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 \sim 3 decades in radius .
In the first part of the paper we demonstrate , using the semi-analytic 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 semi-analytic approaches , we argue that the scale-free nature of \rho / \sigma ^ { 3 } can not 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 } \propto 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 \sim r ^ { -0.78 } , and \rho \sim r ^ { -3.44 } , respectively .