惑星軌道におけるカオスとフラクタル

Abstract

It has been known that there is a chaotic state in the planetary orbit. In this paper, the features of a chaotic state of planetary orbit have been studied focusing on the distribution and characteristics of zero velocity points of the orbit. At first, together with an additional purpose to evaluate the accuracy of numerical calculation, simulations on the planetary orbits were carried out for one-body problem, two-body problem, three-body problem and quasi-three-body problem. Then, the simulations on a relative coordinate system two-body problem and a relative coordinate system quasi-three-body problem were carried out. As the results, a set of zero velocity points was found, which had an attractor assumed to be a chaotic state, in the case of taking a certain initial value in the relative coordinate system quasi-three-body problem. Poincare mapping, Lyapunov exponent and fractal dimension were calculated on this set of zero velocity points. From above results, it was made clear that the planetary orbits have a chaotic state. The details of the calculation were presented.

惑星軌道にはカオス状態が存在することが知られている。本論文では、惑星軌道のカオス状態の特徴を軌道のゼロ速度点の分布とその特性に注目して研究した。まずはじめに数値計算の精度を評価する目的もあって、1体問題、2体問題、3体問題、準3体問題の順に惑星軌道のシミュレーションを行なった。続いて、相対座標系2体問題、相対座標系準3体問題のシミュレーションを行なった。その結果、相対座標系準3体問題においてある初期値を取った場合にカオスと思えるアトラクタを持った0(ゼロ)速度点の集合が見つかった。この0速度点の集合についてポアンカレ写像、リヤプノフ指数、フラクタル次元を求めた。その結果、惑星軌道はカオス状態を持つことが判明した。その計算の詳細を以下に述べる。