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.