38 research outputs found
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
Extracting Multidimensional Phase Space Topology from Periodic Orbits
We establish a hierarchical ordering of periodic orbits in a strongly coupled
multidimensional Hamiltonian system. Phase space structures can be
reconstructed quantitatively from the knowledge of periodic orbits alone. We
illustrate our findings for the hydrogen atom in crossed electric and magnetic
fields.Comment: 4 pages, 5 figures, accepted for publication in Phys. Rev. Let
Hydrogen atom in crossed electric and magnetic fields: Phase space topology and torus quantization via periodic orbits
A hierarchical ordering is demonstrated for the periodic orbits in a strongly
coupled multidimensional Hamiltonian system, namely the hydrogen atom in
crossed electric and magnetic fields. It mirrors the hierarchy of broken
resonant tori and thereby allows one to characterize the periodic orbits by a
set of winding numbers. With this knowledge, we construct the action variables
as functions of the frequency ratios and carry out a semiclassical torus
quantization. The semiclassical energy levels thus obtained agree well with
exact quantum calculations
A Semi-Analytic Algorithm for Constructing Lower Dimensional Elliptic Tori in Planetary Systems
We adapt the Kolmogorov's normalization algorithm (which is the key element
of the original proof scheme of the KAM theorem) to the construction of a
suitable normal form related to an invariant elliptic torus. As a byproduct,
our procedure can also provide some analytic expansions of the motions on
elliptic tori. By extensively using algebraic manipulations on a computer, we
explicitly apply our method to a planar four-body model not too different with
respect to the real Sun--Jupiter--Saturn--Uranus system. The frequency analysis
method allows us to check that our location of the initial conditions on an
invariant elliptic torus is really accurate.Comment: 31 pages, 4 figure
Dynamic Limits on Planar Libration-Orbit Coupling Around an Oblate Primary
This paper explores the dynamic properties of the planar system of an
ellipsoidal satellite in an equatorial orbit about an oblate primary. In
particular, we investigate the conditions for which the satellite is bound in
librational motion or when the satellite will circulate with respect to the
primary. We find the existence of stable equilibrium points about which the
satellite can librate, and explore both the linearized and non-linear dynamics
around these points. Absolute bounds are placed on the phase space of the
libration-orbit coupling through the use of zero-velocity curves that exist in
the system. These zero-velocity curves are used to derive a sufficient
condition for when the satellite's libration is bound to less than 90 degrees.
When this condition is not satisfied so that circulation of the satellite is
possible, the initial conditions at zero libration angle are determined which
lead to circulation of the satellite. Exact analytical conditions for
circulation and the maximum libration angle are derived for the case of a small
satellite in orbits of any eccentricity.Comment: Submitted to Celestial Mechanics and Dynamical Astronom
Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability into the light of Kolmogorov and Nekhoroshev theories
We investigate the long-time stability of the Sun-Jupiter-Saturn-Uranus
system by considering a planar secular model, that can be regarded as a major
refinement of the approach first introduced by Lagrange. Indeed, concerning the
planetary orbital revolutions, we improve the classical circular approximation
by replacing it with a solution that is invariant up to order two in the
masses; therefore, we investigate the stability of the secular system for
rather small values of the eccentricities. First, we explicitly construct a
Kolmogorov normal form, so as to find an invariant KAM torus which approximates
very well the secular orbits. Finally, we adapt the approach that is at basis
of the analytic part of the Nekhoroshev's theorem, so as to show that there is
a neighborhood of that torus for which the estimated stability time is larger
than the lifetime of the Solar System. The size of such a neighborhood,
compared with the uncertainties of the astronomical observations, is about ten
times smaller.Comment: 31 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1010.260
Generalizing the restricted three-body problem. The Bianular and Tricircular coherent problems
In this paper we construct two models for the motion of a
particle under the gravitational attraction of Sun, Jupiter, Saturn
and Uranus, that can be seen as a generalization of the well known
Restricted Three-Body Problem (RTBP). Both models are obtained by
computing quasi-periodic solutions – with two basic frequencies – of a
suitable N-body problem.
The first model is based on a quasi-periodic solution of the planar
Sun-Jupiter-Saturn Three-Body problem, that tries to approach the real
motion of Jupiter. The second model is based on a quasi-periodic
solution of the Sun-Jupiter-Saturn-Uranus Four-Body problem. In both
cases, we derive the equations of motion for a particle under the
gravitational attraction of these bodies as a quasi-periodic
time-dependent perturbation of the well-known RTBP.