2 Relativistic shocks and magnetic fields 17

Transcription

1 Contents I Introduction 13 1 Overview 15 2 Relativistic shocks and magnetic fields 17 3 Basic Physics Shock waves The theory of relativity The light cone Proper time and proper density Relativistic Doppler shift Relativistic momentum and energy Electricity and Magnetism Light Charged Particles in electromagnetic fields Interaction of particles and electromagnetic radiation The Weibel or filamentation instability Gamma-ray Bursts Observational facts Properties derived from observations The sources of Gamma-ray Bursts II Research 43 5 Relativistic Shock Conditions 45 7

2 8 CONTENTS 5.1 Introduction Theory: Magnetohydrodynamics MHD description of the plasma Physics of MHD Plasmas Approximations Shock conditions and their solutions General situation Perpendicular Shocks Non-perpendicular shocks Discussion of the results General results Internal structure of the shock Linear theory of the Weibel instability Introduction Covariant formulation of plasma dispersion Polarization tensor in the fluid approximation Polarization tensor in the kinetic description The cold plasma limit The beam-driven Weibel instability Non-relativistic cold beam limit Weibel instability driven by two symmetric beams The case of ultra-relativistic beams Ultra-relativistic background gas Magnetized Weibel instability When can non-magnetic results be used? The asymmetric case Conclusions Covariant formulation of plasma response Plasma response in the fluid approach Plasma response in the kinetic approach Covariant dispersion relation The components of the dispersion tensor Fluid approximation Kinetic theory Waterbag approximation Magnetized Weibel instability

3 CONTENTS 9 7 Variations on the Weibel instability Introduction Counterstreaming beam model Initial condition Background plasma Maxwell s equations Instability analysis Linear response of the beams Linear response of the background plasma Dispersion relation The Normal Weibel instability Dispersion relation for spatial modes Weibel instability for oblique propagation Discussion Proton Weibel instability Introduction Simple model for a relativistic shock The proton velocity distribution The shock conditions for the electrons The Weibel instability The dispersion relation Stabilization of the Weibel instability The equipartition parameter Discussion Conclusions Asymmetric beams The end of the Weibel instability Introduction Stabilization mechanisms: an overview Fluid model for the beam response Dynamics of the beam particles Reduced set of Maxwell s equations Linear response of the beam plasma The nonlinear response for single modes: the effect of wave breaking Influence of the background plasma Kinetic stabilization due to quiver motion

4 10 CONTENTS Broadening of the momentum distribution Effect on the beam response Current channel coalescence Maximum current for a cylindrical channel: the Alfvén current Simple model for field growth The effect of screening currents Equation of motion for two attracting screened filaments The coalescence time scale Implications for ultra-relativistic shocks Thermalization through the Weibel instability Thermalization of protons Conclusions Computer simulations Introduction The simulated system Basic equations Particle in Cell method Weibel instability theory Linear response of the beam particles Linear response of the background particles Dispersion relation Saturation of the instability Simulation results Conclusions Numerical approach Dimensionless form of the equations Solution method: particles Solution method: electromagnetic fields III Discussion and summary Discussion and conclusions 231

5 CONTENTS Samenvatting in het Nederlands Inleiding Gamma-ray Bursts Schokgolven en magnetische velden Simulaties De sterkte van het magnetische veld Vergelijking met waarnemingen Curriculum Vitae 245

6 12 CONTENTS