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 深空探测学报  2020, Vol. 7 Issue (1): 93-101  DOI: 10.15982/j.issn.2095-7777.2020.20190408001 0

### 引用本文

ZHOU Jing, HU Jun, ZHANG Bin. Approximate Analytical Solutions of Motion near the Collinear Libration-Points in Restricted Three-Body Problem[J]. Journal of Deep Space Exploration, 2020, 7(1): 93-101. DOI: 10.15982/j.issn.2095-7777.2020.20190408001.

### 文章历史

Approximate Analytical Solutions of Motion near the Collinear Libration-Points in Restricted Three-Body Problem
ZHOU Jing, HU Jun, ZHANG Bin
Beijing Institute of Control Engineering，Beijing 100190，China
Abstract: With deep space exploration becoming a research focus in aerospace, corresponding fundamental research on three-body problem is of increasingly significant, especially the motion analysis near the collinear libration-points, which play a leading role in deep space mission design 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 are 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, and the approximate analytical solutions are finally obtained using the linear system theory and compared with the solutions of the circular restricted three-body problem. Simulation results indicated the method is valid and the deduced analytical solutions have higher precision than that of the circular restricted three-body problem.
Key words: circular restricted three-body problem    elliptic restricted three-body problem    nonlinear system    collinear libration-points    analytical solutions

Hight lights:

1. ●　The positions of libration points in elliptic restricted three-body problem are given.
2. ●　The approximate analytical solutions of motion around collinear pibration points in elliptic restricted three-body problem are derived.
3. ●　The solutions for elliptic restricted three-body problem have higher precision than that of the circular restricted three-body problem was proved.

CRTBP是最简单的三体运动模型，其没有考虑主天体公转轨道的偏心率以及各种环境摄动力，主要用来研究三体轨道最基本的动力学特征。目前关于CRTBP平动点及其附近运动的解析研究已经较为全面，Gómez[3]研究了CRTBP中平动点位置、平动点附近轨道的一阶近似解析解，如Lissajous轨道、Lyapunov轨道、halo轨道；Richardson[4]利用Lindstedt-Poincaré方法构造了CRTBP下共线平动点halo轨道的三阶近似解析解；Masdemont[5]研究了CRTBP下共线平动点附近运动的双曲形态和中心形态，将不变流形扩展成双曲振幅和中心振幅的幂级数形式；Jorba等[6]同样采用Lindstedt-Poincaré方法并结合Normal Form Scheme方法半解析地构造了CRTBP下共线平动点动力学方程的解；Rithwik等[7]使用基于优化技术的梯度和非梯度方法研究了halo轨道的设计问题；刘刚等[8]以CRTBP中的周期轨道作为迭代初值，用星历表数据对轨道进行拼接获得了所需的拟周期轨道；陶雪峰等[9]在CRTBP框架内，针对不同类型平动点轨道的特点，通过设定轨道特征点，以轨道闭合程度为目标函数，提出了一种基于优化算法的平动点轨道生成方法；Lara等[10]使用摄动方法和归一化方法来代替Lindstedt-Poincaré方法，获得了平动点附近Hill问题的解析解；郑越等[11]建立了一种改进型庞加莱截面图，结合状态转移矩阵和打靶法，提出了一种计算周期轨道的新方法；Lei等[12] 构建了CRTBP下三角平动点附近运动的高阶解析解，并详细讨论了解的收敛性；Liang等[13]利用改进的参数变分方法，获得了三角平动点附近运动的高阶近似解析解；Qian等[14]受振动动力学中的非线性模态启发，结合多项式展开和微分修正方法获得了三角平动点附近垂直周期轨道的解析解和数值解；翟冠峤等[15]从模态运动的角度出发，分析了三角平动点附近周期轨道，通过多项式展开法构建出了周期轨道3个运动方向之间的渐近关系。

