A Semi-analytic Formulation for Relativistic Blast Waves with a Long-lived Reverse Shock

TL;DR

This paper develops a semi-analytic model for relativistic gamma-ray burst blast waves with a long-lived reverse shock, incorporating a realistic variable adiabatic index EOS, and compares two methods of modeling their dynamics.

Contribution

It introduces a new semi-analytic approach using a variable adiabatic index EOS and compares pressure balance and mechanical models for blast wave evolution.

Findings

Pressure balance method violates energy conservation in adiabatic cases.

Mechanical model accurately conserves energy and predicts blast wave dynamics.

Different modeling approaches lead to significantly different afterglow light curves.

Abstract

This paper performs a semi-analytic study of relativistic blast waves in the context of gamma-ray bursts (GRBs). Although commonly used in a wide range of analytical and numerical studies, the equation of state (EOS) with a constant adiabatic index is a poor approximation for relativistic hydrodynamics. Adopting a more realistic EOS with a variable adiabatic index, we present a simple form of jump conditions for relativistic hydrodynamical shocks. Then we describe in detail our technique of modeling a very general class of GRB blast waves with a long-lived reverse shock. Our technique admits an arbitrary radial stratification of the ejecta and ambient medium. We use two different methods to find dynamics of the blast wave: (1) customary pressure balance across the blast wave and (2) the "mechanical model". Using a simple example model, we demonstrate that the two methods yield significantly different dynamical evolutions of the blast wave. We show that the pressure balance does not satisfy the energy conservation for an adiabatic blast wave while the mechanical model does. We also compare two sets of afterglow light curves obtained with the two different methods.