GRB Energies and Eγ-Epeak Correlation with the Jet Expanding Laterally at the Sound Speed

A Gamma-ray burst (GRB) is generally believed to be a jet with a small opening angle, this opening angle is usually derived with the afterglow light curve break time using an analytical method. Here we show that the method is not accurate. Using the set of equations of hydrodynamic evolution with the sideways expansion at the local sound speed derived by previous authors and the observed light curve break times, we numerically derive the initial opening angles. Then the collimation-corrected energies (E-gamma) for a sample of GRBs are calculated. They are found to show a wide spread, suggesting that the previously declared clustering by some authors may not exist. Also, the E-peak - E-gamma relation, claimed by some other authors (E-peak is the spectral peak energy), is found still to hold, with a slightly stronger correlation.