Problem set 6 for ASTR 323/423: The Local Universe
Due in class Friday April 18
(1) Individual problem: Sparke and Gallagher Problem 2.9
(2) Group problem for undergrads, individual for grads:
Assume that you are observing stars in a square degree at the NGP,
that the thick disk has a scale height of 1 kpc and the thin disk 300
pc, and the halo has an r^-3 density distribution. Furthermore, assume
that the ratio of thin to thick to halo stars at the Sun is
1000:100:1 and that the local density of your tracers (halo giants) is
35 per kpc^3. Draw plots of the number of stars per square degree as a
function of z height out to 15 kpc. At what distance will the counts of thick disk stars peak?
halo stars? (You will need to account for the variation in volume
element along the line of sight as well as the variation in star
density.)
(3) Individual problem: The stellar luminosity function (4) Group problem for undergrads, individual for grads: Work out what color range you should use to identify clump stars
in the 2MASS data by plotting the (J-Ks)0 color range
for clump stars vs their absolute magnitude Mk in 47 Tuc and the following four open clusters:
NGC 2477 ([Fe/H]=+0.07), NGC 2266 ([Fe/H]=-0.38), NGC 2682 (M67,
[Fe/H]=0) and NGC 2506 (Fe/H]=-0.19). You will need to work out the reddening in J-Ks
towards the clusters. Use the compilation of globular cluster properties at
http://physwww.physics.mcmaster.ca/%7Eharris/mwgc.dat
for 47 Tuc and the Diaz et al values for the open clusters. Diaz et
al give transformations between E(B-V) and E(J-Ks). Do you see any
trends with age or metallicity in the color range occupied by the
clump? Do you agree with the value of Mk quoted above? Also estimate
the position of the turnoff for each of these open clusters, and 47
Tuc.
(b) Download some data from the North Galactic Pole to get a feeling
for what a field with almost no reddening looks like. You will need to use
the GATOR server, at
http://irsa.ipac.caltech.edu/applications/Gator/.
(Use the 2MASS all-sky point-source catalog. There is a
tutorial for GATOR.) Use the maximum
radius (3600 arcsec), enter galactic coordinates, and download J
and Ks colors for everything brighter than Ks=14. Plot Ks vs J-Ks. You may
need to download more than one such field to get enough stars in this
low-density region.
(c) With MK=-1.65, and K magnitudes from say 8 to 14, what
distances are probed in the Galaxy? Use this information to help
interpret your NGP figure: are you seeing stars from the thin disk,
thick disk or halo in the different regions of the CMD? Take the
metallicity and age distributions of each population into account,
looking particularly at the turnoff and the giant branches.
(d) Choose a low-latitude line of sight with a low reddening. Use the
maps of Burstein and Heiles (AJ, 87, 1165, 1982) to find a good field.
Give (l,b) and E(B-V) and E(J-Ks) for this field.
Download data from this direction, plot a de-reddened CMD, calculate
distances for each clump star in the sample (justifying your color
range), and plot a histogram of distances for the stars. Justify your
choie of color range to identify clump stars. Once again, use the
distance modulus calculation to work out whether the stars you are
using belong to thin disk, thick disk or halo. Does your answer make
sense? Why or why not?
(e) (Graduate students only) Using a double exponential disk model for the Milky Way thin disk with
realistic estimates for scale length and scale height,
predict how the number of stars should vary with distance along the
line of sight. Give the equations you use to connect l and b and distance to
galactic X,Y,Z coordinates. How good a fit is the model to the data?
You should only worry about the variation along the line of sight, not
the absolute normalization of number of stars.
(f) (Grad students only) Are there any assumptions that we have made
above that are likely to produce significant
errors? Explain why. Can you suggest another region
of the CMD which might work better than the clump?
Download the Preliminary version
of the "Catalog of Nearby Stars 3" (CNS3) of Gliese and Jahreiss,
which you can access from
http://vizier.u-strasbg.fr/viz-bin/VizieR?-source=V/70A. This
catalog gives information on all known stars (in 1991) within 25 pc of
the Sun. Within its limitations, this was one of the best
volume-limited samples available in astronomy at that time. Use this catalog to calculate the local stellar luminosity
function in MV. Make sure you download both the parallax
and its error. Things to think about:
(a) There are roughly 3000 stars in
the catalog, but they are not uniformly spread in MV. What might be a good binsize to use here?
(b) For what absolute magnitude ranges is this catalog likely to be
incomplete? Why?
(c) How might you derive the errors on the luminosity function for
each bin?
(d) What units do we usually use for a stellar luminosity
function?
In
this problem you will use data from the Two Micron All-Sky Survey (an
all-sky survey in the near IR) to model the distribution of stars in
the Milky Way disk. Using the near IR has the advantage that the
effect of dust is much smaller. We will use the "red clump": the
He-burning stage of stellar evolution which is also referred to as the
"red horizontal branch" for older clusters, which is claimed to have
an absolute magnitude which does not change much with age or
metallicity.
There have been a
number of recent papers which have assumed that all stars in a given
J-K color range are clump stars with the same absolute magnitude
(MK=-1.65) and used the magnitudes of these stars to
estimate their distance distribution along the line of sight.
(a) Look at the Hipparcos color-magnitude diagram and identify the red
clump on it; you can see that
red clump stars are quite common. What is the range
of B-V color of the clump stars in this diagram? of absolute magnitude Mv?
Salaris et al (2007: A&A 476, 243) present a Ks vs J-Ks CMD of 47 Tuc,
a globular cluster with age around 12 Gyr and metallicity around -0.7. Hesser et al (1987) present B and V photometry of 47 Tuc.
For younger and more metal-rich open clusters, Dias et al A&A 539, A125 (2012) present CMDs for a number of open clusters and derive ages, reddening values and distances.
When you look at a CMD of the field, stars are found at
different distances and so a cluster CMD is "blurred" in apparent
magnitude in the regions of distance that a given population dominates.
Explain the broad features you see in this diagram, using the information about the color of the clump for stars of different metallicity from part (a), and the positions of the turnoff for both the younger metal-rich open clusters and the old intermediate-metallicity 47 Tuc. What
evolutionary states dominate in each region? You should
be able to find, and mark on the CMD, the giant branch, clump and two different parts of the
main sequence.