An analytical solution for the swing-by problem

The extension of Öpik's theory of close encounters developed in the last decades allows a fully analytical, quantitative treatment of the motion of a small body encountering a massive perturber on a circular orbit. In this paper we derive explicit expressions for the initial values of the angular elements of the small body orbit, of given semimajor axis, eccentricity and inclination, in order to obtain a post-encounter orbit with prescribed values of semimajor axis, eccentricity and inclination. We describe the geometrical aspects of the algorithm, and give two examples of application; the first of them concerns the geometry of the 2029 Earth encounter of Apophis, while the second illustrates a sequence of close encounters with Callisto of the JUICE probe, aimed at changing the inclination of the spacecraft orbit. In the planning of complex space missions involving multiple encounters with planets or satellites, the algorithm described in the paper could provide a reliable initial guess to start the computationally intensive optimization process.