Constraining the Structure of GRB Jets Through the log(N)-log(S) Distribution

Abstract

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 ${\theta }_{min}\le {\theta }_j\le {\theta }_{max}$ 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 $R_{GRB}(z=0)=0.86_{-0.05}^{+0.14}Gp{c}^{-3}y{r}^{-1}$.

9 pages, 5 figures, 1 table; accepted for publication in ApJ