Contributions of Spherical Harmonics to Gravitational Moment

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Contributions of Spherical Harmonics to Gravitational Moment

Roithmayr, Carlos M.

A scalar gravitational potential function expressed as a series of spherical harmonics frequently serves as the basis for a model of an astronomical body’s gravitational field. The contribution of an individual generic spherical harmonic to the gravitational gradient is expressed as a dyadic, which is then used to obtain an analytical expression in vector-dyadic form for the contribution to the moment of gravitational forces about the mass center of a small body such as a spacecraft. The expression developed for a harmonic’s contribution to the gravitational gradient can be applied in areas beyond the scope of the paper; for example, the gravitational gradient plays an important role in the state propagation matrix and the state transition matrix that are used in spacecraft trajectory targeting and Kalman filtering. Additionally, it can be employed in numerical simulations of orbit determination based on measurements obtained with a gradiometer. Contributions of spherical harmonics to the gravitational moment may be of interest in connection with attitude control of a spacecraft in the vicinity of a body with an irregular shape, such as an asteroid. Normalized spherical harmonic coefficients up to degree and order 10 are obtained for the asteroid 216 Kleopatra and used in numerical evaluations of contributions to the gravitational moment.