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

TL;DR

This study investigates how the force exponent $oldsymbol{ extit{ extalpha}}$ influences the equilibrium and stability of radially anisotropic collisionless systems with $r^{- extalpha}$ forces, using phase-space distribution functions and N-body simulations.

Contribution

It constructs self-consistent anisotropic models for $r^{- extalpha}$ forces and determines the minimum anisotropy radius for phase-space consistency across different $ extalpha$ values.

Findings

The minimum anisotropy radius decreases as $ extalpha$ decreases.

Radial kinetic energy capacity increases for lower $ extalpha$ in marginally stable models.

Isotropic systems are stable and consistent within the studied $ extalpha$ range.

Abstract

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.