Two stars of masses $M$ and $2M$ are at a distance $d$ apart. They rotate about their common center of mass in circular orbits. The period of revolution $T$ is proportional to:

Solution

1. The gravitational force provides the centripetal force for each star.
2. The effective mass for the system is the reduced mass $$\begin{array}{rl}\mu & =(M\cdot 2M)/(M+2M)\\ & =2M/3\end{array}$$.
3. Using Kepler's third law for a binary system, $T^2=4{\pi }^2{d}^3/(G(M_1+M_2))$.
4. Thus $T\propto d^{3/2}$.
5. Hence the answer is (A).