You might want to read this article.
Determining the Rotation Curve of the Milky Way Galaxy
The rotation curve of a system is a plot of the orbital speed versus the distance from the center of the system. For different objects, or systems of objects, different rotation curves result. For example, the rotation curve of a disk is a line with a positive slope, because as you move outward on the disk, the angular velocity remains constant. A planetary system has a different kind of rotation curve. The angular velocity is not constant in this case, and the velocity decreases as the inverse of the square root of distance. The third type of rotation curve, and the one exhibited by spiral galaxies, is a flat rotation curve.
A flat rotation curve implies that the mass increases as distance from the center increases. The rotation curve, when plotted, did not agree with previous estimates, which were made based on the luminous matter in the galaxy. It was therefore concluded that there is much more matter in the galaxy than is luminous.
Since the advent of radio astronomy, the 21-cm emission line of HI has been a vital part of the study of the kinematics of the Milky Way. It was predicted in the early 1940’s and was first observed in 1951. This line is produced by a “flip” in the spin of an electron. Most of the Galaxy is optically thin to the 21-cm line, making it an excellent resource for determining the rotation curve of the Galaxy.
I wish they had described the physical implications of a "flat" rotation curve. Evidently I had quite the wrong picture of how galaxies rotate, one corresponding to the non-flat rotation curve of a disk.
I assume the positive slope of that line in that example comes from the increasing linear velocity--the angular being constant. If the line height at a given radius is linear velocity, that would mean that the outer parts of a galaxy (with its flat rotation curve) are indeed rotating much slower (angular velocity) than the inner parts after all, which doesn't sound like the problem befuddling astronomers at all.