ASTR 328/438 - HW 4

1. Collapse of overdensities

Remember our discussion about the turnaround and collapse of initial overdensities in matter-only universes. If you have a little overdense bubble of the universe embedded in a flat universe, show that the density of the overdensity at maximum size is given by

where the "prime" mark means the value of omega0 in the overdense bubble.

2. Mass and luminosity functions

We have developed an expression using the Press-Schecter formalism for the relative numbers of collapsed dark matter halos as a function of mass. In the range of masses 109 - 1012.5 Msun or so, these represent galaxies.

We also know observationally that galaxies follow a luminosity function (relative numbers of galaxies of a given luminosity) called the Schecter luminosity function, with parameters L* = 2x1010 and alpha = -1.

If we assume a mass to light ratio, we can connect the two?

Make some choices for mass-to-light ratio (remember this is total mass-to-light ratio, not stellar mass-to-light ratio) and see if you can make the PS mass function and the Schecter luminosity function match up. Explain why you chose those mass-to-light ratios -- are they (somewhat) physically motivated?

Only worry about mass ranges for galaxies (where this discussion is applicable), and remember that you are allowed to shift the curves up and down arbitrarily to make them match.

If you can succeed in matching them up with a single mass-to-light ratio, that suggests that star formation history is similar for all galaxies regardless of mass. If you can't succeed in matching them up with a single mass-to-light ratio, that argues that star formation is different in low and high mass galaxies. What do your tests suggest? If it fits across all masses, what is the mass-to-light ratio that works, and does that make sense physically? If it doesn't work, what would you have to do to make it work, and what does that suggest physically?

3. Baryon Budgeting

Use the literature to get estimates of the following for the Coma cluster:
 From this calculate a local value of the baryon/dark matter ratio in Coma. Then figure out what you would expect cosmologically given our discussions of OmegaB and OmegaM. Do these two numbers match? If not, give some physical reasons why they might not match.

4. Group Project: Cosmic Flows


We are going to study the motion of the local group on large scales. This involves two conceptual steps: finding peculiar velocities for clusters of galaxies, and then, once a sample is cluster peculiar velocities is built up, solving for the motion. So we'll do this in steps.

First, get a feel for finding peculiar velocities by the following exercises:
 OK, now you have some data. Obviously two data points isn't enough. Now go out and grab somebody else's big catalog of cluster peculiar velocities so you have a big dataset. Then using their data plus yours, solve for the peculiar velocity (speed and direction) of the local group, using the handout from Mihalas and Binney to show you the way. Compare the velocity you get to the local group's velocity with respect to the microwave background. Comment on differences.