Gravity inversion allows us to constrain the interior mass distribution of a planetary body using the observed shape, rotation, and gravity. Traditionally, techniques developed for gravity inversion can be divided into Monte Carlo methods, matrix inv
ersion methods, and spectral methods. Here we employ both matrix inversion and Monte Carlo in order to explore the space of exact solutions, in a method which is particularly suited for arbitrary shape bodies. We expand the mass density function using orthogonal polynomials, and map the contribution of each term to the global gravitational field generated. This map is linear in the density terms, and can be pseudo-inverted in the under-determined regime using QR decomposition, to obtain a basis of the affine space of exact interior structure solutions. As the interior structure solutions are degenerate, assumptions have to be made in order to control their properties, and these assumptions can be transformed into scalar functions and used to explore the solutions space using Monte Carlo techniques. Sample applications show that the range of solutions tend to converge towards the nominal one as long as the generic assumptions made are correct, even in the presence of moderate noise. We present the underlying mathematical formalism and an analysis of how to impose specific features on the global solution, including uniform solutions, gradients, and layered models. Analytical formulas for the computation of the relevant quantities when the shape is represented using several common methods are included in the Appendix.
Dawn is the first NASA mission to operate in the vicinity of the two most massive asteroids in the main belt, Ceres and Vesta. This double-rendezvous mission is enabled by the use of low-thrust solar electric propulsion. Dawn will arrive at Vesta in
2011 and will operate in its vicinity for approximately one year. Vestas mass and non-spherical shape, coupled with its rotational period, presents very interesting challenges to a spacecraft that depends principally upon low-thrust propulsion for trajectory-changing maneuvers. The details of Vestas high-order gravitational terms will not be determined until after Dawns arrival at Vesta, but it is clear that their effect on Dawn operations creates the most complex operational environment for a NASA mission to date. Gravitational perturbations give rise to oscillations in Dawns orbital radius, and it is found that trapping of the spacecraft is possible near the 1:1 resonance between Dawns orbital period and Vestas rotational period, located approximately between 520 and 580 km orbital radius.This resonant trapping can be escaped by thrusting at the appropriate orbital phase. Having passed through the 1:1 resonance, gravitational perturbations ultimately limit the minimum radius for low-altitude operations to about 400 km,in order to safely prevent surface impact. The lowest practical orbit is desirable in order to maximize signal-to-noise and spatial resolution of the Gamma-Ray and Neutron Detector and to provide the highest spatial resolution observations by Dawns Framing Camera and Visible InfraRed mapping spectrometer. Dawn dynamical behavior is modeled in the context of a wide range of Vesta gravity models. Many of these models are distinguishable during Dawns High Altitude Mapping Orbit and the remainder are resolved during Dawns Low Altitude Mapping Orbit, providing insight into Vestas interior structure.
