Fingers of Galaxies

Identify ``fingers'' in the Sloan data. These are groups of galaxies which appear to be on a line pointing at us. One interpretation of such features is that the measured $z$ has a non-cosmological component due to relative internal motion within a cluster of galaxies. This adds and subtracts a redshift to the cosmological redshift, but when $z$ is interpreted as $d$ we cannot remove the non-cosmological part.

Explore a few of the fingers, and use the range of $z$ (and thereby $v$) in them to find a velocity ``dispersion'' $\sigma$ for the members of a cluster of galaxies. It may be defined for $N$ galaxies in the same way as a standard deviation such as

$$\sigma^2 = \Sigma\left(v - v_{av}\right)^2/N$$

The velocity dispersion is related to the gravitational potential energy well in which the cluster galaxies might be bound.

With the virial theorem you can show that for a spherical cluster

$$M\sigma^2 = (3/5) (G M^2/R)$$

or

$$M = (5/3)\sigma^2 R / G$$

$R$ is the apparent radius of the cluster and the mass $M$ determined in this way is termed the ``virial mass'' of the cluster. Relate the velocity dispersion in this formula to cluster mass if the velocity is measured in km/s and the mass in solar masses.

1. Clusters which show in the 3-D maps as fingers should have a distinctive appearance in the original images. Identify at least 3 such clusters by giving the RA, Dec, and distance for each. Locate the clusters in the Sloan Survey and if possible print out an image for each one.

2. Find the virial cluster masses for these clusters based on this technique.

3. Given the $\sigma$ you find here, estimate the crossing time for a galaxy in a cluster to move across its diameter. Compare this to the Hubble time, $1/H_0$.