Precession in a central potential

[noospace]

Homework Statement

Consider the following parametrization of an orbit in polar form,

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

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 $\phi_0$.

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

Setting $u'(0) =0$ we obtain

$(\phi -\phi_0)\Gamma = n \pi$ where n is an integer.

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

Thanks.

 

[lalbatros]

I preffer this:

$\frac{2pi}{\Gamma}-\frac{2pi}{1}$

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 $\Gamma = 1$ the trajectory is periodic and there is no shift.

 

[noospace]

How did you obtain 2Pi/Gamma - 2Pi?

 

[noospace]

On reflection I get $2\pi/\Gamma$. I think the first perihelion occurs at 0, the first aphelion occurs at $\pi/\Gamma$ and the second perihelion at $2\pi/\Gamma$.