Density profiles of collisionless equilibria. I. Spherical isotropic systems

We investigate the connection between collisionless equilibria and the phase-space relation between density rho and velocity dispersion sigma found in simulations of dark matter halo formation, F = rho/sigma(3) proportional to r(-alpha). 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 rho/sigma(3) distributions when in equilibrium. We have also done a preliminary investigation of variables other than r that may match or supersede the correlation with F. In the same vein, we show that rho/sigma(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 rho/sigma(3) distributions of virialized systems.