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Copernicus' geometric model of a heliocentric planetary orbit is illustrated in Fig. 20.
The planet rotates on a circular epicycle whose center moves around the sun on the eccentric circle (only
half of which is shown). The diameter is the effective major axis of the orbit, where is the geometric center of circle , and the fixed position of the sun. When is at or the planet is at its perihelion
or aphelion points, respectively. The radius of circle is the effective major radius, , of the orbit. The distance is
equal to , where is the orbit's effective eccentricity. Moreover, the radius of the epicycle is equal to .
The angle
is identified with the mean anomaly, , and increases linearly in time. In other words, as seen from , the center of
the epicycle
moves uniformly around circle in a counterclockwise direction. The angle , where is
point at which produced meets the epicycle, is equal to
the mean anomaly . In other words, the planet moves uniformly around the epicycle , in an counterclockwise direction, at twice
the speed that point moves around circle .
Finally, is the radial
distance, , of the planet from the sun, and angle is the planet's true anomaly, .
Figure 20:
A Copernican orbit.

Let us draw the straightline parallel to , and passing through point , and then complete the rectangle . Simple geometry reveals that
,
, and
. Let be drawn normal
to , and let it meet produced at point . Simple geometry reveals that ,
, and
. It follows that
, and
.
Moreover,
,
which implies that

(91) 
Now, , where is angle . However,

(92) 
Finally, expanding the previous two equations to secondorder in the small parameter , we obtain
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, , and the true anomaly, , in Copernicus' geometric model of a heliocentric planetary orbit only deviate from those in the (correct) Keplerian model to secondorder in . However, the deviation in the Ptolemaic
model is slightly smaller than that in the Copernican model. To be more exact, the maximum deviation in
is in the former model, and in the latter. On the other hand, the maximum deviation in is
in both models.
Next: The Sun
Up: Geometric Planetary Orbit Models
Previous: Model of Ptolemy
Richard Fitzpatrick
20100721