# Copernicus Epicycles versus Kepler's Ellipses

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Contrary to common belief, the Sun is not precisely at the center of the Copernican universe, nor do the planets orbit around it in perfect circles. In order to describe the angular motion of the planets, Copernicus moved the Sun slightly from the center and let the planets describe oval orbits obtained as a combination of two uniform circular motions. The agreement with observations was good but not perfect. It took all Kepler's genius to understand that the data could be perfectly explained by assuming that the orbits are ellipses with the Sun at one of the foci, traveled in such a way that the line joining the Sun with the planet sweeps out equal areas in equal times.

Contributed by: Paolo Maraner (May 2018)

Partly based on a program by: Sjoerd C. de Vries

Open content licensed under CC BY-NC-SA

## Snapshots

## Details

In astronomy, we measure angles, not distances. What we observe is that planets travel different angles in equal periods of time. Consequently, if their motion were to be circular, it would not be uniform. Since ancient times, astronomers have tried to find the rule describing the variations in angular velocity of the planets.

Copernicus partially succeeded by assuming that each planet uniformly moves on a small circle, called an epicycle, whose center uniformly moves on a bigger circle, called the deferent circle, centered on the "mean Sun." While the center of the epicycle completes a counterclockwise orbit around the mean Sun, the planet completes two counterclockwise orbits around the center of the epicycle. This produces an oval orbit with the planet slowing down at the aphelion, the farthest point from the "real Sun," and accelerating when approaching the perihelion, the closest point to the real Sun, in good agreement with his observations [1].

Kepler tried to fit Tycho Brahe's most accurate data, starting from the physical hypothesis that there should be an inverse relationship between the distance of the planet from the Sun and its velocity. In this way, he discovered that the planetary orbits are ellipses with the Sun at one of the foci, traveled in such a way that the line joining the Sun with the planet sweeps out equal areas in equal times. His ephemeris tables, the *Rudolphine Tables*, proved so precise that they left little doubt as to the correctness of his laws [2, 3].

In this Demonstration, the eccentricity of the orbits and the variations in angular velocity have been exaggerated for illustrative reasons. The text wrapping in the Copernican diagram is generated using a program by Sjoerd C. de Vries [4].

References

[1] A. Pannekoek, "The Planetary Theory of Copernicus," *Popular Astronomy*, 56, 1948 pp. 2–13.

[2] A. Pannekoek, "The Planetary Theory of Kepler," *Popular Astronomy* 56, 1948 pp. 63–75.

[3] A. M. Lombardi, *Keplero: Una biografia scientifica*, Turin: Codice Edizioni, 2008 (in Italian).

[4] Mathematica Stack Exchange. "How Can I Wrap Text around a Circle?" (Apr 23, 2018) mathematica.stackexchange.com/questions/5719/how-can-i-wrap-text-around-a-circle/5727.

## Permanent Citation

"Copernicus Epicycles versus Kepler's Ellipses"

http://demonstrations.wolfram.com/CopernicusEpicyclesVersusKeplersEllipses/

Wolfram Demonstrations Project

Published: May 8 2018