column, and therefore I haveto consider only the border of the polar cap instead of the full area.Following the approach in § 5.1.1, I can calculate the QT effect from the ratioWt/Wem for the radiation coming from the columns, where Wt is the width of theregion in which the QT effect is important, given by eq. 5.5 and Wem is the widthof the emission region. The strength of the effect depends on how far from thestar the light ray crosses the QT region, as the magnetic field scales as the distancefrom the star to the power of −3. For this reason, Wt/Wem depends on both thez coordinate along the column of the emitting region and on the position of thecolumn with respect to the line of sight (which for 1 column is simply indicatedby φ ). Additionally, Wt/Wem decreases with energy to the power of 1/3 (eq. 5.5).These dependencies are shown in Figure 7.15. On both panels, the x axis represents137100 101 102Energy [keV]0255075100125150175200I0.51.01.52.02.5100 101 102Energy [keV]0102030405060I0.51.01.52.02.5100 101 102Energy [keV]0.80.60.40.20.00.20.40.60.8Q/I0.51.01.52.02.5100 101 102Energy [keV]0.80.60.40.20.00.20.40.60.8Q/I0.51.01.52.02.5100 101 102Energy [keV]0.40.30.20.10.00.10.20.3V/I0.51.01.52.02.5100 101 102Energy [keV]0.40.30.20.10.00.10.20.3V/I0.51.01.52.02.5Figure 7.14: Average Stokes parameters for 1 column and different heights.The left panels depict the polarization parameters of radiation from1 column with zmax = 6.6 km, while panels on the right represent acolumn with zmax = 1.4 km. From top to bottom: intensity I, linearpolarization fraction Q/I and circular polarization fraction V/I againstthe energy of the photons. The color code represents the pulsar phaseφ , and it goes from 0 to pi .the phase φ and the y axis is Wt/Wem. The horizontal lines are the same as thevertical lines in Figure 5.4: the beige lines delimit the region where the QT effectis important and the red line indicates the value at which it is the strongest. In theleft panel, radiation is coming from a region of the column at z = 0.4 zmax and the1380.5 1.0 1.5 2.0 2.510 1100101Wt/Wem0.00.20.40.60.81.0Energy [keV]0.5 1.0 1.5 2.0 2.510 1100101Wt/Wem0.00.20.40.60.81.0z/z maxFigure 7.15: Both panels show Wt/Wem versus φ . The horizontal lines arethe same as the vertical lines in Figure 5.4: the beige lines delimitthe region where the QT effect is strong and the red line indicates thevalue at which it is the strongest. Left panel: photons are coming fromz = 0.4zmax = 2.6 km above the stellar surface; the different colorsrepresent different photon energies. Right panel: the photon energy is30 keV; the different colors represent different position in the column,z, of the emitting region.different colors depict photons of different energies; we can see the dependenceon energy, with higher energies being more affected by the QT propagation effect.On the right panel, photons have the same energy, 30 keV, but come from differentaltitudes along the column, with yellow lines representing photons coming fromthe top of the column, and blue lines coming from the bottom. The lower parts ofthe column are more affected by the QT effect, but are blocked by the star at highφ (in the plot, it is shown by Wt/Wem dropping abruptly to zero).I now can finally calculate the effect of QED on the total polarization from thestar. The intensity is not affected by the QT crossing, while the linear and circularpolarization are. In particular, the circular polarization of each photon receivesa random rotation, completely destroying the average circular polarization. Theeffect on linear polarization can be obtained from the calculations above and theresults are shown in Figure 7.16 for a 6.6 km column, for both the 1 column andthe 2 columns case.139100 101 102Energy [keV]0.80.60.40.20.00.20.40.60.8Q/I0.51.01.52.02.5100 101 102Energy [keV]0.60.40.20.00.20.40.60.8Q/I0.51.01.52.02.5100 101 102Energy [keV]0.60.40.20.00.20.40.60.8Q/I0.20.40.60.81.01.21.4100 101 102Energy [keV]0.60.40.20.00.20.40.60.8Q/I0.20.40.60.81.01.21.4Figure 7.16: Average linear polarization fraction for zmax = 6.6 km withoutand with QED. The upper panels depict the 1 column case and thelower panels the 2 column case. The left panels depict the linear po-larization fraction without QED (same as in Figures 7.14 and 7.12),while the right panels show the effect of QT crossing. The color coderepresents the pulsar phase φ , and it goes from 0 to pi for the 1 columncase and from 0 to pi/2 for the 2 columns case.7.2.4 ResultsIn this section, I have calculated the polarization pattern for the radiation of a brightX-ray pulsar as Her X-1, in the context of the Becker and Wolff [20] model. In themodel, accretion occurs via a column (or two columns) at the magnetic pole (poles)of the neutron star. In the accretion column, the opacity is dominated by electronscattering and the photosphere for free-free absorption resides at the bottom of thecolumn, at the top of the so-called thermal mound. The production of polarizedradiation in the column is dominated by the strong magnetic field, but other effectshave to be taken into account to calculate the observed polarization: relativisticbeaming, gravitational lensing and QED.140The very high average number of scatterings per photon (∼ 250,000) that ispredicted by the B&W model leads to an average polarization state inside the col-umn that is determined uniquely by the energy of the photon and by the strength ofthe magnetic field, and that is independent of the initial polarization of the photon(Fig. 7.4, left panels). At low energies (far from the cyclotron line) the averagephoton is nearly 100% polarized in the ordinary mode, except for photons propa-gating in a direction almost parallel to the magnetic field (θ = 0). In the directionparallel to the magnetic field (and to the column axis) there is almost no flux, asthe intensity peaks in the direction perpendicular to the column walls (θ = pi/2).Close to the cyclotron line, there is an inversion in the polarization direction (sothat the linear polarization fraction goes through a zero), and at the cyclotron linephotons are mostly polarized in the extraordinary mode. At low angles, close to thecyclotron line, circular polarization is predominant, as photons traveling parallel tothe magnetic field can resonantly scatter off electrons, that receive enough energyto jump to the second Landau level. The circular polarization fraction decreaseswith θ .As photons escape the column, the difference in scattering cross section be-tween photons polarized in the different modes changes the picture (Fig. 7.4, rightpanels). At low energies, the scattering cross sections for light polarized in the Oor in the X mode differ by several orders of magnitude (except for photons prop-agating parallel to the magnetic field, for which they are equal, see Fig. 7.5). Forthis reason, photons in the extraordinary mode can escape freely, while photonsin the ordinary mode are trapped. This difference causes the linear polarizationdegree to drop to 80% at low energies. Because all the cross sections diverge in asimilar way at the cyclotron energy, photons at the cyclotron energy are completelyunpolarized. The increase of all the cross sections close to the cyclotron line alsoreduces the emission at high energy (the intensity drops sharply above 20 keV).The right plots of Fig. 7.4 show the polarization parameters of photons comingout of the column in the frame of the accreting gas. In order to calculate the param-eters in the frame of the observer, the effect of relativistic aberration and beamingis important, especially at the top of the accretion column, because the electronsin the gas have a bulk downward speed that is as high as half the speed of light atthe top of the column and decreases as the gas approaches the stellar surface. The141radiation scattered by relativistic electrons is strongly beamed downward (Fig. 7.6)and the features described in the previous paragraph are shifted in energy by anamount that increase with the emission angle, (see Fig. 7.7 for an electron velocityof ∼ 0.4 c).The amount of relativistic beaming depends on the velocity of the electronsand therefore changes along the column. Moreover, if two columns are presentat the two magnetic poles, there is always one column that is completely visibleand a column that is partially blocked by the neutron star. In this chapter, I haveconsidered an orthogonal rotator, with the columns always in the same plane withthe line of sight. Fig. 7.8 shows the intensity and the polarization pattern for the twocolumns, without light bending for now, at the different rotation phases. When onlythe front column is visible (θ ∼ 0,pi), the emission is low because the relativisticbeaming in the column is collimating the emission toward the neutron star surface.For the same reason, as soon as the back column starts to be visible (at aboutθ ∼ pi/5), the intensity jumps and the emission is completely dominated by theback column. The same effect can be seen in Fig. 7.9.Fig. 7.9 shows the Stokes parameters as seen by the observer if there were nogravitational lensing. Gravitational lensing bends the path of light and distorts theimage of the star. An important consequence is that part of the