Binary stars, time taken to collide

[OONeo01]

Homework Statement

A binary system consists of two stars of equal mass m orbiting each other in a circular orbit under the influence of gravitational forces. The period of the orbit is τ . At t = 0, the motion is stopped and the stars are allowed to fall towards each other. After what time t, expressed in terms of τ , do they collide?

Homework Equations

T²=4^∏(r1+r2)³/G(M1+M2)²

F=GMm/R²

The Attempt at a Solution

I tried using Kepler's law(T²=(4π²/GM²)(r1+r2)^3)
Assuming r1+r2=R is the distance between them, how do I use it in a force equation(Say like GMm/R^2) and end up getting the required time in terms of τ. (I figure there is an integration somewhere but I can't set up any justifiable equations to begin with !)

I am confused. Any help would be appreciated. Even better if somebody can actually solve this or at least give helpful mathematical hints instead of purely descriptive ones.
Thanks a lot in advance for your time :-)

 

[SteamKing]

If the stars are equal in mass, where is the center of gravity of the system?

 

[OONeo01]

If the stars are equal in mass, where is the center of gravity of the system?

Well, the center of the line joining the two stars.

 

[SteamKing]

Yes, but how far is each star from the center of gravity?

 

[tthtlc]

Conceptually, the two stars can be treated as a point mass + a star with mass = sum of the two stars. Then given the periodicity T, can can calculate the distance between the point mass and the main star -based on the centripetal force should equal the attractive gravitation pull.

And from the calculated distance between the two star, u can then know its full potential energy.

Formulate an equation of kinetic + potential energy at any point in between the original distance and when they meet, and then integrate over the full distance, and other side is integration over time. This part I am not sure, but perhaps someone can continue for me?

 

[OONeo01]

Yes, but how far is each star from the center of gravity?

If the total distance is taken to be R, then at a distance R/2. Or alternately if I take the total distance as 2R, then at a distance of R; whichever makes my equations simpler.

 

[OONeo01]

Conceptually, the two stars can be treated as a point mass + a star with mass = sum of the two stars. Then given the periodicity T, can can calculate the distance between the point mass and the main star -based on the centripetal force should equal the attractive gravitation pull.

Umm.. Point mass AND a Main star ? X-).. I'm not convinced by your method, can you set up some equations for what you have explained ?

 

[tthtlc]

well, these are all classical mechanics, and this method is also a classical assumption in physics, look up textbooks for the details, sorry about that.