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 H_{0}=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
M_{B}=+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 3x10^{14} 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)
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 t_{0}=(2/3)(1/H_{0}). 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 H_{0}?
- A few years later, you decide that the measurements of H_{0} are getting better, so you need to believe them. So take H_{0}=55, and tell me what constraint you can place on Omega. What about if you believe H_{0}=65? H_{0}=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 H_{0}=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 Omega_{L} as well. Given your GC ages and H_{0}, pick values of Omega_{L} and, for those values, place limits on Omega_{M}. Make a plot of your results on an Omega_{L}-Omega_{M} 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=H_{0}d, v=cz, H_{0}=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 H_{0}=72 km/s/Mpc, Omega_{M}=0.3, Omega_{L}=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 Omega_{L} 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?