Orbital inclination

Now, ecliptic longitude, $\lambda$, and latitude, $\beta$, are angular coordinates in a Cartesian system, ($x$, $y$, $z$), centered on the Earth, which is such that the $x$-$y$ plane coincides with the ecliptic plane, the $x$-axis points in the direction of the vernal equinox, and the $z$-axis points in the direction of the northern ecliptic pole (cf., Fig. 13). In fact, $x=\cos\lambda\,\cos \beta$, $y=\sin\lambda\,\cos \beta$, and $z=\sin \beta$. Here, the radial coordinate is given the default value 1, since it cannot be determined via purely angular positional data. Suppose that the plane of the Moon's orbit around the Earth is inclined at an angle $i$ to the ecliptic plane, and that the direction to the ascending node subtends an angle $\theta$ with the $x$-axis--see Fig. 13. We can transform to a new coordinate system, ($x'$, $y'$, $z'$), in which the Moon's orbit lies in the $x'$-$y'$ plane. This is achieved by first rotating anti-clockwise about the $z$-axis (looking down the axis) by an angle $\theta$, and then rotating anti-clockwise about the new $x$-axis (looking down the axis) by an angle $i$. Let $\lambda'$ and $\beta'$ be the new longitude and latitude, respectively, in the ($x'$, $y'$, $z'$) frame. By definition, $\beta'=0$. Hence, standard matrix transformation theory yields

$$\cos\lambda\,\cos\beta = \cos\lambda'\,\cos\theta - \sin\lambda'\,\cos\theta\,\cos i,$$ (71)

$$\cos\lambda\cos\beta = \cos\lambda'\,\sin\theta + \sin\lambda'\,\cos\theta\,\cos i,$$ (72)

$$\sin\beta = \sin\lambda'\,\sin i.$$ (73)

It is reasonable to assume that $\lambda'$ exhibits a similar strong secular increase in time to $\lambda$--see Fig. 43. Hence, it follows, from the above equations, that the maximum and minimum values of the Moon's latitude in the ($x$, $y$, $z$) frame are $\beta_{max} = i$ and $\beta_{min}=-i$, respectively. Moreover, when the Moon passes through the ecliptic plane ($\beta=0$) from below then its longitude satisfies $\lambda=\theta$. Likewise, when it passes through the ecliptic plane from above then $\lambda=\theta+\pi$. Thus, by carefully examining data like that shown in Figs. 43 and 44, and locating the zeros and the maximal values of the ecliptic latitude, $\beta$, we can easily determine the Moon's orbital inclination, $i$, and the longitude of its ascending node, $\theta$.

Figure 45: The inclination of the Moon's orbit to the ecliptic versus time.

Figure 46: The longitude of the ascending node of the Moon's orbit versus time.

Figures 45 and 46 show $i$ and $\theta$, determined via the method discussed above, versus time, for the years 1995-2006. It can be seen that the graph of orbital inclination, $i$, versus time consists of a constant value, plus a small amplitude periodic oscillation. On the other hand, the graph of the longitude of the ascending node versus time consists of a constant value, plus a secular decrease, plus a small amplitude periodic oscillation. It is a straight-forward task to fit curves to these two graphs. We find that

$$i(~^\circ) = 5.146 + 0.1505\,\cos(2.078\,t+118.0),$$ (74)

$$\theta(~^\circ) = 221.8 - 0.05297\,t + 1.678\,\cos(2.078\,t + 22.1).$$ (75)

Thus, the orbital inclination executes small amplitude oscillations about a mean value of $5.146^\circ$, with a period of $T_i = 360/2.078\simeq 173.2$ days. The longitude of the ascending node exhibits a secular decrease, combined with a small amplitude oscillation with the same period. The secular decrease is such as to cause the longitude of the ascending node to execute a complete retrograde (i.e., in the opposite sense to the orbital rotation) rotation in $T_\theta = 360/0.05297\simeq 6796.4$ days, which is equivalent to $18.6$ years.

Let us try to find a physical interpretation for Eqs. (74) and (75). Of course, the constant terms in these expressions require no explanation. As has already been discussed, the secular decrease in $\theta$ corresponds to a retrograde precession of the Moon's ascending node with a period of $18.6$ years. Now, it is well-known that external gravitational perturbations cause the ascending nodes and perihelions of Keplerian orbits to precess. For the case of the planets, these precession rates are sufficiently small that they can be ignored, unless one is following planetary motion over a very long period of time. However, for the case of the Moon, the external perturbation due to the Sun's gravitational influence is so large that the associated precession of the ascending node cannot be ignored. Indeed, this effect causes the Moon's ascending node to rotate by $19.3^\circ$ a year.

We still need to account for the oscillatory components of Eqs. (74) and (75). We propose that both of these terms can be written in the form

$$A\,\cos\left[2\,(\lambda_{\astrosun}-\theta-\phi)\right],$$ (76)

where $A$ is a constant amplitude, $\lambda_{\astrosun}$ the ecliptic longitude of the Sun, $\theta$ the longitude of the Moon's ascending node, and $\phi$ a constant phase-shift. It is sufficient to evaluate the above expression using the longitude of the mean Sun, and the non-oscillatory component of $\theta$: i.e.,

$$\lambda_{\astrosun}(~^\circ) \simeq -79.8 + 0.9857\,t,$$ (77)

$$\theta(~^\circ) \simeq 221.8 - 0.05297\,t.$$ (78)

Hence, expression (76) becomes

$$A\,\cos(2.078\,t + 116.8-2\,\phi).$$ (79)

Note that the frequency in the above expression exactly matches that of the oscillatory components of Eqs. (74) and (76). Thus, these equations can be rewritten in the form

$$i(~^\circ) = 5.146 + 0.1505\,\cos\left[2\,(\lambda_{\astrosun}-\theta+0.6)\right],$$ (80)

$$\theta(~^\circ) = 221.8 - 360.0\,(t/T_\omega) + 1.678\,\sin\left[2\,(\lambda_{\astrosun}-\theta-2.4)\right],$$ (81)

where $T_\omega =18.6$ years. We can now appreciate that the Sun's perturbing influence has two effects on the Moon's orbital inclination. First, it causes a retrograde precession in the ascending node with a period of $18.6$ years. Second, it causes a slight wobble in the orbital plane. The wobble executes two periods as the Sun makes a complete rotation with respect to the Moon's ascending node.

Figure 47: The ($x'$, $y'$, $z'$) and ($x''$, $y''$, $z''$) coordinate systems. Here, $E$ is the Earth, $M$ the Moon, $N$ the ascending node, and ${\vernal}$ the vernal equinox.