Rings in fireball afterglows

We derive the equation for the surface of equal arrival time of radiation from a thin relativistic shell interacting with an external medium, representing the afterglow of a gamma-ray burst produced by a fireball. Due to the deceleration, these surfaces become distorted ellipsoids and, at late times, most of the light comes from a ringlike region whose width depends only on age. We analyze the shape of these surfaces for different dynamic and radiative regimes and homogeneous or power-law external densities. We tabulate the most relevant parameters describing the surfaces and the source brightness distribution, both for bolometric and fixed frequency observations, which are useful for more accurate analytic estimates of the afterglow evolution.