Gravity, Electrostatic and Thermal Fields of Slightly Flattened Triaxial Ellipsoidal Conductors via Spherical Harmonic Expansions with Irrational Polar Distance Powers

We present gravitational, electrostatic and thermal field intensities as series of spherical harmonics with the polar distance raised to irrational powers. Theoretical cases involving slightly flattened oblate spheroids and triaxial ellipsoids are considered. For the electrostatic and thermal transfer cases, triaxial ellipsoids are treated as perfect conductors. The resulting Dirichlet boundary value problems, based on the G-modified Helmholtz equation, are solved outside these surfaces. For each geometry, the expansions are constructed explicitly, with a first-order approximation applied for the oblate spheroid and triaxial ellipsoid by retaining only terms containing the first polar and equatorial eccentricities squared. Finally, we determine Earth’s disturbing gravity potential as a series of spherical harmonics solving a Dirichlet boundary value problem.