Phase-space consistency of stellar dynamical models determined by separable augmented densities

TL;DR

This paper investigates the conditions under which spherical stellar dynamical models with separable augmented densities produce non-negative distribution functions, providing new mathematical criteria and inversion formulas for phase-space consistency.

Contribution

It generalizes known conditions for phase-space consistency to arbitrary separable augmented densities and derives a specialized inversion formula for the distribution function.

Findings

Necessary and sufficient conditions for non-negativity of distribution functions.

Generalization of previous conditions to broader classes of models.

New inversion formula for separable augmented densities.

Abstract

Assuming the separable augmented density, it is always possible to construct a distribution function of a spherical population with any given density and anisotropy. We consider under what conditions the distribution constructed as such is in fact non-negative everywhere in the accessible phase-space. We first generalize known necessary conditions on the augmented density using fractional calculus. The condition on the radius part R(r^2) (whose logarithmic derivative is the anisotropy parameter) is equivalent to the complete monotonicity of R(1/w)/w. The condition on the potential part on the other hand is given by its derivative up to any order not greater than (3/2-beta) being non-negative where beta is the central anisotropy parameter. We also derive a specialized inversion formula for the distribution from the separable augmented density, which leads to sufficient conditions on separable augmented densities for the non-negativity of the distribution. The last generalizes the similar condition derived earlier for the generalized Cuddeford system to arbitrary separable systems.