A new class of symmetric periodic solutions of the spatial elliptic restricted three-body problem

Abstract

We show that there exists a new class of symmetric periodic solutions of the spatial elliptic restricted three-body problem. In such a solution, the infinitesimal body is confined to the vicinity of a primary and moves on a nearly circular orbit. This orbit is almost perpendicular to the orbital plane of the primaries, where the line of symmetry of the orbit lies. The existence is shown by applying a corollary of Arenstorf's fixed point theorem to a periodicity equation system of the problem. And this existence doesn't require any restriction on the mass ratio of the primaries, nor on the eccentricity of their relative elliptic orbit. Potential relevance of this new class of periodic solutions to real celestial body systems and the follow-up studies in this respect are also discussed.periodic orbits, spatial restricted three-body problem, symmetric orbits, averaging Poincaré believed that any (bounded) particular solution of a nearly integrable Hamiltonian system can be approximated by a periodic one for any finite time interval with any precision [1] . This belief established the importance of periodic solutions, and now is known as Poincaré conjecture on periodic solutions. In ref.[2], this conjecture reads "the periodic orbits are dense in the set of bounded orbits" and is listed as the first unsolved conjecture on 3 or N-body problem. For various aspects of the study of periodic solutions of the N-body problem with or without restrictions, the reader can refer to, e.g. Siegel & Moser [3] , Hénon [4] , Meyer [5] , Chenciner [6] , Terracini [7] , Hadjidemetriou [8] .In such kind of study, the model of restricted three-body problem (Szebehely) [9] is useful because it can be served not only as a relatively simple example of non-integrable dynamical systems, but also as a successful model in understanding many almost periodic phenomena in astronomy. To prove the existence of a periodic solution of the restricted three-body problem, the analytical continuation method of Poincaré, which is based on the implicit function theorem for regular functions, is often used. However, there are cases to which this classical theorem does not apply. In some such cases, one may resort to Arenstorf's fixed point theorem [10] . Recent successful applications of this latter theorem to the restricted three-body problem are given by Howison and Meyer [11,12] , Cors, Pinyol and Soler [13,14] , and Brandão and Vidal [15] .Brandão and Vidal [15] considered the periodic solutions of the elliptic collision isosceles restricted threebody problem, where the two primaries with equal mass move, respectively, along collision elliptic orbits lying on the same line, and the infinitesimal body is strictly confined to the plane passing through the mass center of the system and perpendicular to the orbital line of the primaries.In the case when the infinitesimal body moves in the 3-dimensional space, Howison and Meyer [11] proved the existence of two classes of doubly symmetric periodic orbits of circular restricted 3-body problem. One class is

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

“…[5] applied the Lie transformations to the circular restricted three-body problem, and found that there exist equilibria at the cosine value ±1/ √ 5 of the inclination. The existence proof of the Hill-type nearly circular periodic orbits with large inclinations is given by Xu & Fu (2009) [6]. The authors applied double Zeipel transformations to eliminate the short periodic effects in the first-order perturbation system.…”

Effects of Radiation Pressure on the Elliptic Restricted Four-Body Problem

“…Chakraborty and Narayan [14] investigated the equilibrium points, linear stability, zero velocity curves, and fractal trough of the four-body constrained elliptic problem. The model cited in Xu and Fu [15] is useful as a simple example of nonintegral dynamical systems and is successful in understanding many quasiperiodic phenomena in astronomy. Ibrahim [16] studied the effect of solar radiation pressure onto the motion of many-body problem, and applied the study on the Sun-Earth-Moon-spacecraft system.…”