arXiv subjects

Ugo Locatelli

Publications and source records attributed to Ugo Locatelli.

At least 19 recordsLinked to original sources

arXiv2023

Computer-assisted proofs of existence of KAM tori in planetary dynamical models of $\upsilon$-And $\mathbf{b}$

We reconsider the problem of the orbital dynamics of the innermost exoplanet of the $\upsilon$-Andromedae system (i.e., $\upsilon$-And $\mathbf{b}$) into the framework of a Secular Quasi-Periodic Restricted Hamiltonian model. This means that we preassign the orbits of the planets that are expected to be the biggest ones in that extrasolar system (namely, $\upsilon$-And $\mathbf{c}$ and $\upsilon$-And $\mathbf{d}$). The Fourier decompositions of their secular motions are injected in the equations describing the orbital dynamics of $\upsilon$-And $\mathbf{b}$ under the gravitational effects exerted by those two exoplanets. By a computer-assisted procedure, we prove the existence of KAM tori corresponding to orbital motions that we consider to be very robust configurations, according to the analysis and the numerical explorations made in our previous article. The computer-assisted assisted proofs are successfully performed for two variants of the Secular Quasi-Periodic Restricted Hamiltonian model, which differs for what concerns the effects of the relativistic corrections on the orbital motion of $\upsilon$-And $\mathbf{b}$, depending on whether they are considered or not.

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arXiv2023

Secular orbital dynamics of the innermost exoplanet of the $\upsilon$-Andromedæ 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 $\upsilon$-And $b$ which is the innermost planet among those discovered in the extrasolar system orbiting around the $\upsilon$-Andromedae A star. We preassign the orbits of the Super-Jupiter exoplanets $\upsilon$-And $c$ and $\upsilon$-And $d$ 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 $\upsilon$-And $b$ 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 $2+3/2$ degrees of freedom Hamiltonian model; its validity is confirmed by the comparison with several numerical integrations of the complete $4$-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 $\upsilon$-And $b$ 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 $2$ 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 $\upsilon$-And $b$, with respect to the set of the initial orbital configurations that are compatible with the observations.

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arXiv2023

Existence proof of librational invariant tori in an averaged model of HD60532 planetary system

We 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 of 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.

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arXiv2022

A numerical criterion evaluating the robustness of planetary architectures; applications to the $\upsilon$ Andromedæ system

We revisit the problem of the existence of KAM tori in extrasolar planetary systems. Specifically, we consider the $\upsilon$ 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 $\upsilon$ Andromedæ system itself.

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arXiv2022

Invariant KAM tori: from theory to applications to exoplanetary systems

We 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.

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arXiv2021

Librational KAM tori in the secular dynamics of the $\upsilon$ Andromedæ planetary system

We study the planetary system of $\upsilon$~Andromedæ, considering the three-body problem formed by the central star and the two largest planets, $\upsilon$~And~\emph{c} and $\upsilon$~And~\emph{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.

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arXiv2021

Hamiltonian Control of Magnetic Field Lines: Computer Assisted Results Proving the Existence of KAM Barriers

We reconsider a control theory for Hamiltonian systems, that was introduced on the basis of KAM theory and applied to a model of magnetic field in previous articles. By a combination of Frequency Analysis and of a rigorous (Computer Assisted) KAM algorithm we prove that in the phase space of the magnetic field, due to the control term, a set of invariant tori appear, and it acts as a transport barrier. Our analysis, which is common (but often also limited) to Celestial Mechanics, is based on a normal form approach; it is also quite general and can be applied to quasi-integrable Hamiltonian systems satisfying a few additional mild assumptions. As a novelty with respect to the works that in the last two decades applied Computer Assisted Proofs into the framework of KAM theory, we provide all the codes allowing to produce our results. They are collected in a software package that is publicly available from the {\it Mendeley Data} repository. All these codes are designed in such a way to be easy-to-use, also for what concerns eventual adaptations for applications to similar problems.

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arXiv2021

Elliptic tori in FPU non-linear chains with a small number of nodes

We revisit an algorithm constructing elliptic tori, that was originally designed for applications to planetary hamiltonian systems. The scheme is adapted to properly work with models of chains of $N+1$ particles interacting via anharmonic potentials, thus covering also the case of FPU chains. After having preliminarily settled the Hamiltonian in a suitable way, we perform a sequence of canonical transformations removing the undesired perturbative terms by an iterative procedure. This is done by using the Lie series approach, that is explicitly implemented in a programming code with the help of a software package, which is especially designed for computer algebra manipulations. In the cases of FPU chains with $N=4,\, 8$, we successfully apply our new algorithm to the construction of elliptic tori for wide sets of the parameter ruling the size of the perturbation, i.e., the total energy of the system. Moreover, we explore the stability regions surrounding 1D elliptic tori. We compare our semi-analytical results with those provided by numerical explorations of the FPU-model dynamics, where the latter ones are obtained by using techniques based on the so called frequency analysis. We find that our procedure works up to values of the total energy that are of the same order of magnitude with respect to the maximal ones, for which elliptic tori are detected by numerical methods.

