Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability in the light of Kolmogorov and Nekhoroshev theories. (English) Zbl 1390.70020
Summary: 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.
MSC:
| 70F10 | \(n\)-body problems |
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 37N05 | Dynamical systems in classical and celestial mechanics |
| 70-08 | Computational methods for problems pertaining to mechanics of particles and systems |
| 70H08 | Nearly integrable Hamiltonian systems, KAM theory |
Keywords:
\(n\)-body planetary problem; KAM theory; Nekhoroshev theory; normal form methods; exponential stability; Hamiltonian systems; celestial mechanicsReferences:
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