Perturbed Lunar Motion

The perturbed nonlinear lunar equations of motion, (1151)-(1153), take the general form

$$\delta \ddot{X}-2\,\delta \dot{Y} - 3\,(1+m^2/2)\,\delta X \simeq R_X,$$ (1169)

$$\delta \ddot{Y}+2\,\delta \dot{X} \simeq R_Y,$$ (1170)

$$\delta\ddot{Z} + (1+3\,m^2/2)\,\delta Z \simeq R_Z,$$ (1171)

where

$$R_X = a_0 + \sum_{j>0}a_j\,\cos(\omega_j\,T-\alpha_j),$$ (1172)

$$R_Y = \sum_{j>0} b_j\,\sin(\omega_j\,T-\alpha_j),$$ (1173)

$$R_Z = \sum_{j>0} c_j\,\sin(\Omega_j\,T-\gamma_j).$$ (1174)

Let us search for solutions of the general form

$$\delta X = x_0 + \sum_{j>0}x_j\,\cos(\omega_j\,T-\alpha_j),$$ (1175)

$$\delta Y = \sum_{j>0} y_j\,\sin(\omega_j\,T-\alpha_j),$$ (1176)

$$\delta Z = \sum_{j>0}z_j\,\sin(\Omega_j\,T-\gamma_j).$$ (1177)

Substituting expressions (1172)-(1177) into Equations (1169)-(1171), it is easily demonstrated that

$$x_0 = -\frac{a_0}{3\,(1+m^2/2)},$$ (1178)

$$x_j = \frac{\omega_j\,a_j-2\,b_j}{\omega_j\,(1-3\,m^2/2-\omega_j^{\,2})},$$ (1179)

$$y_j = \frac{(\omega_j^{\,2}+ 3+3\,m^2/2)\,b_j - 2\,\omega_j\,a_j}{\omega_j^{\,2}\,(1-3\,m^2/2-\omega_j^{\,2})},$$ (1180)

$$z_j = \frac{c_j}{1+3\,m^2/2-\Omega_j^{\,2}},$$ (1181)

where $j>0$.

Table 3: Angular frequencies and phase-shifts associated with the principal periodic driving terms appearing in the perturbed nonlinear lunar equations of motion.

$j$	$\omega_j$	$\alpha_j$	$\Omega_j$	$\gamma_j$
$1$	$1+c\,m^2$	$\alpha_0$	$1+g\,m^2$	$\gamma_0$
$2$	$2\,(1+c\,m^2)$	$2\,\alpha_0$	$(c-g)\,m^2$	$\alpha_0-\gamma_0$
$3$	$2\,(1+g\,m^2)$	$2\,\gamma_0$	$2+(c+g)\,m^2$	$\alpha_0+\gamma_0$
$4$	$2-2\,m$	$0$
$5$	$1-2\,m-c\,m^2$	$-\alpha_0$	$1-2\,m-g\,m^2$	$-\gamma_0$

The angular frequencies, $\omega_j$, $\Omega_j$, and phase shifts, $\alpha_j$, $\gamma_j$, of the principal periodic driving terms that appear on the right-hand sides of the perturbed nonlinear lunar equations of motion, (1169)-(1171), are specified in Table 3. Here, $c$ and $g$ are, as yet, unspecified ${\cal O}(1)$ constants associated with the precession of the lunar perigee, and the regression of the ascending node, respectively. Note that $\omega_1$ and $\Omega_1$ are the frequencies of the Moon's unforced motion in ecliptic longitude and latitude, respectively. Moreover, $\omega_4$ is the forcing frequency associated with the perturbing influence of the Sun. All other frequencies appearing in Table 3 are combinations of these three fundamental frequencies. In fact, $\omega_2=2\,\omega_1$, $\omega_3=2\,\Omega_1$, $\omega_5=\omega_4-\omega_1$, $\Omega_2=\omega_1-\Omega_1$, $\Omega_3=\omega_1+\Omega_1$, and $\Omega_5=\omega_4-\Omega_1$. Note that there is no $\Omega_4$.

Now, a comparison of Equations (1151)-(1153), (1169)-(1171), and Table 1119 reveals that

$$R_X = \frac{3}{2}\,m^2\,\cos(\omega_4\,T-\alpha_4)+\frac{3}{2}\,m^2\,\cos(\omega_4\,T-\alpha_4)\,\delta X$$

$$-\frac{3}{2}\,m^2\,\sin(\omega_4\,T-\alpha_4)\,\delta Y-3\,\delta X^{\,2} + \frac{3}{2}\,(\delta Y^{\,2}+\delta Z^{\,2}),$$ (1182)

$$R_Y = - \frac{3}{2}\,m^2\,\sin(\omega_4\,T-\alpha_4)-\frac{3}{2}\,m^2\,\sin(\omega_4\,T-\alpha_4)\,\delta X$$

$$-\frac{3}{2}\,m^2\,\cos(\omega_4\,T-\alpha_4)\,\delta Y +3\,\delta X\,\delta Y,$$ (1183)

$$R_Z = 3\,\delta X\,\delta Z.$$ (1184)

