Project/Area Number  01540223 
Research Category 
GrantinAid for Scientific Research (C).

Research Field 
Astronomy

Research Institution  National Astronomical Observatory 
Principal Investigator 
KINOSHITA Hiroshi National Astronomical Observatory Professor, 位置天文・天体力学研究系, 教授 (00012857)

CoInvestigator(Kenkyūbuntansha) 
YOSHIKAWA Makoto Tokyo University Fellowship of JSPS, 理学部天文学教室, 学振特別研究員 (10212309)
NAKAI Hiroshi National Astronomical Observatory Research Assistant, 位置天文・天体力学研究系, 助手 (60155653)
YOSHIDA Haruo National Astronomical Observatory Research Assistant, 位置天文・天体力学研究系, 助手 (70220663)
TANIKAWA Kiyotaka National Astronomical Observatory Associate Professor, 理論天文学研究系, 助教授 (80125210)

Project Fiscal Year 
1989 – 1991

Project Status 
Completed(Fiscal Year 1990)

Budget Amount *help 
¥1,400,000 (Direct Cost : ¥1,400,000)
Fiscal Year 1990 : ¥200,000 (Direct Cost : ¥200,000)
Fiscal Year 1989 : ¥1,200,000 (Direct Cost : ¥1,200,000)

Keywords  Asteroid / Kirkwood Gap / Asteroidal Family / Asteroidal Group / Secular Resonance / Mean Motion Resonance / Symplectic Integrator / Symmetric Integrator / 小惑星 / カクウッド間隙 / 小惑星の族 / 小惑星の群 / シンプレクティック積分法 / 永年共鳴 / 対称型数値積分法 
Research Abstract 
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 multistep 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.
