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Minimum Orbital Radius

  1. Feb 19, 2015 #1
    Hi there,

    I was reading one of my textbooks and I had a thought. For a black hole, there is minimum orbiting radius of ##R_{min}=3R_s## where ##R_s## is the Schwarzschild Radius. This minimum orbit is created by the fact that in order to obtain an orbit of that radius around a black hole, you would need to be traveling at a velocity exceeding ##c##. That's all good and dandy. However, wouldn't it be true that there exists a minimum orbit for any massive body due to the the velocity limit of c? In other words, if I am in an orbit around the earth such that my velocity is equal to ##c##, what is my radius? Just something I thought up.
     
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  3. Feb 19, 2015 #2

    Chronos

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    The short answer is It's the same as the photon sphere around a black hole [hint, less than the radius of earth].
     
  4. Feb 19, 2015 #3
    When you say "photon sphere" do you mean the sphere created by a radius such that ##r=1.5R_s## (for a black hole)?

    Wait. I think I understand. If I think of it in terms of Birkhoff's theorem, if the density of a black hole were expanded to have the density of another object (say, the earth) then the photon sphere and ##R_s## would be unmoved but they would also be inside of the object. Is that kind of the hand-wavy gist of it?
     
  5. Feb 19, 2015 #4

    Chronos

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