o-c calculation for the new binaries gsc 2087-0364, gsc 2083-1870, and v1097 her
TRANSCRIPT
O-C Calculations for the New Binaries
GSC 2087 – 0364, GSC 2083 - 1870
and V1097 Herculis
James Meier
Thesis Advisers: Dr. Richard P. Olenick, Mr. Arthur Sweeney
Submitted in partial fulfillment of the requirements for the Bachelor of Science degree in the
Department of Physics
May 2013
2
Contents Chapter 1: Introduction ................................................................................................................................................. 3
1.1 Abstract ............................................................................................................................................................... 3
1.2 W-Ursae Majoris Type ......................................................................................................................................... 3
1.3 Eclipsing Algol Type ............................................................................................................................................. 4
1.4 Peranso Overview ............................................................................................................................................... 5
1.5 Lomb Scargle Explanation ................................................................................................................................... 6
1.6 Kwee Van Woerden Algorithm Derivation .......................................................................................................... 8
1.7 O – C Introduction and Derivation .................................................................................................................... 10
Chapter 2: Data Aquisition .......................................................................................................................................... 12
2.1 Experimental Setup ........................................................................................................................................... 12
2.3 Telescopes ......................................................................................................................................................... 13
2.4 Cameras ............................................................................................................................................................. 13
2.6 Guiding .............................................................................................................................................................. 15
2.7 Experimental Setup ........................................................................................................................................... 16
2.8 Calibration Frames ............................................................................................................................................ 18
2.9 Flat field dark frames......................................................................................................................................... 19
2.10 Bias Frames ..................................................................................................................................................... 19
2.11 Flat fields ......................................................................................................................................................... 19
2.12 Error Removal .................................................................................................................................................. 20
2.13 Image Reduction Phase I ..................................................................................................................................... 21
2.14 Image Reduction Phase II .................................................................................................................................... 22
2.1 Image Chart ........................................................................................................................................................... 24
3.1 Data Analysis .......................................................................................................................................................... 25
3.2 GSC 2087-0364 Peranso estimate and Phase Diagram ..................................................................................... 25
3.3 O-C data and Calculations for GSC 2087-0364 ................................................... Error! Bookmark not defined.
3.4 Peranso results for 2083-1870 .......................................................................................................................... 29
3.5 Results for GSC 1622-1875 ................................................................................................................................ 32
4.1 Results ................................................................................................................................................................... 35
5 Discussion ................................................................................................................................................................. 37
6 Appendix A ................................................................................................................................................................ 38
7 References ................................................................................................................................................................ 39
3
Chapter 1: Introduction
1.1 Abstract We report the results of time-resolved CCD photometry of two new binaries and V1097 in the
constellation Hercules. Our observations were carried out using a six-inch, wide angle lens
astrograph with a set focal length of 200 mm, 30 field of view and f/1.5 stopped down to an f/2.8
in Pitkin, Colorado in the R band for 35 nights during the early summer of 2012. We used 60
second integration times and captured 300 images per night, obtaining 10,500 images in total.
Using Peranso software, Lomb-Scargle period analysis was carried out for the binaries. We
present the period analysis and O-C calculations for the two new binaries, GSC 2087- 0364,
GSC 2083-1870 as well as for V1097 Her.
1.2 W-Ursae Majoris Type W-Ursae Majoris binaries were first discovered in 1903 by Muller and Kempf [1903] who were
performing observations for the Potsdam Photometric Durchmusterung. They observed a period
of variation of about four hours which was the shortest period discovered to that date and too
short to be characteristic of any known classes of stars at the time. The original light-curve from
their data is shown in Figure 1. W-Ursae Majoris was classified 16 years later and given detailed
parameters. Characteristically, the system has near equal minima and continuous light variation.
The apparent magnitude can vary anywhere from a few tenths to just over one magnitude.
