The Virgo cluster is a fairly nearby cluster of ~2000 galaxies. The distance of the Virgo cluster is 16 Mpc and its bulk radial velocity is approximately 1120 km/s. The cluster is very large on the sky, its angular 'core radius' is approximately 6.5 degrees! a) (5 pts) If -2.2% of Virgo galaxies have negative radial velocities, what is the approximate mass of the cluster? Hint: you will need to determine the velocity dispersion by approximating the distribution of cluster members' velocities as a Gaussian centered on 1120 km/s, with a 'tail' of 2.2% of galaxies extending below O km/s. With this information, you should be able to find the FWHM of the distribution, which is related to the velocity dispersion, Or. Look up the 68-95-99.7 rule for Normal distributions! The mass you are looking for is the 'Virial Mass' which can be derived from the Virial theorem (Try it!!). Or feel free to use equation 3.45 in Sparke and Gallagher, or any equation you think makes sense in this context. Justify your choice. The numerical constant will vary from source to source given what they assume about the gravitational potential...just pick one, and try to justify your answer. b) (4 pts) Use the Tully-Fisher relation for Milky-Way like galaxies (given below) to determine the absolute magnitude of our Galaxy given what you know about its rotation curve. Then, find the apparent brightness the Milky Way would have if it were located at the center of the Virgo cluster. At this apparent brightness, is the Milky Way likely to have been included in Messier's catalog of non-comet nebulae? If the Milky Way were located at the same distance as the Andromeda galaxy (google it!), would it be brighter or fainter than Andromeda? For reference, Andromeda has a B-band apparent magnitude of ~3.4 mag. (9pts) Use the Following Tully Fisher Relation: Mb = –10.2 log10 Vmax + 2.71 c) (4 pts) The Tully-Fisher relation has an intrinsic scatter of around 0.15 magnitudes - i. e. at a fixed rotation speed, there will be roughly a +0.15 magnitude scatter in the absolute magnitude, even for perfect measurements. In other words, even if you were the world's best astronomer, you would never be able to use the Tully-Fisher relationship to derive an exact distance to one particular galaxy, since not every galaxy follows the exact Tully-Fisher relation. What percentage distance uncertainty would this intrinsic scatter correspond to, assuming no error in the measurement of either mg or Vmax? Hint: The distance modulus is your friend. d) (2 pts) Given this uncertainty in the distance for a single galaxy, can the Tully- Fisher relationship be used to determine if a particular spiral galaxy is within a cluster that is 400 Mpc away from us (assuming the cluster has a physical radius in Mpc that you determined in part a)?