Relativistic Runaway Electron Avalanches Inside the High Field Regions of Thunderclouds by Eric Scott Cramer Bachelor of Science, Physics & Applied Mathematics CUNY Queens College 2008 Master of Science, Space Sciences Florida Institute of Technology 2011 A dissertation submitted to Florida Institute of Technology in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Physics Melbourne, Florida May 2015 ⃝c 2015 Eric Scott Cramer All Rights Reserved The author grants permission to make single copies We the undersigned committee hereby recommends that the attached document be accepted as fulfilling in part of the requirements for the degree of Doctor of Philosophy in Physics. \Relativistic Runaway Electron Avalanches Inside the High Field Regions of Thunderclouds" a dissertation by Eric Scott Cramer Major Advisor Hamid K. Rassoul, Ph.D. Dean, College of Science Professor, Physics and Space Sciences Co-Advisor Joseph R. Dwyer, Ph.D. Professor, Physics, University of New Hampshire Peter T. Paul Chair in Space Sciences Ningyu Liu, Ph.D. Associate Professor, Physics and Space Sciences Ming Zhang, Ph.D. Professor, Physics and Space Sciences Ugur Abdulla, Ph.D. Department Head, Mathematical Sciences Professor, Mathematical Sciences Abstract Relativistic Runaway Electron Avalanches Inside the High Field Regions of Thunderclouds by Eric Scott Cramer Dissertation Advisor: Hamid K. Rassoul, Ph.D. In this dissertation, simplified equations describing the transport and energy spec- trum of runaway electrons are derived from the basic kinematics of the continuity equations. These equations are useful in modeling the energy distribution of ener- getic electrons in strong electric fields, such as those found inside thunderstorms. Dwyer and Babich [2011] investigated the generation of low-energy electrons in relativistic runaway electron avalanches. The paper also developed simple ana- lytical expressions to describe the detailed physics of Monte Carlo simulations of relativistic runaway electrons in air. In this work, the energy spectra of the run- away electron population are studied in detail. Dependence of electron avalanche development on properties such as the avalanche length, radiation length, and the effective Møller scattering efficiency factor, are discussed in detail. To describe the shapes of the electron energy spectra for a wide range of electric field strengths, the random deviation of electron energy loss from the mean value is added to the solutions. We find that this effect helps maintain an exponential energy spectrum for electric fields that approach the runaway electron threshold field. We also investigate the source mechanisms of Terrestrial Gamma-ray Flashes, which are a result of relativistic runaway electron avalanches in air. In this study, the bremsstrahlung photons are propagated through the atmosphere, where they undergo Compton scattering, pair production, and photoelectric absorption. We model these interactions with a Monte Carlo simulation from the TGF source lo- cation (assumed to vary between 8 and 20 km) and the edge of the atmosphere (≈ 100 km). We then propagate these photons to a satellite plane at 568 km in order to compare with measurements. In collaboration with the GBM instrument team in Huntsville, AL, we were able to model spectral and temporal properties of observed TGFs. Although the analysis of individual TGF photon spectra was qualitative, we were able to put some constraints, i.e. source altitude and beaming angle, on a sample of observed GBM TGFs. However, assuming a height of 15 km, we were able to model the softening in the spectrum observed as the satel- lite moves off-axis from the TGF source location [Fitzpatrick et al., 2014]. The conclusion of this analysis shows that Compton scattering alone can not explain the temporal dispersion observed. This suggests that an intrinsic time variation exists at the source of the TGF. iii Contents Abstract iii List of Figures vii List of Tables ix Abbreviations x Symbols xi Acknowledgements xiii Dedication xvi 1 Introduction 1 1.1 Motivation ............................... 1 1.2 High-Energy Atmospheric Physics .................. 3 1.2.1 Relativistic Runaway Electron Avalanches (RREAs) . 3 1.2.2 Relativistic Feedback ...................... 6 1.2.3 X-rays from Lightning ..................... 10 1.2.4 Terrestrial Gamma-ray Flashes . 13 1.3 Monte Carlo Simulations ........................ 18 1.4 Scientific Contributions ........................ 23 2 Electron Interactions in Air 24 2.1 Motion in a Uniform Electric Field . 24 2.2 Brownian Motion ............................ 26 2.3 Avalanche Length, λe− ......................... 28 2.4 Bethe Formula for Ionization Energy Loss . 30 iv 2.4.1 Minimum Runaway Electron Energy, "th . 