Derivation of lunar equations of motion

Let ${\bf r}\equiv (x,\,y,\,z)$. It is helpful to define

$$x_1 =\cos(n\,t)\,x+\sin(n\,t)\,y,$$ (11.44)

$$y_1 =-\sin(n\,t)\,x+\cos(n\,t)\,y,$$ (11.45)

$$z_1 =z_1,$$ (11.46)

as well as

$$x_1' =\cos(n\,t)\,x'+\sin(n\,t)\,y',$$ (11.47)

$$y_1' = -\sin(n\,t)\,x' + \cos(n\,t)\,y',$$ (11.48)

$$z_1' =z'.$$ (11.49)

Here, $x_1$, $y_1$, $z_1$ and $x_1'$, $y_1'$, $z_1'$ are the Cartesian coordinates of the Moon (relative to the Earth) and the Sun (relative to the Earth–Moon barycenter), respectively, in a reference frame that rotates at angular velocity $n$ (i.e., the Moon's mean orbital angular velocity) about an axis perpendicular to the ecliptic plane. Note that if the lunar orbit were a circle, centered on the Earth, and lying in the ecliptic plane, then the coordinates $x_1$, $y_1$, and $z_1$ would all be independent of time. In fact, the small eccentricity of the lunar orbit, $e=0.05488$, combined with its slight inclination to the ecliptic plane, $I=5.16^\circ$, as well as the various solar perturbations, generate small-amplitude oscillations in $x_1$, $y_1$, and $z_1$ (Yoder 1995).

Equations (11.41)–(11.43) and (11.47)–(11.49) yield

$$\frac{x_1'}{a} \simeq \cos[(n-n')\,t]-\frac{3}{2}\,e'\,\cos(n\,t) +\frac{1}{2}\,e'\,\cos[(n-2\,n')\,t],$$ (11.50)

$$\frac{y_1'}{a} \simeq -\sin[(n-n')\,t]+\frac{3}{2}\,e'\,\sin(n\,t)-\frac{1}{2}\,e'\,\sin[(n-2\,n')\,t],$$ (11.51)

$$\frac{z_1'}{a'} \simeq 0.$$ (11.52)

It is also easily demonstrated that

$$\frac{r'}{a'} \simeq 1 - e'\,\cos(n'\,t).$$ (11.53)

The Cartesian components of the lunar equation of motion, (11.33), are

$$\ddot{x} + n^{\,2}\,a^{\,3}\,\frac{x}{r^{\,3}} = f_{x}+g_{x},$$ (11.54)

$$\ddot{y} + n^{\,2}\,a^{\,3}\,\frac{y}{r^{\,3}} = f_{y}+g_{y},$$ (11.55)

$$\ddot{z} + n^{\,2}\,a^{\,3}\,\frac{z}{r^{\,3}} = f_{z}+g_{z}.$$ (11.56)

Making use of Equations (11.44)–(11.46), the previous expressions transform to give

$$\ddot{x}_1-2\,n\,\dot{y}_1-n^{\,2}\,x_1 +n^{\,2}\,a^{\,3}\,\frac{x_1}{r^{\,3}} =f_{x_1} + g_{x_1}$$ (11.57)

$$\ddot{y}_1+2\,n\,\dot{x}_1-n^{\,2}\,y_1 +n^{\,2}\,a^{\,3}\,\frac{y_1}{r^{\,3}} =f_{y_1}=g_{y_1},$$ (11.58)

$$\ddot{z}_1+n^{\,2}\,a^{\,3}\,\frac{z_1}{r^{\,3}} =f_{z_1}+g_{z_1}.$$ (11.59)

Here, $f_{x_1}=\cos(n\,t)\,f_x+\sin(n\,t)\,f_y$, $f_{y_1}=-\sin(n\,t)\,f_x+\cos(n\,t)\,f_y$, $f_{z_1}=f_z$, et cetera.

