Anisotropic distribution functions for spherical galaxies

Abstract

A method is presented for finding anisotropic distribution functions for stellar systems with known, spherically symmetric, densities, which depends only on the two classical integrals of the energy and the magnitude of the angular momentum. It requires the density 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. The products of the density and its radial and transverse velocity dispersions can be also expressed as a sum of products of functions of the potential and of the radial coordinate. Several examples are given, including some of new anisotropic distribution functions. This device can be extended further to the related problem of finding two-integral distribution functions for axisymmetric galaxies.Comment: 5 figure

Two-integral distribution functions for axisymmetric systems

“…These DFs depend only on the two integrals of the energy and the magnitude of the angular momentum about the symmetry axis. These formulae are also an extension of those shown by Jiang & Ossipkov (2007) for finding anisotropic DFs for spherical galaxies. A type of two‐integral DF which is a sum of products of functions only of the energy and powers of the magnitude of the angular momentum with respect to the axis of symmetry is derived in and another, which is a sum of products of functions only of a special variable and powers of the magnitude of the angular momentum about the axis of symmetry, in .…”

“…Some formulae of the two‐integral DFs can be obtained for stellar systems with known axisymmetric density as a sum of products of functions only of the potential and a special function (or power) only of the radial coordinate, that is, these DFs are a sum of products of functions only of a special variable (or the energy) and a power only of the magnitude of the angular momentum about the axis of symmetry. They come from a combination of the ideas of Eddington and Fricke and they are also an extension of those shown by Jiang and Ossipkov (2007) for finding anisotropic DFs for spherical galaxies. As an analogue for spherical models, 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 the radial coordinate.…”

“…Camm 1952; Veltmann 1961; Bouvier 1962, 1963; Veltmann 1965; Kuzmin & Veltmann 1967, 1973; Veltmann 1979, 1981; Kent & Gunn 1982; Dejonghe 1986, 1987; Dejonghe & Merritt 1988). Recently, a method was presented by Jiang & Ossipkov (2007) for finding anisotropic DFs for spherical galaxies. This is a combination of Eddington's (1916) formula and Fricke's (1952) expansion idea.…”

Axially Symmetric Post-Newtonian Stellar Systems

“…For galaxies in general, the average time between collisions or individual meetings (mean collision time) is greater than the system's life time. Many of the current models take Newton's law as their field equations [1,2,3,4,5,6,7,8,9,10,11]. However, many models have been developed recently under general relativity [12,13,14,15,16,17,18,19], being one of the main motivations for including corrections made by general relativity the actual incompatibility between the rotation curves of theoretical models and the ones observed.…”

Two-integral distribution functions for axisymmetric stellar systems with separable densities

“…This paper is a continuity of our recent work (Jiang & Ossipkov 2007a,b) to construct a self‐consistent stellar system by means of finding a two‐integral distribution function (DF) for a stellar system with a known gravitational potential. Jiang & Ossipkov (2007a) have shown a method of finding anisotropic DFs for spherical galaxies. This is a combination of Eddington's (1916) formula and Fricke's (1952) expansion idea.…”