Consistency criteria for generalized Cuddeford systems

TL;DR

This paper derives comprehensive criteria for ensuring the physical plausibility of multi-component Cuddeford stellar systems, revealing that the central cusp-anisotropy relation applies throughout the system and potentially reducing mass-anisotropy degeneracy.

Contribution

It extends previous work by providing inversion formulas and necessary and sufficient conditions for phase-space consistency of generalized Cuddeford models with multiple components.

Findings

Derived analytical conditions for system consistency.

Found the cusp-anisotropy relation applies at all radii.

Suggests mass-anisotropy degeneracy may be less problematic.

Abstract

General criteria to check the positivity of the distribution function (phase-space consistency) of stellar systems of assigned density and anisotropy profile are useful starting points in Jeans-based modeling. Here we substantially extend previous results, and we present the inversion formula and the analytical necessary and sufficient conditions for phase-space consistency of the family of multi-component Cuddeford spherical systems: the distribution function of each density component of these systems is defined as the sum of an arbitrary number of Cuddeford distribution functions with arbitrary values of the anisotropy radius, but identical angular momentum exponent. The radial trend of anisotropy that can be realized by these models is therefore very general. As a surprising by-product of our study, we found that the ``central cusp-anisotropy theorem'' (a necessary condition for consistency relating the values of the central density slope and of the anisotropy parameter) holds not only at the center, but at all radii in consistent multi-component generalized Cuddeford systems. This last result suggests that the so--called mass--anisotropy degeneracy could be less severe than what is sometimes feared.