Periods typically range from about 0.25 days to 1.0 day. Both stars are of the same spectral type,
and are assumed to be in a similar evolutionary state. The two stars usually have somewhat
different masses, but very similar temperatures and luminosities. Because of their near equal
luminosities, the light curves are symmetric. They are classified as near or overcontact systems,
which means they are touching and transferring mass. As will be discussed in more detail in the
4
research results section, the period can be variable over time. In 1974, Whelan, Mochanacki, and
Worden considered three possible causes of period change: (1) mass exchange and/or mass loss;
(ii) apsidal motion and (iii) the possibility of a third body.
Figure 1: Light Curve of first Discovered W-Ursae Majoris by Muller and Kempf
1.3 Eclipsing Algol Type Better known by its colloquial name, The Demon Star, Algol is the only eclipsing binary we can
perceive with the naked eye. It has a much longer period than the W-Ursae Majoris (W UMa)
type binaries, and it varies from 2.1 magnitude to 3.4 at full eclipse. Algol actually has three stars
in the system, but the third star does not participate in the eclipsing. The primary star is a B8 and
the secondary star is a less massive and dimmer K2 subgiant, which is a dying giant star.
Because they are so close together with a separation distance less than half the distance from the
earth to the sun, a tremendous amount of mass has been transferred from the K2 to the B8. Due
to the difference in magnitude, the light curve does not have the symmetry of a W U Ma. A
sample of the apparent magnitude versus time is shown in Figure 2. The major difference
5
between W U Ma type and Algol type is that W UMa binaries symmetric light curves due to the
similar magnitudes the stars share, while stars in an Algol type system have magnitudes that are
very different from each other. When a light curve is constructed of these stars, there are two
characteristic minima that occur. A primary minimum occurs when the dimmer star eclipses the
brighter star, creating the change in intensity of light perceived by the naked eye larger. The
second happens when the brighter eclipses in front of the dimmer star, creating a change in
intensity that is only perceivable photometrically.
Figure 2: Geometry of Eclipsing Algol. The primary eclipse shown is the large change in apparent magnitude while the secondary shown is a much smaller magnitude change.
1.4 Peranso Overview Peranso, Period Analysis Software, was developed by Tonny Vanmunster who operates a
privately owned astronomical observatory that is the head observatory for CBA, The Center for
Backyard Astrophysics. CBA is a worldwide network of professional and amateur astronomers
studying cataclysmic variables. Peranso software is very useful in providing a rough period
estimate of binaries. After data were imported into the program, a Lomb-Scargle Periodogram
for each system was calculated to obtain the estimate. The issue with Peranso is that it only
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calculates a period up to 10-4
days, demonstrated in Figure 4, which is not precise enough. The
program was then utilized to calculate each minima and error for each eclipse of the light curve.
To perform these tasks, Peranso implements the Kwee-Van Woerden algorithm which will be
discussed in more detail. From this information provided by Peranso, Observed – Calculated
calculations were made to obtain a much more precise period and ephemeris.
Figure 3: Example of zoomed in portion of periodogram in Peranso
1.5 Lomb Scargle Explanation
Before Lomb and Scargle’s contribution to astronomy, the periodogram had conventionally been
defined as
( )
| ( )| 1.1
7
|∑ ( )
|
1.2
[(∑
)
(∑
)
]
1.3
This is what Scargle refers to in his paper as the classical periodogram (Scargle, 1982). It can be
evaluated for any value of the frequency. The reason for using this is that if contains a
sinusoidal component of frequency , then at and near , the factors ( ) and are
in phase and make a large contribution to the sums in this equation. At other values of , the
terms in the sum flip back and forth from negative to positive and the resulting cancelation yields
a small sum. The presence of a sinusoid is made known by a large value of P near one value of
and will appear as a distinct narrow peak in the spectrum.
Lomb slightly modified the definition’s output by giving it a highly modified expression.
