# Can You Explain Kepler's Third Law?

By means of his first two planetary motion laws, Kepler had derived formulas for the shape and the speed of a planet's orbit. The answer to another perplexing question remained: What relation is there between a planet's distance from the sun and the time it takes to complete a circuit. He knew that planets that are closer to the sun travel at greater speeds than those farther away. After nearly 10 years of labour he discovered a formula that expressed this relationship. This came to be known as his Third Law. This law states that the squares of the periods of revolutions of any two planets are in the same ratio as the cubes of their average distances from the sun.

An example of this relationship can be seen in the case of the planet Jupiter. Jupiter is approximately 5.2 times as far from the sun as is the Earth. Correspondingly, it takes Jupiter about 11.8 earth years to make one orbit around the sun (called its "period" in the chart below), which is one Jupiter year. Let us prove the accuracy of the Third Law by applying it in the case of the planet Jupiter.

To square a number is to multiply it by itself; to cube a number is to multiply this result again by the original number. So going back to the example of Jupiter, what do we find? If we square the period (Jupiter's period of orbit around the sun is 11.8 earth years), we get 11.8 times 11.8, which equals nearly 140. Now, if we cube the distance, we get 5.2 times 5.2 times 5.2, which also equals approximately 140. This equality holds true for each one of the planets. You can easily prove this for yourself by carrying out the same calculation for the rest of the planets.
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