Problem set 1 for ASTR 323/423: The Local Universe
Part (1) is due in class Friday Jan 17, part (2) in class Tuesday Jan
21, part (3) in class Friday Jan 24. I will choose a student at
random on each day to present their solution to the class.
This homework set is to help you review the material on the
color-magnitude diagram, the luminosity function and stellar evolution
which you have encountered in previous classes. These will be very
important as we discuss the properties of galaxies. It will also give
you valuable experience in the use of the state-of-the-art BaSTI
stellar models.
Go to the BaSTI home page at http://albione.oa-teramo.inaf.it/
They also provide models both for stars with the
solar pattern of abundances (Scaled Solar), suitable for younger stars
in the Galaxy's disk, and for an alpha-enhanced pattern of abundance
(α -enhanced); appropriate for stars in our Galaxy's halo and
thick disk.
(1) Evolutionary tracks
To get started, download the evolutionary track for a one
solar mass star with solar metallicity (Z=0.0198) and abundance
patterns (Scaled Solar), using "canonical" assumptions. Because of studies of the Sun using
asteroseismology and other detailed techniques, this is the best
constrained stellar model we have. Use η of 0.4 (this is the
Reimers (1975) mass loss formalism with the standard choice of
parameter). Why do stellar evolutionary models need to worry about
mass loss?
This evolutionary track follow's a star's evolution
from the Zero Age Main Sequence, as a function of time (log age in the
first column; remember astronomers use log10 unless
otherwise noted). It gives the star's mass, luminosity, temperature
and observational quantities such as absolute magnitude and colors in
various filters. The evolutionary tracks here are for the UBVRIJKL
photometric system.
Plot up the evolutionary track on the color-magnitude diagram, both in
theoretical units (Teff vs luminosity in units of LSun) and
experimental units (a color such as B-V or V-I vs absolute magnitude
MV).
Then, in order to see how the evolutionary time varies along the
evolutionary track, plot age against stellar luminosity. Review (from
your 221 notes or equivalent) the different stages of nucleosynthesis
(and other important evolutionary points) that happen to a low mass
star from the main sequence until it leaves the asymptotic giant
branch to become a planetary nebula. On the theoretical
color-magnitude diagram, label the point at which each of these
changes occurs, sketching the interior structure of the star at each
point, including convective and radiative areas and where
nucleosynthesis is happening.
Then demonstrate, using the evolutionary tracks for a one solar mass
star with several different metallicities varying by more than an
order of magnitude, that evolutionary time on the first ascent giant
branch is almost independent of metallicity. What is the theoretical
reason for this?
(2) Isochrones
Isochrones show the evolution of a population of stars of the same metallicity but varying initial mass, "frozen" at a certain instant in time. Since we think that to first order, star clusters were formed at a single time, out of gas whose metallicity does not vary, they are particularly useful to compare with cluster color-magnitude diagrams.
Download the isochrones for the solar metallicity, scaled solar
population. NB: evolutionary tracks are at the top of the BaSTI page for a given metallicity and value of η,
isochrones at the bottom. All the isochrones are contained in a
gzipped tar file which contains isochrones. You will then have files
for all ages from 0.03 to 19 Gyr. The first lines of each file show
its age, metallicity, etc, so you can work out the convention and
meaning of the long filenames, which should start with wz.
Plot up color-magnitude diagrams for 0.5, 2, 5 and 10 Gyr age
populations, showing both in theoretical units (Teff vs luminosity in
units of LSun) and experimental units (a color such as B-V
or V-I vs absolute magnitude MV).
To make sure you are on the right track, compare the isochrones with
the exquisite photometry of Peter Stetson for the "solar twin" open
cluster M67, thought to have an age of about 4 Gyr and solar
abundance. You can download this from Stetson's Photometric Standard
Fields at
http://www3.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/community/STETSON/standards/
as soon as you work out M67's NGC number.
Choose either B-V or V-I as a color, and overplot your best match,
using a reddening of E(B-V) = 0.04 mag and a distance modulus
(m-M)0 = 9.61 for the cluster. If you use V-I, you will need
to use Appendix B of Schlegel et al (1998) to calculate the reddening
in V-I. What age do you find is the best match? Which part of the
isochrones fit best? Which parts don't fit so well? Speculate about causes of the poor fits. The expert eye can spot another "sequence" of stars
about 0.75 mag brighter than the main sequence in this cluster: what
do you think these stars are?
(3) Luminosity functions
These show the number of stars for a population of
stars at a given age at each luminosity. They are derived from the
evolutionary tracks, because one of the main contributors to the
luminosity function is the time that a star will spend at a given
temperature and luminosity.
Download the luminosity function (LF) which corresponds to your best fit
isochrone for M67. For this you will need to use the web tool at the
bottom of the Stellar Evolutionary Models page and upload the
appropriate isochrone. To simplify matters, only download the
luminosity function which follows the evolution to the RGB tip.
Plot up the number of stars in a given interval of absolute magnitude
as a function of absolute magnitude. Using your work from (2), mark on
the absolute V magnitude at the beginning of each evolutionary phase.
Compare this with your data from M67. Do you see what you expect in
terms of the number of stars in each absoute magnitude interval? In
parts of the color magnitude diagram where you do not find agreement
with the LF, give some possible observational reasons why this might
be the case.
Good references for these topics are my 221
notes
http://astroweb.case.edu/heather/221.13/index.html and Carroll
and Ostlie's book, on reserve in the library if you dont own a copy.
Basti provides models both for individual stars and single-age
systems (Stellar Evolutionary Models) which you will use on this
homework, and for more complex stellar populations such as galaxies
(Population Synthesis Models) which will be the subject of a future
homework set.
Why does the evolution speed up for
later evolutionary stages such as giant branch and horizontal branch?