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Specifically in circular orbits of two stars around the center of mass:

How can I show that the periods are equal?

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  • $\begingroup$ From en.wikipedia.org/wiki/Gravitational_two-body_problem "Thus in the differential equation the two occurrences of the reduced mass cancel each other, and we get the same differential equation as for the position of a very small body orbiting a body with a mass equal to the sum of the two masses." Does that help? $\endgroup$
    – PM 2Ring
    Jul 12, 2019 at 12:20
  • $\begingroup$ It doesn't follow from that, or does it? :) $\endgroup$
    – user192234
    Jul 12, 2019 at 12:22
  • $\begingroup$ Yes, it follows. The period of each star equals the period of that "very small body". Also take a look at the Wikipedia page about reduced mass, which explains why it's a valid procedure. $\endgroup$
    – PM 2Ring
    Jul 12, 2019 at 12:59

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Systems oscillating with two different frequencies (or two different periods) undergo a phenomenon called beating, where at times the two oscillators are moving together and at times the two oscillators are moving opposite each other. You mostly hear about beats in acoustics, where they can be audible, but it's a phenomenon that occurs with any oscillator. I usually notice it when I'm waiting to turn at a stoplight, and the turn signals on the cars in front of me move in and out of sync.

If you had two stars orbiting their center of mass with different frequencies (or periods), then some of the time the two stars would be on opposite sides of the center of mass, but at other times they would be on the same side of the center of mass. That isn't how a center of mass works --- the center of mass, by definition, is always between the two stars. So the two orbital periods must be the same.

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  • $\begingroup$ Thanks. I'm trying to prove that adding the two angular velocities is 0 or something like that. Should follow from the constant velocity of the center of mass or vice versa. It's easy to imagine moving the center of mass to the origin and how easily different periods would make it simply move. $\endgroup$
    – user192234
    Jul 12, 2019 at 14:06

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