Optimization of Low-Thrust Gravity-Assist Trajectories under Application of Tisserand's Criterion

Kurzfassung

Abstract Exploring the solar system is becoming more and more challenging and the associated space missions are becoming similarly more demanding concerning Δv, targeted at distant bodies. Recent examples for such missions are New Horizons, which visited the dwarf planet Pluto and Rosetta’s rendezvous with the comet 67P/Churyumov-Gerasimenko. These missions would not have been possible without the application of gravity assists, a technique already used for Pioneer 10 and 11 and the Voyager missions. It is based on transferring energy between a planet or other flyby partner and a spacecraft, which results in trajectory changes without associated propellant consumption. Low-thrust propulsion, which is usually operating continuously for significant amounts of time during a mission, is very efficient and thus often acts as mission enabler. The effort - expressed in used propellant mass fraction – to achieve a certain mission goal is usually smaller for low-thrust propulsion than for a similar mission applying chemical propulsion. This is due to the large specific impulse (typically >3000s) associated with low-thrust propulsion. To further enhance our ability to explore the solar system it is therefore desirable to combine these two techniques. Currently, a number of methods exists to optimize low-thrust trajectories and these have been applied to gravity-assist scenarios as well. However, up to now a method for optimization of the actual sequence of gravity- assist partners is not publically available. Instead, the respective mission analyst has provided the sequence, based on assumptions, estimates and experience. The major goal of this dissertation is to close this gap and investigate if it is possible to reduce the amount of expert information needed and for optimization and thus increase the degree to which the search space is searched in completeness. Reducing the amount of involved experience and trajectory resp. sequence information, improves the chance of finding unforseen and worthwhile new mission sequences. This dissertation provides an analysis of how to incorporate the gravityassist sequence into the optimization process and thus allow a complete, thorough and effective search of useful mission scenarios. First, it is analyzed how gravity-assist sequences are obtained for impulsive missions and whether the same technique can be similarly applied to low-thrust scenarios. A typical approach for gravityassist planning of impulsive mission scenarios is the application of Tisserand’s Criterion. It is an energy relation, derived for observations of comets in the 19th century within the context of the circular restricted three-body system. This criterion can be represented visually in the form of socalled Tisserand graphs, which are used to map possible gravity-assist sequences. Furthermore, two sources of errors are analyzed: First, the error is analysed, which is caused by applying this relation to the actual solar system and thus diverting from the circular restricted three-body system. Second, the error is analysed, which is caused by adding thrust into the equations of motion. It is shown that the thrust force is negligible in the balance of forces for realistic mission designs. However, the thrust’s effect on the specific orbit energy cannot be neglected. Next, a correction term to apply Tisserand’s Criterion in a low-thrust situation is derived. This term however, does not allow an a priori evaluation of the mission sequence and therefore the modified Tisserand’s Criterion cannot be used to map out possible paths of gravity assists similarly as the unmodified Tisserad’s Criteron for impulsive missions. A method, based on a heuristic search, a shape-based trajectory model and the inclusion of possible gravity-assist partners as variable, is set up. The heuristic search circumvents the usage of Tisserand’s Criterion. As a next step to establish the method, the respective variable structure is analyzed. Some of the relevant trajectory variables are shown to be interdepent. For instance, the flight times of the individual trajectory legs between all encountered bodies have to add up to thetotal mission flight time. Consequently, two types of variables are defined: global and local. The first are independent and thus can be used by an evolutionary algorithm to evolve the mission candidates into better (based on solution fitness) solutions during the search. The latter are interdependent. A number of example calculations are conducted to assess the success and usefulness of applying this search method. It is shown that for a single gravity assist in an Earth to Jupiter mission the obtained results are reliable concerning the gravity-assist partner and date and successfully improve the non-gravity assist benchmark by more than 20%. Performance for multi-gravity-assist missions is worse, but still solution improvement and a dominance of the global variables can be observed. Further calculations introduce constraints based on the variable regions of maximum Δv change and show that the solution quality and reliability can thus be improved. Furthermore, the limitations of reproducing existing trajectories are shown and discussed. Finally, the drawbacks of the applied shape-based trajectory model are expressed in a discussion concerning its relation to Tisserand graph diagrams. It is recommended to use a different trajectory model to address these drawbacks in future work.