Three-dimensional periodic motion of three finite masses around collinear equilibrium configurations

Abstract

Three-dimensional periodic motions of three bodies are shown to exist in the infinitesimal neighbourhood of their collinear equilibrium configurations. These configurations and some characteristic quantities of the emanating three-dimensional periodic orbits are given for many values of the two mass parameters, μ=m ₂/(m ₁+m ₂) andm ₃, of the general three-body problem, under the assumption that the straight line containing the bodies at equilibrium rotates with unit angular velocity. The analysis of the small periodic orbits near the equilibrium configurations is carried out to second-order terms in the small quantities describing the deviation from plane motion but the analytical solution obtained for the ‘horizontal’ components of the state vector is valid to third-order terms in those quantities. The families of three-dimensional periodic orbits emanating from two of the collinear equilibrium configurations are continued numerically to large orbits. These families are found to terminate at large vertical-critical orbits of the familym of retrograde periodic orbits ofm ₃ around the ‘primaries’m ₁ andm ₂. The series of these termination orbits, formed when the value ofm ₃ varies, are also given. The three-dimensional orbits are computed form ₃=0.1.