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We investigate the long-time stability of the Sun-Jupiter-Saturn-Uranussystem by considering a planar secular model, that can be regarded as a majorrefinement of the approach first introduced by Lagrange. Indeed, concerning theplanetary orbital revolutions, we improve the classical circular approximationby replacing it with a solution that is invariant up to order two in themasses; therefore, we investigate the stability of the secular system forrather small values of the eccentricities. First, we explicitly construct aKolmogorov normal form, so as to find an invariant KAM torus which approximatesvery well the secular orbits. Finally, we adapt the approach that is at basisof the analytic part of the Nekhoroshev's theorem, so as to show that there isa neighborhood of that torus for which the estimated stability time is largerthan the lifetime of the Solar System. The size of such a neighborhood,compared with the uncertainties of the astronomical observations, is about tentimes smaller.

regular and chaotic dynamics/regular and chaotic dynamics

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