 |  |
|---|---|
| Figure 1: Large Magellanic Cloud | Figure 2:Henrietta Leavitt |
| Image by David Malin |
Goals of this project
You must also have read the IDL primer. This is a data intensive lab and you will not be able to waste much time the first period reading the notes.
Grading Policy
Grading is based on the quality of the data analysis, the discussion of the uncertainties, and the answers to the questions at the end of this page.
| |
|---|---|
| Figure 1: Large Magellanic Cloud | Figure 2:Henrietta Leavitt |
| Image by David Malin |
In 1885 the Harvard College observatory begin surveying
the sky. It did so fairly regularly until about 1950 (with some plates taken up to 1983).
Over half a million of these plates are stored in the
Harvard plate stacks. One of the early
targets was the Large
Magellanic Cloud (LMC; see above left image), a nearby (65 kilo-parsecs)
dwarf irregular galaxy. The LMC is
slowly merging with the Milky Way.
Henrietta Leavitt
(right image above);
also see this
site, who held the position of "computer" at the Harvard College
Observatory, began classifying the variable stars in the LMC in 1905.
Stars were not well understood a century ago, and much of astronomy was
devoted to discovering and characterizing the variables.
The astrophysical significance of studying variable stars in the LMC is that
all the stars in the LMC are roughly the same distance from us. Distances are
one of the most fundamental, but most difficult to measure, quantities in
astronomy. While the distance to the LMC was not well known, it was
appreciated that
all the stars are at the same distance, with a spread in distance (and hence
an uncertainty on any individual star) of less than 10%.
Therefore, one can compare distance dependent quantities (like luminosities),
which one cannot do easily for the brighter stars in our own Milky Way galaxy.
Luminosities are related to the star's size, since to a good approximation
stars radiate like black bodies, with
L= 4πR2σT4, where R and T are the radius and
effective temperature of the star, respectively.
Most, if not all, stars are variable. Some, like the novae, are spectacularly variable,
while others are barely noticeable even upon close inspection. Among the plethora of types
of variables are:
Normally in stars the opacity κ is proportional to T -β,
where T is the local temperature.
The instability strip runs diagonally through the H-R diagram,
with temperatures near 10,000K.
Pulsating stars include the Cepheids (giants to supergiants), RR Lyrae stars
(subgiants to giants), δ Scuti stars (main sequence stars)
and the ZZ Cet white dwarfs.
A second, hotter instability strip, near 40,000K,
driven by the partial ionization of Helium,
includes the β Cep stars and the PG1159 subdwarfs.
All stars are natural pulsators at low amplitudes. The atmosphere of the Sun
oscillates with a fundamental period of about 5 minutes. The study of these periods
(helioseismology) provides a probe of the interior of the Sun, and provides temperature
and density profiles accurate to a few percent within the convective zone.
Asteroseismology, now possible for some of the brighter stars,
reveals their internal characteristics (temperatures, densities, etc).
Ms Leavitt plotted the apparent magnitude of the LMC Cepheids
as a function of
the pulsation period, and found a correlation. Since all the stars are in the
LMC, and are at the same distance from us, the apparent magnitudes are an
accurate measure of the true relative luminosities of the stars. She found a
relation similar to that shown in Figure 7. She actually used apparent
magnitudes; the conversion to absolute magnitudes (shown in Figure 7) requires
an estimate of the distance to the LMC.
The importance of such a relation, once it is calibrated, is that it
provides a simple way to determine the distance to a Cepheid variable and,
hence, to the cluster or galaxy that contains it. One merely determines the
period, and observes the mean magnitude m. One
looks up the absolute magnitude M that corresponds to that period. The
difference between the apparent and absolute magnitudes, m-M, knows
as the distance modulus, is equal to 5 log(D) -5, where D is the distance
in parsecs. This relation is easily derived from the inverse-square law,
knowing that magnitudes are log2.5 of the brightness, and from
the definition that the absolute magnitude equals the apparent magnitude
at a distance of 10 parsecs. Hence a measure of the period directly
yields the distance.
