# Period of revolution of two double stars

1. Nov 10, 2013

### leftnes

1. The problem statement, all variables and given/known data
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?

2. Relevant 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 = $Gm_{1}m_{2}/r$

3. 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.

2. Nov 10, 2013

### haruspex

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?

3. Nov 10, 2013

### 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?

4. Nov 10, 2013

### leftnes

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

5. Nov 10, 2013

### haruspex

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