Astr/Phys 328/428 Homework #3
1. Distance ladder
that a series of four different standard candles are used to step out
along the cosmic distance ladder as far as the Hubble flow, and you are
using distances measured this way to derive the Hubble constant. Assume
the calibration of each of the four standard candles being used carries
an uncertainty of 0.2
magnitudes. What is the fractional uncertainty in the Hubble
constant you would derive?
2. The Difference between "Near" and "Far"
(problem courtesy Heather Morrison)
In order to give you a
feel for the
problems associated with using galaxies which are not distant enough to
be in the Hubble flow for deriving H0
, here is a
slice of the "Virgo consortium universe"
. These data come from a
massive simulation of a cube of the universe measuring 150 Mpc on a
side. The data give the x, y, and z coordinates [in Mpc] of each galaxy
in the simulation, their line-of-sight velocity [in km/s], and their
star formation rate [in Msun
/yr]. The slice has been taken by restricting the x coordinates of the galaxies.
First, make a plot of y vs z to see the large scale structure in the
simulation. Remember, here you are making an actual picture of the
universe (ie, plotting spatial coordinate vs spatial coordinate), so
the axes on your plot should have equal aspect ratios. In
python/matplotlib, use set_aspect('equal')
Assume that the Sun is at the
coordinate (50,0,0), and calculate the inferred Hubble constant from
each of the following samples:
- nearby sample: galaxies closer than 20 Mpc from the Sun,
- local universe sample: galaxies between 20 and 40 Mpc from the Sun, and
- distant sample: galaxies further than 50 Mpc from the Sun
To do this, use the known distance of
the galaxies (calculated from the coordinates) and the line-of-sight
velocity. For each sample, make a Hubble plot (velocity versus distance) and plot your derived Hubble law on the data.
What do you estimate for the value of the Hubble constant
used to produce the simulation (include an errorbar!)?
Comment on the accuracy of using the
two relatively nearby samples: how much of an error do the peculiar
velocities of galaxies add?
Now repeat this using only elliptical
galaxies (star formation rate = 0). Are your results different? Why?
3. The Peculiar Velocity of S639
are Fundamental Plane datasets for two galaxy clusters:
Each data table has the measured velocity dispersions (logsig, in
km/s), observed effective radii (logre, in arcsec), and effective
surface brightnesses (mue, in mags per square arcsec) for a set of
elliptical galaxies in each cluster.
- Using the Coma data given in Table 1 of Jorgensen et al (1993),
derive the zeropoint of its B-band Fundamental Plane: log(re) =
1.24*log(sigma) - 0.82*log<I> + ZP. Note that log<I>=-0.4*(<mu>-26.4).
- Do the same for the S639 data, obtaining the zeropoint of its B-band Fundamental Plane. Note that the S639 data has surface
brightness in r mags, not B mags. Convert those surface brightnesses to
B by adopting a B-r color of 1.1 for the galaxies, and that way they'll
match the magnitude system of the Coma data.
- In both cases, estimate not only the zeropoint but its uncertainty.
the data for each cluster on a Fundamental Plane plot, along with a
line showing the Fundamental Plane with your fitted zeropoints.
- The zeropoints for the clusters will be different, since one of
the clusters is further away than the other. Use the cluster zeropoints
to find the relative distances of the cluster, ie d(S639)/d(Coma). Also
use your zeropoint uncertainties to estimate the uncertainty in the
- Then work out the distance to Coma using its redshift, adopting a
Hubble constant of 72 km/s/Mpc, assuming Coma has no peculiar motion.
- Given that distance, and the relative distance measure you got
from your FP zeropoints, work out the distance to S639, along with an
estimate of its peculiar motion.
4. Integrative Essay: Constraining the Omegas.
Describe the physics behind use of
supernovae to study OmegaM and OmegaL, and what constraints the
supernovae results have placed on OmegaM and OmegaL. Then describe the
physics behind the use of the microwave background fluctuations to
constrain OmegaM, OmegaL, and OmegaB, and what constraints the CMB have
placed on these parameters. Finally, explain physically why the two
methods together give so much tighter constraints on OmegaM and OmegaL than each one does individually.
Target Length: ~ 1.5 pages single spaced, 3.0 pages double spaced.
5 ASTR 428: Calibrating the Tully Fisher Relationship
We are going to use spiral galaxies in the Virgo Cluster to calibrate the Tully-Fisher relationship.
Start with this Tully-Fisher dataset
for Virgo spiral galaxies (from Pierce & Tully 1988
). The dataset has
- the NGC number of each galaxy,
- apparent magnitude of the galaxies in the B, R, and I band,
- the inclination of the galaxy to the line of sight (90o=edge on, 0o=face on).
- the observed rotation speed (W20/2) of the galaxies
We need to make a correction to the data based on inclination in order
to get the true rotation speed of the galaxy. What would this
correction be? Apply this correction to the observed rotation speeds to
get the true rotation speed.
Make a Tully-Fisher plot (apparent mag versus log(V)) in the B, R, and
I bands, and for each case, fit a line of the form m=a*(log(V)-2.5)+b.
Report your fit by giving the parameters a and b as well as their
- What is the slope of the line in each case? When is it closest to the "expected" value?
- What is the dispersion (aka 'scatter') around the line in each case? What do you think the main sources of scatter are?
- Give physical arguments about which band would best define the
Tully-Fisher relationship. Think both about stellar populations and
Now we need to calibrate the Tully-Fisher relationship. We want to know
how absolute magnitude depends on circular velocity, which means we
need to know a distance to the Virgo cluster. Using the Hubble Space
Telescope, we observe Cepheid variables in M100, also known as NGC
4321. Here is a dataset of M100 Cepheids
from Ferrarese et al 1996
, listing period (in days) and magnitudes.
Take the Vave magnitudes and correct them for dust extinction assuming
a typical reddening of E(B-V)=0.05. Then use the periods and corrected
magnitudes, along with a Cepheid period-luminosity relationship that
looks like this: MV= -2.76*[log(P)-1] - 4.16
to calculate a distance estimate to M100 for each Cepheid. Finally
average them to give your best estimate for the M100 distance. Also
give a statistical uncertainty to your distance.
Using this distance, calibrate the Tully-Fisher relationship. You had a
T-F relationship that connected apparent magnitude (m) with logV, and
now you know the distance to Virgo to turn apparent magnitude (m) into
absolute magnitude (M), so you can rewrite your T-F relationship now in
terms of absolute magnitude: M = a*(log(V)-2.5) + b. What are a and b,
and their uncertainties?
What do you feel are the main sources of uncertainty, both systematic and random, in your derivation of the T-F relationship?
Finally, now you are looking at a spiral galaxy in the Coma cluster. It
has an I-band apparent magnitude of 13.5, an observed rotation speed of
180 km/s, and an inclination of 65o . What is the distance to Coma?
What is the uncertainty in your distance?
6. ASTR 428: Project
Note: this will be due with HW #4 but you should start working on it now.
a good, well researched outline of your project, along with a quality reference list.
Let's say that I was doing a project about using the Sunyaev-Zeldovich Effect to get H0
. Here are
examples of outlines: