ASTR 222  Homework #3
1. Population Synthesis
(You can code this one up in Python, or alternatively an Exceltype spreadsheet would
work great for this problem, as well...)
We are going to make galaxies by mixing stars. Here are the four
types of stars we are going to use. For
each type of star, describe in words what kind of star it is: what
evolutionary stage it is in, what kind of lifetime it has, what limits
(if any) you can place on its age. Then calculate its stellar masstolight
ratio (M/L)_{*}.
(Remember that masstolight units are solar units, so the Sun has a masstolight ratio of (M/L)_{*} = 1 Msun/Lsun. And also remember
that the Sun has an absolute magnitude of Mv=4.82 and a BV color of
0.65.)

Star 1

Star 2

Star 3

Star 4

Spectral Type

A2V

G2V

K5V

K2III

Mv

1.3

4.82

7.35

0.5

BV

0.05

0.65

1.15

1.16

Mass (Msun)

2

1

0.67

1.1

(M/L)_{*} (fill this in!)





Now let's build some galaxies. The galaxies should each have a
total V luminosity of L_{v}=10^{10} Lsun. The fraction
of V light
each star contributes to each galaxy is given in the table below. Calculate the total (aka "integrated")
BV color and stellar Vband (M/L)_{*} ratio, as well as
the fraction
of each
star by number for each model galaxy. Show your work, by walking
through one example by hand in exquisite detail.

Fraction
of Vband light from each star


Star 1

Star 2

Star 3

Star 4

Galaxy 1

15%

40%

25%

20%

Galaxy 2

30%

0%

0%

70%

Galaxy 3

45%

25%

20%

10%

Galaxy 4

0%

30%

70%

0%

Galaxy 5

0%

30%

50%

20%

Now, a "typical" color for a spiral galaxy like the Milky Way is
BV=0.7, an elliptical might have a color of BV=1.0, and a starburst
galaxy might have BV=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:
(In this spreadsheet, masses are given in solar masses (M
_{sun}), and luminosities are in solar luminosities in the V filter (L
_{sun,V})
2. Disk Galaxies: Surface Brightness and Luminosity
Problem 2 is no longer part of HW3. It will be restructured and included as part of HW4.
 If a galaxy is observed face on (with no dust) and has a
surface brightness equivalent to one solartype star per square parsec,
show that the surface brightness in
magnitudes per square arcseconds is mu=27.05 mag/arcsec^{2} 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=I_{0}exp(r/h), where h is the scale
length and I_{0} is the central luminosity density, calculate:
 the total luminosity of the galaxy (in terms of I_{0} and h)
 the halflight radius of the galaxy
 the radius containing 98% of the
total light.
3. The TullyFisher Relationship
 In class, we made arguments about why we might
expect L ~ v^{4} 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 TullyFisher 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 inclination of the
galaxy to the line of sight (90^{o}=edge on, 0^{o}=face
on).
 the observed rotation speed
(W20/2) of the galaxies
 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
TullyFisher plot (apparent mag versus log(V)) in the B,
R, and I bands, and
for each case, fit a line of the form m=a*(log(V)2.5)+b. Report your fit
by giving the parameters a and b as well as their uncertainties.
 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 TullyFisher relationship. Think both about stellar populations and dust.
Now we need to calibrate the
TullyFisher 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 yaxis and time
in days ("phi") on the xaxis. The derived period is shown in each
frame as "P=xx".
Using the Cepheid
periodluminosity relationship given
in class  (M=2.43*log(P)1.62)  calculate a distance estimate to
M100 for each Cepheid and then average them to give your best estimate
for the M100 distance. Also give a statistical uncertainty to your
distance.
 Using
this distance,
calibrate the TullyFisher relationship. You had a TF relationship
that connected apparent magnitude (m) with logV, and now you know the
distance to Virgo to turn apparent magnitude (m) into absolute
magnitude (M), so you can rewrite your TF relationship now in terms of
absolute magnitude: M = a*(log(V)2.5) + b. What are a and b, and their uncertainties?
 What do you feel
are the main sources of uncertainty, both systematic and random, in your derivation of the TF
relationship?
 Now you are looking at a
spiral galaxy in the Coma cluster. It has an Iband 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?