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Solution of lunar equations of motion

The lunar equations of motion, Equations (11.86)-(11.93), take the general form

    $\displaystyle X_0$ $\displaystyle \simeq 1-\frac{1}{6}\,m^{\,2}+\frac{1}{18}\,m^{\,4},$ (11.106)
    $\displaystyle \delta \ddot{X}-2\,\delta \dot{Y} - 3\left(1+\frac{1}{2}\,m^{\,2}\right)\,\delta X$ $\displaystyle \simeq R_X,$ (11.107)
    $\displaystyle \delta \ddot{Y}+2\,\delta \dot{X}$ $\displaystyle \simeq R_Y,$ (11.108)
and   $\displaystyle \delta\ddot{Z} + \left(1+\frac{3}{2}\,m^{\,2}\right)\,\delta Z$ $\displaystyle \simeq R_Z,$     (11.109)

where

    $\displaystyle R_X$ $\displaystyle \simeq \frac{3}{2}\,m^{\,2}\left(1-\frac{m^{\,2}}{6}\right)\cos(2...
...}\,m^{\,2}\,e'\,\cos {\cal M}' +\frac{21}{4}\,m^{\,2}\,e'\,\cos(2\,D-{\cal M}')$    
      $\displaystyle \phantom{=} - \frac{3}{4}\,m^{\,2}\,e'\,\cos(2\,D+{\cal M}') + \frac{9}{8}\,m^{\,2}\,\zeta\,\cos D + \frac{15}{8}\,m^{\,2}\,\zeta\,\cos(3\,D)$    
      $\displaystyle \phantom{=} +\frac{3}{2}\,m^{\,2}\,\cos(2\,D)\,\delta X -\frac{3}...
...lta Y -3\,\delta X^{\,2}+\frac{3}{2}\left(\delta Y^{\,2}+\delta Z^{\,2}\right),$ (11.110)
    $\displaystyle R_Y$ $\displaystyle \simeq - \frac{3}{2}\,m^{\,2}\left(1-\frac{m^{\,2}}{6}\right)\sin(2\,D) -\frac{21}{4}\,m^{\,2}\,e'\,\sin(2\,D-{\cal M}')$    
      $\displaystyle \phantom{=}+ \frac{3}{4}\,m^{\,2}\,e'\,\sin(2\,D+{\cal M}') - \frac{3}{8}\,m^{\,2}\,\zeta\,\sin D - \frac{15}{8}\,m^{\,2}\,\zeta\,\sin(3\,D)$    
      $\displaystyle \phantom{=} -\frac{3}{2}\,m^{\,2}\,\sin(2\,D)\,\delta X -\frac{3}{2}\,m^{\,2}\,\cos(2\,D)\,\delta Y +3\,\delta X\,\delta Y,$ (11.111)
and   $\displaystyle R_Z$ $\displaystyle \simeq 3\,\delta X\,\delta Z.$     (11.112)

Here, use has been made of the definitions (11.99)-(11.102). Furthermore, it follows from Equations (11.94)-(11.96), (11.103)-(11.105), and (11.106) that

    $\displaystyle \delta R$ $\displaystyle \simeq -\frac{1}{6}\,m^{\,2} + \frac{1}{18} \,m^{\,4}+\delta X + \frac{1}{2}\,\delta Y^{\,2}+\frac{1}{2}\,\delta Z^{\,2},$ (11.113)
    $\displaystyle \delta \lambda$ $\displaystyle \simeq \left(1+\frac{m^{\,2}}{6}\right)\delta Y-\delta X\,\delta Y,$ (11.114)
and   $\displaystyle \delta \beta$ $\displaystyle \simeq \left(1+\frac{m^{\,2}}{6}\right)\delta Z -\delta X\,\delta Y.$ (11.115)

As before, we have neglected terms that are third order, or greater, in the small parameters $ m^{\,2}$ , $ e'$ , $ \zeta$ , $ \delta X$ , $ \delta Y$ , and $ \delta Z$ .

