The dark matter distribution function and halo thermalization from the Eddington equation in galaxies

We find the distribution function f(E) for dark matter (DM) halos in galaxies and the corresponding equation of state from the (empirical) DM density profiles derived from observations. We solve for DM in galaxies the analogous of the Eddington equation originally used for the gas of stars in globular clusters. The observed density profiles are a good realistic starting point and the distribution functions 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 dark matter (WDM). With these methods we find: (i) Cored density profiles behaving quadratically for small distances rho(r) =(r -> 0) rho(0) - Kr-2 produce distribution functions which are finite and positive at the halo center while cusped density profiles always produce divergent distribution functions at the center. (ii) Cored density profiles produce approximate thermal Boltzmann distribution functions for r less than or similar to 3r(h) where r(h) is the halo radius. (iii) 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) = mv(2)(r)/3. T(r) turns out to be constant in the same region where the distribution function is thermal and exhibits the same temperature within the percent. The self-gravitating DM gas can thermalize despite being collisionless because it is an ergodic system. (iv) The DM halo can be consistently considered at local thermal equilibrium with: (a) a constant temperature T(r) = T-0 for r less than or similar to 3r(h), (b) a space dependent temperature T(r) for 3rh < r less than or similar to R-virial, which slowly decreases with r. That is, the DM halo is realistically a collisionless self-gravitating thermal gas for r less than or similar to R-virial. (v) T(r) outside the halo radius nicely follows the decrease of the circular velocity squared.