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Next: Description of Lunar Motion Up: Lunar Motion Previous: Unperturbed Lunar Motion

Perturbed Lunar Motion

The perturbed nonlinear lunar equations of motion, (1151)-(1153), take the general form
$\displaystyle \delta \ddot{X}-2\,\delta \dot{Y} - 3\,(1+m^2/2)\,\delta X$ $\textstyle \simeq$ $\displaystyle R_X,$ (1169)
$\displaystyle \delta \ddot{Y}+2\,\delta \dot{X}$ $\textstyle \simeq$ $\displaystyle R_Y,$ (1170)
$\displaystyle \delta\ddot{Z} + (1+3\,m^2/2)\,\delta Z$ $\textstyle \simeq$ $\displaystyle R_Z,$ (1171)

where
$\displaystyle R_X$ $\textstyle =$ $\displaystyle a_0 + \sum_{j>0}a_j\,\cos(\omega_j\,T-\alpha_j),$ (1172)
$\displaystyle R_Y$ $\textstyle =$ $\displaystyle \sum_{j>0} b_j\,\sin(\omega_j\,T-\alpha_j),$ (1173)
$\displaystyle R_Z$ $\textstyle =$ $\displaystyle \sum_{j>0} c_j\,\sin(\Omega_j\,T-\gamma_j).$ (1174)

Let us search for solutions of the general form
$\displaystyle \delta X$ $\textstyle =$ $\displaystyle x_0 + \sum_{j>0}x_j\,\cos(\omega_j\,T-\alpha_j),$ (1175)
$\displaystyle \delta Y$ $\textstyle =$ $\displaystyle \sum_{j>0} y_j\,\sin(\omega_j\,T-\alpha_j),$ (1176)
$\displaystyle \delta Z$ $\textstyle =$ $\displaystyle \sum_{j>0}z_j\,\sin(\Omega_j\,T-\gamma_j).$ (1177)

Substituting expressions (1172)-(1177) into Equations (1169)-(1171), it is easily demonstrated that
$\displaystyle x_0$ $\textstyle =$ $\displaystyle -\frac{a_0}{3\,(1+m^2/2)},$ (1178)
$\displaystyle x_j$ $\textstyle =$ $\displaystyle \frac{\omega_j\,a_j-2\,b_j}{\omega_j\,(1-3\,m^2/2-\omega_j^{\,2})},$ (1179)
$\displaystyle y_j$ $\textstyle =$ $\displaystyle \frac{(\omega_j^{\,2}+ 3+3\,m^2/2)\,b_j - 2\,\omega_j\,a_j}{\omega_j^{\,2}\,(1-3\,m^2/2-\omega_j^{\,2})},$ (1180)
$\displaystyle z_j$ $\textstyle =$ $\displaystyle \frac{c_j}{1+3\,m^2/2-\Omega_j^{\,2}},$ (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

$\displaystyle R_X$ $\textstyle =$ $\displaystyle \frac{3}{2}\,m^2\,\cos(\omega_4\,T-\alpha_4)+\frac{3}{2}\,m^2\,\cos(\omega_4\,T-\alpha_4)\,\delta X$  
    $\displaystyle -\frac{3}{2}\,m^2\,\sin(\omega_4\,T-\alpha_4)\,\delta Y-3\,\delta X^{\,2} + \frac{3}{2}\,(\delta Y^{\,2}+\delta Z^{\,2}),$ (1182)
$\displaystyle R_Y$ $\textstyle =$ $\displaystyle - \frac{3}{2}\,m^2\,\sin(\omega_4\,T-\alpha_4)-\frac{3}{2}\,m^2\,\sin(\omega_4\,T-\alpha_4)\,\delta X$  
    $\displaystyle -\frac{3}{2}\,m^2\,\cos(\omega_4\,T-\alpha_4)\,\delta Y +3\,\delta X\,\delta Y,$ (1183)
$\displaystyle R_Z$ $\textstyle =$ $\displaystyle 3\,\delta X\,\delta Z.$ (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

