The work of Vera Rubin
astrobites.org
astrobites.org
and the link on the "rather beautifully" points to spherical shell calculations which disk galaxies are clearly not. Reminds that joke about spherical cow in vacuum.
The mass outside of the star's orbit does matter. That outer ring mass' gravity doesn't cancel itself out. The outer ring's segment closest to the star pulls stronger than the total of the segments across the galaxy and that itself provides for stars' velocities not being distributed according to the naive spherical shell calculations. And the farther you're from the massive galaxy center the less that naive spherical calculation reflects your situation, and the more the disk based gravity calculation matters.
Links to this that I have found:
- Deriving the Galactic Mass from the Rotation Curve [ http://www.astronomynotes.com/ismnotes/s7.htm ]
- Measuring the mass of the Milky Way (PDF) [ http://jila.colorado.edu/~pja/astr1120/lecture20.pdf ]
1. The bulge, which is represented by a spheroidal potential
2. The stellar disk, which is represented by two exponential potentials (one radial, the other vertical)
3. The interstellar medium, represented by a single exponential potential
4. A bar, which generally doesn't have a simple analytic representation, but can be represented by a series of ellipsoidal potentials
5. A dark matter halo, represented by a two-component power law. The orbits of stars require the existence of this component of the potential.
Interestingly, the dark matter halo is quite spherically symmetric and comprises nearly all of the Galaxy's mass, so a spherical potential is more correct the further away you get from the center!