Origin of Chaos and Kirkwood Gaps in Astreoidal Motions

小行星的轨道元素分布并不均匀，最显著的特征是半长轴的分布存在缺口。这些间隙(柯克伍德间隙)位于小行星的平均运动与木星的平均运动可公度的区域，如3：1、5：2、7：3和2：1。我们在二维和三维情况下研究了这些可公度区域(共振区)的动力学结构，并检验了假设，即小行星进入共振区的混乱区，其偏心率变得足够大，足以与内行星相遇，这颗小行星被从可公度区中移除，这是Wisity最先提出的。由于本研究需要较长的数值积分，因此我们还研究了不会产生能量长期截断误差的各种积分器。在3：1和5：2共振中，这些共振区大部分小行星的偏心率变化很大，足以跨越内行星轨道。在7：3共振时，小行星的偏心率变化较大，而在2：1共振时，较大部分的偏心率变化不大。我们给出了辛积分器不会在哈密顿系统的能量中产生长期截断误差的数学证明，这意味着在经向的截断误差只随时间线性增加，并构造了用于精确轨道计算的高阶辛积分器。我们还从数学上证明了对称多步积分器确实只产生经向的线性截断误差，并表明这种新型积分器耗时较少，非常适合于长时间的轨道积分。

The distribution of orbital elements of asteroids is not uniform, and the most remarkable feature is the gaps in the distribution of the semimajor axis. The gaps (Kirkwood gaps) are located at the region where the mean motion of asteroids is commensurable with that of Jupiter, such as 3:1, 5:2, 7:3, and 2:1. We investigated the dynamical structures of these commensurable regions (resonance regions) both for two and three dimensional cases, and examined the hypothesis that an asteroid enters in a chaotic area in the resonance region and its eccentricity becomes large enough to encounter with inner planets, and this asteroid is removed from the commensurable region, which was first proposed by Wisdom. Since a long numerical integration is necessary for this research, we also investigated various integrators which do not produce a secular truncation error in the energy.Our results are summarized in the followings.1. In the 3:1 and 5:2 resonances, the eccentricity of asteroids in the most part of these resonance regions changes largely enough to cross inner planet orbits. In the 7:3 resonance the eccentricity of asteroids mainly in the central part of the resonance region changes considerably and in the 2:1 resonance the eccentricity in the rather large part of the resonance does not show a large change.2. We gave a mathematical proof that symplectic integrators do not produce a secular truncation error in the energy of a Hamiltonian system, which means that the truncation error in longitude increases only linearly with time, and constructed higher order symplectic integrators for a precise orbital computation. We also proved mathematically that symmetric multi-step integrators do produce only a linear truncation error in longitude and showed that this new type integrators are less time consuming and are quite suitable for a long time orbital integration.