On the stability of the secular evolution of the planar Sun-Jupiter-Saturn-Uranus system

TL;DR

This paper analyzes the long-term stability of the Sun-Jupiter-Saturn-Uranus system using an improved secular model, estimating stability over millions of years and extending classical theories.

Contribution

It extends Lagrange's secular theory by replacing the circular approximation with an invariant torus, enhancing the understanding of planetary stability.

Findings

Estimated stability time of 10 million years for current planetary data

Improved secular model accounts for elliptic equilibrium points

Stability estimate is significant compared to Solar System lifetime

Abstract

We investigate the long time stability of the Sun-Jupiter-Saturn-Uranus system by considering the planar, secular model. Our method may be considered as an extension of Lagrange's theory for the secular motions. Indeed, concerning the planetary orbital revolutions, we improve the classical circular approximation by replacing it with a torus which is invariant up to order two in the masses; therefore, we investigate the stability of the elliptic equilibrium point of the secular system for small values of the eccentricities. For the initial data corresponding to a real set of astronomical observations, we find an estimated stability time of $10^7$ years, which is not extremely smaller than the lifetime of the Solar System ($\sim 5$ Gyr).