Astr/Phys 328/428 Homework #3

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Suppose 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 also that 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?

(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 M_sun/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?

Here are Fundamental Plane datasets for two galaxy clusters:

• Coma (cz=7008 km/s): Jorgensen et al (1993)
• S639 (cz=6545 km/s): Jorgensen & Jonch-Sorensen (1998)

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.

1. Using the Coma data given in Table 1 of Jorgensen et al (1993), derive the zeropoint of its B-band Fundamental Plane: log(r_e) = 1.24*log(sigma) - 0.82*log<I> + ZP. Note that log<I>=-0.4*(<mu>-26.4).
2. 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.
   
3. In both cases, estimate not only the zeropoint but its uncertainty.
   
4. Plot the data for each cluster on a Fundamental Plane plot, along with a line showing the Fundamental Plane with your fitted zeropoints.
5. 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 relative distances.
6. 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.
7. 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.
   

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.

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 (90^o=edge on, 0^o=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 uncertainties.

• 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 dust.

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:  M_V= -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?

Note: this will be due with HW #4 but you should start working on it now.

I want 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:

• A bad outline I do not want to see.
• A good outline, like what I want.