It is convenient, at this stage, to adopt the following normalization scheme:

$$X = x_1/a, Y =y_1/a, Z = z_1/a,$$ (11.60)

$$X' = x_1'/a', Y' =y_1'/a', Z' = z_1'/a',$$ (11.61)

with $R=(X^{\,2}+Y^{\,2}+Z^{\,2})^{1/2}$, $R'=(X'^{\,2}+Y'^{\,2}+Z'^{\,2})^{1/2}$, and $T = n\,t$. In normalized form, Equation (11.50)–(11.53) become

$$X' \simeq \cos[(1-m)\,T] -\frac{3}{2}\,e'\,\cos T + \frac{1}{2}\,e'\,\cos[(1-2\,m)\,T],$$ (11.62)

$$Y' \simeq -\sin[(1-m)\,T] +\frac{3}{2}\,e'\,\sin T- \frac{1}{2}\,e'\,\sin[(1-2\,m)\,T],$$ (11.63)

$$Z' \simeq 0,$$ (11.64)

$$R' \simeq 1-e'\,\cos(m\,T),$$ (11.65)

whereas Equations (11.57)–(11.59) yield

$$\ddot{X} -2\,\dot{Y}-\,X+\frac{X}{R^{\,3}} = F_{X}+ G_{X},$$ (11.66)

$$\ddot{Y}+2\,\dot{X}-Y+\frac{Y}{R^{\,3}} =F_{Y}+G_{Y},$$ (11.67)

$$\ddot{Z} +\frac{Z}{R^{\,3}} = F_{Z}+G_{Z}.$$ (11.68)

Here,

$$F_{X} = m^{\,2}\left[\left(\frac{3\,X'^{\,2}}{R'^{\,5}}-\frac{1}{R'^{\,3}}\right)X +\left( \frac{3\,X'\,Y'}{R'^{\,5}}\right)Y\right],$$ (11.69)

$$F_{Y} = m^{\,2}\left[\left(\frac{3\,Y'^{\,2}}{R'^{\,5}}-\frac{1}{R'^{\,3}}\right)Y +\left( \frac{3\,X'\,Y'}{R'^{\,5}}\right)X\right],$$ (11.70)

$$F_{Z} = m^{\,2}\left(-\frac{Z}{R'^{\,3}}\right),$$ (11.71)

$$G_{X} =m^{\,2}\,\zeta\left[\left(\frac{15}{2}\,\frac{X'^{\,3}}{R'^{\,7}... ...c{Y'^{\,2}\,X'}{R'^{\,7}}-\frac{3}{2}\,\frac{X'}{R'^{\,5}}\right)Y^{\,2}\right.$$

$$\phantom{=}\left.+\left(\frac{15}{2}\,\frac{X'^{\,2}\,Y'}{R'^{\,7... ...}}\right)2\,X\,Y +\left(-\frac{3}{2}\,\frac{X'}{R'^{\,5}}\right)Z^{\,2}\right],$$ (11.72)

$$G_{Y} =m^{\,2}\,\zeta\left[\left(\frac{15}{2}\,\frac{X'^{\,2}\,Y'}{R'^{... ...\frac{Y'^{\,3}}{R'^{\,7}}-\frac{9}{2}\,\frac{Y'}{R'^{\,5}}\right)Y^{\,2}\right.$$

$$\phantom{=}\left.+\left(\frac{15}{2}\,\frac{X'\,Y'^{\,2}}{R'^{\,7... ...}}\right)2\,X\,Y +\left(-\frac{3}{2}\,\frac{Y'}{R'^{\,5}}\right)Z^{\,2}\right],$$ (11.73)

$$G_{Z} = m^{\,2}\,\zeta\left(\frac{3\,X'}{R'^{\,5}}\,X\,Z+ \frac{3\,Y'}{R'^{\,5}}\,Y\,Z\right),$$ (11.74)

where

$$m= \frac{n'}{n}= 0.07480$$ (11.75)

and

$$\zeta=\left(\frac{m_1-m_2}{m_1+m_2}\right)\left(\frac{a}{a'}\right)= 0.002507.$$ (11.76)

Furthermore, $F_X=f_{x_1}/(n^{\,2}\,a)$, $F_Y=f_{y_1}/(n^{\,2}\,a)$, et cetera. Finally, $\ddot{\phantom{X}}\equiv d^{\,2}/dT^{\,2}$ and $\dot{\phantom{X}}\equiv d/dT$.

