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Volume 13
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10.3390/math13132115
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Article

Spherical Harmonics and Gravity Intensity Modeling Related to a Special Class of Triaxial Ellipsoids

by
Gerassimos Manoussakis
1,* and
Panayiotis Vafeas
2
1
Division of Geometry and Algebra, Department of Mathematics, School of Applied Mathematical and Physical Sciences, National Technical University of Athens, 15780 Zografos, Greece
2
Department of Chemical Engineering, University of Patras, 26504 Patras, Greece
*
Author to whom correspondence should be addressed.
Mathematics 2025, 13(13), 2115; https://doi.org/10.3390/math13132115 (registering DOI)
Submission received: 2 June 2025 / Revised: 21 June 2025 / Accepted: 24 June 2025 / Published: 27 June 2025
(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 2nd Edition)
Versions Notes

Abstract

The G-modified Helmholtz equation is a partial differential equation that allows gravity intensity g to be expressed as a series of spherical harmonics, with the radial distance r raised to irrational powers. In this study, we consider a non-rotating triaxial ellipsoid parameterized by the geodetic latitude φ and geodetic longitude λ, and eccentricities ee, ex, ey. On its surface, the value of gravity potential has a constant value, defining a level triaxial ellipsoid. In addition, the gravity intensity is known on the surface, which allows us to formulate a Dirichlet boundary value problem for determining the gravity intensity as a series of spherical harmonics. This expression for gravity intensity is presented here for the first time, filling a gap in the study of triaxial ellipsoids and spheroids. Given that the triaxial ellipsoid has very small eccentricities, a first order approximation can be made by retaining only the terms containing ee2 and ex2. The resulting expression in spherical harmonics contains even degree and even order harmonic coefficients, along with the associated Legendre functions. The maximum degree and order that occurs is four. Finally, as a special case, we present the geometrical degeneration of an oblate spheroid.
Keywords: gravity; triaxial ellipsoid; Dirichlet problem; spherical harmonics gravity; triaxial ellipsoid; Dirichlet problem; spherical harmonics

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MDPI and ACS Style

Manoussakis, G.; Vafeas, P. Spherical Harmonics and Gravity Intensity Modeling Related to a Special Class of Triaxial Ellipsoids. Mathematics 2025, 13, 2115. https://doi.org/10.3390/math13132115

AMA Style

Manoussakis G, Vafeas P. Spherical Harmonics and Gravity Intensity Modeling Related to a Special Class of Triaxial Ellipsoids. Mathematics. 2025; 13(13):2115. https://doi.org/10.3390/math13132115

Chicago/Turabian Style

Manoussakis, Gerassimos, and Panayiotis Vafeas. 2025. "Spherical Harmonics and Gravity Intensity Modeling Related to a Special Class of Triaxial Ellipsoids" Mathematics 13, no. 13: 2115. https://doi.org/10.3390/math13132115

APA Style

Manoussakis, G., & Vafeas, P. (2025). Spherical Harmonics and Gravity Intensity Modeling Related to a Special Class of Triaxial Ellipsoids. Mathematics, 13(13), 2115. https://doi.org/10.3390/math13132115

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