Analytical solution of orthogonal similar oblate spheroidal coordinate system

Satisfactory description of gravitational and gravity potential is needed for a proper modelling of a wide spectrum of physical problems on various size scales, ranging from atmosphere dynamics up to the movement of stars in a galaxy. The presented orthogonal similar oblate spheroidal (SOS) coordinate system can be a modelling tool applicable for a broad variety of objects exhibiting density, gravity or gravitation potential levels resembling similar oblate spheroids. This can be the case inside or in the vicinity of various oblate spheroidal objects (planets, stars, elliptical galaxies, disk galaxies) exhibiting broad range of oblateness. Although the solution of the relevant expressions for the SOS system cannot be written in a closed form, they are derived as analytical expressions-convergent infinite power series employing generalized binomial coefficients. Transformations of SOS coordinates to and from the Cartesian coordinates are shown. 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 the SOS coordinate system are expressed. The Jacobian determinant is derived as well.