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  • "Minimal" orbital dynamics

    Paper ID



    • A.J. Sarnecki
    • R.H. Gooding


    Royal Aircraft Establishment, Space Department


    United Kingdom




    Relations between position and velocity vectors at different points on a trajectory in a pure inverse-square field of force are derived without the use of geometrical descriptors of the orbit. An along-track 'minimal' transformation variable is found, which permits the direct integration of the equation of motion. The result is equally applicable to elliptic, hyperbolic, parabolic and rectilinear trajectories. The relationship between the transformation variable and time constitutes an archetype of Kepler's equation, conventional forms of that equation appearing as special cases. The results allow a further simplification for rectilinear motion, with the velocity used as the along- track variable. The 'minimalist' approach is also applied to the rendezvous problem: Lambert's celebrated theorem reduces to an obvious observation. Application of the theorem to the rectilinear trajectory allows the physical interpretation of parameters introduced by other authors through a purely mathematical analysis. The Appendix gives further material on the Lambert problem, including a procedure for its numerical solution.