An isothermal star of 100 solar radii and 10 solar masses will have a
pulse period of about 5.8 days.
One can derive the period-luminosity law as follows:
Assume a sample of stars of the same mass (classical Cepheids have masses
of 5-10 solar masses), and the same temperature (the instability strip is
approximately vertical in the H-R diagram). The luuminosity then scales as
R2. The pulse period scales as R3/2 (see above), or
as the luminosity L3/4. Luminosity is proportional to
2.5-V, where V is the magnitude. Therefore one expects to
find that log P ≈ 0.3 V
Beyond distances of a few tens of parsecs, one must cobble together
distances from a variety of techniques.
One can use the trigonometric parallaxes to determine the
luminosities of stars on the main sequence. Then, the color or spectral type
and observed magnitude of a main sequence
star is sufficient to determine its true distance (this is called the
spectroscopic parallax). Cepheids are not main sequence stars.
However, some galactic Cepheids are found in clusters of stars. Like the
stars in the LMC, all the stars in these clusters are at the same distance
from us. One can us the spectroscopic parallax of the main sequence
stars in the cluster to determine the distance to the cluster, and the Cepheid.
The distance and observed magnitude then directly give the luminosity of the
Cepheid, and a calibration of the period-luminosity relation.
There are of course many complications, most of which are beyond the scope
of this introduction. However, there is one very important caveat.
The galactic calibration of the Cepheids is inapplicable to the LMC
Cepheids. This is because there is a large difference in mean metallicity
between the two galaxies: stars in the LMC are metal-deficient relative to
our Galaxy. This affects the opacity, and the periods. Technically, the
stars in the LMC are known as W Virginis stars, or population II Cepheids.
Population I consists of metal-rich stars, including the Sun. Population II
is metal poor, representing a population of stars that formed from a less
enriched interstellar medium. At a given period, W Vir stars are less
luminous than are classical Cepheids. Inadvertent application of the
classical Cepheid P-L relation to W Vir stars leads to a
to a large overestimate of the distances.
The study of the phenomenology of variable stars reached its zenith in the
early 20th century. Of what use is the study of Cepheids today?
Simply put, because of the elegance and simplicity of the period-luminosity
relation, they form the principle rung in the cosmic distance ladder.
The Cepheids are giants and supergiants which lie in the instability strip.
Because they are intrinsically luminous, they can be seen to great distances.
One of the key projects in the early years of the Hubble Space Telescope was
to find and measure the periods of Cepheids in nearby galaxies, out to
distances of about 20 Mpc (6 x 1020 km). With an accurate
calibration of the period-luminosity relation, these observations pinned
down the Hubble constant H0 to be 72 ± 2 km/s/Mpc.
Cepheids have allowed us to determine the scale of the universe.
In the fall of 2003 we obtained a series of V and I band images of two fields
in the Large Magellanic Cloud. The data were taken with the
SMARTS
0.9m
reflector at the Cerro Tololo Interamerican Observatory by service observers.
The detector is a 2048 x 2048 pixel
CCD array.
We obtained images in both the
V and I filters. The exposures were one minute in each filter. Two exposures
were taken in each filter, with the second offset from the first by 10 arcsec
East and North to reduce the chance that any point on the sky will be lost
to a chip defect.
The data have been bias-corrected, trimmed, and flat-fielded. The two
images in each filter have been co-aligned and summed.
The CCD plate scale is 0.4 arcsec/pixel; the full field of view is 13.6 arcmin
on a side. The fields (Table 1) were chosen to maximize the number
of Cepheid variables in the field. Because the pointing of the telescope is
not exact, and because there was little to be gained by spending the time to
tweak the positions to match perfectly, we coaligned the images after the
fact and trimmed off those areas that were not observed on every night.
The resulting image fields, and the number of known Cepheids contained therein,
are given in Table 1.
