1,178 research outputs found
High order symplectic integrators for perturbed Hamiltonian systems
We present a class of symplectic integrators adapted for the integration of
perturbed Hamiltonian systems of the form . We give a
constructive proof that for all integer , there exists an integrator with
positive steps with a remainder of order ,
where is the stepsize of the integrator. The analytical expressions of
the leading terms of the remainders are given at all orders. In many cases, a
corrector step can be performed such that the remainder becomes
. The performances of these integrators
are compared for the simple pendulum and the planetary 3-Body problem of
Sun-Jupiter-Saturn.Comment: 24 pages, 6 figurre
A ring as a model of the main belt in planetary ephemerides
We assess the ability of a solid ring to model a global perturbation induced
by several thousands of main-belt asteroids. The ring is first studied in an
analytical framework that provides an estimate of all the ring's parameters
excepting mass. In the second part, numerically estimated perturbations on the
Earth-Mars, Earth-Venus, and Earth-Mercury distances induced by various subsets
of the main-belt population are compared with perturbations induced by a ring.
To account for large uncertainties in the asteroid masses, we obtain results
from Monte Carlo experiments based on asteroid masses randomly generated
according to available data and the statistical asteroid model. The radius of
the ring is analytically estimated at 2.8 AU. A systematic comparison of the
ring with subsets of the main belt shows that, after removing the 300 most
perturbing asteroids, the total main-belt perturbation of the Earth-Mars
distance reaches on average 246 m on the 1969-2010 time interval. A ring with
appropriate mass is able to reduce this effect to 38 m. We show that, by
removing from the main belt ~240 asteroids that are not necessarily the most
perturbing ones, the corresponding total perturbation reaches on average 472 m,
but the ring is able to reduce it down to a few meters, thus accounting for
more than 99% of the total effect.Comment: 18 pages, accepted in A&
Tests of General relativity with planetary orbits and Monte Carlo simulations
Based on the new developped planetary ephemerides INPOP13c, determinations of
acceptable intervals of General Relativity violation in considering
simultaneously the PPN parameters , PPN , the flattening of the
sun and time variation of the gravitational mass of the sun
are obtained in using Monte Carlo simulation coupled with basic genetic
algorithm. Possible time variations of the gravitational constant G are also
deduced. Discussions are lead about the better choice of indicators for the
goodness-of-fit for each run and limits consistent with general relativity are
obtained simultaneously.Comment: submitte
Planets in Mean-Motion Resonances and the System Around HD45364
In some planetary systems, the orbital periods of two of its members present
a commensurability, usually known by mean-motion resonance. These resonances
greatly enhance the mutual gravitational influence of the planets. As a
consequence, these systems present uncommon behaviors, and their motions need
to be studied with specific methods. Some features are unique and allow us a
better understanding and characterization of these systems. Moreover,
mean-motion resonances are a result of an early migration of the orbits in an
accretion disk, so it is possible to derive constraints on their formation.
Here we review the dynamics of a pair of resonant planets and explain how their
orbits evolve in time. We apply our results to the HD 45365 planetary system.Comment: invited review, 17 pages, 6 figure
An Overview of the 13:8 Mean Motion Resonance between Venus and Earth
It is known since the seminal study of Laskar (1989) that the inner planetary
system is chaotic with respect to its orbits and even escapes are not
impossible, although in time scales of billions of years. The aim of this
investigation is to locate the orbits of Venus and Earth in phase space,
respectively to see how close their orbits are to chaotic motion which would
lead to unstable orbits for the inner planets on much shorter time scales.
Therefore we did numerical experiments in different dynamical models with
different initial conditions -- on one hand the couple Venus-Earth was set
close to different mean motion resonances (MMR), and on the other hand Venus'
orbital eccentricity (or inclination) was set to values as large as e = 0.36 (i
= 40deg). The couple Venus-Earth is almost exactly in the 13:8 mean motion
resonance. The stronger acting 8:5 MMR inside, and the 5:3 MMR outside the 13:8
resonance are within a small shift in the Earth's semimajor axis (only 1.5
percent). Especially Mercury is strongly affected by relatively small changes
in eccentricity and/or inclination of Venus in these resonances. Even escapes
for the innermost planet are possible which may happen quite rapidly.Comment: 14 pages, 11 figures, submitted to CMD
Long-Term Stability of Horseshoe Orbits
Unlike Trojans, horseshoe coorbitals are not generally considered to be
long-term stable (Dermott and Murray, 1981; Murray and Dermott, 1999). As the
lifetime of Earth's and Venus's horseshoe coorbitals is expected to be about a
Gyr, we investigated the possible contribution of late-escaping inner planet
coorbitals to the lunar Late Heavy Bombardment. Contrary to analytical
estimates, we do not find many horseshoe objects escaping after first 100 Myr.
In order to understand this behaviour, we ran a second set of simulations
featuring idealized planets on circular orbits with a range of masses. We find
that horseshoe coorbitals are generally long lived (and potentially stable) for
systems with primary-to-secondary mass ratios larger than about 1200. This is
consistent with results of Laughlin and Chambers (2002) for equal-mass pairs or
coorbital planets and the instability of Jupiter's horseshoe companions (Stacey
and Connors, 2008). Horseshoe orbits at smaller mass ratios are unstable
because they must approach within 5 Hill radii of the secondary. In contrast,
tadpole orbits are more robust and can remain stable even when approaching
within 4 Hill radii of the secondary.Comment: Accepted for MNRA
Note on the generalized Hansen and Laplace coefficients
Recently, Breiter et al (2004) reported the computation of Hansen
coefficients for non integer values of . In fact, the
Hansen coefficients are closely related to the Laplace , and
generalized Laplace coefficients (Laskar and Robutel, 1995)
that do not require to be integers. In particular, the coefficients
X_0^{\g,m} have very simple expressions in terms of the usual Laplace
coefficients b_{\g+2}^{(m)}, and all their properties derive easily from the
known properties of the Laplace coefficients.Comment: 9/11/200
Detecting chaos in particle accelerators through the frequency map analysis method
The motion of beams in particle accelerators is dominated by a plethora of
non-linear effects which can enhance chaotic motion and limit their
performance. The application of advanced non-linear dynamics methods for
detecting and correcting these effects and thereby increasing the region of
beam stability plays an essential role during the accelerator design phase but
also their operation. After describing the nature of non-linear effects and
their impact on performance parameters of different particle accelerator
categories, the theory of non-linear particle motion is outlined. The recent
developments on the methods employed for the analysis of chaotic beam motion
are detailed. In particular, the ability of the frequency map analysis method
to detect chaotic motion and guide the correction of non-linear effects is
demonstrated in particle tracking simulations but also experimental data.Comment: Submitted for publication in Chaos, Focus Issue: Chaos Detection
Methods and Predictabilit
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