Analytical Solution of Similar Oblate Spheroidal Coordinate System
  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.
