In other words, a disk-like distribution of matter does not have a uniform gravitational pull at all points of distance R from its center. You'll find it is stronger near the edge of the disk than at the same distance perpendicular to it.
But stars (and gas clouds) orbiting in a disk are essentially always "near the edge" of the disk (that is, the disk defined as matter interior to the orbit in question), so the approximation is usually pretty good.
reference to the "radial velocity" or speed of stars where I believe they are actually talking about the orbital or tangential velocity.
The (understandable!) confusion originates because "radial" and "tangential" refer, in an observational sense, to velocities along or perpendicular to the line-of-sight vector. To simplify a bit: for observations of astronomical objects, velocities along the line of sight from you to the object (detectable from Doppler shifts in its light) are "radial", while velocities perpendicular to the line of sight (which require that you wait long enough to see things change position on the sky) are "tangential". For edge-on spiral galaxies, the orbital motions that we can observe are those producing Doppler shifts along the line of sight (which are, in the frame of the galaxy, "orbital or tangential", as you suggest).
And lastly, if the dark matter obeys the same laws of gravitation, there must be a reason for its distribution to be different than the other matter.
The usual argument is that dark matter is "non-dissipative" -- that is, it can't lose orbital energy via things like radiation, the way ordinary matter in the form of gas can. (And, unlike gas, it doesn't feel pressure forces.) Stars are also non-dissipative, but stars form out of gas, so their distribution reflects the effects of dissipation in ways that dark matter can't.
>> But stars (and gas clouds) orbiting in a disk are essentially always "near the edge" of the disk (that is, the disk defined as matter interior to the orbit in question), so the approximation is usually pretty good.
Not sure we're on the same page. For a star orbiting at radius r on a disk of radius R, some folks will lump all mass inside r into a point and apply keplers laws. They will reject all the mass between r and R because of the same surface integral theorems. The problem is that neither of those assumptions are anywhere near correct. The pull from the interior <r mass is not the same as though it were a point, and the pull from the outer mass >r is very much non-zero. For an example using electric charge instead of gravity:
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.
But stars (and gas clouds) orbiting in a disk are essentially always "near the edge" of the disk (that is, the disk defined as matter interior to the orbit in question), so the approximation is usually pretty good.
reference to the "radial velocity" or speed of stars where I believe they are actually talking about the orbital or tangential velocity.
The (understandable!) confusion originates because "radial" and "tangential" refer, in an observational sense, to velocities along or perpendicular to the line-of-sight vector. To simplify a bit: for observations of astronomical objects, velocities along the line of sight from you to the object (detectable from Doppler shifts in its light) are "radial", while velocities perpendicular to the line of sight (which require that you wait long enough to see things change position on the sky) are "tangential". For edge-on spiral galaxies, the orbital motions that we can observe are those producing Doppler shifts along the line of sight (which are, in the frame of the galaxy, "orbital or tangential", as you suggest).
And lastly, if the dark matter obeys the same laws of gravitation, there must be a reason for its distribution to be different than the other matter.
The usual argument is that dark matter is "non-dissipative" -- that is, it can't lose orbital energy via things like radiation, the way ordinary matter in the form of gas can. (And, unlike gas, it doesn't feel pressure forces.) Stars are also non-dissipative, but stars form out of gas, so their distribution reflects the effects of dissipation in ways that dark matter can't.