Problem set 1 for ASTR 323/423: The Local Universe
Part (1) is due in class Wed Aug 30,
part (2) in class Wednesday Sept 6, part (3) at 5pm Friday Sept 8.
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://basti.oa-teramo.inaf.it/index.html
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. It will be useful to look at the README file which is at
the top of the page for scaled solar models with canonical
assumptions.
(Part 1) Evolutionary tracks
Q1) Why do we need to know all the different element abundances
in order to calculate the stellar evolution and the observational
quantities such as colors and magnitudes which the Basti models give
us? Answer with about a paragraph.
To get started, download the evolutionary track for a one
solar mass star with solar metallicity (Z=0.0198 on this site) and abundance
patterns (Scaled Solar), using "canonical" assumptions.
Because of studies of the Sun using
helioseismology 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).
Q2) (a) 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.
(b)
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).
(c) Then, in order to see how the evolutionary time varies along
the evolutionary track, plot age against stellar luminosity. Given
what you know about stellar evolution and how the properties of stars
change as they evolve, would a linear or log scale be suitable for
these two quantities?
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 for 2(b) and the age vs. luminosity plot for 2(c), 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.
Q3) Why does the evolution speed up for
later evolutionary stages such as the giant branch and horizontal branch?
Q4) Then investigate, using the evolutionary tracks for a one solar
mass star with several different metallicities varying by more than an
order of magnitude, how evolutionary time on the Zero Age Main
Sequence (up to the exhaustion of the hydrogen core) and also the time on
the first ascent giant branch varies with metallicity.
Summarize the
behavior you see, and make a plot or plots. Note that you will almost
certainly have to mess with the axes to see what you want to
display. Also note that Basti has a list of particular evolutionary
points as line numbers in the files; see the README file. What is the
theoretical expectation here? You should consider the amount of available fuel for nucleosynthesis and also the structure of the star.
Q4) (graduate students only) What is the difference between the AB
magnitude system used by SDSS and Pan-STARRS and the more traditional
Vega magnitude system? Write about 2 paragraphs.
(Part 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.
Q1) 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. They are also described in the README file.
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).
Q2) 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://www.cadc-ccda.hia-iha.nrc-cnrc.gc.ca/en/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. Remember to subtract extinction from V as well as reddening from B-V! If you use V-I, you will need
to use Appendix B of Schlegel et al (1998) to calculate the reddening
in V-I.
Q3) (a) What age do you find is the best match? Use more isochrones to zero in on your best estimate, and plot them up on the CMD.
(b)
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? Can you see a sequence of stars from the
foreground disk? Which evolutionary stage do they likely correspond to?
Q4) (graduate students only) Do the same for one of the oldest, most
metal-rich open clusters, NGC 6791, which has [Fe/H]=+0.4 and
E(B-V)=0.10. You might consider varying the reddening and
metallicity a bit to get a better fit .....
(Part 3) Luminosity functions
These show the number of stars at each luminosity for a population of
stars at a given age. 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.
Q1)
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 this luminosity function.
Q2) By counting stars in bins along the M67 sequence, plot up the number of stars in a given interval of absolute magnitude
as a function of absolute magnitude.
Here you will need to work out any contribution from foreground stars from the disk, which should be excluded from your star counts. You should also exclude blue stragglers. Explain how you did this.
Using your work from (Part 2), mark on
the absolute V magnitude at the beginning of each evolutionary phase.
Q3) Compare the theoretical LF from Basti with your observational
luminosity function of M67. Do you see what you expect in terms of the
number of stars in each absolute 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.
Q5) (graduate students only) In a similar way, compare observational and theoretical luminosity functions for NGC 6791.
Good references for these topics are my 221
notes
http://astroweb.case.edu/heather/221.14/index.html, Section 2 on Stars; and Carroll
and Ostlie's book, on reserve in the library if you don't 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.