I remember being in fifth grade science class (in the early 1980s) and hearing first about Kepler's law and that planets further out circle the sun less quickly than those closer in, and then next hearing that we had spiral galaxies.
I raised my hand and asked "how do the galaxies maintain their spiral shape? Shouldn't the inner stars rotate much more quickly than the outer ones?"
The teacher didn't understand my question (this was one of my first inklings that teachers didn't tend to be that bright).
The last 10-15 years with headlines in the popular press about dark matter, MOND, etc. are quite satisfying - 10 year old me is thrilled that my question is mainstream now!
It's a good question, but that is not the same issue as the one the article is about.
The spiral arms maintain their shape not because of an anomaly in orbital velocities, but because they are density waves[1]. Kepler doesn't work because the arms themselves and the rest of the disk exerts significant gravity.
To emphasize what wcoenen said: the appearance of spiral arms persists even in the interior of galaxies where there is no discrepancy from Kepler's laws. Moreover, in order to explain fixed spiral structure using modified orbital mechanics, you would need to assert that outers stars have larger velocities than inner ones (in order that their angular speed is the same, to preserve disk structure), but in fact the rotation-curve conundrum is based on the observation that outer star have roughly constant velocity, and so reduced angular speed.
I raised my hand and asked "how do the galaxies maintain their spiral shape? Shouldn't the inner stars rotate much more quickly than the outer ones?"
The teacher didn't understand my question (this was one of my first inklings that teachers didn't tend to be that bright).
The last 10-15 years with headlines in the popular press about dark matter, MOND, etc. are quite satisfying - 10 year old me is thrilled that my question is mainstream now!