back of the star’ssurface becomes visible, and, in the case of the two columns, the back column isvisible even close to φ ∼ 0, pi . If the height of the column is ∼ 7 km, as predictedby the B&W07 model, the back column is seen at all phases, and when it is exactlyin the opposite direction with respect to the line of sight (φ = 0, pi), gravitationallensing generates a huge magnification, as it projects the column in an Einsteinring around the star (see Fig. 7.11). Of course, if the back column is not perfectlyaligned with the line of sight, the effect is reduced, and therefore, depending on thegeometry of the system, very large pulse fractions can be achieved.In § 7.2.1, I showed how a different assumption on the velocity profile insidethe column predicts a shorter column: zmax = 1.4 km instead of zmax = 6.6 km.The effect of having a smaller column is principally seen in the pulse fraction, as asmaller column cannot be seen at φ = pi and therefore the huge magnification of theback column does not occur (see Fig. 7.13). However, the effect on the polarizationfraction is subtle, and it mainly consists of a reduction of the energy shift due to142relativistic beaming (see Fig. 7.14).The final effect to be considered is the effect of QED, and specifically theso-called quasi-tangential effect. Due to the birefringence of the vacuum, whena photon crosses a region where its momentum is nearly aligned with the localmagnetic field (the QT region), the polarization direction of the photon can rotate.The overall effect is a reduction of the linear polarization fraction and a completedestruction of the circular polarization. The final linear polarization fraction forthe 1 and 2 columns possibilities is shown in Fig. 7.16.143Chapter 8Polarization of Magnetars SoftEmissionMagnetars are isolated neutron stars that are powered by their magnetic field. Asseen in § 1.1.3, magnetars were discovered because of their strong bursting activity;however, many magnetars are seen in a quiescent state as persistent X-ray sources,and their spectra are interpreted as a mixture of thermal and magnetospheric emis-sion. In this chapter, I derive the polarization in the soft X-rays (0.5–10 keV) ofpersistent magnetars in quiescence, as for example 4U 0142+61, in the context ofdifferent physical models.As mentioned in § 1.1.3, between outbursts, magnetars display persistent orslowly decaying X-ray emission, with luminosities L ∼ 1034− 1035 erg s−1. TheX-ray spectrum is characterized by two peaks, with similar luminosities. The softspectrum peaks at about 1 keV and is usually interpreted as thermal emission fromthe neutron star surface. The thermal emission is thought to be reprocessed in theatmosphere and/or in the magnetosphere, because the soft emission can be wellfitted with an absorbed blackbody with an excess above the peak (between 1 and10 keV). The excess can be described by a steep power law with photon index∼ 2− 4, or by a second, hotter blackbody component, usually associated withemission from a hot-spot. The second, higher peak is above 100 keV. The emissionbetween 10 and 100 keV has a positive small spectral index, of about 1-1.5, and isvery pulsed. Fig. 8.1 shows the spectrum of the magnetar 4U 0142+61, with data144100 101 102Energy [keV]10 410 310 210 1Ef E[keV2 cm2 s1 keV1 ]BB (abs)pl1 (abs)pl2 (abs)totalFigure 8.1: The figure shows the spectrum of the magnetar 4U 0142+61. Thespectral data was acquired by Tendulkar et al. [221] with Swift and NuS-TAR. A phenomenological fit is shown, as the sum of an absorbed black-body and two absorbed powerlaws.from Swift and NuSTAR [221]. 4U 0142+61 presents clearly in its spectrum thethree main features of a magnetar spectrum, which are shown as phenomenologicalfits.8.1 Thermal emission: Lloyd’s atmospheresThe soft X-ray emission is interpreted as thermal emission coming from the hotneutron star surface, heated by the magnetic field decay inside the neutron star[92, 227]. The thermal peak, at about 0.5 keV, is usually fitted by a blackbody.However, many studies have shown that the emission from a real atmosphere canbe quite different from a blackbody, and that the temperature of the atmosphereis usually overestimated by blackbody-fitting [132]. In this chapter, I will use themodels by Lloyd [95, 96, 131] to simulate the spectrum and polarization of thethermal emission.Lloyd’s method is very efficient at computing light-element (hydrogen and/orhelium), plane-parallel atmospheres in radiative equilibrium in the limit of com-145plete ionization, and is extensible to partial ionization. Also, in the code, the di-rection of the magnetic field is allowed to vary from the vertical direction, and theeffects of vacuum and plasma birefringence are included self-consistently.In the models, the atmosphere is assumed to be in hydrostatic equilibrium (anybulk motion is neglected). The pressure at any depth is the sum of ideal gas pres-sure, radiation pressure and non-ideal effects rising from Coulomb interactions inthe ionized plasma. The ideal gas pressure includes the contribution from the de-generate pressure of electrons. In a strongly magnetized plasma though, electronsare forced into Landau levels, and the phase space volume occupied by the electrondistribution is small; therefore, the onset of degeneracy occurs at higher densitiescompared to a weakly-magnetized plasma and the degeneracy pressure contributesfor less than ∼ 4% even in the deepest layer.The principal opacity sources in the ionized plasma are Thomson scattering andfree-free absorption. The presence of a strong magnetic field creates a preferredorientation to scattering and absorption processes, modifying the cross sections inthe two modes and generating a finite polarization in the propagating radiation,as described in § 3.2.1. Cyclotron resonances are treated by the self-consistentinclusion of ions and vacuum in the plasma dielectric.The X-ray spectrum and polarization are found by iteratively solving the ra-diative transfer equations over a mesh in energy, polar angles and depth. The codeassumes a plane-parallel atmosphere, which is a very good approximation sincethe atmosphere is incredibly thin compared to the radius of the star (centimeterscompared to kilometers). However, the magnetic field varies in magnitude and di-rection across the surface of the neutron star, and therefore the surface of the star isdivided in small patches in which the direction and strength of the magnetic fieldcan be considered constant. In order to get the total spectrum and polarization, oneneeds to sum over all the different patches.8.1.1 Thermal structure and the angular dependence of the effectivetemperatureThe difference between the patches is not only given by the direction and strengthof the magnetic field, but also by a difference in effective temperature. The thermalstructure of neutron stars is affected by the presence of the strong magnetic field146and this results in an angular dependence of the effective temperature across thesurface. As I have mentioned before, in the strong magnetic field, the energy ofthe electrons is quantized, and their thermal energy is typically lower than theirLandau energy (h¯ωc). This quantization determines the structure of the electronphase-space and must be taken into account in calculating the thermodynamicsof the electron gas. Heyl and Hernquist [91] calculated the thermal conductionin the thin region, the envelope, which insulates the bulk of the neutron star, forthe low-temperature, strong-field regime (when only one Landau level is filled).They found that the flux from the surface depends on the angle between the localdirection of the magnetic field and the normal to the surface (ψ) with a cos2ψdependence, and on the strength of the local magnetic field as ∼ B0.4.In order to calculate the thermal emission from the surface, I therefore createseveral atmosphere patches, using Lloyd’s code, to simulate the emission fromdifferent regions on the neutron star surface. Specifically, for a patch at colatitudeθ , the local magnetic field strength is given by (dipolar field)B = Bp√3cos2 θ +14, (8.1)where Bp is the magnetic field at the pole; the angle between the local vertical andthe magnetic field is given bycos2ψ =4cos2 θ3cos2 θ +1; (8.2)while the flux is given byF = Fp(BBp)0.4cos2ψ = Fp[√3cos2 θ +14]0.44cos2 θ3cos2 θ +1, (8.3)where Fp is the flux at the pole. The effective temperature of the patch is thereforegiven byTeff = Teffp[(3cos2 θ +14)0.2]1/4( 4cos2 θ3cos2 θ +1)1/4, (8.4)147Figure 8.2: Intensity map for the thermal emission of a magnetar with Teffp =3.0×106 K and Bp = 1.3×1014 G, including light bending. Left: view-ing angle 30◦; right: viewing angle 90◦. Black circles indicate contoursof equal colatitude, The colormap shows the intensity of the emissionfor 2 keV photons.where Teffp is the effective temperature at the pole.Fig. 8.2 shows the intensity map for the thermal emission at 2 keV from a mag-netar with the magnetic pole at 30◦ (left) and at 90◦ (right) from the line of sight.The black circles are drawn at constant colatitude. Gravitational light bending isincluded in the calculation, and that is why both magnetic poles are visible in the90◦ case. To calculate the intensity, I divided the surface