Relating quasi-stationary one zone emission models to expanding relativistic shocks

Abstract

For an expanding spherical relativistic shock, we derive relations between the parameters of downstream emitting zone and the quantities measured by a distant observer. These relations are formulated in terms of dimensionless effective coefficients combined with self-evident dimensional estimates. Our calculations take into account evolution of the shock's Lorentz factor, geometrical delay due to the shock's front curvature, and angular dependence of Lorentz boost for frequency and brightness. The relations are designed primarily for application in Gamma-Ray Burst afterglow studies, although they may have a broader use.