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arXiv2021

Computer-assisted estimates for Birkhoff normal forms

Birkhoff normal forms are commonly used in order to ensure the so called "effective stability" in the neighborhood of elliptic equilibrium points for Hamiltonian systems. From a theoretical point of view, this means that the eventual diffusion can be bounded for time intervals that are exponentially large with respect to the inverse of the distance of the initial conditions from such equilibrium points. Here, we focus on an approach that is suitable for practical applications: we extend a rather classical scheme of estimates for both the Birkhoff normal forms to any finite order and their remainders. This is made for providing explicit lower bounds of the stability time (that are valid for initial conditions in a fixed open ball), by using a fully rigorous computer-assisted procedure. We apply our approach in two simple contexts that are widely studied in Celestial Mechanics: the Hénon-Heiles model and the Circular Planar Restricted Three-Body Problem. In the latter case, we adapt our scheme of estimates for covering also the case of resonant Birkhoff normal forms and, in some concrete models about the motion of the Trojan asteroids, we show that it can be more advantageous with respect to the usual non-resonant ones.

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arXiv2017

Improved convergence estimates for the Schröder-Siegel problem

We reconsider the Schröder-Siegel problem of conjugating an analytic map in $\mathbb{C}$ 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 $λ_1,\ldots,λ_n$ of the linear part we show that the convergence radius $ρ$ of the conjugating transformation satisfies $\ln ρ(λ)\geq -CΓ(λ)+C'$ 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 $n=1$.

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arXiv2017

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 Poincaré canonical variables and in the small parameter $D_2$, which represents the normalised Angular Momentum Deficit. The value of the mutual inclination is deduced from $D_2$ 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.

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arXiv2017

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.

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arXiv2016

New Hamiltonian expansions adapted to the Trojan problem

A number of studies, referring to the observed Trojan asteroids of various planets in our Solar System, or to hypothetical Trojan bodies in extrasolar planetary systems, have emphasized the importance of so-called secondary resonances in the problem of the long term stability of Trojan motions. Such resonances describe commensurabilities between the fast, synodic, and secular frequency of the Trojan body, and, possibly, additional slow frequencies produced by more than one perturbing bodies. The presence of secondary resonances sculpts the dynamical structure of the phase space. Hence, identifying their location is a relevant task for theoretical studies. In the present paper we combine the methods introduced in two recent papers (Paez & Efthymiopoulos, 2015, Paez & Locatelli, 2015) in order to analytically predict the location of secondary resonances in the Trojan problem (SEE FILE FOR COMPLETE ABSTRACT)

astro-ph.EP↗

arXiv2015

Design of maneuvers based on new normal form approximations: The case study of the CPRTBP

In this work, we study the motions in the region around the equilateral Lagrangian equilibrium points L4 and L5, in the framework of the Circular Planar Restricted Three-Body Problem (hereafter, CPRTBP). We design a semi-analytic approach based on some ideas by Garfinkel in [4]: the Hamiltonian is expanded in Poincaré-Delaunay coordinates and a suitable average is performed. This allows us to construct (quasi) invariant tori that are moderately far from the Lagrangian points L4-L5 and approximate wide tadpole orbits. This construction provides the tools for studying optimal transfers in the neighborhood of the equilateral points, when instantaneous impulses are considered. We show some applications of the new averaged Hamiltonian for the Earth-Moon system, applied to the setting-up of some transfers which allow to enter in the stability region filled by tadpole orbits.

astro-ph.EP↗

arXiv2015

Trojan dynamics well approximated by a new Hamiltonian normal form

We revisit a classical perturbative approach to the Hamiltonian related to the motions of Trojan bodies, in the framework of the Planar Circular Restricted Three-Body Problem (PCRTBP), by introducing a number of key new ideas in the formulation. In some sense, we adapt the approach of Garfinkel (1977) to the context of the normal form theory and its modern techniques. First, we make use of Delaunay variables for a physically accurate representation of the system. Therefore, we introduce a novel manipulation of the variables so as to respect the natural behavior of the model. We develop a normalization procedure over the fast angle which exploits the fact that singularities in this model are essentially related to the slow angle. Thus, we produce a new normal form, i.e. an integrable approximation to the Hamiltonian. We emphasize some practical examples of the applicability of our normalizing scheme, e.g. the estimation of the stable libration region. Finally, we compare the level curves produced by our normal form with surfaces of section provided by the integration of the non--normalized Hamiltonian, with very good agreement. Further precision tests are also provided. In addition, we give a step-by-step description of the algorithm, allowing for extensions to more complicated models.

astro-ph.EP↗

arXiv2014

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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arXiv2014

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.

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arXiv2013

Su un'estensione della teoria di Lagrange per i moti secolari

La teoria di Lagrange per i moti secolari delle eccentricità ed inclinazioni delle orbite planetarie si fondava su un'approssimazione, dettata in larga misura dalla complessità dei calcoli necessari, che consisteva nel considerare solo equazioni lineari. In questa memoria riprendiamo in considerazione i metodi di Lagrange alla luce della teoria della stabilità esponenziale di Nekhoroshev. Grazie agli algoritmi sviluppati negli ultimi anni e alle tecniche di manipolazione algebrica possiamo tener conto anche dei contributi non lineari alle equazioni. Come applicazione cerchiamo di determinare i tempi di stabilità per il problema dei tre corpi nel caso del Sole e dei due pianeti maggiori, Giove e Saturno, mostrando che si possono ottenere risultati realistici, ancorché non ottimali. Lagrange's theory for the secular motion of perihelia and nodes of the planetary orbits was based on consideration of a linear approssimation of the dynamical equations, compatible with the complexity of the calculations. We extend Lagrange's investigations in the light of Nekhoroshev's theory of exponential stability. Using effective algorithms recently developed and computer algebra we investigate the non linear problem. We apply our methods to the problem of three bodies in the Sun-Jupiter-Saturn case, thus showing that realistic results, although not optimal, can be obtained.

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