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 = [itex]\mu\dot{r}^{2}/2[/itex] + [itex]\vec{L}^{2}/2\mu r^{2}[/itex] - [itex]Gm_{1}m_{2}/r[/itex]
F = ma = m[itex]\omega^{2}[/itex]r = [itex]Gm_{1}m_{2}/r[/itex]

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[itex]\omega^{2}[/itex]r = [itex]Gm_{1}m_{2}/r[/itex] and solving for the period using [itex]\omega[/itex] = [itex]2\pi r/T[/itex] 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 [itex]\mu[/itex] = [itex](2m_{1})(2m_{2})/(2m_{1} + 2m_{2})[/itex] = [itex]4m^{2}/4m[/itex] = 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[itex]\omega^{2}[/itex]r = [itex]Gm_{1}m_{2}/r[/itex]

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[itex]\omega^{2}[/itex]r = [itex]Gm_{1}m_{2}/r^{2}[/itex]

I believe? Since [itex]\omega^{2}[/itex] = [itex]a/r[/itex], 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?