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

We continue the study of collisionless systems governed by additive $r^{-\alpha}$ interparticle forces by focusing on the influence of the force exponent $\alpha$ 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 $r^{-\alpha}$ forces and $1\leq\alpha<3$. The resulting systems are isotropic at the center and increasingly dominated by radial orbits at radii larger than the anisotropy radius $r_a$. For radially anisotropic models we determine the minimum value of the anisotropy radius $r_{ac}$ as a function of $\alpha$ for phase-space consistency (such that the phase-space distribution function is nowhere negative for $r_a\geq r_{ac}$). We find that $r_{ac}$ decreases for decreasing $\alpha$, 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 $\alpha$. In particular, we find that isotropic systems are consistent in the explored range of $\alpha$. By means of direct $N$-body simulations we finally verify that the isotropic systems are also stable.