Solution of lunar equations of motion

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

$$X_0 \simeq 1-\frac{1}{6}\,m^{\,2}+\frac{1}{18}\,m^{\,4},$$ (11.106)

$$\delta \ddot{X}-2\,\delta \dot{Y} - 3\left(1+\frac{1}{2}\,m^{\,2}\right)\,\delta X \simeq R_X,$$ (11.107)

$$\delta \ddot{Y}+2\,\delta \dot{X} \simeq R_Y,$$ (11.108)

$$\delta\ddot{Z} + \left(1+\frac{3}{2}\,m^{\,2}\right)\,\delta Z \simeq R_Z,$$ (11.109)

where

$$R_X \simeq \frac{3}{2}\,m^{\,2}\left(1-\frac{m^{\,2}}{6}\right)\cos(2... ...}\,m^{\,2}\,e'\,\cos {\cal M}' +\frac{21}{4}\,m^{\,2}\,e'\,\cos(2\,D-{\cal M}')$$

$$\phantom{=} - \frac{3}{4}\,m^{\,2}\,e'\,\cos(2\,D+{\cal M}') + \frac{9}{8}\,m^{\,2}\,\zeta\,\cos D + \frac{15}{8}\,m^{\,2}\,\zeta\,\cos(3\,D)$$

$$\phantom{=} +\frac{3}{2}\,m^{\,2}\,\cos(2\,D)\,\delta X -\frac{3}... ...lta Y -3\,\delta X^{\,2}+\frac{3}{2}\left(\delta Y^{\,2}+\delta Z^{\,2}\right),$$ (11.110)

$$R_Y \simeq - \frac{3}{2}\,m^{\,2}\left(1-\frac{m^{\,2}}{6}\right)\sin(2\,D) -\frac{21}{4}\,m^{\,2}\,e'\,\sin(2\,D-{\cal M}')$$

$$\phantom{=}+ \frac{3}{4}\,m^{\,2}\,e'\,\sin(2\,D+{\cal M}') - \frac{3}{8}\,m^{\,2}\,\zeta\,\sin D - \frac{15}{8}\,m^{\,2}\,\zeta\,\sin(3\,D)$$

$$\phantom{=} -\frac{3}{2}\,m^{\,2}\,\sin(2\,D)\,\delta X -\frac{3}{2}\,m^{\,2}\,\cos(2\,D)\,\delta Y +3\,\delta X\,\delta Y,$$ (11.111)

$$R_Z \simeq 3\,\delta X\,\delta Z.$$ (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

$$\delta R \simeq -\frac{1}{6}\,m^{\,2} + \frac{1}{18} \,m^{\,4}+\delta X + \frac{1}{2}\,\delta Y^{\,2}+\frac{1}{2}\,\delta Z^{\,2}$$ (11.113)

$$\delta \lambda \simeq \left(1+\frac{m^{\,2}}{6}\right)\delta Y-\delta X\,\delta Y,$$ (11.114)

$$\delta \beta \simeq \left(1+\frac{m^{\,2}}{6}\right)\delta Z -\delta X\,\delta Y.$$ (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

$$\delta X = x_{01}\,m^{\,4}+ x_{02}\,e^{\,2}+x_{03}\,I^{\,2} + x_1\,e\,\cos{\cal M} +x_2\,e^{\,2}\,\cos(2\,{\cal M}) + x_3\,I^{\,2}\,\cos(2\,F)$$

$$\phantom{=} +x_4\,m^{\,2}\,\cos(2\,D) + x_5\,m^{\,4}\,\cos(4\,D) + x_6\,m\,e\,\cos(2\,D-{\cal M})$$

$$\phantom{=}+x_7\,m^{\,2}\,e\,\cos(2\,D+{\cal M})+x_8\,m\,e'\,\cos {\cal M}' + x_9\,m^{\,2}\,e'\,\cos(2\,D-{\cal M}')$$

$$\phantom{=}+x_{10}\,m^{\,2}\,e'\,\cos(2\,D+{\cal M}')+x_{11}\,m\,\zeta\,\cos D + x_{12}\,m^{\,2}\,\zeta\,\cos(3\,D),$$ (11.116)

