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Stability of Orbits

  1. May 7, 2012 #1
    So I saw a video on youtube: http://youtu.be/uhS8K4gFu4s

    And so I thought I'd try to understand the whole stable orbit thing.

    So first you have a simple energy equation of E = K + U.

    K = (1/2)mv^2 (kinetic energy)
    U = -GMm/r^2 (gravitational energy)

    r = radius of orbit
    v = velocity of orbiting object
    M = mass of center object
    m = mass of orbiting object

    And we want it to be a stable orbit in the first place, so we have:
    v=sqrt(GM/r)

    Now, let's say we slightly bump the object in orbit. So r -> r + s, where s is much less than r.
    (also w is much less than v)

    So we can adjust the stable orbit equation:

    v=sqrt(GM/r)

    v+w = sqrt(GM/(r+s))

    approximate for small distances:

    v+w = sqrt(GM/r)(1-s/(2r))

    So then, you can subtract the original equation out and have:
    w = -sqrt(GM/r)s/(2r)
    This makes sense directionally, if you bump the orbiting object inwards, the velocity with increase.

    So now, back to the energy equation: E = K + U
    Since it's a stable orbit being bumped only slightly, you expect it to be able to eventually return to the same state, so E can't change.

    So initially you have:

    E = (1/2)mv^2 - GMm/r^2

    Then apply the bump:

    E' = ((1/2)mv^2 )(1+2w/v) - (GMm/r^2 )(1-s/r)

    This then reduces to the form:

    E' = E + mvw + (GMm/r^2 )s

    Since E' must equal E, the two extra terms should add to zero:

    E' = E + (mv)sqrt(GM/r)(-s/(2r)) + (GMm/r^2 )s

    E' = E + (-1/2)(GMm/r^2 )s + (GMm/r^2 )s

    So I am doing something wrong. I have a missing factor of two somewhere, but I don't know why.

    If you guys can help, it will be appreciated. Thanks.
     
  2. jcsd
  3. May 8, 2012 #2

    K^2

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    Science Advisor

    Stable orbit need not be circular. So if you bump it a little, it can become slightly elliptic and v=sqrt(GM/r) no longer holds. I think that's where things go wrong with your derivation.

    Best check of orbit stability is looking at effective potential.
     
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