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限制性三体问题研究的是一个小天体或航天器在两个大天体(主天体)引力场中的运动规律,其中小天体或航天器的质量相比于两主天体可忽略不计,不会对两主天体之间的相互运动产生影响。当主天体围绕公共质心作圆轨道运动时,称为CRTBP;当主天体围绕公共质心作椭圆轨道运动时,称为ERTBP,下面分别进行介绍。
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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) 其中:
${m_1}$ 、${m_2}$ 为两主天体质量;${L_{12}}$ 为两主天体之间的距离;$G$ 为万有引力常数。为了更直观、更清晰地描述航天器在CRTBP中的运动,通常选择在会合坐标系(也称为质心旋转坐标系)下进行研究。如图1所示,会合坐标系的原点位于两主天体的公共质心,x轴由较大主天体指向较小主天体,z轴与主天体系统角动量方向平行,y轴满足右手定则[19]。
图 1质心旋转坐标系
$O - XYZ$ 和平动点旋转坐标系${L_2} - xyz$ Figure 1.Barycentric rotating coordinate system
$O - XYZ$ and libration-point centered rotating coordinate system${L_2} - xyz$ 假设
${{x}} = {[x,y,z]^{\rm T}}$ 为航天器在会合坐标系中的位置矢量,则CRTBP下航天器运动的动力学模型为$$\left\{ \begin{split} & \ddot x - 2\dot y = \dfrac{{\partial \varOmega }}{{\partial x}} \\ & \ddot y + 2\dot x = \dfrac{{\partial \varOmega }}{{\partial y}} \\ & \ddot z = \dfrac{{\partial \varOmega }}{{\partial z}} \\ \end{split} \right.$$ (2) 根据文献[36],CRTBP下共线平动点附近运动
$\delta {{x}} = {[\delta x,\delta y,\delta z]^{\rm T}}$ 的近似解析解为$$\left\{ \begin{split} \delta x(t) = & {A_1}{{\rm e}^{{\lambda _1}t}} + {A_2}{\rm e}^{ - {\lambda _1}t} + {A_3}\cos ({\rm Im} ({\lambda _3}))t + \\ & {A_4}\sin ({\rm Im} ({\lambda _3})t) \\ \delta y(t) = & {k_1}{A_1}{{\rm e}^{{\lambda _1}}}t - {k_1}{A_2}{\rm e}^{ - {\lambda _1}t} - {k_2}{A_3}\sin ({\rm Im} ({\lambda _3}))t + \\ & {k_2}{A_4}\cos ({\rm Im} ({\lambda _3})) t\\ \delta z(t) = & {A_5}\cos ({\rm Im} ({\lambda _5})t) + {A_6}\sin ({\rm Im} ({\lambda _5})) t \end{split} \right.$$ (3) 式中,
${A_i}(i = 1, \cdots , 6)$ 是由初始条件确定的积分常数,参数${k_1},{k_2},{\lambda _1},{\lambda _3},{\lambda _5}$ 的详细计算公式可以参考文献[36]。 -
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) 其中:
$\hat R$ 为当前时刻主天体之间的瞬时距离;$a$ 为主天体公转轨道的半长轴;$e$ 为公转轨道的偏心率;$f$ 为公转轨道的真近点角;$\dot f$ 为公转轨道的真近点角变化率。在ERTBP中,为研究和计算的方便,通常选择在脉动坐标系下进行研究。脉动坐标系的原点同样位于两主天体的质心,三轴指向与会合坐标系的三轴指向相同,但由于此坐标系的旋转角速率是变化的,且单位长度对应的真实物理长度也是时变的,为体现该坐标系的这种脉动特性,故称为脉动坐标系[19]。因此,在脉动坐标系下的航天器运动的动力学模型为
$$\left\{ \begin{split} & \ddot x - 2\dot y = \dfrac{1}{{1 + e\cos f}}{\varOmega _x} \\ &\ddot y + 2\dot x = \dfrac{1}{{1 + e\cos f}}{\varOmega _y} \\ &\ddot z + z = \dfrac{1}{{1 + e\cos f}}{\varOmega _z} \\ \end{split} \right.$$ (5) 其中:
$\varOmega $ 为ERTBP下的有效势函数,定义如下$$\varOmega = \dfrac{1}{2}({x^2} + {y^2} + {z^2}) + \dfrac{{1 - \mu }}{{{r_1}}} + \dfrac{\mu }{{{r_2}}}$$ (6) 而
