A generalization of Kuzmin's theorem

Abstract

In a triaxial mass model with a gravitational potential of Stäckel form in ellipsoidal coordinates the density at a general point is related in a simple way to the density on the short axis. The density is nowhere negative if and only if it is non-negative on the short axis. This is the generalization of a theorem derived by Kuzmin for oblate axisymmetric mass models. For a given short axis density profile and choice of ellipsoidal coordinates one can find, by straightforward integration, not only the complete mass model that has this short axis profile and a Stäckel potential in these coordinates, but also the potential itself.