Dark Matter in the Coma Cluster

The Coma cluster has radius and radial velocity dispersion .

1. What is the mean square velocity of galaxies in the Coma cluster, in units of ? Explain your answer.
2. Write down the virial theorem for the Coma cluster. Use it to deduce the mass of the cluster, in units of .

Figure 4 shows a sample plot of the surface brightness of the Coma cluster, , as a function of angular distance from the centre of the cluster.

Figure: 4

3. Convert Figure 4 to a plot of surface brightness as a function of linear distance r from the centre of the cluster. Assume the cluster is from Earth.
4. Fit a de Vaucouleurs profile to your new Figure 4. Hence estimate the central surface brightness (in units of ) and characteristic radius (in units of Mpc).
5. Estimate the total luminosity of the cluster using the information from 4. Express your answer in units of . You can do this exactly by either graphical or analytic methods. If you need to, you may assume by way of simplification that the de Vaucouleurs profile extends to infinity. (Hint: if you become stuck, make an approximate estimate to obtain partial credit.)
6. Compute the mass-to-light ratio for the Coma cluster in units of . How does it compare to the ratio in the Solar neighbourhood? In the Milky Way overall? (Look up these numbers in the lecture notes and textbooks.)
7. Write a brief paragraph discussing the implications of your answer to 6.