Dark Matter Halos: the dynamical basis of effective empirical models

We investigate the dynamical basis of the classic empirical models (specifically, Sérsic-Einasto and generalized NFW) that are widely used to describe the distributions of collisionless matter in galaxies. We submit that such a basis is provided by our α-profiles, shown to constitute solutions of the Jeans dynamical equilibrium with physical boundary conditions. We show how to set the parameters of the empirical in terms of the dynamical models; we find the empirical models, and specifically Sérsic-Einasto, to constitute a simple and close approximation to the dynamical models. Finally, we discuss how these provide a useful baseline for assessing the impact of the small-scale dynamics that may modulate the density slope in the central galaxy regions.