In the dark matter ( DM ) halos embedding galaxies and galaxy systems the ‘ entropy ’ K \equiv \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 \propto r ^ { \alpha } throughout the halos ’ bulk , with \alpha around 1.25 .
Taking up from phenomenology just that \alpha \approx 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 _ { \mathrm { crit } } ( \alpha ) 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 ; these correspond to halos ’ current masses in the range 10 ^ { 11 } -10 ^ { 15 } M _ { \odot } .
Our range of \alpha applies since the transition time that - both in our semianalytic 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 .
In our range of \alpha we provide a close fit for the relation \kappa _ { \mathrm { crit } } ( \alpha ) , and discuss a related physical interpretation in terms of incomplete randomization of the infall kinetic energy through dynamical mixing .
We also give an accurate analytic representation of the \alpha -profiles with parameters derived from the Jeans equation ; this provides straightforward precision fits to recent detailed data from gravitational lensing in and around massive galaxy clusters , and thus replaces the empirical NFW formula relieving the related problems of high concentration and old age .
We finally stress how our findings and predictions as to \alpha and \kappa _ { \mathrm { crit } } contribute to understand hitherto unsolved issues concerning the fundamental structure of DM halos .