Three-dimensional periodic orbits and their stability

Abstract

Nineteen families of periodic orbits in the restricted three-body problem in three-dimensional space have been discovered by numerical processes using the “Mercury” Electronic Computer of Manchester University; 9 of them are discussed here. Their orbits are doubly symmetrical. In order that the results could be applicable to problems relating to the solar system or double stars, most of the families we computed for several values of μ (μ is the mass of the minor component, 1-μ being the mass of the other). The stability of each orbit has also been examined by evaluating the eigenvalues of the Jacobian of the orbit. There exist no stable orbits. The least unstable ones are those for which all the eigenvalues are equal to unity. Such orbits are called quasi-stable. We prove that particles moving along such orbits when perturbed by O(ε) do not deviate from the periodic track by more than O(ε) for at least [ Aε −2] periods, where A is a suitable positive constant. Three important conclusions have been drawn: (a) The quantity μ has only a quantitative effect on the problem in the sense that its variation within the whole range of permissible values (0,1), does not produce new types of periodic orbits. (b) Almost all the periodic orbits which are highly inclined to the plane of motion of the two primaries are very unstable. (c) All the periodic orbits going round the straight-line equilibrium points of Lagrange are also very unstable, whatever their inclination. The second conclusion suggests that systems for which the assumptions of the restricted problem hold to a good approximation, must be or will become flat. The third conclusion suggests that no accumulation of stellar matter ejected by the components of binary stars can take place at the equilibrium points. Abhayankar (1959) has drawn the same conclusion from computations on planar orbits.