From that point of view， many of the previous long-term numerical integrations included only the outer five planets (Sussman Wisdom 1988; Kinoshita Nakai 1996). This is because the orbital periods of the outer planets are so much longer than those of the inner four planets that it is much easier to follow the system for a given integration period. At present， the longest numerical integrations published in journals are those of Duncan Lissauer (1998). Although their main target was the effect of post-main-sequence solar mass loss on the stability of planetary orbits， they performed many integrations covering up to ～1011 yr of the orbital motions of the four jovian planets. The initial orbital elements and masses of planets are the same as those of our Solar system in Duncan Lissauer's paper， but they decrease the mass of the Sun gradually in their numerical experiments. This is because they consider the effect of post-main-sequence solar mass loss in the paper. Consequently， they found that the crossing time-scale of planetary orbits， which can be a typical indicator of the instability time-scale， is quite sensitive to the rate of mass decrease of the Sun. When the mass of the Sun is close to its present value， the jovian planets remain stable over 1010 yr， or perhaps longer. Duncan Lissauer also performed four similar experiments on the orbital motion of seven planets (Venus to Neptune)， which cover a span of ～109 yr. Their experiments on the seven planets are not yet prehensive， but it seems that the terrestrial planets also remain stable during the integration period， maintaining almost regular oscillations.
On the other hand， in his accurate semi-analytical secular perturbation theory (Laskar 1988)， Laskar finds that large and irregular variations can appear in the eccentricities and inclinations of the terrestrial planets， especially of Mercury and Mars on a time-scale of several 109 yr (Laskar 1996). The results of Laskar's secular perturbation theory should be confirmed and investigated by fully numerical integrations.
小主，这个章节后面还有哦，请点击下一页继续阅读，后面更精彩！