We obtained images on 10 nights, from 25 November 2003 through 12 December
2003. There are 14 I band images, and 15 V band images, for each field.
The data are stored on one CD which should be easy to locate in the lab
room. There are two main directories, field1 and field2. Each of these
directories contains 29 fits images, named lmcfx_yymmdd_z#.fits,
where x is the fie1d number (1 or 2), z is the band (v or i), and yymmdd
is the date of observation. # is the number of the observation for that night
(1, 2, or 3), and is absent if there is only one observation.
All the images are stored in FITS format.
You should probably create a subdirectory on your PC, and then
copy the contents of the CD into that directory.
A third directory on the CD contains the backup versions of the
finding charts.
These data are also provided as ASCII text files
f1ceph.list and f2ceph.list, in the respective data directories
on the CD.
To read in the images using IDL, you would use the READFITS function. For example,
d=readfits(filename,header), where filename is the name of the fits file,
d is the output data array, and h is the fits header.
Display the image using the TV command.
The TV command is primitive: it does
not scale the data, or resize the TV window.
If the image is larger than the size of the monitor, you should probably
rebin it (see this section) to a
manageable size.
Use the TVPLOT command or,
for more control, size the plot window with the XSIZE and YSIZE keywords and
do the flux scaling manually (see the
"Using the Cursor to Measure Positions"
section in the IDL primer).
The fits header is a string array. You can print it
(e.g., print,header). There is a lot of information
in the header. For the purposes of this lab, the most important information is
In the case at hand, for all practical purposes the axes are oriented with +Y heading
north and -X heading east. The b and d coefficients are zero.
The cross terms cij and fij are zero.
The distortion is minimal; the tangent-plane projection is effectively
flat, so terms with i or j > 1 are negligble. The solution reduces to
When you convert seconds of time to seconds of arc, remember that there is
a cos(dec) factor in the conversion.
If you rebin the image, remember to change the plate scale accordingly.
If you'd like to try this technique, be wary of the following:
Remember that there are lots of kinds of variable stars. In fact,
all stars vary at some level. Henrietta Leavitt found 20 Cepheids
in the LMC - and 1750 other variable stars!
To confirm the identity, check the finding chart (linked from Tables
2 and 3). The Cepheid is identified by the cross-marks.
That said, there are some stars that are not worth doing. Some are very
close to the edge of the field. If your extraction aperture goes off the
image, your results will suffer. Some stars are in clusters or small
asterisms (groups of stars). If you cannot place an aperture cleanly around
the Cepheid, it may be worth skipping it. However, a faint
star in the extraction aperture is no big deal. All you are interested in
is the period, and a faint star will just add a small bit of noise.
Repeat this process for all the Cepheids in each image, and for each night.
You may use the same set of reference stars for all the Cepheids in a given
image. You MUST use exactly the same reference stars when comparing between
different nights.
Use either the V or I band images; you need not do both. They should give
the same result.
Figure 3: The location of the upper
part of the instability strip in the H-R diagram
Figure 4: Artist's conception of a Polar, showing
the disruption of the accretion stream by the 10-100 MG magnetic field
of the white dwarf
Image copyright M. Garlick
Figure 5: Doppler image of AE Phe at four phases,
from Barnes et al., 2004 MNRAS, 348, 1321 
Figure 6: The light curve of δ Cephei
Figure 7: The Cepheid period-luminosity relation
A fluid object (including a star) of radius R has a fundamental pulsation
period P ≈ 2R/vs
where vs is
the sound speed.
This is simply the time it takes a sound (or pressure)
wave to cross the stellar diameter.
vs2=dP/dρ,
where P and ρ are the pressure
and density, respectively. In an isothermal gas P=ργ,
so vs2=γP/ρ. The virial theorem tells us
that
GM2/R (the potential energy) = 3 M kT/μH
(twice the kinetic energy).
Using the ideal gas law, P/ρ = kT/μH,
we find that
GM2/R = 3 vs2 M/γ.
γ is 5/3.
The pulsation period is then
P ≈ √(R3/GM).
Plugging in numbers,
P ≈ 1600 (R/R0)1.5/(M/M0)½ seconds
where R0 and M0 refer to the
radius and mass of the Sun.
