Analytical solution of orthogonal similar oblate spheroidal coordinate system

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.

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

“…The SOS coordinates were introduced by White et al [3]. The full description of the analytical solution of SOS coordinates can be found in [4]. A shortened, but more overviewable summary is reported in [8].…”

“…The full description of the analytical solution of SOS coordinates can be found in [4]. A shortened, but more overviewable summary is reported in [8]. It is recommended to find details of the system in the abovementioned references.…”

“…For the SOS coordinate system (R, ν, λ) [4], the basic coordinate surfaces of the R coordinate are similar oblate spheroids given in 3D Cartesian coordinates (x3D, y3D, z3D) by the formula…”

“…From Eqs. ( 3) and ( 17) of [4], the relation for the power function coordinate surfaces in 3D (2) in dependence on the coordinate ν can be detailed as…”

“…The similar oblate spheroidal coordinates are distinct from all the standard orthogonal coordinate systems [2,7], as one coordinate surfaces family is not of the second degree (and even not of the fourth degree), but it is formed by general power functions 𝑧 ≈ 𝑥 1+𝜇 (i.e. with a real-number exponent 1+μ) rotated around the z-axis [3,4]. They are orthogonal to the similar oblate spheroids representing the first set of the coordinate surfaces.…”