Analytic Light-Curves of Gamma-Ray Burst Afterglows: Homogeneous versus Wind External Media

Assuming an adiabatic evolution of a Gamma-Ray Burst (GRB) remnant interacting with an external medium, we calculate the injection, cooling, and absorption break frequencies, and the afterglow flux for plausible orderings of the break and observing frequencies. The analytical calculations are restricted to a relativistic remnant and, in the case of collimated ejecta, to the phase where there is an insignificant lateral expansion. Results are given for both a homogeneous external medium and for a wind ejected by the GRB progenitor. We compare the afterglow emission at different observing frequencies, for each type of external medium. It is found that observations at sub-millimeter frequencies during the first day provide the best way of discriminating between the two models. By taking into account the effect of inverse Compton scatterings on the electron cooling, a new possible time-dependence of the cooling break is identified. The signature of the up-scattering losses could be seen in the optical synchrotron emission from a GRB remnant interacting with a pre-ejected wind, as a temporary mild flattening of the afterglow decay. The up-scattered radiation itself should be detected in the soft X-ray emission from GRB remnants running into denser external media, starting few hours after the main event.