# Math Tutorials and Moreby George

## Math Tutorials and Moreby George

### Kepler's Laws

The model we use today is not the one introduced by Copernicus, but is essentially the one introduced later by Johannes Kepler (1571–1630). Kepler was a protege of Tycho Brahe and used Tycho's measurements to arrive at his model. One of the first things that Kepler did was to actually plot the motion of Mars as predicted by the Ptolemaic model. This motion is shown in Figure 9.

Kepler found it hard to believe that Mars would follow such a strange path with all the loops. He was attracted to the Copernican model, but he was not satisfied with the accuracy of its results. He finally arrived at a model based on the following three laws.

• Kepler's First Law: A planet moves in a plane along an elliptical orbit with the sun at one focus.
• Kepler's Second Law: The position vector from the sun to a planet sweeps out area at a constant rate.
• Kepler's Third Law: The square of the period of a planet around the sun is proportional to the cube of the semi-major axis length.

These laws are an amazing achievement. Kepler was able to give up the long standing ideas of circular orbits and uniform speed. His laws were not derived from some theory, but were obtained by carefully examining empirical data. This was the first model that didn't require epicycles to produce accurate results. It should be noted that Kepler and Galileo were contemporaries and corresponded frequently. Although Galileo was familiar with Kepler's work, he refused to give up the idea of circular orbits. Kepler's laws were later derived by Newton from his laws of motion and his law of universal gravitation. The success of Newton's laws in many areas of mechanics added additional support to Kepler's model. The derivation of Kepler's laws from Newton's laws is contained in the Appendix. The final nail in the coffin of the earth-centered view was the measurement of stellar parallax. In 1838 Friedrich Bessel made the first successful parallax measurement, for the star 61 Cygni, using a Fraunhofer heliometer at Königsberg Observatory.