Dynamical Systems, the Three-Body Problem and Space Mission Design

This paper concerns heteroclinic connections and resonance transitions in the planar circular restricted 3-body problem, with applications to the dynamics of comets and asteroids and the design of space missions such as the Genesis Discovery Mission and low energy Earth to Moon transfers.The existence of a heteroclinic connection between pairs of equal energy periodic orbits around two of the libration points is shown numerically.This is applied to resonance transition and the construction of orbits with prescribed itineraries.Invariant manifold structures are relevant for transport between the interior and exterior Hill's regions, and other resonant phenomena throughout the solar system. Introduction.Resonant Transition in Comet Orbits.Some Jupiter comets such as Oterma and Gehrels 3 make a rapid transition from heliocentric orbits outside the orbit of Jupiter to orbits inside that of Jupiter and vice versa.During this transition, the comet may be captured temporarily by Jupiter for several orbits.The interior orbit is typically close to the 3:2 resonance while the exterior orbit is near the 2:3 resonance.During the transition, the orbit passes close to the libration points L 1 and L 2 , two of the five equilibrium points (in a rotating frame) for the planar circular restricted 3-body problem (PCR3BP) for the Sun-Jupiter system.The equilibrium points L 1 and L 2 are the ones closest to Jupiter, lying on either side of Jupiter along the Sun-Jupiter line, with L 1 being between Jupiter and the sun.The Relevance of Invariant Manifolds.Belbruno and B. Marsden [1997] considered the comet transitions using the "fuzzy boundary" concept.Lo and Ross [1997] used the PCR3BP as the underlying model and related it to invariant manifolds, noticing that the orbits of Oterma and Gehrels 3 (in the Sun-Jupiter rotating frame) closely follow the invariant manifolds of L 1 and L 2 .We develop this viewpoint along with another key ingredient, a heteroclinic connection between periodic orbits around L 1 and L 2 with the same Jacobi constant (a multiple of the energy for the PCR3BP) and the dynamical consequences of such an orbit.Invariant manifold structures associated with L 1 and L 2 periodic orbits and the heteroclinic connections assist in the understanding of transport throughout the solar system.We have drawn upon work of the Barcelona group on the PCR3BP, in particular, Llibre, Martinez and Simó [1985] as well as works of Moser, Conley and McGehee.Specific citations are given later.Heteroclinic Connections.A numerical demonstration is given of a heteroclinic connection between pairs of equal energy periodic orbits, one around L 1 , the other around L 2 .This heteroclinic connection augments the previously known homoclinic orbits associated with the L 1 and L 2 periodic orbits.Linking these heteroclinic connections and homoclinic orbits leads to dynamical chains which form the backbone for temporary capture and rapid resonance transition of Jupiter comets.See Figure 1.1.