On the augmented density of a spherical anisotropic dynamic system

TL;DR

This paper establishes new mathematical conditions on the augmented density of spherical anisotropic systems, ensuring the physical plausibility of the underlying distribution function through non-negativity constraints.

Contribution

It introduces necessary conditions involving Abel transformations for the augmented density, extending previous results to more general cases of spherical anisotropic systems.

Findings

Partial derivatives of Abel transforms must be non-negative

Conditions recover known results for separable densities

Provides a framework for verifying physical consistency of models

Abstract

This paper presents a set of new conditions on the augmented density of a spherical anisotropic system that is necessary for the underlying two-integral phase-space distribution function to be non-negative. In particular, it is shown that the partial derivatives of the Abel transformations of the augmented density must be non-negative. Applied for the separable augmented densities, this recovers the result of van Hese et al. (2011).