2-INTEGRAL DISTRIBUTION-FUNCTIONS FOR AXISYMMETRICAL GALAXIES

We present a new method for finding distribution functions, which depend only on the classical integrals of energy and angular momentum, for stellar systems with known axisymmetric densities. Our method is the analogue for the axisymmetric case of Eddington's classical solution for the isotropic distribution function, depending only on energy, of a known spherical density. Like his method, ours requires that the density be expressed as a function of the potential, and now also of a radial coordinate. Our solution is also an integral which is derived directly from the density, and hence can be used with complicated densities. Unlike Eddington's solution, ours is a contour integral. A numerical quadrature is generally required to evaluate this solution, but contour integrals can be computed accurately by numerical quadrature. This is a simpler and much more accurate procedure than direct solution of the integral equation for the distribution function, and is even preferable to an explicit evaluation if the latter is an infinite series, such as is obtained using Fricke's method. We give several examples, including some for which our distribution functions are new. Our method can be extended simply to the related problems of finding anisotropic distribution functions for spherical or disc systems.