Distribution functions for spheroids

TL;DR

This paper discusses the construction of galaxy models using distribution functions based on action integrals, highlighting physical constraints, stability issues, and methods for various system geometries.

Contribution

It provides new methods for constructing distribution functions for spherical and flattened galaxy systems with specified properties.

Findings

Unphysical velocity distributions occur unless certain conditions are met as J_phi -> 0.

Constructed DFs for isotropic and radially biased spherical systems with specified f(J).

Radially-biased spherical systems are generally unstable to quadrupolar perturbations.

Abstract

Galaxy models comprising several components (including dark matter) that are bound by the self-consistently generated gravitational field are readily constructed from distribution functions (DFs) that are analytic functions of the action integrals J. We explain why such models have unphysical velocity distributions unless the DFs of hot components satisfy certain conditions as J_\phi -> 0. We show how DFs for both isotropic and radially biased spherical systems can be constructed with specified f(J). We show how to construct DFs for flattened systems with significant velocity anisotropy. Construction of self-consistent models rather than populations that are confined by an external potential leads to the conclusion that radially-biased spherical systems are generically unstable to quadrupolar perturbations. Chaos is likely key to maintenance of these constraints during adiabatic disc growth.