in colatitude in 18 patches(from one pole to the other), and for each patch I calculated the atmosphere emis-sion using Lloyd’s code. The effective temperature and magnetic field at the poleare taken to be Teffp = 3.0× 106 K and Bp = 1.3× 1014 G, and the temperatureand magnetic field for each patch are calculated using Eqs. 8.1 and 8.4. The cos2dependence can be seen in the map. The dark region close to the pole in the leftimage and the two dark regions at mirrored positions with respect to the center inthe right image correspond to the regions where the local magnetic field is pointingin the direction of the line of sight. This seems counter-intuitive, as photons can es-cape more easily when streaming along the magnetic field line; however, I cannotresolve the very small region in angle around the magnetic field direction wherethe intensity peaks, and the region around the peak is depleted of photons because148they can easily get scattered in the magnetic field direction. This behavior can beseen in Fig. 3.4, where the intensity as a function of angle is shown for photons at0.32 keV, but the peak is much finer (and harder to resolve) at 2 keV.8.2 Non-thermal emission: a twisted magnetosphereThe non-thermal emission is thought to be fueled by the energy stored in the mag-netosphere of the neutron star. Similar to the Sun, magnetars are believed to possesstwisted magnetospheres. Inside the star, magnetic fields can reach values close to∼ 1017 G and the poloidal and toroidal components are expected to be roughly inequipartition [225, 226]. The internal toroidal field creates strong stresses on thesurface layers, causing occasional starquakes or slow plastic flowing of the crust[227]. The magnetosphere is anchored to the crust and, similar to the Sun’s corona,gets twisted by the motions of the crust [228]. As a result, the magnetosphere be-comes non-potential (∇×B 6= 0) and is threaded by force-free electric currents,that flow along the magnetic field lines ( j×B = 0) [28]. Twisted, force-free mag-netospheres in axial symmetry can be described by the Grad-Schlu¨ter-Shafrenov(GSS) equation, and numerical solutions have been found for a self-similar twisteddipole [245] and self-similar multipoles [176]. Realistic, non-self-similar solutionswere only explored for the case of weak twists [24, 25].The first solution for an axisymmetrical, force-free and self-similar magnetarmagnetosphere was proposed by Thompson et al. [228]. They express the flux ofthe poloidal component of the field with the function P =P(r,θ), independentof the azimuth φ because of the symmetry. The magnetic field can be thereforeexpressed as a sum of the poloidal and the toroidal component:B =∇P(r,θ)× eˆφr sinθ+Bφ (r,θ)eˆφ (8.5)where eˆφ is the unit vector in the φ direction. The force-free condition imposesBφ to be a function ofP , and therefore, by writing Bφ = F(P)/(r sinθ), one canobtain the GSS equation∂ 2P∂ r2+sin2 θr2∂ 2P∂ cos2 θ+F(P)dFdP= 0 (8.6)149Thompson et al. [228] assume a self-similar solution, that makes eq. 8.6 separable:P =P0(rR∗)−pf (cosθ) (8.7)where R∗ is the radius of the neutron star and the constant is set to P0 = BpR2∗/2in order to find the right value of the magnetic field at the pole (Bp). In thisway, the GSS equation becomes a second order eigenvalue differential equationfor f (cosθ), that can be solved for 0 ≤ p ≤ 1. If the boundary conditions arechosen for a dipolar field, one finds the polar components of the magnetic field tobeBr =−Bp2(rR∗)−2−p d fdcosθ(8.8)Bθ =Bp2(rR∗)−2−p p fsinθ(8.9)Bφ =Bp2(rR∗)−2−p√ Cpp+1f 1+1/psinθ(8.10)where C is an eigenvalue that depends only on p. The parameter p controls theradial decrease of the magnetic field, but also the net amount of twist. The twistangle, defined as the displacement between the north and south footprints of themagnetic field lines, is found as∆φN−S =∫fieldlineBφBθdcosθsin2 θ= 2√Cp(p+1)limθ0→0∫ pi/2θ0f 1/psinθdθ (8.11)From the magnetic structure, one can find the current density induced along thetwisted field linesj =∑iZieniβ ic =(p+1)c4pirBφBθB (8.12)where Zie is the electric charge of the particles species i carrying the current, ni isthe particle density, and β i is the particle velocity in units of c in the direction ofthe local magnetic field.The twisted magnetosphere stores energy in the currents, and tends to dissipate150it over time and to “untwist”. The plasma that fills the magnetosphere convertsthis energy into radiation, generating the non-thermal emission. Beloborodov [24]studied the electrodynamics of untwisting, and found that whenever the crustalmotion that causes the twist stops or slows down, the electric currents flowing onmagnetic field lines close to the neutron star surface (with apex radii Rmax . 2R∗,where R∗ is the radius of the neutron star) are quickly removed and absorbed bythe crust. The lifetimes of the currents on field lines with Rmax  R∗ are muchlonger, and therefore the persistent emission has to be created by the plasma on theextended field lines, which form what he calls a “j-bundle”. The plasma that fillsthe j-bundle is responsible for carrying the electric currents j = c/4pi∇×B andis continually created by electron-positron pairs (e±) discharge near the surface.Because the discharge has a threshold voltage of about 109 V [28], the boundarybetween the depleted region where j = 0 and the j-bundle is sharp, and it movestoward the magnetic dipole axis with time.8.2.1 Non-thermal models: the 5-10 keV rangeThe twisted magnetosphere around the neutron star is bound to affect the thermalemission from the surface, and to cause the observed non-thermal power laws.Several models have been proposed to explain the power law or excess emissionobserved above the thermal peak in the 2-10 keV range. I will briefly introducesome of the models here that I will employ later to calculate the spectral shape andpolarization.The hot-spot modelThe impact of the particles carrying the current on the neutron star surface couldbe a source of heating, as a fraction of the kinetic energy could be deposited inthe surface layer of the neutron star, and could create a hot-spot at the magneticpoles. In particular, the transient X-ray emission from the magnetar XTE J1810-197 is best fitted with a double blackbody (one for the surface emission and onefor the hot-spot). The luminosity of the hot-spot was observed to decrease [96],implying a shrinking of the surface area, which is in agreement with the picture ofa shrinking j-bundle. In § 8.3.1, I will analyze the spectrum and polarization in this151context by modeling the thermal emission and the excess above the peak with anhydrogen atmosphere model for the entire surface and coupled with a hotter modelfor the hot spot.The resonant Compton scattering modelIn the twisted magnetosphere picture, an electric current j = c/4pi∇×B must flowalong the magnetic field lines and has to be provided by the e± pairs generatedfrom the discharge in the magnetosphere, or, alternatively, by a flow of electronsand ions ripped from the neutron star surface. In the model first introduced byThompson et al. [228], the positive and negative charges are counter-streamingalong the magnetic field lines and possess similar number densities n+ ∼ n−; inthis way, the coronal plasma is nearly neutral and the required current is provided:j = e(v+n+− v−n−), where v+v− < 0.In the Thompson et al. [228] model, the thermal (∼ 1 keV) photons emittedfrom the stellar surface are resonantly Compton scattered by the coronal plasmaat large radii (∼ 100 R∗), where the dipolar magnetic field has a strength B ∼1011− 1012 G. In order to be able to resonantly scatter photons in this region, theflowing charges have to be mildly relativistic (see § 3.2.3). In the Thompson et al.[228] model, the velocities of the charges v+ and v− are considered free parameters,which may be adjusted so that the scattering of the thermal photons can reproducethe observed spectrum.Ferna´ndez and Thompson [62] have performed Monte Carlo simulations todetermine the basic types of non-thermal spectra and pulse profile that can be ob-tained by varying the free parameters of this model, namely the twist angle ∆φN−S(eq. 8.12), the spectral distribution of seed photons (they assume a blackbody distri-bution), the polarization of the seed photons (they assume either 100% X or 100%O), the angular distribution of the seed photons and the momentum distribution ofthe charge carriers. They obtain a variety of spectral shapes, some of which re-semble the magnetar soft X-ray emission, although they never fit them to observedX-ray spectra. When they employ a broad, relativistic distribution of charge mo-menta, they obtain a spectrum that dips above the thermal peak and then rises againas a power law to a maximum frequency that increases with the maximum Lorentz152factor of their distribution. This would suggest that RCS could be at the originof the hard power law (20–100 keV) as well. A similar model was employed byNobili et al. [165], and in [250] they successful fit the obtained spectra to severalmagnetars, with a few exceptions.The RCS model is very successful at fitting the soft spectrum of magnetars [250];however, Beloborodov [27] points out that the RCS picture for the soft spectrum,in