Terms in the analytic expansion of the doubly averaged disturbing function for the circular restricted three-body problem using the Legendre polynomial are explicitly calculated up to the fourteenth order of semimajor axis ratio between perturbed and perturbing bodies in the inner case , and up to the fifteenth order in the outer case . The expansion outcome is compared with results from numerical quadrature on an equipotential surface. Comparison with direct numerical integration of equations of motion is also presented. Overall, the high-order analytic expansion of the doubly averaged disturbing function yields a result that agrees well with the numerical quadrature and with the numerical integration. Local extremums of the doubly averaged disturbing function are quantitatively reproduced by the high-order analytic expansion even when is large. Although the analytic expansion is not applicable in some circumstances such as when orbits of perturbed and perturbing bodies cross or when strong mean motion resonance is at work, our expansion result will be useful for analytically understanding the long-term dynamical behavior of perturbed bodies in circular restricted three-body systems.

中文翻译：

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双重平均圆约束三体问题的扰动函数的高阶解析展开

明确地计算了使用勒让德多项式的圆形受限三体问题的双平均扰动函数的解析展开的项，直到在内壳中被摄动体与被摄动体之间的半长轴比的第十四阶，以及第十五阶外壳顺序。将扩展结果与等电位面上的数字正交结果进行比较。还提出了与运动方程的直接数值积分的比较。总体而言，双重平均扰动函数的高阶解析展开式得出的结果与数值正交和数值积分非常吻合。甚至在以下情况下，通过高阶解析展开式也可以定量地再现双重平均干扰函数的局部极值。大。尽管解析扩展在某些情况下不适用，例如当被摄体和被摄体的轨道交叉时或当强平均运动共振起作用时，我们的扩展结果将有助于分析理解被摄体在圆形中的长期动力学行为受限制的三体系统。