Density Profiles of Collisionless Equilibria. I. Spherical Isotropic Systems

We investigate the connection between collisionless equilibria and thephase-space relation between density $\rho$ and velocity dispersion $\sigma$found in simulations of dark matter halo formation, $F=\psd \proptor^{-\alpha}$. Understanding this relation will shed light on the physicsrelevant to collisionless collapse and on the subsequent structures formed. Weshow that empirical density profiles that provide good fits to N-body halosalso happen to have nearly scale-free \psd distributions when in equilibrium.We have also done a preliminary investigation of variables other than $r$ thatmay match or supercede the correlation with $F$. In the same vein, we show that$\rho/\sigma^m$, where $m=3$ is the most appropriate combination to use indiscussions of the power-law relationship. Since the mechanical equilibriumcondition that characterizes the final systems does not by itself lead topower-law $F$ distributions, our findings prompt us to posit that dynamicalcollapse processes (such as violent relaxation) are responsible for the radialpower-law nature of the \psd distributions of virialized systems.