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

We investigate the long time stability of the Sun–Jupiter–Saturn–Uranus system by considering the planar, secular model. Our ethod may be considered as an extension of Lagrange’s theory for the secular motions. Indeed, concerning the planetary orbital evolutions, we improve the classical circular approximation by replacing it with a torus which is invariant up to order two in he masses; therefore, we investigate the stability of the elliptic equilibrium point of the secular system for small values of the ccentricities. For the initial data corresponding to a real set of astronomical observations, we find an estimated stability time of 107 ears, which is not extremely smaller than the lifetime of the Solar System (∼5 Gyr). 2011 IMACS. Published by Elsevier B.V. All rights reserved.