Kepler's third law

Kepler's third law states that

“

The squares of the orbital periods of planets are directly proportional to the cubes of the semi-major axes.

”

To be exact,

$T^2 = \frac{4\pi^2}{G(M+m)} a^3\,$ .

so that the constant of proportionality depends on the mass of the planet. Normally $M \gg m\,$ , and we approximate

$T^2 \approx \frac{4\pi^2}{GM} a^3\,$ ,

which was the original statement of Kepler's third law.

Motivation

This can be seen in the case of circular orbits with $a = R\,$ , by identifying the centripetal force with the gravitational force

$\frac{\mu v^2}{R} = \frac{GMm}{R^2}\,$ ,

where $\mu\,$ is the reduced mass. This expression, using $v = \tfrac{2\pi R}{T}\,$ and $\mu = \tfrac{M m}{M+m}\,$ , becomes

$\frac{4\pi^2 R}{T^2}\frac{Mm}{M+m} = \frac{GMm}{R^2}\,$ .

Thus

$T^2 = \frac{4\pi^2}{G(M+m)} R^3\,$ .