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

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

$\displaystyle x_2$ $\textstyle \simeq$ $\displaystyle \frac{1}{2}\,e^2,$ (1186)
$\displaystyle y_2$ $\textstyle \simeq$ $\displaystyle \frac{1}{4}\,e^2.$ (1187)

Likewise, making the approximation $\Omega_2\simeq 0$ (see Table 3), it follows from Equation (1181) and Table 4 that
\begin{displaymath}
z_2\simeq \frac{3}{2}\,e\,\iota.
\end{displaymath} (1188)

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

$\displaystyle x_3$ $\textstyle \simeq$ $\displaystyle \frac{1}{4}\,\iota^2,$ (1189)
$\displaystyle y_3$ $\textstyle \simeq$ $\displaystyle -\frac{1}{4}\,\iota^2.$ (1190)

Likewise, making the approximation $\Omega_3\simeq 2$ (see Table 3), it follows from Equation (1181) and Table 4 that
\begin{displaymath}
z_3\simeq \frac{1}{2}\,e\,\iota.
\end{displaymath} (1191)

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

$\displaystyle x_4$ $\textstyle \simeq$ $\displaystyle -m^2,$ (1192)
$\displaystyle y_4$ $\textstyle \simeq$ $\displaystyle \frac{11}{8}\,m^2.$ (1193)

Thus, according to Table 4,
$\displaystyle a_5$ $\textstyle \simeq$ $\displaystyle -\frac{9}{8}\,m^2\,e,$ (1194)
$\displaystyle b_5$ $\textstyle \simeq$ $\displaystyle \frac{51}{16}\,m^2\,e,$ (1195)
$\displaystyle c_5$ $\textstyle \simeq$ $\displaystyle \frac{3}{2}\,m^2\,\iota.$ (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

$\displaystyle x_5$ $\textstyle \simeq$ $\displaystyle -\frac{15}{8}\,m\,e,$ (1197)
$\displaystyle y_5$ $\textstyle \simeq$ $\displaystyle \frac{15}{4}\,m\,e.$ (1198)

Likewise, making the approximation $\Omega_5\simeq 1-2\,m$ (see Table 3), it follows from Equations (1181) and (1196) that
\begin{displaymath}
z_5\simeq \frac{3}{8}\ m\,\iota.
\end{displaymath} (1199)

Thus, according to Table 4,
$\displaystyle a_1$ $\textstyle =$ $\displaystyle -\frac{135}{64}\,m^3\,e,$ (1200)
$\displaystyle b_1$ $\textstyle =$ $\displaystyle \frac{765}{128}\,m^3\,e,$ (1201)
$\displaystyle c_1$ $\textstyle =$ $\displaystyle \frac{9}{16}\,m^3\,\iota.$ (1202)

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

$\displaystyle a_1$ $\textstyle =$ $\displaystyle -e,$ (1203)
$\displaystyle b_1$ $\textstyle =$ $\displaystyle 2\,e,$ (1204)
$\displaystyle c_1$ $\textstyle =$ $\displaystyle \iota.$ (1205)

Thus, since $\omega_1=1+c\,m^2$ (see Table 3), it follows from Equations (1179), (1200), (1201), and (1203) that
\begin{displaymath}
-e \simeq \frac{ - (225/16)\,m^3\,e}{-(3/2)\,m^2-2\,c\,m^2},
\end{displaymath} (1206)

which yields
\begin{displaymath}
c \simeq- \frac{3}{4} - \frac{225}{32}\,m + {\cal O}(m^2).
\end{displaymath} (1207)

Likewise, since $\Omega_1=1+g\,m^2$ (see Table 3), it follows from Equations (1181), (1202), and (1205) that
\begin{displaymath}
\iota \simeq \frac{ (9/16)\,m^3\,\iota}{(3/2)\,m^2-2\,g\,m^2},
\end{displaymath} (1208)

which yields
\begin{displaymath}
g \simeq \frac{3}{4} + \frac{9}{32}\,m + {\cal O}(m^2).
\end{displaymath} (1209)