$$\delta Y = y_1\,e\,\sin{\cal M} +y_2\,e^{\,2}\,\sin(2\,{\cal M}) + y_3\,I^{\,2}\,\sin(2\,F)$$

$$\phantom{=} +y_4\,m^{\,2}\,\sin(2\,D) + y_5\,m^{\,4}\,\sin(4\,D) + y_6\,m\,e\,\sin(2\,D-{\cal M})$$

$$\phantom{=}+y_7\,m^{\,2}\,e\,\sin(2\,D+{\cal M})+y_8\,m\,e'\,\sin {\cal M}' + y_9\,m^{\,2}\,e'\,\sin(2\,D-{\cal M}')$$

$$\phantom{=}+y_{10}\,m^{\,2}\,e'\,\sin(2\,D+{\cal M}')+y_{11}\,m\,\zeta\,\sin D + y_{12}\,m^{\,2}\,\zeta\,\sin(3\,D),$$ (11.117)

$$\delta Z = z_1\,I\,\sin F + z_2\,e\,I\,\sin (F-{\cal M}) + z_3\,e\,I\,\sin(F+{\cal M}) + z_4\,m\,I\,\sin(2\,D-F)$$

$$\phantom{=} +z_5\,m^{\,2}\,I\,\sin(2\,D+F),$$ (11.118)

$$R_X = a_{01}\,m^{\,4}+ a_{02}\,e^{\,2}+a_{03}\,I^{\,2} + a_1\,m^{\,3}\,e\,\cos{\cal M} +a_2\,e^{\,2}\,\cos(2\,{\cal M}) + a_3\,I^{\,2}\,\cos(2\,F)$$

$$\phantom{=} +a_4\,m^{\,2}\,\cos(2\,D) + a_5\,m^{\,4}\,\cos(4\,D) + a_6\,m^{\,2}\,e\,\cos(2\,D-{\cal M})$$

$$\phantom{=}+a_7\,m^{\,2}\,e\,\cos(2\,D+{\cal M})+a_8\,m^{\,2}\,e'\,\cos {\cal M}' + a_9\,m^{\,2}\,e'\,\cos(2\,D-{\cal M}')$$

$$\phantom{=}+a_{10}\,m^{\,2}\,e'\,\cos(2\,D+{\cal M}')+a_{11}\,m^{\,2}\,\zeta\,\cos D + a_{12}\,m^{\,2}\,\zeta\,\cos(3\,D),$$ (11.119)

$$R_Y = b_1\,m^{\,3}\,e\,\sin{\cal M} +b_2\,e^{\,2}\,\sin(2\,{\cal M}) + b_3\,I^{\,2}\,\sin(2\,F)$$

$$\phantom{=} +b_4\,m^{\,2}\,\sin(2\,D) + b_5\,m^{\,4}\,\sin(4\,D) + b_6\,m^{\,2}\,e\,\sin(2\,D-{\cal M})$$

$$\phantom{=}+b_7\,m^{\,2}\,e\,\sin(2\,D+{\cal M})+b_8\,m^{\,2}\,e'\,\sin {\cal M}' + b_9\,m^{\,2}\,e'\,\sin(2\,D-{\cal M}')$$

$$\phantom{=}+b_{10}\,m^{\,2}\,e'\,\sin(2\,D+{\cal M}')+b_{11}\,m^{\,2}\,\zeta\,\sin D + b_{12}\,m^{\,2}\,\zeta\,\sin(3\,D),$$ (11.120)

$$R_Z = c_1\,m^{\,3}\,I\,\sin F + c_2\,e\,I\,\sin (F-{\cal M}) + c_3\,e\,I\,\sin(F+{\cal M}) + c_4\,m^{\,2}\,I\,\sin(2\,D-F)$$

$$\phantom{=} +c_5\,m^{\,2}\,I\,\sin(2\,D+F),$$ (11.121)

