Determination of the Distance to M13 Using A Short Period Cepheid Variable
Justin Finke
2004 June 9
Abstract
V band photometry of M13 was performed on three nights with
the Great Ohio Telescope. A light curve for
the Type II Cepheid variable
v1 was obtained but without enough data points to find a period. The
light curve is, however, consistent with previous observations.
1. Introduction
Variable stars such as Cepheid and RR Lyrae stars vary due to pulsations that
are intrinsic to the star itself. Often, such stars obey Period-Luminosity
(P-L) relations which can be extremely useful in determining the distance to
galaxies and clusters in which these stars are found. Once the period is known
from observing a star's light curve, its absolute magnitude can be
inferred from this P-L relation. Combining this with measurements of the star's
apparent magnitude gives its distance from the relation
d = 10(m-M+5)/5 pc(1)
where m is the apparent magnitude of the star, M is the absolute
magnitude of the star, and d is its distance in parsecs.
One type of regularly varying star is the Cepheid variable, which can be
subdivided into Classical Cepheids (or Population I Cepheids) and Population II
Cepheids. Population II Cepheids are often further subdivided into W Virginis
stars (with period > 10 days) and BL Hercules stars (period < 10 days).
There has been a bit of controversy concerning whether W Vir and BL Her stars
obey the same P-L relation; the most recent analysis (McNamara 1995) indicates
that a difference does indeed exist, and gives separate P-L relations for these
two types. Given the time constraints for Great Ohio Telescope (GOT) projects,
BL Her stars with periods on the order of a day seemed the best suited for study.
Globular clusters are interesting objects because of their extremely old age which
can place limits on the age of the universe. In doing this, an accurate distance,
measured with variable stars or by some other method (e.g., parallax)
is useful for determining the exact absolute magnitude of the main sequence turnoff.
Another use for Cepheids was the HST key project which used the HST's high resolving power
to observe Cepheids in
nearby galaxies for calibration and higher redshift galaxies to determine a
value of H0, as well as test whether Cepheids' period-luminosity
relation is universal (Freedman et al. 2001).
The globular cluster M13 (NGC 6205; α=16h41m42s,
δ=36°27'37") contains three identified BL Her stars, two
of which, according to the Catalogue of Variable Stars in Globular Clusters
(CVSGC, Clement et al. 2001), have periods of around two days or less (for v1,
P=1.46 days; for v6, P=2.11 days) and thus are possible targets for the GOT.
Cepheid variables may pulsate in one of two different modes, a
fundamental mode and a first overtone mode (Arp 1955), each mode having its own
P-L relation. According to McNamara (1995), all population II Cepheids are
fundamental pulsators, although Nemec et al (1994) identifies v6 as a first
overtone pulsator. I will assume the former, since 1) Nemec was unsure about
this identification, 2) McNamara (1995) is more recent, and 3) he was aware of
Nemec et al's (1994) work. P-L relations are usually written in terms of
B and V absolute magnitudes; the GOT only has V filters, so
only this relation is relevant. Thus, the corresponding P-L relation is
MV = -1.61 log P -0.05 (2)
from McNamara (1995), where P is the Cepheid's period and
MV is its average absolute V magnitude.
For M13, Bailey (1902) discovered v1, and Shapley (1915) discovered
v6. Astronomers have had ample time to study these objects,
so their periods are well determined (Clement et al. 2001), as is
M13's distance modulus from a Hipparcos satellite parallax measurement
(Grundhal et al. 1998) which found it to be 14.38 ±0.10. It was,
therefore, the goal of this project to take photometric observations of v1 and v6 in M13
for two or more nights in a row; and using this data, to interpolate a light
curve and deduce a period for each; and using equation (2), to infer the
absolute V magnitude for them; and finally, to use equation (1) to
determine the distance to this globular cluster. However, only v1 proved
observable, for reasons explained in section 3 below.
