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Physics 410/510 Last Homework Assignment
Due: March 14

1. Okay this time I will do it right. For some idiotic reason
I had a/r the last time instead of r/a - duh ... sorry.

Numerically show that for a galaxy with an exponential mass density
profile:

* M(r)=M(o)exp*^{(-r/a)} * *
that the total mass is = 2*pi*M(o)*a^{2}.

Next show, by integration as a function of scale length (a)
that the maximum circular velocity is reached at 2.2 scale lengths.

2. This is another real data excercise to show you how difficult
it is to determine substructure in clusters of galaxies.

The following references may help.

Geller and Beers 1982, Pub. Astro. Society of Pacific 94 421

West \etal 1988, Astrophysical Journal 327 1

West and Bothun 1990 Astrophysical Journal 350 36

Fitchett and Webster 1987 Astrophysical Journal 317 653

The issue of substructure in clusters remains a volatile one. The main
problem is the lack of a rigid, unambiguous test for determining the small
clustering attributes of an N-point distribution. The eye is very good at
seeing structure even when none is present.

For this exercise, you are to analyze real X Y (on the plane of
the sky) positional data for three clusters of galaxies. The
data can be retrieved below:

Cluster 1
Cluster 2
Cluster 3
For each of these clusters attempt to do the following using the
data.

- Determine the cluster density profile. That is the projected surface
density
of points as a function of radius from the cluster center. This is best done by counting
points in rings of some specified width.

- Develop your own statistical tests (or pirate some from the literature) to
determine if any of these clusters has substructure. That is, are there multiple
lumps in the galaxy distribution or is it smooth. Try and assess the significance of
your tests for substructure

- Determine if the galaxy distribution in each of the
three clusters is flattened or spherical.

3. Using the Abstract Service or the Library, briefly summarize one
recent paper which uses data to suggest that the Cosmological Constant
is non-zero. Do not consider papers which make this argument on the
basis of the age problem we have been talking about. Instead, I am
looking for you to find papers dealing with large scale structure
and/or geometrical issues that are best resolved by using a
positive Cosmological Constant.
Bonus Question: Can you think of a direct
observational test that would constrain the value of the Cosmological Constant.
?