Dark Matter Equilibria in Galaxies and Galaxy Systems

abridged In the dark matter (DM) halos embedding galaxies and galaxy systems the `entropy' K = ² / ^2/3 (a quantity that combines the radial velocity dispersion with the density) is found from intensive N-body simulations to follow a powerlaw run K ~ r^ throughout the halos' bulk, with around 1. 25. Taking up from phenomenology just that ~ const applies, we cut through the rich analytic contents of the Jeans equation describing the self-gravitating equilibria of the DM; we specifically focus on computing and discussing a set of novel physical solutions that we name -profiles, marked by the entropy slope itself, and by the maximal gravitational pull crit required for a viable equilibrium to hold. We then use an advanced semianalytic description for the cosmological buildup of halos to constrain the values of to within the narrow range 1. 25-1. 29 from galaxies to galaxy systems. Our range of applies since the transition time that - both in oursemianalytic description and in state-of-the-art numerical simulations - separates two development stages: an early violent collapse that comprises a few major mergers and enforces dynamical mixing, followed by smoother mass addition through slow accretion. We also give an accurate analytic representation of the -profiles with parameters derived from the Jeans equation. We finally stress how our findings and predictions as to and crit contribute to understand hitherto unsolved issues concerning the fundamental structure of DM halos.