ar X iv : a st ro - p h / 97 06 07 8 v 1 9 J un 1 99 7 Hydrodynamics of GRB

Received ; accepted – 2 – ABSTRACT The detection of delayed emission at X-ray optical and radio wavelengths, " after-glow " , suggests that the relativistic shell which emitted the initial GRB due to internal shocks decelerates on encountering an external medium, giving rise to the after-glow. We explore the interaction of a relativistic shell with a uniform inter stellar medium (ISM), up to the non relativistic 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 t 2. On longer time scales (more than ∼ 10 sec), 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 after-glow overlaps the main burst, otherwise there is a time separation between the two. On the long time scale, the flow decelerate in a self similar way, reaching non relativistic velocities after ∼ 30 days. Explicit expressions for the radial profiles of this self similar deceleration are given. Due to 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/16γ 2 c which is a factor of 8 different from the usual expression. The majority of particles are those of the original ejecta (and not the ISM) up to about ∼ 900s. These particles reach sub-relativistic velocities on a time scale of ∼ 2hours, well before the flow becomes sub-relativistic. Therefore the ejecta particles are probably unimportant for most of the after-glow radiation. We show that even though only a small fraction of the energy is given to the electrons, most of the energy can be radiated over time. If this fraction is greater than ∼ 10% radiation losses will significantly influence the hydrodynamical evolution at early times (less than 1day).