ANALYTIC SOLUTION FOR SATELLITE RELATIVE MOTION: THE COMPLETE ZONAL GRAVITATIONAL PROBLEM
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A state transition matrix for satellite relative motion for the complete zonal gravitational problem is presented. Using the canonical perturbation methods the generalized analytic formulae, closed-form in eccentricity, for second-order secular and short-period effects with first-order long-period effects for an arbitrary zonal harmonic are computed. This approach avoids symbolic manipulations required for the Delaunay normalizations of the zonal gravity potential in addition to significant savings in storage requirements by having generalized formulae valid for any arbitrary zonal harmonic. Using differential perturbations, the secular and periodic effects are included in the analytic solution in the form of a state transition matrix of the satellite relative motion. The perturbation effects are computed for the Equinoctial elements to avoid singularities for circular and equatorial reference orbits. Verification of the proposed analytic solution for the perturbed relative motion is carried out by comparing propagation results with a numerically propagated formation of two satellites using the GMAT simulation software.
author list (cited authors)
Mahajan, B., Vadali, S. R., & Alfriend, K. T.