Let us write

    $\displaystyle \delta X$ $\displaystyle = x_{01}\,m^{\,4}+ x_{02}\,e^{\,2}+x_{03}\,I^{\,2} + x_1\,e\,\cos{\cal M} +x_2\,e^{\,2}\,\cos(2\,{\cal M}) + x_3\,I^{\,2}\,\cos(2\,F)$    
      $\displaystyle \phantom{=} +x_4\,m^{\,2}\,\cos(2\,D) + x_5\,m^{\,4}\,\cos(4\,D) + x_6\,m\,e\,\cos(2\,D-{\cal M})$    
      $\displaystyle \phantom{=}+x_7\,m^{\,2}\,e\,\cos(2\,D+{\cal M})+x_8\,m\,e'\,\cos {\cal M}' + x_9\,m^{\,2}\,e'\,\cos(2\,D-{\cal M}')$    
      $\displaystyle \phantom{=}+x_{10}\,m^{\,2}\,e'\,\cos(2\,D+{\cal M}')+x_{11}\,m\,\zeta\,\cos D + x_{12}\,m^{\,2}\,\zeta\,\cos(3\,D),$ (11.116)
    $\displaystyle \delta Y$ $\displaystyle = y_1\,e\,\sin{\cal M} +y_2\,e^{\,2}\,\sin(2\,{\cal M}) + y_3\,I^{\,2}\,\sin(2\,F)$    
      $\displaystyle \phantom{=} +y_4\,m^{\,2}\,\sin(2\,D) + y_5\,m^{\,4}\,\sin(4\,D) + y_6\,m\,e\,\sin(2\,D-{\cal M})$    
      $\displaystyle \phantom{=}+y_7\,m^{\,2}\,e\,\sin(2\,D+{\cal M})+y_8\,m\,e'\,\sin {\cal M}' + y_9\,m^{\,2}\,e'\,\sin(2\,D-{\cal M}')$    
      $\displaystyle \phantom{=}+y_{10}\,m^{\,2}\,e'\,\sin(2\,D+{\cal M}')+y_{11}\,m\,\zeta\,\sin D + y_{12}\,m^{\,2}\,\zeta\,\sin(3\,D),$ (11.117)
    $\displaystyle \delta Z$ $\displaystyle = z_1\,I\,\sin F + z_2\,e\,I\,\sin (F-{\cal M}) + z_3\,e\,I\,\sin(F+{\cal M}) + z_4\,m\,I\,\sin(2\,D-F)$    
      $\displaystyle \phantom{=} +z_5\,m^{\,2}\,I\,\sin(2\,D+F),$ (11.118)
    $\displaystyle R_X$ $\displaystyle = a_{01}\,m^{\,4}+ a_{02}\,e^{\,2}+a_{03}\,I^{\,2} + a_1\,m^{\,3}\,e\,\cos{\cal M} +a_2\,e^{\,2}\,\cos(2\,{\cal M}) + a_3\,I^{\,2}\,\cos(2\,F)$    
      $\displaystyle \phantom{=} +a_4\,m^{\,2}\,\cos(2\,D) + a_5\,m^{\,4}\,\cos(4\,D) + a_6\,m^{\,2}\,e\,\cos(2\,D-{\cal M})$    
      $\displaystyle \phantom{=}+a_7\,m^{\,2}\,e\,\cos(2\,D+{\cal M})+a_8\,m^{\,2}\,e'\,\cos {\cal M}' + a_9\,m^{\,2}\,e'\,\cos(2\,D-{\cal M}')$    
      $\displaystyle \phantom{=}+a_{10}\,m^{\,2}\,e'\,\cos(2\,D+{\cal M}')+a_{11}\,m^{\,2}\,\zeta\,\cos D + a_{12}\,m^{\,2}\,\zeta\,\cos(3\,D),$ (11.119)
    $\displaystyle R_Y$ $\displaystyle = b_1\,m^{\,3}\,e\,\sin{\cal M} +b_2\,e^{\,2}\,\sin(2\,{\cal M}) + b_3\,I^{\,2}\,\sin(2\,F)$    
