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Stability Condition for Circular Orbit

  1. Apr 24, 2014 #1

    cpburris

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    Gold Member

    1. The problem statement, all variables and given/known data

    Show that the stability condition for a circular orbit of radius a, i.e.

    [itex]f(a) + \frac{a}{3} (\frac{df}{dr})_{r=a} < 0 [/itex]

    is equivalent to the condition

    [itex] \frac{d^2V(r)}{dr^2} > 0 [/itex]

    for r=a where V(r) is the effective potential given by

    [itex] V(r) = U(r) + \frac{ml^2}{2r^2} [/itex]

    3. The attempt at a solution

    I understand fully why they are equivalent, and I would have no problem proving individually how each is a condition for stability, but analytically I really don't know how to show the two are equivalent. I'm not even sure what the question is asking. I tried just setting

    [itex] -\frac{d^2V(r)}{dr^2} = f(a) + \frac{a}{3} (\frac{df}{dr})_{r=a} [/itex]

    and do something from there, but it didn't get me anywhere.
     
  2. jcsd
  3. Apr 28, 2014 #2

    BruceW

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    Homework Helper

    I'm guessing that you already know what ##f(r)## is as a function of the potential? So if you use this definition, you could write out ##f(a) + \frac{a}{3} (\frac{df}{dr})_{r=a} < 0## in terms of the potential instead, and start to see how it could be similar to the other equation.
     
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