${\varOmega _x},{\varOmega _y},{\varOmega _z}$ 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) 将上述偏导数代入动力学方程(5)可得
$$ \left\{ {\begin{split} & {\ddot x - 2\dot y = \dfrac{1}{{1 + e\cos f}}\left[ {x - (1 - \mu )\dfrac{{x + \mu }}{{r_1^3}} - \mu \dfrac{{x + \mu - 1}}{{r_2^3}}} \right]}\\ & {\ddot y + 2\dot x = \dfrac{1}{{1 + e\cos f}}\left[ {y - (1 - \mu )\dfrac{y}{{r_1^3}} - \mu \dfrac{y}{{r_2^3}}} \right]}\\ & {\ddot z + z = \dfrac{1}{{1 + e\cos f}}\left[ {z - (1 - \mu )\dfrac{z}{{r_1^3}} - \mu \dfrac{z}{{r_2^3}}} \right]} \end{split}} \right. $$ (8) 为获得ERTBP下平动点附近的航天器运动近似解析解,首先需要计算ERTBP下平动点的位置。由于平动点为三体系统的动平衡点,在平动点处满足速度、加速度为零,即
$$\dot x = \dot y = \dot z = \ddot x = \ddot y = \ddot z = 0$$ (9) 将此条件代入式(8),可得
$$ \left\{ {\begin{split} & {\dfrac{1}{{1 + e\cos f}}\left[ {x - (1 - \mu )\dfrac{{x + \mu }}{{r_1^3}} - \mu \dfrac{{x + \mu - 1}}{{r_2^3}}} \right] = 0}\\ & {\dfrac{1}{{1 + e\cos f}}\left[ {y - (1 - \mu )\dfrac{y}{{r_1^3}} - \mu \dfrac{y}{{r_2^3}}} \right] = 0}\\ & {\dfrac{1}{{1 + e\cos f}}\left[ {z - (1 - \mu )\dfrac{z}{{r_1^3}} - \mu \dfrac{z}{{r_2^3}}} \right] - z = 0} \end{split}} \right. $$ (10) 由于在Z轴上存在如下约束关系
$$\left\{ \begin{split} & z(e\cos f + \dfrac{{1 - \mu }}{{r_1^3}} + \dfrac{\mu }{{r_2^3}}) = 0 \\ & e\cos f + \dfrac{{1 - \mu }}{{r_1^3}} + \dfrac{\mu }{{r_2^3}} \ne 0 \\ \end{split} \right.$$ (11) 因此必有
$$z = 0$$ (12) 故ERTBP中的平动点位置和CRTBP一样,均在XY平面内,更具体的有
$$\left\{ \begin{split} & x - (1 - \mu )\dfrac{{x + \mu }}{{r_1^3}} - \mu \dfrac{{x + \mu - 1}}{{r_2^3}} = 0 \\ & y - (1 - \mu )\dfrac{y}{{r_1^3}} - \mu \dfrac{y}{{r_2^3}} = 0 \\ \end{split} \right.$$ (13) 这与CRTBP下平动点的计算公式完全相同。据此可知,在每一瞬时,ERTBP中的真近点角
$f$ 为一定值时,在脉动坐标系下的ERTBP平动点位置与会合坐标系下的CRTBP平动点位置完全相同,这也与文献[16]中的论述相符合,具体的平动点位置计算公式可以参考文献[36],本文不再赘述。为获得ERTBP下平动点附近运动的近似解析解,需要在平动点
${x_{Li}}$ 处将ERTBP下的航天器运动动力学模型进行线性化$$ \delta \dot {{x}} = {\left. {\dfrac{{\partial f({{x}})}}{{\partial {{x}}}}} \right|_{{{x}} = {{{x}}_{{L_i}}}}}\delta {{x}} = {D_{{L_i}}}\delta {{x}} + {\rm H.O.T} $$ (14) 其中,H.O.T(High-Order-Terms)表示线性化过程中的高阶项。将线性化动力学模型式(14)按照三轴位置、速度展开可得式(15)。其中,
${\varOmega _{i,j}}(i,j = x,y,z)$ 是势函数$\varOmega $ 关于三轴的二阶偏导数,具体表达式为$$\delta \dot x = \left[ {\begin{array}{cccccccc} 0 & 0 &0&1&0&0 \\ 0 & 0 &0&0&1&0 \\ 0& 0&0&0&0&1 \\ {{\varOmega _{x,x}}}&{{\varOmega _{x,y}}}&{{\varOmega _{x,z}}}&0&2&0 \\ {{\varOmega _{y,x}}}&{{\varOmega _{y,y}}}&{{\varOmega _{y,z}}}&{ - 2}&0&0 \\ {{\varOmega _{z,x}}}&{{\varOmega _{z,y}}}&{{\varOmega _{z,z}}}&0&0&0 \end{array}} \right]\delta x$$ (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) 对于共线平动点,满足
