Semianalytical Dark Matter Halos and the Jeans Equation

Although N-body studies of dark matter halos show that the density profiles, (rho)(r), are not simple power laws, the quantity (rho)/(sigma)^3, where (sigma)(r) is the velocity dispersion, is in fact a featureless power law over ~3 decades in radius. In the first part of the paper we demonstrate, using the semianalytic Extended Secondary Infall Model (ESIM), that the nearly scale-free nature of (rho)/(sigma)^ 3 is a robust feature of virialized halos in equilibrium. By examining the processes in common between numerical N-body and semianalytic approaches, we argue that the scale-free nature of (rho)/(sigma)^3 cannot be the result of hierarchical merging; rather it must be an outcome of violent relaxation. The empirical results of the first part of the paper motivate the analytical work of the second part of the paper, where we use (rho)/(sigma)^ 3 (alpha)r^-(alpha) as an additional constraint in the isotropic Jeans equation of hydrostatic equilibrium. Our analysis shows that the constrained Jeans equation has different types of solutions, and in particular, it admits a unique "periodic" solution with (alpha)= 1.9444. We derive the analytic expression for this density profile, which asymptotes to inner and outer profiles of (rho)~ r^-0.78 and (rho)~ r^-3.44, respectively.