Methods for calculating the gravitational potential of a homogeneous triaxial ellipsoid and their application to the analysis of asteroid's satellite dynamics

For the study of the asteroid’s satellite dynamics, the problem of calculating the exterior gravitational potential and the attraction of an asteroid approximated by a homogeneous triaxial ellipsoid is considered. Three algorithms for determining the gravitational potential of an ellipsoid are constructed. It is, firstly, an algorithm for approximate determination of the potential using its usual expansion in a series of spherical functions with harmonics up to 4$^{\mathrm{th}}$ order inclusive. The other two ones determine the potential using the Dirichlet formula containing elliptic integrals. One calculates these integrals numerically by quadrature formulas. Another, analytical one uses the expansions of integrals developed in the work into rapidly convergent series. The application of these algorithms to determine the potential and the attraction of the asteroid Apophis is considered.