There has recently been considerable interest in sending a spacecraft to orbit Europa, the smallest of the four Galilean moons of Jupiter. The trajectory design involved in effecting a capture by Europa presents formidable challenges to traditional conic analysis since the regimes of motion involved depend heavily on three-body dynamics. New three-body perspectives are required to design successful and efficient missions which take full advantage of the natural dynamics. Not only does a three-body approach provide low-fuel trajectories, but it also increases the flexibility and versatility of missions. We apply this approach to design a new mission concept wherein a spacecraft “leap-frogs ” between moons, orbiting each for a desired duration in a temporary capture orbit. We call this concept the “Petit Grand Tour.” For this application, we apply dynamical systems techniques developed in a previous paper to design a Europa capture orbit. We show how it is possible, using a gravitional boost from Ganymede, to go from a jovicentric orbit beyond the orbit of Ganymede to a ballistic capture orbit around Europa. The main new technical result is the employment of dynamical channels in the phase space — tubes in the energy surface which naturally link the vicinity of Ganymede to the vicinity of Europa.

The invariant manifold structures of the collinear libration points for the restricted three-body problem provide the framework for understanding transport phenomena from a geometrical point of view. In particular, the stable and unstable invariant manifold tubes associated with libration point orbits are the phase space conduits transporting material between primary bodies for separate three-body systems. These tubes can be used to construct new spacecraft trajectories, such as a ‘Petit Grand Tour ’ of the moons of Jupiter. Previous work focused on the planar circular restricted three-body problem. This work extends the results to the three-dimensional case. Besides providing a full description of different kinds of libration motions in a large vicinity of these points, this paper numerically demonstrates the existence of heteroclinic connections between pairs of libration orbits, one around the libration point L1 and the other around L2. Since these connections are asymptotic orbits, no manoeuvre is needed to perform the transfer from

We combine the techniques of almost invariant sets (using tree structured box elimination and graph partitioning algorithms) with invariant manifold and lobe dynamics techniques. The result is a new computational technique for computing key dynamical features, including almost invariant sets, resonance regions as well as transport rates and bottlenecks between regions in dynamical systems. This methodology can be applied to a variety of multibody problems, including those in molecular modeling, chemical reaction rates and dynamical astronomy. In this paper we focus on problems in dynamical astronomy to illustrate the power of the combination of these different numerical tools and their applicability. In particular, we compute transport rates between two resonance regions for the three body system consisting of the Sun, Jupiter and a third body (such as an asteroid). These resonance regions are appropriate for certain comets and asteroids. Key words. Three-body problem, transport rates, dynamical systems, almost

Abstract. A number of Jupiter family comets such as Oterma and Gehrels 3 make a rapid transition from heliocentric orbits outside the orbit of Jupiter to heliocentric orbits inside the orbit of Jupiter and vice versa. During this transition, the comet can be captured temporarily by Jupiter for one to several orbits around Jupiter. The interior heliocentric orbit is typically close to the 3:2 resonance while the exterior heliocentric orbit is near the 2:3 resonance. An important feature of the dynamics of these comets is that during the transition, the orbit passes close to the libration points L 1 and L 2, two of the equilibrium points for the restricted three-body problem for the Sun-Jupiter system. Studying the libration point invariant manifold structures for L1 and L2 is a starting point for understanding the capture and resonance transition of these comets. For example, the recently discovered heteroclinic connection between pairs of unstable periodic orbits (one around the L1 and the other around L2) implies a complicated dynamics for comets in a certain energy range. Furthermore, the stable and unstable invariant manifold ‘tubes ’ associated to libration point periodic orbits, of which the heteroclinic connections are a part, are phase space conduits transporting material to and from Jupiter and between the interior and exterior of Jupiter’s orbit.

The computation, starting from basic principles, of chemical reaction rates in realistic systems (with three or more degrees of freedom) has been a longstanding goal of the chemistry community. Our current work, which merges tube dynamics with Monte Carlo methods provides some key theoretical and computational tools for achieving this goal. We use basic tools of dynamical systems theory, merging the ideas of Koon et al. [Chaos 10, 427 (2000)] and De Leon et al. [J. Chem. Phys. 94, 8310 (1991)], particularly the use of invariant manifold tubes that mediate the reaction, into the start of a comprehensive theory of lifetime distributions and rates of chemical reactions and scattering phenomena, even in systems that exhibit non-statistical behavior. Previously, the main problem with the application of tube dynamics has been with the analytical evaluation of volumes in phase spaces of arbitrary dimension. The present work overcomes this hurdle with some new ideas and implements them numerically. Specifically, an efficient algorithm that uses tube dynamics to provide the initial bounding box for a Monte Carlo volume determination is used. The combination of a fine scale method for understanding the phase space structure (invariant manifold theory) with statistical methods for practical

We present a new method based on set oriented computations for the calculation of reaction rates in chemical systems. The method is demonstrated with the Rydberg atom, an example for which traditional Transition State Theory fails. Coupled with dynamical systems theory, the set oriented approach provides a global description of the dynamics. The main idea of the method is as follows. We construct a box covering of a Poincaré section under consideration, use the Poincaré first return time for the identification of those regions relevant for transport and then we apply an adaptation of recently developed techniques for the computation of transport rates ([12], [27]). The reaction rates in chemical systems are of great interest in chemistry, especially for realistic three and higher dimensional systems. Our approach is applied to the Rydberg atom in crossed electric and magnetic fields. Our methods are complementary to, but in common problems considered, agree with, the results of [14]. For the Rydberg atom, we consider the half and full scattering problems in both the 2- and the 3-degree of freedom systems. The ionization of such atoms is a system on which many experiments have been done and it serves to illustrate the elegance of our method. Contents To the memory of Henri Poincaré,

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. 1 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

Abstract. The collinear relative equilibrium solutions are among the few explicitly known periodic solutions of the Newtonian three-body problem. When the energy and angular momentum constants are varied slightly these unstable periodic orbits become normally hyperbolic invariant spheres whose stable and unstable manifolds form separatrices in the integral manifolds. The goal of this paper is to construct simple isolating blocks for these invariant spheres analogous to those introduced by Conley in the restricted three-body problem. This allows continuation of the invariant set and the separatrices to energies and angular momenta far from those of the relative equilibrium. 1.

Abstract. Because of the Jacobi integral, solutions of the planar, circular restricted three-body problem are confined to certain subsets of the plane called Hill’s regions. For certain values of the integral, one component of the Hill’s region consists of disklike regions around each of the two primary masses, connected by a tunnel near the collinear Lagrange point, L2. A transit orbit is a solution which crosses the tunnel, in a sense which can be made precise using Conley’s isolating block construction. For values of the Jacobi integral sufficiently close to its value at L2, Conley found transit orbits by linearizing near the equilibrium point. The goal of this paper is to develop a method for proving existence of transit orbits for values of the Jacobi constant far from equilbrium. The method is based on the Maupertuis variational principle but isolating blocks turn out to play an important role. 1. The Restricted Three-Body Problem The paper is divided into four sections. This section contains the equations

Abstract. In 1991, the Japanese Hiten mission used a low energy transfer with a ballistic capture at the Moon which required less �V than a standard Hohmann transfer. In this paper, we apply the dynamical systems techniques developed in our earlier work to reproduce systematically a Hiten-like mission. We approximate the Sun–Earth–Moon-spacecraft 4-body system as two 3-body systems. Using the invariant manifold structures of the Lagrange points of the 3-body systems, we are able to construct low energy transfer trajectories from the Earth which execute ballistic capture at the Moon. The techniques used in the design and construction of this trajectory may be applied in many situations.