(a) Consider a binary star system in which the two stars have masses M1 and M2 and the stars move on circular orbits separated by a distance R. Derive the formula for the period of revolution.
(b) Suppose M1= 1.22M and M2= 0.64M (where M = mass of the sun) and R= 0.63 AU. Calculate the period of revolution.
Centripetal Force: F_c = (mv^2)/r
Gravitational Force: F_g = (GM1M2)/r^2
v = (w^2)r
The Attempt at a Solution
I set the gravitational force equal to the centripetal force, then used the centripetal forces of each star to solve an equation for the R and r. The overall equation so far is:
(GM1M2)/R^2 = M1(w^2)*((rM2)/(M1+M2))
Solving for w^2, I get: w^2 = (G(M1+M2)/R^3)
Now, taking the period to be T = 2∏/w, the equation for period is:
T = √((4∏^2R^3)/(G(M1+M2)))
This is not correct, and I can't seem to figure out why. I'm sure my derivation went wrong somewhere, any help would be appreciated.