( )
{ ∑ ( )
∑ ( )
∑ ( )
∑ ( ) }
1.4
Where is defined by
( )
(∑ )
(∑ )
1.5
This method is preferable for two reasons: it has a simpler statistical behavior than the classical
periodogram and it is equivalent to the reduction of the sum of squares in least-squares fitting of
sin waves to the data. Like the classical periodogram, it reduces to
8
( )
|∑
|
1.6
If the spacing between data points is even, and has time-translation invariance. The computing of
the new definition is not too much more difficult than the old, a point that is obsolete due to
modern computing technology. Peranso fits a series of sine waves to the data and finds the best
fit. The period with the highest theta value for example, as shown in Figure 2 has the highest
probability of fitting the variation in the data set.
1.6 Kwee Van Woerden Algorithm
Let N be the total number of observations in the phase interval used. Form ( ) magnitudes
spaced by equal time intervals , by linear interpolation between consecutive data points. To
prevent a use of weights of the observed magnitudes that is too unequal, take ( ) to be
about equal to 1. One of the equidistant times, for example, should act as a preliminary time
of minimum. An estimate from the light curve will suffice for determining . Take the time
as reflection axis and reflect the interpolated magnitudes of one upon the other, giving two
magnitudes for every equidistant time on the latter branch. Due to the assumed symmetry, and
also assuming was a relatively accurate guess, the two points should be close in magnitude.
Take the differences of these magnitude pairs
( ) 1.7
and compute the sum of their squares
( ) ∑( )
1.8
9
Then shift the symmetry of the axis to
(
) (
)
1.9
Successively, and proceed in the same manner to compute the sums
(
) (
)
1.10
If was properly chosen, ( ) will be smaller than both other sums. If this isn’t true, i.e. if
(
) ( )
1.11
The subsequent sum
( ) 1.12
Must be computed. N must be the same in all the summations. The function ( ) is represented
by a quadratic formula:
( ) 1.13
the parabola represented by ( ) has a minimum value
( )
1.14
at
1.15
where is the time of minimum sought.
The mean error of epoch is given by
10
( )
1.16
Here, Z is the maximum number of independent magnitude pairs.
1.7 O – C Introduction and Derivation From rough period and minimum calculated in Peranso, the O-C essentially subtracts the
calculated minima from the observed minima i.e. the time of minima we actually obtained minus
the time of minima we should have obtained based on the epoch and period calculated in
Peranso. As time goes on, O-C values stray further from zero if the period is not exact. This
happens because the estimate for the period is imprecise. By graphing the O-C versus cycle, we
can refine the value of the period. The goal is to get the lowest slope and y-intercept possible by
tweaking the period and epoch of the data.
To perform the calculation, we let
O = time of observed minimum
C = calculated time of minimum based on measured superhump period
JD0 = time of maximum at epoch E = 0
Pest = estimated period
P(E) = true period
We can write the following equation relating the quantities:
0
0
( )
est
est
O JD P E E
C JD P E
O C P E P E
1.17
1.18
1.19
Next we assume that the O-C residuals can be fit by a parabola, that is,
2O C aE bE c 1.20
.
Equating this with our relation for O-C we get
2
est
est
P E P E aE bE c
P E E P E aE bE c
1.21
1.22
11
Differentiating with respect to E, we obtain.
2
dPE P E P aE b
dE
1.23
Equating like coefficients of powers of E on both sides of the last equation, we obtain
2
est
dPa
dE
P E P b
1.24
1.25
We see that if the period is indeed constant, then dP/dE = 0 and so a = 0 and the residuals as a
function of E would be a straight line with slope b = (P – Pest). In other words, the correct period
would be.
estP P b 1.26
Shown in Appendix A is a portion of an O-C calculations chart.
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Chapter 2: Data Acquisition
2.1 Experimental Setup The experimental setup consisted of one primary telescope was used for data acquisition with a
CCD camera connected to it, which communicated with the computer. On top of the astrograph,
a guide scope with a camera was mounted as our tracking telescope, which was also connected to
the computer to provide feedback for tracking. The guidescope was responsible for minute
adjustments throughout the night to keep the frame centered on our “guide star”, SAO 85182. In
addition, we had heaters attached to both cameras that prevented frost from building up on the
lens as the temperatures dropped below freezing almost every night. All of this was set up on a
German Equatorial Mount. Figure 4 shows the experimental setup.
