Astr 498V Fall 2001

Galaxies Sylvain Veilleux

September 11, 2001 DUE: September 18, 2001

**Problem 1.**

Estimate the average density of the Local Group of galaxies and
compare it with the critical density g cm^{-3}. What are the implications of your
result? Mention the references you use for your calculations.

**Problem 2.**

The time taken for a galaxy or cluster to grow to density must
be at least as great as the free-fall time (*G* = gravitational constant), since cosmic expansion must first be
halted locally. What is *t*_{ff} for (a) the Milky Way () and (b) a cluster of galaxies (). Compare with the Hubble time *H ^{-1}*

**Problem 3.**

We saw in class that the power spectrum

- (a)
- Prove the last equality in the previous equation. [Hint: one
method is to write the volume integral for
*P*(*k*) in spherical polar coordinates and then set . - (b)
- Show that because describes departures from the mean density, the above equation implies , and hence as .
- (c)
- Show that the power spectrum corresponds to a correlation function . Hence implies , approximately as observed.

**Problem 4.**

Linder's ``? Think About 1.3'' on p. 16

**Problem 5.**

Linder's ``? Think About 1.4'' on p. 17