Numerical aspects of gravitational field modelling using spheroidal harmonic functions

The determination of gravitational fields generated by planetary bodies represents a fundamental task in modern geodesy. To facilitate the computation of gravitational field functionals, we approximate the shapes of individual planetary bodies. The spherical approximation is the most popular, as it conveniently employs numerous symmetries of the sphere. Generally, however, planetary bodies are flattened at the poles or even at equators. Therefore, a conceptual framework on the spheroidal approximation should be analysed.

In this contribution, we develop a new mathematical theory for modelling gravitational fields generated by irregular bodies. Specifically, the gravitational potential, the components of the gravitational gradient and the second- and third-order gravitational tensor components are parametrised using spheroidal harmonic functions defined within the minimal Brillouin spheroid.

To enable global calculation, especially near the poles, the original spheroidal harmonic expansions are transformed into their non-singular counterparts.  Additionally, we investigate selected numerical aspects of the Legendre functions of the first and second kind. Numerical experiments are performed to validate the proposed approach. Both singular and non-singular formulations are systematically evaluated.