  [abridged] In the dark matter (DM) halos embedding galaxies and galaxy
systems the `entropy' K = \sigma^2 / \rho^{2/3} (a quantity that combines the
radial velocity dispersion \sigma with the density \rho) is found from
intensive N-body simulations to follow a powerlaw run K ~ r^{\alpha} throughout
the halos' bulk, with \alpha around 1.25. Taking up from phenomenology just
that \alpha ~ 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 \alpha-profiles, marked by the entropy slope \alpha
itself, and by the maximal gravitational pull \kappa_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 \alpha to within the
narrow range 1.25-1.29 from galaxies to galaxy systems. Our range of \alpha
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
\alpha-profiles with parameters derived from the Jeans equation. We finally
stress how our findings and predictions as to \alpha and \kappa_crit contribute
to understand hitherto unsolved issues concerning the fundamental structure of
DM halos.
