A numerical study on the integration radius separating convergent and divergent spherical harmonic series of topography-
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ORIGINAL ARTICLE
A numerical study on the integration radius separating convergent and divergent spherical harmonic series of topography-implied gravity Blažej Bucha1 · Michael Kuhn2 Received: 19 February 2020 / Accepted: 26 September 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
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 ∼ 20 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. Keywords Convergence/divergence of spherical harmonics · Gravity forward modelling · Topography-implied gravitational field · Spherical harmonics · Integration radius · Molodensky’s truncation coefficients
1 Introduction Several studies have reported divergent behaviour of external harmonic expansions when used to represent external gravitational fields in proximity to gravitating bodies. This effect has been observed with the gravitational fields implied by the Earth (Hirt et al. 2016; Rexer 2017), the Earth’s Moon (Hirt and Kuhn 2017; Šprlák et al. 2018), and by other irregularly shaped small celestial bodies (Garmier and Barriot 2001; Takahashi and Scheeres 2014; Hu and Jekeli 2015;
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Blažej Bucha [email protected] Michael Kuhn [email protected]
1
Department of Theoretical Geodesy, Slovak University of Technology in Bratislava, Radlinského 11, 81005 Bratislava, Slovak Republic
2
Western Australian Geodesy Group, School of Earth and Planetary Sciences, Curtin University, GPO Box U1987, Perth, WA 6845, Australia
Reimond and Baur 2016; Sebera et al. 2016; Šprlák et al. 2020). The findings are in agreement w
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