Binary stars, time taken to collide

OONeo01
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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 :-)
 
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If the stars are equal in mass, where is the center of gravity of the system?
 
SteamKing said:
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.
 
Yes, but how far is each star from the center of gravity?
 
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?
 
SteamKing said:
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.
 
tthtlc said:
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 ?
 
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.
 
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