The Luminosity and Angular Distributions of Long GRBs

The realization that the total energy of 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, $L\propto\theta^{-2}$. Such a relation implies that the GRB luminosity function determines the angular distribution. We re-derive 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 $4.4 \times 10^{51}$ erg/sec, a typical jet angle of 0.12 rad and a local GRB rate of $0.44 h_{65}^{3}$ Gpc$^{-3}$yr$^{-1}$. The angular distribution implied by $L\propto\theta^{-2}$ agrees well with the observed one, and implies a correction factor to the local rate due to beaming of $75\pm25$ (instead of 500 as commonly used). The inferred overall local GRB rate is $33\pm11 h_{65}^{3}$ Gpc$^{-3}$yr$^{-1}$. The luminosity function and angle distribution obtained within the universal structured jet model, where the angular distribution is essentially $\propto \theta$ and hence the luminosity function must be $\propto L^{-2}$, deviate from the observations at low peak fluxes and, correspondingly, at large angles. The corresponding correction factor for the universal structure jet is $\sim 20 \pm 10$.