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 (tobs) 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 tobs ∼> 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 (ǫe) are each less than about one order of magnitude and the FWHM for the electron energy index is 0.6 or less. Subject headings: gamma rays: bursts – gamma-rays: theory