The luminosity and angular distributions of long-duration gamma-ray bursts

The realization that the total energy of gamma-ray bursts (GRBs) is correlated with their jet break angles motivates the search for a similar relation between the peak luminosity L and the jet break angles, L ∝ θ^-2. Such a relation implies that the GRB luminosity function determines the angular distribution. We rederive the GRB luminosity function using the BATSE peak flux distribution and compare the predicted distribution with the observed redshift distribution. The luminosity function can be approximated by a broken power law with a break peak luminosity of 4.4 × 10⁵¹ ergs s^-1, a typical jet angle of 0.12 rad, and a local GRB rate of 0.44 h₆₅³ Gpc^-3 yr ^-1. The angular distribution implied by L ∝ θ ^-2 agrees well with the observed one and implies a correction factor to the local rate due to beaming of 75 ± 25 (instead of 500, as commonly used). The inferred overall local GRB rate is 33 ± 11 h₆₅ ³ Gpc^-3 yr^-1. The luminosity function and angle distribution obtained within the universal structured jet model, where the angular distribution is essentially ∝θ and hence the luminosity function must be ∝L^-2, deviate from the observations at low peak fluxes and, correspondingly, at large angles. The corresponding correction factor for the universal structure jet is ∼20 ± 10.