We continue the study of collisionless systems governed by additive$r^{-{\it\alpha}}$interparticle forces by focusing on the influence of the force exponent${\it\alpha}$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$r^{-{\it\alpha}}$forces and$1\leqslant {\it\alpha}<3$. The resulting systems are isotropic at the centre 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${\it\alpha}$for phase-space consistency (such that the phase-space distribution function is nowhere negative for$r_{a}\geqslant r_{ac}$). We find that$r_{ac}$decreases for decreasing${\it\alpha}$, 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${\it\alpha}$. In particular, we find that isotropic systems are consistent in the explored range of${\it\alpha}$. By means of direct$N$-body simulations, we finally verify that the isotropic systems are also stable.
Non-existence of positive phase-space distribution functions in quantum mechanics
