The Dark Matter Universe
Last time we saw that their might be evidence for non-luminous matter in the solar neighborhood. On the scale of the Galaxy however, there is rather direct evidence for nun-luminous material. To wit:
If light traces mass in a galaxy then we expect a kelperian fall off in rotational velocity as a function of radial distance. This is never observed in any galaxy:
Some sample rotation curves:
In most all observed cases, the rotation curve is either flat or rising within the optically observed part of the galaxy. Note that, in general, spirals have an exponential light distribution:
The optical portion of a galaxy usually encompasses about 3 scale lengths (a) of light. If the mass distribution is also exponential and remembering that the circular velocity goes as (M/R)(1/2) , then one can show numerically that for an exponential mass distrubtion the quantity M/R reaches a maximum at 2.15 scale lengths which is about the optical extent of a galaxy. Thus, optical rotation curves could be somewhat flat, they are not unambiguous probes of the Halo.
However, there do exist some galaxies which have large gaseous extents relative to their optical extent. In some cases, this gas has a radial extent of 10 scale lengths and hence the circular velocity is dominated by an extended halo. If not, the circular velocity would fall off rapidly once the optical edge of the galaxy is reached. An excellent example of this is shown below for NGC 3109.
In the above example, the data is fitted against mass contributions from the gas, stars and dark matter in the halo. For this case its obvious that the halo dominates but is this Universal?
Caveats: Perhaps only galaxies with extended dark matter halos are capable of retaining large gaseous extents?
Modified Newtonian Dynamics? --> let's assume that there is no dark matter, what you see is what you get, and use rotation curves to learn about gravity on large scales
In any event, the rotation curve data, in general, suggests that 90\% of the mass of a galaxy is distributed in some spherical halo which contains very little light.
Clusters of Galaxies:
Historically, this was the first set of observatoins that hinted at dark matter. This is the subject of problem 3 in the homework.
However, substructure can fool you and next time we will consider the interesting case of the Cancer cluster.