The Dark Matter distribution function and Halo Thermalization from the Eddington equation in Galaxies
Résumé
We find the distribution function (DF) f(E) for dark matter (DM) halos in galaxies and the corresponding equation of state from the observed DM halo density profiles. We solve for DM in galaxies the analogous of the Eddington equation used for the gas of stars in globular clusters. The observed density profiles are a good realistic starting point and the DF's derived from them are realistic. We do not make any assumption about the DM nature, the methods developed here apply to any DM kind, though all results are consistent with Warm DM (WDM). Cored density profiles behaving quadratically for small distances rho(r->0) = rho(0)-K r^2 produce DF's which are finite and positive at the halo center while cusped density profiles always produce divergent DF's at the center. Cored density profiles produce thermal Boltzmann DF's functions for r < 3 r_h where r_h is the halo radius.Analytic expressions for the dispersion velocity and the pressure are derived yielding at each halo point an ideal DM gas equation of state with local temperature T(r) = m v^2(r)/3. T(r) turns to be constant in the same region where the DF is thermal. The DM halo can be consistently considered at local thermal equilibrium with: (i) a constant temperature for r < 3 r_h, (ii) a space dependent temperature T(r) for 3 r_h < r < R_{virial}, which slowly decreases with r. That is, the DM halo is realistically a self-gravitating thermal gas for r < R_{virial}. T(r) outside the halo radius nicely follows the decrease of the circular velocity squared. When observational data on density profiles of smaller (dwarf) galaxies will become available, the analytic framework provided here will allow to find the phase--space DF f(E) for such compact galaxies (abridged).