Abstract—

The inner (or satellite) version of the restricted elliptic three-body problem is considered. The terms up to the fourth order inclusive in small parameter are retained in the expansion of the perturbing function for the problem. The ratio of the orbital semimajor axes of perturbed and perturbing bodies is such a parameter, while their mean longitudes are the fastest variables. The Gauss scheme of independent double averaging over fast variables is used to analyze the orbital evolution of a body of negligible mass. Explicit analytical expressions for the doubly averaged perturbing function and its derivatives with respect to the elements on the right-hand sides of the evolution equations are presented. The integrable cases of the doubly averaged problem are studied in detail: planar and orthogonal apsidal orbits. The evolution system is numerically integrated in the general (nonintegrable) case for some special values of the problem parameters and initial conditions, in particular, for a set of orbital elements in which the so-called “flips”, i.e., transitions of the orbit from prograde to retrograde and vice versa, manifest themselves. In the Sun–Jupiter–asteroid model using some special asteroid orbits as an example, we show the influence of the retained fourth-order terms and the ellipticity of Jupiter’s orbit on their evolution.

中文翻译：

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卫星限制椭圆双平均三体问题中轨道演化的一些特殊性

摘要-

考虑了受限椭圆三体问题的内部（或卫星）版本。在此问题的摄动函数的扩展中保留了小参数中直到四阶（包括四阶）的项。被摄体和被摄体的轨道半长轴之比就是这样一个参数，而它们的平均经度是最快的变量。快速变量独立两次平均的高斯方案用于分析可忽略质量物体的轨道演化。给出了相对于演化方程右侧元素的双重平均摄动函数及其导数的显式解析表达式。详细研究了双平均问题的可积情况：平面和正交的两栖轨道。对于某些特殊的问题参数值和初始条件值，特别是对于其中所谓的“翻转”即所谓的轨道翻转的一组轨道元素，通常在（不可积分）情况下将演化系统数值集成。从前进到逆行，反之亦然。在以某些特殊小行星轨道为例的太阳木星小行星模型中，我们显示了保留的四阶项和木星轨道的椭圆度对其演化的影响。