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

TL;DR

This paper derives explicit formulas for two-integral distribution functions in axisymmetric stellar systems with separable densities, simplifying calculations and providing examples for specific models like Jaffe and Plummer.

Contribution

It introduces a computationally efficient method to obtain two-integral DFs for axisymmetric systems with separable densities, connecting contour integral and Laplace-Mellin approaches.

Findings

Derived explicit expressions for even and odd parts of the two-integral DF.

Demonstrated the method with Jaffe and Plummer models.

Connected Hunter-Qian and Dejonghe formalisms.

Abstract

We show different expressions of distribution functions (DFs) which depend 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. The density of the system is required to be a product of functions separable in the potential and the radial coordinate, where the functions of the radial coordinate are powers of a sum of a square of the radial coordinate and its unit scale. The even part of the two-integral DF corresponding to this type of density is in turn a sum or an infinite series of products of functions of the energy and of the magnitude of the angular momentum about the axis of symmetry. A similar expression of its odd part can be also obtained under the assumption of the rotation laws. It can be further shown that these expressions are in fact equivalent to those obtained by using Hunter and Qian's contour integral formulae for the system. This method is generally computationally preferable to the contour integral method. Two examples are given to obtain the even and odd parts of their two-integral DFs. One is for the prolate Jaffe model and the other for the prolate Plummer model. It can be also found that the Hunter-Qian contour integral formulae of the two-integral even DF for axisymmetric systems can be recovered by use of the Laplace-Mellin integral transformation originally developed by Dejonghe.