2. Observations
M13 was observed with the 0.25 m Great Ohio Telescope and an ST-8 CCD detector
on the nights of 2004 April 28, April 29, and May 6 UT. All observations were
done with the V filter and darks, flats, and zeros were taken each night
in the morning and evening. On the first night, conditions were photometric and
the CCD was at -25°C;
2 x 90 s and 2 x 300 s of M13 were taken; 5 flats, 11 zeros and 3 x 90 s and
3 x 300 s darks were taken in the evening and 12 flats, 2 zeros, and 1 x 90 s
and 1 x 300 s darks were taken in the morning. On April 29th, the CCD temperature
was -15°C and conditions were not
photometric as clouds drifted in and out of the field and there was a moderate wind;
11 x 300 s were taken of M13 along with 9 flats, 9 zeros and 1 x 300 s dark in
the evening, and 6 flats, 4 zeros, and 3 x 300 s darks in the morning. On May 6
the sky was again slightly cloudy and the CCD temperature was again -15°C;
2 x 300 s of M13 were taken, 8 flats,
11 zeros, and 3 x 300 s darks were taken in the evening, and 5 flats, 4 zeros and
1 x 300 s dark were taken in the morning.
It was hoped that a photometric night would be made available so that
observations of standard stars for photometric calibration could be made; however,
no such photometric nights occurred, and I have not observed any standard stars other
than those in the frame with M13.
3. Reductions
The data were reduced using packages from the Image Reduction and Analysis
Facility (IRAF). Throughout the reduction process I used the values of
2.9 e-/ADU and 11.8 e- for the gain and readnoise of
the CCD, respectively, as stated in Statler (2004).
The basic reductions roughly followed Massey (1997), and were done for
each frame as follows:
- The zeros were combined, and darks of the same exposure time were combined.
- The combined darks were zero corrected with the combined zero and combined
into one dark.
- The flats were zero and dark corrected with the combined zero and combined dark,
then combined into a normalized flat.
- The object frames were dark and zero corrected with the respective zero-corrected
combined darks.
- The object frames were divided by the combined flat.
The morning and evening flats taken on April 28 were found to be substantially
different from each other, so only the evening flats were used, as it was thought
the difference was caused by moisture collecting on the telescope in the morning.
One flat taken
in the evening of April 29 was substantially different from the others, and
was thrown out. Two of the evening zeros, one morning zero and one morning
flat taken on May 6 were thrown out for similar reasons. In general the
image quality was best for April 28 due to photometric conditions and a
cold CCD. On April 29 the images tended to be blurry, presumably because
the wind was shaking the telescope during exposures. The frames taken on
May 6 were still a little blurry but not a bad as April 29, however they did
have a higher background than the data taken on the other two nights.
To get instrumental magnitudes for
the variable stars and for comparison stars, crowded field photometry was
performed on the reduced images with the IRAF package DAOPHOT, following
the procedure of Massey et al (1992). This involved the following:
- The function DAOFIND was used to search for stars that are above a certain
threshold, and these were fit with a Gaussian of a predetermined Full Width Half
Maximum (FWHM). Preliminary aperture photometry was performed on all the found stars
with the task PHOT.
- Several stars (between three and seven) were used to construct a zeroth order
model Point Spread Function (PSF). Once this was done, stars near these
"PSF stars" were subtracted in case they interfered with the PSF stars. Then a final
PSF was made from the same stars, but this time from the image with the neighbor stars
subtracted.
- The final PSF was used to simultaneously fit all of the stars found in step 1
with the task ALLSTAR to get their instrumental magnitudes and corresponding errors.
Three frames taken on April 29 were thrown out, one because it had imaged stars
twice, and two because they had hot pixels on v1. For most of the frames v6 was
not found with DAOFIND as it was too close to a neighboring bright star. v1 was
found in all of the frames except the ones taken on May 6; on those frames
it was added by hand with the function TVMARK after step 1.
4. Results
The best image of M13 obtained on April 28 can be seen in Figure 1, in false
color, with the variables of interest marked. Note v6's proximity to a brighter
star, and v1's proximity to a less bright star.
It was impossible to resolve v6 from its neighbor
on all but the best of images. V1 may be closer to its neighbor, but its
neighbor is less bright, making it easier for IRAF to find. This is exactly opposite of
what was expected; it was thought that v1 would be harder to distinguish from its
neighbor because its neighbor is closer.
|
Figure 1. This V band false color image of the globular cluster M13, with
the Cepheid variables v1 and v6 marked, was taken on April 28.
|
Because a photometric night was not available, obtaining standard V magnitudes
was done in a rather crude fashion. The instrumental magnitudes for v1 and the
standard stars were used along with the known magnitudes of the standard stars
from Stetson (2000) in a simple flux ratio to calculate the standard
V magnitude of the variable:
mV,var - mV,comp = minst,var - minst,comp
(3)
where mV,var>, mV,comp, minst,var and
minst,comp are the V band variable and comparison star magnitudes,
and instrumental variable and comparison star magnitudes, respectively. A list of
comparison stars from Stetson (2000) that were used can be seen in Table 1. Three
of these (s578, s1145, s1493) were chosen for colors that were approximately
close to the color range for v1 (a B-V ≅ 0.2 to 0.4), and six where
chosen for their brightness (s566, s568, s1204, s1412, s1407 and s572).
