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Laws of Planetary Motion


Johannes Kepler (1571-1630) was a German astronomer and mathematician. After Tycho Brahe (1546-1601) settled in Prague in 1599 (then the site of the court of the German emperor Rudolf) and became court astronomer, he hired Johannes Kepler to carry out his calculations. Kepler had studied astronomy long before he met Tycho; he favored the Copernican world-view and corresponded with Galileo.

Kepler's Task

Kepler was assigned the task by Tycho Brahe to analyze the observations that Tycho had made of Mars. Tycho's observations included some very accurate measurements of the position of the planet Mars, which did not agree with either Ptolemy or Copernicus. Of all the planets, the predicted position of Mars had the largest errors and therefore posed the greatest problem. Tycho's data were the best available before the invention of the telescope. While paying Kepler for his assistance, Brahe guarded his data jealously.

Accurate Data

When Tycho died, Kepler was able to obtain Brahe's observations and attempted to puzzle them out. In 1609, the same magic year when Galileo first turned his telescope towards the heavens, Kepler caught a glimpse of what he thought might be the answer. The accuracy of the observations was good enough for Kepler to show that Mars' orbit would precisely fit an ellipse.

Shape of the Path

Johannes Kepler was the first to understand that the planets in our solar system move in ellipses, not circles. He then continued his investigations, finally arriving at three principles of planetary motion. Known as Kepler's Laws, these principles revolutionized planetary astronomy. Many years after Kepler, Sir Isaac Newton proved that all three of Kepler's Laws are a direct result of the laws of gravitation and physics which govern the forces at work between various massive bodies.

Here, then are Kepler's Three Laws of Planetary Motion:

1. Planets move in ellipses with the Sun at one focus

Circular and Elliptical Orbits Having the Same Period and Focus
Kepler's first law states "all planets move in elliptical orbits with the Sun at one focus and the other focus empty". Applied to Earth satellites, the center of the Earth becomes one focus, with the other focus empty. For circular orbits, the two foci coincide.

2. The radius vector describes equal areas in equal times

Illustrating Kepler's 2nd law: Segments AB and CD take equal times to cover.
Nick Greene
Kepler's 2nd law, the law of areas, states "the line joining the planet to the Sun sweeps over equal areas in equal time intervals". When a satellite orbits, the line joining it to the Earth sweeps over equal areas in equal periods of time. Segments AB and CD take equal times to cover. Therefore, the speed of the satellite changes, depending on its distance from the center of the Earth. Speed is greatest at the point in the orbit closest to the Earth, called perigee, and is slowest at the point farthest from the Earth, called apogee. It is important to note that the orbit followed by a satellite is not dependent on its mass.

3. Squares of periodic times are to each other as cubes of the mean distances

Kepler's Third Law: The Hohmann Transfer Orbit
Kepler's 3rd law, the law of periods, relates time required for a planet to make 1 complete trip around the Sun to its mean distance from the Sun. "For any planet, the square of its period of revolution is directly proportional to the cube of its mean distance from the Sun." Applied to Earth satellites, Kepler's 3rd law explains that the farther a satellite is from Earth, the longer it will take to complete and orbit, the greater the distance it will travel to complete an orbit, and the slower its average speed will be. One problem which can be calculated using the third law involves the question "How long does it take to reach Mars, in the most efficient orbit?" The answer is called the "Hohmann Transfer Orbit" (Wolfgang Hohmann, 1925).

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