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# Kepler's third law

We have seen that the radius vector connecting our planet to the origin sweeps out area at the constant rate . [See Equation (4.24).] We have also seen that the planetary orbit is an ellipse. Now, the major and minor radii of such an ellipse are and , respectively. [See Equations (A.107) and (A.108).] The area of the ellipse is . We expect the radius vector to sweep out the whole area of the ellipse in a single orbital period, . Hence,

 (4.32)

It follows from Equation (4.31) that

 (4.33)

In other words, the square of the orbital period of our planet is proportional to the cube of its orbital major radius--this is Kepler's third law of planetary motion.

Richard Fitzpatrick 2016-03-31