Heteroclinic connections between periodic orbits and resonance transitions in celestial mechanics

In this paper we apply dynamical systems techniques to the problem of heteroclinic connections and resonance transitions in the planar circular restricted three-body problem. These related phenomena have been of concern for some time in topics such as the capture of comets and asteroids and with the design of trajectories for space missions such as the Genesis Discovery Mission. The main new technical result in this paper is the numerical demonstration of the existence of a heteroclinic connection between pairs of periodic orbits: one around the libration point L1 and the other around L2, with the two periodic orbits having the same energy. This result is applied to the resonance transition problem and to the explicit numerical construction of interesting orbits with prescribed itineraries. The point of view developed in this paper is that the invariant manifold structures associated to L1 and L2 as well as the aforementioned heteroclinic connection are fundamental tools that can aid in understanding dynamical channels throughout the solar system as well as transport between the "interior" and "exterior" Hill's regions and other resonant phenomena

Heteroclinic Connections between Periodic Orbits in Planar Restricted Circular Three Body Problem - Part II

We present a method for proving the existence of symmetric periodic, heteroclinic or homoclinic orbits in dynamical systems with the reversing symmetry. As an application we show that the Planar Restricted Circular Three Body Problem (PCR3BP) corresponding to the Sun-Jupiter-Oterma system possesses an infinite number of symmetric periodic orbits and homoclinic orbits to the Lyapunov orbits. Moreover, we show the existence of symbolic dynamics on six symbols for PCR3BP and the possibility of resonance transitions of the comet. This extends earlier results by Wilczak and Zgliczynski - Heteroclinic Connections between Periodic Orbits in Planar Restricted Circular Three Body Problem - A Computer Assisted Proof, Commun. Math. Phys. 234, 37-75 (2003).Comment: 18 pages, 5 figure

Constructing a Low Energy Transfer Between Jovian Moons

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 transfer V necessary to jump from one moon to another is less than half that required by a standard Hohmann transfer

Geometry of Weak Stability Boundaries

The notion of a weak stability boundary has been successfully used to design low energy trajectories from the Earth to the Moon. The structure of this boundary has been investigated in a number of studies, where partial results have been obtained. We propose a generalization of the weak stability boundary. We prove analytically that, in the context of the planar circular restricted three-body problem, under certain conditions on the mass ratio of the primaries and on the energy, the weak stability boundary about the heavier primary coincides with a branch of the global stable manifold of the Lyapunov orbit about one of the Lagrange points

Trajectory design in the spatial circular restricted three-body problem exploiting higher-dimensional Poincare maps

In this investigation, the role of higher-dimensional Poincaré maps in facilitating trajectory design is explored for a variety of applications. To begin, existing strategies to implement Poincaré maps for trajectory design applications in the spatial CR3BP are evaluated. New applications for these strategies are explored, including an analysis of the natural motion of Jupiter-family comets that experience temporary capture about Jupiter, and the search for periodic orbits in the vicinity of the primary bodies in the spatial problem. Because current strategies to represent higher-dimensional maps, generally, lead to a loss of information, new approaches to represent all information contained in higher-dimensional Poincaré maps are sought. ^ The field of data visualization offers many options to visually represent multivariate data sets, including the use of glyphs. A glyph is any graphical object whose physical attributes are determined by the variables of a data set. In this investigation, the role of glyphs in representing higher-dimensional Poincaré maps is explored, and the resulting map representations are demonstrated to search for maneuver-free and low-cost transfers between libration point orbits. A catalog of libration point orbit transfers is developed in the Earth-Moon system, and observations about the catalog solutions yields insight into the existence of these transfers. The application of Poincaré maps to compute transfers between libration point orbits in different three-body systems is additionally considered. Finally, interactive trajectory design environments that incorporate Poincaré maps into the design process are demonstrated. Such design environments offer a unique opportunity to explore the available trajectory options and to gain intuition about the solution space