Density Profiles of Collisionless Equilibria. I. Spherical Isotropic Systems

Abstract

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