Kepler's Laws of Planetary Motion/Third Law

Physical Law
Kepler's third law of planetary motion is one of the three physical laws of celestial mechanics deduced by :
 * The square of the period of the orbit of a planet around the sun is proportional to the cube of its average distance from the sun.

Proof
Consider a planet $p$ of mass $m$ orbiting a star $S$ of mass $M$ under the influence of the gravitational field which the two bodies give rise to.

From Kepler's First Law of Planetary Motion, $p$ travels in an elliptical orbit around $S$:
 * $(1): \quad r = \dfrac {h^2 / k} {1 + e \cos \theta}$

where $k = G M$.

From Equation of Ellipse in Reduced Form: Cartesian Frame, the equation of the orbit can also be given as:
 * $(2): \quad \dfrac {x^2} {a^2} + \dfrac {y^2} {b^2} = 1$

where the foci are placed at $\tuple {\pm c, 0}$.

From Focus of Ellipse from Major and Minor Axis:
 * $a^2 - b^2 = c^2$

and also:
 * $e = \dfrac c a$

Thus:


 * EllipsePolarCartesian.png

Then mean distance $a$ of $p$ from the focus $F$ is half the sum of the least and greatest values of $r$.

So $(1)$ and $(3)$ give:

Let $T$ be the orbital period of $p$.

From Area of Ellipse, the area $\mathcal A$ of the orbit is given by:
 * $\mathcal A = \pi a b$

From Kepler's Second Law of Planetary Motion it follows that:
 * $\dfrac {h T} 2 = \pi a b$

and so from $(4)$:

We have that:
 * $k = G M$

where $G$ is the gravitational constant and $M$ is the mass of $S$.

Hence the result.

Also see

 * Kepler's First Law of Planetary Motion
 * Kepler's Second Law of Planetary Motion