      $\displaystyle \phantom{=} +b_4\,m^{\,2}\,\sin(2\,D) + b_5\,m^{\,4}\,\sin(4\,D) + b_6\,m^{\,2}\,e\,\sin(2\,D-{\cal M})$    
      $\displaystyle \phantom{=}+b_7\,m^{\,2}\,e\,\sin(2\,D+{\cal M})+b_8\,m^{\,2}\,e'\,\sin {\cal M}' + b_9\,m^{\,2}\,e'\,\sin(2\,D-{\cal M}')$    
      $\displaystyle \phantom{=}+b_{10}\,m^{\,2}\,e'\,\sin(2\,D+{\cal M}')+b_{11}\,m^{\,2}\,\zeta\,\sin D + b_{12}\,m^{\,2}\,\zeta\,\sin(3\,D),$ (11.120)
    $\displaystyle R_Z$ $\displaystyle = c_1\,m^{\,3}\,I\,\sin F + c_2\,e\,I\,\sin (F-{\cal M}) + c_3\,e\,I\,\sin(F+{\cal M}) + c_4\,m^{\,2}\,I\,\sin(2\,D-F)$    
      $\displaystyle \phantom{=} +c_5\,m^{\,2}\,I\,\sin(2\,D+F),$ (11.121)
    $\displaystyle \delta R$ $\displaystyle = -\frac{1}{6}\,m^{\,2}+ r_{01}\,m^{\,4}+ r_{02}\,e^{\,2}+r_{03}\...
..._1\,e\,\cos{\cal M} +r_2\,e^{\,2}\,\cos(2\,{\cal M}) + r_3\,I^{\,2}\,\cos(2\,F)$    
      $\displaystyle \phantom{=} +r_4\,m^{\,2}\,\cos(2\,D) + r_5\,m^{\,4}\,\cos(4\,D) + r_6\,m\,e\,\cos(2\,D-{\cal M})$    
      $\displaystyle \phantom{=}+r_7\,m^{\,2}\,e\,\cos(2\,D+{\cal M})+r_8\,m\,e'\,\cos {\cal M}' + r_9\,m^{\,2}\,e'\,\cos(2\,D-{\cal M}')$    
      $\displaystyle \phantom{=}+r_{10}\,m^{\,2}\,e'\,\cos(2\,D+{\cal M}')+r_{11}\,m\,\zeta\,\cos D + r_{12}\,m^{\,2}\,\zeta\,\cos(3\,D),$ (11.122)
    $\displaystyle \delta \lambda$ $\displaystyle = l_1\,e\,\sin{\cal M} +l_2\,\,e^{\,2}\,\sin(2\,{\cal M}) + l_3\,I^{\,2}\,\sin(2\,F)$    
      $\displaystyle \phantom{=} +l_4\,m^{\,2}\,\sin(2\,D) + l_5\,m^{\,4}\,\sin(4\,D) + l_6\,m\,e\,\sin(2\,D-{\cal M})$    
      $\displaystyle \phantom{=}+l_7\,m^{\,2}\,e\,\sin(2\,D+{\cal M})+l_8\,m\,e'\,\sin {\cal M}' + l_9\,m^{\,2}\,e'\,\sin(2\,D-{\cal M}')$    
      $\displaystyle \phantom{=}+l_{10}\,m^{\,2}\,e'\,\sin(2\,D+{\cal M}')+l_{11}\,m\,\zeta\,\sin D + l_{12}\,m^{\,2}\,\zeta\,\sin(3\,D),$ (11.123)
and   $\displaystyle \delta \beta$ $\displaystyle = d_1\,I\,\sin F + d_2\,e\,I\,\sin (F-{\cal M}) + d_3\,e\,I\,\sin(F+{\cal M}) + d_4\,m\,I\,\sin(2\,D-F)$    
      $\displaystyle \phantom{=} +d_5\,m^{\,2}\,I\,\sin(2\,D+F).$     (11.124)