$$\delta R = -\frac{1}{6}\,m^{\,2}+ r_{01}\,m^{\,4}+ r_{02}\,e^{\,2}+r_{03}\... ..._1\,e\,\cos{\cal M} +r_2\,e^{\,2}\,\cos(2\,{\cal M}) + r_3\,I^{\,2}\,\cos(2\,F)$$

$$\phantom{=} +r_4\,m^{\,2}\,\cos(2\,D) + r_5\,m^{\,4}\,\cos(4\,D) + r_6\,m\,e\,\cos(2\,D-{\cal M})$$

$$\phantom{=}+r_7\,m^{\,2}\,e\,\cos(2\,D+{\cal M})+r_8\,m\,e'\,\cos {\cal M}' + r_9\,m^{\,2}\,e'\,\cos(2\,D-{\cal M}')$$

$$\phantom{=}+r_{10}\,m^{\,2}\,e'\,\cos(2\,D+{\cal M}')+r_{11}\,m\,\zeta\,\cos D + r_{12}\,m^{\,2}\,\zeta\,\cos(3\,D),$$ (11.122)

$$\delta \lambda = l_1\,e\,\sin{\cal M} +l_2\,\,e^{\,2}\,\sin(2\,{\cal M}) + l_3\,I^{\,2}\,\sin(2\,F)$$

$$\phantom{=} +l_4\,m^{\,2}\,\sin(2\,D) + l_5\,m^{\,4}\,\sin(4\,D) + l_6\,m\,e\,\sin(2\,D-{\cal M})$$

$$\phantom{=}+l_7\,m^{\,2}\,e\,\sin(2\,D+{\cal M})+l_8\,m\,e'\,\sin {\cal M}' + l_9\,m^{\,2}\,e'\,\sin(2\,D-{\cal M}')$$

$$\phantom{=}+l_{10}\,m^{\,2}\,e'\,\sin(2\,D+{\cal M}')+l_{11}\,m\,\zeta\,\sin D + l_{12}\,m^{\,2}\,\zeta\,\sin(3\,D),$$ (11.123)

$$\delta \beta = d_1\,I\,\sin F + d_2\,e\,I\,\sin (F-{\cal M}) + d_3\,e\,I\,\sin(F+{\cal M}) + d_4\,m\,I\,\sin(2\,D-F)$$

$$\phantom{=} +d_5\,m^{\,2}\,I\,\sin(2\,D+F).$$ (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

$$a_{01} = \frac{3}{4}\,x_4-\frac{3}{4}\,y_4 -\frac{3}{2}\,x_4^{\,2} + \frac{3}{4}\,y_4^{\,2},$$ (11.125)

$$a_{02} = -\frac{3}{2}\,x_1^{\,2}+\frac{3}{4}\,y_1^{\,2},$$ (11.126)

$$a_{03} = \frac{3}{4}\,z_1^{\,2},$$ (11.127)

$$a_1 =\frac{3}{4}\,x_6 -\frac{3}{4}\,y_6-3\,x_4\,x_6+\frac{3}{2}\,y_4\,y_6,$$ (11.128)

$$a_2 = -\frac{3}{2}\,x_1^{\,2}-\frac{3}{4}\,y_1^{\,2},$$ (11.129)

$$a_3 = -\frac{3}{4}\,z_1^{\,2},$$ (11.130)

$$a_4 = \frac{3}{2}-\frac{1}{4}\,m^{\,2},$$ (11.131)

$$a_5 = \frac{3}{4}\,x_4+\frac{3}{4}\,y_4-\frac{3}{2}\,x_4^{\,2}-\frac{3}{4}\,y_4^{\,2},$$ (11.132)

$$a_6 = \frac{3}{4}\,x_1-\frac{3}{4}\,y_1-3\,x_1\,x_4+\frac{3}{2}\,y_1\,y_4,$$ (11.133)

$$a_7 = \frac{3}{4}\,x_1+\frac{3}{4}\,y_1-3\,x_1\,x_4-\frac{3}{2}\,y_1\,y_4,$$ (11.134)

$$a_8 = \frac{3}{2},$$ (11.135)

$$a_9 = \frac{21}{4},$$ (11.136)