1 限制性三体问题

1.1 圆型限制性三体问题（CRTBP）

CRTBP是最简单的三体运动模型，其没有考虑主天体公转轨道的偏心率以及各种环境摄动力，主要用来研究三体轨道最基本的动力学特征。为描述航天器运动的方便和计算的简化，在CRTBP中通常需要对相关物理量进行无量纲化处理，并以时间作为独立变量。相应的质量 $[M]$ 、长度 $[L]$ 和时间 $[T]$ 的归一化单位取为

 $\left\{ \begin{split} & \left[ M \right] = {m_1} + {m_2}\\ & \left[ L \right] = {L_{12}}\\ & \left[ T \right] = {(\dfrac{{{L_{12}}^3}}{{G({m_1} + {m_2})}})^{1/2}} \end{split} \right.$ (1)

 图 1 质心旋转坐标系 $O - XYZ$ 和平动点旋转坐标系 ${L_2} - xyz$ Fig. 1 Barycentric rotating coordinate system $O - XYZ$ and libration-point centered rotating coordinate system ${L_2} - xyz$

 $\left\{ \begin{split} & \ddot x - 2\dot y = \dfrac{{\partial \varOmega }}{{\partial x}} \\ & \ddot y + 2\dot x = \dfrac{{\partial \varOmega }}{{\partial y}} \\ & \ddot z = \dfrac{{\partial \varOmega }}{{\partial z}} \\ \end{split} \right.$ (2)

 $\left\{ \begin{split} \delta x(t) = & {A_1}{{\rm e}^{{\lambda _1}t}} + {A_2}{\rm e}^{ - {\lambda _1}t} + {A_3}\cos ({\rm Im} ({\lambda _3}))t + \\ & {A_4}\sin ({\rm Im} ({\lambda _3})t) \\ \delta y(t) = & {k_1}{A_1}{{\rm e}^{{\lambda _1}}}t - {k_1}{A_2}{\rm e}^{ - {\lambda _1}t} - {k_2}{A_3}\sin ({\rm Im} ({\lambda _3}))t + \\ & {k_2}{A_4}\cos ({\rm Im} ({\lambda _3})) t\\ \delta z(t) = & {A_5}\cos ({\rm Im} ({\lambda _5})t) + {A_6}\sin ({\rm Im} ({\lambda _5})) t \end{split} \right.$ (3)

1.2 椭圆型限制性三体问题（ERTBP）

ERTBP考虑了主天体公转轨道的偏心率，相比CRTBP可以更精确地描述三体系统内航天器的运动情况。当两主天体围绕公共质心作椭圆轨道运动时，二者之间的距离呈现周期性变化，不再为一固定值，此时仍以时间作为独立变量不再适合，转而以主天体公转轨道的真近点角 $f$ 作为独立变量。相应的质量 $[M]$ 、长度 $[L]$ 和时间 $[T]$ 的归一化单位取为

 $\left\{\begin{split} & {[M]=m_{1}+m_{2}} \\ & {[L]=L_{12}=\dfrac{a\left(1-e^{2}\right)}{1+e \cos f}} \\ & {[T]=\sqrt{\dfrac{L_{12}^{3}}{G\left(m_{1}+m_{2}\right)}}=\dfrac{\sqrt{1+e \cos f}}{\dot{f}}} \end{split}\right.$ (4)

 $\left\{ \begin{split} & \ddot x - 2\dot y = \dfrac{1}{{1 + e\cos f}}{\varOmega _x} \\ &\ddot y + 2\dot x = \dfrac{1}{{1 + e\cos f}}{\varOmega _y} \\ &\ddot z + z = \dfrac{1}{{1 + e\cos f}}{\varOmega _z} \\ \end{split} \right.$ (5)

 $\varOmega = \dfrac{1}{2}({x^2} + {y^2} + {z^2}) + \dfrac{{1 - \mu }}{{{r_1}}} + \dfrac{\mu }{{{r_2}}}$ (6)

