Interior solution of azimuthally symmetric case of Laplace equation in orthogonal similar oblate spheroidal coordinates

TL;DR

This paper derives the interior solution of the azimuthally symmetric Laplace equation within the orthogonal similar oblate spheroidal coordinate system, providing explicit harmonic functions involving generalized Legendre functions.

Contribution

It introduces a method to solve the Laplace equation in SOS coordinates, including separation procedures, generalized Legendre functions, and recursion formulas for higher degrees.

Findings

Derived the interior solution of the Laplace equation in SOS coordinates.

Expressed harmonic functions using generalized Legendre functions.

Provided recursion formulas for higher-degree solutions.

Abstract

Curvilinear coordinate systems distinct from the rectangular Cartesian coordinate system are particularly valuable in the field calculations as they facilitate the expression of boundary conditions of differential equations in a reasonably simple way when the coordinate surfaces fit the physical boundaries of the problem. The recently finalized orthogonal similar oblate spheroidal (SOS) coordinate system can be particularly useful for a physical processes description inside or in the vicinity of the bodies with the geometry of an oblate spheroid. Such shape is aproximating well objects investigated within astrophysics. The solution of the azimuthally symmetric case of the Laplace equation was found for the interior space in the orthogonal SOS coordinates. In the frame of the derivation of the harmonic functions, the Laplace equation was separated by a special separation procedure. A generalized Legendre equation was introduced as the equation for the angular part of the separated Laplace equation. The harmonic functions were determined as relations involving generalized Legendre functions of the first and of the second kind. Several lower-degree functions are reported. Recursion formula facilitating determination of the higher-degree harmonic functions was found. The general solution of the azimuthally symmetric Laplace equation for the interior space in the SOS coordinates is reported.