Astr 328 Homework #2

1. Distance ladder

(this is Problem 7.6 from Binney & Merrifield's "Galactic Astronomy", on reserve in the library)

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 that the calibration of each standard candle carries an uncertainty of 0.2 magnitudes. Show that, by changing the calibration fo each step within the range allowed by its uncertainty, it is possible to derive values for the Hubble constant that lie anywhere between ~ 0.7 and ~ 1.4 of the nominal value.


2. Calibrating Tully-Fisher
(problem courtesy Heather Morrison)

To use the Tully-Fisher relationship to estimate distances to distant galaxies, we need to first calibrate it using nearby spiral galaxies with known distances. Here is a list of the galaxies generally used to calibrate TF:

 

Galaxy Absolute I magnitude Apparent I magnitude Observed Log10(W20)
NGC 224 -23.11 +/- 0.18
2.734 +/- 0.028
NGC 598 -20.31 +/- 0.11
2.305 +/- 0.11
NGC 925
9.17 +/- 0.27 2.339 +/- 0.049
NGC 1365 -23.40 +/- 0.12
2.628 +/- 0.035
NGC 1425
9.42 +/- 0.05 2.575 +/- 0.041
NGC 2090
9.15 +/- 0.07 2.501 +/- 0.035
NGC 2403 -20.38 +/- 0.28
2.408 +/- 0.059
NGC 2541
10.60 +/- 0.10 2.298 +/- 0.049
NGC 3031
5.30 +/- 0.15 2.647 +/- 0.034
NGC 3198 -21.56 +/- 0.08
2.507 +/- 0.032
NGC 3319
10.45 +/- 0.07 2.342 +/- 0.048
NGC 3351 -21.64 +/- 0.09
2.435 +/- 0.047
NGC 3368 -22.31 +/- 0.11
2.564 +/- 0.036
NGC 3621
8.07 +/- 0.05 2.453 +/- 0.035
NGC 3627 -22.66 +/- 0.18
2.583 +/- 0.026
NGC 4414 -22.61 +/- 0.11
2.600 +/- 0.039
NGC 4535 -22.28 +/- 0.08
2.464 +/- 0.038
NGC 4536 -21.95 +/- 0.13
2.529 +/- 0.030
NGC 4548
8.87 +/- 0.05 2.406 +/- 0.046
NGC 4725 -22.77 +/- 0.09
2.562 +/- 0.026
NGC 7331 -23.38 +/- 0.11
2.713 +/- 0.021

Step 1. Get distances:

For galaxies with only apparent I magnitudes, go to the home page of the HST Distance Scale Key Project Team and grab the Cepheid properties for each galaxy. For each galaxy, plot a period-luminosity diagram for its Cepheids, and fit it to the calibrated Cepheid period-luminosity relationship given in class. Fitting the zero point of the relationship gives you the distance modulus m-M which will let you convert apparent I mag to absolute I mag.
Keep track of and propogate your errors! Errors in the distance modulus lead to errors in the absoute magnitude.


Step 2. Correct velocities for inclination:

For each galaxy, look at its image in the NASA Extragalactic Database. From the images, estimate the inclination by measuring the ratio of the major to minor axis of the galaxy, and then using the relationship i=cos-1(b/a). The use the inclination to correct the velocity width.

Keep track of and propogate your errors! Errors in the inclination lead to errors in the velocity width.


Step 3. Plot a Tully-Fisher relationship of absolute I mag vs logW20:

Fit the slope and the zeropoint of the Tully Fisher relationship. Give your fit parameters, plus errors.


Step 4: Think and discuss:


3. The Difference between "Near" and "Far"

(problem also 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, so plot y vs z to see the large scale structure in the simulation slice.

Assume that the Sun is at the coordinate (50,0,0), and calculate the inferred Hubble constant from each of the following samples:
To do this, use the known distance of the galaxies (calculated from the coordinates) and the line-of-sight velocity. What do you estimate for the value of the Hubble constant used to produce the simulation?

Comment on the accuracy of using the two relatively nearby samples: how much of an error does the peculiar velocity of each galaxy add?

Now repeat this using only elliptical galaxies (star formation rate = 0). Are your results different? Why?

4. Grad Student Project: Bias in Tully-Fisher estimates of H0.
Write a code to simulate the effects of Mahlmquist bias. Assume a value for the Hubble constant, and for the Tully-Fisher relationship. Lay down galaxies randomly in space and in circular velocity, then assign them a luminosity based on their circular velocity, and an apparent magnitude and a recession velocity based on their distance. Then "build" a sample of galaxies out to some limiting apparent magnitude, and use it to derive H0. Did you get out what you put in (hopefully, yes).

Now do the same thing, but this time add a random dispersion in the calibration of the T-F relationship of 0.2 magnitudes. Then 0.4, then 0.7. How does your derived H0 change?
Now, using a fixed dispersion of 0.4 magnitude in your T-F relationship, vary your limiting magnitude. How does your derived H0 change?

Explain why you see these effects.


Groupwork

(The rules of this project are the same as in HW #1)

Use the fundamental plane to derive the distance to the Coma cluster. Make sure you demonstrate both calibrating and using the fundamental plane, and that you include an uncertainty calculation. Then do a literature search and find distances to the Coma cluster derived using at least two independant methods. Describe those other methods and compare their answer to your result. Which method (yours or others) do you find most reliable, and why?