Model of Copernicus

Copernicus' geometric model of a heliocentric planetary orbit is illustrated in Fig. 20. The planet $P$ rotates on a circular epicycle $YP$ whose center $X$ moves around the sun on the eccentric circle $\Pi X D A$ (only half of which is shown). The diameter $\Pi S C A$ is the effective major axis of the orbit, where $C$ is the geometric center of circle $\Pi X D A$, and $S$ the fixed position of the sun. When $X$ is at $\Pi$ or $A$ the planet is at its perihelion or aphelion points, respectively. The radius $CX$ of circle $\Pi X D A$ is the effective major radius, $a$, of the orbit. The distance $SC$ is equal to $(3/2)\,e\,a$, where $e$ is the orbit's effective eccentricity. Moreover, the radius $XP$ of the epicycle is equal to $(1/2)\,e\,a$. The angle $XC\Pi$ is identified with the mean anomaly, $M$, and increases linearly in time. In other words, as seen from $C$, the center of the epicycle $X$ moves uniformly around circle $\Pi X D A$ in a counterclockwise direction. The angle $PXY$, where $Y$ is point at which $CX$ produced meets the epicycle, is equal to the mean anomaly $M$. In other words, the planet $P$ moves uniformly around the epicycle $YP$, in an counterclockwise direction, at twice the speed that point $X$ moves around circle $\Pi X D A$. 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 20: A Copernican orbit.

Let us draw the straight-line $KSL$ parallel to $CX$, and passing through point $S$, and then complete the rectangle $XCKL$. Simple geometry reveals that $CK = XL = (3/2)\,e\,a\,\sin M$, $KS=(3/2)\,e\,a\,\cos M$, and $SL = a-(3/2)\,e\,a\,\cos M$. Let $PZ$ be drawn normal to $XY$, and let it meet $KSL$ produced at point $W$. Simple geometry reveals that $ZW=XL$, $ZP=(1/2)\,e\,a\,\sin M$, and $XZ =LW = (1/2)\,e\,a\,\cos M$. It follows that $WP = ZW+ZP = XL+ZP = 2\,e\,a\,\sin M$, and $SW = SL+LW=SL+XZ= a-e\,a\,\cos\,M$. Moreover, $SP^2 = SW^2+ WP^2$, which implies that

$$\frac{r}{a} = (1-2\,e\,\cos M +e^2+3\,e^2\,\sin^2 M)^{1/2}.$$ (91)

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

$$\sin q = \frac{WP}{SP} = \frac{2\,e\,\sin M}{(1-2\,e\,\cos M +e^2+3\,e^2\,\sin^2 M)^{1/2}}.$$ (92)

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

$$\frac{r}{a} = 1 -e\,\cos M + 2\,e^2\,\sin^2 M,$$ (93)

$$T = M + 2\,e\,\sin M + e^2\,\sin 2M.$$ (94)

It can be seen, by comparison with Eqs. (81)-(82) and (89)-(90), that, as is the case for Ptolemy's model, both the relative radial distance, $r/a$, and the true anomaly, $T$, in Copernicus' geometric model of a heliocentric planetary orbit only deviate from those in the (correct) Keplerian model to second-order in $e$. However, the deviation in the Ptolemaic model is slightly smaller than that in the Copernican model. To be more exact, the maximum deviation in $r/a$ is $(1/2)\,e^2$ in the former model, and $e^2$ in the latter. On the other hand, the maximum deviation in $T$ is $(1/4)\,e^2$ in both models.