In a binary star system, two stars $A$ and $B$ follow circular orbits, of radius $R$ and $r$ respectively, centred on their common centre of mass $O$. The mass of star $A$ is $M$, and that of star $B$ is $m$. I am having trouble with the following problem:
Explain why the period of rotation of star $A$ is equal to the period of rotation of star $B$.
By using Kepler's Third Law, we know that $$r^3\propto T^2.$$ In this question, however, we want to show that they are the same. How should I approach this question?
I only notice that the two stars are always on the straight line joining them and the centre $O$.