ASTR 222 - Homework #3

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1. Population Synthesis

Now, a "typical" color for a spiral galaxy like the Milky Way is B-V=0.7, an elliptical might have a color of B-V=1.0, and a starburst galaxy might have B-V=0.4. Which of these galaxies is a good match for an elliptical, which for a spiral, and which for a starburst? Which two galaxies don't make sense? Argue your answer both from integrated colors and from the mix of stellar types.


To give you a push in the right direction, here is what my spreadsheet looks like if I put in a mix of 25% of each type of star. So use this as a test case, if you can get these numbers for that kind of mix, you've got it right:

2. Surface Brightness calculations

• If a galaxy is observed face on (with no dust) and has a surface brightness equivalent to one solar-type star per square parsec, show that the surface brightness in magnitudes per square arcseconds is mu=27.05 mag/arcsec² in the B band. From this, show that the relationship between surface brightness and luminosity density is mu=27.05 - 2.5 log I, where I is the luminosity density of the galaxy in solar luminosities per square parsec. If a galaxy has a central surface brightness of 21.65 mag/arcsec ², what is the luminosity density in the center of the galaxy?
• If a disk galaxy has an exponential surface brightness profile like this: I=I₀exp(-r/h), where h is the scale length and I ₀ is the central luminosity density, calculate:
• the total luminosity of the galaxy
• the half-light radius of the galaxy
• the radius containing 98% of the total light.

3. The Tully-Fisher Relationship

• In class, we made arguments about why we might expect L ~ v⁴ for spiral galaxies. Show analytically that if we plotted absolute magnitude against log(v), we would expect this line to have a  slope of 10.

 
• In general, unless we know the distances of galaxies, we can't make a Tully-Fisher plot. But we can be crafty and realize that if we look at galaxies in a cluster we can plot apparent magnitude against log(v) and get the same slope. Why?  

So here is a Tully Fisher dataset for galaxies in the Virgo Cluster (from Pierce & Tully 1988, ApJ, 330, 579). The dataset has

• the NGC number of each galaxy,
• apparent magnitude of the galaxies in the B, R, and I band,
• the observed rotation speed of the galaxies
• the inclination of the galaxy to the line of sight (90^o=edge on, 0^o=face on).

• 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 true rotation speed) in the B, R, and I bands, and for each case, fit a line of the form m=a*(logV-2.5)+b
• What is the slope of the line in each case? When is it closest to the "expected" value?
• What is the dispersion around the line in each case?
• 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 are the reduced light curves (from Freedman etal 1994, Nature, 371, 757). These are plots (one for each detected Cepheid) of apparent magnitude on the y-axis and time in days ("phi") on the x-axis. The derived period is shown in each frame as "P=xx".

Using the Cepheid period-luminosity relationship given in class -- (M=-2.76*log(P)-1.18) -- calculate a distance estimate to M101 for each Cepheid and then average them to give your best estimate for the M101 distance. Also give a statistical uncertainty to your distance.
 

• Using this distance, calibrate the Tully-Fisher relationship: M = a*(logV-2.5) + b. What are a and b?
  
 
• What do you feel are the main sources of uncertainty in your derivation of the T-F relationship?

• 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 65^o . What is the distance to Coma? What is the uncertainty in your distance?