98 research outputs found
A reverse KAM method to estimate unknown mutual inclinations in exoplanetary systems
The inclinations of exoplanets detected via radial velocity method are
essentially unknown. We aim to provide estimations of the ranges of mutual
inclinations that are compatible with the long-term stability of the system.
Focusing on the skeleton of an extrasolar system, i.e., considering only the
two most massive planets, we study the Hamiltonian of the three-body problem
after the reduction of the angular momentum. Such a Hamiltonian is expanded
both in Poincar\'e canonical variables and in the small parameter , which
represents the normalised Angular Momentum Deficit. The value of the mutual
inclination is deduced from and, thanks to the use of interval
arithmetic, we are able to consider open sets of initial conditions instead of
single values. Looking at the convergence radius of the Kolmogorov normal form,
we develop a reverse KAM approach in order to estimate the ranges of mutual
inclinations that are compatible with the long-term stability in a KAM sense.
Our method is successfully applied to the extrasolar systems HD 141399, HD
143761 and HD 40307.Comment: 19 pages, 3 figure
On the convergence of an algorithm constructing the normal form for lower dimensional elliptic tori in planetary systems
We give a constructive proof of the existence of lower dimensional elliptic
tori in nearly integrable Hamiltonian systems. In particular we adapt the
classical Kolmogorov's normalization algorithm to the case of planetary
systems, for which elliptic tori may be used as replacements of elliptic
keplerian orbits in Lagrange-Laplace theory. With this paper we support with
rigorous convergence estimates the semi-analytical work in our previous article
(2011), where an explicit calculation of an invariant torus for a planar model
of the Sun-Jupiter-Saturn-Uranus system has been made. With respect to previous
works on the same subject we exploit the characteristic of Lie series giving a
precise control of all terms generated by our algorithm. This allows us to
slightly relax the non-resonance conditions on the frequencies.Comment: 45 page
Improved convergence estimates for the Schr\"oder-Siegel problem
We reconsider the Schr\"oder-Siegel problem of conjugating an analytic map in
in the neighborhood of a fixed point to its linear part, extending
it to the case of dimension . Assuming a condition which is equivalent to
Bruno's one on the eigenvalues of the linear part
we show that the convergence radius of the conjugating transformation
satisfies with
characterizing the eigenvalues , a constant not depending on
and . This improves the previous results for , where the
known proofs give . We also recall that is known to be the optimal
value for .Comment: 21 page
Secular orbital dynamics of the innermost exoplanet of the -Andromed{\ae} system
We introduce a quasi-periodic restricted Hamiltonian to describe the secular
motion of a small-mass planet in a multi-planetary system. In particular, we
refer to the motion of -And which is the innermost planet among
those discovered in the extrasolar system orbiting around the
-Andromedae A star. We preassign the orbits of the Super-Jupiter
exoplanets -And and -And in a stable configuration.
The Fourier decompositions of their secular motions are reconstructed by using
the Frequency Analysis and are injected in the equations describing the orbital
dynamics of -And under the gravitational effects exerted by those
two external exoplanets (expected to be major ones in such an extrasolar
system). We end up with a degrees of freedom Hamiltonian model; its
validity is confirmed by the comparison with several numerical integrations of
the complete -body problem. Furthermore, the model is enriched by taking
into account also the relativistic effects on the secular motion of the
innermost exoplanet. We focus on the problem of the stability of -And
as a function of the parameters that mostly impact on its orbit, i.e. the
initial values of its inclination and the longitude of its node. We study the
evolution of its eccentricity, crucial to exclude orbital configurations with
high probability of (quasi)collision with the central star in the long-time
evolution of the system. Moreover, we also introduce a normal form approach,
that further reduces our Hamiltonian model to a system with degrees of
freedom, which is integrable because it admits a constant of motion related to
the total angular momentum. This allows us to quickly preselect the domains of
stability for -And , with respect to the set of the initial
orbital configurations that are compatible with the observations
Quasi-periodic motions in a special class of dynamical equations with dissipative effects: a pair of detection methods
We consider a particular class of equations of motion, generalizing to n
degrees of freedom the "dissipative spin--orbit problem", commonly studied in
Celestial Mechanics. Those equations are formulated in a pseudo-Hamiltonian
framework with action-angle coordinates; they contain a quasi-integrable
conservative part and friction terms, assumed to be linear and isotropic with
respect to the action variables. In such a context, we transfer two methods
determining quasi-periodic solutions, which were originally designed to analyze
purely Hamiltonian quasi-integrable problems.
First, we show how the frequency map analysis can be adapted to this kind of
dissipative models. Our approach is based on a key remark: the method can work
as usual, by studying the behavior of the angular velocities of the motions as
a function of the so called "external frequencies", instead of the actions.
Moreover, we explicitly implement the Kolmogorov's normalization algorithm for
the dissipative systems considered here. In a previous article, we proved a
theoretical result: such a constructing procedure is convergent under the
hypotheses usually assumed in KAM theory. In the present work, we show that it
can be translated to a code making algebraic manipulations on a computer, so to
calculate effectively quasi-periodic solutions on invariant tori. Both the
methods are carefully tested, by checking that their predictions are in
agreement, in the case of the so called "dissipative forced pendulum".
Furthermore, the results obtained by applying our adaptation of the frequency
analysis method to the dissipative standard map are compared with some existing
ones in the literature
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