Geometric model

It now remains to give a geometric interpretation of our Moon model, along the lines suggested by the Almagest.

Figure 55: Geometric model of the precession of the Moon's orbital plane (not to scale). Here, ${\bf l}$ is the normal to the ecliptic plane, and ${\bf n}$ the normal to the Moon's orbital plane. The ascending node lies in the direction ${\bf l}\times{\bf n}$.

Let us, first of all, consider the inclination of the Moon's orbital plane. This is fully specified by giving the normal vector to this plane, ${\bf n}$. As illustrated in Fig. 56, our geometric model has ${\bf n}$ precessing about some vector ${\bf k}$, which, in turn, precesses about the normal vector to the ecliptic plane, ${\bf l}$. Vector ${\bf k}$ has a constant inclination to vector ${\bf l}$ of $i_0=5.146^\circ$. This, of course, is the mean inclination of the Moon's orbital plane to the ecliptic. Moreover, ${\bf k}$ precesses about ${\bf l}$, in a retrograde direction, with a period of $18.6$ years. This effect gives rise to the retrograde precession of the Moon's ascending node. Vector ${\bf n}$ has a constant inclination to vector ${\bf k}$ of $i_1=0.1505^\circ$, and precesses about ${\bf k}$, in a prograde sense, with a period of $173.2$ days (i.e., half the Sun's orbital period relative to the Moon's ascending node). This effect generates the periodic oscillations in the Moon's orbital inclination, $i$, and the longitude of its ascending node, $\theta$. In fact, it is easily demonstrated, via a little geometry, that the oscillations in $i$ and $\theta$ should be in phase-quadrature. Moreover, the amplitude of the oscillation in $i$ should equal $i_1=0.1505^\circ$, whereas that of the oscillation in $\theta$ should equal $i_1/\sin i_0= 1.678^\circ$. It can be seen, from Eqs. (80) and (81), that the oscillations in $i$ and $\theta$ satisfy these requirements very well.

Figure 56: Geometric model of the Moon's orbital rotation (not to scale). Here, $C$ is the geometric center of the deferent, $E$ the Earth, $Q$ the equant, $P$ the perigee, and $M$ the Moon.

Let us, now, consider the rotation of the Moon in its instantaneous orbital plane. This plane is defined in such a fashion that the direction to the vernal equinox remains fixed. Our geometric model consists of a circular deferent, $CP$, which is given the nominal radius unity. Here, unity represents the Moon's mean orbital radius. (Unfortunately, it is not possible to determine this radius from purely angular measurements.) The Earth, $E$, is displaced a distance $e=0.0548$ from the center, $C$, of the deferent, in the direction of the perigee, $P$. The equant, $Q$, is displaced the same distance in the opposite direction. The perigee, $P$, rotates steadily around the deferent with a period of $8.85$ years. This effect gives rise to the precession of the Moon's perigee. The radius vector $QF$ rotates uniformly about the equant, $Q$, with a period of $27.3$ days. This, of course, is the Moon's mean sidereal period. The guide-point $F$ is located a distance $1.86$ along $QF$ from point $Q$. Point $F$ is the center of a circular epicycle, $FG$, of radius $0.39\,e$. The guide-point $G$ rotates uniformly about this epicycle with a period of $9.6$ days (i.e., the period associated with the sum of the angular velocities of radius vectors $QF$ and $HM$). The guide-point $H$ lies at the point of intersection of the radius vector $QG$ and the deferent $CP$. Point $H$ is the center of a second circular epicycle, $HM$, with the same radius as the first. Finally, the Moon, $M$, rotates uniformly about this epicycle with a period of $14.8$ days (i.e., half the Moon's synodic period). As is easily demonstrated, via a little geometry, the equant in our model gives rise to the so-called first anomaly of the Moon, the epicycle $HM$ to the second anomaly--which is also called evection--and the epicycle $FG$ to the third anomaly--which is also called variation. Note, finally, that all rotation in our orbital model in in a prograde sense.