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 is-proportional-to C(gamma) with gamma = -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 is-proportional-to N(-p) with p = 1 - gamma = 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 approximately 1/H0, provides the volume limit, and the burst distribution on the sky is isotropic by fiat. It is demonstrated that gamma --> 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 less than or similar to 0.11 ("1-sigma") or less than or similar to 0.33 ("3-sigma"); these limits should only decrease for bursts that are distributed nonuniformly in a nonspherical (e.g., ellipsoidal) halo. If x0 R is identified with the distance from the solar system to the Galactic center (x0 R almost-equal-to 8.5 kpc), then these limits imply R greater than or similar to 79 kpc (1-sigma) or R greater than or similar to 26 kpc (3-sigma). If gamma-ray bursts are Galactic phenomena, then their largest possible luminosities are greater than or similar to 10(42) ergs s-1, and if they are extragalactic phenomena, greater than or similar to 10(53) ergs s-1, uncorrected for possible relativistic beaming.