Two Celestial Bodies in Orbit (Centripetal & Gravitational Forces)

So I did set the two above equations equal for the solution. But is that not in effect ignoring one of the m's in the orbital motion? I feel like it should be F_c = 2ma. How should I think about it?

Thank you.

Please show your actual work

amandela said:

I feel like it should be F_c = 2ma. How should I think about it?

What you feel is not what each of the stars "feels". Draw a free body diagram of only one of the stars. Put in the force(s) acting on it and only it. Find the net force and set it equal to its mass time its acceleration.

By the way, where are the "following" options from which you have to choose one?