The only directly measurable distances in astronomy are those made by
trigonometric parallax. As observed from the Earth, stars trace appear to trace
out the motion of the Earth's orbit around the Sun. The semi-major axis of
the ellipse has an angular size of 1/D arcsec, where D is the distance in
parsecs.
Trigonometric parallax is useful to distance of
about 50 pc for ground-based optical observations, and a few hundred
pc for space-based, or radio VLBI, observations. The accuracy is limited by
smallness of the motions. There are no Cepheids within this distance.
II. The Tasks at Hand
II. The Data
target RA Dec Field Cepheids LMC Field 1 5 27 27.3 -69 50 30 10.5 x 10.5 22 LMC Field 2 5 40 31.9 -70 21 00 10.5 x 10.1 33
The 55 known Cepheids in these images are presented in Tables 2 and 3.
Cepheids in Field 1
Cepheid RA Dec Type I V 35307 5 27 40.14 -69 55 45.7 FO 14.90 15.597 44256 5 27 36.82 -69 53 42.8 FU 14.11 14.878 44391 5 27 57.84 -69 53 48.5 FU 15.79 16.548 53226 5 27 34.14 -69 51 22.4 FU 13.94 14.667 53242 5 27 32.77 -69 49 14.8 FO 14.15 14.846 62624 5 27 55.13 -69 48 3.9 FU 14.11 14.826 62742 5 27 49.69 -69 46 17.6 FO 15.84 16.461 162132 5 28 29.78 -69 52 36.7 FU 14.19 14.967 162135 5 28 45.59 -69 52 6.6 FU 14.40 15.188 162180 5 28 23.25 -69 52 3.8 FO 14.81 15.422 170203 5 28 35.24 -69 51 23.4 FU 14.28 14.974 170205 5 28 32.05 -69 50 54.2 FU 14.28 15.135 170223 5 28 23.03 -69 51 36.5 FO 15.26 15.899 170246 5 28 44.56 -69 50 5.0 FO 14.91 15.567 170248 5 28 43.69 -69 50 3.2 FU 14.71 15.546 295932 5 26 48.88 -69 51 33.6 FO 14.33 14.996 305691 5 26 50.13 -69 45 52.9 FU 14.17 14.970 408742 5 27 21.41 -69 52 19.4 FU 14.25 15.029 417848 5 26 59.59 -69 51 10.4 FU 14.37 15.114 417850 5 27 23.10 -69 50 57.9 FU 14.35 15.193 417853 5 27 5.12 -69 50 44.4 FU 14.04 14.821 427313 5 27 0.88 -69 47 46.7 FO 16.88 17.476
Cepheids in Field 2
Cepheid RA Dec Type I V 20975 5 40 23.77 -70 25 44.2 FU 15.49 16.261 24980 5 40 31.29 -70 23 55.1 FO 15.04 15.746 24988 5 40 32.66 -70 23 19.1 FU 15.08 15.852 25039 5 40 26.56 -70 21 46.7 FU 15.57 16.290 29186 5 40 31.94 -70 21 0.9 FU 15.17 15.942 29194 5 40 33.18 -70 20 27.6 FU 14.88 15.776 29197 5 40 26.82 -70 19 58.1 FU 14.56 15.414 29199 5 40 15.70 -70 19 53.7 FU 15.30 16.120 29260 5 40 32.47 -70 18 2.4 FU 15.30 16.024 81414 5 41 9.25 -70 24 10.6 FU 14.78 15.589 81430 5 40 43.10 -70 21 54.4 FU 15.20 15.949 81432 5 41 11.87 -70 21 55.0 FU 15.18 16.035 81440 5 41 6.62 -70 24 19.1 FU 15.59 16.458 81454 5 41 7.96 -70 22 55.3 FO 15.48 16.256 81459 5 40 52.09 -70 22 36.8 FU 15.91 16.681 85244 5 41 5.45 -70 19 36.7 FU 14.99 15.744 85260 5 40 43.90 -70 20 53.3 FU 15.64 16.398 85282 5 41 16.54 -70 18 26.1 FO 15.59 16.428 89131 5 40 58.13 -70 17 4.7 FO 15.43 16.573 130469 5 41 50.76 -70 25 5.9 FU 15.06 15.963 133771 5 41 50.57 -70 23 36.4 FU 15.30 16.074 133782 5 41 19.09 -70 22 50.7 FU 15.26 16.133 137407 5 41 26.59 -70 19 34.6 FU 15.69 16.553 137424 5 41 24.44 -70 17 48.0 FU 16.10 17.382 137680 5 41 47.74 -70 17 51.2 FO 17.33 17.980 140993 5 41 23.73 -70 16 58.0 FU 14.98 15.835 141015 5 41 35.32 -70 17 34.3 FU 15.51 16.303 141019 5 41 34.35 -70 17 22.6 FO 15.44 16.286 203767 5 40 11.00 -70 24 30.9 FO 15.45 16.181 207480 5 40 6.46 -70 23 31.7 FU 15.30 16.116 207506 5 40 8.33 -70 21 36.2 FU 15.43 16.347 211310 5 39 59.41 -70 19 46.6 FO 15.51 16.284 214843 5 40 2.02 -70 16 13.3 FO 15.69 17.313
