A numerical study on the integration radius separating convergent and divergent spherical harmonic series of topography-implied gravity

Abstract

We show that far-zone topography-implied gravitational effects may be accurately computed via external spherical harmonics not only above the limit sphere encompassing all the masses, but also inside it on planetary topographies. Although a rigorous mathematical proof is still missing, our numerical experiments indicate that this is possible, provided that near-zone masses within a certain spherical cap centred at the evaluation point are omitted from gravity forward modelling. We formulate and numerically examine a hypothesis, saying that in order to achieve convergence, the cap size needs to be larger than the highest topographical height. The hypothesis relies on the spherical arrangement of field-generating topographic masses and strictly positive topographic heights. To put our hypothesis to a test, we gravity forward model lunar degree-2160 topography using a constant mass density and expand the far-zone gravitational effects up to degree 10,800. The results are compared with respect to divergence-free reference values from spatial-domain gravity forward modelling. By systematically increasing the cap radius from 2.5 km up to 100.0 km (the maximum topographic height is $${\sim }\,20\,\mathrm {km}$$ ), we obtained results that appear to be in line with our hypothesis. Nonetheless, a rigorous mathematical proof still needs to be found to prove whether the hypothesis is true or false. The outcomes of the paper could be beneficial for the study of convergence/divergence of spherical harmonics on planetary surfaces and for geoid computations based on spherical harmonic expansion of far-zone gravitational effects.