Orbital rotation

It can be seen, from Fig. 48, that the longitude, $\lambda''$, exhibits a strong secular increase in time. We can easily separate out this effect, to obtain

$$\lambda''(~^\circ) = 49.85 + 13.176\,t + \delta\lambda'',$$ (87)

where the residual, $\delta\lambda''$, is periodic in time. We can identify $13.176$ as the mean orbital rotation rate of the Moon in degrees per day. This rotation rate corresponds to a mean orbital period of $T_{\leftmoon} = 360.0/13.176 = 27.321$ days. This is the Moon's so-called sidereal period: i.e., its orbital period with respect to the fixed stars. The synodic period, which is the mean period between successive full moons, corresponds to the Moon's mean orbital period with respect to the Sun, and is given by $T_{\leftmoon}'= (1/T_{\leftmoon}-1/T_{\astrosun})^{-1} = 29.530$ days, since $T_{\astrosun}=365.24$ days.

Figure 50: The residual in the Moon's longitude versus time.

Figure 50 shows the residual in the Moon's longitude, $\delta\lambda''$, versus time. This residual clearly consists of a superposition of oscillations with different frequencies. However, we can easily determine these frequencies by Fourier transforming $\delta\lambda''$. Let us define

$$\Lambda(n) = \frac{2}{t_{\max}}\int_0^{t_{max}} \delta\lambda''(t)\,{\rm e}^{\,{\rm i}\,n\,t}\,dt.$$ (88)

Figure 51: The modulus of the Fourier transformed residual in the Moon's longitude (degrees) versus frequency (degrees per day).

Figure 51 shows the modulus of $\Lambda$ versus the angular frequency, $n$. It can be seen that the residual is made up predominately of a combination of oscillations at three different frequencies--$11.318$, $13.065$ and $24.383$ degrees per day. The amplitudes and phases of these oscillations correspond to the amplitudes and phases of $\Lambda(n)$ evaluated at the appropriate frequencies. Hence, we obtain

$$\delta\lambda''(~^\circ) = 6.281\,\cos(13.065\,t-58.3) + 1.228\,\cos(11.318\,t-138.4)$$

$$+\, 0.665\,\cos(24.383\,t-109.9).$$ (89)

Adopting the nomenclature of the Almagest, the terms on the right-hand side of the above expression are known as the first, second, and third anomalies of the Moon, respectively. An anomaly is simply a deviation from uniform rotation. The first anomaly was discovered by Hipparchus, the second by Ptolemy, and the third was only discovered in the modern era. We now need to find physical interpretations for the three lunar anomalies.

Now, we would expect the Moon's orbit to be slightly elliptical, like those of the planets around the Sun. It, therefore, follows, by analogy with our earlier analysis of planetary orbits, that there should be a lunar anomaly of the form

$$2\,e\,\sin(\lambda''-\omega),$$ (90)

where $e$ is the ellipticity of the Moon's orbit, and $\omega$ the longitude of the perigee (i.e., the point of closest approach to the Earth)--cf., Eq. (15). The physical origin of the above expression is the fact that the Moon rotates uniformly about its equant, which is displaced a distance $2\,e$ from the Earth (normalizing the semi-major radius of the orbit to unity, for the sake of simplicity) in the direction of the apogee--see Fig. 6. It is sufficient to evaluate expression (90) using the secular component of $\lambda''$: i.e.,

$$\lambda''(~^\circ) \simeq 49.85 + 13.176\,t.$$ (91)

Since we expect to find a substantial precession in the Moon's perigee (because we already know that there is a substantial precession in its ascending node, and these two effects generally go hand in hand), we write

$$\omega = \omega_0 + n_\omega\,t,$$ (92)

where $\omega_0$ is a constant, and $n_\omega$ the precession rate. Thus, expression (90) reduces to

$$2\,e\,\cos\left[(13.176-n_\omega)\,t - 40.15 -\omega_0)\right].$$ (93)

It can be seen, from Eq. (89), that this expression matches the first anomaly provided that $e=0.0548$ (note that the amplitude must be converted into radians before calculating $e$), $n_\omega = 13.176-13.065=0.1114$, and $\omega_0=18.2^\circ$. Hence, we conclude that the Moon's orbital ellipticity is $0.0548$, and that its perigee has a prograde precession with a period of $T_\omega=360/0.1114=3231.6$ days, which is equivalent to $8.85$ years.