which two mildly relativistic fluids of opposite charges scatter photons far awayfrom the star, presents a problem. In the region where the magnetic field is aboutB ∼ 1011− 1012 G, the radiation pressure pushes the plasma away from the starand a strong electric field parallel to the magnetic field line needs to be present tocounteract the radiative drag and bring the particles back toward the star. The sameelectric field, however, will push the particles with opposite sign away, increasingthe effect of the drag, and bringing the particle velocity to be highly, and not mildly,relativistic.In § 8.3.3 I will present a heuristic model to calculate the number of scatteringsneeded to reproduce the spectrum in case of a partial Comptonization due to mildlyrelativistic electrons, and I calculate the polarization expected in this case.Saturated Comptonization modelAn alternative explanation for the excess above the thermal peak, that has not beenproposed before, is a saturated Comptonization of the thermal photons by a non-relativistic population of electrons close to the stellar surface. This is the mostconservative model, as it does not require relativistic or ultra-relativistic electrons,or a specific distance from the star. However, because the majority of thermalphotons are being emitted in the X-mode, and the scattering cross section for X-mode photons is very small, resonant scattering has been invoked in order to buildenough optical depth to explain the power law excess with Compton scattering ofX-mode photons. This is not necessary though in the case in which at least a smallfraction of thermal photons are emitted in the O-mode; from Lloyd’s atmospheremodels, the amount of photons emitted in the O-mode is of the order of 2%.The difference in scattering cross section between the two modes at these ener-gies is several orders of magnitude (the O-mode cross section is simply the Thom-153100 101 102Energy [keV]10 610 510 410 310 210 1100/T,/T0.00.20.40.60.81.01.21.4Figure 8.3: O-mode (σ‖) and X-mode (σ⊥) cross section in the X-rays for amagnetar magnetic field. The different colors show the O-mode crosssection for different angles α (in radians) between the incident photondirection and the magnetic field, while the black solid line shows the X-mode cross section, which is equal to the O-mode cross section for α =0. Except for photons propagating along the magnetic field direction,the X-mode cross section is several order of magnitudes smaller thanthe O-mode cross section. Same as Fig. 7.5 but for B = 1.3× 1014 Ginstead of B = 4.3×1012 G.son cross section, see Fig. 8.3), and therefore, for O-mode photons, it is mucheasier to build enough optical depth to fully Comptonize the population. More-over, whenever an X-mode photon happens to scatter, it is immediately convertedinto an O-mode photon, because we are considering energies far below the electroncyclotron line, and afterwards, its scattering cross section is hugely increased.The evolution of the spectrum in the presence of repeated scatterings off non-relativistic or mildly relativistic electrons can be calculated with the Kompaneetsequation [113] (see § 8.3.2) and in case of saturated scattering, the equation leadsto an approximated Wien law. This is because, when photons undergo many scat-terings, they reach a thermal equilibrium with the electrons, and get “scattered up”into a Bose-Einstein distribution [195]. Even if only 2% photons are in the O-mode, this is enough to explain the observed excess above the thermal peak if thescattering plasma has a temperature of ∼ 2 keV.1548.2.2 Non-thermal models: the 20-100 keV rangeResonant Compton scattering has been also invoked to explain the high-energy tailobserved in several magnetars. A similar model to the one introduced in § 8.2.1,with thermal photons from the surface scattering off a counter-streaming flow ofopposite charges in the magnetosphere at about 100 km from the surface, wasemployed by Ferna´ndez and Thompson [62] and by Wadiasingh et al. [236] to tryand fit the high-energy tail. The high-energy emission is explained in this model byassuming a broader, highly-relativistic (γ > 20) distribution of scattering centers inthe magnetosphere, compared to the mildly relativistic flow assumed in the soft-emission model.An alternative model was proposed by Beloborodov [26, 27], who also invokesRCS, but off a different plasma flow in the magnetar corona: instead of having twocounter-streaming flows of opposite charge, in Beloborodov’s model the requiredcurrent for the twist is provided by e± flowing in the same direction away fromthe star but with a small difference in velocity. The density of the plasma is muchhigher than the density required by the current j/ec, and the current is sustainedin the outflow by a moderate electric field parallel to the magnetic field lines. Theflow of e± is the same from the two magnetic poles and the particles accumulate atthe equatorial plane, where they annihilate.In this picture, e± are created at the footprints of the magnetic field lines andaccelerated to high Lorentz factor, until they can resonantly scatter in the strongmagnetic field, at about γ ∼ 103. The upscattered photons immediately producemore e±. Some of the e± flow out along the magnetic field lines; as they moveto larger radii, the strength of the magnetic field decreases, and the charges canscatter photons of lower energy, which are more abundant. In the region betweenthe surface of the star and where B. BQED, all the scattered photons convert to e±and the pair density goes up by a factor of ∼ 100. In this region, the particles slowdown but cannot radiate away their energy, and the total kinetic energy is conservedbut shared by more particles. Near the region where B∼ 1013 G, pair creation ends,and outside this surface the resonantly scattered photons can escape. The outflowdecelerates and annihilates at the top of the magnetic loop, in the equatorial plane;here it becomes very opaque to the thermal keV photons flowing from the star.155Photons reflected from this region have the best chance of being upscattered bythe relativistic outflow in the high magnetic field region, because of their incidentangles, and control its deceleration.Since both models employ RCS to explain the high-energy emission, a similarpolarization signal is expected, which is about 33% in the X-mode (see below).Other models have been proposed that predict a different polarization signal. Forexample, Thompson and Beloborodov [224] suggest the origin of the high-energytail to be thermal bremsstrahlung in the surface layers of the star, heated by thereturning charges flowing in the twisted magnetosphere. In their model, the energyis deposited in a shallow layer on the surface, and therefore they assume that pho-tons emerging from the atmosphere, mainly in the X-mode, cannot scatter and cooldown the layer, which has to cool by emitting O-mode photons.8.3 Modeling the spectrum and polarization of 4U0142+61The spectrum of magnetar 4U 0142+61 presents clearly the three main featuresobserved in magnetar spectra: the thermal emission peaking at ∼ 1 keV, the hardexcess above the thermal peak and below 10 keV, and the hard tail above 10 keV.I will therefore use its spectrum, observed by NuSTAR and Swift [221], as a ref-erence for modeling the spectrum and polarization of magnetars X-ray emission.In this section, I will always use Lloyd’s hydrogen atmospheres to model the ther-mal emission, but with different effective temperatures as required by the differentmodels.For the hard-energy tail, I will not derive the spectral shape from a theoreticalphysical model, and I will simply employ a power law with Γ ∼ 1.3. If one as-sumes a RCS model (either the counter-streaming or the Beloborodov model), thepolarization is determined by the number of scatterings in the magnetosphere (seethe discussion in § 3.2.3): the probability of a photon to resonantly scatter into aX-photon or into a O-photon is given by the ratio of the resonant cross sections andis, respectively, 3/4 and 1/4, and an X-photon is three times more likely to scatter.Therefore, if the incident photons are all in X, the polarization after one scatter-ing is about 50% in X and it is subsequently reduced for further scatterings. The156analysis of Ferna´ndez and Davis [61] shows that the average number of scatter-ings at energies above ∼ 10 keV converges to about 2, and the polarization degreeconsequently converges to ∼ 33% in X.In the following sections, I will focus on the soft emission and I will derivethe spectral shape and polarization in the context of the different models proposed.In order to compare the calculated spectrum with the observed one, I model theabsorption in the X-ray from the interstellar medium using the model developed byWilms et al. [243] for a neutral hydrogen column density NH ∼ 1022 cm−2 (this isincluded in XSPEC as the tbabs model).8.3.1 The hot spotThe spectrum of magnetar 4U 0142+61 can be fitted with a double black body,which hints to the possibility of the excess above the thermal peak to be caused bya hot spot. In [221], the two black bodies used to fit the spectrum have tempera-tures kTeff = 0.422±0.004 keV for the whole surface and kTeff = 0.93±0.02 keVfor