Shane D. Ross

Connecting orbits and invariant manifolds in the spatial restricted three-body problem

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 L_1 and the other around L_2. Since these connections are asymptotic orbits, no manoeuvre is needed to perform the transfer from one libration point orbit to the other. A knowledge of these orbits can be very useful in the design of missions such as the Genesis Discovery Mission, and may provide the backbone for other interesting orbits in the future.

The Lunar L1 Gateway : portal to the stars and beyond

Our Solar System is interconnected by a vast system of winding tunnels generated by the Lagrange Points of all the planets and their moons. These passageways are identified by portals around L1 and L2, the halo orbits. By passing through a halo orbit portal, one enters the ancient and colossal labyrinth of the Sun. This natural Interplanetary Supher highway System (IPS) provides ultra-low energy transport throughout the Earths Neighborhood, the region between Earths L1 and L2. This is enabled by an accident: the current energy levels of the Earth L1 and L2 Lagrange points differ from that of the Earth-Moon by only about 50 rn/s (as measured by AV). The significance of this happy coincidence to the development of space cannot be overstated. For example, this implies that Lunar L1 halo orbits are connected to halo orbits around Earths L1 or L2 via low energy pathways...

The computation of finite-time Lyapunov exponents on unstructured meshes and for non-Euclidean manifolds.

We generalize the concepts of finite-time Lyapunov exponent (FTLE) and Lagrangian coherent structures to arbitrary Riemannian manifolds. The methods are illustrated for convection cells on cylinders and Mobius strips, as well as for the splitting of the Antarctic polar vortex in the spherical stratosphere and a related point vortex model. We modify the FTLE computational method and accommodate unstructured meshes of triangles and tetrahedra to fit manifolds of arbitrary shape, as well as to facilitate dynamic refinement of the FTLE mesh.

Transport in Dynamical Astronomy and Multibody Problems

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.

New methods in celestial mechanics and mission design

The title of this paper is inspired by the work of Poincare [1890, 1892], who introduced many key dynamical systems methods during his research on celestial mechanics and especially the three-body problem. Since then, many researchers have contributed to his legacy by developing and applying these methods to problems in celestial mechanics and, more recently, with the design of space missions. This paper will give a survey of some of these exciting ideas, and we would especially like to acknowledge the work of Michael Dellnitz, Frederic Gabern, Katalin Grubits, Oliver Junge, Wang-Sang Koon, Francois Lekien, Martin Lo, Sina Ober-Blobaum, Kathrin Padberg, Robert Preis, and Bianca Thiere. One of the purposes of the AMS Current Events session is to discuss work of others. Even though we were involved in the research reported on here, this short paper is intended to survey many ideas due to our collaborators and others. This survey is by no means complete, and we apologize for not having time or space to do justice to many important and fundamental works. In fact, the results reported on here rely on and were inspired by important preceding work of many others in celestial mechanics, mission design and in dynamical systems. We mention just a few whose work had a positive influence on what is reported here: Brian Barden, Ed Belbruno, Robert Farquhar, Gerard Gomez, George Haller, Charles Jaffe, Kathleen Howell, Linda Petzold, Josep Masdemont, Vered Rom-Kedar, Radu Serban, Carles Simo, Turgay Uzer, Steve Wiggins, and Roby Wilson. In an upcoming monograph (see Koon, Lo, Marsden, and Ross [2005]), the dynamical systems and computational approach and its application to mission design are discussed in detail. One of the key ideas is that the competing gravitational pull between celestial bodies creates a vast array of passageways that wind around the Sun, planets and moons. The boundaries of these passageways are realized geometrically as invariant manifolds attached to equilibrium points and periodic orbits in interlinked three-body problems. In particular, tube-like structures form an interplanetary transport network which will facilitate the exploration of Mercury, the Moon, the asteroids, and the outer solar system, including a mission to assess the possibility of life on Jupiters icy moons. The use of these methods in problems in molecular dynamics of interest in chemistry is also briefly discussed.

