Numerical Simulation of Dynamical Evolution of the Solar System over 5 Billion Years

从最大李亚普诺夫指数为正的意义上说，行星运动是混沌的。到目前为止，迄今为止最长的整合，超过8.45亿年（冥王星的李亚普诺夫时间的42倍），没有显示出冥王星运动的任何不稳定迹象。我们对冥王星在太阳系的年龄（从过去到未来55亿年）进行了数值积分。这种整合也没有给出冥王星混沌演化的任何迹象。附近轨道的开普勒元素的散度起初随时间呈线性增长，然后开始呈指数增长。指数发散在4.2亿年左右停止并达到饱和。指数发散被冥王星的以下三个共振所抑制：1)冥王星与海王星处于3:2的平均运动共振中；2)近日点的角在90度左右摆动，周期为3.8 Myr。偏心率和倾角的主导周期变化与近日点角的摆动同步。3)而且冥王星交点的经度参照海王星交点的经度循环，这个循环的周期等于近日点的自转周期（二次共振）。为了研究冥王星是如何进化到现在的三个共振锁的稳定状态的，我们必须考虑太阳系早期的一个非保守机制。

Planetary motions are said to be chaotic in the sense that the maximum Lyapunov exponent is positive.So far the longest integration up to now, over 845 million years (42 Pluto's Lyapunov times) , does not show any indication of a gross instability in the motion of Pluto.We carried out the numerical integration of Pluto over the age of the solar system (5.5 billion years both towards the past and the future) . This integration also did not give any indication of chaotic evolution of Pluto.The divergences of Keplerian elements of a nearby trajectory at first grow limearly with the time and then start to increase exponentially.The exponential divergences stop at about 420 million years and saturate.The exponential divergneces are suppressed by the following three resoncnaces that Pluto has :1) Pulto is in the 3 : 2 mean motion resonance with Neptune.2) The argument of perihelion librates around 90 degrees and its period is 3.8 Myr.The dominant periodic variations of the eccentricity and the inclination are synchronized with the libration of the argument of perihelion.3) Moreover the longitude of Pluto's node referred to the longitude of Neptune's node circulates and the period of this circulation is equal to the period of the libration of the argument of Perihelion (a secondary resonance) .In order to investigate how Pluto evolves to the present stable state of three resonance lockings, we have to take account of a non-conservative mechanism in the earlsy stage of the solar system.