the hot spot. However, Lloyd et al. [132] have shown that using a blackbodyfunction leads to an overestimation of the surface temperature, and that detailedatmosphere calculations are needed to obtain a proper fit of the spectral shapeof neutron stars. I now use the equations introduced in § 8.1.1 to calculate theemission from the whole surface using Lloyd’s hydrogen atmospheres, assuming atemperature at the pole kTeffp = 0.26 keV and a magnetic field Bp = 1.3×1014 G.For the patches within θ = 2◦ from the magnetic pole, I use a different tempera-ture kTeffp = 1.33 keV. In this way, I obtain a hot-spot at the magnetic pole, and aspectral shape that compares well with the observed spectrum. The intensity mapobtained is shown in Fig. 8.6 for photons with energy 5 keV; since the small hotspot is much hotter than the rest of the star, the logarithm of the intensity is shown(a linear map would only show the hot spot).The upper panel of Fig. 8.4 shows the observed spectral data from [221], to-gether with the calculated emission. The black lines show the contribution fromthe hydrogen atmosphere (dashes), which includes the hot spot, and from the high-energy power-law (dots). The red line shows the sum of the contributions as ob-served at infinity with an inclination angle of 30◦, including the effects of light157100 101 102Energy [keV]10 410 310 210 1Ef E[keV2 cm2 s1 keV1 ]H atmo + HSPLTotal100 101 102Energy [keV]1.00.80.60.40.20.0Q/IH atmo + HSPLTotal w/o QEDTotal w/ QEDFigure 8.4: Spectral shape and polarization for the hot spot model. Upperpanel: the figure shows the spectral data of 4U 0142+61 [221]; on top,the different components are shown in black: the absorbed hydrogenatmosphere and the hot spot (dashed) and the high-energy power-law,with spectral index Γ= 1.4 (dotted). The red line shows the sum of the2 components and it is plotted on top of the spectral data for comparison.Lower panel: linear polarization fraction; the components in black arethe same as for the upper panel. The total polarization fraction is shownwith (red solid line) and without (blue solid line) taking into accountQED.bending and gravitational redshift. I did not attempt spectral fitting, which is left asa task for future work; however, Fig. 8.4 shows that the emission from the surfaceand the hot spot can describe well the spectral shape observed below 10 keV.The lower panel of Fig. 8.4 shows the linear polarization degree as function of158Figure 8.5: Polarization map for a magnetar with and without QED. Blackcircles indicate contours of equal colatitude, while red lines indicatethe direction of polarization. Upper images: viewing angle 30◦; lowerimages: viewing angle 75◦. Left images: polarization map at sur-face assuming 100% X-mode photons, Bp = 1.3× 1014 G and noQED. Right images: polarization map at the polarization-limiting ra-dius (and therefore at the observer) assuming 100% X-mode photons,Bp = 1.3×1014 G, and including QED.the photon energy, where a positive Q/I corresponds to O-mode polarization anda negative Q/I corresponds to X-mode polarization. The viewing angle is again30◦. The components in black are the same as for the upper panel: the multi-temperature atmosphere as dashes and the power-law as dots, and they were both159calculated including QED. The total is shown including the QED effect of vacuumbirefringence in red and without including QED in blue. If QED is not included,the degree of polarization at the observer is dramatically reduced. This is because,even if the polarization degree at surface is more than 90% in the X-mode for thethermal emission, when the contributions from different parts of the surface aresummed over, the total polarization degree is reduced because the magnetic fieldpoints in all different directions. The left images of Fig. 8.5 show the polarizationmap for the emission at surface, assuming it is all polarized perpendicularly tothe local field lines, in the X-mode, and taking into account gravitational lensing.The upper images show the surface map of the star for an inclination angle of30◦, while the lower images for an inclination angle of 75◦. The black circlesindicate contours of equal colatitude, while the red lines indicate the direction ofpolarization locally. If QED is not included, the polarization observed at infinityis the sum of the contribution from the whole surface of the left images, where themagnetic field is pointed in many different directions and so is the polarization.Even if the surface emission is 100% polarized, the total observed polarization isvery low, as can be seen from the blue line in Fig. 8.4 at low energies. The blueline starts getting more polarized around 4 keV because at this point the emissionfrom the hot spot becomes predominant. The reason can be seen in Fig. 8.6: if werestrict to the emission from the hot spot (yellow dot), where the magnetic field isquite uniform in direction, the polarization is aligned in also in the left image.The effect of QED in presence of a high magnetic field, as shown in Chapter 5,is to preserve the polarization degree at emission. This effect is shown in the rightimages of Fig. 8.5: if one includes QED, the polarization direction is not frozen tothe value at the surface (shown in the left images), but keeps changing followingthe local magnetic field to the polarization-limiting radius, tens of stellar radii awayfrom the surface in the case of magnetars, where the magnetic field through whichthe radiation passes is uniform. If a photon is emitted in the X-mode, its polariza-tion will rotate so that it keeps being perpendicular to the local magnetic field (thephotons keeps staying in the X-mode) and the same for an O-mode photon. In theright images of Fig. 8.5 the polarization at the rpl is almost completely aligned inthe map, and therefore by summing over the entire map one re-obtain the polar-ization at emission. The treatment for the high-energy power law is similar. It is160Figure 8.6: Polarization and intensity map for the hot spot model. Same asthe upper images of Fig. 8.5: viewing angle 30◦, without QED (left)and with QED (right). The yellow dot is the hot spot, and the greenellipses are regions where circular polarization is generated through thequasi-tagential effect (§ 5.1.1). Black circles indicate contours of equalcolatitude, while green lines indicate the direction of polarization. Thecolormap shows the logarithm of the intensity for 5 keV photons.emitted in the magnetosphere with a 33% polarization degree in X, from a sphereof about 100 km in radius, where the magnetic field is pointed in many directions.Therefore, without QED, the observed polarization is almost zero. Since 100 kmis well within the polarization-limiting radius for a magnetar, QED has the sameeffect of preserving the polarization, and the observed polarization is the same asat emission, about 33%.The red line of Fig. 8.4 shows the polarization at the observer for a viewingangle of 30◦. At low energy, the contribution from the atmosphere (and the hotspot) is predominant, and the emission is highly polarized in the X-mode. As thehigh-energy power law becomes more important, around 6 keV, the polarizationdegree diminishes and reaches the 33% of the power law around 10 keV.8.3.2 The saturated ComptonizationIn this section, I will assume that the excess above the thermal peak is caused byinverse Compton scattering off a population of hot electrons in a plasma above161the atmosphere. The evolution of the photon phase space density, n(ω), due toscattering off electrons is described by the Boltzmann equation [195]∂n(ω)∂ t= c∫d3 p∫ dσdΩdΩ[ fe(p1)n(ω1)(1+n(ω))− fe(p)n(ω)(1+n(ω1))](8.13)where fe(p) is the phase density of electrons of momentum p. The first term of thesum represent photons scattering into the frequency ω and the second term repre-sents photons scattering out of the frequency ω , in the scattering events genericallydescribed asp+ω ↔ p1+ω1 (8.14)This is the standard Boltzmann equation, with the addition of the quantum correc-tions (1+ n(ω)) for stimulated emission. In the case of a non-relativistic thermaldistribution of electrons ( fe(p) = ne(2pimekTe)−3/2e−p2/2mekTe), and a small energytransfer per scattering (h¯(ω1−ω)/kTe 1), eq. 8.13 can be approximated by theKompaneets equation∂n∂ tc=(kTmec2)1x2∂∂x[x4(n′+n+n2)] (8.15)where x = h¯ω/kTe, n′ = ∂n/∂x and wheretc ≡ (neσT c)t (8.16)is the time measured in units of mean time between scatterings. When photons arescattered many times, the spectrum reaches a steady state solution, which is, thephotons tend to be in equilibrium with the electrons and assume a Bose-Einsteindistribution with a chemical potential, because photons cannot be created or de-stroyed by scatteringn(x) = (eµkTe+x−1)−1 . (8.17)In order to calculate the emission spectrum from the magnetar in the case offull Comptonization, I will therefore take the thermal emission from the surfaceand divide the photons in X-mode and O-mode photons. The X-mode photons es-cape freely and preserve the original distribution, while for the O-mode photons I162Figure 8.7: Intensity map for the saturated Comptonization model, for aviewing angle 30◦. The left image shows the intensity for the X-modephotons, while the right image is intensity