Abstract

We consider the dynamical evolution of planetary systems whose structure is nearly circular and coplanar. The analysis is performed by the Hori–Deprit averaging method within the theory of the first order in planetary masses. A convenient set of canonical elements and a rarely employed variety of astrocentric coordinates are used to derive the equations of motion. Owing to the use of the chosen system of canonical elements, the expansions of the right-hand sides of the averaged equations contain a relatively small number of terms. Compared to other widespread coordinate systems, the astrocentric coordinates used by us allow a more convenient representation of the disturbing function to be obtained and do not require its expansion into a series in powers of a small parameter. On time scales \({\sim}10^{5}{-}10^{7}\) years we have studied the long-term evolution of the planetary systems HD 12661, \(\upsilon\) Andromedae, and some model systems by numerical integration of the averaged equations. Possible secular resonances have been revealed in the systems considered.

中文翻译：

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大同时间尺度上行星系统的稳定性分析

摘要

我们考虑结构近似于圆形和共面的行星系统的动力学演化。该分析是通过行星质量一阶理论内的Hori-Deprit平均方法进行的。使用一组方便的规范元素和很少使用的各种天体坐标来导出运动方程。由于使用了所选择的典范元素系统，平均方程右侧的展开包含相对较少数量的项。与其他广泛使用的坐标系相比，我们使用的天心坐标可以更方便地表示干扰函数，并且不需要以小参数的幂形式将其扩展为一系列。按时间比例\（{\ sim} 10 ^ {5} {-} 10 ^ {7} \）几年来，我们通过平均方程的数值积分研究了HD 12661行星系统，\（\ upsilon \） Andromedae以及一些模型系统的长期演化。在所考虑的系统中已经揭示出可能的长期共振。