Model of Hipparchus

Hipparchus' geometric model of the apparent orbit of the sun around the earth can also be used to describe a heliocentric planetary orbit. The model is illustrated in Fig. 18. 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 $\Pi S C A$ is the effective major axis of the orbit (to be more exact, it is the line of apsides), where $C$ is the geometric center of circle $\Pi P D A$, and $S$ the fixed position of the sun. The radius $CP$ of circle $\Pi P D A$ is the effective major radius, $a$, of the orbit. The distance $SC$ is equal to $2\,e\,a$, where $e$ is the orbit's effective eccentricity. The angle $PC\Pi$ is identified with the mean anomaly, $M$, and increases linearly in time. In other words, as seen from $C$, 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 18: A Hipparchian orbit.

Let us draw the straight-line $KSL$ parallel to $CP$, and passing through point $S$, and then complete the rectangle $PCKL$. Simple geometry reveals that $CK = PL = 2\,e\,a\,\sin M$, $KS=2\,e\,a\,\cos M$, and $SL = a-2\,e\,a\,\cos M$. Moreover, $SP^2 = SL^2+ PL^2$, which implies that

$$\frac{r}{a} = (1-4\,e\,\cos M + 4\,e^2)^{1/2}.$$ (83)

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

$$\sin q = \frac{PL}{SP} = \frac{2\,e\,\sin M}{(1-4\,e\,\cos M + 4\,e^2)^{1/2}}.$$ (84)

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

$$\frac{r}{a} = 1 -2\,e\,\cos M + 2\,e^2\,\sin^2 M,$$ (85)

$$T = M + 2\,e\,\sin M + 2\,e^2\,\sin 2M.$$ (86)

It can be seen, by comparison with Eqs. (81) and (82), that the relative radial distance, $r/a$, in the Hipparchian model deviates from that in the (correct) Keplerian model to first-order in $e$ (in fact, the variation of $r/a$ is greater by a factor of $2$ in the former model), whereas the true anomaly, $T$, only deviates to second-order in $e$. We conclude that Hipparchus' geometric model of a heliocentric planetary orbit does a reasonably good job at predicting the angular position of the planet, relative to the sun, but significantly exaggerates (by a factor of $2$) the variation in the radial distance between the two during the course of a complete orbital rotation.