The Distribution of Burst Energy and Shock Parameters for Gamma-ray Bursts

We calculate the statistical distribution of observed afterglow flux, in some fixed observed frequency band, and at some fixed observer time after the explosion ($t_{obs}$) in two models - one where the explosion takes place in a uniform density medium and the other where the surrounding medium has a power-law stratification such as is expected for a stellar wind. For photon energies greater than about 500 electron-volt and $t_{obs}\gta 10^3$ sec the afterglow flux distribution functions for the uniform ISM and the wind models are nearly identical. We compare the width of the theoretical distribution with the observed x-ray afterglow flux and find that the FWHM of the distribution for energy in explosion and the fractional energy in electrons ($\epsilon_e$) are each less than about one order of magnitude and the FWHM for the electron energy index is 0.6 or less.