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

We calculate the luminosity function for gamma-ray burst afterglows in some fixed observed frequency band and at some fixed elapsed time in observer frame ( t obs ) in two models—one in which the explosion takes place in a uniform density medium and another in which the density falls off as inverse square (expected for stellar winds). For photon energies greater than about 500 eV and t obs ≳ 10 3 s, the afterglow flux is independent of interstellar medium (ISM) density and luminosity functions for wind and uniform ISM are identical. We deduce from the width of the observed X-ray afterglow distribution, 5 hr after the burst, that the FWHM of the distribution for isotropic energy in explosion and the fractional energy in electrons ( e ) are each less than about 1 order of magnitude and the FWHM for the electron energy index is 0.6 or less.