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 inte
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