Dynamical insight into dark matter haloes

We investigate, using the spherical Jeans equation, self-gravitating dynamical equilibria satisfying a relation rho/sigma(3)(r) alpha r(-alpha), which holds for simulated dark matter haloes over their whole resolved radial range. Considering first the case of velocity isotropy, we find that this problem has only one solution for which the density profile is not truncated or otherwise unrealistic. This solution occurs only for a critical value of alpha(crit) = 35/18 = 1.9 (4) over bar, which is consistent with the empirical value of 1.9 +/- 0.05. We extend our analysis in two ways: first, we introduce a parameter epsilon to allow for a more general relation rho/sigma(epsilon)(r) alpha r(-alpha); and secondly, we consider velocity anisotropy parametrized by Binney's beta( r) equivalent to 1- sigma(2)(theta)/sigma(2)(r). If we assume beta to be linearly related to the logarithmic density slope gamma(r) equivalent to -(d ln rho / d ln r), which is in agreement with simulations, the problem remains analytically tractable and is equivalent to the simpler isotropic case: there exists only one physical solution, which occurs at a critical a value. Remarkably, this value of a, and the density and velocity-dispersion profiles, depend only on epsilon and the value beta(0) = beta(r = 0), but not on the value beta infinity = beta(r ->infinity) ( or, equivalently, the slope d beta/d gamma of the adopted linear beta-gamma relation). For epsilon = 3, alpha(crit) = 35/18 - 2 beta(0)/ 9 and the resulting density profile is fully analytic ( as are the velocity dispersion and circular speed) with an inner cusp rho alpha r-(( 7+ 10 beta 0)/ 9) and a very smooth transition to a steeper outer power-law asymptote. These models are in excellent agreement with the density, velocity-dispersion and anisotropy profiles of simulated dark matter haloes over their full resolved radial range. If epsilon = 3 is a universal constant, some scatter in beta(0) approximate to 0 may account for some diversity in the density profiles, provided a relation rho/sigma(3)(r) alpha r(-alpha crit) always holds.