${\varOmega _x},{\varOmega _y},{\varOmega _z}$ 为势函数 $\varOmega$ 关于三轴的偏导数，具体表达式为

 $\left\{ \begin{split} & {\varOmega _x} = x - (1 - \mu )\dfrac{{x + \mu }}{{r_1^3}} - \mu \dfrac{{x + \mu - 1}}{{r_2^3}} \\ & {\varOmega _y} = y - (1 - \mu )\dfrac{y}{{r_1^3}} - \mu \dfrac{y}{{r_2^3}} \\ & {\varOmega _z} = z - (1 - \mu )\dfrac{z}{{r_1^3}} - \mu \dfrac{z}{{r_2^3}} \\ \end{split} \right.$ (7)

 $\left\{ {\begin{split} & {\ddot x - 2\dot y = \dfrac{1}{{1 + e\cos f}}\left[ {x - (1 - \mu )\dfrac{{x + \mu }}{{r_1^3}} - \mu \dfrac{{x + \mu - 1}}{{r_2^3}}} \right]}\\ & {\ddot y + 2\dot x = \dfrac{1}{{1 + e\cos f}}\left[ {y - (1 - \mu )\dfrac{y}{{r_1^3}} - \mu \dfrac{y}{{r_2^3}}} \right]}\\ & {\ddot z + z = \dfrac{1}{{1 + e\cos f}}\left[ {z - (1 - \mu )\dfrac{z}{{r_1^3}} - \mu \dfrac{z}{{r_2^3}}} \right]} \end{split}} \right.$ (8)

 $\dot x = \dot y = \dot z = \ddot x = \ddot y = \ddot z = 0$ (9)

 $\left\{ {\begin{split} & {\dfrac{1}{{1 + e\cos f}}\left[ {x - (1 - \mu )\dfrac{{x + \mu }}{{r_1^3}} - \mu \dfrac{{x + \mu - 1}}{{r_2^3}}} \right] = 0}\\ & {\dfrac{1}{{1 + e\cos f}}\left[ {y - (1 - \mu )\dfrac{y}{{r_1^3}} - \mu \dfrac{y}{{r_2^3}}} \right] = 0}\\ & {\dfrac{1}{{1 + e\cos f}}\left[ {z - (1 - \mu )\dfrac{z}{{r_1^3}} - \mu \dfrac{z}{{r_2^3}}} \right] - z = 0} \end{split}} \right.$ (10)

 $\left\{ \begin{split} & z(e\cos f + \dfrac{{1 - \mu }}{{r_1^3}} + \dfrac{\mu }{{r_2^3}}) = 0 \\ & e\cos f + \dfrac{{1 - \mu }}{{r_1^3}} + \dfrac{\mu }{{r_2^3}} \ne 0 \\ \end{split} \right.$ (11)

 $z = 0$ (12)

 $\left\{ \begin{split} & x - (1 - \mu )\dfrac{{x + \mu }}{{r_1^3}} - \mu \dfrac{{x + \mu - 1}}{{r_2^3}} = 0 \\ & y - (1 - \mu )\dfrac{y}{{r_1^3}} - \mu \dfrac{y}{{r_2^3}} = 0 \\ \end{split} \right.$ (13)

 $\delta \dot {{x}} = {\left. {\dfrac{{\partial f({{x}})}}{{\partial {{x}}}}} \right|_{{{x}} = {{{x}}_{{L_i}}}}}\delta {{x}} = {D_{{L_i}}}\delta {{x}} + {\rm H.O.T}$ (14)

