Self-consistent axisymmetric Sridhar--Touma models

Abstract

We construct phase-space distribution functions for the oblate, cuspy mass models of Sridhar & Touma (ST), which may contain a central point mass (black hole) and have potentials of Stäckel form in parabolic coordinates. The density in the ST models is proportional to a power r−γ of the radius, with . We derive distribution functions f (E, Lz) for the scale-free ST models (no black hole) using a power series of the energy E and the component Lz of the angular momentum parallel to the symmetry axis. We use the contour integral method of Hunter & Qian to construct f (E, Lz) for ST models with central black holes, and employ the scheme introduced by Dejonghe & de Zeeuw to derive more general distribution functions that depend on E, Lz and the exact third integral I3. We find that self-consistent two- and three-integral distribution functions exist for all values .