Multiple Gravity Assists, Capture, and Escape in the Restricted Three-Body Problem ∗

For low energy spacecraft trajectories such as multimoon orbiters for the Jupiter system, multiple gravity assists by moons could be used in conjunction with ballistic capture to drastically decrease fuel usage. In this paper, we investigate a special class of multiple gravity assists which can occur outside of the perturbing bodys sphere of influence (the Hill sphere) and which is dynamically connected to orbits that get captured by the perturber and orbits which escape to infinity. We proceed by deriving a family of symplectic twist maps to approximate a particles motion in the planar circular restricted three-body problem. The maps capture well the dynamics of the full equations of motion; the phase space contains a connected chaotic zone where intersections between unstable resonant orbit manifolds provide the template for lanes of fast migration between orbits of different semimajor axes. Within the chaotic zone, the concept of a set of reachable orbits is useful. This set can be considered bounded...

Resonance and Capture of Jupiter Comets

A number of Jupiter family comets such as Otermaand Gehrels 3make 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 L1and L2, two of the equilibrium points for the restricted three-body problem for the Sun-Jupiter system. Studying the libration point invariant manifold structures for L1and L2is 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 L1and 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 Jupiters orbit.

Halo orbit mission correction maneuvers using optimal control

This paper addresses the computation of the required trajectory correction maneuvers for a halo orbit space mission to compensate for the launch velocity errors introduced by inaccuracies of the launch vehicle. By combining dynamical systems theory with optimal control techniques, we are able to provide a compelling portrait of the complex landscape of the trajectory design space. This approach enables automation of the analysis to perform parametric studies that simply were not available to mission designers a few years ago, such as how the magnitude of the errors and the timing of the first trajectory correction maneuver affects the correction @DV. The impetus for combining dynamical systems theory and optimal control in this problem arises from design issues for the Genesis Discovery Mission being developed for NASA by the Jet Propulsion Laboratory.

Geometric Mechanics and the Dynamics of Asteroid Pairs

Abstract: The purpose of this paper is to describe the general setting for the application of techniques from geometric mechanics and dynamical systems to the problem of asteroid pairs. The paper also gives some preliminary results on transport calculations and the associated problem of calculating binary asteroid escape rates. The dynamics of an asteroid pair, consisting of two irregularly shaped asteroids interacting through their gravitational potential is an example of a full‐body problem or FBP in which two or more extended bodies interact. One of the interesting features of the binary asteroid problem is that there is coupling between their translational and rotational degrees of freedom. General FBPs have a wide range of other interesting aspects as well, including the 6‐DOF guidance, control, and dynamics of vehicles, the dynamics of interacting or ionizing molecules, the evolution of small body, planetary, or stellar systems, and almost any other problem in which distributed bodies interact with each other or with an external field. This paper focuses on the specific case of asteroid pairs using techniques that are generally applicable to many other FBPs. This particular full two‐body problem (F2BP) concerns the dynamical evolution of two rigid bodies mutually interacting via a gravitational field. Motivation comes from planetary science, where these interactions play a key role in the evolution of asteroid rotation states and binary asteroid systems. The techniques that are applied to this problem fall into two main categories. The first is the use of geometric mechanics to obtain a description of the reduced phase space, which opens the door to a number of powerful techniques, such as the energy‐momentum method for determining the stability of equilibria and the use of variational integrators for greater accuracy in simulation. Second, techniques from computational dynamic systems are used to determine phase space structures that are important for transport phenomena and dynamic evolution.

Monitoring the Long-Distance Transport of Fusarium graminearum from Field-Scale Sources of Inoculum

The fungus Fusarium graminearum causes Fusarium head blight (FHB) of wheat. Little is known about dispersal of the fungus from field-scale sources of inoculum. We monitored the movement of a clonal isolate of F. graminearum from a 3,716 m2 (0.372 ha) source of inoculum over two field seasons. Ground-based collection devices were placed at distances of 0 (in the source), 100, 250, 500, 750, and 1,000 m from the center of the clonal sources of inoculum. Three polymorphic microsatellites were used to identify the released clone from 1,027 isolates (790 in 2011 and 237 in 2012) of the fungus. Results demonstrated that the recovery of the released clone decreased at greater distances from the source. The majority (87%, 152/175 in 2011; 77%, 74/96 in 2012) of the released clone was recaptured during the night (1900 to 0700). The released clone was recovered up to 750 m from the source. Recovery of the released clone followed a logistic regression model and was significant (P < 0.041 for all slope term scenarios) as a function of distance from the source of inoculum. This work offers a means to experimentally determine the dispersal kernel of a plant pathogen, and could be integrated into management strategies for FHB.