Radiation Transport Around Kerr Black Holes by Jeremy David Schnittman B. A. Physics Harvard University (1999) Submitted to the Department of Physics in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY February 2005 c Jeremy David Schnittman, MMV. All rights reserved. The author hereby grants to MIT permission to reproduce and distribute publicly paper and electronic copies of this thesis document in whole or in part. Author . Department of Physics January 20, 2005 Certified by. Edmund Bertschinger Professor of Physics Thesis Supervisor Accepted by . Thomas J. Greytak Professor of Physics Associate Department Head for Education 2 3 Radiation Transport Around Kerr Black Holes by Jeremy David Schnittman Submitted to the Department of Physics on January 20, 2005, in partial fulfillment of the requirements for the degree of Doctor of Philosophy Abstract This Thesis describes the basic framework of a relativistic ray-tracing code for ana- lyzing accretion processes around Kerr black holes. We begin in Chapter 1 with a brief historical summary of the major advances in black hole astrophysics over the past few decades. In Chapter 2 we present a detailed description of the ray-tracing code, which can be used to calculate the transfer function between the plane of the accretion disk and the detector plane, an important tool for modeling relativistically broadened emission lines. Observations from the Rossi X-Ray Timing Explorer have shown the existence of high frequency quasi-periodic oscillations (HFQPOs) in a number of black hole binary systems. In Chapter 3, we employ a simple \hot spot" model to explain the position and amplitude of these HFQPO peaks. The power spectrum of the periodic X-ray light curve consists of multiple peaks located at integral combinations of the black hole coordinate frequencies, with the relative amplitude of each peak determined by the orbital inclination, eccentricity, and hot spot arc length. In Chapter 4, we introduce additional features to the model to explain the broadening of the QPO peaks as well as the damping of higher frequency harmonics in the power spectrum. The complete model is used to fit the power spectra observed in XTE J1550{564, giving confidence limits on each of the model parameters. In Chapter 5 we present a description of the structure of a relativistic alpha-disk around a Kerr black hole. Given the surface temperature of the disk, the observed spectrum is calculated using the transfer function mentioned above. The features of this modified thermal spectrum may be used to infer the physical properties of the accretion disk and the central black hole. In Chapter 6 we develop a Monte Carlo code to calculate the detailed propagation of photons from a hot spot emitter scattering through a corona surrounding the black hole. The coronal scattering has two major observable effects: the inverse-Compton process alters the photon spectrum by adding a high energy power-law tail, and the random scattering of each photon effectively damps out the highest frequency modulations in the X-ray light curve. Thesis Supervisor: Edmund Bertschinger Title: Professor of Physics 4 5 Acknowledgments Gravity can not be held responsible for people falling in love. No, this trick won't work...How on earth are you ever going to explain in terms of chemistry and physics so important a biological phenomenon as first love? -Albert Einstein For Nomi The love of my life, my closest friend, my foundation, my joy, my α and my Ω. And sometimes even my υ. This work is the cumulative result of so many people and so much effort that I am almost certain to omit important names, places, and events. My apologies in advance. In roughly chronological order, I would like to thank the following: The Almighty, the Creator, for making a world so beautiful and full of wonder. Albert Einstein, my childhood hero, who only gained in stature and esteem as I grew older and was fortunate enough to learn more about his unequaled accomplishments. He was a Giant standing on the shoulders of giants, seeing farther than anyone else has before or since. My parents, Michael and Suzanne Schnittman, for giving me a lifetime full of love and encouragement, and always reminding me to get back to work. Thank you dad for teaching me the value of a sense of humor. Thank you mom for teaching me the value of an education, whether formal or informal. My brother Aaron for keeping me in my place and always giving me someone to look up to. My superb teachers and classmates at Wilson Magnet High School, for igniting my passion