Reversed-Series Solution to the Universal Kepler Equation
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Abstract
Introduction P REVIOUS work involvinga variationof the epochstate vector using the so-calleduniversal functions2 has shown the need for an approximate analytic solution to the universal form of Kepler’s time equation. The appealing feature of the universal function formulation is that a single expression relates orbital position and time for any orbit type, unlike the universal variable formulations,3;4 which employ a single position variable but separate position– time relations for elliptical, parabolic, and hyperbolic trajectories.Combining these three via the universalfunctionsavoidsdif cultieswith differentiating the constituent relations in problems where the trajectory is changingorbital state, e.g., a low-thrust spacecraft transitioning from an elliptical to a hyperbolic path. Our approach makes use of the universal functions as described by Battin2 (for convenience some of the analysis is repeated here). In the universal function representation,Kepler’s equation relating time and position (the universal anomaly  ) is given by