Precession in a central potential

  • Thread starter noospace
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  • #1
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Homework Statement



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.

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]?

Thanks.
 

Answers and Replies

  • #2
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I preffer this:

[itex]\frac{2pi}{\Gamma}-\frac{2pi}{1}[/itex]

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.
 
  • #3
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How did you obtain 2Pi/Gamma - 2Pi?
 
  • #4
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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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