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) greater than or equal to 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 (epsilon(e)) are each less than about 1 order of magnitude and the FWHM for the electron energy index is 0.6 or less.