$$a_{10} = -\frac{3}{4},$$ (11.137)

$$a_{11} = \frac{9}{8}+\frac{3}{4}\,x_{11}\,m-\frac{3}{4}\,y_{11}\,m-3\,x_4\,x_{11}\,m+\frac{3}{2}\,y_4\,y_{11}\,m,$$ (11.138)

$$a_{12} = \frac{15}{8},$$ (11.139)

as well as

$$b_1 =-\frac{3}{4}\,x_6+ \frac{3}{4}\,y_6 -\frac{3}{2}\,x_4\,y_6 +\frac{3}{2}\,y_4\,x_6,$$ (11.140)

$$b_2 = \frac{3}{2}\,x_1\,y_1,$$ (11.141)

$$b_3 = 0,$$ (11.142)

$$b_4 = -\frac{3}{2}+\frac{1}{4}\,m^{\,2},$$ (11.143)

$$b_5 =- \frac{3}{4}\,x_4-\frac{3}{4}\,y_4+\frac{3}{2}\,x_4\,y_4,$$ (11.144)

$$b_6 =- \frac{3}{4}\,x_1+\frac{3}{4}\,y_1+\frac{3}{2}\,x_1\,y_4-\frac{3}{2}\,y_1\,x_4,$$ (11.145)

$$b_7 = -\frac{3}{4}\,x_1-\frac{3}{4}\,y_1+\frac{3}{2}\,x_1\,y_4+\frac{3}{2}\,y_1\,x_4,$$ (11.146)

$$b_8 = 0,$$ (11.147)

$$b_9 =- \frac{21}{4},$$ (11.148)

$$b_{10} = \frac{3}{4},$$ (11.149)

$$b_{11} = -\frac{3}{8}-\frac{3}{4}\,x_{11}\,m+\frac{3}{4}\,y_{11}\,m-\frac{3}{2}\,x_4\,y_{11}\,m+\frac{3}{2}\,y_4\,x_{11}\,m,$$ (11.150)

$$b_{12} =- \frac{15}{8},$$ (11.151)

as well as

$$c_1 =-\frac{3}{2}\,x_4\,z_4,$$ (11.152)

$$c_2 = \frac{3}{2}\,x_1\,z_1,$$ (11.153)

$$c_3 = \frac{3}{2}\,x_1\,z_1,$$ (11.154)

$$c_4 = -\frac{3}{2}\,x_4\,z_1,$$ (11.155)

$$c_5 =\frac{3}{2}\,x_4\,z_1,$$ (11.156)

as well as

$$r_{01} = x_{01} + \frac{1}{18}+\frac{1}{4}\,y_4^{\,2},$$ (11.157)

$$r_{02} = x_{02} + \frac{1}{4}\,y_1^{\,2},$$ (11.158)

$$r_{03} = x_{03}+\frac{1}{4}\,z_1^{\,2},$$ (11.159)

$$r_1 =x_1,$$ (11.160)

$$r_2 = x_2-\frac{1}{4}\,y_1^{\,2},$$ (11.161)

$$r_3 = x_3-\frac{1}{4}\,z_1^{\,2},$$ (11.162)

$$r_4 = x_4,$$ (11.163)

$$r_5 =x_5-\frac{1}{4}\,y_4^{\,2},$$ (11.164)

$$r_6 = x_6+\frac{1}{2}\,y_1\,y_4\,m,$$ (11.165)

$$r_7 =x_7-\frac{1}{2}\,y_1\,y_4,$$ (11.166)

$$r_8 =x_8,$$ (11.167)

$$r_9 = x_9,$$ (11.168)

$$r_{10} = x_{10},$$ (11.169)

$$r_{11} = x_{11},$$ (11.170)

$$r_{12} = x_{12},$$ (11.171)

as well as

$$l_1 =\left(1+\frac{m^{\,2}}{6}\right)\,y_1,$$ (11.172)

$$l_2 = y_2-\frac{1}{2}\,x_1\,y_1,$$ (11.173)

$$l_3 = y_3,$$ (11.174)

$$l_4 =\left(1+\frac{m^{\,2}}{6}\right)\,y_4 ,$$ (11.175)

