An automatic tree search algorithm for the tisserand graph

The Tisserand graph (TG) is a graphical tool used in the preliminary design of gravity-assisted trajectories. The TG is a bi-dimensional map showing essential orbital information regarding the Keplerian orbits resulting from the hyperbolic passage by a set of massive bodies, given the magnitude of the hyperbolic excess speed (v_∞) and the minimum allowed pericenter height for each passage. Contours of constant v_∞ populate the TG. Intersections between such curves allow to link consecutive flybys and build sequences of encounters en route to a selected destination. When the number of perturbing bodies is large and many v_∞ levels are considered, the identification of all the possible sequences of encounters through the visual inspection of the TG becomes a hard task. Besides, if the determination of the encounter sequences constitutes the input to a numerical code for trajectory design and optimization, an automated examination of the TG is desirable. This contribution describes the development and implementation of an automatic technique to traverse the TG and determine all the sequences of encounters that it contains. The technique is based on a tree search method, and the computation of intersections between contours is approximated with the regula-falsi scheme. The method is applied to a number of interplanetary mission scenarios and the results obtained are processed by a trajectory optimizer.