Astr 222 - Homework #5

1. Coma-tose (20 points)

We are going to find the mass of the Coma cluster. For this problem, adopt a Hubble constant of H0=72 km/s/Mpc.

2. Donkey Dark Matter (15 points)

One of my favorite ideas about the infamous "dark matter" surrounding galaxies is that it is made up of a population of free floating space donkeys (FFSDs). FFSDs would not radiate in the optical (have you ever seen a donkey shine?) but would emit light in the infrared (since they would have little heat generators in the donkey space suits to keep them warm). So let's see if we can rule out this model. Say the dark matter halo of a bright spiral galaxy like the Milky Way has a mass of 1012 Msun. If it was made of FFSDs, what would the bolometric (total) luminosity of the dark matter halo be? (Assume FFSD are blackbody radiators.) What would the peak wavelength of this light be? Compare this to the bolometric luminosity of the Milky Way stars (~5x1010 Lsun) -- how many times brighter or fainter are the FFSDs than the stars? Do you think we'd be able to detect them?

3. The age of a flat, matter-only universe (10 points)

Start with the Friedman equation:

Integrate the Friedman equation for a flat, matter-only universe (ie no cosmological constant), to show that the age of the universe in this model is t0=(2/3)(1/H0). Describe simplifying assumptions, and show your work step by step.

4. Early and Late Cosmological Expansion (15 points)
Start again with the Friedman equation, and assume that we have a universe that is spatially flat. In the following two situations, explain which terms on the Friedman equation are important and which you can ignore or set to zero. Then integrate the Friedman equation to solve for how the scale factor of the universe changes with time -- in other words, solve for R(t).

5. Cosmological Effects on Size and Brightness (15 points)

Let's work out where cosmological effects start to become important for studying distant galaxies. Assume a galaxy has an absolute magnitude of Mv=-21 and a size of r=20 kpc.

First, let's be dumb and say Hubble's law (v=H0d, v=cz, H0=72 km/s/Mpc) works no matter what, and that the distance we derive using Hubble's law can be applied to work out apparent magnitudes and angular sizes. Work out the "distance", apparent magnitude, and angular radius (in arcseconds) of the galaxy as a function of redshift from log(z)=-2 to 0 (in other words, z=0.01 to 1.0) in steps of dlogz=0.2. Like this: first do a logz=np.linspace(-2.0,0.0,10) then do a z=10**logz

Next, let's be smart and realize that we need to do the cosmological observables properly for H0=72 km/s/Mpc, OmegaM=0.3, OmegaL=0.7. Use Ned Wright's Cosmology Calculator to calculate the luminosity distance and angular size distance to the galaxy for the same redshifts as before, and use those values to properly calculate the galaxy's apparent magnitude and angular size. (Note that the online calculator refers to OmegaL as "Omega_vac") You can then put them into an array to plot by saying, for example: dL=np.array([1.0, 2.0, 3.0, 4.0, etc]) where the numbers would be the luminosity distance you get for each of the redshifts in your redshift array.

Make the following plots from this data (all plots should have log(z) as the x-axis):

Finally, plot the observed surface brightness (mu=m+2.5*log(area)) as a function of log(z) in each case. In the nearby universe, we know that surface brightness is independent of distance. How badly has this broken down by a redshift of z=0.1 -- how much has "cosmological surface brightness dimming" changed the surface brightness at that redshift?