 $\delta \dot x = \left[ {\begin{array}{cccccccc} 0 & 0 &0&1&0&0 \\ 0 & 0 &0&0&1&0 \\ 0& 0&0&0&0&1 \\ {{\varOmega _{x,x}}}&{{\varOmega _{x,y}}}&{{\varOmega _{x,z}}}&0&2&0 \\ {{\varOmega _{y,x}}}&{{\varOmega _{y,y}}}&{{\varOmega _{y,z}}}&{ - 2}&0&0 \\ {{\varOmega _{z,x}}}&{{\varOmega _{z,y}}}&{{\varOmega _{z,z}}}&0&0&0 \end{array}} \right]\delta x$ (15)
 $\left\{ \begin{array}{l} {\varOmega _{x,x}} = \dfrac{1}{{1 + e\cos f}}\Bigg[1 - \dfrac{{1 - \mu }}{{r_1^3}} + 3\dfrac{{(1 - \mu ){{(x + \mu )}^2}}}{{r_1^5}} - \dfrac{\mu }{{r_2^3}} + 3\dfrac{{\mu {{(x + \mu - 1)}^2}}}{{r_2^5}}\Bigg] \\ {\varOmega _{y,y}} = \dfrac{1}{{1 + e\cos f}}\Bigg[1 - \dfrac{{1 - \mu }}{{r_1^3}} + 3\dfrac{{(1 - \mu ){y^2}}}{{r_1^5}} - \dfrac{\mu }{{r_2^3}} + 3\dfrac{{\mu {y^2}}}{{r_2^5}}\Bigg] \\ {\varOmega _{z,z}} = \dfrac{1}{{1 + e\cos f}}\Bigg[ - \dfrac{{1 - \mu }}{{r_1^3}} + 3\dfrac{{(1 - \mu ){z^2}}}{{r_1^5}} - \dfrac{\mu }{{r_2^3}} + 3\dfrac{{\mu {z^2}}}{{r_2^5}}\Bigg] - \dfrac{{e\cos f}}{{1 + e\cos f}} \\ {\varOmega _{x,y}} = {\varOmega _{y,x}} = \dfrac{1}{{1 + e\cos f}}\Bigg[3\dfrac{{(1 - \mu )(x + \mu )y}}{{r_1^5}} + 3\dfrac{{\mu (x + \mu - 1)y}}{{r_2^5}}\Bigg] \\ {\varOmega _{x,z}} = {\varOmega _{z,x}} = \dfrac{1}{{1 + e\cos f}}\Bigg[3\dfrac{{(1 - \mu )(x + \mu )z}}{{r_1^5}} + 3\dfrac{{\mu (x + \mu - 1)z}}{{r_2^5}}\Bigg] \\ {\varOmega _{y,z}} = {\varOmega _{z,y}} = \dfrac{1}{{1 + e\cos f}}\Bigg[3\dfrac{{(1 - \mu )yz}}{{r_1^5}} + 3\dfrac{{\mu yz}}{{r_2^5}}\Bigg] \\ \end{array} \right.$ (16)

 $\delta \dot x = \left[ {\begin{array}{cccccccc} 0&0&0&1&0&0 \\ 0&0&0&0&1&0 \\ 0&0&0&0&0&1 \\ {g(1 + 2\bar \mu )}&0&0&0&2&0 \\ 0&{g(1 - \bar \mu )}&0&{ - 2}&0&0 \\ 0&0&{g( - \bar \mu ) + k}&0&0&0 \end{array}} \right]\delta x$ (17)

 $\left\{ \begin{array}{l} g = \dfrac{1}{{1 + e\cos f}} \\ \bar \mu = \dfrac{{1 - \mu }}{{{{\left| {x + \mu } \right|}^3}}} + \dfrac{\mu }{{{{\left| {x + \mu - 1} \right|}^3}}} > 1 \\ k = \dfrac{{ - e\cos f}}{{1 + e\cos f}} \\ \end{array} \right.$ (18)

 $\left\{ \begin{array}{l} {\lambda _1} = \sqrt {\dfrac{{2g + g\bar \mu - 4 + \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} }}{2}} \\ {\lambda _2} = - {\lambda _1} \\ {\lambda _3} = i\sqrt {\dfrac{{ - 2g - g\bar \mu + 4 + \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} }}{2}} \\ {\lambda _4} = - {\lambda _3} \\ {\lambda _5} = i\sqrt {g\bar u - k} \\ {\lambda _6} = - {\lambda _5} \\ \end{array} \right.$ (19)

