137 research outputs found
Analytical method for perturbed frozen orbit around an Asteroid in highly inhomogeneous gravitational fields : A first approach
This article provides a method for nding initial conditions for perturbed frozen orbits around inhomogeneous fast rotating asteroids. These orbits can be used as reference trajectories in missions that require close inspection of any rigid body. The generalized perturbative procedure followed exploits the analytical methods of relegation of the argument of node and Delaunay normalisation to arbitrary order. These analytical methods are extremely powerful but highly computational. The gravitational potential of the heterogeneous body is rstly stated, in polar-nodal coordinates, which takes into account the coecients of the spherical harmonics up to an arbitrary order. Through the relegation of the argument of node and the Delaunay normalization, a series of canonical transformations of coordinates is found, which reduces the Hamiltonian describing the system to a integrable, two degrees of freedom Hamiltonian plus a truncated reminder of higher order. Setting eccentricity, argument of pericenter and inclination of the orbit of the truncated system to be constant, initial conditions are found, which evolve into frozen orbits for the truncated system. Using the same initial conditions yields perturbed frozen orbits for the full system, whose perturbation decreases with the consideration of arbitrary homologic equations in the relegation and normalization procedures. Such procedure can be automated for the first homologic equation up to the consideration of any arbitrary number of spherical harmonics coefficients. The project has been developed in collaboration with the European Space Agency (ESA)
Production of trans-Neptunian binaries through chaos-assisted capture
The recent discovery of binary objects in the Kuiper-belt opens an invaluable
window into past and present conditions in the trans-Neptunian part of the
Solar System. For example, knowledge of how these objects formed can be used to
impose constraints on planetary formation theories. We have recently proposed a
binary-object formation model based on the notion of chaos-assisted capture.
Here we present a more detailed analysis with calculations performed in the
spatial (three-dimensional) three- and four-body Hill approximations. It is
assumed that the potential binary partners are initially following heliocentric
Keplerian orbits and that their relative motion becomes perturbed as these
objects undergo close encounters. First, the mass, velocity, and orbital
element distribu- tions which favour binary formation are identified in the
circular and elliptical Hill limits. We then consider intruder scattering in
the circular Hill four-body problem and find that the chaos-assisted capture
mechanism is consistent with observed, apparently randomly distributed, binary
mutual orbit inclinations. It also predicts asymmetric distributions of
retrograde versus prograde orbits. The time-delay induced by chaos on particle
transport through the Hill sphere is analogous to the formation of a resonance
in a chemical reaction. Implications for binary formation rates are considered
and the 'fine-tuning' problem recently identified by Noll et al. (2007) is also
addressed.Comment: submitted to MNRA
Families of Canonical Transformations by Hamilton-Jacobi-Poincar\'e equation. Application to Rotational and Orbital Motion
The Hamilton-Jacobi equation in the sense of Poincar\'e, i.e. formulated in
the extended phase space and including regularization, is revisited building
canonical transformations with the purpose of Hamiltonian reduction. We
illustrate our approach dealing with orbital and attitude dynamics. Based on
the use of Whittaker and Andoyer symplectic charts, for which all but one
coordinates are cyclic in the Hamilton-Jacobi equation, we provide whole
families of canonical transformations, among which one recognizes the familiar
ones used in orbital and attitude dynamics. In addition, new canonical
transformations are demonstrated.Comment: 21 page
The theory of canonical perturbations applied to attitude dynamics and to the Earth rotation. Osculating and nonosculating Andoyer variables
The Hamiltonian theory of Earth rotation, known as the Kinoshita-Souchay
theory, operates with nonosculating Andoyer elements. This situation parallels
a similar phenomenon that often happens (but seldom gets noticed) in orbital
dynamics, when the standard Lagrange-type or Delaunay-type planetary equations
unexpectedly render nonosculating orbital elements. In orbital mechanics,
osculation loss happens when a velocity-dependent perturbation is plugged into
the standard planetary equations. In attitude mechanics, osculation is lost
when an angular-velocity-dependent disturbance is plugged in the standard
dynamical equations for the Andoyer elements. We encounter exactly this
situation in the theory of Earth rotation, because this theory contains an
angular-velocity-dependent perturbation (the switch from an inertial frame to
that associated with the precessing ecliptic of date).
While the osculation loss does not influence the predictions for the figure
axis of the planet, it considerably alters the predictions for the
instantaneous spin-axis' orientation. We explore this issue in great detail
The Moyal-Lie Theory of Phase Space Quantum Mechanics
A Lie algebraic approach to the unitary transformations in Weyl quantization
is discussed. This approach, being formally equivalent to the
-quantization, is an extension of the classical Poisson-Lie formalism
which can be used as an efficient tool in the quantum phase space
transformation theory.Comment: 15 pages, no figures, to appear in J. Phys. A (2001
A new dynamical model for the study of galactic structure
In the present article, we present a new gravitational galactic model,
describing motion in elliptical as well as in disk galaxies, by suitably
choosing the dynamical parameters. Moreover, a new dynamical parameter, the
S(g) spectrum, is introduced and used, in order to detect islandic motion of
resonant orbits and the evolution of the sticky regions. We investigate the
regular or chaotic character of motion, with emphasis in the different
dynamical models and make an extensive study of the sticky regions of the
system. We use the classical method of the Poincare (r-pr) phase plane and the
new dynamical parameter of the S(g) spectrum. The LCE is used, in order to make
an estimation of the degree of chaos in our galactic model. In both cases, the
numerical calculations, suggest that our new model, displays a wide variety of
families of regular orbits, compared to other galactic models. In addition to
the regular motion, this new model displays also chaotic regions. Furthermore,
the extent of the chaotic regions increases, as the value of the flatness
parameter b of the model increases. Moreover, our simulations indicate, that
the degree of chaos in elliptical galaxies, is much smaller than that in dense
disk galaxies. In both cases numerical calculations show, that the degree of
chaos increases linearly, as the flatness parameter b increases. In addition, a
linear relationship between the critical value of angular momentum and the b
parameter if found, in both cases (elliptical and disk galaxies). Some
theoretical arguments to support the numerical outcomes are presented.
