Binary Asteroid Observation Orbits from a Global Dynamical Perspective

We study spacecraft motion near a binary asteroid by means of theoretical and computational tools from geometric mechanics and dynamical systems. We model the system assuming that one of the asteroids is a rigid body (ellipsoid) and the other a sphere. In particular, we are interested in finding periodic and quasi-periodic orbits for the spacecraft near the asteroid pair that are suitable for observations and measurements. First, using reduction theory, we study the full two body problem (gravitational interaction between the ellipsoid and the sphere) and use the energy-momentum method to prove nonlinear stability of certain relative equilibria. This study allows us to construct the restricted full three-body problem (RF3BP) for the spacecraft motion around the binary, assuming that the asteroid pair is in relative equilibrium. Then, we compute the modified Lagrangian fixed points and study their spectral stability. The fixed points of the restricted three-body problem are modified in the RF3BP because one of the primaries is a rigid body and not a point mass. A systematic studydepending on the parameters of the problem is performed in an effort to understand the rigid body effects on the Lagrangian stability regions. Finally, using frequency analysis, we study the global dynamics near these modified Lagrangian points. From this global picture, we are able to identify (almost-) invariant tori in the stability region near the modified Lagrangian points. Quasi-periodic trajectories on these invariant tori are potentially convenient places to park the spacecraft while it is observing the asteroid pair

Frequency map analysis and quasiperiodic decompositions

Frequency Map Analysis is a numerical method based on refined Fourier techniques which provides a clear representation of the global dynamics of many multi-dimensional systems, and which is particularly adapted for systems of 3-degrees of freedom and more. This method relies heavily on the possibility of making accurate quasiperiodic approximations of of quasiperiodic signal given in a numerical way. In the present paper, we will describe the basis of the frequency analysis method, focussing on the quasi periodic approximation techniques. Application of these methods for the study of the global dynamics and chaotic diffusion of Hamiltonian systems and symplectic maps in different domains can be found in (Laskar, 1988, 1990, Laskar and Robutel, 1993, Robutel and Laskar, 2001, Nesvorny and Ferraz-Mello, 1997) for solar system dynamics, and in (Papaphilippou and Laskar, 1996, 1998, Laskar, 2000, Wachlin and Ferraz-Mello, 1998, Valluri and Merritt, 1998, Merritt and Valluri, 1999) for galactic dynamics. The method has been particularly successful for its application in particle accelerators (Dumas and Laskar, 1993, Laskar and Robin, 1996, Robin et al., 2000, Comunian et al., 2001, Papaphilippou and Zimmermann, 2002, Steier et al., 2002), and was also used for the understanding of atomic physics (Milczewski et al., 1997), or more general dynamical system issues (Laskar et al., 1992, Laskar, 1993, 1999, Chandre et al., 2001).Comment: 13 march 200

Elliptical Galaxy Dynamics

A review of elliptical galaxy dynamics, with a focus on nonintegrable models. Topics covered include torus construction; modelling axisymmetric galaxies; triaxiality; collisionless relaxation; and collective instabilities.Comment: 97 Latex pages, 14 Postscript figures, uses aastex. To appear in Publications of the Astronomical Society of the Pacific, February 199

On the relevance of chaos for halo stars in the Solar Neighbourhood

We show that diffusion due to chaotic mixing in the Neighbourhood of the Sun may not be as relevant as previously suggested in erasing phase space signatures of past Galactic accretion events. For this purpose, we analyse Solar Neighbourhood-like volumes extracted from cosmological simulations that naturally account for chaotic orbital behaviour induced by the strongly triaxial and cuspy shape of the resulting dark matter haloes, among other factors. In the approximation of an analytical static triaxial model, our results show that a large fraction of stellar halo particles in such local volumes have chaos onset times (i.e., the timescale at which stars commonly associated with chaotic orbits will exhibit their chaotic behaviour) significantly larger than a Hubble time. Furthermore, particles that do present a chaotic behaviour within a Hubble time do not exhibit significant diffusion in phase space.Comment: 20 pages, 16 figures. Accepted for publication in MNRA