Linear stability of periodic three-body orbits with zero angular momentum and topological dependence of Kepler’s third law: a numerical test
Linear stability of periodic three-body orbits with zero angular momentum and topological dependence of Kepler’s third law: a numerical test
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DOI：
10.1088/1751-8121/aaca41
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发表时间：
2017-05
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通讯作者：
V. Dmitrasinovic;Ana Hudomal;M. Shibayama;A. Sugita
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作者：
V. Dmitrasinovic;Ana Hudomal;M. Shibayama;A. Sugita
我们测试数值最近提出的线性关系的标度不变的周期，和一个轨道的拓扑结构，在几百个平面牛顿周期性三体轨道。这里T是轨道的周期，E是它的能量，所以这是标度不变周期，或者等价地，单位能量下的周期。所有这些轨道都有消失的角动量，并通过一个线性，等距配置至少一次。这样的轨道被分类在10代数定义良好的序列。每个序列中的轨道遵循近似的线性依赖性，尽管斜率和截距略有不同。在它的序列中具有最短周期的轨道被称为“祖先”：六个不同的轨道是这十个序列的祖先。我们研究了这些轨道的线性稳定性，得到了21个线性稳定的轨道，其中包括了所有的祖先轨道。这与Birkhoff-Lewis定理是一致的，该定理意味着每个稳定的祖先存在无穷多个周期轨道，并且以这种方式解释了每个序列的存在并确保了每个序列的无限扩展。
We test numerically the recently proposed linear relationship between the scale-invariant period , and the topology of an orbit, on several hundred planar Newtonian periodic three-body orbits. Here T is the period of an orbit, E is its energy, so that is the scale-invariant period, or, equivalently, the period at unit energy . All of these orbits have vanishing angular momentum and pass through a linear, equidistant configuration at least once. Such orbits are classified in ten algebraically well-defined sequences. Orbits in each sequence follow an approximate linear dependence of , albeit with slightly different slopes and intercepts. The orbit with the shortest period in its sequence is called the ‘progenitor’: six distinct orbits are the progenitors of these ten sequences. We have studied linear stability of these orbits, with the result that 21 orbits are linearly stable, which includes all of the progenitors. This is consistent with the Birkhoff–Lewis theorem, which implies existence of infinitely many periodic orbits for each stable progenitor, and in this way explains the existence and ensures infinite extension of each sequence.