Kepler’s Third Law

Kepler’s third law states the orbital period as a simple relationship to its semi-major axis and mass of the system. Most commonly, this relationship is stated as a proportion to the semi-major axis,

\[ P \propto a^{3/2} \]

More useful is the full expression,

\[ P = 2 \pi \sqrt{\frac{a^3}{G(m_1 + m_2)}} \]

Alternatively expressed with \(\mu = G(m_1 + m_2)\)

Proof - Using Specific Angular Momentum

Following the proof of Kepler’s 2nd law,

\[ \mathrm dA = \frac{h}{2} \mathrm dt \]

To get the period we need to solve for \(\mathrm dt\) and recall that the area of an ellipse is \(A = \pi a b\).

\[\begin{split} P = \int_A \mathrm dt = \int_A \frac{2}{h}\mathrm dA\\ P = \frac{2\pi a b}{h} \end{split}\]

Recall that \(b = \sqrt{ap}\) and \(h = \sqrt{\mu p}\) gives,

\[ P = 2 \pi \sqrt{\frac{a^3}{\mu}} \]