Evolution with the mass parameter of families of asymmetric periodic solutions of the restricted three body problem

Abstract

The two asymmetric bifurcations associated with the exterior commensurabilities of the formq+1: 1 are found to exist forq=1, 2, 3, 4 throughout the entire range of the mass parameter μ. The asymmetric families emanating from these bifurcation orbits for values of μ=0.05, 0.15, 0.25, 0.35, and 0.45 are completely determined. For μ=0.05 and μ=0.15 for each of the four pairs of critical orbits the family of asymmetric periodic orbits evolves continuously from one bifurcation orbit to the other. The families evolving from each pair of bifurcations for μ=0.25, 0.35, and 0.45 terminate on an asymptotic orbit to one of the two equilateral equilibrium points. The evolution of these four families of asymmetric periodic orbits therefore changes significantly for some value of μ between 0.15 and 0.25. An indication is given on how the evolution with μ of familyl and its branches can be studied.