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 ∼20km\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sim }\,20\,\mathrm {km}$$\end{document}), 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.

中文翻译：

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地形隐含重力收敛发散球谐级数积分半径的数值研究

我们表明，远区地形隐含的引力效应可以通过外部球谐函数精确计算，不仅在包含所有质量的极限球体之上，而且在行星地形学上也可以在极限球体内部进行计算。尽管仍然缺少严格的数学证明，但我们的数值实验表明这是可能的，前提是在重力前向建模中省略了以评估点为中心的某个球冠内的近区质量。我们制定并数值检验了一个假设，即为了实现收敛，帽盖尺寸需要大于最高地形高度。该假设依赖于场产生地形质量的球形排列和严格的正地形高度。为了检验我们的假设，我们使用恒定质量密度对月球 2160 度地形进行重力前推模型，并将远区引力效应扩展到 10,800 度。将结果与来自空间域重力正演建模的无发散参考值进行比较。通过系统地将上限半径从 2.5 公里增加到 100.0 公里（最大地形高度为 ~20km\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \ usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\sim }\,20\,\mathrm {km}$$\end{文档}），我们得到的结果似乎与我们的假设一致。尽管如此，仍然需要找到严格的数学证明来证明假设是真还是假。