This work is part of a research project I'm carrying out in collaboration
with A. Giorgilli and M. Sansottera.

We study the stability of the planetary problem including Sun,
Jupiter, Saturn and Uranus (SJSU, respectively). We first make
classical expansions of all the interaction terms between each pair of
planets appearing in the Hamiltonian of the SJSU system.  Therefore,
we remove the main perturbing terms depending on the mean motion
angles by a couple of canonical transformations similar to those
usually adopted in the Kolmogorov's scheme for the construction of KAM
tori. All these expansions are explicitly performed by algebraic
manipulation on a computer. In order to reduce the huge number of
terms to handle in the expansions, we limit ourselves to study the
SJSU problem in the planar case.

Since we want to investigate the results we can produce in the
framework of the normal form theory, as a first stressing test, we
furtherly simplify our model in a drastic way, by averaging the
Hamiltonian with respect to the mean motion angles. Thus, we obtain a
three degrees of freedom model, which is able to approximate the
secular motions up to order 2 in the masses (and to high degree,
i.e. 12, in the eccentricities).

In a previous work of us (i.e., "On the stability of the secular
evolution of the planar SJSU system", submitted to Mathematics
and Computers in Simulation), we applied a rather standard scheme of
estimates to the partial construction of the Birkhoff normal form
about the origin of the phase space which is an elliptic equilibrium
point. This allowed us to show that our secular Hamiltonian model is
"effectively stable" for times larger than the age of the solar
system "just" for all the initial conditions corresponding to
eccentricities smaller than (about) 1/2 of the real values.

In the present talk, we reconsider the same problem in order to extend
the result to the "real" initial conditions. Therefore, we changed
our approach by firstly looking for a KAM torus in the vicinity of the
"real" orbit. Such a KAM torus is explicitly constructed up to a
high order of approximation, by using algebraic manipulations on a
computer; this allowed us to compare the flow on the torus with the
numerical integration of the secular system and to show the good
agreement between them. As a final step of our new approach, we
evaluate the stability time related to the Birkhoff normal form about
the KAM torus. This strongly improves our previous result.