of the O-mode photons, bothat 5 keV. The total intensity looks like the image to the left for ener-gies below 6-7 keV; for higher energies, the O-mode photons begin todominate and the total intensity starts looking like the right image.calculate the Comptonized spectrum assuming they reach a Bose-Einstein distribu-tion.For the thermal emission, I now assume a pole temperature of kTeffp = 0.32 keVand a magnetic field at the pole Bp = 1.3× 1014 G, to calculate the atmospheremodels at the surface, following the equations in § 8.1.1. For each patch, theamount of O-mode radiation (which is about 2%) gets scattered up by an electron-dominated plasma at a temperature of kTe = 2.1 keV, and reaches a Bose-Einsteindistribution with a chemical potential µ/kTe ∼ 10. I calculate the chemical poten-tial by making sure that the total number of O-photons remain the same before andafter scattering. Fig. 8.7 shows the intensity map for the X-mode and the O-modeon the surface of the neutron star for a viewing angle of 30◦. The left panel showsthe intensity map for the X-mode photons at 2 keV, which escape directly fromthe atmosphere without scattering. As most of the atmosphere photons are in theX-mode, the intensity shown in the left panel is very similar to the total thermalemission shown in Fig. 8.2. The right panel shows the intensity map for the O-mode photons, which scatter many times in a hot corona right above the neutron163100 101 102Energy [keV]10 410 310 210 1Ef E[keV2 cm2 s1 keV1 ]H atmo in XComptonized OPLTotal100 101 102Energy [keV]1.000.750.500.250.000.250.500.751.00Q/IH atmo in XComptonized OPLTotal w/o QEDTotal w/ QEDFigure 8.8: Spectral shape and polarization for the saturated Comptoniza-tion model. Upper panel: the figure shows the spectral data of4U 0142+61 [221]; the different components are shown in black:the absorbed hydrogen atmosphere subtracted of the O-mode photons(dashed), the Comptonized O-mode photons (dots and dashes) and thehigh-energy powerl-law, with spectral index Γ = 1.4 (dotted). The redline shows the sum of the 3 components and it is plotted on top of thespectral data for comparison.Lower panel: linear polarization fraction;the components in black are the same as for the upper panel. The to-tal polarization fraction is shown with and without taking into accountQED.star surface. Because of the inclination of the magnetic field with respect to the sur-face, the atmosphere patch at 45◦ produces a higher fraction of O-mode photons,and a bright ring is shown at about 45◦ in the right panel.The total emission as function of energy is shown in Figure 8.8: the X-mode164photons of the hydrogen atmosphere in black dashes, the Comptonized O-modephotons in black dashes and dots, and the power law in black dots. The red solidline is the sum of the three components, and it is plotted on top of the spectral dataof 4U 0142+61 [221] for comparison. The comparison is aimed to show that thismodel produces a spectral shape that can explain the observed spectrum; however,no fitting attempt was made, and proper fitting has been left as a task for futurework.The lower panel of Fig. 8.8 shows the polarization at the observer, with andwithout QED. The effect of QED, as discussed in the previous section, is to con-serve the polarization degree of radiation at emission, while without QED, the sumof the contributions over the entire surface brings the polarization degree to lessthan a few percent. The polarization at low energies is dominated by the thermalemission and the polarization is almost 100% X. At intermediate energies, the O-photons in the Wien spectrum make the polarization swing to positive values, untilthe power-law starts to dominate and brings the polarization degree down to the−33% assumed for the high-energy tail.8.3.3 The resonant Compton scatteringIn order to simulate the RCS model, I have developed a simple estimate of theeffect of RCS on the thermal polarization coming from the neutron star surface. Inthe previous section, I have considered a case in which the Comptonized spectrumsaturates to the Wien spectrum for most photons. In the RCS model, the Comptonprocess is not saturated and I have to analyze partial Comptonization. A steady-state form of the Kompaneets equations (eq. 8.15) in presence of a photon sourceQ(x) is given by [195]0 =(kTmec2)1x2∂∂x[x4(n′+n)]+Q(x)− nMax(τes,τ2es)(8.18)where the small n2 term of eq. 8.15 has been dropped. Since the medium in consid-eration is finite, both incoming (Q(x)) and escaping photons have to be taken intoaccount. The probability for a photon to escape is proportional to the inverse of themaximum value between τes and τ2es, where τes is the scattering optical depth, and165therefore the term n/Max(τes,τ2es) is included to consider the escaping photons.In the RCS model for the soft emission, the electrons are mildly relativistic.I therefore assume a Maxwellian distribution with kTe = 150 keV, which means atypical Lorentz factor of γ ∼ 1.2. In the soft region, with h¯ω < 10 keV, x is smalland thus I can neglect the n term in eq. 8.18. A power-law solution to eq. 8.18 canthen be found for Q(x) = 0:n(x) ∝ xm (8.19a)m(m+3) =4y(8.19b)m± =−32 ±√94+4y(8.19c)wherey =4kTemec2Max(τes,τ2es) (8.20)is the Compton y parameter, which is given by the product of the average energygained per scattering (4kTe/mec2) times the average number of scatterings, andgives an estimate of the total average energy gain per photon: ε f = εiey. Fromthese two power-law solutions, a kernel can be built to solve eq. 8.18 including thephoton source function. If I define the functionf ′(x) ={xm+ for x < 1xm− for x > 1(8.21)then the normalized functionf (x) =f ′(x)∫ ∞−∞ f ′(x′)dx′(8.22)acts as the Green’s function for eq. 8.18 if the initial energy is equal to 1, with theright boundary conditions.A solution of the steady state Kompaneets equation (eq. 8.18) that includes thesource function Q(x) can therefore be found through the integraln(x) =∫ ∞−∞f( xx′)Q(x′)dx′ . (8.23)166The Kompaneets equation redistribute the same number of photons to a newscattered distribution in energy, where the average energy gain per photon is regu-lated by the y parameter. In our case, we are redistributing the photons in energythrough resonant Compton scattering, which means that the angular distributionof the scattering cross section is different (in order for a photon to be scattered, itneeds to be beamed toward the electron); the redistribution formalism, however,is still valid. Since I do not know the optical depth of the electron population, Ichoose the right y parameter by picking the value that reproduces the soft spectrumpower law and I find that the best value is y ∼ 0.15. This sets the optical depthof the plasma. However, I can change both the temperature of the electrons andthe optical depth of the plasma keeping y fixed and I will obtain the same spectralshape.As described in § 3.2.3, the linear polarization degree of the resonantly scat-tered radiation depends on the average number of scatterings per photon, and itdecreases by 50% for each scattering. Therefore, I need to find how many times onaverage a photon of a certain final energy has scattered to reach that energy. Themean relative energy gained by a photon of final energy x can be obtained as〈∣∣∣∣ln( xxi)∣∣∣∣〉= 1n(x)∫ ∞−∞f( xx′)Q(x′)∣∣∣ln xx′∣∣∣dx′ (8.24)where xi = h¯ωi/kTe is the initial energy of the photon divided by kTe. If εi is theenergy of a photon before scattering, the ratio of its energy after scattering ε f andthe initial energy is on average〈ε fεi〉= 1− εimec2+4kTemec2∼ 1+ 4kTemec2(8.25)The average number of scatterings of a photon with final energy x can be there-fore found as the ratio between the average total energy gain per photon with finalenergy x divided by the average gain per scatteringnsc(x) =〈∣∣∣ln( xxi)∣∣∣〉ln〈ε fεi〉 =〈∣∣∣ln( xxi)∣∣∣〉ln(1+ 4kTemec2) ; (8.26)167100 101 102Photon energy [keV]0.00.51.01.52.02.53.03.54.0n sc0.00.20.40.60.81.0Qf/QiFigure 8.9: Mean number of scattering in the RCS model as function of en-ergy (left axis, blue line) and resulting depolarization (right axis, redline).the amount of depolarization can be found directly from the average number ofscatterings.I now take the seed photons Q(x) to be thermal photons coming from the sur-face, with kTeffp = 0.26 keV and Bp = 1.3× 1014 G. The complete treatment forthis model would imply assuming a geometry for the magnetospheric plasma anddetermining the angular dependence of the scattering between the radiation comingfrom the surface and the plasma at 100 km from the neutron star. This is beyondthe scope of this work and it is left for future work. For illustration, I take theemission from a surface patch at colatitude 49◦, and I calculate the effect of thepartial Comptonization on the final polarization.Fig. 8.9 shows the average number of scatterings nsc as function of the finalphoton energy (h¯ω = xkTe) in the frame of the star (left y-axis, in blue), and thedepolarization effect as Q f /Qi (right y-axis, in red). At low energies, the averagenumber of scattering per photon is low, as expected, and the polarization fractionis lowered only by a few percent. In order for a thermal photon (∼ 1 keV) to reacha high final energy greater than 7 keV or