Period of revolution of two double stars

[leftnes]

Homework Statement

Two double stars of the same mass as the sun rotate about their common center of mass. Their separation is 4 light years. What is their period of revolution?

Homework Equations

Lagranian = T - U = \mu\dot{r}^{2}/2 + \vec{L}^{2}/2\mu r^{2} - Gm_{1}m_{2}/r
F = ma = m\omega^{2}r = Gm_{1}m_{2}/r

The Attempt at a Solution

Tried to solve this using the orbital equation , but I'm off by a power of 10. I've also tried using F = m\omega^{2}r = Gm_{1}m_{2}/r and solving for the period using \omega = 2\pi r/T but I'm not sure where I'm going wrong. Since the question asks for the period of two double stars, does this mean that the reduced mass is \mu = (2m_{1})(2m_{2})/(2m_{1} + 2m_{2}) = 4m^{2}/4m = m since all the masses are the same? I'm assuming that two double stars means 4 separate stars acting in pairs. I'm not really sure where to go with this problem.

 

[haruspex]

F = m\omega^{2}r = Gm_{1}m_{2}/r

Aren't there a couple of things wrong with the RHS? It's dimensionally wrong for a force, no? And is r standing for the same distance each side?

 

[leftnes]

Oops, yeah.

F = m\omega^{2}r = Gm_{1}m_{2}/r^{2}

I believe? Since \omega^{2} = a/r, I substituted for acceleration and set the only acting force on the stars as their gravitational attraction towards each other. Am I missing something else?

 

[leftnes]

And assuming a circular orbit, r = .5d, where d is the separation between the stars.

 

[haruspex]

And assuming a circular orbit, r = .5d, where d is the separation between the stars.

If the separation is d, what force does each experience?