$y = z = 0$ ,将此条件代入上述线性化动力学模型式(15),化简后可得$$\delta \dot x = \left[ {\begin{array}{cccccccc} 0&0&0&1&0&0 \\ 0&0&0&0&1&0 \\ 0&0&0&0&0&1 \\ {g(1 + 2\bar \mu )}&0&0&0&2&0 \\ 0&{g(1 - \bar \mu )}&0&{ - 2}&0&0 \\ 0&0&{g( - \bar \mu ) + k}&0&0&0 \end{array}} \right]\delta x$$ (17) 其中:参数
$g$ ,$\bar \mu $ ,$k$ 的表达式如式(18)所示$$\left\{ \begin{array}{l} g = \dfrac{1}{{1 + e\cos f}} \\ \bar \mu = \dfrac{{1 - \mu }}{{{{\left| {x + \mu } \right|}^3}}} + \dfrac{\mu }{{{{\left| {x + \mu - 1} \right|}^3}}} > 1 \\ k = \dfrac{{ - e\cos f}}{{1 + e\cos f}} \\ \end{array} \right.$$ (18) 通过线性系统理论方法,计算得到线性化常微分方程组式(15)的6个特征值
${\lambda _i}(i = 1,\cdots,6)$ 为$$\left\{ \begin{array}{l} {\lambda _1} = \sqrt {\dfrac{{2g + g\bar \mu - 4 + \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} }}{2}} \\ {\lambda _2} = - {\lambda _1} \\ {\lambda _3} = i\sqrt {\dfrac{{ - 2g - g\bar \mu + 4 + \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} }}{2}} \\ {\lambda _4} = - {\lambda _3} \\ {\lambda _5} = i\sqrt {g\bar u - k} \\ {\lambda _6} = - {\lambda _5} \\ \end{array} \right.$$ (19) 对应的6个特征向量
${\vec v_i}(i = 1,\cdots,6)$ 为$$\left\{ \begin{array}{l} {{ v}_1} = \left[ {\begin{array}{*{20}{c}} 1&{{\eta _1}}&0&{{\lambda _1}}&{{\eta _2}}&0 \end{array}} \right] \\ {{ v}_2} = \left[ {\begin{array}{*{20}{c}} 1&{ - {\eta _1}}&0&{ - {\lambda _1}}&{{\eta _2}}&0 \end{array}} \right] \\ {{ v}_3} = \left[ {\begin{array}{*{20}{c}} 1&{{\eta _3}}&0&{{\lambda _3}}&{{\eta _4}}&0 \end{array}} \right] \\ {{ v}_4} = \left[ {\begin{array}{*{20}{c}} 1&{ - {\eta _3}}&0&{ - {\lambda _3}}&{{\eta _4}}&0 \end{array}} \right] \\ {{ v}_5} = \left[ {\begin{array}{*{20}{c}} 0&0&1&0&0&{{\lambda _5}} \end{array}} \right] \\ {{ v}_6} = \left[ {\begin{array}{*{20}{c}} 0&0&1&0&0&{ - {\lambda _5}} \end{array}} \right] \\ \end{array} \right.$$ (20) 其中,
