Problem

Derive Keplar’s third law of planetary orbits for comparing the periods and orbital radius of two planets. This law is sometimes called the harmonic law.

Assume the orbits are circles.

Orbits.jpeg

Solution

The net force in orbital motion is the gravitational force

.

Since the net force is the centripetal force for planetary orbits,

.

For a planet with an orbital radius

.

For a planet with an orbital radius

.

Divide the two equations to discover that

.

Note

Most planetary orbits are elliptical, not circular. Although the derivation uses more advanced mathematics, the final result is not that different.

Here, is the semi-major axis of the ellipse. Thus the Harmonic law would be

.
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