Abstract
The spherical harmonic coefficients of the gravitational potential of an homogeneous body are analytically derived from the harmonics describing its shape. General formulas are given as well as detailed expressions up to the fifth order of the topography harmonics. The volume, surface and inertia tensor of the body are obtained as by-products. The case of a triaxial ellipsoid is given as example and used for numerical checking. Another numerical scheme for verification is provided. The application to Phobos is made and the convergence of the expressions for the harmonics is numerically established.
Résumé
Les harmoniques du champ de gravitation d'un corps homogène de forme donnée sont calculés analytiquement à partir des harmoniques du développement en série du rayon vecteur exprimant la forme de la surface du corps. Outre la formule générale, des expressions détaillées, au cinquième ordre des harmoniques du rayon vecteur, sont données sous une forme bien adaptée à la programmation. Le volume, la surface et le tenseur d'inertie du corps sont calculés analytiquement `a partir des formules générales. Le cas de l'ellipsoide triaxial est pris comme test des formules établies. Un autre test numérique est fourni dans le cas le plus général. Ceci est appliqué à Phobos, et la convergence des expressions fournissant les harmoniques est numériquement démontrée.
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Balmino, G. Gravitational potential harmonics from the shape of an homogeneous body. Celestial Mech Dyn Astr 60, 331–364 (1994). https://doi.org/10.1007/BF00691901
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DOI: https://doi.org/10.1007/BF00691901