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 Regul. Chaotic Dyn., 2017, Volume 22, Issue 1, Pages 54–77 (Mi rcd243)

Secular Dynamics of a Planar Model of the Sun-Jupiter-Saturn-Uranus System; Effective Stability in the Light of Kolmogorov and Nekhoroshev Theories

Antonio Giorgillia, Ugo Locatellib, Marco Sansotteraa

a Dipartimento di Matematica, Università degli Studi di Milano, via Saldini 50, 20133, Milano, Italy
b Dipartimento di Matematica, Università degli Studi di Roma ''Tor Vergata'', via della Ricerca Scientifica 1, 00133, Roma, Italy

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

 Funding Agency Grant Number Italian Ministry of Education, University and Research PRIN 2010JJ4KPA009 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.

DOI: https://doi.org/10.1134/S156035471701004X

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Bibliographic databases:

MSC: 70F10, 37J40, 37N05, 70-08, 70H08
Accepted:20.12.2016
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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

Citation in format AMSBIB
\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} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000394354800004} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85012180658} 

• http://mi.mathnet.ru/eng/rcd243
• http://mi.mathnet.ru/eng/rcd/v22/i1/p54

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Ma D.-Zh., Fu Ya.-N., Wang X.-L., “The Orbital Configuration of the Two Interacting Jupiters in Hd 155358 System”, Mon. Not. Roy. Astron. Soc., 470:1 (2017), 706–712
2. D. Bambusi, A. Fusè, M. Sansottera, “Exponential Stability in the Perturbed Central Force Problem”, Regul. Chaotic Dyn., 23:7-8 (2018), 821–841
3. F. Paita, A. Celletti, G. Pucacco, “Element history of the Laplace resonance: a dynamical approach”, Astron. Astrophys., 617 (2018), A35
4. M. Volpi, U. Locatelli, M. Sansottera, “A reverse KAM method to estimate unknown mutual inclinations in exoplanetary systems”, Celest. Mech. Dyn. Astron., 130:5 (2018), 36