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Vis-viva equation (Orbital Velocity) with massive satellite?

  1. Aug 28, 2014 #1
    Wikipedia states that "In the vis-viva equation the mass m of the orbiting body (e.g., a spacecraft) is taken to be negligible in comparison to the mass M of the central body."
    I'm wondering how the velocity is determined if the satellite's mass is non-negligible. For example, in a binary star system where both stars have comparable masses, would the vis-viva equation be accurate?
    Thanks.
     
  2. jcsd
  3. Aug 30, 2014 #2
    Have you found an answer to your question yet? I was waiting for someone to reply to you because I found the vis-viva equation interesting. So don't take my post as an authoritative answer.

    v is defined as the relative speed of the two bodies. If the difference in mass of the two bodies is not significant then simply adding the two masses should work.

    gif.latex?v_o=\sqrt{G(M_1+M_2)\left(\frac{2}{r}-\frac{1}{a}&space;\right&space;)}.gif

    But since I could not find a reference of the equation being used this way, I am not sure that it is correct.

    Keep in mind that the relative speed of the two bodies is not the same thing as the speed of one body about the common center of mass of the two bodies. In the latter case, the speed of one body will be different from the speed of the other body if there masses are different.
     
  4. Aug 30, 2014 #3
    Thank you! That was the problem.
    Because Body1's speed = Body2's speed * (Mass2/Mass1), and v = Body1's speed + Body2's speed, I found Body2's speed = (Mass1 * v) / (Mass1 + Mass2).
    The problem was that I thought v represented Body2's speed, not the combined speed of both.
    Thanks
     
  5. Aug 30, 2014 #4
    Glad I could help. If you write out the equation using your method, you will find that the (M1+M2) cancels out:

    gif.latex?v_1=\sqrt{GM_2\left(\frac{2}{r}-\frac{1}{a}&space;\right&space;)}.gif

    gif.latex?v_2=\sqrt{GM_1\left(\frac{2}{r}-\frac{1}{a}&space;\right&space;)}.gif

    This is the speed of each body relative to the barycentre of the two bodies.
     
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