You can extract values from the fits header using the SXPAR function.
For example, ra=sxpar(header,'CRVAL1') will extract
the value of CRVAL1 and place it in the variable ra
You measure the locations of things in the images in terms of pixels.
If you want to convert the pixels to (RA,Dec), or if you know the (RA, Dec)
of something and want to know what pixel it is in, you need the
astrometric solution. This is in general a pair of multi-order polynomial
equation of the form
Dec = diΣXi + eiΣYj +
fijΣXiYj
Dec = c0 + c1Y
These images were taken under a variety of sky condition. Atmospheric
transparency can vary significantly even on clear nights. The
fluxes, or count rates, cannot be directly compared. Therefore, we employ
a technique called relative photometry. The premise is that the true mean
brightness of a large (N>>1) number of randomly-selected stars will not
vary significantly with time.
The effects
of sky transparency (differing airmass, thin clouds, etc.) will affect
all the stars in the image equally.
Therefore, the ratio of the brightness (of the difference
in magnitudes) between a target star and this mean of a large number of stars
will represent the true brightness variation of the target star.
Differential photometry entails the following steps:
Given a table of times and corresponding magnitudes, how do you find the period?
Basically, you need to either fold the period on a number of trial periods, or
do a Fourier decomposition of the data. You can find some illustrated examples
on the AST 443/PHY 517
Timing Analysis page.
FFT techniques are best for continuous data sets, and are most sensitive to sinusoidal variations. For non-sinusoidal variations, some power will be diverted into sidelobes. For unevenly sampled data, power is diverted into aliases of the true period with that of the sampling function. To find out the power of the sampling function, take an FFT of a uniform data set (all points with the same amplitude) at the actual observation times. This is called the windowing function. You should probably ignore any periods that stand out strongly in the windowing function. For example, if you observe every night at exactly the same time, the windowing data (and the FFT of your data) will have a strong peak at a period of 24 hours.
The power I as a function of frequency ω is given by
You cannot find a period longer than your sampling range. Ideally, you will detect at least 2 complete periods (though 1.5 can be convincing).
You cannot find a period shorter than your shortest sampling interval. In other words, the frequency cannot exceed the Nyquist, or sampling, frequency. There are an infinite number of shorter periods that can fit your data exactly!
Aliasing is often a problem. Observations taken from one site on the Earth often show a 1 day period, because the data are always taken at night. (The period will be one sidereal day if the object is always observed when overhead). This is why we obtained 2 or 3 observations on some nights.
In all cases, some of the power appears in the harmonics of the true frequency ωN, where ω is the frequency and N is an integer. If there is aliasing, there will be beating between the true period and the alias, giving power at 1/(1/P ± 1/A), where P is the true period and A is the alias period.
You report should address the following
The data reductions were performed by Rustum Nyquist.
