The evolution of GRB remnants

The detection of the delayed emission in X-ray, optical and radio band, i.e. the afterglow of $\gamma$-ray bursts (GRBs), suggestes that the sources of GRBs are likely to be at cosmological distances. Here we explore the interaction of a relativistic shell with a uniform interstellar medium (ISM) and obtain the exact solution of the evolution of $\gamma$-ray burst remnants, including the radiative losses. We show that in general the evolution of bulk Lorentz factor $\gamma$ satisfies $\gamma \propto t^{-\alpha_{t}}$ when $\gamma \gg 1$, here $\alpha_{t}$ is mainly in the range $9/22 \sim 3/8$, the latter corresponds to adiabatic expansion. So it is clear that adiabatic expansion is a good approximation even when radiative loss is considered. However, in fact, $\alpha_{t}$ is slightly larger than 3/8, which may have some effects on a detailed data analysis. Synchrotron-self-absorption is also calculated and it is demonstrated that the radio emission may become optically thin during the afterglow. Our solution can also apply to the nonrelativistic case ($\gamma \sim 1$), at that time the observed flux decrease more rapidly than that in the relativistic case.