Three-dimensional, axisymmetric cusps without chaos

Abstract

We construct three-dimensional axisymmetric, cuspy density distributions, whose potentials are of Stackel form in parabolic coordinates. As in a recent paper by Sridhar BL Touma, a black hole of arbitrary mass may be added at the centre, without destroying the Stackel form of the potentials. The construction uses a classic method, originally owing to Kuzmin, which is extended to parabolic coordinates here. The models are highly oblate, and the cusps are 'weak', with the density, rho proportional to 1/r(k), where 0<k<1.