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Homework Help: Precession in a central potential

  1. Aug 23, 2007 #1
    1. The problem statement, all variables and given/known data

    Consider the following parametrization of an orbit in polar form,

    [itex] \ell u = 1 + e \cos[(\phi -\phi_0)\Gamma][/itex]

    where u = 1/r.

    I'm trying to find the shift in the angular position of the periapsis after one complete orbit.

    3. The attempt at a solution

    Choose axes so that the point of first closest approach is [itex]\phi_0[/itex].

    [itex] u'(\phi) = - \Gamma e \sin[(\phi -\phi_0)\Gamma][/itex]

    Setting [itex]u'(0) =0[/itex] we obtain

    [itex] (\phi -\phi_0)\Gamma = n \pi[/itex] where n is an integer.

    So after one complete orbit I guess the shift is [itex]\Delta \phi = \phi - \phi_0 = \frac{\pi}{\Gamma}[/itex], or should that be [itex]\frac{2pi}{\Gamma}[/itex]?

  2. jcsd
  3. Aug 23, 2007 #2
    I preffer this:


    To solve this problem, you even don't need to consider the exact trajectory.
    You simply need to check how much (an angle) the trajectory deviates from periodicity.
    For [itex]\Gamma = 1[/itex] the trajectory is periodic and there is no shift.
  4. Aug 23, 2007 #3
    How did you obtain 2Pi/Gamma - 2Pi?
  5. Aug 23, 2007 #4
    On reflection I get [itex]2\pi/\Gamma[/itex]. I think the first perihelion occurs at 0, the first aphelion occurs at [itex]\pi/\Gamma[/itex] and the second perihelion at [itex]2\pi/\Gamma[/itex].
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