Origin of Chaos and Kirkwood Gaps in Astreoidal Motions

混沌的起源和星状运动中的柯克伍德间隙

基本信息

批准号：
01540223
负责人：
KINOSHITA Hiroshi
金额：
$ 0.9万
依托单位：
National Astronomical Observatory
依托单位国家：
日本
项目类别：
Grant-in-Aid for General Scientific Research (C)
财政年份：
1989
资助国家：
日本
起止时间：
1989 至 1991
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-01540223/
关键词：
Asteroid Kirkwood Gap Asteroidal Family Asteroidal Group Secular Resonance Mean Motion Resonance Symplectic Integrator Symmetric Integrator

项目摘要

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.

小行星轨道元素的分布并不均匀，最显着的特征是半轴轴分布的差距。差距（柯克伍德差距）位于小行星平均运动与木星（例如3：1、5：2、7：3和2：1）相称的区域。我们研究了两个和三维情况的这些相称区域（共振区域）的动力结构，并检查了以下假设：一个小行星进入共振区域的混乱区域，其偏心率足够大，足以与内部行星相遇，并且这是从可观的区域中删除的，这是最初由Commenturable Andive Suse dons提出的。由于这项研究需要长时间的数值集成，因此我们还研究了各种集成符，这些集成剂不会在能量中产生世俗的截断误差。您的结果在以下内容中总结了1。在3：1和5：2共鸣中，大多数这些共振区域中小行星的偏心率在很大程度上变化以跨越内行星轨道。在7：3共振中，小行星的偏心率主要在共振区域的中心发生很大变化，而在2：1共振中，相当大部分共振的偏心率并没有显示出很大的变化。2。我们给出了一个数学证据，表明符合性积分器不会在汉密尔顿系统的能量中产生世俗的截断误差，这意味着经度中的截断误差仅随时间线性增加，并为精确的轨道计算构建了高阶符号积分器。我们还在数学上证明了对称的多步积分器确实在经度上只会产生线性截断误差，并表明这种新型集成器的耗时较小，并且非常适合长时间的轨道集成。