Rotation Curves

The shape of the rotation curves of galaxies tells us a lot about the distribution of mass inside of galaxies.

Keplerian Motions

• Keplerian motions arise for orbits in centrally concentrated systems. For example, a very centrally condensed system is the Solar System where 99.9 % of the mass is in the Sun and all of the other bodies orbit around the Sun in orbits whose sizes are much greater than the radius of the Sun.

  

  For galaxies, Keplerian orbits would arise for objects which were outside of the main concentration of mass of the galaxy. That is, things which orbit at very large distances are expected to show Keplerian motion. The point where Keplerian motion starts roughly marks the edge of the galaxy.

• Motion inside of a galaxy shows a distinctly different character:
  • we still have: mv**2/R - GM(R)m/R**2
  • however, M(R) now represents the amount of mass contained within the sphere of radius R. M(R) < M, the total mass of the galaxy
  • Consider M(R) = [4 pi / 3] rho R**3, i.e., a uniform density sphere
  • ==> v**2 = 4 pi G rho R**2 / 3 ===> v ~ R

  

We can gas motions far beyond the ends of the visible disks of galaxies (stars generally define the extent of the disk). In all cases, the rotation curves of spiral galaxies are roughly flat with no evidence of Keplerian fall-offs. We do not see the edges of any spiral galaxy. There is evidence for dark mass in all spiral galaxies studied. The amount of dark matter is uncertain, however. It is clear though that at least of the total mass of the Milky Way is dark matter.