Equations (11.62)–(11.65) and (11.69)–(11.71) yield

$$F_{X} \simeq m^{\,2}\left\{ \frac{1}{2}+\frac{3}{2}\,\cos[2\,(1-m)\,T]+\frac{3}{2}\,e'\,\cos(m\,T) -\frac{3}{4}\,e'\,\cos[(2-m)\,T]\right.$$

$$\phantom{=}\left.+\frac{21}{4}\,e'\,\cos[(2-3\,m)\,T]\right\}X+m^{\,2}\left\{-\frac{3}{2}\,\sin[2\,(1-m)\,T]\right.$$

$$\phantom{=}\left. +\frac{3}{4}\,e'\,\sin[(2-m)\,T]-\frac{21}{4}\,e'\,\sin[(2-3\,m)\,T]\right\}Y,$$ (11.77)

$$F_{Y} \simeq m^{\,2}\left\{ \frac{1}{2}-\frac{3}{2}\,\cos[2\,(1-m)\,T]+\frac{3}{2}\,e'\,\cos(m\,T) +\frac{3}{4}\,e'\,\cos[(2-m)\,T]\right.$$

$$\phantom{=}\left.-\frac{21}{4}\,e'\,\cos[(2-3\,m)\,T]\right\}Y+m^{\,2}\left\{-\frac{3}{2}\,\sin[2\,(1-m)\,T]\right.$$

$$\phantom{=}\left.+\frac{3}{4}\,e'\,\sin[(2-m)\,T] -\frac{21}{4}\,e'\,\sin[(2-3\,m)\,T]\right\}X,$$ (11.78)

$$F_{Z} \simeq m^{\,2}\left[-1-3\,e'\,\cos(m\,T)\right]Z.$$ (11.79)

Likewise, (11.62)–(11.65) and (11.72)–(11.74) give

$$G_{X} \simeq m^{\,2}\,\zeta \left\{\left(\frac{9}{8}\,\cos[(1-m)\,T]+\frac{15}{8}\,\cos[3\,(1-m)\,T]\right)X^{\,2}\right.$$

$$\phantom{=} + \left(\frac{3}{8}\,\cos[(1-m)\,T]-\frac{15}{8}\,\cos[3\,(1-m)\,T]\right)Y^{\,2}$$

$$\phantom{=} \left.- \left(\frac{3}{4}\,\sin[(1-m)\,T]+\frac{15}{4... ...,(1-m)\,T]\right)X\,Y -\left(\frac{3}{2}\,\cos[(1-m)\,T]\right)Z^{\,2}\right\},$$ (11.80)

$$G_{Y} \simeq m^{\,2}\,\zeta \left\{-\left(\frac{3}{8}\,\sin[(1-m)\,T]+\frac{15}{8}\,\sin[3\,(1-m)\,T]\right)X^{\,2}\right.$$

$$\phantom{=} + \left(-\frac{9}{8}\,\sin[(1-m)\,T]+\frac{15}{8}\,\sin[3\,(1-m)\,T]\right)Y^{\,2}$$

$$\phantom{=} \left.+ \left(\frac{3}{4}\,\cos[(1-m)\,T]-\frac{15}{4... ...,(1-m)\,T]\right)X\,Y +\left(\frac{3}{2}\,\sin[(1-m)\,T]\right)Z^{\,2}\right\},$$ (11.81)

$$G_{Z} \simeq m^{\,2}\,\zeta \left(-3\,\cos[(1-m)\,T]\,X\,Z + 3\,\sin[(1-m)\,T]\, Y\,Z \right).$$ (11.82)

Here, we have we have neglected terms that are third order, or greater, in the small parameters $m^{\,2}$, $e'$, and $\zeta$.