Figure 4: Pictured is the entire experimental setup including the computer, various connecting wires and power supply, telescopes, cameras, and heaters.
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2.3 Telescopes The primary telescope was a 15.24 cm diameter, wide angle lens astrograph with a 200mm focal
length which gave us a 3 degree field of view. The telescope is originally an f/1.5 stopped down
to an f/2.8. Adjusting the aperture to a smaller opening gave us a greater depth of field to ensure
that more stars would be in focus. If we stopped it down more, we would not have been able to
get enough light in our images, so this was the optimal f stop. The guide scope was an 80mm
Orion refracting telescope.
2.4 Cameras For the camera mounted to our primary telescope, we used a SBIG ST-10XME CCD. This
camera was designed and developed Santa Barbara Instrument Group based out of Santa
Barbara, CA. This camera’s sensor contains 3.2 megapixels for a Full Frame Resolution of 2184
x 1472 pixels at 6.8 microns, which makes it an ideal camera for wide field refracting telescopes
like the one we were using. Communication to the computer is through the USB port at up to
425,000 pixels per second and a schematic is shown in Figure 5.
Figure 5: Flow chart of CCD sensor to computer
The quantum efficiency across the visible spectrum is shown in Figure 6.
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Figure 6: Quantum Efficiency of SBIG ST-10XME CCD. Note the peak in Quantum Efficiency at 600nm.
Ultra-low dark current of less than 1e-/pixel/second at 0oC, which is typical, allows moderate
cooling for applications involving extended exposures, which was useful for us since we were
taking 60 second integration times. The CCD is cooled with a solid-state a thermoelectric (TE)
cooler. The TE cooler pumps heat out of the CCD and dissipate it into a heat sink, which forms
part of the optical head's mechanical housing. With the proficient liquid cooling design, we were
able to set the temperature of the sensor to about -200 C to minimize the thermal noise in our
images. To accommodate for the quantum efficiency of the SBIG ST-10 imaging CCD (which
was 85 % at 600nm) we used an R-band filter. The advantage of using this filter is that we were
able to maximize on the transmitted light at 600nm as shown in the graph for Johnson Filters in
Figure 6. The end result of matching the peak transmittance wavelength of the filter to the peak
quantum efficiency for the KAF-3200ME is a minimal amount of signal loss.
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Figure 7: Graph of % Transmittance of different UB/VRI filters
2.6 Guiding Our entire apparatus was guided by an Orion StarShoot AutoGuider, shown in Figure 7, which
keeps track of a selected guide star and communicates constantly with connected equatorial
telescope mount to correct any tracking errors throughout long exposure astrophotography
sessions. It uses a 1/2" format 1.3MP CMOS chip, with tiny 5.2 x 5.2 micron pixels for highly
accurate autoguiding. Our guide star was a bright enough magnitude to allow us to have very
short integration times with the camera, which made more minute adjustments easier.
Figure 8: StarShoot AutoGuider camera
16
2.7 Experimental Setup In preparation for data acquisition, after mounting the scopes to the mount and hooking up all of
the electronics, the telescope was polar aligned. After that, we would perform the following
protocol every night.
1) Uncover and turn on telescopes, connect to computer.
2) Open the SkyX Pro (screenshot shown below), which is a planetarium, simulation,
telescope control, planning and logging application. 9. The program would always tell us
our scope was at a slightly different location (no more than 2-3 degrees) than it actually
was at the beginning of the night, so we centered the guide scope on Vega, and synced it
with the SkyX Pro.
Figure 9: Screenshot of the SkyX Pro software used for telescope control. Here, it is synced on star we used to focus our telescopes every night.
