Constraints on velocity anisotropy of spherical systems with separable augmented densities

TL;DR

This paper investigates how the assumption of separable augmented densities in spherical systems constrains velocity anisotropy profiles, revealing that not all anisotropy variations are physically consistent with this assumption.

Contribution

It establishes specific constraints on anisotropy parameters derived from the separability assumption, linking the behavior of higher-order velocity moments to the augmented density form.

Findings

Certain anisotropy profiles are incompatible with separable augmented densities.

The radial function is fully determined by the anisotropy parameter behavior.

Non-negativity of the distribution function imposes restrictions on anisotropy variations.

Abstract

If the augmented density of a spherical anisotropic system is assumed to be multiplicatively separable to functions of the potential and the radius, the radial function, which can be completely specified by the behavior of the anisotropy parameter alone, also fixes the anisotropic ratios of every higher-order velocity moment. It is inferred from this that the non-negativity of the distribution function necessarily limits the allowed behaviors of the radial function. This restriction is translated into the constraints on the behavior of the anisotropy parameter. We find that not all radial variations of the anisotropy parameter satisfy these constraints and thus that there exist anisotropy profiles that cannot be consistent with any separable augmented density.