The purpose of this project is to study the properties of groups of galaxies. We will measure several global properties of groups including size and mass and mass-to-light ratios.

The data file contains the galaxies in groups of galaxies identified in the 2 Micron All-Sky Survey (2MASS) Redshift Survey. The columns in the ascii file are labeled with their content There are 1604 groups and over 11,000 galaxies in this catalog.

The Viral estimator, after corrections for projection effects both in the projected radius (to go from 2-D to 3-D, and in the radial velocity (to go from 1-D to 3-D), is:

where R_{ij} is the separation between the ith and jth particle (the limitation
i < j is meant to eliminate double counting) and v_{i} is the velocity
if the ith galaxy w.r.t. the average cluster velocity.

The Projected Mass estimator of Heisler, Bahcall & Tremaine is:

where R_{i,c} is the projected separation between the galaxy and the
cluster center, and v_{i} is as above. The projection factor, f_{p},
depends on the average orbital eccentricity of the system,

How different are the PM and Virial estimates of Virgo's mass?

Well now you have it. With the group mass and the group luminosity you can compute the group mass-to-light ratio.

We do have a determination of the absolute luminosity density from the whole survey. This is about to be published by Mark Hartmann et al.

Given an M/L and an integrated luminosity density, we can calculate the mean mass density:

The critical mass density depends only on the Hubble Constant and the gravitational constant. The formula for the critical density is in section 5 of our book,

So now you can calculate &Omega by comparing the mass density you have measured --- assuming all galaxies have the same mass-to-light ratio as the integral of the Virgo Cluster --- to the critical mass density. Go for it!

After you do that, however, write down what you think are all the assumptions that have gone into this determination. Are they reasonable? Again, how does your value compare to those in the literature (for gravitating matter, since that's what you measure when you measure the mass of Virgo via either the PM or Virial estimators).

The last thing to look at is the structure of the cluster. Can you find a program for making contour diagrams? If so, try making these spatial plots:

You may need to estimate morphological types for the cluster members or possibly look them up in the NASA Extragalactic Database .

Then split the sample into 3 equal bins in radial velocity and plot

What do you see?

Copyright John P. Huchra <huchra@cfa.harvard.edu> 2005