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

$\displaystyle \delta X$ $\textstyle =$ $\displaystyle -\frac{1}{2}\,e^2-\frac{1}{4}\,\iota^2 - e\,\cos[(1+c\,m^2)\,T-\alpha_0] + \frac{1}{2}\,e^2\,\cos[2\,(1+c\,m^2)\,T-2\,\alpha_0]$  
    $\displaystyle + \frac{1}{4}\,\iota^2\,\cos[2\,(1+g\,m^2)\,T-2\,\gamma_0] - m^2\,\cos[2\,(1-m)\,T]$  
    $\displaystyle -\frac{15}{8}\,m\,e\,\cos[(1-2\,m-c\,m^2)\,T+\alpha_0],$ (1210)
$\displaystyle \delta Y$ $\textstyle =$ $\displaystyle 2\,e\,\sin[(1+c\,m^2)\,T-\alpha_0] + \frac{1}{4}\,e^2\,\sin[2\,(1+c\,m^2)\,T-2\,\alpha_0]$  
    $\displaystyle - \frac{1}{4}\,\iota^2\,\sin[2\,(1+g\,m^2)\,T-2\,\gamma_0] + \frac{11}{8}\,m^2\,\cos[2\,(1-m)\,T]$  
    $\displaystyle +\frac{15}{4}\,m\,e\,\sin[(1-2\,m-c\,m^2)\,T+\alpha_0],$ (1211)
$\displaystyle \delta Z$ $\textstyle =$ $\displaystyle \iota\,\sin[(1+g\,m^2)\,T-\gamma_0] + \frac{3}{2}\,e\,\iota\,\sin[(c-g)\,m^2\,T-\alpha_0+\gamma_0]$  
    $\displaystyle +\frac{1}{2}\,e\,\iota\,\sin [(2+c\,m^2+g\,m^2)\,T-\alpha_0-\gamma_0]$  
    $\displaystyle +\frac{3}{8}\,m\,\iota\,\sin[(1-2\,m-g\,m^2)\,T+\gamma_0].$ (1212)

Thus, making use of Equations (1154)-(1156), we find that
$\displaystyle \frac{r}{a}$ $\textstyle =$ $\displaystyle 1 - e\,\cos[(1+c\,m^2)\,T-\alpha_0] +\frac{1}{2}\,e^2 - \frac{1}{6}\,m^2- \frac{1}{2}\,e^2\,\cos[2\,(1+c\,m^2)\,T-2\,\alpha_0]$  
    $\displaystyle - m^2\,\cos[2\,(1-m)\,T]-\frac{15}{8}\,m\,e\,\cos[(1-2\,m-c\,m^2)\,T+\alpha_0],$ (1213)
$\displaystyle \theta$ $\textstyle =$ $\displaystyle T + 2\,e\,\sin[(1+c\,m^2)\,T-\alpha_0] + \frac{5}{4}\,e^2\,\sin[2\,(1+c\,m^2)\,T-2\,\alpha_0]$  
    $\displaystyle - \frac{1}{4}\,\iota^2\,\sin[2\,(1+g\,m^2)\,T-2\,\gamma_0] + \frac{11}{8}\,m^2\,\sin[2\,(1-m)\,T]$  
    $\displaystyle +\frac{15}{4}\,m\,e\,\sin[(1-2\,m-c\,m^2)\,T+\alpha_0],$ (1214)
$\displaystyle \beta$ $\textstyle =$ $\displaystyle \iota\,\sin[(1+g\,m^2)\,T-\gamma_0] + e\,\iota\,\sin[(c-g)\,m^2\,T-\alpha_0+\gamma_0]$  
    $\displaystyle +e\,\iota\,\sin [(2+c\,m^2+g\,m^2)\,T-\alpha_0-\gamma_0]$  
    $\displaystyle +\frac{3}{8}\,m\,\iota\,\sin[(1-2\,m-g\,m^2)\,T+\gamma_0].$ (1215)

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


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Next: Description of Lunar Motion Up: Lunar Motion Previous: Unperturbed Lunar Motion
Richard Fitzpatrick 2011-03-31