Abstract:
We investigate the long-time stability of the Sun-Jupiter-Saturn-Uranus system by considering a planar secular model, which 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 to find an invariant KAM torus which approximates very well the secular orbits. Finally, we adapt the approach that underlies the analytic part of Nekhoroshev’s theorem 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.
Keywords:n-body planetary problem, KAM theory, Nekhoroshev theory, normal form methods, exponential stability, Hamiltonian systems, celestial mechanics.
This work has been supported by the research program “Teorie geometriche e analitiche dei sistemi Hamiltoniani in dimensioni finite e infinite”, PRIN 2010JJ4KPA009, financed by MIUR.
Citation:
Antonio Giorgilli, Ugo Locatelli, Marco Sansottera, “Secular Dynamics of a Planar Model of the Sun-Jupiter-Saturn-Uranus System; Effective Stability in the Light of Kolmogorov and Nekhoroshev Theories”, Regul. Chaotic Dyn., 22:1 (2017), 54–77
\Bibitem{GioLocSan17}
\by Antonio Giorgilli, Ugo Locatelli, Marco Sansottera
\paper Secular Dynamics of a Planar Model of the Sun-Jupiter-Saturn-Uranus System; Effective Stability in the Light of Kolmogorov and Nekhoroshev Theories
\jour Regul. Chaotic Dyn.
\yr 2017
\vol 22
\issue 1
\pages 54--77
\mathnet{http://mi.mathnet.ru/rcd243}
\crossref{https://doi.org/10.1134/S156035471701004X}
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This publication is cited in the following 26 articles:
A. N. Prokopenya, M. Zh. Minglibayev, M. R. Saparova, “Symbolic Calculations in the Study of Secular Perturbations in the Many-Body Problem with Variable Masses”, Program Comput Soft, 51:1 (2025), 32
Rita Mastroianni, Ugo Locatelli, “Computer-assisted proofs of existence of KAM tori in planetary dynamical models of υ-And b”, Communications in Nonlinear Science and Numerical Simulation, 130 (2024), 107706
Chiara Caracciolo, Ugo Locatelli, Marco Sansottera, Mara Volpi, “3D Orbital Architecture of Exoplanetary Systems: KAM-Stability Analysis”, Regul. Chaotic Dyn., 29:4 (2024), 565–582
Veronica Danesi, Ugo Locatelli, Marco Sansottera, “Existence proof of librational invariant tori in an averaged model of HD60532 planetary system”, Celest Mech Dyn Astron, 135:3 (2023)
Alessandra Celletti, “From infinite to finite time stability in Celestial Mechanics and Astrodynamics”, Astrophys Space Sci, 368:12 (2023)
Rita Mastroianni, Ugo Locatelli, “Secular orbital dynamics of the innermost exoplanet of the $\upsilon $-Andromedæ system”, Celest Mech Dyn Astron, 135:3 (2023)
Marco Sansottera, Veronica Danesi, “Kolmogorov variation: KAM with knobs (à la Kolmogorov)”, MINE, 5:5 (2023), 1
Caracciolo Ch., Locatelli U., Sansottera M., Volpi M., “Librational Kam Tori in the Secular Dynamics of the Nu Andromedae Planetary System”, Mon. Not. Roy. Astron. Soc., 510:2 (2022), 2147–2166
Ugo Locatelli, Chiara Caracciolo, Marco Sansottera, Mara Volpi, Springer Proceedings in Mathematics & Statistics, 399, New Frontiers of Celestial Mechanics: Theory and Applications, 2022, 1
Lorenzo Valvo, Ugo Locatelli, “Hamiltonian control of magnetic field lines: Computer assisted results proving the existence of KAM barriers”, JCD, 9:4 (2022), 505
Àngel Jorba, Encyclopedia of Complexity and Systems Science Series, Perturbation Theory, 2022, 153
Àngel Jorba, Encyclopedia of Complexity and Systems Science, 2022, 1
Jérôme Daquin, Sara Di Ruzza, Gabriella Pinzari, Springer Proceedings in Mathematics & Statistics, 399, New Frontiers of Celestial Mechanics: Theory and Applications, 2022, 47
Ch. Caracciolo, U. Locatelli, “Elliptic tori in FPU non-linear chains with a small number of nodes”, Commun. Nonlinear Sci. Numer. Simul., 97 (2021), 105759
S. Di Ruzza, J. Daquin, G. Pinzari, “Symbolic dynamics in a binary asteroid system”, Commun. Nonlinear Sci. Numer. Simul., 91 (2020), 105414
A. Perminov, E. Kuznetsov, “The orbital evolution of the sun-jupiter-saturn-uranus-neptune system on long time scales”, Astrophys. Space Sci., 365:8 (2020), 144
A. Yalinewich, C. Petrovich, “Nekhoroshev estimates for the survival time of tightly packed planetary systems”, Astrophys. J. Lett., 892:1 (2020), L11
S. Barbieri, L. Niederman, “Sharp nekhoroshev estimates for the three-body problem around periodic orbits”, J. Differ. Equ., 268:7 (2020), 3749–3780
Ivan I. Shevchenko, Astrophysics and Space Science Library, 463, Dynamical Chaos in Planetary Systems, 2020, 235
M. Sansottera, A.-S. Libert, “Resonant Laplace-Lagrange theory for extrasolar systems in mean-motion resonance”, Celest. Mech. Dyn. Astron., 131:8 (2019), 38
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