The gravity of an irregularly shaped body, computed from a harmonic expansion of Legendre polynomials and associated functions, diverges inside a sphere of maximum radius that circumscribes the body. A spacecraft that is attempting a landing on an irregularly shaped body at a landing site that is near the minimum radius of the body will traverse the region from the sphere of maximum radius to the landing site. The spacecraft will experience considerable error in the computed gravitational acceleration if an harmonic expansion gravity model is used.

In this paper, a gravity model is described that computes the acceleration of the spacecraft directly from the mass distribution. Included in this model is provision for a variable density defined on the surface of the body and extending uniformly to the center of mass. The variable density is computed as a function of Legendre polynomials and associated functions. The principal advantage of this model is that the external gravity field is exact for the given shape of the body. When the model described in this paper is applied to the Earth, gravity anomalies are detected. Further analysis indicates that these anomalies explain the velocity perturbations observed in the orbit determination solutions obtained as the Near Earth Asteroid Rendezvous (NEAR) and Galileo spacecraft flew by the Earth.