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限制性三体问题共线平动点轨道近似解析解
周敬, 胡军, 张斌
北京控制工程研究所
摘要:
随着深空探测成为航天领域的研究热点,与其密切相关的三体问题基础研究也日益重要,尤其是在深空探测任务设计中处于基础地位的共线平动点附近运动的研究,更是具有重要的工程应用价值。在圆型限制性三体问题下,对共线平动点附近运动近似解析解的研究已经较为全面,但在更接近真实情况、更具一般性的椭圆型限制性三体问题下,相应的研究却相对较少。针对此背景,参考借鉴圆型限制性三体问题的研究方法,首先根据平动点的特性计算出平动点的位置,然后将非线性三体动力学模型在共线平动点处线性化,最后结合线性系统理论,获得了椭圆型限制性三体问题下共线平动点附近运动的近似解析解,并将其与经典的圆型限制性三体问题下的近似解析解进行对比分析,仿真结果证明了方法的有效性,同时也表明所推导的椭圆型限制性三体问题解析解相比圆型限制性三体问题解析解具有更高的精度。
关键词:  圆型限制性三体问题;椭圆型限制性三体问题;非线性系统;共线平动点;解析解
DOI:
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基金项目:国家自然科学基金项目(面上项目,重点项目,重大项目)
Approximate Analytical Solutions of Motion near the Collinear Libration-Points in Restricted Three-body Problem
Zhou Jing
Beijing Institute of Control Engineering
Abstract:
With the deep space exploration becoming a research focus in aerospace, corresponding fundamental research on three-body problem gets increasingly significant, especially the motions analysis near the collinear libration-points, which play a leading role in deep space mission design, has more important engineering application value. The approximate analytical solutions of motion near the collinear libration-points in circular restricted three-body problem has been obtained, however, there are relatively fewer studies about the solutions in elliptic restricted three-body problem, although it is more realistic and general than circular restricted three-body problem. Based on it, the approximate analytical solutions of motion near the collinear libration-points in elliptic restricted three-body problem was deduced by referencing the method used in circular restricted three-body problem, the positions of libration-points are obtained according to its characteristic, then the nonlinear dynamic model is linearized at the collinear libration-points, the approximate analytical solutions are finally obtained using the linear system theory and compared to solutions in circular restricted three-body problem. Simulation results indicated the method is valid and the deduced analytical solutions have higher precision than solutions for circular restricted three-body problem.
Key words:  Circular Restricted Three-body Problem; Elliptic Restricted Three-body Problem; Nonlinear System;Collinear Libration-Points;Analytical Solutions