A mathematical study of the tethered slingshot maneuver using the elliptic restricted problem

The main objective of the present paper is to find the modifications that a tethered slingshot maneuver (TSSM) can make in the orbit of a spacecraft, both in terms of energy and inclination. The TSSM is a maneuver where a tether fixed in a celestial body, like a moon or an asteroid, makes a rotation in the velocity vector of a spacecraft to modify its orbit. In particular, the present paper concentrates in showing the potential savings in fuel consumption for orbital maneuvers that use this technique in an elliptic system of primaries, which gives advantages over the circular problem. To make this study more complete, analytical approximations are derived to provide a general view of the behavior of the maneuver in terms of variations of energy and inclination as a function of different conditions for the geometry, length, and location of the tether. Among the main results, it is showed the best location to place the tether and the best moment and duration to perform the maneuver, as a function of the parameters involved, like the orbits of the primaries, the incoming velocity of the spacecraft, etc. The solutions are shown in maps giving the variations of energy and inclination for different locations of this device and assuming different incoming orbits for the spacecraft. Regions that maximize those variations are indicated. Based on those results, it is possible to find the best solution for several particular problems. The results show that this maneuver has a large potential to be explored, helping a spacecraft to make journeys to the exterior planets and out of the Solar System. Those results are arguments in favor of developing efforts to solve the technological problems involved in real applications of this technique. The main advantage of the proposed technique is the energy gain given by the maneuver, in particular when using the higher velocity of the asteroid, at the periapsis of its orbit around the Sun. The main disadvantages are the technical challenges involved in the implementation of the maneuver and the fact that to get maximum benefits, there are time restrictions to apply the maneuver, because the asteroid must be passing by its periapsis. (AU)