Lunar Equations of Motion

It is convenient to solve the lunar equation of motion, (1123), in a geocentric frame of reference, $S_1$ (say), which rotates with respect to $S$ at the fixed angular velocity $\mbox{\boldmath$\omega$}$. Thus, if the lunar orbit were a circle, centered on the Earth, and lying in the ecliptic plane, then the Moon would appear stationary in $S_1$. In fact, the small eccentricity of the lunar orbit, $e=0.05488$, combined with its slight inclination to the ecliptic plane, $\iota=5.161^\circ$, causes the Moon to execute a small periodic orbit about the stationary point.

Let $x$, $y$, $z$ and $x_1$, $y_1$, $z_1$ be the Cartesian coordinates of the Moon in $S$ and $S_1$, respectively. It is easily demonstrated that (see Section A.16)

$$x = x_1\,\cos(n\,t) -y_1\,\sin(n\,t),$$ (1128)

$$y = x_1\,\sin(n\,t)+y_1\,\cos(n\,t),$$ (1129)

$$z = z_1.$$ (1130)

Moreover, if $x_1'$, $y_1'$, $z_1'$ are the Cartesian components of the Sun in $S_1$ then (see Section A.5)

$$x_1' = x'\,\cos(n\,t)+y'\,\sin(n\,t),$$ (1131)

$$y_1' = -x'\,\sin(n\,t) + y'\,\cos(n\,t),$$ (1132)

$$z_1' = z',$$ (1133)

giving

$$x_1' = a'\,\cos[(n-n')\,t],$$ (1134)

$$y_1' = -a'\,\sin[(n-n')\,t],$$ (1135)

$$z_1' = 0,$$ (1136)

where use has been made of Equations (1125)-(1127).

Now, in the rotating frame $S_1$, the lunar equation of motion (1123) transforms to (see Chapter 7)

$$\ddot{\bf r}+ 2\,\mbox{\boldmath$\omega$}\times\dot{\bf r} ... ...rt^{\,3}} -\frac{{\bf r}'}{\vert{\bf r}'\vert^{\,3}}\right],$$ (1137)

where $\dot{~}\equiv d/dt$. Furthermore, expanding the final term on the right-hand side of (1137) to lowest order in the small parameter $a/a'=0.00257$, we obtain

$$\ddot{\bf r} +2\,\mbox{\boldmath$\omega$}\times\dot{\bf r}+... ...f r}')\,{\bf r}'}{\vert{\bf r}'\vert^{\,2}} - {\bf r}\right].$$ (1138)

When written in terms of Cartesian coordinates, the above equation yields

$$\ddot{x}_1 -2\,n\,\dot{y}_1-\left(n^2+n'^{\,2}/2\right)x_1 \simeq -n^2\,a^3\,\frac{x_1}{r^3}+\frac{3}{2}\,n'^{\,2}\,\cos[2\,(n-n')\,t]\,x_1$$

$$-\frac{3}{2}\,n'^{\,2}\, \sin[2\,(n-n')\,t]\,y_1,$$ (1139)

$$\ddot{y}_1+2\,n\,\dot{x}_1-\left(n^2+n'^{\,2}/2\right)y_1 \simeq -n^2\,a^3\,\frac{y_1}{r^3}-\frac{3}{2}\,n'^{\,2}\,\sin[2\,(n-n')\,t]\,x_1$$

$$-\frac{3}{2}\,n'^{\,2}\, \cos[2\,(n-n')\,t]\,y_1,$$ (1140)

$$\ddot{z}_1 + n'^{\,2}\,z_1 \simeq - n^2\,a^3\,\frac{z_1}{r^3},$$ (1141)

where $r=(x_1^{\,2}+y_1^{\,2}+z_1^{\,2})^{1/2}$, and use has been made of Equations (1134)-(1136).

