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1 Introduction
of the problem
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.
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