|
Table 1. The comparison stars names (from the Stetson catalogue), with their
V magnitude and B-V color.
|
When the magnitudes of v1 were calculated with the different comparison stars,
there was very little discrepancy between them. The magnitudes calculated from
123.859 (see Table 2) seemed unphysical as it would have required a sudden one magnitude
flare in ten minutes, regardless of the comparison star used, so this frame was thrown out.
The magnitudes obtained with the different comparison stars were averaged, and the
results can seen in Table 2.
|
Table 2. The observations of v1 with their V magnitude averaged
for the different comparison stars and the respective
errors.
|
It does not appear from looking at Figure 2 that it
would be possible to determine the period of this variable. However it
does seem consistent with the basic shape of the light curve from other
observations (Figure 3). The curve decreases slowly then increases abruptly,
just as our data do. Our data indicate that the maximum occurs at a magnitude
less than 13.6 and the minimum at greater than 14.5, both of which are consistent
with Figure 3 as well. Finally the time scale does seem to match up as the the dip
on days 124 and 125 takes about a day.
|
Figure 2 The data points in Table 2.
|
|
Figure 3 The light curve of v1 from Pike & Meston (1977) in B and
V filters.
|
I tried using the IDL package EXP (Schwarzenberg-Czerny 1996, software is
freely available at
http://cow.physics.wisc.edu/~craigm/idl/idl.html)
to fit a simple sine function of the form
m = A +B sin(ωt+φ)(4)
to
the data points by minimizing the χ2 statistic. This software
is capable of doing this for uneven sampled observations; however, it appears
out observations were too sparse to get a good result. The best fit was
independent of φ and had A=0.0 which is clearly unphysical. A
parameter space involving the other two parameters, the amplitude B
and the frequence ω can be seen in Figure 4, with the contour lines giving
the χ2. The best fit appears to be independent of B
and to prefer an ω of less than 4 day-1 which corresponds
to a period greater than 1.57 days. This fit is clearly inconsistent with previous
observations, which give an average magnitude-i.e., A in (4)-of 14.1 and a
period of 1.46 days (e.g., Kopacki et al 2003).
|
Figure 4 The phase space of the parameters B and ω from
equation (4), and χ2 as the contours.
|
As I am unable to find a period and average magnitude, I am unable to use
equations (1) and (2) to find the distance to M13.
5. Discussion
Unfortunately, time and weather constraints limited the number of data
points taken, which resulted in a failure to obtain a result for
the period. However these data are generally consistent with other
published values which reassures me that a good period could be obtained
if enough data points are taken. In fact, when the first "correct" (i.e., consistent
with modern observations) period of v1 was found it was measured with about 130
images over several years, and found with an accuracy of thousandths of a day (Sawyer 1942).
Certainly this many points are not needed
to obtain a reasonable period, but more than 13 are probably necessary! Enough frames
could probably be collected to obtain at least tenth of a day accuracy with the GOT,
although collecting and reducing the data would be quite a challenge with time resources
of less than ten weeks.
Bernard (1909) published a period of 6 days for v1 although he did not publish his data,
which is unfortunate, as it is now not known how many points he used,
and it would be interesting to compare this project to his data.
The conversion from instrumental to standard V magnitudes turned out rather
good without photometric calibration. The
conversion is certainly color-dependent, and to really do the calibration correctly
would require observations of M13 as well as standard stars through
a variety of airmasses and in several bands. Systematic errors probably
change the magnitudes I obtain, although all of the magnitudes are
reasonable, indicating that this naive conversion may be an acceptable substitute
for absolute photometry. Of course more experimentation is needed to
determine whether this is so.
Acknowledgements
I'd like to thank Tom Statler for help with learning to use the GOT, and
with guidance throughout all phases of the project. Mangala Sharma
was a great help with learning IRAF, and I am grateful to Steven Diehl for his
help with IDL, especially curve fitting. I'd also like to thank the TAC comitee for
their many useful comments. And of course this project would not be
possible without George Eberts allowing us to use his premises for observing and
his coffee that helped us get through the night.
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