Estimating asteroid density distributions from shape and gravity information

A least-squares approach for estimating the internal density distribution of an asteroid is presented and applied to a simple polyhedron asteroid shape. The method assumes that the asteroid gravity field is measured to a specified degree and order and that a polyhedral model of the asteroid is available and has been discretized into a finite number of constant density polyhedra. The approach is derived using several basic properties of spherical harmonic gravitational expansions and can explicitly accommodate a fully correlated covariance matrix for the estimated gravity field. For an asteroid shape discretized into M constant density polyhedra and a gravity field measured to degree and order N, the least-squares problem is under-determined if M>(N+1)2 and is over-determined if M<(N+1)2. For both cases a singular-value decomposition (SVD) approach will yield solutions. We apply our approach to a number of ideal test situations using an asteroid shape consisting of 508 tetrahedra. We show that the under-determined case is sensitive to non-uniform density distributions. The over-determined case shows very good performance independent of the initial density distribution guess.