Confused about binary star systems?

In summary, the conversation discusses the motion of two stars in a binary star system, specifically their center of mass and velocity as they move closer together. The use of conservation of energy is suggested for calculating velocities and the importance of defining the reduced mass is emphasized. It is also noted that for circular orbits, the expression connecting force and velocity is only applicable for that specific case.
  • #1
21joanna12
126
2
I was thinking about the motion of two stars in a binary star system, but there is something I cannot quite figure out. Suppose you have a binary star system with two stars masses m1 and m2 with m2>m1 so that m2 is closer to the centre of mass of the system. Then when the two stars are as far away from each other as possible, their centre of mass satisfies [itex]\frac{r_1m_1 + r_2m_2}{m_1+m_2}[/itex], so at this position, the velocity of star 1 would be found by [itex]\frac{m_1v_1^2}{r_1}=\frac{Gm_1m_2}{(r_1+r_2)^2}[/itex]
But then as the two stars move closer together, both their centre of mass, and thus their distances from the centre of mass r1 and r2, and the gravitational attraction between them, change. So I can't quite figure out what their eventual motion will be...

Thank you in advance for any help :)
 
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  • #2
First of all, the expression ##F=mV^2/r## is in conflict with this sentence
21joanna12 said:
as the two stars move closer together
can you see why?

As to how to calculate velocities, best to use conservation of energy. You may also want to check the following page:
http://en.wikipedia.org/wiki/Two-body_problem
 
  • #3
Strangely, that Wiki article does not define the reduced mass ("mu") that it uses, you have to follow the link to the "center of mass frame" to get that defined. If you put those two articles together, you will see how to do 2-body problems with a central force.
 
  • #4
Oh! I see that I made a wrong assumption that the centre of mass moves, but it cannot because there is not net force ( so we put the centre of mass in the rest frame) and hence the two bodies must always be opposite each other in their orbits!
 
  • #5
Yes, but remember that formulation you used only works for the special case of circular orbits.
 
  • #6
21joanna12 said:
Oh! I see that I made a wrong assumption that the centre of mass moves, but it cannot because there is not net force ( so we put the centre of mass in the rest frame) and hence the two bodies must always be opposite each other in their orbits!
isn't this true generally? Even for elliptical, parabolic and hyperbolic orbits?
 
  • #7
Yes that much is always true. It was the expression that connects the force to v2 that is only true for circular orbits.
 
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FAQ: Confused about binary star systems?

What is a binary star system?

A binary star system is a system of two stars that are gravitationally bound to each other and orbit around a common center of mass. These stars are called binary stars because they appear as a single point of light when viewed from Earth, but they are actually two separate stars orbiting each other.

How are binary star systems formed?

Binary star systems are formed through the same process as single stars – the gravitational collapse of a cloud of gas and dust. However, in binary systems, there are two distinct regions of the cloud that collapse and form two separate stars instead of one.

What types of binary star systems exist?

There are three main types of binary star systems: visual binaries, spectroscopic binaries, and eclipsing binaries. Visual binaries can be seen with a telescope and appear as two distinct stars. Spectroscopic binaries are detected through their Doppler shift in their spectral lines. Eclipsing binaries are detected through variations in their brightness as one star passes in front of the other.

Can binary star systems support life?

It is possible for binary star systems to support life, but it depends on the distance between the stars and their characteristics. If the stars are too close together, the gravitational pull can disrupt the orbits of any planets in the system. However, if the stars are far enough apart, their combined energy could provide enough heat and light for a planet to sustain life.

Do binary star systems eventually merge into one star?

In some cases, binary star systems can merge into one star, but it is not a common occurrence. This usually happens when one star expands into a red giant and engulfs its companion star, causing them to merge. However, most binary star systems remain stable and continue to orbit each other for billions of years.

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