Two-integral distribution functions for axisymmetric systems

TL;DR

This paper develops formulae for calculating two-integral distribution functions in axisymmetric stellar systems, extending previous methods and enabling application to models like those of Binney and Lynden-Bell.

Contribution

It introduces new formulae for two-integral DFs based on density and velocity dispersions, extending prior spherical models to axisymmetric systems.

Findings

Applicable to Binney and Lynden-Bell models

Allows calculation of odd distribution functions

Extends spherical models to axisymmetric systems

Abstract

Some formulae are presented for finding two-integral distribution functions (DFs) which depends only on the two classical integrals of the energy and the magnitude of the angular momentum with respect to the axis of symmetry for stellar systems with known axisymmetric densities. They come from an combination of the ideas of Eddington and Fricke and they are also an extension of those shown by Jiang and Ossipkov for finding anisotropic DFs for spherical galaxies. The density of the system is required to be expressed as a sum of products of functions of the potential and of the radial coordinate. The solution corresponding to this type of density is in turn a sum of products of functions of the energy and of the magnitude of the angular momentum about the axis of symmetry. The product of the density and its radial velocity dispersion can be also expressed as a sum of products of functions of the potential and of the radial coordinate. It can be further known that the density multipied by its rotational velocity dispersion is equal to a sum of products of functions of the potential and of the radial coordinate minus the product of the density and the square of its mean rotational velocity. These formulae can be applied to the Binney and the Lynden-Bell models. An infinity of the odd DFs for the Binney model can be also found under the assumption of the laws of the rotational velocity.