Endgame Problem Part 1: V-Infinity-Leveraging Technique and the Leveraging Graph

Abstract

Renewed interest by ESA and NASA in missions to Europa, Ganymede, Enceladus, and Titan poses the question of how to best solve the endgame problem. Endgames typically aim at an inexpensive insertion maneuver into the science orbit and can be designed using either V ∞ -leveraging maneuvers or the multibody dynamics. Although historically linked to insertion maneuvers, the endgame problem is symmetric and equally applies to departure. In this two-part series, we analyze and draw connections between the two apparently separate approaches, providing insight into the dynamics of the multibody gravity-assist problem. In this paper we derive new formulas for the V ∞ -leveraging maneuver and build the leveraging graph to be used as a reference guide for designing endgame tours. We prove that the cost of a V ∞ -leveraging-maneuver sequence decreases when using high-altitude flybys (as done in the multibody technique). Finally, we find a simple quadrature formula to compute the minimum ΔV transfer between moons using V ∞ -leveraging maneuvers, which is the main result of the paper, and a method to estimate transfer times. The leveraging graphs and associated formulas are derived in canonical units and therefore apply to any celestial system with a smaller body in a circular orbit around a primary. Specifically, we demonstrate the new method to provide rapid calculations of the theoretical boundary values for ΔV requirements and estimated transfer times for moon tours in the Saturn and Jupiter systems using the V ∞ -leveraging-maneuver model.