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

• Published in: Astrophysics and space science Vol. 368; no. 3; p. 13
• Main Authors: Xu, Xing-Bo, Song, Ye-Zhi
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.03.2023; Springer Nature B.V
• Subjects: Astrobiology; Astronomy; Astrophysics; Astrophysics and Astroparticles; Cosmology; Inclination; Mars; Mercury; Observations and Techniques; Orbits; Physics; Physics and Astronomy; Space Exploration and Astronautics; Space Sciences (including Extraterrestrial Physics; Stability index; Symmetry; Three body problem; Restricted three-body problem; Periodic orbits; Spatial resonance; Numerical continuation
• ISSN: 0004-640X; 1572-946X
• DOI: 10.1007/s10509-023-04169-3

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 ≤ j ≤ 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 μ = 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 μ ≠ 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 μ are supplied. For μ = 0.5 , we set j / k = 1 / 2 , 1 / 4 , 1 / 6 , 1 / 8 and e p ∈ [ 0 , 0.95 ] . For e p = 0.05 and 0.0489, we fix j / k = 1 / 8 and set μ ∈ [ 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.