Statistics – Computation
Scientific paper
Jan 1996
adsabs.harvard.edu/cgi-bin/nph-data_query?bibcode=1996phdt.........6w&link_type=abstract
Thesis (PH.D.)--THE UNIVERSITY OF TEXAS AT AUSTIN, 1996.Source: Dissertation Abstracts International, Volume: 57-06, Section: B,
Statistics
Computation
Scientific paper
We describe a method of approximating the gravity of a small celestial body by representing it as a polyhedron. Large and small faces on the polyhedron can model surface features of an asteroid, comet nucleus, or small planetary satellite such as craters, ridges, and grooves. The gravity expressions do not require that the polyhedron be regular or convex. Faces (which are planar polygons) can differ in size, shape, and edge count, and need not be convex. The entire body does not have to be modeled at a uniform resolution. We derive closed-form analytic expressions for the exterior gravity of a constant-density polyhedron with the aid of the Gauss Divergence theorem and Green's theorem. The derivation leads to coordinate-free expressions for the gravitational potential, the attraction, the gravity gradient matrix, and the Laplacian of potential. As a measure of computational effort, the polyhedron gravity expressions require only one logarithm term per edge and one arctangent term per face. Corresponding expressions derived elsewhere often require two logarithms per edge and/or several arctangents per face. An inverse problem is also investigated. A linear least squares estimator adjusts vertex locations of an a priori polyhedron in order to reproduce gravity observations. The estimator employs the Levenberg-Marquardt algorithm to assist convergence of the nonlinear problem. The singular value decomposition algorithm is used to deal with ill -conditioned matrices. Several polyhedron configurations cause estimation problems by creating ill conditioned matrices. At least three geometries make vertex motion unobservable from gravitation observations. Numerical difficulties also occur when using a high-resolution a priori polyhedron, and when only weak gravitational observations are available.
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