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 $\mu \in [0,1/2]$ 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 $\mu \approx 10^{-3}$ and give a brief description of its evolution as μ increases up to μ = 1/2.