Abstract

A new approach to the study of long-period and secular perturbations in celestial mechanics is developed. In contrast to the traditional use of the apparatus of the perturbing Lagrange function, we rely on the mutual potential energy of elliptic Gaussian rings. This approach is important due to the fact that instead of averaging the expression for the perturbing Lagrange function obtained in a very complicated way, it is methodologically simpler to immediately calculate the mutual energy of Gaussian rings. In this paper, we consider the problem for two Gaussian rings with one common focus, with small eccentricities, a small angle of mutual inclination, and an arbitrary angle between the lines of apsides. An expression for the mutual energy of such a system of rings is obtained in the form of a series up to the terms of the 4th order of smallness inclusively. This expression is used to derive and solve a system of differential equations describing the evolution of rings in an ecliptic reference frame. The method is used for a detailed study of the two-planetary Sun–Jupiter–Saturn problem. The results complement and refine the results of other authors. The new expression of the perturbing function can be applied not only to the planetary problem, in which all inclinations should be small, but also to the problem with nonplanetary rings found around small celestial bodies.

中文翻译：

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高斯环的相互引力和天体力学中的摄动问题

摘要

提出了一种研究天体力学中长期和长期扰动的新方法。与传统的扰动拉格朗日函数装置的使用相反，我们依赖于椭圆高斯环的互势能。该方法之所以重要，是因为它不是对以非常复杂的方式获得的扰动Lagrange函数的表达式求平均值，而是在方法上更简单的立即计算高斯环的互能。在本文中，我们考虑两个高斯环具有一个共同焦点的问题，这些高斯环具有较小的偏心率，较小的相互倾角和后沿线之间的任意角度。这样的环系统的互能的表达式以一系列的形式获得，直至包括小四阶的项。该表达式用于导出和求解微分方程组，该系统描述了黄道参考系中环的演化。该方法用于对两行星太阳-木星-土星问题的详细研究。结果补充并完善了其他作者的结果。摄动函数的新表达不仅可以应用于所有倾角都应该较小的行星问题，而且可以应用于围绕小天体的非行星环问题。该方法用于对两行星太阳-木星-土星问题的详细研究。结果补充并完善了其他作者的结果。摄动函数的新表达不仅可以应用于所有倾角都应较小的行星问题，而且还可以应用于围绕小天体的非行星环问题。该方法用于对两行星太阳-木星-土星问题的详细研究。结果补充并完善了其他作者的结果。摄动函数的新表达不仅可以应用于所有倾角都应该较小的行星问题，而且可以应用于围绕小天体的非行星环问题。