more, it has to undergo more scatteringson average, and therefore its final polarization decreases with energy.The upper panel of Fig. 8.10 shows the spectral shape of the Comptonizedatmosphere obtained with eq. 8.23, in the frame of the observer (after gravitationalredshift, black dashes and dots). The red line shows the sum of the contributions168100 101Energy [keV]10 410 310 210 1EfE [keV2  cm2  s1  keV1H atmoH atmo + RCSPLTotal100 1011.00.80.60.40.20.00.20.4H atmoH + RCS, kTe=100 K H + RCS, kTe=150 K PLTotal, kTe=100 K Total, kTe=150 K Figure 8.10: Spectral shape and polarization for the RCS model. Upperpanel: the figure shows the spectral data of 4U 0142+61 [221]; thedifferent components are shown in black: the absorbed hydrogen at-mosphere (dashed), the Comptonized atmosphere (dots and dashes)and the high-energy power-law, with spectral index Γ = 1.3 (dotted).The red line shows the sum of the 3 components and it is plotted on topof the spectral data for comparison. Lower panel: linear polarizationfraction; the components in black and red are the same as for the upperpanel. In blue: Comptonized atmosphere (dashes and dots) and total(solid line) for an electron temperature kTe = 100 Kfor the Comptonized atmosphere and the power law at high energy. The spectraldata of 4U 0142+61 is plotted in blue for comparison. Again, no spectral fittingwas attempted, and the comparison is only to show that this model can reproducethe observed spectral shape as well. The polarization signal is shown in the lower169panel of Fig. 8.10 for the hydrogen atmosphere (black dashes), the Comptonizedatmosphere (black dashes and dots), for the power law (black dots) and for the total(red solid line). The Comptonized atmosphere is dominant at low energy and it isless polarized than the thermal emission because of the scatterings. As the energyincreases the Comptonized atmosphere becomes less and less polarized becauseof the increase in the number of scatterings, but at about 10 keV the power lawbecomes dominant and brings Q/I to −1/3.The polarization degree is also shown in blue for a different electron tempera-ture, kTe = 100 K. Since the Compton y parameter is kept the same, y = 0.15, thespectral shape for this choice of electron temperature is identical, but the opticaldepth for scattering is higher. This results in an higher number of scatterings at allenergies, and can be seen in Fig. 8.10 as a bigger depolarization.170Chapter 9Conclusions and FuturePerspectivesNeutron stars and black holes emit a good fraction of their energy in the X-rays;for accreting or highly-magnetized objects such fraction is dominant, and the de-velopment of X-ray astronomy in the second half of last century has brought manydiscoveries and a deeper understanding of compact objects. The detection of X-ray polarization from compact objects will provide two additional observables, thepolarization degree and angle, which carry information on the geometry of thesources and on the strong magnetic and gravitational fields.9.1 The developement of X-ray polarimetryX-ray polarimetry is an “old” field of astronomy, as the first evidence of polariza-tion in the X-rays from a celestial source was obtained by the rocket experimenton-board of Aerobee 350, in 1971, which detected the polarized emission fromthe Crab nebula [167, 239]. However, performing polarimetry in the X-rays withenough sensitivity is a hard task, and until recently, the Crab was the only celestialsource for which a positive measurement was taken, by the polarimeter on boardof the OSO-8 satellite in 1978 [240].The classical X-ray polarimeters were mainly based on two physical processes:Bragg diffraction and Thomson scattering. For the latter, a Thomson polarimeter171exploits the fact that the majority of photons that are scattered at about 90◦ de-gree from the initial direction of propagation end up preferentially in a directionperpendicular to the electric field of the incident photon, and therefore a Thomsonpolatimeter measures the angular distribution of photons scattered at 90◦ from theincoming beam. Bragg diffraction, on the other hand, occurs when radiation with asimilar wavelength to the atomic spacing of a crystal gets reflected by the differentlattice points in the crystal with a positive interference. In X-ray polarimetry, thecrystal is oriented at 45◦ with respect to the incoming beam, and only light with apolarisation vector normal to the incidence plane, that is the plane containing theincoming direction and the normal to the crystal plane, gets reflected; the radiationwith the opposite polarisation is, instead, absorbed or passes through the crystal.Achieving good sensitivity and addressing all the systematics and calibrationproblems with the classical techniques is difficult; moreover, these past 50 yearshave shown that X-ray polarimetry cannot be done as a by-product of a differentcore science mission, because the instrument has to be optimized for polarimetry toreach a reasonable sensitivity. For this reason, it has been hard for the communityto have a polarimetry mission accepted by a space agency. Recent developmentsof new techniques have given a new push to the field, in particular the developmentof Gas Pixel Detectors, based on the photoelectric effect, in the early 2000s [47,209, 210]. The photoelectric effect consists of a material emitting electrons, calledphotoelectrons, when shone on by light. In the case of X-rays, the photons areenergetic enough to strip the most bound electron, or K-shell electron, from thenucleus. The direction of emission of a K-shell photoelectron for 100% linearlypolarized incident radiation, is modulated around the polarization direction witha cos2 function of the azimuth angle. For this reason, the photoelectric effect isan ideal tracer of polarization. However, the path length of the photoelectron isshort, of the order of a few hundreds of microns even for a light gas, and thereconstruction of its direction of emission is very hard. In the Gas Pixel Detectors,this problem is solved by using a finely pixelized collecting anode, which allows adetailed imaging of the photoelectron track in a gas chamber, and of a Gas ElectronMultiplier (GEM) to amplify the electron current.Several observatories with an X-ray polarimeter on board are now at differentstages of development: in the 1–10 keV range, the NASA SMEX mission IXPE172[241] and the Chinese–European eXTP [253]; in the hard-X-ray range, 15–150keV, the balloon-borne X-Calibur [22] and PoGO+ [43]; and, in the sub-keV range,the narrow band (250 eV) LAMP [208] and the broad band (0.2–0.8 keV) rocket-based REDSox [64]. A broadband polarimeter has also been proposed as a secondgeneration after IXPE in the 0.2–60 keV band: XPP [104]. Of these observatories,IXPE and eXTP are currently at the most advanced stage of development, and theyboth employ Gas Pixel Detectors to perform imaging polarimetry in the 1–10 keVrange. IXPE will be the first mission entirely devoted to X-ray polarimetry, andis planned for launch in April 2021. The scientific payload of eXTP consists offour instruments: the Spectroscopic Focusing Array (SFA), the Large Area De-tector (LAD), the Polarimetric Focusing Array (PFA) and the Wide Field Monitor(WFM), of which the PFA is composed of 4 identical telescopes for sensitive X-rayimaging and polarimetry. The extended phase A was completed for eXTP at theend of 2018 and the planned launch date is around 2025.9.2 Neutron stars and black holes polarization studieswith the IXPE and eXTPX-ray spectral and timing analysis in the past few decades has broadened our un-derstanding of neutron stars and black holes, but several open questions remainwhich can be answered by observing their polarization in the X-rays. Specifically,in this work I have shown that the polarization signal is sensitive to the physics andgeometry of accretion onto black holes and X-ray pulsars, and on the shape andstrength of the magnetic field threading black-hole accretion disks and magnetarmagnetospheres. In the following sections, I will focus on the 1–10 keV energyrange of the upcoming polarimeters IXPE and eXTP, and I will show some simu-lations performed using the code XIMPOL1 [10] for these two instruments.9.2.1 Black holesA core science case of both IXPE and eXTP is to study the polarized radiation fromaccreting black holes. The effect of vacuum birefringence, that is dramatic forhighly magnetized objects like magnetars, is more subtle for accreting black holes,1https://github.com/lucabaldini/ximpol173Figure 9.1: Simulated polarization degree for the black-hole binary GRS1915+105. Left panel: Model with a? = 0.99 with QED (blueline) and without QED (red line); model with a? = 0.95 with QED(yellow line) and without QED (green line). Blue dots are a sim-ulated 100 ks observation with eXTP (approximately 300 ks withIXPE) for the blue line model. Right panel: Model with a? = 0.99with QED and the minimum magnetic field (blue line) and withoutQED (red line); model with a? = 0.99 with QED and 2.5 times theminimum magnetic field (black line). Black dots are a simulated 1 Msobservation with eXTP for the black line model.where magnetic fields are expected to be several orders of magnitudes below thecritical QED field