1. A Few Short Math Problems

A star cluster 5 kpc away is made up of 5,000 G stars with absolute magnitude M_V=+4.6. What is the star cluster's total apparent magnitude?

A spiral galaxy has a recession velocity of 4,500 km/s, a total apparent magnitude of m_v=12 and a size of 1.2 arcminutes. How far away is it (in Mpc), what is its total V luminosity (in solar luminosities), and what is its size (in kpc)? What is its redshift?

If we want to measure the properties of an elliptical galaxy in its inner 100pc, from the ground (with resolution 1"), how far away can it be? What if we do it from space using the Hubble Space Telescope (resolution ~ 0.1")?

2. Rotation Curves and Dark Matter

We said that the distribution of stars in the Galaxy's disk was given by an exponential function. If stars dominate the mass distribution of the disk, we can also say that the mass distribution of the disk follows an exponential function, like this:

Using this function, 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). Remember that for a disk of material, the mass interior to a radius r is given by

For your expression for the rotation curve, I want something that looks like V_c(r)=f(Sigma₀, h, and r).

Okay, now if Sigma₀=780 M_sun/pc² 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

In class, I wondered if the infamous "dark matter" surrounding galaxies could be made up 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 M_sun. If it was made of FFSDs, what would the bolometric luminosity of the dark matter halo be? What would the peak wavelength of this light be? How much brighter or fainter is that than the luminosity due to the stars? Do you think we'd be able to detect it?

4. Stickman Lives!

Okay here is redshift data from the Second CfA Redshift Survey. Use it to make Stickman. Tell me how you did it.

Estimate the size of the voids (you know, underneath Stickman's armpits). Let's say that I told you that galaxies formed in the voids early on in the Universe, but have since moved out due to peculiar velocities. If typical peculiar velocities are ~ 600 km/s (like that of our galaxy with respect to the CMB), how long would it take for galaxies to clear the void (express your answer both in years and in terms of the Hubble time [ie 1/H₀])? Show how both your answers depend on the Hubble Constant (in other words, if you decided to use a different value for the Hubble constant, how do your answers change?). Is my idea of galaxy formation in voids any good?

5. The Fundamental Plane

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 (r_e) in kpc
the velocity dispersion ( sigma) in km/s
the mean surface brightness (I) in L_sun/kpc²

Plot, fit a straight line, and calculate the scatter in the relationship:

log(r_e) vs log(sigma)
log(r_e) vs log(I)

(You should always have log(r_e) on the y-axis and the other variable on the x-axis, so that in every case you are measuring the scatter in log(r_e). 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 r_e~ sigma^xI^y, where x=1.24 and y=-0.82. Plot log(r_e) vs log(sigma^xI^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) ~ L^aI^b. 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.

Astr 222 - Homework #4

1. A Few Short Math Problems

2. Rotation Curves and Dark Matter

3. Donkey Dark Matter

4. Stickman Lives!

5. The Fundamental Plane