Analytical Solution of Similar Oblate Spheroidal Coordinate System

TL;DR

This paper derives explicit analytical expressions for the Similar Oblate Spheroidal coordinate system, enabling better modeling of gravitational fields around oblate spheroidal objects like planets and galaxies using convergent power series.

Contribution

It provides the first explicit analytical power series solutions for Cartesian coordinates, derivatives, metric scale factors, and Jacobian in the SOS coordinate system.

Findings

Derived analytical expressions for Cartesian coordinates in SOS

Formulated partial derivatives and metric scale factors

Presented the Jacobian determinant for SOS coordinates

Abstract

Satisfactory description of gravitational and gravity potentials is needed for a proper modelling of a wide spectrum of physical problems on various size scales, ranging from atmosphere dynamics up to the movements of stars in a galaxy. In certain cases, Similar Oblate Spheroidal (SOS) coordinate system can be of advantage for such modelling tasks, mainly inside or in the vicinity of oblate spheroidal objects (planets, stars, galaxies). Although the solution of the relevant expressions for the SOS system cannot be written in a closed form, it can be derived as analytical expressions -- convergent infinite power series. Explicit analytical expressions for the Cartesian coordinates in terms of the curvilinear Similar Oblate Spheroidal coordinates are derived in the form of infinite power series with generalized binomial coefficients. The corresponding partial derivatives are found in a suitable form, further enabling derivation of the metric scale factors necessary for differential operations. The terms containing derivatives of the metric scale factors in the velocity advection term of the momentum equation in SOS coordinate system are expressed. The Jacobian determinant is derived as well. The presented analytical solution of SOS coordinate system solution is a tool applicable for a broad variety of objects exhibiting density, gravity or gravitation levels resembling similar oblate spheroids. Such objects range from the bodies with small oblateness (the Earth itself on the first place), through elliptical galaxies up to significantly flattened objects like disk galaxies.