The discovery of the chaotic behavior of the planetary orbits in the Solar System [Laskar, J., 1989. Nature 338, 237-238; Laskar, J., 1990. Icarus 88, 266-291] was obtained using numerical integration of averaged equations. In [Laskar, J., 1994. Astron. Astrophys. 287, L9-L12], these same equations are integrated over several Gyr and show the evidence of very large possible increase of the eccentricity of Mercury through chaotic diffusion. On the other hand, in the direct numerical integration of Ito and Tanikawa [Ito, T., Tanikawa, K., 2002. Mon. Not. R. Astron. Soc. 336, 483-500] performed without general relativity over +/- 4 Gyr, the eccentricity of Mercury presented some chaotic diffusion, but with a maximal excursion smaller than about 0.35. In the present work, a statistical analysis is performed over more than 1001 different integrations of the secular equations over 5 Gyr. This allows to obtain for each planet, the probability for the eccentricity to reach large values. In particular, we obtain that the probability of the eccentricity of Mercury to increase beyond 0.6 in 5 Gyr is about 1 to 2%, which is relatively large. In order to compare with Ito and Tanikawa [Ito, T., Tanikawa, K., 2002. Mon. Not. R. Astron. Soc. 336, 483-500], we have performed the same analysis without general relativity, and obtained even more orbits of large eccentricity for Mercury. In order to clarify these differences, we have performed as well a direct integration of the planetary orbits, without averaging, for a dynamical model that do not include the Moon or general relativity with 10 very close initial conditions over 3 Gyr. The statistics obtained with this reduced set are comparable to the statistics of the secular equations, and in particular we obtain two trajectories for which the eccentricity of Mercury increases beyond 0.8 in less than 1.3 and 2.8 Gyr, respectively. These strong instabilities in the orbital motion of Mercury results from secular resonance between the perihelion of Jupiter and Mercury that are facilitated by the absence of general relativity. The statistical analysis of the 1001 orbits of the secular equations also provides probability density functions (PDF) for the eccentricity and inclination of the terrestrial planets (Mercury, Venus, the Earth and Mars) that are very well approximated by Rice PDF. This provides a very simple representation of the planetary PDF over 5 Gyr. On this time-scale the evolution of the PDF of the terrestrial planets is found to be similar to the one of a diffusive process. As shown in Laskar [Laskar, J., 1994. Astron. Astrophys. 287, L9-L12], the outer planets orbital elements do not present significant diffusion, and the PDFs of their eccentricities and inclinations are well represented by the PDF of quasiperiodic motions with a few periodic terms. (c) 2008 Elsevier Inc. All rights reserved.