Finally, let us write

$$X =X_0 + \delta X,$$ (11.83)

$$Y = \delta Y,$$ (11.84)

$$Z = \delta Z.$$ (11.85)

Here, $X_0$ is a constant, and $\vert X_0-1\vert$, $\vert\delta X\vert$, $\vert\delta Y\vert$, $\vert\delta Z\vert\ll 1$. Expanding Equations (11.66)–(11.68) and (11.77)–(11.82), and neglecting terms that are third order, or greater, in the small parameters $m^{\,2}$, $e'$, $\zeta$, $X_0-1$, $\delta X$, $\delta Y$, and $\delta Z$, we obtain

$$\delta \ddot{X}-2\,\delta \dot{Y} - \left(2\,X_0^{\,-3}+1+\frac{1}{2}\,m^{\,2}\right)\delta X =R_X,$$ (11.86)

$$\delta \ddot{Y}+2\,\delta \dot{X} +\left(X_0^{\,-3}-1-\frac{1}{2}\,m^{\,2}\right)\delta Y =R_Y,$$ (11.87)

$$\delta\ddot{Z} + \left(X_0^{\,-3}+m^{\,2}\right)\delta Z =R_Z,$$ (11.88)

where

$$X_0^{\,-3} = 1+\frac{1}{2}\,m^{\,2},$$ (11.89)

$$R_X \simeq \frac{3}{2}\,m^{\,2}\,X_0\,\cos[2\,(1-m)\,T]$$

$$\phantom{=} +\frac{3}{2}\,m^{\,2}\,e'\,\cos(m\,T) + \frac{21}{4}\,m^{\,2}\,e'\,\cos[(2-3\,m)\,T]-\frac{3}{4}\,m^{\,2}\,e'\,\cos[(2-m)\,T]$$

$$\phantom{=} +\frac{9}{8}\,m^{\,2}\,\zeta\,\cos[(1-m)\,T] + \frac{15}{8}\,m^{\,2}\,\zeta\,\cos[3\,(1-m)\,T]$$

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

$$\phantom{=}-3\,\delta X^{\,2}+\frac{3}{2}\left(\delta Y^{\,2}+\delta Z^{\,2}\right),$$ (11.90)

$$R_Y \simeq -\frac{3}{2}\,m^{\,2}\,X_0\,\sin[2\,(1-m)\,T]$$

$$\phantom{=} - \frac{21}{4}\,m^{\,2}\,e'\,\sin[(2-3\,m)\,T]+\frac{3}{4}\,m^{\,2}\,e'\,\sin[(2-m)\,T]$$

$$\phantom{=} -\frac{3}{8}\,m^{\,2}\,\zeta\,\sin[(1-m)\,T] - \frac{15}{8}\,m^{\,2}\,\zeta\,\sin[3\,(1-m)\,T],$$ (11.91)

$$\phantom{=}-\frac{3}{2}\,m^{\,2}\,\sin[2\,(1-m)\,T]\,\delta X -\frac{3}{2}\,m^{\,2}\,\cos[2\,(1-m)\,T]\,\delta Y+3\,\delta X\,\delta Y,$$ (11.92)

$$R_Z \simeq 3\,\delta X\,\delta Z.$$ (11.93)

After Equations (11.86)–(11.93) have been solved for $X_0$, $\delta X$, $\delta Y$, and $\delta Z$, the geocentric Cartesian coordinates, ($x$, $y$, $z$), of the Moon in the non-rotating reference frame are obtained from Equations (11.44)–(11.46), (11.60)–(11.61), and (11.83)–(11.85). However, it is more convenient to write $x=r\,\cos\beta\,\cos\lambda$, $y=r\,\cos\beta\,\sin\lambda$, and $z=r\,\sin\beta$, where $r$ is the radial distance between the Earth and Moon, and $\lambda$ and $\beta$ are termed the Moon's geocentric (i.e., centered on the Earth) ecliptic longitude and ecliptic latitude, respectively. Moreover, it is easily seen that, neglecting terms that are third order, or greater, in the small parameters $X_0-1$, $\delta X$, $\delta Y$, and $\delta Z$,

$$\frac{r}{a} \simeq X_0+\delta X +\frac{1}{2}\,\delta Y^{\,2}+\frac{1}{2}\,\delta Z^{\,2},$$ (11.94)

$$\lambda \simeq n\,t+X_0^{\,-1}\,\delta Y - \delta X\,\delta Y,$$ (11.95)

$$\beta \simeq X_0^{\,-1}\,\delta Z-\delta X\,\delta Z.$$ (11.96)