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

## TL;DR

This paper analyzes the long-term stability of the Sun-Jupiter-Saturn-Uranus system using advanced mathematical models, demonstrating that the system remains stable over timescales exceeding the Solar System's lifetime within a refined secular framework.

## Contribution

It introduces a refined secular model with second-order mass effects and constructs a Kolmogorov normal form to identify a stable KAM torus, extending stability estimates via Nekhoroshev theory.

## Key findings

- Identifies an invariant KAM torus closely approximating the secular orbits.
- Shows the stability time exceeds the Solar System's lifetime within a specific neighborhood.
- Determines the neighborhood size is about ten times smaller than observational uncertainties.

## 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.