Optimizing expansions of the Earth’s gravity potential and its derivatives over ellipsoidal harmonics

A set of spherical harmonics is the most widely used representation of the Earth’s gravity potential. This series converges outside and on the surface of a reference sphere enveloping the Earth. However, the Earth’s surface is better approximated by the reference ellipsoid—a compressed ellipsoid of revolution that covers the entire Earth. The gravity potential can be expanded in a series over ellipsoidal harmonics on the surface of the reference ellipsoid and on the surface of other external confocal ellipsoids of revolution. In contrast to spherical harmonics, depending on the associated Legendre functions of the first kind, ellipsoidal harmonics depend also on the associated Legendre functions of the second kind. The latter contain the very slowly converging hypergeometric Gauss series. The number of series increases with increasing the order of their derivatives. In this work, we derived new series for the gravitational potential of the Earth and its derivatives over ellipsoidal harmonics. Starting from the first order derivative, all the series corresponding to higher order derivatives depend on the same two hypergeometric Gauss series. The latter converges considerably faster than that for the hypergeometric series previously used when computing the gravity potential and its derivatives.