We propose that the second anomaly of the Moon can be written in the form

$$A\,\sin\left[2\,(\lambda'' - \lambda_{\astrosun}'') -(\lambda''-\omega)\right],$$ (94)

where $A$ is a constant amplitude, and $\lambda_{\astrosun}''\simeq \lambda_{\astrosun}-\theta_0$ the longitude of the Sun in the ($x''$, $y''$, $z''$) frame. It is sufficient to evaluate the above expression using the longitude of the mean Sun, and the secular components of $\lambda''$ and $\omega$: i.e.,

$$\lambda_{\astrosun}''(~^\circ) = 58.4 + 0.9857\,t,$$ (95)

$$\lambda''(~^\circ) = 49.85 + 13.176\,t,$$ (96)

$$\omega (~^\circ) = 18.2+0.1114\,t.$$ (97)

Hence, (94) becomes

$$A\,\cos(11.317\,t-138.8).$$ (98)

Note that the frequency and phase in the above expression match those of the second term on the right-hand side of Eq. (89) almost exactly.

We propose that the third anomaly of the Moon can be written in the form

$$A\,\sin\left[2\,(\lambda'' - \lambda_{\astrosun}'')\right].$$ (99)

It is again sufficient to evaluate this expression using longitude of the mean Sun, and the secular component of $\lambda''$. We obtain

$$A\,\cos(24.382\,t-107.1).$$ (100)

Note that the frequency and phase in the above expression match those of the third term on the right-hand side of Eq. (89) almost exactly.

We can now give a more physical description of the rotation of the Moon in the instantaneous plane of its orbit. Let $\lambda_{\astrosun}$, $\lambda_{\leftmoon}$, and $\omega$ represent the longitude of the Sun, the Moon, and the Moon's perigee, in this frame, measured relative to some convenient fixed meridian, such as that of the vernal equinox. We find that

$$\lambda_{\leftmoon} = \lambda_0 + n_{\leftmoon}\,t + 2\,e\,\sin(\lambda_{\leftmoon}-\om... ...,(\lambda_{\leftmoon}-\lambda_{\astrosun})- (\lambda_{\leftmoon}-\omega)\right]$$

$$+ \,0.21\,e\,\sin\left[2\,(\lambda_{\leftmoon}-\lambda_{\astrosun})\right],$$ (101)

where

$$\omega =\omega_0 + n_\omega\,t.$$ (102)

Here, $e=0.058$ is the eccentricity of the Moon's orbit. Moreover, $n_{\leftmoon}=2\pi/T_{\leftmoon}$ and $n_\omega = 2\pi/T_{\omega}$, where $T_{\leftmoon}=27.3$ days is the Moon's mean sidereal period, and $T_\omega = 8.85$ years is the precession frequency of the Moon's perigee. Finally, $\lambda_0$ and $\omega_0$ are constants. We can see that the perturbing influence of the Sun has two main effects on the Moon's orbital rotation. First, it induces a prograde precession in the perigee of $40.7^\circ$ a year. Second, it generates the final two terms on the right-hand side of Eq. (101). These terms describe small nonuniformities in the Moon's orbital rotation. They are smaller than the conventional nonuniformity arising from the eccentricity of the Moon's orbit (i.e., the first term on the right-hand side), but are by no means negligible. The first of these new terms is generally called evection, whilst the second is called variation. Both evection and variation describe processes by which the gravitational influence of the Sun causes the Moon to slightly speed up or slow down as it rotates about the Earth. Naturally, both effects exhibit a strong dependence on the difference in longitude between the Moon and the Sun.

Figure 52: The ecliptic latitude of the Moon versus its ecliptic longitude. The blue and red curves indicate the prediction of the updated Almagest model, and the original data, respectively.