The Luminosity and Angular Distributions of Long GRBs

The realization that the total energy of GRBs is correlated to 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 two broken power laws 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 $\sim 20 \pm 10$.