Notes on the stability threshold for radially anisotropic polytrope

TL;DR

This paper investigates the stability of radially anisotropic polytropic stellar systems, reconciling conflicting results by showing that growth rates decrease exponentially as anisotropy approaches isotropy, implying stability for systems with finite lifetime.

Contribution

The authors introduce a new method to determine eigenmodes, demonstrating that growth rates diminish exponentially near isotropy, resolving previous contradictions about system stability.

Findings

Growth rates decrease exponentially as anisotropy approaches isotropy.

Systems with finite lifetime are stable despite weak instability.

Reconciliation of previous conflicting stability results.

Abstract

We discuss some contradictions found in the literature concerning the problem of stability of collisionless spherical stellar systems which are the simplest anisotropic generalization of the well-known polytrope models. Their distribution function $F(E,L)$ is a product of power-low functions of the energy $E$ and the angular momentum $L$, i.e. $F\propto L^{-s}(-E)^q$. On the one hand, calculation of the growth rates in the framework of linear stability theory and N-body simulations show that these systems become stable when the parameter $s$ characterizing the velocity anisotropy of the stellar distribution is lower than some finite threshold value, $s<s_\textrm{crit}$. On the other hand Palmer & Papaloizou (1987) showed that the instability remained up to the isotropic limit $s=0$. Using our method of determining the eigenmodes for stellar systems, we show that the growth rates in weakly radially-anisotropic systems are indeed positive, but decrease exponentially as the parameter $s$ approaches zero, i.e. $\gamma\propto \exp(-s_{\ast}/s)$. In fact, for the systems with finite lifetime this means stability.