Here, the $ a_{0i}$ , $ a_j$ , $ b_j$ , $ c_j$ , et cetera, are $ {\cal O}(1)$ constants.

Equations (11.110)-(11.124) can be combined to give

    $\displaystyle a_{01}$ $\displaystyle = \frac{3}{4}\,x_4-\frac{3}{4}\,y_4 -\frac{3}{2}\,x_4^{\,2} + \frac{3}{4}\,y_4^{\,2},$ (11.125)
    $\displaystyle a_{02}$ $\displaystyle = -\frac{3}{2}\,x_1^{\,2}+\frac{3}{4}\,y_1^{\,2},$ (11.126)
    $\displaystyle a_{03}$ $\displaystyle = \frac{3}{4}\,z_1^{\,2},$ (11.127)
    $\displaystyle a_1$ $\displaystyle =\frac{3}{4}\,x_6 -\frac{3}{4}\,y_6-3\,x_4\,x_6+\frac{3}{2}\,y_4\,y_6,$ (11.128)
    $\displaystyle a_2$ $\displaystyle = -\frac{3}{2}\,x_1^{\,2}-\frac{3}{4}\,y_1^{\,2},$ (11.129)
    $\displaystyle a_3$ $\displaystyle = -\frac{3}{4}\,z_1^{\,2},$ (11.130)
    $\displaystyle a_4$ $\displaystyle = \frac{3}{2}-\frac{1}{4}\,m^{\,2},$ (11.131)
    $\displaystyle a_5$ $\displaystyle = \frac{3}{4}\,x_4+\frac{3}{4}\,y_4-\frac{3}{2}\,x_4^{\,2}-\frac{3}{4}\,y_4^{\,2},$ (11.132)
    $\displaystyle a_6$ $\displaystyle = \frac{3}{4}\,x_1-\frac{3}{4}\,y_1-3\,x_1\,x_4+\frac{3}{2}\,y_1\,y_4,$ (11.133)
    $\displaystyle a_7$ $\displaystyle = \frac{3}{4}\,x_1+\frac{3}{4}\,y_1-3\,x_1\,x_4-\frac{3}{2}\,y_1\,y_4,$ (11.134)
    $\displaystyle a_8$ $\displaystyle = \frac{3}{2},$ (11.135)
    $\displaystyle a_9$ $\displaystyle = \frac{21}{4},$ (11.136)
    $\displaystyle a_{10}$ $\displaystyle = -\frac{3}{4},$ (11.137)
    $\displaystyle a_{11}$ $\displaystyle = \frac{9}{8}+\frac{3}{4}\,x_{11}\,m-\frac{3}{4}\,y_{11}\,m-3\,x_4\,x_{11}\,m+\frac{3}{2}\,y_4\,y_{11}\,m,$ (11.138)
and   $\displaystyle a_{12}$ $\displaystyle = \frac{15}{8},$     (11.139)

as well as

    $\displaystyle b_1$ $\displaystyle =-\frac{3}{4}\,x_6+ \frac{3}{4}\,y_6 -\frac{3}{2}\,x_4\,y_6 +\frac{3}{2}\,y_4\,x_6,$ (11.140)
    $\displaystyle b_2$ $\displaystyle = \frac{3}{2}\,x_1\,y_1,$ (11.141)
    $\displaystyle b_3$ $\displaystyle = 0,$ (11.142)
    $\displaystyle b_4$ $\displaystyle = -\frac{3}{2}+\frac{1}{4}\,m^{\,2},$ (11.143)
    $\displaystyle b_5$ $\displaystyle =- \frac{3}{4}\,x_4-\frac{3}{4}\,y_4+\frac{3}{2}\,x_4\,y_4,$ (11.144)
    $\displaystyle b_6$ $\displaystyle =- \frac{3}{4}\,x_1+\frac{3}{4}\,y_1+\frac{3}{2}\,x_1\,y_4-\frac{3}{2}\,y_1\,x_4,$ (11.145)
    $\displaystyle b_7$ $\displaystyle = -\frac{3}{4}\,x_1-\frac{3}{4}\,y_1+\frac{3}{2}\,x_1\,y_4+\frac{3}{2}\,y_1\,x_4,$ (11.146)
    $\displaystyle b_8$ $\displaystyle = 0,$ (11.147)
    $\displaystyle b_9$ $\displaystyle =- \frac{21}{4},$ (11.148)
    $\displaystyle b_{10}$ $\displaystyle = \frac{3}{4},$ (11.149)
    $\displaystyle b_{11}$ $\displaystyle = -\frac{3}{8}-\frac{3}{4}\,x_{11}\,m+\frac{3}{4}\,y_{11}\,m-\frac{3}{2}\,x_4\,y_{11}\,m+\frac{3}{2}\,y_4\,x_{11}\,m,$ (11.150)
and   $\displaystyle b_{12}$ $\displaystyle =- \frac{15}{8},$     (11.151)

