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 $O$.
Answer
The centre of mass of the binary system cannot move because there are no external forces acting.
The line joining the two stars must always pass through the centre of mass, because by definition the centre of mass lies on the line between the two stars.
That means the two stars must orbit with the same period. If their periods weren't the same they could not remain on opposite sides of the COM.
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