It is convenient, at this stage, to normalize all lengths to $a$, and all times to $n^{-1}$. Accordingly, let

$$X = x_1/a,$$ (1142)

$$Y = y_1/a,$$ (1143)

$$Z = z_1/a,$$ (1144)

and $r/a=R=(X^2+Y^2+Z^2)^{1/2}$, and $T = n\,t$. In normalized form, Equations (1139)-(1141) become

$$\ddot{X} -2\,\dot{Y}-(1+m^2/2)\,X \simeq -\frac{X}{R^3}+\frac{3}{2}\,m^{2}\,\cos[2\,(1-m)\,T]\,X$$

$$-\frac{3}{2}\,m^{2}\, \sin[2\,(1-m)\,T]\,Y,$$ (1145)

$$\ddot{Y}+2\,\dot{X}-(1+m^2/2)\,Y \simeq -\frac{Y}{R^3}-\frac{3}{2}\,m^{2}\,\sin[2\,(1-m)\,T]\,X$$

$$-\frac{3}{2}\,m^{2}\, \cos[2\,(1-m)\,T]\,Y,$$ (1146)

$$\ddot{Z} + m^2\,Z \simeq -\frac{Z}{R^3},$$ (1147)

respectively, where $m=n'/n=0.07480$ is a measure of the perturbing influence of the Sun on the lunar orbit. Here, $\ddot{~}\equiv d^2/dT^2$ and $\dot{~}\equiv d/dT$.

Finally, let us write

$$X = X_0 + \delta X,$$ (1148)

$$Y = \delta Y,$$ (1149)

$$Z = \delta Z,$$ (1150)

where $X_0=(1+m^2/2)^{-1/3}$, and $\vert\delta X\vert$, $\vert\delta Y\vert$, $\vert\delta Z\vert\ll X_0$. Thus, if the lunar orbit were a circle, centered on the Earth, and lying in the ecliptic plane, then, in the rotating frame $S_1$, the Moon would appear stationary at the point $(X_0$, $0,$ $0)$. Expanding Equations (1145)-(1147) to second-order in $\delta X$, $\delta Y$, $\delta Z$, and neglecting terms of order $m^4$ and $m^2\,\delta X^{\,2}$, etc., we obtain

$$\delta \ddot{X}-2\,\delta \dot{Y} - 3\,(1+m^2/2)\,\delta X \simeq \frac{3}{2}\,m^2\,\cos[2\,(1-m)\,T]+\frac{3}{2}\,m^2\,\cos[2\,(1-m)\,T]\,\delta X$$

$$-\frac{3}{2}\,m^2\,\sin[2\,(1-m)\,T]\,\delta Y-3\,\delta X^{\,2} + \frac{3}{2}\,(\delta Y^{\,2} + \delta Z^{\,2}),$$ (1151)

$$\delta \ddot{Y}+2\,\delta \dot{X} \simeq -\frac{3}{2}\,m^2\,\sin[2\,(1-m)\,T]-\frac{3}{2}\,m^2\,\sin[2\,(1-m)\,T]\,\delta X$$

$$-\frac{3}{2}\,m^2\,\cos[2\,(1-m)\,T]\,\delta Y +3\,\delta X\,\delta Y,$$ (1152)

$$\delta\ddot{Z} + (1+3\,m^2/2)\,\delta Z \simeq 3\,\delta X\,\delta Z.$$ (1153)

Now, once the above three equations have been solved for $\delta X$, $\delta Y$, and $\delta Z$, the Cartesian coordinates, $x$, $y$, $z$, of the Moon in the non-rotating geocentric frame $S$ are obtained from Equations (1128)-(1130), (1142)-(1144), and (1148)-(1150). However, it is more convenient to write $x=r\,\cos\theta$, $y=r\,\sin\theta$, and $z=r\,\sin\beta$, where $r$ is the radial distance between the Earth and Moon, and $\theta$ and $\beta$ are termed the Moon's ecliptic longitude and ecliptic latitude, respectively. Moreover, it is easily seen that, to second-order in $\delta X$, $\delta Y$, $\delta Z$, and neglecting terms of order $m^4$,

$$\frac{r}{a}-1+\frac{m^2}{6} \simeq \delta X +\frac{1}{2}\,\delta Y^{\,2}+\frac{1}{2}\,\delta Z^{\,2},$$ (1154)

$$\theta - n\,t \simeq \delta Y - \delta X\,\delta Y,$$ (1155)

$$\beta \simeq \delta Z-\delta X\,\delta Z.$$ (1156)