Model of Ptolemy

Ptolemy's geometric model of the motion of the center of an epicycle around a deferent can also be used to describe a heliocentric planetary orbit. The model is illustrated in Fig. 19. The orbit of the planet corresponds to the circle $\Pi P D A$ (only half of which is shown), where $\Pi$ is the perihelion point, $P$ the planet's instantaneous position, and $A$ the aphelion point. The diameter $\Pi S C Q A$ is the effective major axis of the orbit, where $C$ is the geometric center of circle $\Pi P D A$, $S$ the fixed position of the sun, and $Q$ the location of the so-called equant. The radius $CP$ of circle $\Pi P D A$ is the effective major radius, $a$, of the orbit. The distances $SC$ and $CQ$ are both equal to $e\,a$, where $e$ is the orbit's effective eccentricity. The angle $PQ\Pi$ is identified with the mean anomaly, $M$, and increases linearly in time. In other words, as seen from $Q$, the planet $P$ moves uniformly around circle $\Pi P D A$ in a counterclockwise direction. Finally, $SP$ is the radial distance, $r$, of the planet from the sun, and angle $P S \Pi$ is the planet's true anomaly, $T$.

Figure 19: A Ptolemaic orbit.

Let us draw the straight-line $KSL$ parallel to $QP$, and passing through point $S$, and then complete the rectangle $PQKL$. Simple geometry reveals that $QK = PL = 2\,e\,a\,\sin M$, $KS=2\,e\,a\,\cos M$, and $SL = \rho-2\,e\,a\,\cos M$, where $\rho=QP$. The cosine rule applied to triangle $CQP$ yields $CP^2 = CQ^2+QP^2-2\,CQ\,QP\,\cos M$, or $\rho^2-2\,e\,a\,\cos M\,\rho -a^2\,(1-e^2)=0$, which can be solved to give $\rho/a = e\,\cos M +(1-e^2\,\sin^2 M)^{1/2}$. Moreover, $SP^2 = SL^2+ PL^2$, which implies that

$$\frac{r}{a} = [1-2\,e\,\cos M\,(1-e^2\,\sin^2 M)^{1/2}+e^2+2\,e^2\,\sin^2 M]^{1/2}.$$ (87)

Now, $T = M + q$, where $q$ is angle $PSL$. However,

$$\sin q = \frac{PL}{SP} = \frac{2\,e\,\sin M}{[1-2\,e\,\cos M\,(1-e^2\,\sin^2 M)^{1/2}+e^2+2\,e^2\,\sin^2 M]^{1/2}}.$$ (88)

Finally, expanding the previous two equations to second-order in the small parameter $e$, we obtain

$$\frac{r}{a} = 1 -e\,\cos M + (3/2)\,e^2\,\sin^2 M,$$ (89)

$$T = M + 2\,e\,\sin M + e^2\,\sin 2M.$$ (90)

It can be seen, by comparison with Eqs. (81)-(82) and (85)-(86), that Ptolemy's geometric model of a heliocentric planetary orbit is significantly more accurate than Hipparchus' model, since the relative radial distance, $r/a$, and the true anomaly, $T$, in the former model both only deviate from those in the (correct) Keplerian model to second-order in $e$.