$$l_5 =y_5-\frac{1}{2}\,x_4\,y_4,$$ (11.176)

$$l_6 =y_6-\frac{1}{2}\,x_1\,y_4\,m+\frac{1}{2}\,y_1\,x_4\,m,$$ (11.177)

$$l_7 =y_7-\frac{1}{2}\,x_1\,y_4-\frac{1}{2}\,y_1\,x_4,$$ (11.178)

$$l_8 = y_8,$$ (11.179)

$$l_9 =y_9,$$ (11.180)

$$l_{10} = y_{10},$$ (11.181)

$$l_{11} = y_{11},$$ (11.182)

$$l_{12} =y_{12},$$ (11.183)

as well as

$$d_1 =\left(1+\frac{1}{6}m^{\,2}\right)\,z_1,$$ (11.184)

$$d_2 =z_2-\frac{1}{2}\,x_1\,z_1,$$ (11.185)

$$d_3 =z_3-\frac{1}{2}\,x_1\,z_1,$$ (11.186)

$$d_4 = z_4+\frac{1}{2}\,z_1\,x_4\,m,$$ (11.187)

$$d_5 =z_5-\frac{1}{2}\,z_1\,x_4.$$ (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

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

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

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

$$2\,\omega_1\,x_1+\omega_1^{\,2}\,y_1 = -m^{\,3}\,b_1,$$ (11.191)

as well as

$$x_j = \frac{m^{\,\alpha}\,(\omega_j\,a_j-2\,b_j)}{\omega_j\,(1-3\,m^{\,2}/2-\omega_j^{\,2})},$$ (11.192)

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

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

$$\omega_1 = \skew{5}\dot{{\cal M}}= 1-c\,m^{\,2},$$ (11.194)

$$\omega_2 = 2\,\skew{5}\dot{{\cal M}}= 2-2\,c\,m^{\,2},$$ (11.195)

$$\omega_3 = 2\,\skew{5}\dot{F}= 2+2\,g\,m^{\,2},$$ (11.196)

$$\omega_4 = 2\,\skew{5}\dot{D}=2-2\,m,$$ (11.197)

$$\omega_5 = 4\,\skew{5}\dot{D}=4-4\,m,$$ (11.198)

$$\omega_6 = 2\,\skew{5}\dot{D}-\skew{5}\dot{{\cal M}}=1-2\,m+c\,m^{\,2},$$ (11.199)

$$\omega_7 = 2\,\skew{5}\dot{D}+\skew{5}\dot{{\cal M}}=3-2\,m-c\,m^{\,2},$$ (11.200)

$$\omega_8 = \skew{5}\dot{{\cal M}}'=m,$$ (11.201)

$$\omega_9 = 2\,\skew{5}\dot{D}-\skew{5}\dot{{\cal M}}'=2-3\,m,$$ (11.202)

$$\omega_{10} = 2\,\skew{5}\dot{D}+\skew{5}\dot{{\cal M}}'=2-m,$$ (11.203)

$$\omega_{11} = \skew{5}\dot{D}=1-m,$$ (11.204)

$$\omega_{12} = 3\,\skew{5}\dot{D}=3-3\,m.$$ (11.205)

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

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

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

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

and

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

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

$${\mit\Omega}_1 = \skew{5}\dot{F} = 1+g\,m^{\,2},$$ (11.210)

$${\mit\Omega}_2 = \skew{5}\dot{F}-\skew{5}\dot{{\cal M}} = (g+c)\,m^{\,2},$$ (11.211)

$${\mit\Omega}_3 = \skew{5}\dot{F}+\skew{5}\dot{{\cal M}} = 2+(g-c)\,m^{\,2},$$ (11.212)

$${\mit\Omega}_4 = 2\,\skew{5}\dot{D}-\skew{5}\dot{F} = 1-2\,m-g\,m^{\,2},$$ (11.213)

$${\mit\Omega}_5 = 2\,\skew{5}\dot{D}+\skew{5}\dot{F} = 3-2\,m+g\,m^{\,2}.$$ (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.