## Three Laws

Kepler had predicted that with Tycho's data, he would solve the problem of planetary motion in a matter of days. After almost eight years of study, trial, and error, Kepler had a stroke of genius. He concluded that the planets must orbit the sun not in perfect circles, but in elliptical orbits (an ellipse is a flattened circle). He wrote to a friend: "I have the answer ... The orbit of the planet is a perfect ellipse."

Kepler was able to state the fundamentals of planetary motion in three basic laws. That planets move in elliptical orbits, with the sun at one focus of the ellipse, is known as Kepler's First Law. It resolved the discrepancies in observed planetary motion that both Ptolemy and Copernicus had failed to explain adequately. Both of those great minds had been convinced that in a perfect universe, the orbits of planets had to be circular.

Star Words

An ellipse is an oblong circle drawn around two foci instead of a single center point.

Star Words

An ellipse is an oblong circle drawn around two foci instead of a single center point.

### Close Encounter

The nature of an ellipse and Kepler's Second Law are both difficult to explain, but easy to visualize. You can draw an ellipse with the aid of a couple of tacks, string, a piece of cardboard, and a pencil. Push the two tacks into a piece of cardboard and loop the string around them. The tacks are the foci of the ellipse. The closer together the foci, the less elliptical (more perfectly circular) the ellipse. To draw an ellipse, simply place the pencil inside the loop of the string, and pull it tautâ€”now, pull the pencil around the tacks all the way. Depending on the separation of the tacks, you will draw different looking ellipses.

To illustrate Kepler's Second Law, imagine the sun as being located at the focus nearest a (that is, the tack f nearest a). The pencil point is a planet, and the portion of the string connecting the pencil point to the tack nearest a is the imaginary line connecting the sun to a planet.

If you would accurately model the orbit of the planet around the sun, you would have to be able to draw the ellipse such that the pencil would sweep out equal areas in equal intervals of time. This would not be an easy trick, because you would have to draw faster when the pencil/planet was closest to the tack/sun (the position called parahelion) than when it was farthest from the sun (aphelion). To make an area swept near parahelion equal to an area swept at aphelion you have to cover more of the circumference of the ellipse nearer the sun-tack than you have to cover farther from the sun-tack.

Another apparent attribute of elliptical orbits determined Kepler's Second Law. It states that an imaginary line connecting the sun to any planet would sweep out equal areas of the ellipse in equal intervals of time. (See the preceding "Close Encounter" sidebar for an explanation of focus and for a demonstration of Kepler's Second Law.) The Second Law explained the variation in speed with which planets travel. They will move faster when they are closer to the sun. Kepler did not say why this was so, just that it was apparently so. Later minds would confront that why.

The first two of Kepler's laws were published in 1609. The third did not appear until ten years later and is slightly more complex. It states that the square of a planet's orbital period (the time needed to complete one orbit around the sun) is proportional to the cube of its semi-major axis (see the following figure and the preceding "Close Encounter" sidebar for an illustration and explanation of the semi-major axis). Since the planets' orbits, while elliptical, are very nearly circular, the semi-major axis can be considered to be a planet's average distance from the sun.

Drawing an ellipse to illustrate Kepler's Second Law. See the "Close

Encounter" sidebar on the previous page for an explanation.

(Image from the authors' collection)

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