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**1. W _{m} from the Coma Cluster**

This is
the famous ``Slice of the Universe'' picture
(de Lapparent, Geller & Huchra 1986, ApJ Letters 302, L1).
It is a redshift map of galaxies brighter than apparent magnitude 15.5
in a slice 6^{°} thick passing through the north galactic pole.
The Milky Way is at the vertex of the wedge.
The prominent `finger of god' at 13 hours Right Ascension centered about
7,000 km/s away is the Coma cluster.

Clicking on the picture will give you a PostScript version of it;
but be warned, the resolution is lower than the image handed out in class.
You will be better off using the image handed out,
or else photocopying it from the Astrophysical Journal.

**(a) Velocity dispersion, radius**

From the diagram measure *approximately*
(i) the velocity dispersion *v* of the Coma cluster in km/s,
(ii) the radius (not distance!) *H*_{0} *r* of the Coma cluster in km/s,
and hence determine
(iii) the ratio
*v* / ( *H*_{0} *r* ).
Explain what you did.
[Hint:
The velocity dispersion is the root mean square orbital velocity
of galaxies in Coma, relative to the center of Coma.
You can estimate the velocity dispersion from the
length of the finger-of-god.
You can estimate the radius from the transverse size of the
finger-of-god.
Question: How far out should I measure?
Answer: We are interested in the collapsed part of the Coma cluster,
so that we can apply the virial theorem.
Clearly the inner parts of the Coma cluster have a higher velocity
dispersion than the outer parts, so your answer depends on where
exactly you choose to do your measurement.
However, it turns out that the final answer we are looking for,
W_{m} in part (d), is (or should be) insensitive to where you choose
to cut the cluster, as long as you are consistent all the way through.]

**(b) Density of Coma relative to the mean mass density**

(i) Estimate the number density *n*_{Coma},
in galaxies (km/s)^{-3},
of galaxies in the Coma cluster

| (1.1) |

| (1.2) |

**(c) Density of Coma relative to the critical density**

The velocity dispersion *v* of Coma is related to its mass *M*
and radius *r* by
(this is the virial theorem)

| (1.3) |

| (1.4) |

| (1.5) |

**(d) Omega in matter**

Assume that mass densities are in proportion to galaxy number densities,
r_{Coma} / r = *n*_{Coma} / *n*.
Use your results from parts (a)-(c) to infer W_{m}
from the defining formula

| (1.6) |

**2. Relation between Horizon and Flatness Problems**

Show that Friedmann's equation can be written in the form (compare Problem Set 5, Question 3a)

| (2.1) |

File translated from T

On 3 May 2000, 13:54.