Analytic Expressions for the Surface Brightness Profile of GRB Afterglow Images

TL;DR

This paper derives analytic expressions for the surface brightness profile of GRB afterglow images, aiding in understanding the physics of the shock and improving measurements of afterglow properties.

Contribution

It provides new analytic and semi-analytic formulas for GRB afterglow images across different spectral segments and density profiles, enhancing modeling accuracy.

Findings

Derived exact profiles for thin-layer emission regions.

Provided semi-analytic expressions for bulk emission regions.

Applicable to general external density profiles.

Abstract

The exact profile of a gamma-ray burst (GRB) afterglow image on the plane of the sky can provide important constraints on the underlying physics. In particular, it can test whether the magnetic field in the emitting shocked external medium decreases significantly with the distance behind the shock front, or remains roughly constant. Moreover, it enables more accurate measurements of the afterglow image size and the expected scintillation properties. In this work analytic expressions are derived for the afterglow image in power law segments (PLSs) of the afterglow synchrotron spectrum in which the emission originates from a very thin layer just behind the shock, while simple semi-analytic expressions are derived for the remaining PLSs in which the emission arises from the bulk of the shocked fluid. In all cases the expressions are for a general power law external density profile, and are convenient to use in afterglow studies.