Universal method for designing periodic orbits by homotopy classes in the elliptic restricted three-body problem

Abstract

The current methods for designing periodic orbits in the elliptic restricted three-body problem (ERTBP) have the disadvantages of targeting limited orbits and ergodic searches and considering only symmetric orbits. A universal method for designing periodic orbits is proposed in this paper. First, the homotopy classes of orbits are structured based on their topological structures. Second, a dynamic model based on homotopy classes, ranging from the circular restricted three-body problem (CRTBP) to the ERTBP, can be built using the homotopy method. Third, a multi- and a single-period orbit were selected based on the resonance ratios. Finally, the corresponding orbit in the ERTBP was computed by modifying the initial condition of the orbit in the CRTBP. This method, without an ergodic search, can extend to any orbit, including an asymmetric orbit in the CRTBP, to the ERTBP model, and the two orbits are of the same homotopy class. Examples of the Earth–Moon ERTBP are presented to verify the efficiency of this method.