 $\left\{ \begin{array}{l} {{ v}_1} = \left[ {\begin{array}{*{20}{c}} 1&{{\eta _1}}&0&{{\lambda _1}}&{{\eta _2}}&0 \end{array}} \right] \\ {{ v}_2} = \left[ {\begin{array}{*{20}{c}} 1&{ - {\eta _1}}&0&{ - {\lambda _1}}&{{\eta _2}}&0 \end{array}} \right] \\ {{ v}_3} = \left[ {\begin{array}{*{20}{c}} 1&{{\eta _3}}&0&{{\lambda _3}}&{{\eta _4}}&0 \end{array}} \right] \\ {{ v}_4} = \left[ {\begin{array}{*{20}{c}} 1&{ - {\eta _3}}&0&{ - {\lambda _3}}&{{\eta _4}}&0 \end{array}} \right] \\ {{ v}_5} = \left[ {\begin{array}{*{20}{c}} 0&0&1&0&0&{{\lambda _5}} \end{array}} \right] \\ {{ v}_6} = \left[ {\begin{array}{*{20}{c}} 0&0&1&0&0&{ - {\lambda _5}} \end{array}} \right] \\ \end{array} \right.$ (20)

 $\left\{ \begin{array}{l} {\eta _1} = \dfrac{{\sqrt 2 \sqrt {2g + g\bar u + \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} - 4} \cdot [2g + g\bar u - \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} - 4]}}{{g(\bar u - 1)(3g\bar u + \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} + 4)}}\\ {\eta _2} = - \dfrac{{{\rm{4}}g{\rm{(2}}\bar u{\rm{ + 1)}}}}{{3g\bar u + \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} + 4}}\\ {\eta _3} = \dfrac{{\sqrt 2 \sqrt {2g + g\bar u - \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} - 4} \cdot (2g + g\bar u + \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} - 4)}}{{g(\bar u - 1)(3g\bar u - \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} + 4)}}\\ {\eta _4} = - \dfrac{{{\rm{4}}g{\rm{(2}}\bar u{\rm{ + 1)}}}}{{3g\bar u + \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} + 4}} \end{array} \right.$ (21)

 $\left\{ \begin{array}{l} \delta x(t) = {A_1}{e^{{\lambda _1}t}} + {A_2}{e^{ - {\lambda _1}t}} + {A_3}\cos (\rm{Im} ({\lambda _3})t) +\\ \quad\quad\quad {A_4}\sin (\rm{Im} ({\lambda _3})t) \\ \delta y(t) = {k_1}{A_1}{e^{{\lambda _1}t}} - {k_1}{A_2}{e^{ - {\lambda _1}t}} - {k_2}{A_3}\sin (\rm{Im} ({\lambda _3})t)+ \\ \quad\quad\quad {k_2}{A_4}\cos (\rm{Im} ({\lambda _3})t) \\ \delta z(t) = {A_5}\cos (\rm{Im} ({\lambda _5})t) + {A_6}\sin (\rm{Im} ({\lambda _5})t) \\ \end{array} \right.$ (22)

 ${k_1} = {m_1},{k_2} = \rm{Im} ({m_3})$ (23)

2 仿真分析

 图 2 ERTBP下的近似解析解与CRTBP下的近似解析解生成的Lissajous轨道对比 Fig. 2 The comparison of Lissajous orbits generated by solutions in ERTBP and CRTBP
 图 3 ERTBP下的近似解析解相对CRTBP下的近似解析解的精度 Fig. 3 The precision of the solutions in ERTBP relative to the solutions in CRTBP

 图 4 CRTBP解析解精度 Fig. 4 The precision of CRTBP analytical solutions
 图 5 ERTBP解析解精度 Fig. 5 The precision of ERTBP analytical solutions

3 结　论