Radially anisotropic systems with $r^{-alpha}$ forces: equilibrium states

We continue the study of collisionless systems governed by additive r-α interparticle forces by focusing on the influence of the force exponent α on radial orbital anisotropy. In this preparatory work, we construct the radially anisotropic Osipkov-Merritt phase-space distribution functions for self-consistent spherical Hernquist models with r-α forces and 1 ≤ α < 3. The resulting systems are isotropic at the centre and increasingly dominated by radial orbits at radii larger than the anisotropy radius ra. For radially anisotropic models we determine the minimum value of the anisotropy radius rac as a function of α for phase-space consistency (such that the phase-space distribution function is nowhere negative for ra ≥ rac). We find that rac decreases for decreasing α, and that the amount of kinetic energy that can be stored in the radial direction relative to that stored in the tangential directions for marginally consistent models increases for decreasing α. In particular, we find that isotropic systems are consistent in the explored range of α. By means of direct N-body simulations, we finally verify that the isotropic systems are also stable