PLANAR PERIODIC ORBIT CONSTRUCTION AROUND THE TRIANGULAR LIBRATION POINTS BASED ON POLYNOMIAL CONSTRAINTS
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摘要: 平动点是圆型限制性三体问题中的五个平衡解.其中，三角平动点在平面问题中具有“中心×中心”的动力学特性，其附近存在着大量的周期轨道，研究这些周期轨道的构建方法在深空探测中具有理论及工程意义.本文从振动角度分析周期轨道，通过多项式展开法构建出主坐标下周期轨道两个运动方向之间的渐近关系.从新的角度分析了系统的动力学特性和平面周期运动两个方向内在关联以及物理规律.这种多项式形式的关系式，可以作为约束条件用于数值微分修正算法中，通过迭代的方式寻找周期轨道.数值仿真算例验证了方法的正确性及精确性.文章从振动的角度对周期轨道进行分析，改进了微分修正算法.提出的方法可以被拓展至圆型/椭圆型限制性三体问题的三维周期轨道构建中.
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关键词:
- 平面圆型限制性三体问题 /
- 平动点 /
- 多项式展开法
Abstract: Libration points are the five equilibrium solutions in the circular restricted three-body problem (CRTBP).The linearized motions around triangular libration points are typical center×center type.Studies about probes moving around orbits in the vicinity of the libration points have theoretical significance.From the vibrational point of view, the polynomial series are used to derive approximately the relations in different directions during periodic motions, which provides a new point of view to exploring the dynamics and analyzing the overall characteristics of the whole system with general rules.The nonlinear relations in polynomial form between the directions of the planar motions can be treated as constraints to obtain the solutions by numerical integration.Numerical simulations verify the efficiency of the proposed method.The methodology of deriving topological relations has the potential to be extended to circular/elliptical R3BP in three dimensional cases. -
图 2 短周期轨道微分修正结果图
Figure 2. Differential correction for the short-period periodic motion
图 3 微分修正前后短周期轨道图
Figure 3. Comparison of the trajectories numerically integrates with the linearized initial condition and corrected initial condition for the short-period motion
图 4 长周期轨道微分修正结果图
Figure 4. Differential correction for the long-period periodic motion
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