Accurate Calculation of Geodetic Heights of a Celestial Body’s Surface Points Relative to the Triaxial Ellipsoid

Abstract

The approximation of the Earth’s physical surface by a mathematical surface is commonly carried out by a sphere or an ellipsoid of revolution. A triaxial ellipsoid can be used in some cases. The geodetic heights of points of the Earth’s surface are commonly calculated by approximate methods using the relation between spatial rectangular coordinates x, y, z and geodetic coordinates B, L, H. Meanwhile, such first approximation variants are incorrect for small Solar System bodies , for example, Asteroid 433 Eros, because both first approximations are not small values in this case. The proposed fundamentally new approach to calculation of a geodesic height relative to a triaxial ellipsoid is based on the joint use of the equation of the normal to the surface, passing through a given point, and the surface equation proper. The method is reduced to solving the sixth-degree equation by the Sturm method and the fourth-degree equation by the Ferrari method.