17
3) Open CCDSoft Version 5 to control the temperature of the CCD camera and take images.
The cooling takes about 15 minutes, so always perform this step a few minutes before
stellar twilight so as not to lose any precious data time.
4) Move to a star near Vega of relatively bright magnitude to focus on. Using
“Nebulosity3”, shown in figure 9, adjust the focus on the lens until the star in the image
was as sharp. Re-focus the camera every night in case there were vibrations/wind during
the day that shook it loose.
Figure 10: Screenshot of Nebulosity3. When using this program, we aimed for the highest max and lowest HFR we could obtain for the star we centered on. This would give us the sharpest, narrowest peak.
5) Next, open PHD guiding, shown in Figure 10, which is a program synced with the guide
camera. Center the frame on SAO 85182 and begin tracking.
18
Figure 11: Our two most popular programs throughout the night. CCD soft is pictured on the left and PHD Guiding to the right. When we would check the equipment every 5-10 minutes, we would verify the image had not noticably changed and
that our guide telescope had not lost tracking.
6) Check every 5-10 minutes continually throughout the night to make sure the tracking is
still going.
7) When the guide star crossed the meridian, flip the telescope to the other side of the
meridian. Repeat steps 2-6, using Arcturus to sync the telescope with the SkyX Pro.
2.8 Calibration Frames When using a CCD camera, there are three sources of unwanted noise in our images. They arise
from voltage offset (bias), thermal emission of electrons along with the photosite sensitivity (10).
Each time the camera is used these signals change slightly, so we cannot assume enough
similarity to use the same set of calibration frames for every night. We have to take new ones
each night. To correct for the three sources, we take three types of images: flat field dark frames,
bias frames, and flat fields. We then take the average of each set of images by stacking them and
thus creating a master calibration frame for the night. The creation of master calibration frames,
19
removal of poor images, and reduction of raw images was performed with the astronomical
image processing software, AIP4WIN.
2.9 Flat field dark frames The objective here is to remove the CCD Dark Current noise from each Light Frame image.
Dark Frames are images taken with no light falling on the CCD, so that the only values present
in the pixels are due to Bias and Dark Current (+ or - some noise). This was achieved by
covering up the telescope so that no light could enter in. The camera was always kept at the
same temperature and exposure time so calibration image exposures would be consistent with the
data. Every night sixteen images were taken and a master dark image compiled which was
subtracted from the light images from that night.
2.10 Bias Frames The objective here is to capture the bias level. These images are 0 s integration time exposures to
the camera in the dark. Theoretically, the bias level should not fluctuate at all for each pixel, but
a few factors such as interference from nearby electronics along with CCD read out signal.
Combining the bias images by taking the average gives us a master bias frame which can be
subtracted from the raw data to minimize the noise in each image from bias.
2.11 Flat fields The objective here is to minimize the effects of various imperfections such as: CCD
inhomogeneities (pixels more or less sensitive to light), optical vignetting (image is darker away
from the centre), dust on the CCD window (seen as 'donut' shaped spots on the image), and
Internal reflections. A Flat Field is an image of a perfectly plain flat white surface. Any deviation
from absolute homogeneity across the image is due to CCD or optical imperfections. For this
work, a light source designed for flat field images was used, which had a very uniform field of
20
light that was controlled from the computer. In order to minimize the effects of the light being
slightly different intensities at different points on the frame, the light 450
was rotated after every
two pictures, obtaining 16 in total. Like the other calibration images, the average was used to
produce a master Flat Field image.
2.12 Error Removal With the calibration frames, SYSREM, systematic error removal, was implemented to remove
systematic error in images. Figures 12 and 13 show an data pipeline of how our images went
from the camera to our database. Figure 15 shows an uncalibrated image as it would appear on
the computer screen moments after it was captured.