${\eta _1},{\eta _2},{\eta _3},{\eta _4}$ 的计算公式如式(21)所示$$\left\{ \begin{array}{l} {\eta _1} = \dfrac{{\sqrt 2 \sqrt {2g + g\bar u + \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} - 4} \cdot [2g + g\bar u - \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} - 4]}}{{g(\bar u - 1)(3g\bar u + \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} + 4)}}\\ {\eta _2} = - \dfrac{{{\rm{4}}g{\rm{(2}}\bar u{\rm{ + 1)}}}}{{3g\bar u + \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} + 4}}\\ {\eta _3} = \dfrac{{\sqrt 2 \sqrt {2g + g\bar u - \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} - 4} \cdot (2g + g\bar u + \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} - 4)}}{{g(\bar u - 1)(3g\bar u - \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} + 4)}}\\ {\eta _4} = - \dfrac{{{\rm{4}}g{\rm{(2}}\bar u{\rm{ + 1)}}}}{{3g\bar u + \sqrt {9{g^2}{{\bar u}^2} - 8g\bar u - 16g + 16} + 4}} \end{array} \right.$$ (21) 由于上述6个特征向量线性无关,由此得到了线性化常微分方程组式(15)的6个线性无关的解,结合6个特征值,构成了线性化动力学模型的基本解集,方程组的通解可表示为
$$\left\{ \begin{array}{l} \delta x(t) = {A_1}{e^{{\lambda _1}t}} + {A_2}{e^{ - {\lambda _1}t}} + {A_3}\cos (\rm{Im} ({\lambda _3})t) +\\ \quad\quad\quad {A_4}\sin (\rm{Im} ({\lambda _3})t) \\ \delta y(t) = {k_1}{A_1}{e^{{\lambda _1}t}} - {k_1}{A_2}{e^{ - {\lambda _1}t}} - {k_2}{A_3}\sin (\rm{Im} ({\lambda _3})t)+ \\ \quad\quad\quad {k_2}{A_4}\cos (\rm{Im} ({\lambda _3})t) \\ \delta z(t) = {A_5}\cos (\rm{Im} ({\lambda _5})t) + {A_6}\sin (\rm{Im} ({\lambda _5})t) \\ \end{array} \right.$$ (22) 其中,
${A_i}(i = 1, \cdots,6)$ 同样是由初始条件确定的积分常数,参数${k_1}$ ,${k_2}$ 的取值为$${k_1} = {m_1},{k_2} = \rm{Im} ({m_3})$$ (23) 式(22)即为本文研究的ERTBP下共线平动点附近运动的近似解析解。
通过对比发现ERTBP下平动点附近运动的近似解析解与CRTBP下的近似解析解形式完全相同,二者的区别仅仅是特征值
${\lambda _1},{\lambda _3},{\lambda _5}$ 和相关系数${k_1},{k_2}$ 的计算公式不同而已,这也在某种程度上体现了CRTBP与ERTBP的一致性。
Approximate Analytical Solutions of Motion near the Collinear Libration-Points in Restricted Three-Body Problem
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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。
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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
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.Highlights● The positions of libration points in elliptic restricted three-body problem are given. ● The approximate analytical solutions of motion around collinear pibration points in elliptic restricted three-body problem are derived. ● The solutions for elliptic restricted three-body problem have higher precision than that of the circular restricted three-body problem was proved. -
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