Dependence of the spectra of gamma-ray bursts on the initial energy distribution in the ultrarelativistically expanding shell

Abstract

The initial stage of a gamma-ray burst can be described by the diffusion of radiation in the shell which is ultrarelativistically widening. We have herein considered the case when the initial energy distribution in the shell depends on the depth according to a linear law. At the same time, the slope of the initial energy distribution function determines the effective temperature and the instantaneous spectrum of the gamma-ray burst at the initial time moment, as well as the time-integrated spectrum at high frequencies. Moreover, we investigated the dependence of the instantaneous spectrum of the gamma-ray burst on the time of arrival: if it increases, the flux maximum is observed for a smaller frequency independently of the initial energy distribution in the shell. It the time of arrival is larger than 1 s, the instantaneous spectrum is independent of the initial energy distribution in the shell. The slope of the power-law part of the time-integrated spectrum at lower frequencies is the same for any initial energy distribution in the shell and it is equal to 1.98; at higher frequencies, it becomes steeper with increasing the slope of the function corresponding to the energy distribution in the shell at the initial instant of time, and it has values between –2.38 and –13.73.This allows us to explain with the help of our model a large number of gamma-ray bursts that have a typical Band spectrum.