第一百日（1）作屎的老猫（第3/10页）

The question of the stability of our Solar system has been debated over several hundred years， since the era of Newton. The problem has attracted many famous mathematicians over the years and has played a central role in the development of non-linear dynamics and chaos theory. However， we do not yet have a definite answer to the question of whether our Solar system is stable or not. This is partly a result of the fact that the definition of the term ‘stability’ is vague when it is used in relation to the problem of planetary motion in the Solar system. Actually it is not easy to give a clear， rigorous and physically meaningful definition of the stability of our Solar system.

Among many definitions of stability， here we adopt the Hill definition (Gladman 1993): actually this is not a definition of stability， but of instability. We define a system as becoming unstable when a close encounter occurs somewhere in the system， starting from a certain initial configuration (Chambers， Wetherill & Boss 1996; Ito & Tanikawa 1999). A system is defined as experiencing a close encounter when two bodies approach one another within an area of the larger Hill radius. Otherwise the system is defined as being stable. Henceforward we state that our planetary system is dynamically stable if no close encounter happens during the age of our Solar system， about ±5 Gyr. Incidentally， this definition may be replaced by one in which an occurrence of any orbital crossing between either of a pair of planets takes place. This is because we know from experience that an orbital crossing is very likely to lead to a close encounter in planetary and protoplanetary systems (Yoshinaga， Kokubo & Makino 1999). Of course this statement cannot be simply applied to systems with stable orbital resonances such as the Neptune–Pluto system.

1.2Previous studies and aims of this research

In addition to the vagueness of the concept of stability， the planets in our Solar system show a character typical of dynamical chaos (Sussman & Wisdom 1988， 1992). The cause of this chaotic behaviour is now partly understood as being a result of resonance overlapping (Murray & Holman 1999; Lecar， Franklin & Holman 2001). However， it would require integrating over an ensemble of planetary systems including all nine planets for a period covering several 10 Gyr to thoroughly understand the long-term evolution of planetary orbits， since chaotic dynamical systems are characterized by their strong dependence on initial conditions.

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 comprehensive， but it seems that the terrestrial planets also remain stable during the integration period， maintaining almost regular oscillations.