Unperturbed Lunar Motion

Let us, first of all, neglect the perturbing influence of the Sun on the Moon's orbit by setting $m =0$ in the lunar equations of motion (1151)-(1153). For the sake of simplicity, let us also neglect nonlinear effects in these equations by setting $\delta X^2=\delta Y^2=\delta Z^2=\delta X\,\delta Y=\delta X\,\delta Z=0$. In this case, the equations reduce to

$$\delta \ddot{X}-2\,\delta \dot{Y} - 3\,\delta X \simeq 0,$$ (1157)

$$\delta \ddot{Y}+2\,\delta \dot{X} \simeq 0,$$ (1158)

$$\delta\ddot{Z} + \delta Z \simeq 0.$$ (1159)

By inspection, appropriate solutions are

$$\delta X \simeq -e\,\cos(T-\alpha_0),$$ (1160)

$$\delta Y \simeq 2\,e\,\sin(T-\alpha_0),$$ (1161)

$$\delta Z \simeq i\,\sin(T-\gamma_0),$$ (1162)

where $e$, $\alpha_0$, $i$, and $\gamma_0$ are arbitrary constants. Recalling that $T = n\,t$, it follows from (1154)-(1156) that

$$r \simeq a\left[1-e\,\cos(n\,t-\alpha_0)\right],$$ (1163)

$$\theta \simeq n\,t+2\,e\,\sin(n\,t-\alpha_0),$$ (1164)

$$\beta \simeq \iota\,\sin(n\,t-\gamma_0).$$ (1165)

However, Equations (1163) and (1164) are simply first-order (in $e$) approximations to the familiar Keplerian laws (see Chapter 5)

$$r = \frac{a\,(1-e^2)}{1+e\,\cos(\theta-\alpha_0)},$$ (1166)

$$r^2\,\dot{\theta} = (1-e^2)^{1/2}\,n\,a^2,$$ (1167)

where $\dot{~}\equiv d/dt$. Of course, the above two laws describe a body which executes an elliptical orbit, confocal with the Earth, of major radius $a$, mean angular velocity $n$, and eccentricity $e$, such that the radius vector connecting the body to the Earth sweeps out equal areas in equal time intervals. We conclude, unsurprisingly, that the unperturbed lunar orbit is a Keplerian ellipse. Note that the lunar perigee lies at the fixed ecliptic longitude $\theta=\alpha_0$. Equation (1165) is the first-order approximation to

$$\beta= \iota\,\sin(\theta-\gamma_0).$$ (1168)

This expression implies that the unperturbed lunar orbit is co-planar, but is inclined at an angle $\iota$ to the ecliptic plane. Moreover, the ascending node lies at the fixed ecliptic longitude $\theta=\gamma_0$. Incidentally, the neglect of nonlinear terms in Equations (1157)-(1159) is only valid as long as $e$, $\iota\ll 1$: i.e., provided that the unperturbed lunar orbit is only slightly elliptical, and slightly inclined to the ecliptic plane. In fact, the observed values of $e$ and $\iota$ are $0.05488$ and $0.09008$ radians, respectively, so this is indeed the case.