Constraining the Structure of GRB Jets Through the log(N)-log(S)   Distribution
  A general formalism is developed for calculating the luminosity function and the expected number $N$ of observed GRBs above a peak photon flux $S$ for any GRB jet structure. This new formalism directly provides the true GRB rate without the need for a `correction factor'. We apply it to the uniform jet (UJ) and universal structured jet (USJ) models for the structure of GRB jets and perform fits to the observed log(N)-log(S) distribution from the GUSBAD catalog which contains 2204 BATSE bursts. A core angle $\theta_c$ and an outer edge at $\theta_{max}$ are introduced for the structured jet, and a finite range of half-opening angles $\theta_{min}\leq\theta_j\leq\theta_{max}$ is assumed for the uniform jets. The efficiency $\epsilon_\gamma$ for producing gamma-rays, and the energy per solid angle $\epsilon$ in the jet are allowed to vary with $\theta_j$ (the viewing angle $\theta_{obs}$) in the UJ (USJ) model, $\epsilon_\gamma\propto\theta^{-b}$ and $\epsilon\propto\theta^{-a}$. We find that a single power-law luminosity function provides a good fit to the data. Such a luminosity function arises naturally in the USJ model, while in the UJ model it implies a power-law probability distribution for $\theta_j$, $P(\theta_j)\propto\theta_j^{-q}$. The value of $q$ cannot be directly determined from the fit to the observed log(N)-log(S) distribution, and an additional assumption on the value of $a$ or $b$ is required. Alternatively, an independent estimate of the true GRB rate would enable one to determine $a$, $b$ and $q$. The implied values of $\theta_c$ (or $\theta_{min}$) and $\theta_{max}$ are close to the current observational limits. The true GRB rate for the USJ model is found to be $R_{GRB}(z=0)=0.86^{+0.14}_{-0.05} Gpc^{-3} yr^{-1}$. 
Dafne Guetta, Jonathan Granot, and Mitchell C. Begelman