Recent results from BATSE device onboard GRO observatory have shown full isotropy of gamma-ray bursts locations upon the sky and their unhomogeneous space density ) (Meegan et al. 1992), where is the angle between the source and the galactic center, b is the galactic latitude and signifies averaging upon the whole sample. In the case of pure homogeneous isotropic distribution these quantities would be, respectively, 1/2, 1/3, 0. These observed statistical features of gamma-bursts greatly reinforced interest to the nature of its possible progenitors. An attractive hypothesis of their being old galactic neutron stars begins loosing its credibility, especially in view of negative results of searching for any specral feature in gamma-burst spectra obtained by BATSE. Some indications begins appearing that the gamma-ray bursts can be devided at least into two types - the strong bursts having hard spectra, with >1/2 and the weaker ones with softer spectra and <1/2 (APEX experiment results, Mitrofanov et al 1992; see also Lingenfelter & Higdon 1992). These results might indicate that two populations of gamma-ray burst progenitors distinctive by their space distributions actually exist, viz. that producing weak bursts and lying outside galactic disc, and that producing stronger bursts and hosting in the galactic disc. Brainerd (1992) has shown that the BATSE results can be reconciliated with the old neutron star paradigma if one considers an extended ( kpc) halo made from old neutron stars. The question arises is there any mechanism of natural populating such a halo by old neutron stars?
Space distribution of old galactic neutron stars population as putative gamma-ray burst progenitors has been studied earlier (e.g. Paczynski 1990, Hartmann et al. 1990). By using direct integration of equations of motion they found distribution of old neutron star population, taking as an initial velocity that deduced from observations of radiopulsars. Essentially their results show that it seems difficult for a neutron star to run out too far away from the galactic plane (some kiloparsecs) unless very fast tail of their initial velocity distribution exists.
In this paper we consider another method for obtaining such distributions based essentially on searching for stationary solutions of Boltzmann kinetic equation for the distribution fumction in 6-dimensional phase space (here are 3-dimensional coordinate and velocity, respectively). As an initial space position we take ordinary exponential distribution on r- and z- coordinates, whereas as the intial condition for neutron star velocities a distribution following from contemporal binary star evolutionary scenario is taken.
We show that if one can use the ergodic hypothesis for motion in the galactic potential, then the population of old neutron stars can be extended very far from the galactic plane under reasonable initial conditions.