Determination of the doubly symmetric periodic orbits in the restricted three-body problem and Hill’s lunar problem

Abstract

We review some recent progress on the research of the periodic orbits of the N-body problem, and numerically study the spatial doubly symmetric periodic orbits (SDSPs for short). Both comet- and lunar-type SDSPs in the circular restricted three-body problem are computed, as well as the Hill-type SDSPs in Hill’s lunar problem. Double symmetries are exploited so that the SDSPs can be computed efficiently. The monodromy matrix can be calculated by the information of one fourth period. The periodicity conditions are solved by Broyden’s method with a line-search, and some numerical examples show that the scheme is very efficient. For a fixed period ratio and a given acute angle, there exist sixteen cases of initial values. For the restricted three-body problem, the cases of “Copenhagen problem” and the Sun–Jupiter–asteroid model are considered. New SDSPs are also numerically found in Hill’s lunar problem. Though the period ratio should be small theoretically, some new periodic orbits are found when the ratio is not too small, and the linear stability of the searched SDSPs is numerically determined.