Relativistic Conic Beams and Spatial Distribution of Gamma‐Ray Bursts

We study the statistics of gamma-ray bursts, assuming that gamma-ray burstsare cosmological and they are beamed in the form of a conical jet with a largebulk Lorentz factor $\sim 100$. In such a conic beam, the relativistic ejectamay have a spatial variation in the bulk Lorentz factor and the densitydistribution of gamma-ray emitting jet material. An apparent luminosityfunction arises because the axis of the cone is randomly oriented with respectto the observer's line of sight. The width and the shape of the luminosityfunction are determined by the ratio of the beam opening angle of the conicaljet to the inverse of the bulk Lorentz factor, when the bulk Lorentz factor andthe jet material density is uniform on the photon emitting jet surface. Wecalculate effects of spatial variation of the Lorentz factor and the spatialdensity fluctuations within the cone on the luminosity function and thestatistics of gamma-ray bursts. In particular, we focus on the redshiftdistribution of the observed gamma-ray bursts. The maximum distance to and theaverage redshift of the gamma-ray bursts are strongly affected by thebeaming-induced luminosity function. The bursts with the angle-dependentLorentz factor which peaks at the center of the cone have substantially higheraverage gamma-ray burst redshifts. When both the jet material density and theLorentz factor are inhomogeneous in the conical beam, the average redshift ofthe bursts could be 5 times higher than that of the case in which relativisticjet is completely homogeneous and structureless. Even the simplest models forthe gamma-ray burst jets and their apparent luminosity distributions have asignificant effect on the redshift distribution of the gamma-ray bursts.Comment: 15 pages, 4 figures, submitted to ApJ