Some Analytic Integrals of the Averaged Variational Equations for a Thrusting Spacecraft

Starting with a numerical analysis of various thrust profiles for continuous-thrust escape from geostationary transfer orbit, we develop an averaging analysis that is valid for the full range of initial eccentricities (zero to almost one) and thrust-toweight ratios. From the numerical analysis, it is found that shorter escape times are obtained when the phasing of the final revolution before escape yields an “escape from apoapsis” condition. This phasing is sensitive to slight variations in initial orbit and the exact thrust profile. In the averaging analysis, which bypasses this sensitivity, analytic integrals are found for the averaged variational equations for energy and eccentricity, based on elliptic integrals and series expansions thereof. Reasonably accurate explicit relations between mean energy and mean eccentricity, as well as time and these two quantities, are obtained for the full range of eccentricities and thrust ratios.

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