Substitution of the solutions (1175)-(1177) into the above equations, followed by a comparison with expressions (1172)-(1174), yields the amplitudes $a_j$, $b_j$, and $c_j$ specified in Table 4. Note that, in calculating these amplitudes, we have neglected all contributions to the periodic driving terms, appearing in Equations (1169)-(1171) which involve cubic, or higher order, combinations of $e$, $\iota$, $m^2$, $x_j$, $y_j$, and $z_j$, since we only expanded Equations (1151)-(1153) to second-order in $\delta X$, $\delta Y$, and $\delta Z$.

Table 4: Amplitudes of the periodic driving terms appearing in the perturbed nonlinear lunar equations of motion.

$j$	$a_j$	$b_j$	$c_j$
$0$	$\frac{3}{2}\,e^2+\frac{3}{4}\,\iota^2$
$1$	$\frac{3}{4}\,m^2\,x_5 - \frac{3}{4}\,m^2\,y_5-3\,x_4\,x_5+\frac{3}{2}\,y_4\,y_5$	$-\frac{3}{4}\,m^2\,x_5 +\frac{3}{4}\,m^2\,y_5+\frac{3}{2}\,y_4\,x_5-\frac{3}{2}\,y_5\,x_4$	$-\frac{3}{2}\,x_4\,z_5$
$2$	$-\frac{9}{2}\,e^2$	$-3\,e^2$	$\frac{3}{2}\,e\,\iota$
$3$	$-\frac{3}{4}\,\iota^2$	$0$	$-\frac{3}{2}\,e\,\iota$
$4$	$\frac{3}{2}\,m^2$	$-\frac{3}{2}\,m^2$	0
$5$	$-\frac{9}{4}\,m^2\,e + 3\,e\,x_4+3\,e\,y_4$	$\frac{9}{4}\,m^2\,e-3\,e\,x_4 - \frac{3}{2}\,e\,y_4$	$-\frac{3}{2}\,\iota\,x_4$

For $j=0$, it follows from Equation (1178) and Table 4 that

$$x_0 = -\frac{1}{2}\,e^2 -\frac{1}{4}\,\iota^{\,2}.$$ (1185)

For $j=2$, making the approximation $\omega_2\simeq 2$ (see Table 3), it follows from Equations (1179), (1180) and Table 4 that

$$x_2 \simeq \frac{1}{2}\,e^2,$$ (1186)

$$y_2 \simeq \frac{1}{4}\,e^2.$$ (1187)

Likewise, making the approximation $\Omega_2\simeq 0$ (see Table 3), it follows from Equation (1181) and Table 4 that

$$z_2\simeq \frac{3}{2}\,e\,\iota.$$ (1188)

For $j=3$, making the approximation $\omega_3\simeq 2$ (see Table 3), it follows from Equations (1179), (1180) and Table 4 that

$$x_3 \simeq \frac{1}{4}\,\iota^2,$$ (1189)

$$y_3 \simeq -\frac{1}{4}\,\iota^2.$$ (1190)

Likewise, making the approximation $\Omega_3\simeq 2$ (see Table 3), it follows from Equation (1181) and Table 4 that

$$z_3\simeq \frac{1}{2}\,e\,\iota.$$ (1191)

For $j=4$, making the approximation $\omega_4\simeq 2$ (see Table 3), it follows from Equations (1179), (1180) and Table 4 that

$$x_4 \simeq -m^2,$$ (1192)

$$y_4 \simeq \frac{11}{8}\,m^2.$$ (1193)

Thus, according to Table 4,

$$a_5 \simeq -\frac{9}{8}\,m^2\,e,$$ (1194)

$$b_5 \simeq \frac{51}{16}\,m^2\,e,$$ (1195)

$$c_5 \simeq \frac{3}{2}\,m^2\,\iota.$$ (1196)

For $j=5$, making the approximation $\omega_5\simeq 1-2\,m$ (see Table 3), it follows from Equations (1179), (1180), (1194), and (1195) that

$$x_5 \simeq -\frac{15}{8}\,m\,e,$$ (1197)

$$y_5 \simeq \frac{15}{4}\,m\,e.$$ (1198)

Likewise, making the approximation $\Omega_5\simeq 1-2\,m$ (see Table 3), it follows from Equations (1181) and (1196) that

$$z_5\simeq \frac{3}{8}\ m\,\iota.$$ (1199)

Thus, according to Table 4,

$$a_1 = -\frac{135}{64}\,m^3\,e,$$ (1200)

$$b_1 = \frac{765}{128}\,m^3\,e,$$ (1201)

$$c_1 = \frac{9}{16}\,m^3\,\iota.$$ (1202)

