The energy distribution of GRBs

We analyze the distribution of total energy of bursts with optical afterglows. Our sample contains eleven bursts for which there are BATSE data allowing detailed fits to the spectra. Six of these bursts have measured redshifts while five have host galaxies whose redshifts are unknown. Using a new technique based on the distribution of magnitudes of observed high redshift galaxies, we bracket the expected redshifts of these host galaxies. We perform a maximum likelihood fit for the energy distribution of these GRBs. Assuming that the total energy emitted has a log-normal distribution, we find that the average isotropic energy emitted is E-gamma = 1.5(-0.6)(+1.5) x 10(53)ergs (for H-0 = 65 km s(-1) Mpc(-1), Omega (m) = 0.3 and Omega (Lambda) = 0.7) with a standard deviation sigma (gamma) = 1.2(-0.4)(+0.6). The corresponding distribution of X-ray afterglow energy (for seven bursts) is significantly narrower with sigma (x) = 0.5(-0.25)(+0.4) and E-x = 3.8(-1.5)(+2) x 10(51)ergs, in agreement with the prediction of the patchy shell model of Kumar and Piran [1]. We also give a table with the detailed fits to the spectra of the 11 GRBs with optical afterglow with BATSE data.