# Astr 222 - Homework #4

## 1. Surface Brightness (10 points)

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=26.3 mag/arcsec2 in the V band. From this, we can express the relationship between surface brightness and luminosity density as mu=26.3 - 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 muV=21.0 mag/arcsec2, what is the luminosity density in the center of the galaxy? We are now discovering really faint "ultradiffuse galaxies" that have central surface brightnesses of muV=27.0 mag/arcsec2 -- what does that correspond to in terms of luminosity density?

## 2. Disk Galaxies: Luminosity, Rotation Curves and Dark Matter (20 points)

If a disk galaxy has an exponential surface brightness profile like this: I=I0exp(-r/h), where h is the scale length and I0 is the central luminosity density (in Lsun/pc2), calculate:
1. the total luminosity of the galaxy (in terms of I0 and h)
2. the half-light radius of the galaxy (in terms of h)
3. the radius containing 98% of the total light (in terms of h).
If the galaxy has a constant stellar mass-to-light ratio (M/L)*, derive an analytic expression for what the rotation curve of the galaxy should look like (ignore the bulge and halo of the Galaxy for this calculation). For your expression for the rotation curve, I want something that looks like Vc(r)=f(I0, (M/L)*, h, and r).

Okay, now if the galaxy has a central surface brightness mu0=19.2 mag/arcsec2, (M/L)*=1 Msun/Lsun, and h=3.5 kpc, plot what the rotation curve of the galaxy should look like from r=0 to r=30 kpc. (Hint: if it looks flat, you screwed up.)

Now, the observed rotation curve is flat; at r=30 kpc, the circular velocity is still ~ 220 km/s. What is the mass needed to give this circular velocity? How much disk mass is there inside r=30 kpc? So how much dark matter do we need? So inside 30 kpc, what percentage of the Galaxy's mass is in dark matter?

Some studies suggest that the rotation curve of the Galaxy remains flat out to 150 kpc (or further!). If so, what fraction of the Galaxy's mass (inside 150 kpc) is in the form of dark matter?

## 3. Donkey Dark Matter (15 points)

One of my favorite ideas about the infamous "dark matter" surrounding galaxies is that it is made up of a population of free floating space donkeys (FFSDs). FFSDs would not radiate in the optical (have you ever seen a donkey shine?) but would emit light in the infrared (since they would have little heat generators in the donkey space suits to keep them warm). So let's see if we can rule out this model.

Say the dark matter halo of a bright spiral galaxy like the Milky Way has a mass of 1012 Msun. If it was made of FFSDs, what would the bolometric (total) luminosity of the dark matter halo be? (Assume FFSD are blackbody radiators.) What would the peak wavelength of this light be? Compare this to the bolometric luminosity of the Milky Way stars (~5x1010 Lsun) -- how many times brighter or fainter are the FFSDs than the stars? Do you think we'd be able to detect them?

## 4. The Fundamental Plane (20 points)

• Here is a table of data for elliptical galaxies (from Bender, Burstein, & Faber 1992, ApJ, 399, 462) :
• the catalog number of the galaxy (ngc)
• the effective radius (re) in kpc
• the velocity dispersion (sigma) in km/s
• the mean surface brightness (I), converted to linear units of Lsun/pc2
• Plot, fit a straight line, and calculate the scatter in the relationship:
• log(re) vs log(sigma)
• log(re) vs log(I)
(You should always have log(re) on the y-axis and the other variable on the x-axis, so that in every case you are measuring the scatter in log(re). Also remember that "log" means "log10".)
• Now we want to show that the fundamental plane relating the three quantities is a much better fit than any of the above relations which involve just two of the quantities.
• The expression (#1) for the fundamental plane given in class was re ~ sigmaxI y, where x=1.24 and y=-0.82. Plot  log(re) vs log(sigmaxI y) , fit a straight line, and calculate the dispersion around the Fundamental Plane. Compare that dispersion to those in the previous plots involving just two parameters.
• Okay, in the rest of this we are going to ignore constants and work simply with variables - r, M, sigma, I, L, etc. That means that I don't want to see any pi's, 2's, G's or anything else running around your work (except for the exponents on the variables). Don't worry about the fundamental plane for this part:
• Write down an expression (#2) for how mass depends on velocity and size.
• Write down an expression (#3) for how luminosity depends on surface brightness and size.
• Combine #2 and #3 to show how the mass-to-light ratio of the galaxy (M/L) depends on sigma, I, and R (#4)
• Now bring the fundamental plane in. Use expressions #1 and #4 to derive the expression (M/L) ~ LaIb. b should turn out to be a very small number (<0.05), meaning that the mass-to-light ratio is very insensitive to surface brightness, and that to a very good approximation, the mass-to-light ratio of an elliptical galaxy depends primarily on its luminosity.
• Make some (Astr222-educated) arguments about why elliptical galaxies have different mass-to-light ratios and why more luminous ellipticals might have higher mass-to-light ratios. Think about both stellar populations and dark matter.