for science and mathematics. Steve Craxton, my first research advisor, who introduced me to scientific computing, proper writing style (may he forgive me for the pages to come), ray-tracing and radiation transport, and the world of professional physics research. Rabbis Blau, Brovender, Ebner, Kilimnick, Schrader, and Walk, for teaching me about what is really important in this life. And my MIT office-mates, for distracting me with what is really unimportant in life. From the early days in 37-644, Tesla Jel- tema, Josh Winn, and Jon Miller; from the mezzanine clubhouse, Jamie Portsmouth, Nick Morgan, Josh Faber, and John Fregeau; from 37-638, Adrienne Juett, Ed Boyce, Alex Shirokov, Kristin Burgess, Dave Pooley, and Adam Bolton; from the silly putty foosball madhouse, Matt Muterspaugh, Jake Hartman, Eric Pfahl, and my replace- ments Allyn Dullighan and Will Farr; from the antechamber, Justin Kasper, Miriam Krauss, Judd Bowman, Molly Swanson, and Bobby Cohanim. Thanks to the D. Samuel Gottesman Library of the Albert Einstein College of Medicine, for being my office away from home. 6 Thank you to Paul Schechter, who took me under his wing my first day on the job; Fred Rasio, who introduced me to black holes and general relativity; Jack Wisdom, who trained me in the art of chaotic dynamics; Scott Hughes, whose office was always wide open for me to come in and bounce ideas off the white board; Jackie Hewitt, for offering an outsider's point of view and fitting me into her crazy schedule. Arlyn Hertz for fitting me into Jackie's schedule and helping make the defense run so smoothly. Martha Bezzat for answering all my random questions and requests for help with the mail, fax machine, and MIT bureaucratic regulations, always promptly and with a smile. And of course, many many thanks to my advisor, Ed Bertschinger, who provided the inspiration and direction for much of this work. His careful, critical eye kept me in line, and his encouragement this past year brought the final product to completion. Much of this work was supported by his NASA ATP grant NAG5-13306. Again, the biggest accolades are for my wife Nomi, for putting up with me and supporting me, especially these past few months. I couldn't have done it without you. ¢¡¤£¦¥¨§©¥ ¦¥ ¥ § ¥¥ ! "#§©¦$% #'&() *,+-¥ /.0"§1+-¥ § 2 The heavens proclaim the glory of God; the sky declares His handiwork | Psalms 19:2 )3'¥&4$%)/ 56¥ 798#)¦ :0"3;*<3!)3'¥ &¨!$% §*<5= 9)3;§§> ? @ A¥ §B¥ 3;./ C §B)3;¥&D+*<5 $¤E))5=&FEG)5H&I HEGG "&49EGG "&4J K§L)&4©$% *< *9 ¥ 2§L¢EG¥3M*N &¨G O$ 7¥) " : 3 £¡P Q) &R¥ .B¦S§&4 F 6 )§©T$¤¥ S#§ U&4 9/§©)&R#§ V&4 W U § ? The wise man does not speak in the presence of one who is greater than he in wisdom; he does not interrupt the speech of his companion; he is not hasty to answer; he questions and answers properly, to the point; he speaks on the first point first, and on the last point last; regarding that which he has not learned he says: \I have not learned"; and he acknowledges the truth. | Ethics of the Fathers 5:9 Contents 1 Introduction and Outline 15 1.1 Motivation . 16 1.2 Historical Background . 17 1.2.1 Theory . 17 1.2.2 Observations . 20 1.3 Outline of Methods and Results . 22 1.3.1 Ray-tracing in the Kerr Metric . 22 1.3.2 The Hot Spot Model . 25 1.3.3 Steady-state Disks . 27 1.3.4 Electron Scattering . 30 1.4 Alternative QPO Models . 31 2 Ray-Tracing in the Kerr Metric 35 2.1 Equations of Motion . 35 2.1.1 Boyer-Lindquist Coordinates . 38 2.1.2 Doran Coordinates . 40 2.1.3 Analytic Methods . 42 2.2 Geodesic Ray-tracing . 44 2.2.1 Tetrads . 48 2.2.2 The Radiative Transfer Equation . 50 2.3 Numerical Methods . 56 2.4 Broadened Emission Lines from Thin Disks . 59 2.4.1 Transfer Function . 59 2.4.2 Observations of Iron Emission Lines . 63 3 The Geodesic Hot Spot Model 69 3.1 Hot Spot Emission . 69 3.1.1 Overbrightness and QPO Amplitudes . 71 3.1.2 Harmonic Dependence on Inclination and Spin . 73 3.2 Non-circular Orbits . 77 3.3 Non-planar Orbits . 87 3.4 Summary . 90 7 8 CONTENTS 4 Features of the QPO Spectrum 93 4.1 Introduction . 93 4.2 Parameters for the Basic Hot Spot Model . 95 4.3 Peak Broadening from Hot Spots with Finite Lifetimes . 96 4.4 Distribution of Coordinate Frequencies . 100 4.5 Electron Scattering in the Corona . 107 4.6 Fitting QPO Data from XTE J1550{564 . 112 4.7 Higher Order Statistics . 117 4.7.1 The Bispectrum and Bicoherence . 117 4.7.2 The Bicoherence of the Simulated Data . 118 4.7.3 Simulations with Poisson Noise . 122 4.8 Summary . 124 5 Steady-state α-disks 127 5.1 Steady-state Disks Outside the ISCO .