Contributions of Spherical Harmonics to Gravitational Moment

A scalar gravitational potentialfunction expressed as a series of spherical harmonicsfrequently serves as the basis for a model of an astronomical body’s gravitational field. The contribution of an individual generic spherical harmonicto the gravitationalgradient is expressed as a dyadic, which is then used to obtain an analytical expression in vector- dyadicform for the contribution to the moment of gravitationalforces about the mass center of a small body such as a spacecraft. The expression developed for a harmonic’s contribution to the gravitationalgradient can be applied in areas beyond the scope of the paper; for example, the gravitationalgradient plays an important role in the state propagation matrix and the state transition matrixthat are used in spacecraft trajectory targeting and Kalman filtering. Additionally, it can be employed in numerical simulations of orbit determinationbased on measurements obtained with a gradiometer. Contributions of spherical harmonicsto the gravitationalmo...