# Astr 222 - Homework #5

## 1. Coma-tose (20 points)

We are going to find the mass of the Coma cluster of galaxies. For this problem, adopt a Hubble constant of H0=72 km/s/Mpc.
First, here's an image of Coma, so that you know what you are studying.

And here is a dataset of galaxies in a 6x6 degree field around the Coma cluster (from Doi etal 1995, ApJS, 97, 77). It contains
• name: galaxy ID number
• x & y: position relative to the center of Coma (defined by NGC 4886), measured in arcminutes
• cz: the observed radial velocity in km/s. cz=0 means no measurement.
• Bt: apparent blue (B) magnitude
• Make an x,y plot of the galaxy distribution. (make sure the axes on your plot have a square aspect ratio) See -- it's a cluster!
• Make a histogram of the radial velocity of all the galaxies (use bins with width of 250 km/s, and don't include galaxies without velocity measurements when making your histogram!) . How would this help you decide which galaxies actually were part of the Coma cluster?
• Looking only at galaxies with 4000 < cz < 10000 km/s (why?), calculate the mean velocity and velocity dispersion of the galaxy sample.
• From your data, how far away is Coma?
• Calculate the total blue luminosity (in solar luminosities) for the Coma cluster. Ignore galaxies with cz < 4000 and cz >10000, but include galaxies with unmeasured redshifts. For reference, the absolute blue magnitude of the Sun is MB=+5.5.
• Figure out the radius (in arcminutes) which contains roughly half the total blue luminosity (it doesn't have to be an exact solution, but you should get it to +/- 20 arcminutes or so). This is called the half-light radius, which is our estimate of the size of the Coma cluster. What is the half-light radius of Coma in Mpc?
• Now calculate the virial mass of Coma (in solar masses).
• If the stars in the Coma cluster galaxies have a stellar mass-to-light ratio of (M/L)=3 Msun/Lsun, what is the total mass of stars in the Coma cluster?
• If the galaxies in the Coma cluster have a total mass-to-light ratio of (M/L)=20 Msun/Lsun, what is the total galaxy mass of Coma (this will consist of stellar mass and the mass of any dark matter and gas which is contained inside galaxies).
• X-ray measurements indicate that Coma has a hot gas mass of 3x1014 Msun. What fraction of Coma is dark matter unassociated with galaxies (i.e., is dark matter distributed smoothly throughout the cluster)?

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

3. 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).
• In the early universe, the energy density of radiation exceeded that of matter, and at this point in time (referred to as the "radiation era") radiation drove the expansion. Solve for R(t) under these conditions.
• Now consider very late times in a universe with a non-zero cosmological constant. Again solve for R(t) under these conditions.

4. Ages, Lookback times, and Cosmological constraints (15 points)

Use Ned Wright's Cosmology Calculator to get t(z) estimates. Always use Lambda=0 universes, until you get to the last step!

• It's 1990. You are a good, party-line cosmologist and "know" that Omega=1, and that there is no such thing as the cosmological constant. You also "know" that globular clusters are 14-16 Gyr old. What constraints can you place on H0?
• A few years later, you decide that the measurements of H0 are getting better, so you need to believe them. So take H0=55, and tell me what constraint you can place on Omega. What about if you believe H0=65? H0=75?
• And then Hipparcos comes along (around 1997) and tell us that globular clusters are further away -- this means the ages of the globular clusters get revised downwards to 11-13 Gyr. Why does the further distance mean a younger age? At the same time, you decide, for better or worse, that you like H0=65. Now what constraints can you place on Omega?
• Later, in 1998 the high redshift galaxy LBD53W091 was discovered. It lives at a redshift of z=1.55, and its was determined to have a minimum age of 3.5 Gyr. How does this change your constraints on Omega?
• Six months later, the age of LBD53W091 was revised downwards, and became 1.7 +/- 0.3 Gyr. NOW how do your constraints on Omega change?
• Finally, this wacky Lambda idea starts to take off, so you have to consider OmegaL as well. Given your GC ages and H0, pick values of OmegaL and, for those values, place limits on OmegaM. Make a plot of your results on an OmegaL-OmegaM plane, showing regions of "allowed cosmology" given your GC age constraints.

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

This problem has been taken off the assignment.

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):

• A plot of apparent magnitude (y-axis) versus log(z), plotting both cases on the same plot for easy comparison.
• A plot of log(size) versus log(z), again plotting both cases on the same plot.
• A plot of the magnitude difference (mdumb-mcosmo) as a function of log(z). At what redshift is the dumb way wrong by 0.1 mag? By 0.5 mag?
• A plot of the relative size error [(rdumb-rcosmo)/rcosmo] between the two assumptions, as a function of log(z). At what redshift is the dumb way wrong by 10%? By 50%?
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?