Statistics of gamma-ray bursts - Homogeneous spherical models

The BATSE team has reported that gamma-ray bursts are distributed highly isotropically on the sky, and that the rate of detection of bursts with peak count rates greater than C is proportional to Cgamma^ with γ = -0. 8 (Meegan et al; Fishman). Here, it is shown that these observations may be understood if (1) the bursts occupy a limited spherical volume in space, nearly if not exactly centered on the solar system; (2) the bursts with largest maximum photon emission rates are bright enough to be seen to distances beyond the boundary of the volume; (3) the bursts with smallest photon emission rates can be detected only to distances well within the boundary of the volume; and (4) the distribution of maximum photon emission rates N is proportional to N^-p^ with p = 1 - γ = 1. 8. Two different models are considered in which the burst population is uniformly distributed in a bounded spherical region of space. In the first model, bursts are assumed to be extragalactic, with a comoving density that is independent of time. In this model, the particle horizon, at a distance ~1/H₀_, provides the volume limit, and the burst distribution on the sky is isotropic by fiat. It is demonstrated that γ approaches 1 - p asymptotically in a k = 0 = 1 - OMEGA universe even when color corrections are included. In the second model, bursts are assumed to reside in a nearby (possibly Galactic) halo of radius R whose center is a distance x₀_ R from the solar system. In the context of this (uniform) halo model, observational bounds on the dipole moment of the burst distribution on the sky require that x₀_ ~ 79 kpc (1 σ) or R ~> 26 kpc (3 σ). If gamma-ray bursts are Galactic phenomena, then their largest possible luminosities are >~ 10⁴2^ ergs s^-1^, and if they are extragalactic phenomena, >~ 10⁵3^ ergs s^-1^, uncorrected for possible relativistic beaming.