This problem set due on the last day of classes, Jun 6. This assignment will require a great deal of calculation. I strongly urge you to start working on these problems soonest. Watch your units, you probably want to keep things in meters and kilograms.

1) The following data is for a cluster of galaxies with 20 members. The redshift of each galaxy is given in km/sec, R is the distance from the geometric center of the cluster in arcsecs, and m is the apparent magnitude of the galaxy.

gal   z    R   m
 #  km/s  (")
-----------------
 1 60235  10 18.4 
 2 60446  13 18.3 
 3 60157  26 17.7 
 4 60197  28 17.4 
 5 60534  31 17.7 
 6 60747  75 17.9 
 7 59935  77 17.7 
 8 58607  85 17.8 
 9 60262  90 18.4 
10 60371  93 17.2 
11 60346  98 18.9 
12 60390  98 16.8 
13 57181 103 17.6 
14 57674 116 18.0 
15 60813 119 18.7 
16 60457 144 17.7 
17 59379 163 18.0 
18 60377   5 16.5 
19 60256 343 18.8 
20 60789 346 17.9 

b) Determine the mass of the cluster in solar masses. To do this you must calculate the velocity dispersion, i.e. the sigma on the mean of the velocities, and use it for the v^2 term in the gravity equation. In addition, you must determine the mean radius for the same formula (in the appropriate units i.e. converting arcsecs to meters).

c) Determine the luminosity of the cluster (i.e. the sum of the galaxy light).

d) Determine the M/L of the cluster in solar units.

e) If clusters like this one are evenly distributed in the Universe with mean distances of 10 Mpc, calculate Omega_matter (i.e. find the mean density and compare this to the critical density of the Universe, you should get a value near 0.33).

2) Using the Omega_matter from above and calculate Omega_Lambda for each of the following SN data. You will need to use the following web site (http://www.astro.ucla.edu/~wright/CosmoCalc.html) to try various values of Omega_Lambda to match the distance modulus. Assume that all SN have a absolute mag of -22.

SN  z     m
-------------
 1 0.65 20.77
 2 0.80 21.32
 3 0.83 21.41
 4 0.90 21.63
 5 0.95 21.77

a) Find Omega_Lambda for each SN. What is the average Omega_Lambda value for the sample.

b) Make a Hubble diagram (distance modulus versus redshift) for this sample. Plot a curve for your Omega_matter and Omega_Lambda on the same chart, comment on the fit to the data.