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# Model of Copernicus

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, . Let us draw the straight-line 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 second-order in the small parameter , we obtain   (93)   (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, , and the true anomaly, , in Copernicus' geometric model of a heliocentric planetary orbit only deviate from those in the (correct) Keplerian model to second-order 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 2010-07-21