Astr 222 - Homework #5
1. Coma-tose
We are going to find the mass of the Coma
cluster. In this, assume that H0=65 km/s/Mpc.
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 your plot has justified axes...)
See -- it's a cluster!
- Make a histogram of
the radial
velocity of all the galaxies (don't include
non-measurements!)
. How would this help you decide which
galaxies
actually were part of the Coma cluster?
- Looking only at galaxies with 4000 < cz <
10000
(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.
- If the stars in the Coma cluster galaxies have
a 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
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 (ie distributed smoothly
throughout the cluster)?
2. The age of a flat, matter-dominated universe.
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)
.
3. The Future w/ a cosmological constant
Use the Friedman equation
with
a cosmological constant to show how the expansion of the Universe will
evolve
in the very distant future if the universe is flat. In other words, derive the form of R(t) as
t becomes large. (Hint:
in the very distant future, which terms of the Friedman equation
dominate?
Which ones can you ignore? Why can you ignore them?)
4. Short math problem
If a
galaxy cluster has a mass of 1015 Msun, and a
characteristic size of 3 Mpc, estimate how long it would take a typical
galaxy to orbit the cluster. If this galaxy fell into the
cluster at a redshift of
z=1, how long
has it been in the cluster (assume a flat OmegaM=1
and work this out analytically)? So how many times has
the galaxy orbited the
cluster during its time in the cluster?
5. Cosmological Effects on Size and Brightness
Use Ned Wright's
Cosmology Calculator to help you with this one.
Assume a galaxy has an absolute magnitude of Mv=-21 and a size of 20
kpc. Plot its
apparent magnitude and angular size as a function of redshift from
z=0.01 to z=1 under two different assumptions:
- h: The naive and incorrect assumption that space is flat and
static, and that you can use Hubble's law to get the distance, no
matter what the redshift.
- true: The correct use of a proper, modern day cosmological
model.
Make the redshift axis on your plot logarithmic, and sample
redshift logarithmically when you calculate magnitude and size.
Also plot the
difference in magnitude [(mh-mtrue)] and the
relative size error [(rh-rtrue)/rtrue]
between the two
assumptions. In terms of size and magnitude, at what redshift is the
naive assumption wrong by 10%? By 50%?
6. The density of the universe
Given the expression for the critical density:
Evaluate the
critical density today (in solar masses per cubic megaparsec). Compare
this
to the following:
- the density of the Milky Way galaxy
- the density of matter in the Local
Group
- the density of the Coma cluster
Describe any
assumptions you made to get these densities, and cite any references
you used. Which of those
three densities do you feel is
most accurately determined (if any)? Which of them are the best
estimate
of the overall density of the universe? Are any of them a good estimate
of
the overall density of the universe? Why or why not?
