On radial anisotropy limits in stellar systems

Following earlier authors, we re-examine constraints on the radial velocity anisotropy of generic stellar systems using arguments for phase space density positivity, stability, and separability. It is known that although the majority of commonly used systems have an maximum anisotropy of less than half of the logarithmic density slope \emph{i.e.} $\beta < \gamma/2$, there are exceptions for separable models with large central anisotropy. Here we present a new exceptional case with above-threshold anisotropy locally but with an isotropic center nevertheless. These models are non-separable and we maintain positivity. Our analysis suggests that regions of above-threshold anisotropy are more related to regions of possible secular instability, which might be observed in self-consistent galaxies in a short-lived phase. s an equilibrium solution for a collisionless system.