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2.13 Image Reduction Phase I
Process one night’s set of images
Raw images
Calibrate image set
Dark framesBias framesFlat frame
Calibrated images for one night
For each image
Perform astrometry using SExtractor
Identify each star by position
GSC
Cat
alo
gstart
Is star in GSC?
Assign unique
name to star
NO
Assign GSC name to
star
YES
Last star?
NOGet next star
Raw data into Master data
base for night
Next night
Master Data Base
Figure 12: Transition from raw data to Master Data Base
22
2.14 Image Reduction Phase II
Master Data Base
start
Process one night’s set of images for
each star
For each night
For each star
Build array of all data for this
star on this night
Set “all there” flag if 75% of nights images have a stellar
mag.
star data array
Calculate “Sysrem”
magnitude for each image
Flag outlier mags.
Update master data base
Next star
Next night
End
Figure 13: In the second phase of the data pipeline, systematic error is removed
23
Figure 14: Sample image from data acquisition.
24
2.1 Observation Times Data was obtained from May 21, 2012 to June 25, 2012 in Pitkin, Colorado which is located in
the Rocky mountains about 200 miles southeast of Denver. Pitkin is at an elevation of 9240 feet.
The team began each night by centering on SAO 85182, and taking 60 second integration times
starting at stellar twilight and ending around 5 A.M.
Month Day Year First image last image Number of images
5 21 2012 0 3:50 220
5 22 2012 10:56 5:28 321
5 23 2012 12:39 5:11 205
5 24 2012 10:48 5:20 213
5 25 2012 10:38 4:55 308
5 27 2012 10:59 5:01 297
5 28 2012 10:55 4:59 303
5 29 2012 11:45 5:40 292
5 30 2012 11:12 5:39 309
5 31 2012 11:12 5:29 177
6 1 2012 11:04 5:51 337
6 5 2012 11:06 5:27 317
6 6 2012 10:45 4:05 266
6 7 2012 10:26 5:09 343
6 8 2012 10:50 4:58 306
6 9 2012 10:32 5:27 344
6 10 2012 10:38 5:23 339
6 11 2012 10:31 5:21 342
6 12 2012 10:30 4:50 317
6 13 2012 10:27 5:31 352
6 14 2012 11:25 5:03 273
6 15 2012 11:02 4:59 298
6 16 2012 10:33 5:05 323
6 17 2012 11:28 5:00 286
6 18 2012 10:33 5:00 324
6 19 2012 10:50 5:07 315
6 20 2012 10:54 5:18 320
6 21 2012 11:15 5:36 318
6 22 2012 11:07 4:59 292
6 23 2012 11:06 5:40 306
6 24 2012 11:57 4:48 243
6 25 2012 10:46 5:31 331 Figure 15: Record of observation times for summer 2012
25
3.1 Data Analysis From the thousands of stars in each image, three definite binaries were selected for this study.
Light curve data for each star was inserted into Peranso to find the rough period estimate and the
time for minima using the methods discussed in sections 1.5 and 1.6, and then performed the O-
C calculations to obtain a more precise period and ephemeris. For each star, screenshots of the
Peranso periodogram and phase diagram, along with the O-C data and graph will be shown.
3.2 GSC 2087-0364 Peranso period estimate and Phase Diagram, O-C data and
Calculations
Figure 16: Lomb-Scargle Periodogram for GSC 2087 – 0364
The Peranso calculation only has the precision of 10-4
days. It is difficult to make this out
because the periodogram is not zoomed in, but the dotted line does rest on the maximum theta
value of any of the five major humps on the graph. Occasionally, Peranso’s estimate would be
slightly off of the maximum, but this window is interactive, which allowed small adjustments to
be made so as to determine the maximum as better. This produced the most precise period
Peranso could offer. The period of 0.2630 is then doubled to 0.526 to find the true period
because Peranso calculates half of the true period.
26
Figure 17: Screenshot of phase diagram in Peranso
Once the periodogram was obtained, a phase diagram was plotted to make sure that the
calculated period was indeed close to the actual period. When the phase diagram looked
relatively clean, a viable period estimate was obtained.
