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

*2017, Volume 22, Number 1, pp. 54-77*

Author(s):

**Giorgilli A., Locatelli U., Sansottera M.**

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 Nekhoroshevâ€™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.

Access to the full text on the Springer website |