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How are the orbital speed of binary stars balanced

SonLight
I saw an interesting post on physics.stackexchange about two stars orbiting each other, and the way the question was written, together with the illustration, made it seem that Kepler's laws would prohibit the two stars from orbiting in the same time period. So far no one's made it clear why Kepler's third law does not apply in this case.

http://physics.stackexchange.com/questions/268631/why-is-the-period-of-rotation-the-same-for-two-stars-orbiting-the-same-centre

 Quote: Explain why the period of rotation of star AA is equal to the period of rotation of star BB. [...] By using Keppler's Third law, we know that r3r3 is directly proportional to T2T2. But in this question, we want to show that they are the same. How to approach this question?

As you can see, the questioner clearly thought of the conditions for Kepler's third being satisfied; if you look at the illustration it sort of reinforces the analogy with the solar system. I read the answers which stated that the two stars had to move equal and opposite, and agreed, but got caught in the trap of thinking the analogy of planets around the sun still held.

My argument for Kepler still being valid was that the stars both orbited the barycenter, so why wouldn't they move as if small masses around a single massive center object? So I plugged in the Earth and Jupiter, and asked why the relative mass mattered? Clearly the masses of the two objects determined where the barycenter was. Ok, so scale the masses so the barycenter was 1 AU from Earth, and 5 AU from Jupiter, and we have two planets revolving around the barycenter.

I finally realized that while we say, "The stars revolve around the barycenter" and "The planets revolve around the Sun", we do not mean the mechanically same thing in the two cases. Each star moves only due to the force of the other; the barycenter never applies a force on a moment-by-moment basis. Each star revolves about the other, but the force comes from a moving object. Only the result of the motion is about the barycenter. In the case of the orbits around the sun, the sun produces a field of force which results in all planets, regardless of their own sizes, following the Kepler rule.

For experts, yes I know there's actually a barycenter in the sun's case, but we can ignore that fact to get a very close approximation.

I hope someone can state this more clearly than I was able to. Also, I wondered if I should have used spoiler tags and treated this as a puzzle to solve.
kelseymh
SonLight wrote:
I saw an interesting post on physics.stackexchange about two stars orbiting each other, and the way the question was written, together with the illustration, made it seem that Kepler's laws would prohibit the two stars from orbiting in the same time period. So far no one's made it clear why Kepler's third law does not apply in this case.

http://physics.stackexchange.com/questions/268631/why-is-the-period-of-rotation-the-same-for-two-stars-orbiting-the-same-centre

 Quote: Explain why the period of rotation of star AA is equal to the period of rotation of star BB. [...] By using Keppler's Third law, we know that r3r3 is directly proportional to T2T2. But in this question, we want to show that they are the same. How to approach this question?

I'm very confused. Kepler's laws do apply, and in fact for a binary system, the two stars _must_ have exactly the same period! Graphically, if you draw a line between the stars, the barycenter lines along that line, and as the stars orbit, they stay connected at the end of that line (which may get longer or shorter, like a spring, but never bends).

Maybe the point isn't that Kepler's laws "don't apply," but rather that the system is so simple (central force, mutual rotation about barycenter), that you can solve everything directly from Newtonian gravity, without _needing_ Kepler's laws (which, of course, are consequences of Newton, not independent!).

Here's a nice summary of the force balancing algebra involved:

http://voyager.egglescliffe.org.uk/physics/gravitation/binary/binary.html
SonLight
Thanks, kelseymh for replying. It's clear that either I haven't understood the problem well enough yet, or didn't explain my reasoning adequately. My thinking about Kepler's laws are that they apply when there is one central mass and much smaller objects surrounding them. Doubtless there are generalizations that work in the binary star case.

Looking at the diagram in the linked question, one is tempted to think the two stars, at varying distance from the barycenter because they are of different mass, should orbit in different time intervals. Yet it is much more obvious that they will orbit in the same time interval, in order that the barycenter can stay in the same place. I don't see how to apply Kepler's laws correctly to the binary star case. Since it is a two-body problem, it is actually simpler mathematically than a planetary system.
kelseymh
 SonLight wrote: Thanks, kelseymh for replying. It's clear that either I haven't understood the problem well enough yet, or didn't explain my reasoning adequately. My thinking about Kepler's laws are that they apply when there is one central mass and much smaller objects surrounding them. Doubtless there are generalizations that work in the binary star case. Looking at the diagram in the linked question, one is tempted to think the two stars, at varying distance from the barycenter because they are of different mass, should orbit in different time intervals. Yet it is much more obvious that they will orbit in the same time interval, in order that the barycenter can stay in the same place. I don't see how to apply Kepler's laws correctly to the binary star case. Since it is a two-body problem, it is actually simpler mathematically than a planetary system.

Treat each star independently, orbiting around the barycenter. You've already deduced (above) that the barycenter is a true fixed point in the system, around which each star orbits. At that point you have exactly the same situation as with a very high mass central object.

Keep in mind that Kepler's laws are just phenomenological relationships. They don't have anything to say (directly!) about forces, or potentials, or anything else. They just give you proportionalities between the bodies in the system.