ABSTRACT WARREN III, DONALD CAMERON. Monte Carlo Simulations of Efficient Shock Acceleration during the Afterglow Phase of Gamma-Ray Bursts. (Under the direction of Donald Ellison.) Gamma-ray bursts (GRBs) signal the violent death of massive stars, and are the brightest ex- plosions in the Universe since the Big Bang itself. Their afterglows are relics of the phenomenal amounts of energy released in the blast, and are visible from radio to X-ray wavelengths up to years after the event. The relativistic jet that is responsible for the GRB drives a strong shock into the circumburst medium that gives rise to the afterglow. The afterglows are thus intimately related to the GRB and its mechanism of origin, so studying the afterglow can offer a great deal of insight into the physics of these extraordinary objects. Afterglows are studied using their photon emission, which cannot be understood without a model for how they generate cosmic rays (CRs)—subatomic particles at energies much higher than the local plasma temperature. The current leading mechanism for converting the bulk energy of shock fronts into energetic particles is diffusive shock acceleration (DSA), in which charged particles gain energy by randomly scattering back and forth across the shock many times. DSA is well-understood in the non-relativistic case—where the shock speed is much lower than the speed of light—and thoroughly-studied (but with greater difficulty) in the relativistic case. At both limits of speed, DSA can be extremely efficient, placing significant amounts of energy into CRs. This must, in turn, affect the structure of the shock, as the presence of the CRs upstream of the shock acts to modify the incoming plasma flow. The modified plasma flow affects the acceleration process responsible for generating CRs, creating a highly nonlinear feedback loop between the shock profile and the CRs accelerated by that shock. Any study of efficient DSA must include this process, but such studies must be numerical due to the inherent nonlinearity. In this dissertation, I undertake a Monte Carlo simulation of the afterglow phase of a GRB. I follow the shock from early on, while it is still relativistic, through the trans-relativistic phase at later times. The trans-relativistic regime must be approached numerically, as it defies approximations usually made in both the non-relativistic and fully relativistic limits. I will discuss the necessary extensions to the existing Monte Carlo code needed to examine trans-relativistic shocks and the CRs they produce, as well as the parameters and tools introduced to study the acceleration of protons, heavier ions, and much lighter electrons. I will also explain how the code takes the resultant CR spectra and produces observable photon emission. The most significant part of this dissertation, though, is the multi-stage simulation of a GRB afterglow. I will show that the nature of DSA is radically different in the fully relativistic and trans- relativistic regimes of shock speed, which in turn requires different assumptions about the parame- ters critical to DSA (such as the strength of the turbulent magnetic field or the amount of energy present in CR electrons). Starting from physically plausible external conditions and instituting just two new parameters— a compressed magnetic field for producing synchrotron emission, and a transfer of energy from ions to electrons in the shock precursor—motivated by first-principles particle-in-cell simulations, I will show that the Monte Carlo model can adequately capture the light curve of a GRB afterglow during the main decay stage. The brightness of any particular afterglow will constrain the parameter space available to the Monte Carlo code, allowing for the future possibility of working backwards to determine the conditions at the shock based on observations. © Copyright 2015 by Donald Cameron Warren III All Rights Reserved Monte Carlo Simulations of Efficient Shock Acceleration during the Afterglow Phase of Gamma-Ray Bursts by Donald Cameron Warren III A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Physics Raleigh, North Carolina 2015 APPROVED BY: Stephen Reynolds James Kneller John Griggs Donald Ellison Chair of Advisory Committee BIOGRAPHY The author was born. He went to college, graduated, and hared off to Japan for a break before graduate school. While in Japan he met the woman who would become his wife, who graciously agreed to join him back in North Carolina while he undertook the fool’s errand of earning a Ph.D. There was a marriage, a pregnancy, and a child, all of which the author recommends against (at least while still in graduate school) to any future graduate students reading this work. He regrets none of those decisions, but completing the dissertation you are now reading was rather significantly delayed by them. He is currently, as of this writing, writing a biography that is required by the NCSU dissertation format. ii ACKNOWLEDGEMENTS There are many people who deserve thanks for their role in the production of this work. In no particular order, I would like to thank all of the following people. • My wife Celine and daughter Aurora, for their (mostly silent) patience while I was generally emotionally, sometimes physically, and often mentally unavailable writing this dissertation. • My mother, who (like all good Jewish mothers) wanted her child to grow up and become a doctor. Done. • My father, who I’m mostly certain is just thankful that I grew up. • My advisor, Don Ellison, for his unflinching support in the face of a constant barrage of questions while I became familiar with this material, as well as for doing more work than anyone besides myself (which point may be arguable) in producing the manuscript you are currently reading. • The gaming crew, for keeping me sane all these years. Any omissions are unintentional, and are due to the aforementioned mental unavailability. This work made use of data supplied by the UK Swift Science Data Centre at the University of Leicester. Numerous other researchers allowed me to reproduce figures from their work for comparison or clarification; they are credited in the captions of those figures, but thanked here. iii TABLE OF CONTENTS LIST OF TABLES ......................................................... vii LIST OF FIGURES ........................................................ viii CHAPTER 1 INTRODUCTION TO COSMIC RAYS AND SHOCK ACCELERATION ......... 1 1.1 Cosmic Rays ..................................................... 1 1.1.1 Discovery.................................................. 1 1.1.2 Spectrum at Earth............................................ 2 1.2 Diffusive Shock Acceleration ......................................... 5 1.2.1 The macroscopic method...................................... 6 1.2.2 The microscopic method ...................................... 9 1.2.3 Comments and Consequences .................................. 12 1.3 DSA at Relativistic Shocks............................................ 14 1.4 DSA at Nonlinear Shocks ............................................ 16 1.5 Approaches to DSA ................................................ 18 1.5.1 Analytic and Semi-analytic ..................................... 18 1.5.2 Particle-in-Cell.............................................. 19 1.5.3 Monte Carlo................................................ 20 1.6 Summary........................................................ 21 CHAPTER 2 GAMMA-RAY BURSTS AND THEIR AFTERGLOWS ...................... 23 2.1 History of GRBs................................................... 23 2.2 Afterglow theory and methods ........................................ 32 2.2.1 Hydrodynamical scaling relations................................ 33 2.2.2 Photon spectrum scaling relations................................ 34 2.2.3 Hydrodynamics of the jet ...................................... 37 2.2.4 Consequences of a collimated jet................................. 39 2.3 Observations of afterglows........................................... 43 2.3.1 Pre-Swift era................................................ 43 2.3.2 Swift era................................................... 44 2.4 DSA parameters................................................... 46 2.5 Summary........................................................ 50 2.6 Goals of this Dissertation............................................ 51 CHAPTER 3 THE CODE ................................................... 52 3.1 Assumptions..................................................... 53 3.2 Injection........................................................ 56 3.3 The Particle Distribution and Fluxes.................................... 57 3.4 Propagation...................................................... 61 3.4.1 Scattering.................................................. 62 iv 3.4.2 Momentum Splitting.......................................... 65 3.5 The Upper Limit on the Particle Spectrum................................ 66 3.6 Smoothed Shocks ................................................. 68 3.7 Speedup ........................................................ 72 3.8 Summary........................................................ 73 CHAPTER 4 TRANSRELATIVISTIC SMOOTHED SHOCKS .......................... 74 4.1 Abstract......................................................... 74 4.2 Introduction ..................................................... 75 4.3 Model.........................................................