Finally, for $j=1$, by analogy with Equations (1160)-(1162), we expect

$$a_1 = -e,$$ (1203)

$$b_1 = 2\,e,$$ (1204)

$$c_1 = \iota.$$ (1205)

Thus, since $\omega_1=1+c\,m^2$ (see Table 3), it follows from Equations (1179), (1200), (1201), and (1203) that

$$-e \simeq \frac{ - (225/16)\,m^3\,e}{-(3/2)\,m^2-2\,c\,m^2},$$ (1206)

which yields

$$c \simeq- \frac{3}{4} - \frac{225}{32}\,m + {\cal O}(m^2).$$ (1207)

Likewise, since $\Omega_1=1+g\,m^2$ (see Table 3), it follows from Equations (1181), (1202), and (1205) that

$$\iota \simeq \frac{ (9/16)\,m^3\,\iota}{(3/2)\,m^2-2\,g\,m^2},$$ (1208)

which yields

$$g \simeq \frac{3}{4} + \frac{9}{32}\,m + {\cal O}(m^2).$$ (1209)

According to the above analysis, our final expressions for $\delta X$, $\delta Y$, and $\delta Z$ are

$$\delta X = -\frac{1}{2}\,e^2-\frac{1}{4}\,\iota^2 - e\,\cos[(1+c\,m^2)\,T-\alpha_0] + \frac{1}{2}\,e^2\,\cos[2\,(1+c\,m^2)\,T-2\,\alpha_0]$$

$$+ \frac{1}{4}\,\iota^2\,\cos[2\,(1+g\,m^2)\,T-2\,\gamma_0] - m^2\,\cos[2\,(1-m)\,T]$$

$$-\frac{15}{8}\,m\,e\,\cos[(1-2\,m-c\,m^2)\,T+\alpha_0],$$ (1210)

$$\delta Y = 2\,e\,\sin[(1+c\,m^2)\,T-\alpha_0] + \frac{1}{4}\,e^2\,\sin[2\,(1+c\,m^2)\,T-2\,\alpha_0]$$

$$- \frac{1}{4}\,\iota^2\,\sin[2\,(1+g\,m^2)\,T-2\,\gamma_0] + \frac{11}{8}\,m^2\,\cos[2\,(1-m)\,T]$$

$$+\frac{15}{4}\,m\,e\,\sin[(1-2\,m-c\,m^2)\,T+\alpha_0],$$ (1211)

$$\delta Z = \iota\,\sin[(1+g\,m^2)\,T-\gamma_0] + \frac{3}{2}\,e\,\iota\,\sin[(c-g)\,m^2\,T-\alpha_0+\gamma_0]$$

$$+\frac{1}{2}\,e\,\iota\,\sin [(2+c\,m^2+g\,m^2)\,T-\alpha_0-\gamma_0]$$

$$+\frac{3}{8}\,m\,\iota\,\sin[(1-2\,m-g\,m^2)\,T+\gamma_0].$$ (1212)

Thus, making use of Equations (1154)-(1156), we find that

$$\frac{r}{a} = 1 - e\,\cos[(1+c\,m^2)\,T-\alpha_0] +\frac{1}{2}\,e^2 - \frac{1}{6}\,m^2- \frac{1}{2}\,e^2\,\cos[2\,(1+c\,m^2)\,T-2\,\alpha_0]$$

$$- m^2\,\cos[2\,(1-m)\,T]-\frac{15}{8}\,m\,e\,\cos[(1-2\,m-c\,m^2)\,T+\alpha_0],$$ (1213)

$$\theta = T + 2\,e\,\sin[(1+c\,m^2)\,T-\alpha_0] + \frac{5}{4}\,e^2\,\sin[2\,(1+c\,m^2)\,T-2\,\alpha_0]$$

$$- \frac{1}{4}\,\iota^2\,\sin[2\,(1+g\,m^2)\,T-2\,\gamma_0] + \frac{11}{8}\,m^2\,\sin[2\,(1-m)\,T]$$

$$+\frac{15}{4}\,m\,e\,\sin[(1-2\,m-c\,m^2)\,T+\alpha_0],$$ (1214)

$$\beta = \iota\,\sin[(1+g\,m^2)\,T-\gamma_0] + e\,\iota\,\sin[(c-g)\,m^2\,T-\alpha_0+\gamma_0]$$

$$+e\,\iota\,\sin [(2+c\,m^2+g\,m^2)\,T-\alpha_0-\gamma_0]$$

$$+\frac{3}{8}\,m\,\iota\,\sin[(1-2\,m-g\,m^2)\,T+\gamma_0].$$ (1215)

The above expressions are accurate up to second-order in the small parameters $e$, $\iota$, and $m$.