Self-consistent axisymmetric Sridhar-Touma models

Abstract

We construct phase-space distribution functions for the oblate, cuspy mass models of Sridhar & Touma, which may contain a central point mass (black hole) and have potentials of St\"ackel form in parabolic coordinates. The density in the ST models is proportional to a power $r^{-\gamma}$ of the radius, with $0<\gamma<1$. We derive distribution functions $f(E, L_z)$ for the scale-free ST models (no black hole) using a power series of the energy $E$ and the component $L_z$ of the angular momentum parallel to the symmetry axis. We use the contour integral method of Hunter & Qian to construct $f(E, L_z)$ for ST models with central black holes, and employ the scheme introduced by Dejonghe & de Zeeuw to derive more general distribution functions which depend on $E$, $L_z$ and the exact third integral $I_3$. We find that self-consistent two- and three-integral distribution functions exist for all values $0 < \gamma < 1$.