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Binary stars, time taken to collide

  1. Feb 1, 2013 #1
    1. The problem statement, all variables and given/known data
    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?


    2. Relevant equations
    T2=4(r1+r2)3/G(M1+M2)2

    F=GMm/R2
    3. 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 :-)
     
  2. jcsd
  3. Feb 1, 2013 #2

    SteamKing

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    If the stars are equal in mass, where is the center of gravity of the system?
     
  4. Feb 1, 2013 #3
    Well, the center of the line joining the two stars.
     
  5. Feb 1, 2013 #4

    SteamKing

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    Yes, but how far is each star from the center of gravity?
     
  6. Feb 1, 2013 #5
    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?
     
  7. Feb 1, 2013 #6
    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.
     
  8. Feb 1, 2013 #7
    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 ?
     
  9. Feb 1, 2013 #8
    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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