On families of periodic solutions of the restricted three-body problem

Abstract

We consider the plane circular restricted three-body problem. It is described by an autonomous Hamiltonian system with two degrees of freedom and one parameter μ ∈ [0, 1/2] which is the mass ratio of the two massive bodies. Periodic solutions of this problem form two-parameter families. We propose methods of computation of symmetric periodic solutions for all values of the parameter μ. Each solution has a period and two traces, namely, the plane and the vertical one. Two characteristics of a family, i.e., its intersection with the symmetry plane, are plotted in the four coordinate systems used in the investigations: two global and two local ones related to the two massive bodies. We also describe generating families, i.e., the limits of families as μ → 0, which are known almost explicitly. As an example, we consider the family h, which begins with retrograde circular orbits of infinitely small radius around the primary P 1 of bigger mass. For this family, we cite detailed data for μ = 0 and μ ≈ 10 −3 and give a brief description of its evolution as μ increases up to μ = 1/2.Keywords Restricted three-body problem · Periodic solutions · Two-parameter families IntroductionLet three point bodies P 1 , P 2 , and P 3 move in a single plane under the Newton law of gravitation. The bodies P 1 and P 2 have masses m and m 2 respectively, and the mass of the body P 3 is so small that its influence on the bodies P 1 and P 2 can be ignored. We say that the mass of the body P 3 is equal to zero. Then the body P 2 moves along

Vertical bifurcation families from the long and short period families around the equilateral equilibrium points

“…These families are often called bridges and have been widely studied (Deprit and Henrard 1968;Henrard 2002;Bruno and Varin 2007). However, seldom work has been done about the vertical bifurcation orbits (Perdios and Zagouras 1991;Bruno and Varin 2006). We wonder whether these vertical bifurcation orbits are connected with some bifurcation orbits in a special periodic family, as what happens in the planar case.…”

Extension of fast periodic transfer orbits from the Earth–Moon RTBP to the Sun–Earth–Moon Quasi-Bicircular Problem

“…It is worth mentioning that there are several recent works dealing with the search for families of POs in the RTBP and the study of their properties (Bolotin and MacKay 2006;Llibre and Paşca 2006;Perdios 2007). However, as far as we know the only work making reference to periodic transfer orbits is the one by Bruno and Varin (2006), which although dealing only with symmetric orbits in the RTBP, explores the full range of the mass parameter µ, from 0 to 1/2.…”

Families of first kind periodic solutions of the spatial and planar elliptic restricted three-body problem with collision

“…In these previous works, the orbit of each primary is elliptic but without collision, and the parameter is obviously the eccentricity of the primaries. Families of periodic solutions for the planar circular restricted three-body problem have been calculated by several authors, we cite here only Bruno and Varin (2006). On the other hand, in Palacian et al (2006), symmetric periodic orbits were obtained by the use of the averaging method for the spatial elliptic restricted three-body problem.…”