BQED = 4.4×1013 G. In Chapter 6, however, I have shown thatQED affects the propagation of polarized light in the black hole magnetosphere, sothat the observed polarization becomes sensitive to the strength and shape of themagnetic field threading the accretion disk.Fig 9.1 shows a simulated 100 ks observation with eXTP of the black-hole bi-nary GRS 1915+105, which would correspond to an observation of approximately300 ks with IXPE. The simulated data (blue and black dots) is plotted on top ofthe models shown in Fig. 6.4 of Chapter 6. As I already stressed in § 6.4.4, thesefigures show preliminary calculations, and further work is required to model theexpected polarization degree, including the contribution of photons coming frommore distant regions than the ISCO and a more realistic structure for the magneticfield. However, Fig 9.1 shows the importance of including QED in modeling theexpected polarization: the left panel shows the polarization degree for a black holerotating at 99% the critical velocity (blue solid line and blue dots); if QED was not174included in the calculation, the signal would be mistaken for a black hole rotatingat a? = 0.95.Another important effect of QED is to make the polarization signal sensitive tothe strength of the magnetic field. The magnetic field in the accretion disk is notstrong enough to modify the scattering cross sections, and polarization of light atemission derives from regular Thomson scattering, following the Chandrasekharresult shown in § 3.1, and doesn’t carry information on the local magnetic field.The depolarization effect of vacuum birefringence, on the other hand, dependsstrongly on the strength of the magnetic field and the right panel of Fig 9.1 showsthe how the polarization signal changes with the strength of the magnetic field.The difference between the two models would be easily detected in a 1000 ksobservation with eXTP.These results present both a challenge and a promise for the upcoming po-larimeters. Including the effect of QED in the modeling of the polarization signalis hard, because it is important to keep track of the polarization direction and howit changes due to the local magnetic field while calculating light tracing in the Kerrmetric. An additional complication rises when one considers photons coming outof the plane of the disk: the structure of the magnetic field is here expected to beordered on a large scale, and therefore a magnetic field structure needs to be as-sumed and included in the calculation. Nonetheless, if properly modeled includingQED, the polarization signal from accreting black holes can provide a measure ofthe black hole spin, independent of other techniques as spectral fitting or reverber-ation, and of the magnetic field close to the ISCO.9.2.2 X-ray pulsarsThe geometry of accretion and the physics abehind the X-ray spectra in accretingX-ray pulsars is still debated. The upcoming polarimeters will provide new observ-ables that can help constrain the different models. In Chapter 7, I have calculatedthe polarization signal, including general relativity and QED, for the existing mod-els by Me´sza´ros and Nagel [150] and by Kii [111], and I presented a new modelfor polarization based on the accretion model by Becker and Wolff [20].Fig. 9.2 shows the predicted polarization signal in the two models in the band175●●●●● ● ● ●●−0.8−0.6−0.4−0.20.0Q/I[PolarizationDegree]2 3 4 5 6 7 8Energy [keV]Meszaros/Kii:10 km15 kmBecker/Wolff:Two Columns: 1kmTwo Columns: 7kmeXTP 100ksFigure 9.2: Simulated polarization degree for Her X-1. Positive Q/I hereindicates X-mode polarization. Solid lines: models calculated includ-ing QED; dashed lines: without QED. The green and orange lines arefor Me´szaros and Nagel and Kii models with a stellar radius of 10 km(green) and 15 km (orange). Green dots are a simulated 100 ks obser-vation with eXTP (approximately 300 ks with IXPE) for the green linemodel. Blue and black models are for a two-accretion-column model,with zmax = 6.6 km (blue) and zmax = 1.4 km (black).of the upcoming polarimeters, together with a simulation for a 100 ks (300 ks)observation with eXTP (IXPE). The predicted polarization signal is very differentin the two models, as a short observation would easily detect: the old slab modelpredicts a small polarization degree, which goes through a zero in the band, whilethe new column model is very polarized in the O-mode. Also, the effect of QEDwould be easily detected in both models.9.2.3 MagnetarsMagnetars are amongst the prime targets for the upcoming polarimeters becausethey are bright in the X-rays and because they can provide the first test of theQED effect of vacuum birefringence. Indeed, in Chapter 8 I have shown that thedifference in polarization signal is huge between the models calculated with QEDand the models without QED. Apart from testing QED, the polarization signal176100 101 102Energy [keV]1.00.80.60.40.20.00.20.4Q/I1 2 3 4 5 6 7 8 9 10Energy [keV]1.00.80.60.40.20.00.20.4Q/IFigure 9.3: The polarization fraction for the three models. The left panelshows the full energy range, and the right panel depicts the energy rangeof IXPE and eXTP. The blue dashed curve depicts the hotspot model,the orange dot-dashed curve depicts the full Comptonization of the ordi-nary polarization model and the green dotted curve depicts the resonantComptonization model for the emission at about 5 keV against non-relativistic electrons. In all cases, the highest energy emission is gener-ated through resonant Comptonization against relativistic electrons. Thetypical measurement uncertainty in Q/I for a 100 ks eXTP observationfor 4U 0142+61 is five percent with twenty energy bins between 2 and8 keV.can provide insights on the nature of the non-thermal processes happening in themagnetosphere of the stars. In Chapter 8, I have calculated the polarization of thethermal emission from persistent magnetars employing a realistic model for thehydrogen atmosphere, previously developed by Lloyd [95, 96, 131]. For the non-thermal emission, I have analyzed different proposed emission models (the hot-spotand the RCS models) together with a new model (the saturated Comptonizationmodel), and calculated the polarization signal for each one of them.Fig. 9.3 shows the polarization fraction for the three models: at low energy thepolarization fraction is high in the X-mode (negative Q/I) and at high energies itis dictated by the polarization fraction of the high-energy power law (that here isassumed to be 1/3 in X). The energy range of the upcoming polarimeters, between1 and 10 keV, is where the models are mostly different (right panel). Magnetarsare very bright in this range, and both IXPE and eXTP will be able to resolve thedifference between the models with short exposure times. The typical uncertaintyin Q/I is five percent with twenty energy bins.1779.3 Future workSome of the calculations shown in this work are still preliminary, and further anal-ysis is needed to make robust predictions for what the upcoming polarimeters aregoing to detect.For the black hole case, my analysis is restricted to edge-on photons, travelingclose to the disk plane, where the magnetic field is expected to be partially orga-nized on small scales. Further studies are needed to calculate the effect of QEDfor photons coming out of the disk plane, where the magnetic field is expected tobe organized on large scales. In this case, the effect of QED could be the oppositeof what happens for edge-on photons: the organized magnetic field could align thepolarization of photons traveling through the magnetosphere, resulting in a largernet observed polarization. This analysis will be crucial to predict the polarizationsignal for system that are observed at high inclination angles. Additionally, thecalculations shown in Fig. 9.1 only consider photons coming from a region closeto the ISCO, which is a good approximation only for high energy photons, whosecontribution is mainly due to the region very close to the ISCO. The next stepwould be to add the contribution from photons coming from more distant regions.The spectral formation model by Becker and Wolff [20] is very promising be-cause it is based on a robust theoretical model and it fits well the observed spectraof accreting X-ray pulsars. The polarization signal prediction that I obtained inthe context of the B&W model is also robust, and it is worth further analysis. Inparticular, it will be important to create a full suite of models with predictions onthe polarization degree and angle as a function of energy and phase for differentpossible viewing angles and rotation geometries.Similarly, a full suite of predictions for the different emission models will beneeded for the polarization degree and angle of magnetars. Also, the predictedspectral shapes should be fitted to the observed spectra. In order to make realisticpredictions on the polarization signal in the context of the RCS model, a correctgeometry for the magnetospheric plasma has to be taken into account. Regardingthe high-energy power law, several models have been proposed for its origin of butin Chapter 8 I have not calculated the spectral emission and polarization for thedifferent models. 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