A satellite relative motion model including \(J_2\) and \(J_3\) via Vinti’s intermediary

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Abstract

Vinti’s potential is revisited for analytical propagation of the main satellite problem, this time in the context of relative motion. A particular version of Vinti’s spheroidal method is chosen that is valid for arbitrary elliptical orbits, encapsulating \(J_2\), \(J_3\), and generally a partial \(J_4\) in an orbit propagation theory without recourse to perturbation methods. As a child of Vinti’s solution, the proposed relative motion model inherits these properties. Furthermore, the problem is solved in oblate spheroidal elements, leading to large regions of validity for the linearization approximation. After offering several enhancements to Vinti’s solution, including boosts in accuracy and removal of some singularities, the proposed model is derived and subsequently reformulated so that Vinti’s solution is piecewise differentiable. While the model is valid for the critical inclination and nonsingular in the element space, singularities remain in the linear transformation from Earth-centered inertial coordinates to spheroidal elements when the eccentricity is zero or for nearly equatorial orbits. The new state transition matrix is evaluated against numerical solutions including the \(J_2\) through \(J_5\) terms for a wide range of chief orbits and separation distances. The solution is also compared with side-by-side simulations of the original Gim–Alfriend state transition matrix, which considers the \(J_2\) perturbation. Code for computing the resulting state transition matrix and associated reference frame and coordinate transformations is provided online as supplementary material.

Keywords

Relative motion Vinti potential Spheroidal method Intermediary Formation flying State transition matrix 

Notes

Acknowledgements

This material is based upon work supported in part by the Space Vehicles Directorate of the Air Force Research Laboratory (AFRL) under Contract \(\#\)FA9453-13-C-0201. The authors are grateful for their interest and support. The authors also thank Gim Der for multiple fruitful discussions.

Supplementary material

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Authors and Affiliations

  1. 1.The University of Texas at AustinAustinUSA

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