Exact versus approximate solutions in Gamma-Ray Burst afterglows

Abstract

We have recently obtained the exact analytic solutions of the relativistic equations relating the radial and time coordinate of a relativistic thin uniform shell expanding in the interstellar medium in the fully radiative and fully adiabatic regimes. We here re-examine the validity of the constant-index power-law relations between the Lorentz gamma factor and its radial coordinate, usually adopted in the current Gamma-Ray Burst (GRB) literature on the grounds of an "ultrarelativistic" approximation. Such expressions are found to be mathematically correct but only approximately valid in a very limited range of the physical and astrophysical parameters and in an asymptotic regime which is reached only for a very short time, if any, and are shown to be not applicable to GRBs.