Comparison with earlier work is also made.Comment: Published in New Astronomy journa
Geometrical Models of the Phase Space Structures Governing Reaction Dynamics
Hamiltonian dynamical systems possessing equilibria of stability type display \emph{reaction-type
dynamics} for energies close to the energy of such equilibria; entrance and
exit from certain regions of the phase space is only possible via narrow
\emph{bottlenecks} created by the influence of the equilibrium points. In this
paper we provide a thorough pedagogical description of the phase space
structures that are responsible for controlling transport in these problems. Of
central importance is the existence of a \emph{Normally Hyperbolic Invariant
Manifold (NHIM)}, whose \emph{stable and unstable manifolds} have sufficient
dimensionality to act as separatrices, partitioning energy surfaces into
regions of qualitatively distinct behavior. This NHIM forms the natural
(dynamical) equator of a (spherical) \emph{dividing surface} which locally
divides an energy surface into two components (`reactants' and `products'), one
on either side of the bottleneck. This dividing surface has all the desired
properties sought for in \emph{transition state theory} where reaction rates
are computed from the flux through a dividing surface. In fact, the dividing
surface that we construct is crossed exactly once by reactive trajectories, and
not crossed by nonreactive trajectories, and related to these properties,
minimizes the flux upon variation of the dividing surface.
We discuss three presentations of the energy surface and the phase space
structures contained in it for 2-degree-of-freedom (DoF) systems in the
threedimensional space , and two schematic models which capture many of
the essential features of the dynamics for -DoF systems. In addition, we
elucidate the structure of the NHIM.Comment: 44 pages, 38 figures, PDFLaTe
Investigating the nature of motion in 3D perturbed elliptic oscillators displaying exact periodic orbits
We study the nature of motion in a 3D potential composed of perturbed
elliptic oscillators. Our technique is to use the results obtained from the 2D
potential in order to find the initial conditions generating regular or chaotic
orbits in the 3D potential. Both 2D and 3D potentials display exact periodic
orbits together with extended chaotic regions. Numerical experiments suggest,
that the degree of chaos increases rapidly, as the energy of the test particle
increases. About 97% of the phase plane of the 2D system is covered by chaotic
orbits for large energies. The regular or chaotic character of the 2D orbits is
checked using the S(c) dynamical spectrum, while for the 3D potential we use
the S(c) spectrum, along with the P(f) spectral method. Comparison with other
dynamical indicators shows that the S(c) spectrum gives fast and reliable
information about the character of motion.Comment: Published in Nonlinear Dynamics (NODY) journa
Planetary Dynamics and Habitable Planet Formation In Binary Star Systems
Whether binaries can harbor potentially habitable planets depends on several
factors including the physical properties and the orbital characteristics of
the binary system. While the former determines the location of the habitable
zone (HZ), the latter affects the dynamics of the material from which
terrestrial planets are formed (i.e., planetesimals and planetary embryos), and
drives the final architecture of the planets assembly. In order for a habitable
planet to form in a binary star system, these two factors have to work in
harmony. That is, the orbital dynamics of the two stars and their interactions
with the planet-forming material have to allow terrestrial planet formation in
the habitable zone, and ensure that the orbit of a potentially habitable planet
will be stable for long times. We have organized this chapter with the same
order in mind. We begin by presenting a general discussion on the motion of
planets in binary stars and their stability. We then discuss the stability of
terrestrial planets, and the formation of potentially habitable planets in a
binary-planetary system.Comment: 56 pages, 29 figures, chapter to appear in the book: Planets in
Binary Star Systems (Ed. N. Haghighipour, Springer publishing company
A new set of integrals of motion to propagate the perturbed two-body problem
A formulation of the perturbed two-body problem that relies on a new set of orbital elements is presented. The proposed method represents a generalization of the special perturbation method published by Peláez et al. (Celest Mech Dyn Astron 97(2):131?150,2007) for the case of a perturbing force that is partially or totally derivable from a potential. We accomplish this result by employing a generalized Sundman time transformation in the framework of the projective decomposition, which is a known approach for transforming the two-body problem into a set of linear and regular differential equations of motion. Numerical tests, carried out with examples extensively used in the literature, show the remarkable improvement of the performance of the new method for different kinds of perturbations and eccentricities. In particular, one notable result is that the quadratic dependence of the position error on the time-like argument exhibited by Peláez?s method for near-circular motion under the J2 perturbation is transformed into linear.Moreover, themethod reveals to be competitive with two very popular elementmethods derived from theKustaanheimo-Stiefel and Sperling-Burdet regularizations
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