4. Collision rate and X-ray flux
Let us now estimate the number of HNS and their consequent X-ray luminosity. We assume a number of neutron stars spherically distributed in the galaxy (up to the radius kpc) and a number of dark clusters (assumed for simplicity of the same mass ). Here is the fraction of dark matter in the form of molecular clouds. According to the standard halo model (see e.g. De Paolis, Ingrosso & Jetzer 1996d ), the dark cluster number density is given by
where pc-3 is the local density parameter, kpc is the core radius and kpc is the local galactocentric distance. In the present case Eq. (1) holds for which, in our scenario, is the minimum distance at which dark clusters form.
As regard the spatial distribution of the neutron stars, for the sake of simplicity, we assume the same distribution law as for the dark matter in Eq. (1), but with a different core radius b which we take as a free parameter in the range 5-150 kpc (the latter value giving a near uniform distribution). Correspondingly, the local neutron star number density results in the range pc-3. Another free parameter of our calculation is the velocity distribution of the neutron stars which we take to be maxwellian with dispersion velocity in the range 200-600 km s-1.
where is the effective cross-section for the collision and the relative velocity between neutron stars and dark clusters. The obtained values of are in the range
correspondingly to the extreme cases of kpc and kpc, respectively.
The crossing time of a HNS in a dark cluster is yr, so that the total number of HNS crossing today dark clusters 4 is . This quantity is found to be independent from , being .
where the local number density of HNS is given by and results to be in the range pc-3 for kpc and kpc, respectively. As one expects, if dark clusters and neutron stars are distributed similarly, crossings happen prevalently in the inner part of the galaxy.
The instantaneous accretion rate into the surface of a neutron star is described by the Bondi formula (Shapiro & Teukolsky 1983 )
where (in the range 0.1-10 km s-1) is the sound velocity in the accreting medium (a molecular cloud at temperature near to that of the CBR), is an efficiency constant of order unity, is the mass of the neutron star and is the accreting matter density. The latter quantity is expected to vary in the range g cm-3, for an internal number density between cm-3. It is also well known that the accreting matter corresponds to the luminosity
where km is the radius of the neutron star. Therefore, according to we have and in the range and , respectively.
The luminosity in Eq. (6) strongly depends on the value of which enters with the inverse cube power. Thus, for the same values of the density , slower neutron stars have higher X-ray luminosities. In the framework of our scenario, we expect in particular that for the DCNS is of the order of the velocity dispersion ( km s-1) inside dark clusters. Therefore, DCNS continuously accrete and emit X-rays with a luminosity .
The effective temperature of the emitting region on the surface of the neutron star can be estimated (assuming a black body emission) by using the relation , where A is the area of the emitting region (Shapiro & Teukolsky 1983 ). In the case of magnetized neutron stars, the incoming material is channeled to the poles along the magnetic lines and , where is the Alfvén radius. Therefore, the effective temperature of the emitting region can be expressed in the form
which implies substantial emission in the X-ray band, for a large range of values of and B.
© European Southern Observatory (ESO) 1997
Online publication: June 30, 1998