Continuation of some nearly circular symmetric periodic orbits in the elliptic restricted three-body problem

Abstract

Some comet- and Hill-type families of nearly circular symmetric periodic orbits of the elliptic restricted three-body problem in the inertial frame are numerically explored by Broyden’s method with a line search. Some basic knowledge is introduced for self-consistency. Set $j/k$ as the period ratio between the inner and the outer orbits. The values of $j/k$ are mainly $1/j$ with $2\leq j\leq 10$ and $j=15,20,98,100,102$. Many sets of the initial values of these periodic orbits are given when the orbital eccentricity $e_{p}$ of the primaries equals 0.05. When the mass ratio $\mu =0.5$, both spatial and planar doubly-symmetric periodic orbits are numerically investigated. The spacial orbits are almost perpendicular to the orbital plane of the primaries. Generally, these orbits are linearly stable when the $j/k$ is small enough, and there exist linearly stable orbits when $j/k$ is not small. If $\mu \neq 0.5$, there is only one symmetry for the high-inclination periodic orbits, and the accuracy of the periodic orbits is determined after one period. Some diagrams between the stability index and $e_{p}$ or $\mu$ are supplied. For $\mu =0.5$, we set $j/k=1/2,1/4,1/6,1/8$ and $e_{p}\in [0,0.95]$. For $e_{p}=0.05$ and 0.0489, we fix $j/k=1/8$ and set $\mu \in [0,0.5]$. Some Hill-type high-inclination periodic orbits are numerically studied. When the mass of the central primary is very small, the elliptic Hill lunar model is suggested, and some numerical examples are given.