as well as

    $\displaystyle c_1$ $\displaystyle =-\frac{3}{2}\,x_4\,z_4,$ (11.152)
    $\displaystyle c_2$ $\displaystyle = \frac{3}{2}\,x_1\,z_1,$ (11.153)
    $\displaystyle c_3$ $\displaystyle = \frac{3}{2}\,x_1\,z_1,$ (11.154)
    $\displaystyle c_4$ $\displaystyle = -\frac{3}{2}\,x_4\,z_1,$ (11.155)
and   $\displaystyle c_5$ $\displaystyle =\frac{3}{2}\,x_4\,z_1,$     (11.156)

as well as

    $\displaystyle r_{01}$ $\displaystyle = x_{01} + \frac{1}{18}+\frac{1}{4}\,y_4^{\,2},$ (11.157)
    $\displaystyle r_{02}$ $\displaystyle = x_{02} + \frac{1}{4}\,y_1^{\,2},$ (11.158)
    $\displaystyle r_{03}$ $\displaystyle = x_{03}+\frac{1}{4}\,z_1^{\,2},$ (11.159)
    $\displaystyle r_1$ $\displaystyle =x_1,$ (11.160)
    $\displaystyle r_2$ $\displaystyle = x_2-\frac{1}{4}\,y_1^{\,2},$ (11.161)
    $\displaystyle r_3$ $\displaystyle = x_3-\frac{1}{4}\,z_1^{\,2},$ (11.162)
    $\displaystyle r_4$ $\displaystyle = x_4,$ (11.163)
    $\displaystyle r_5$ $\displaystyle =x_5-\frac{1}{4}\,y_4^{\,2},$ (11.164)
    $\displaystyle r_6$ $\displaystyle = x_6+\frac{1}{2}\,y_1\,y_4\,m,$ (11.165)
    $\displaystyle r_7$ $\displaystyle =x_7-\frac{1}{2}\,y_1\,y_4,$ (11.166)
    $\displaystyle r_8$ $\displaystyle =x_8,$ (11.167)
    $\displaystyle r_9$ $\displaystyle = x_9,$ (11.168)
    $\displaystyle r_{10}$ $\displaystyle = x_{10},$ (11.169)
    $\displaystyle r_{11}$ $\displaystyle = x_{11},$ (11.170)
and   $\displaystyle r_{12}$ $\displaystyle = x_{12},$ (11.171)

as well as

    $\displaystyle l_1$ $\displaystyle =\left(1+\frac{m^{\,2}}{6}\right)\,y_1,$ (11.172)
    $\displaystyle l_2$ $\displaystyle = y_2-\frac{1}{2}\,x_1\,y_1,$ (11.173)
    $\displaystyle l_3$ $\displaystyle = y_3,$ (11.174)
    $\displaystyle l_4$ $\displaystyle =\left(1+\frac{m^{\,2}}{6}\right)\,y_4 ,$ (11.175)
    $\displaystyle l_5$ $\displaystyle =y_5-\frac{1}{2}\,x_4\,y_4,$ (11.176)
    $\displaystyle l_6$ $\displaystyle =y_6-\frac{1}{2}\,x_1\,y_4\,m+\frac{1}{2}\,y_1\,x_4\,m,$ (11.177)
    $\displaystyle l_7$ $\displaystyle =y_7-\frac{1}{2}\,x_1\,y_4-\frac{1}{2}\,y_1\,x_4,$ (11.178)
    $\displaystyle l_8$ $\displaystyle = y_8,$ (11.179)
    $\displaystyle l_9$ $\displaystyle =y_9,$ (11.180)
    $\displaystyle l_{10}$ $\displaystyle = y_{10},$ (11.181)
    $\displaystyle l_{11}$ $\displaystyle = y_{11},$ (11.182)
and   $\displaystyle l_{12}$ $\displaystyle =y_{12},$ (11.183)

