The deferent-epicycle system

Figure 7 illustrates the Copernican and Ptolemaic models of the Earth-Sun system. Of course, in the former model the Earth, $E$, orbits the Sun, $S$, whereas in the latter the Sun orbits the Earth. Note, however, than in both cases the Earth-Sun displacement vector, ${\bf S}$, is the same.

Figure 7: Copernican and Ptolemaic models of the Earth-Sun system. Here, $S$ is the Sun, and $E$ the Earth.

Figure 8 illustrates the Copernican and Ptolemaic models of the motion of a superior planet, $P$, as seen from the Earth, $E$. The Sun is at $S$. In the Copernican model, we can write the Earth-Planet displacement vector, ${\bf P}$, as the sum of the Earth-Sun displacement vector, ${\bf S}$, and the Sun-Planet displacement vector, ${\bf P}'$. The Ptolemaic model relies on the simple vector identity

$${\bf P} = {\bf S}+{\bf P}' \equiv{\bf P}'+{\bf S}.$$ (16)

In other words, we can get from the Earth to the planet by one of two different routes. The first route corresponds to the Copernican model, and the second to the Ptolemaic model. In the latter model, ${\bf P}'$ gives the displacement of the so-called guide-point, $G$, from the Earth. Since the length of vector ${\bf P}'$ is equal to the radius of the planetary orbit, and the vector rotates through one revolution every orbital period of the planet, it is clear that $G$ will execute a circle, whose radius is the same as that of the planetary orbit, about the Earth. Moreover, the orbital period of $G$ will be the same as that of the planet about the Sun. The circle traced out by $G$ is known as the deferent. The vector ${\bf S}$ gives the displacement of the planet from the guide-point. Since the length of ${\bf S}$ is equal to 1 AU (i.e., the mean radius of the Earth's orbit), and the vector rotates through one revolution every year, it is also clear that the planet, $P$, will execute a circle, whose radius is 1 AU, about the guide-point, $G$. Moreover, the orbital period of $P$ about $G$ will be 1 year. The circle traced out by $P$ about $G$ is known as the epicycle. Note, finally, that $G$ circles the deferent, and $P$ circles the epicycle, in the same direction that the planet circles the Sun.

Figure 8: Copernican and Ptolemaic models of the motion of a superior planet seen from Earth. Here, $S$ is the Sun, $E$ the Earth, and $P$ the planet.

The deferent-epicycle model illustrated in Fig. 8 can be applied to all three (visible) superior planets. It is easily seen that the deferent of Saturn is larger than that of Jupiter, which, in turn, is larger than that of Mars. However, the epicycles of all three planets are the same size. Moreover, the epicycle radius vectors, ${\bf S}$, of all the superior planets point in the same direction (i.e., the direction of the Sun relative to the Earth) at all times.

Figure 9: Copernican and Ptolemaic models of the motion of an inferior planet seen from Earth. See previous caption.

Figure 9 illustrates the Copernican and Ptolemaic models of the motion of an inferior planet, $P$, seen from the Earth, $E$. The Sun is at $S$. As before, in the Copernican model, the Earth-Planet displacement vector, ${\bf P}$, is the sum of the Earth-Sun displacement vector, ${\bf S}$, and the Sun-Planet displacement vector, ${\bf P}'$. In the Ptolemaic model, ${\bf S}$ gives the displacement of the guide-point, $G$, from the Earth. Since the length of vector ${\bf S}$ is equal to 1 AU, and the vector rotates through one revolution every year, it is clear that $G$ will execute a circle, whose radius is 1 AU, about the Earth. Moreover, the orbital period of $G$ will be 1 year. The circle traced out by $G$ is again known as the deferent. The vector ${\bf P}'$ gives the displacement of the planet from the guide-point. Since the length of ${\bf P}'$ is equal to the radius of the planetary orbit, and the vector rotates through one revolution every orbital period of the planet, it is also clear that the planet, $P$, will execute a circle, whose radius is the same as that of the planetary orbit, about the guide-point, $G$. Moreover, the orbital period of $P$ about $G$ will be the same as that of the planet about the Sun. The circle traced out by $P$ about $G$ is again known as the epicycle.

The deferent-epicycle model illustrated in Fig. 9 can be applied to both inferior planets. It is easily seen that Venus and Mercury share the same deferent and guide-point. Moreover, the common guide-point corresponds to the position of the Sun. However, the epicycle of Mercury is smaller than that of Venus. Furthermore, the common deferent radius vector, ${\bf S}$, points in the same direction as the epicycle radius vectors of the superior planets at all times.

The overall Ptolemaic model of the solar system (excluding the Moon) is illustrated in Fig. 10.

Figure: The Ptolemaic model of the solar system (excluding the Moon) seen from the direction of the northern ecliptic pole. (not to scale). The key to the various symbols is as follows: Earth, $\earth$; Sun, $\astrosun$; Mercury, $\mercury$; Venus, $\venus$; Mars, $\mars$; Jupiter, $\jupiter$; Saturn, $\saturn$. Note that the common deferent for Mercury and Venus should actually be the same size as the epicycles for Mars, Jupiter, and Saturn. All points rotate anti-clockwise around their respective circles with the periods shown.