In star count lingo, N(m) is defined to be the number of stars observed with apparent magnitude brighter than m. As we saw in class, if the Galaxy had a uniform density of stars and is infinite in extent, we should see the star counts go as logN(m) ~ 0.6m (even if stars dont all have the same brightness).

Here is a table of integrated star counts logN(m) (again, N(m)=number of stars brighter than apparent magnitude m) from Allen's "Astrophysical Quantities." The columns are

With this data, make a single plot showing log N vs m in the two different directions. Also include on the plot the expected logN-vs-m relation for an infinite, uniform distribution of stars (given above). Explain qualitatively why the star counts in different directions are different from each other, and also why they are different from the uniform model.

- mag: apparent magnitude (m)
- logN_up: log N(m) in a direction up out of the galactic plane
- logN_in: log N(m) in a direction towards the galactic center

Note: when I say "plot this versus that", it means that "this" goes on the y-axis and "that" goes on the x-axis. So in the plot I'm asking you to make for this problem, log N goes on the y-axis and m goes on the x-axis.

Use differential calculus to
show that if you are using distance modulus (m-M) to get the
distance to an object, if you have a magnitude uncertainty of dm
(where dm is small), you get a fractional uncertainty in
distance of approximately 0.5*dm. In other words, as an example,
if your distance modulus error is 0.1 magnitudes, your distance
uncertainty is about 5%.

Hipparcos was a satellite
mission which obtained parallax data for a large number of
stars. Using this data, we can construct a calibrated
Herzprung-Russell diagram (meaning absolute magnitude versus B-V color) for nearby stars. Here is a table of parallax data for all
stars in the Hipparcos catalog which

- are classified as main sequence stars
- have parallax > 100 milli-arcseconds (ie relatively nearby)

- have parallax uncertainty < 1 milli-arcsecond (ie good parallax data)
- have color uncertainty < 0.2 magnitudes (ie good photometric data)

The columns in the datafile are

- p: parallax [milli-arcseconds]
- sig_p: parallax uncertainty [milli-arcseconds]
- v: apparent V magnitude [mags]
- bv: B-V color [mags]
- sig_bv: B-V uncertainty [mags]

Use
the data to make a calibrated color magnitude diagram of these stars,
again remembering that "calibrated" means we are plotting absolute
magnitude versus B-V color. And remember -- color magnitude diagrams
must always have bright blue stars in the upper left!

Now, overplot a theoretical zero age main sequence for solar metallicity (Z=0.02) stars. The data file has:

Describe and explain differences between the ZAMS and your Hipparcos dataset.

Now, overplot a theoretical zero age main sequence for solar metallicity (Z=0.02) stars. The data file has:

- mass: stellar mass [Msun]
- M_V: absolute V magnitude
- B-V: B-V color

Describe and explain differences between the ZAMS and your Hipparcos dataset.

Here is a dataset for an open cluster known as
"The Stable" (columns: apparent V mag and B-V color). The Stable has a
reddening of E(B-V)=0.25 magnitudes and roughly solar
metallicity. Correct the colors and magnitudes for the dust
(explain how you did this!) and the plot an observed color magnitude
diagram (apparent
mag vs color) for the Stable.

Then take the solar metallicity ZAMS, derive a distance to the Stable using main sequence fitting. Using your derived distance, convert the ZAMS absolute magnitudes to ZAMS apparent magnitudes ,and overplot the apparent magnitude ZAMS on your Stable data to show how good your match is.

Estimate the error in your distance to the Stable, and explain what the sources of error are.

Then take the solar metallicity ZAMS, derive a distance to the Stable using main sequence fitting. Using your derived distance, convert the ZAMS absolute magnitudes to ZAMS apparent magnitudes ,and overplot the apparent magnitude ZAMS on your Stable data to show how good your match is.

Estimate the error in your distance to the Stable, and explain what the sources of error are.

What is the B-V color of the stars at the main sequence turnoff? What is the age of this star cluster? (Use Figure 13.19 from Carroll and Ostlie to help you with this question.)

Here is photometry of Laungheer 413, a globular cluster. Again, you have apparent V magnitude and observed B-V color (note, however, that this dataset does not include red giant branch stars, only main sequence stars and stars just starting to evolve off the MS). Laungheer 413 has a reddening of E(B-V)=0.1 and a metallicity of [Fe/H]=-0.76. Figure out the distance (and distance uncertainty) of Laungheer 413, the same way you did for the Stable. Since it is metal-poor, you'll want to compare it to this metal-poor (Z=0.004) ZAMS. Also use the main sequence turnoff point to get a rough age estimate, like you did for the Stable.

Laungheer 413 has one RR Lyrae variable star in it: V9, with a mean V apparent magnitude of 14.685 and period of 0.737 days. What is the mean absolute magnitude of V9 (remember to correct for the dust!)?If you mistakenly thought it was a Cepheid, what would you have derived for its mean absolute magnitude given the Cepheid period-luminosity relationship? Under that (mistaken) assumption, what would you then estimate of the distance to Laungheer 413 to be?

- A (very strange!) star cluster is made up of 10
^{6}stars identical to the Sun (M_{V}=+4.82, B-V=0.65). What is its total V-band absolute magnitude? What is its B-V color? - Another very strange star cluster is made up of 10
^{6}Suns and another 10^{5}red giants (M_{V}=+1.00, B-V=1.0). What is its total V band absolute magnitude? What is its B-V color? What fraction of the total V band light of the cluster is coming from the red giants?

*The Stable is a real cluster in disguise -- it's data for the Hyades open cluster, shifted to a different distance.

**Similiarly, Laungheer 413 is also a cluster in disguise, this time it's the globular cluster 47 Tuc.