as well as

    $\displaystyle d_1$ $\displaystyle =\left(1+\frac{1}{6}m^{\,2}\right)\,z_1,$ (11.184)
    $\displaystyle d_2$ $\displaystyle =z_2-\frac{1}{2}\,x_1\,z_1,$ (11.185)
    $\displaystyle d_3$ $\displaystyle =z_3-\frac{1}{2}\,x_1\,z_1,$ (11.186)
    $\displaystyle d_4$ $\displaystyle = z_4+\frac{1}{2}\,z_1\,x_4\,m,$ (11.187)
and   $\displaystyle d_5$ $\displaystyle =z_5-\frac{1}{2}\,z_1\,x_4.$     (11.188)

Here, and in the following, we have retained $ {\cal O}(m)$ and $ {\cal O}(m^{\,2})$ corrections to the parameters $ x_1$ , $ y_1$ , $ r_1$ , $ l_1$ , $ a_4$ , $ b_4$ , $ x_4$ , $ y_4$ , $ r_4$ , $ l_4$ , $ z_1$ , and $ d_1$ , and $ {\cal O}(m)$ corrections to the parameters $ a_6$ , $ b_6$ , $ x_6$ , $ y_6$ , $ r_6$ , $ l_6$ , $ x_8$ , $ r_8$ , $ a_{11}$ , $ b_{11}$ , $ x_{11}$ , $ y_{11}$ , $ r_{11}$ , $ l_{11}$ , $ z_4$ , and $ d_4$ , while neglecting similar corrections for all of the other parameters appearing in Equations (11.116)-(11.124).

Substitution of Equations (11.116), (11.117), (11.119), and (11.120) into Equations (11.107) and (11.108) yields

$\displaystyle x_{0i} = -\frac{a_{0i}}{3\,(1+m^{\,2}/2)},$ (11.189)

for $ i=1, 3$ , as well as

    $\displaystyle \left(\omega_1^{\,2}+3+3\,m^{\,2}/2\right)x_1+2\,\omega_1\,y_1$ $\displaystyle = -m^{\,3}\,a_1,$ (11.190)
and   $\displaystyle 2\,\omega_1\,x_1+\omega_1^{\,2}\,y_1$ $\displaystyle = -m^{\,3}\,b_1,$ (11.191)

as well as

    $\displaystyle x_j$ $\displaystyle = \frac{m^{\,\alpha}\,(\omega_j\,a_j-2\,b_j)}{\omega_j\,(1-3\,m^{\,2}/2-\omega_j^{\,2})},$ (11.192)
and   $\displaystyle y_j$ $\displaystyle = \frac{m^{\,\alpha}\left[(\omega_j^{\,2}+3+3\,m^{\,2}/2)\,b_j-2\,\omega_j\,a_j\right]}{\omega_j^{\,2}\,(1-3\,m^{\,2}/2-\omega_j^{\,2})},$ (11.193)

for $ j=2,11$ . In the previous two equations, $ \alpha= 1$ for $ j = 6,8,11$ , and $ \alpha=0$ otherwise. Moreover,

