Hydrodynamics of Gamma-Ray Burst Afterglow

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Abstract
The detection of delayed emission at X-ray optical and radio wavelengths ("afterglow") following gamma-ray bursts (GRBs) suggests that the relativistic shell that emitted the initial GRB as the result of internal shocks decelerates on encountering an external medium, giving rise to the afterglow. We explore the interaction of a relativistic shell with a uniform interstellar medium (ISM) up to the nonrelativistic stage. We demonstrate the importance of several effects that were previously ignored and must be included in a detailed radiation analysis. At a very early stage (few seconds), the observed bolometric luminosity increases as t2. On longer timescales (more than ~10 s), the luminosity drops as t−1. If the main burst is long enough, an intermediate stage of constant luminosity will form. In this case, the afterglow overlaps the main burst; otherwise there is a time separation between the two. On the long timescale, the flow decelerates in a self-similar way, reaching nonrelativistic velocities after ~30 days. Explicit expressions for the radial profiles of this self-similar deceleration are given. As a result of the deceleration and the accumulation of ISM material, the relation between the observed time, the shock radius, and its Lorentz factor is given by t=R/16γ2c, which is a factor of 8 different from the usual expression. We show that even though only a small fraction of the internal energy is given to the electrons, most of the energy can be radiated over time. If the fraction of energy in electrons is greater than ~10%, radiation losses will significantly influence the hydrodynamical evolution at early times (less than ~1 day).