Pareto Frontier for the time–energy cost vector to an Earth–Moon transfer orbit using the patched-conic approximation

Abstract

In this work, we present a study about the determination of the optimal time–energy cost vector, i.e., flight time and total \({\Delta }V\) (velocity change) spent in an orbital transfer of a spacecraft from an Earth circular parking orbit to a circular orbit around the Moon. The method used to determine the flight time and total \({\Delta }V\) is based on the well-known approach of patched conic in which the three-body problem that involves Earth, Moon and spacecraft is decomposed into two ‘two bodies’ problems, i.e., Earth–spacecraft and Moon–spacecraft. Thus, the trajectory followed by the spacecraft is a composition of two parts: The first one, when the spacecraft is within the Earth’s sphere of influence; The second one, when the spacecraft enters into the Moon’s sphere of influence. Therefore, the flight time and total \({\Delta }V\) to inject the spacecraft into the lunar trajectory and place it around the Moon can be determined using the expressions for the two-body problem. In this study, we use the concept of Pareto Frontier to find a set of parameters in the geometry of patched-conic solution that minimizes simultaneously the flight time and total \({\Delta }V\) of the mission. These results present different possibilities for performing an Earth–Moon transfer where two conflicting objectives are optimized.