Ah, I thought you were making some special argument about stars orbiting with very tilted orbits or something like that.
Yes, the orbits within the inner part of a galaxy don't follow a Keplerian curve. However, at large radii most of the visible matter is well inside the orbit and very little is outside (galaxies are quite centrally concentrated), so the rotation curve should approach the Keplerian limit as you go to larger and larger radii. The fact that it doesn't is one of the primary pieces of evidence for dark matter in galaxies (though not the only one).
(I'm not sure why you linked to that slide, since it's not calculating the field in the plane of the ring, nor is it dealing with a continuous charge distribution. Regardless, you'll note that E_x goes to 1/x^2 for x >> a, which is a "Keplerian" limit.)
>> so the rotation curve should approach the Keplerian limit...
I don't agree. If we treat the galaxy as a bunch of concentric rings, many of them are nowhere near far enough away to reach that "Keplerian" limit.
As an experiment I modeled a flat disk of uniformly distributed stars. Without doing a dynamic simulation one can do the n-body calculation to determine the total pull on each star and determine what its orbital velocity must be to go in a circular orbit. You find that the "galactic rotation curve" will have a velocity that actually increases all the way to the edge. Of course a uniform distribution is not what a real galaxy looks like, but I think we can all agree reality is somewhere between the uniform flat disk and the highly concentrated central mass. And the actual rotation curves are somewhere between Keplers and mine. I'd expect people to work backward from the rotation curves to determine the mass distribution and then try to understand why that isn't what is observed. I suppose that's just a different way to get at the same mystery.
Yes, the orbits within the inner part of a galaxy don't follow a Keplerian curve. However, at large radii most of the visible matter is well inside the orbit and very little is outside (galaxies are quite centrally concentrated), so the rotation curve should approach the Keplerian limit as you go to larger and larger radii. The fact that it doesn't is one of the primary pieces of evidence for dark matter in galaxies (though not the only one).
(I'm not sure why you linked to that slide, since it's not calculating the field in the plane of the ring, nor is it dealing with a continuous charge distribution. Regardless, you'll note that E_x goes to 1/x^2 for x >> a, which is a "Keplerian" limit.)