Asymptotic orbits in the restricted four-body problem

This paper studies the asymptotic solutions of the restricted planar problem of four bodies, three of which are finite, moving in circular orbits around their center of masses, while the fourth is infinitesimal. Two of the primaries have equal mass and the most-massive primary is located at the origin of the system. We found the invariant unstable and stable manifolds around the hyperbolic Lyapunov periodic orbits which emanate from the collinear equilibrium points Li,i=1,…,4, as well as the invariant manifolds from the Lagrangian critical points L5L5 and L6L6. We construct numerically, applying forward and backward integration from the intersection points of the appropriate Poincaré cuts, homo- and hetero-clinic, symmetric and non-symmetric asymptotic orbits. We present the characteristic curves of the 24 families which consist of symmetric simple-periodic orbits of the problem for a fixed value of the mass parameter b. The stability of the families is computed and also presented. Sixteen families contain as terminal points asymptotic periodic orbits which intersect the x  -axis perpendicularly and tend asymptotically to L5L5 for t→+∞t→+∞ and to L6L6 for t→-∞t→-∞, spiralling into (and out of) these points. The corresponding 16 terminating heteroclinic asymptotic orbits, for b=2b=2, are illustrated.