In this paper, we develop an analytical stability criterion for a five-body symmetrical system, called the Caledonian Symmetric Five-Body Problem (CS5BP), which has two pairs of equal masses and a fifth mass located at the centre of mass. The CS5BP is a planar problem that is configured to utilise past–future symmetry and dynamical symmetry. The introduction of symmetries greatly reduces the dimensions of the five-body problem. Sundman’s inequality is applied to derive boundary surfaces to the allowed real motion of the system. This enables the derivation of a stability criterion valid for all time for the hierarchical stability of the CS5BP. We show that the hierarchical stability depends solely on the Szebehely constant $$C_0$$ which is a dimensionless function involving the total energy and angular momentum. We then explore the effect on the stability of the whole system of varying the relative sizes of the masses. The CS5BP is hierarchically stable for $$C_0 > 0.065946$$ . This criterion can be applied in the investigation of the stability of quintuple hierarchical stellar systems and symmetrical planetary systems.

中文翻译：

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Caledonian 对称五体问题的解析稳定性

在本文中，我们为五体对称系统开发了一个解析稳定性准则，称为 Caledonian 对称五体问题 (CS5BP)，它具有两对相等的质量和位于质心的第五个质量。CS5BP 是一个平面问题，它被配置为利用过去-未来对称性和动态对称性。对称性的引入大大降低了五体问题的维度。Sundman 不等式用于推导出系统允许的实际运动的边界面。这使得能够推导出对 CS5BP 的分层稳定性始终有效的稳定性标准。我们表明层次稳定性仅取决于 Szebehely 常数 $$C_0$$，这是一个涉及总能量和角动量的无量纲函数。然后，我们探讨了改变质量的相对大小对整个系统稳定性的影响。CS5BP 在 $$C_0 > 0.065946$$ 的层次上是稳定的。该判据可用于研究五重分级恒星系统和对称行星系统的稳定性。