The steady state distribution function of dark matter particles in galaxy clusters: The perseus cluster as an illustrative example

High resolution X-ray data on the distribution of hot gas inside galaxy clusters, together with optical data on the galaxy distribution inside a cluster, provide information about the radial distributions of luminous and dark matter densities on scales of at least several cluster core radii. Assuming that dark matter (DM) particles form a collisionless self-gravitating system, we use these two distributions to derive the steady state distribution function of dark matter particles and study some properties of the ensemble. We show the following: (1) The distribution function of DM particles does not look Maxwellian. It depends on the distance to the center of the cluster, and it vanishes at finite values of particle velocity. (2) The ensemble of DM particles cannot be described by an ''isothermal'' sphere model. The velocity dispersion of DM particles decreases at large distance from the center of the cluster. (3) DM particles are distributed in a broad interval of energies, with the maximum of this distribution occurring at an energy approximately equal to twice the potential energy at the center of the cluster. (4) The ensemble of DM particles associated with the Perseus galaxy cluster is stable with respect to small short-wavelength perturbations.