Approximate third integrals for axisymmetric potentials using local Stackel fits

We use a set of Stackel potentials to obtain a local approximation for an effective third integral in axisymmetric systems. We present a study on the feasibility and effectiveness of this approach. We apply it to three trial potentials of various flattenings, corresponding to nearly ellipsoidal, discy and boxy density isophotes. In all three cases, a good fit to the potential requires only a small set of Stackel potentials, and the associated Stackel third integral provides a very satisfactory, yet analytically simple, approximation to the trial potentials effective third integral.