Harmonic Function Expansion of Nearly Oblate Systems

Abstract

We show how to develop an expansion of nearly oblate systems in terms of a set of potential-density pairs. A harmonic (multipole) structure is imposed on the potential set at infinity, and the density can be made everywhere regular. We concentrate on a set whose zeroth order functions describe the perfect oblate spheroid of de Zeeuw (1985). This set is not bi-orthogonal, but it can be shown to be complete in a weak sense. Poisson's equation can be solved approximately by truncating the expansion of the potential in such a set. A simple example of a potential which is not one of the basis functions is expanded using the symmetric members of the basis set up to fourth order. The basis functions up to first order are reconstructed approximately using 10,000 particles to show that this set could be used as part of an $N$-body code.