This article is cited in 4 scientific papers (total in 4 papers)
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
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.
$n$-body planetary problem, KAM theory, Nekhoroshev theory, normal form methods, exponential stability, Hamiltonian systems, celestial mechanics
|Italian Ministry of Education, University and Research
|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.
MSC: 70F10, 37J40, 37N05, 70-08, 70H08
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
\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.
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