Binary star system distance between stars

[frostking]

Homework Statement

A binary star system has a period of 90 days. It consists of two equally massed stars each with a mass of twice that of the sun, that rotate like a dumbbell about the center of mass at the midpoint between them. How far apart are these stars?

Homework Equations

F = ma v = 2 pi r/T force of g = GMm/r^2 distance = 2r

The Attempt at a Solution

I set GMm/r^2 = m v^2/r then solve for r and d is 2 r. My problem is that my solution sheet uses force of g = GM^2/(2r)^2 and then force = to M v^2/r I know this has something to do with the fact that we have two stars and therefore, I guess M times M but I am confused... Could someone please explain why the equations used in the solution sheet are the correct ones and why in turn mine are incorrect? Thanks, Frostking

 

[Jasso]

The Attempt at a Solution

I set GMm/r^2 ...my solution sheet uses force of g = GM^2/(2r)^2

You used r^2 where the distance between them is 2r, they are orbiting about a common center of mass.

 

[ehild]

Both stars revolve around their common centre of mass which is halfway between them, at distance r, so the centripetal force is mv²/r, but the other star is 2r distance apart, so the gravitational interaction between the stars is Gm²/(2r)².

ehild