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And third, Kepler found a universal relationship between the orbital properties of all the planets orbiting the sun. For each planet, the cube of the planet's distance from the sun, measured in astronomical units (AU), is equal to the square of the planet's orbital period, measured in Earth years. Jupiter, for example, is approximately 5.2 AU from the sun and its orbital period is 11.86 Earth years. So 5.2 cubed equals 11.86 squared, as predicted.
Kepler's third law, which is often called the harmonic law, is a mathematical relationship between the time it takes the planet to orbit the Sun and the distance between the planet and the Sun. The time it takes for a planet to orbit the Sun is its orbital period, which is often simply called its period. For the average distance between the planet and the Sun, Kepler used what we call the semi-major axis of the ellipse. The semi-major axis is half the major axis, which is the longest distance across the ellipse. Think of it as the longest radius of the ellipse.
Kepler's third law states that the square of the period, P, is proportional to the cube of the semi-major axis, a. In equation form Kepler expressed the third law as: P^2=ka^3. k is the proportionality constant. To Kepler it was just a number that he determined from the data. Kepler did not know why this law worked. He found it by playing with the numbers.
Newton's Form of Kepler's Third Law
Using Newton's laws it is possible to show why Kepler's third law works. For circular orbits, the centripetal force required to keep the planet moving in a circular path equals the gravitational force between the Sun and planet. For elliptical orbits, the idea is similar but a little more complex. Because the gravitational force depends on the mass it turns out that the proportionality constant in Kepler's third law involves the mass of the Sun or other object being orbited. See the figure for the equation for Newton's form of Kepler's third law. In the case of a planet and a star, the mass of the planet is negligible and can be dropped from the equation. In the case of two stars or other orbiting objects of similar mass, both masses must be included.
Read more: mechanical-physics.suite101.com...&C
Originally posted by jkrog08
reply to post by Soylent Green Is People
Good explanation, that is a common misconception of orbits. That is an interesting thought about orbiting one cm from the surface of a perfect sphere. Thanks for the addition.
PS: May I ask what is up with your screen name? lol, I have ben wondering that seeing you on the boards.