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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.<\/jats:p>","DOI":"10.1088\/1361-6633\/aea024","type":"journal-article","created":{"date-parts":[[2026,8,28]],"date-time":"2026-08-28T22:51:16Z","timestamp":1787957476000},"page":"098301","update-policy":"https:\/\/doi.org\/10.1088\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Singularities and the divergence of spherical harmonic expansions: axially symmetric planets"],"prefix":"10.1088","volume":"89","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1338-339X","authenticated-orcid":true,"given":"Gerald V","family":"Dunne","sequence":"first","affiliation":[{"name":"University of Connecticut"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"formal-analysis","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"methodology","vocabulary":"credit"},{"role":"project-administration","vocabulary":"credit"},{"role":"writing-original-draft","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"}]},{"given":"Ovidiu","family":"Costin","sequence":"additional","affiliation":[{"name":"The Ohio State University"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"formal-analysis","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"methodology","vocabulary":"credit"},{"role":"validation","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"}]},{"given":"Crichton","family":"Ogle","sequence":"additional","affiliation":[{"name":"The Ohio State University"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"formal-analysis","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"methodology","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-4944-8186","authenticated-orcid":true,"given":"Michael","family":"Bevis","sequence":"additional","affiliation":[{"name":"The Ohio State University"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"formal-analysis","vocabulary":"credit"},{"role":"funding-acquisition","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"methodology","vocabulary":"credit"},{"role":"visualization","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"}]},{"given":"Robert A","family":"Ashcraft","sequence":"additional","affiliation":[{"name":"National Geospatial-Intelligence Agency"}],"role":[{"vocabulary":"crossref","role":"author"},{"role":"conceptualization","vocabulary":"credit"},{"role":"formal-analysis","vocabulary":"credit"},{"role":"investigation","vocabulary":"credit"},{"role":"methodology","vocabulary":"credit"},{"role":"validation","vocabulary":"credit"},{"role":"writing-review-editing","vocabulary":"credit"}]}],"member":"266","published-online":{"date-parts":[[2026,9,23]]},"reference":[{"key":"ropaea024bib1","author":"Isaac Newton","type":"other"},{"key":"ropaea024bib2","author":"Todhunter","year":"1873","type":"other"},{"key":"ropaea024bib3","author":"Chandrasekhar","year":"2003","type":"book"},{"key":"ropaea024bib4","author":"Poincar\u00e9","year":"1899","type":"book"},{"key":"ropaea024bib5","first-page":"p 384","author":"Kellogg","year":"1929","type":"book"},{"key":"ropaea024bib6","author":"Moritz","year":"1980","type":"book"},{"key":"ropaea024bib7","doi-asserted-by":"publisher","DOI":"10.1063\/5.0044930","type":"journal-article","article-title":"Non-convergence of the spherical harmonic expansion of gravitational potential below the Brillouin sphere: the continuous case","volume":"62","author":"Ogle","year":"2021","journal-title":"J. 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