Inspired by the original developments of recursive power series by means of Lagrange invariants for the classical two-body problem, a new analytic continuation algorithm is presented and studied. The method utilizes kinematic transformation scalar variables differentiated to arbitrary order to generate the required power series coefficients. The present formulation is extended to accommodate the spherical harmonics gravity potential model. The scalar variable transformation essentially eliminates any divisions in the analytic continuation and introduces a set of variables that are closed with respect to differentiation, allowing for arbitrary-order time derivatives to be computed recursively. Leibniz product rule is used to produce the needed arbitrary-order expansion variables. With arbitrary-order time derivatives available, Taylor series-based analytic continuation is applied to propagate the position and velocity vectors for the nonlinear two-body problem. This foundational method has been extended to also demonstrate an effective variable step size control for the Taylor series expansion. The analytic power series approach is demonstrated using trajectory calculations for the main problem in satellite orbit mechanics including high-order spherical harmonics gravity perturbation terms. Numerical results are presented to demonstrate the high accuracy and computational efficiency of the produced solutions. It is shown that the present method is highly accurate for all types of studied orbits achieving 12–16 digits of accuracy (the extent of double precision). While this double-precision accuracy exceeds typical engineering accuracy, the results address the precision versus computational cost issue and also implicitly demonstrate the process to optimize efficiency for any desired accuracy. We comment on the shortcomings of existing power series-based general numerical solver to highlight the benefits of the present algorithm, directly tailored for solving astrodynamics problems. Such efficient low-cost algorithms are highly needed in long-term propagation of cataloged RSOs for space situational awareness applications. The present analytic continuation algorithm is very simple to implement and efficiently provides highly accurate results for orbit propagation problems. The methodology is also extendable to consider a wide variety of perturbations, such as third body, atmospheric drag and solar radiation pressure.

中文翻译：

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扰动二体问题的一种新的解析连续幂级数解

受经典二体问题拉格朗日不变量递归幂级数原始发展的启发，提出并研究了一种新的解析延拓算法。该方法利用微分到任意阶数的运动变换标量变量来生成所需的幂级数系数。本公式被扩展以适应球谐重力势模型。标量变量变换基本上消除了解析延拓中的任何除法，并引入了一组关于微分封闭的变量，允许递归计算任意阶时间导数。莱布尼茨乘积规则用于产生所需的任意阶展开变量。随着任意阶时间导数可用，应用基于泰勒级数的解析延拓来传播非线性二体问题的位置和速度向量。这种基本方法已经扩展到还演示了对泰勒级数展开的有效可变步长控制。解析幂级数方法使用轨迹计算来演示卫星轨道力学中的主要问题，包括高阶球谐重力扰动项。数值结果显示了所产生的解决方案的高精度和计算效率。结果表明，本方法对所有类型的研究轨道都具有很高的精度，达到 12-16 位精度（双精度的程度）。虽然这种双精度精度超过了典型的工程精度，结果解决了精度与计算成本的问题，并隐含地展示了优化效率以获得任何所需精度的过程。我们评论了现有的基于幂级数的通用数值求解器的缺点，以突出本算法的优点，直接为解决天体动力学问题而量身定制。在用于空间态势感知应用的编目 RSO 的长期传播中，非常需要这种高效的低成本算法。本解析延拓算法实施起来非常简单，并且有效地为轨道传播问题提供了高度准确的结果。该方法还可以扩展以考虑各种扰动，例如第三体、大气阻力和太阳辐射压力。