1,720,972 research outputs found
Existence proof of librational invariant tori in an averaged model of HD60532 planetary system
Full text linkWe investigate the long-term dynamics of HD60532, an extrasolar system hosting two giant planets orbiting in a 3:1 mean motion resonance. We consider an average approximation at order one in the masses which results (after the reduction in the constants of motion) in a resonant Hamiltonian with two libration angles. In this framework, the usual algorithms constructing the Kolmogorov normal form approach do not easily apply and we need to perform some untrivial preliminary operations, in order to adapt the method to this kind of problems. First, we perform an average over the fast angle of libration which provides an integrable approximation of the Hamiltonian. Then, we introduce action-angle variables that are adapted to such an integrable approximation. This sequence of preliminary operations brings the Hamiltonian in a suitable form to successfully start the Kolmogorov normalization scheme. The convergence of the KAM algorithm is proved by applying a technique based on a computer-assisted proof. This allows us to reconstruct the quasi-periodic motion of the system, with initial conditions that are compatible with the observations
Invariant KAM Tori: From Theory to Applications to Exoplanetary Systems
No full textWe consider the classical problem of the construction of invariant tori exploiting suitable Hamiltonian normal forms. This kind of approach can be translated by means of the Lie series method into explicit computational algorithms, which are particularly suitable for applications in the field of Celestial Mechanics. First, the algorithm constructing the Kolmogorov normal form is described in detail. Then, the extension to lower-dimensional elliptic tori is provided. We adopt the same formalism and notations in both cases, with the aim of making the latter easier to understand. Finally, they are both used in a combined way in order to approximate carefully the secular dynamics of the extrasolar system hosting two planets orbiting around the HD 4732 star
A numerical criterion evaluating the robustness of planetary architectures; applications to the υ Andromedæ system
Full text linkWe revisit the problem of the existence of KAM tori in extrasolar planetary systems. Specifically, we consider the υ Andromedæ system, by modelling it with a three-body problem. This preliminary study allows us to introduce a natural way to evaluate the robustness of the planetary orbits, which can be very easily implemented in numerical explorations. We apply our criterion to the problem of the choice of a suitable orbital configuration which exhibits strong stability properties and is compatible with the observational data that are available for the υ Andromedæ system itself
Librational KAM tori in the secular dynamics of the υ Andromedæ planetary system
No full textThe nu Andromed oe system is the first extrasolar system where the mutual inclination between exoplanets has been determined using astrometric methods. We study a model of the.Andromedaeplanetary system considering the three-body problem formed by the central star and the two largest planets, nu And c and nu And d. We adopt a secular, three-dimensional model and initial conditions within the range of the observed values. The numerical integrations highlight that the system is orbiting around a one-dimensional elliptic torus (i.e. a periodic orbit that is linearly stable). This invariant object is used as a seed for an algorithm based on a sequence of canonical transformations. The algorithm determines the normal form related to a KAM torus, whose shape is in excellent agreement with the orbits of the secular model. We rigorously prove that the algorithm constructing the final KAM invariant torus is convergent, by adopting a suitable technique based on a computer-assisted proof. Thus, we are able to prove the stability of the secular dynamics for a set of values of the orbital elements that are in agreement with the observed ones. As a by-product, we can also extract a numerical indicator of robustness for the constructed invariant KAM tori. This allows us to identify ranges of the inclinations that are the most likely candidates according to the KAM stability prescription. In this context, we conclude that the most robust orbital configurations are those with large values of nu And c's mass, that is about 16 time bigger than Jupiter's one
3D Orbital Architecture of Exoplanetary Systems: KAM-Stability Analysis
Full text linkWe study the KAM-stability of several single star two-planetnonresonant extrasolar systems. It is likely that the observedexoplanets are the most massive of the system considered. Therefore,their robust stability is a crucial and necessary condition for thelong-term survival of the system when considering potentialadditional exoplanets yet to be seen. Our study is based on theconstruction of a combination of lower-dimensional elliptic and KAMtori, so as to better approximate the dynamics in the framework ofaccurate secular models. For each extrasolar system, we explore theparameter space of both inclinations: the one with respect to theline of sight and the mutual inclination between the planets. Ourapproach shows that remarkable inclinations, resulting inthree-dimensional architectures that are far from being coplanar,can be compatible with the KAM stability of the system. We findthat the highest values of the mutual inclinations are comparable tothose of the few systems for which the said inclinations are determinedby the observations
Improved convergence estimates for the Schröder-Siegel problem
Full text linkWe reconsider the Schröder–Siegel problem of conjugating an analytic map in ℂ in the neighborhood of a fixed point to its linear part, extending it to the case of dimension n>1 . Assuming a condition which is equivalent to Bruno’s one on the eigenvalues λ1,…,λn of the linear part, we show that the convergence radius ρ of the conjugating transformation satisfies lnρ(λ)≥−CΓ(λ)+C′ with Γ(λ) characterizing the eigenvalues λ , a constant C′ not depending on λ and C=1 . This improves the previous results for n>1 , where the known proofs give C=2 . We also recall that C=1 is known to be the optimal value for n=1
On the stability of the secular evolution of the planar Sun–Jupiter–Saturn–Uranus system
No full textWe investigate the long time stability of the Sun-Jupiter-Saturn-Uranus system by considering the planar, secular model. Our method may be considered as an extension of Lagrange's theory for the secular motions. Indeed, concerning the planetary orbital revolutions, we improve the classical circular approximation by replacing it with a torus which is invariant up to order two in the masses; therefore, we investigate the stability of the elliptic equilibrium point of the secular system for small values of the eccentricities. For the initial data corresponding to a real set of astronomical observations, we find an estimated stability time of 107 years, which is not extremely smaller than the lifetime of the Solar System (∼5 Gyr)
Kolmogorov and Nekhoroshev theory for the problem of three bodies
No full textWe investigate the long time stability in Nekhoroshev’s sense for the Sun–
Jupiter–Saturn problem in the framework of the problem of three bodies. Using computer
algebra in order to perform huge perturbation expansions we show that the stability for a time
comparable with the age of the universe is actually reached, but with some strong truncations
on the perturbation expansion of the Hamiltonian at some stage. An improvement of such
results is currently under investigation
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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