Binary stars, time taken to collide

In summary, two stars of equal mass m in a binary system are in circular orbit with a period of τ. When the motion is stopped and the stars fall towards each other, they will collide after a time t, expressed in terms of τ. This can be calculated using Kepler's law and the concept of center of gravity, and by formulating equations for the kinetic and potential energy of the system at different points in their movement. However, the method for solving this problem is based on classical mechanics and may not be entirely convincing.
  • #1
OONeo01
18
0

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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  • #2
If the stars are equal in mass, where is the center of gravity of the system?
 
  • #3
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.
 
  • #4
Yes, but how far is each star from the center of gravity?
 
  • #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?
 
  • #6
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.
 
  • #7
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 ?
 
  • #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.
 

Related to Binary stars, time taken to collide

1. How do binary stars form?

Binary stars form from the same cloud of gas and dust that collapses under its own gravity. As the cloud collapses, it breaks into smaller clumps, which eventually become individual stars. If two of these clumps are close enough together, they can form a binary star system.

2. What determines the time it takes for binary stars to collide?

The time it takes for binary stars to collide depends on several factors, including the distance between the stars, their masses, and their orbital speed. The closer the stars are and the more massive they are, the shorter the time it takes for them to collide. The orbital speed also plays a role, as faster-moving stars will collide sooner than slower-moving ones.

3. Can binary stars collide multiple times?

Yes, binary stars can collide multiple times. After the initial collision, the stars may continue to orbit each other and eventually collide again. This process can repeat several times until the stars merge into a single, larger star or form a new binary system.

4. How do scientists detect binary stars?

Scientists can detect binary stars through a variety of methods, including observing their gravitational interactions, changes in their brightness, and their spectral lines. The most common method is through radial velocity measurements, which track the stars' movements towards and away from us as they orbit each other.

5. What happens when binary stars collide?

When binary stars collide, they release a tremendous amount of energy, causing a bright explosion known as a supernova. The explosion can also result in the formation of new elements, making binary star collisions important sources of chemical enrichment in the universe. Depending on the masses of the stars, the resulting remnant can be a neutron star, a black hole, or a more massive star.

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