31 2.5 Møller Scattering ............................ 32 2.5.1 Differential Cross Section ................... 32 2.5.2 Efficiency Factor, ξeff ..................... 34 2.6 Electron-Nuclear Bremsstrahlung ................... 37 2.6.1 Triply Differential Cross Section . 37 2.6.2 Radiation Length, X0 ..................... 40 2.6.3 Energy Loss Formula ...................... 42 2.7 Coulomb Scattering .......................... 42 2.7.1 Differential Cross Section ................... 42 2.7.2 Root Mean Square Scattering Angle, θrms . 45 3 Runaway Electron Energy Spectra 48 3.1 Derivation of the Kinetic Equation . 48 3.2 An Analytical Approach ........................ 55 3.2.1 Simple Exponential Model ................... 57 3.2.2 Møller Scattering Source Term . 58 3.2.3 Radiative Losses ........................ 61 3.2.4 Random Walk Model ...................... 65 3.3 Average Kinetic Energy ........................ 74 3.4 Discussion ................................ 75 4 TGF Source Mechanisms 78 4.1 GBM Observations ........................... 78 4.2 Photon Interactions .......................... 83 4.2.1 Compton Scattering ...................... 84 4.2.2 Photo-Electric Absorption ................... 86 4.2.3 Pair Production ........................ 87 4.3 Source Properties of Energetic Photons . 88 4.3.1 Photon Energy Spectrum ................... 88 4.3.2 Angular Distribution ...................... 90 4.3.3 Particle Composition of the Beam . 92 4.4 γ-Ray Propagation in the Atmosphere . 93 4.5 Monte Carlo Modeling of GBM Data . 98 4.5.1 Spectral Analysis . 100 4.5.2 Temporal Analysis . 106 4.6 Discussion ................................ 111 5 Conclusions and Suggestions for Future Work 114 5.1 Conclusions ............................... 114 5.2 Suggestions for Future Work . 116 v A Green's Function Calculation 117 B Calculation of Electron Energy Loss Fluctuations 119 C The Diffusion Term 127 C.1 Derivation of the Fokker Planck Equation . 127 C.2 Method of Determining the Diffusion Coefficients . 130 C.3 Sturm-Liouville Form . 132 Bibliography 133 vi List of Figures 1.1 Effective frictional force experienced by a free electron in air . 4 1.2 Graphic of a runaway electron avalanche ............... 6 1.3 Relativistic feedback from a Monte Carlo simulation . 8 1.4 Summary of production methods for runaway electrons . 9 1.5 X-rays from lightning as observed by Moore et al. [2001] . 11 1.6 X-rays from lightning as observed by Dwyer et al. [2004] . 12 1.7 BATSE observation of a TGF ..................... 14 1.8 RHESSI TGF spectrum and model fits . 15 1.9 Cross section of the REAM Monte Carlo simulation . 20 1.10 REAM simulation of an electron avalanche . 22 2.1 Graphic of an electron in a uniform electric field . 25 2.2 Electron avalanche length ....................... 29 2.3 Minimum runaway electron energy, "th . 32 2.4 Møller scattering Feynman diagram . 33 2.5 Møller efficiency factor ......................... 36 2.6 Radiation length ............................ 41 2.7 Coulomb effects on electron path of travel . 44 2.8 Empirical fits for Coulomb scattering . 45 3.1 Solutions to Equation (3.36) using the approaches in sections 3.2.1 & 3.2.2 .................................. 60 3.2 Maximum runaway electron energy, "max . 63 3.3 Solutions to Equation (3.36) using the approach in section 3.2.3 . 65 3.4 Width of the energy fluctuations used in the random walk model . 68 3.5 Gaussian distributions of runaway electron energy fluctuations as a function of primary electron energy . 70 3.6 Gaussian distributions of runaway electron energy fluctuations as a function of applied electric field .................... 72 3.7 Solutions to Equation (3.27) using the approaches in sections 3.2.3 & 3.2.4 .................................. 73 vii 3.8 Average runaway electron energy as a function of applied electric field ................................... 75 4.1 GBM detected TGF from Briggs et al. [2010] . 79 4.2 Symmetric and asymmetric time profiles of two separate TGF events. Figure taken from Briggs et al. [2010] . 80 4.3 GBM spectral data and model fits from Briggs et al. [2011] . 82 4.4 Total cross sections for photon interactions in nitrogen . 83 4.5 Graphic of Compton scattering .................... 84 4.6 Photon differential energy spectra produced by the REAM Monte Carlo simulation ............................ 89 4.7 Simulated γ-ray angular distribution . 90 4.8 Simulated γ-ray rms angular distribution . 91 4.9 Ratio of the number of γ-rays to the number of electrons inside the avalanche region ............................ 93 4.10 Cartoon of γ-ray propagation using REAM . 95 4.11 Intrinsic γ-ray beam shape from the REAM simulation at the top of the atmosphere ........................... 97 4.12 Particle concentration at the top of the atmosphere for a wide