Singularities and the divergence of spherical harmonic expansions: Axially symmetric planets

The region of convergence of the spherical harmonic expansion is determined by the (generally complex) singularities of the gravitational potential. This complex analysis perspective is at the heart of recent rigorous results concerning the divergence properties of the spherical harmonic expansion. In this paper we build physical intuition for these general mathematical results using illustrative examples (some familiar and some new) of idealized planets for which the analysis is particularly explicit. This approach provides new methods to determine the region of convergence, without computing and analyzing expansion coefficients, and gives a novel geometric understanding of the divergence phenomenon. It also explains the fundamental origin of the numerical instabilities inherent to polyhedral models of planets. For the sake of clarity, we illustrate this new approach for the special case of axisymmetric planets of constant density, with explicit comparisons, but the key ideas do not rely on these restrictions.