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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.

小行星轨道元的分布是不均匀的，最显著的特征是半长轴分布的间隙。这些间隙（Kirkwood gap）位于小行星的平均运动与木星的平均运动可通约的区域，如3:1、5:2、7:3和2:1。我们研究了二维和三维情况下这些可公度区域（共振区域）的动力学结构，并检验了Wisdom首次提出的小行星进入共振区域的混沌区域，其偏心率大到足以与内行星相遇，该小行星被移出可公度区域的假设。由于本研究需要长时间的数值积分，我们还研究了各种不会产生能量长期截断误差的积分器。我们的研究结果总结如下：在3:1和5:2共振区，这些共振区的大部分小行星的离心率变化很大，足以跨越内行星轨道。在7:3的共振中，以共振区中心部分为主的小行星的偏心率变化较大，而在2:1的共振中，共振区较大部分的偏心率变化不大。我们给出了辛积分器在哈密顿系统的能量中不会产生长期截断误差的数学证明，这意味着经度上的截断误差仅随时间线性增加，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在经度上只产生线性截断误差，并表明这种新型积分器耗时更少，非常适合于长时间的轨道积分。