Astr 222 - Homework #4
1. A Few Short Math Problems (10 points)
- A star cluster 5 kpc away is
made up of 5,000 G stars, each of which has an
absolute magnitude MV=+4.6. What is the star cluster's total
- 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. 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 21.0 mag/arcsec2, what is the luminosity density in the center of the galaxy?
3. 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:
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
- the total luminosity of the galaxy (in terms of I0 and h)
- the half-light radius of the galaxy (in terms of h)
- the radius containing 98% of the total light (in terms of h).
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
the mass needed to give this circular velocity? How much disk mass is
inside r=30 kpc? So how much dark matter do we need? So inside 30 kpc,
percentage of the Galaxy's mass is in dark matter?
Some studies suggest that
curve of the Galaxy remains flat out to 150 kpc (or further!). If so,
fraction of the Galaxy's mass (inside 150 kpc) is in the form of dark
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)
- 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:
(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
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
around your work (except for the exponents on the variables). Don't
worry about the fundamental plane for this part:
- Write down an
(#2) for how mass depends on velocity and size.
- Write down an
expression (#3) for how luminosity depends on surface brightness and
- Combine #2 and #3 to
show how the mass-to-light ratio of the galaxy (M/L) depends on sigma,
I, and R
- 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
to surface brightness, and that to a very good approximation, the mass-to-light ratio of an elliptical galaxy
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.