Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability into the light of Kolmogorov and Nekhoroshev theories

Abstract

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.

arXiv comment31 pages, 2 figures. arXiv admin note: text overlap with arXiv:1010.2609

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• Librational KAM tori in the secular dynamics of the $\upsilon$ Andromedæ planetary system
• Elliptic tori in FPU non-linear chains with a small number of nodes
• Exponential stability in the perturbed central force problem
• Secular orbital dynamics of the innermost exoplanet of the $\upsilon$-Andromedæ system
• A numerical criterion evaluating the robustness of planetary architectures; applications to the $\upsilon$ Andromedæ system
• The construction of averaged planetary motion theory by means of computer algebra system Piranha
• Existence proof of librational invariant tori in an averaged model of HD60532 planetary system
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