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/arcsec^{2} 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 mu_{V}=21.0 mag/arcsec^{2},
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 mu_{V}=27.0 mag/arcsec^{2} -- 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=I_{0}exp(-r/h), where h is the scale length and I_{0} is the central luminosity density (in Lsun/pc^{2}), calculate:
- the total luminosity of the galaxy (in terms of I_{0} 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).
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 V_{c}(r)=f(I_{0}, (M/L)_{*}, h, and
r).
Okay, now if the galaxy has a central surface brightness mu_{0}=19.2 mag/arcsec^{2}, (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 10^{12} M_{sun}.
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 (~5x10^{10} L_{sun}) -- 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 (r_{e})
in kpc
- the velocity dispersion (sigma) in km/s
- the mean surface brightness (I), converted to linear units of L_{sun}/pc^{2}
- 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^{x}I^{
y}, where x=1.24 and y=-0.82. Plot log(r_{e}) vs
log(sigma^{x}I^{ 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^{a}I^{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.