Dark matter does not need to be distributed in a "faucet washer" shape in order to explain why the outer stars in galaxies orbit so fast. DM is evenly distributed throughout the galaxy. It could be in disc shape or spherical, depending on the shape of the galaxy.
Here is how differential rotation occurs:
1) At one extreme we have a galaxy in which all mass is entirely at a point in the center. In this case, the classical orbit equations work perfectly. Anything close to the center orbits at a high speed. Anything far away goes much slower.
2) At the other extreme we have all of the mass of the galaxy as a disc shaped or spherical, homogenous cloud with no concentrations of mass anywhere. This cloud will not experience differential rotation, it will rotate as a solid would, same angular speed everywhere. A solid disc or sphere. You could well say "It is immune to gravity".
Here is the problem. Most of the universe's galaxies fall somewhere in between and when we go looking for the mass in that cloud, we can't find it.
Thanks you,
You are causing me to sharpen my thinking.
Excellent brain exercise.
The galaxy is probably more of a web work so i probably need to back off over emphasizing the 'faucet washer',
DM at a minimum must be a disk shape.
A sphere would have stars wandering out of the elliptic plane.
Remember DM is supposed to be 4 times the mass of baryonic matter.
For any/all orbits to be sustained there must be a circular gravity differential ('slope').
This must be concentrically consistent.
It could be some bare minimum.
If a galaxy's web work of gravity is the barest minimum the first passing galaxy would peel off many of the outer stars.
The Roche limit.
Just a thought.
According to reports the outer stars are orbiting too fast.
So there would need to be a sharper differential ring, 'slope' of gravity to sustain that.
That forces some source of gravity to be inside the orbits of the outer stars.
Bring on 'dark matter'.
Dark matter's outer edge has DM and its gravity on the inside and nothing much on the outside,
yet it doesn't migrate inward over a galaxy's lifetime.
So it's immune to gravity there.
Elsewhere in the disk of DM it doesn't pile up around stars and planets and amplify their gravity.
So it's immune to gravity there.
It doesn't seem to be cascading into the central black hole so gravity immunity once again.
DM does exactly one thing.
It produces gravity.
So what is the distinction between DM and simply saying gravity?
Nothing...
but neurotic reflex.