    $\displaystyle \omega_1$ $\displaystyle = \skew{5}\dot{{\cal M}}= 1-c\,m^{\,2},$ (11.194)
    $\displaystyle \omega_2$ $\displaystyle = 2\,\skew{5}\dot{{\cal M}}= 2-2\,c\,m^{\,2},$ (11.195)
    $\displaystyle \omega_3$ $\displaystyle = 2\,\skew{5}\dot{F}= 2+2\,g\,m^{\,2},$ (11.196)
    $\displaystyle \omega_4$ $\displaystyle = 2\,\skew{5}\dot{D}=2-2\,m,$ (11.197)
    $\displaystyle \omega_5$ $\displaystyle = 4\,\skew{5}\dot{D}=4-4\,m,$ (11.198)
    $\displaystyle \omega_6$ $\displaystyle = 2\,\skew{5}\dot{D}-\skew{5}\dot{{\cal M}}=1-2\,m+c\,m^{\,2},$ (11.199)
    $\displaystyle \omega_7$ $\displaystyle = 2\,\skew{5}\dot{D}+\skew{5}\dot{{\cal M}}=3-2\,m-c\,m^{\,2},$ (11.200)
    $\displaystyle \omega_8$ $\displaystyle = \skew{5}\dot{{\cal M}}'=m,$ (11.201)
    $\displaystyle \omega_9$ $\displaystyle = 2\,\skew{5}\dot{D}-\skew{5}\dot{{\cal M}}'=2-3\,m,$ (11.202)
    $\displaystyle \omega_{10}$ $\displaystyle = 2\,\skew{5}\dot{D}+\skew{5}\dot{{\cal M}}'=2-m,$ (11.203)
    $\displaystyle \omega_{11}$ $\displaystyle = \skew{5}\dot{D}=1-m,$ (11.204)
and   $\displaystyle \omega_{12}$ $\displaystyle = 3\,\skew{5}\dot{D}=3-3\,m.$ (11.205)

In calculating the previous frequencies, we have assumed that the ecliptic longitude of the lunar perigee takes the form

$\displaystyle \alpha = \alpha_0 + c\,m^{\,2}\,T = \alpha_0+c\,m\,n'\,t,$ (11.206)

where $ \alpha_0$ is an arbitrary constant, and $ c$ is an, as yet, unknown $ {\cal O}(1)$ constant that parameterizes the precession of the lunar perigee. Here, use has been made of the facts that $ T = n\,t$ and $ m=n'/n$ . Likewise, we have assumed that the ecliptic longitude of the lunar ascending mode takes the form

$\displaystyle \gamma = \gamma_0 - g\,m^{\,2}\,T=\gamma_0-g\,m\,n'\,t,$ (11.207)

where $ \gamma_0$ is an arbitrary constant, and $ g$ is an, as yet, unknown $ {\cal O}(1)$ constant that parameterizes the regression of the lunar ascending node.

Substituting Equations (11.118) and (11.121) into Equation (11.109), we obtain

$\displaystyle \left(1-{\mit\Omega}_1^{\,2}+3\,m^{\,2}/2\right)z_1 = c_1\,m^{\,3},$ (11.208)

and

$\displaystyle z_j = \frac{m^{\,\alpha}\,c_j}{1+3\,m^{\,2}/2-{\mit\Omega}_j^{\,2}},$ (11.209)

for $ j=2,5$ . In the previous equation, $ \alpha= 1$ for $ j=4$ , and $ \alpha=0$ otherwise. Moreover,

    $\displaystyle {\mit\Omega}_1$ $\displaystyle = \skew{5}\dot{F} = 1+g\,m^{\,2},$ (11.210)
    $\displaystyle {\mit\Omega}_2$ $\displaystyle = \skew{5}\dot{F}-\skew{5}\dot{{\cal M}} = (g+c)\,m^{\,2},$ (11.211)
    $\displaystyle {\mit\Omega}_3$ $\displaystyle = \skew{5}\dot{F}+\skew{5}\dot{{\cal M}} = 2+(g-c)\,m^{\,2},$ (11.212)
    $\displaystyle {\mit\Omega}_4$ $\displaystyle = 2\,\skew{5}\dot{D}-\skew{5}\dot{F} = 1-2\,m-g\,m^{\,2},$ (11.213)
and   $\displaystyle {\mit\Omega}_5$ $\displaystyle = 2\,\skew{5}\dot{D}+\skew{5}\dot{F} = 3-2\,m+g\,m^{\,2}.$     (11.214)

In the following few sections, we shall develop our solution of the lunar equations of motion in a systematic fashion by considering groups of similar terms separately.


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Next: Variation Up: Lunar motion Previous: Useful definitions
Richard Fitzpatrick 2016-03-31