Two-integral distribution functions for axisymmetric stellar systems with separable densities
Article first published online: 26 NOV 2007
Monthly Notices of the Royal Astronomical Society
Volume 382, Issue 4, pages 1971–1981, December 2007
How to Cite
Jiang, Z. and Ossipkov, L. (2007), Two-integral distribution functions for axisymmetric stellar systems with separable densities. Monthly Notices of the Royal Astronomical Society, 382: 1971–1981. doi: 10.1111/j.1365-2966.2007.12514.x
- Issue published online: 26 NOV 2007
- Article first published online: 26 NOV 2007
- Accepted 2007 September 25. Received 2007 September 15; in original form 2007 July 14
- stellar dynamics;
- celestial mechanics
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 coordinates, 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 & 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.