Chapter 13.6, Problem 2E

To determine

To show:

Using Kepler’s Third Law that if a planet revolves around a star with period T and semimajor axis a, then the mass of the star is $M=\left(\frac{4{\pi }^2}{G}\right)\left(\frac{a^3}{T^2}\right)$

Proof:

Using Kepler’s Third Law, we can state that $T^2\propto a^3$.

A more general form of Kepler’s Third Law using Newton’s Law of Motion and Newton’s Law of gravity is:

$\Rightarrow T^2=\left(\frac{4{\pi }^2}{G\left(M+m\right)}\right)a^3$

The mass of the planet will be much larger, thus, m can be omitted from the equation to rewrite it as:

$\begin{array}{l}\Rightarrow T^2=\left(\frac{4{\pi }^2}{G\left(M\right)}\right)a^3\\ \Rightarrow T^2=\left(\frac{4{\pi }^2}{GM}\right)a^3\\ \Rightarrow M=\left(\frac{4{\pi }^2}{G}\right)\left(\frac{a^3}{T^2}\right)\end{array}$