Confused about binary star systems?

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Discussion Overview

The discussion revolves around the dynamics of binary star systems, specifically focusing on the motion of two stars with different masses and their center of mass. Participants explore the implications of gravitational forces and orbital mechanics in various scenarios, including circular and elliptical orbits.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant describes the motion of two stars in a binary system and expresses confusion about the changing distances from the center of mass as the stars move closer together.
  • Another participant points out a potential conflict in the use of a specific formula related to the stars' motion as they approach each other.
  • A different participant notes the lack of definition for "reduced mass" in a referenced article and suggests combining resources to better understand two-body problems with central forces.
  • One participant acknowledges a misunderstanding regarding the center of mass, asserting that it does not move due to the absence of net force, leading to the conclusion that the bodies must remain opposite each other in their orbits.
  • Another participant clarifies that the initial formulation discussed is only applicable to circular orbits.
  • A subsequent post questions whether the assertion about the center of mass being stationary holds true for other types of orbits, such as elliptical, parabolic, and hyperbolic orbits.
  • A participant confirms that the statement about the center of mass being stationary is generally true, but reiterates that the specific force-velocity relationship discussed is only valid for circular orbits.

Areas of Agreement / Disagreement

Participants generally agree on the behavior of the center of mass in a binary star system, but there is disagreement regarding the applicability of certain formulas to different types of orbits. The discussion remains unresolved on the implications of these formulas in non-circular orbits.

Contextual Notes

Some assumptions about the motion of the center of mass and the conditions under which specific formulas apply are not fully explored, leaving room for further clarification on the dynamics involved in various orbital configurations.

21joanna12
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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 \frac{r_1m_1 + r_2m_2}{m_1+m_2}, so at this position, the velocity of star 1 would be found by \frac{m_1v_1^2}{r_1}=\frac{Gm_1m_2}{(r_1+r_2)^2}
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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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
 
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.
 
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!
 
Yes, but remember that formulation you used only works for the special case of circular orbits.
 
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?
 
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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