  In this paper, we present evidence of the stability of a simplified model of
the Solar System, a flat (Newtonian) Sun-Jupiter-Saturn system with realistic
data: masses of the Sun and the planets, their semi-axes, eccentricities and
(apsidal) precessions of the planets close to the real ones. The evidence is
based on convincing numerics that a KAM theorem can be applied to the
Hamiltonian equations of the model to produce quasiperiodic motion (on an
invariant torus) with the appropriate frequencies. To do so, we first use KAM
numerical schemes to compute translated tori to continue from the Kepler
approximation (two uncoupled two-body problems) up to the actual Hamiltonian of
the system, for which the translated torus is an invariant torus. Second, we
use KAM numerical schemes for invariant tori to refine the solution giving the
desired torus. Lastly, the convergence of the KAM scheme for the invariant
torus is (numerically) checked by applying several times a KAM iterative lemma,
from which we obtain that the final torus (numerically) satisfies the existence
conditions given by a KAM theorem.
