Exact versus approximate beaming formulas in Gamma-Ray Burst afterglows

Abstract

We present the exact analytic expressions to compute, assuming the emitted Gamma-Ray Burst (GRB) radiation is not spherically symmetric but is confined into a narrow jet, the value of the detector arrival time at which we start to "see" the sides of the jet, both in the fully radiative and adiabatic regimes. We obtain this result using our exact analytic expressions for the EQuiTemporal Surfaces (EQTSs) in GRB afterglows. We re-examine the validity of three different approximate formulas currently adopted for the adiabatic regime in the GRB literature. We also present an empirical fit of the numerical solutions of the exact equations, compared and contrasted with the three above approximate formulas. The extent of the differences is such as to require a reassessment on the existence and entity of beaming in the cases considered in the current literature, as well as on its consequences on the GRB energetics.