27
O-C data for GSC 2087 - 0364
n O Error (+/-) C O-C
0 56076.195750 0.000175 56076.198641 -0.002891
2 56077.251420 0.000384 56077.250868 0.000552
4 56078.305332 0.000130 56078.303096 0.002236
6 56079.35921 0.000484 56079.355323 0.003890
8 56080.414633 0.000247 56080.407550 0.007083
15.5 56084.346303 0.000177 56084.353403 -0.007100
19 56086.191551 0.000210 56086.194800 -0.003249
21 56087.243413 0.000184 56087.247028 -0.003615
23 56088.299039 0.000256 56088.299255 -0.000216
25 56089.348210 0.000188 56089.351482 -0.003272
27 56090.406613 0.000233 56090.403710 0.002903
28.5 56091.197100 0.000245 56091.192880 0.004220
30.5 56092.243924 0.000258 56092.245108 -0.001184
32.5 56093.298713 0.000232 56093.297335 0.001378
34.5 56094.356604 0.000376 56094.349562 0.007042
36.5 56095.398932 0.000174 56095.401790 -0.002858
40 56097.236620 0.000146 56097.243187 -0.006567
42 56098.293638 0.000163 56098.295415 -0.001777
44 56099.344878 0.000373 56099.347642 -0.002764
46 56100.39113 0.000175 56100.399869 -0.008743
49.5 56102.238849 0.000962 56102.241267 -0.002418
51.5 56103.317981 0.000209 56103.293494 0.024487
53.5 56104.338586 0.000189 56104.345722 -0.007136 Figure 18: O-C data for GSC 2087 - 0364. Each one of these data points is a calculated minima with the error in time.
For this chart, n is the cycle number. O is the observed minima in MJD calculated in Peranso.
Error is the error in days calculated in Peranso. C is the calculated minima from the O-C method.
And finally, O-C is the observed data minus the calculated data.
28
Figure 19: O-C values with error bars for GSC 2087 – 0364
Most of the error bars are relatively the same size. However, the error bars around the n = 50 is
much larger than the rest. This could have been due to very minor cloud cover which slightly
skewed some of the data points that night. A period of 0.526113662 d was found, which was
much more precise than Peranso offered.
-0.010000
-0.005000
0.000000
0.005000
0.010000
0 10 20 30 40 50 60
O-C values
Cycle Number
O-C Calculations for GSC 2087-0364
29
3.4 GSC 2083-1870 Peranso estimate and Phase Diagram, O-C data and
Calculations
Figure 20: Lomb-Scargle Periodogram for GSC 2083 – 1870. Estimated period = 0.361 d.
Figure 21: Respective Phase Diagram for GSC 2083 – 1870 for a period of 0.361 d.
The phase diagram again is very clean and shows the periods overlapped in a neat manner,
showing me that the estimated period is close to the actual period.
30
O-C data for GSC 2083 – 1870
n O Error (+/-) C O-C 0 56075.350312 0.000147 56075.351310 -0.000998
2.5 56076.254767 0.000114 56076.253433 0.001334
5.5 56077.3358 0.000102 56077.335982 -0.000179
8 56078.237923 0.000115 56078.238105 -0.000182
8.5 56078.41985 0.000188 56078.418530 0.001319
11 56079.321021 0.000133 56079.320653 0.000368
13.5 56080.22274 0.00022 56080.222777 -0.000034
14 56080.4034 0.000126 56080.403201 0.000200
24.5 56084.19359 0.000445 56084.192120 0.001466
25 56084.3726 0.000547 56084.372545 0.000051
27.5 56085.275346 0.000116 56085.274668 0.000678
30 56086.17689 0.000175 56086.176792 0.000093
30.5 56086.358275 0.000149 56086.357216 0.001059
33 56087.25899 0.000219 56087.259340 -0.000347
35.5 56088.16358 0.000181 56088.161463 0.002120
36 56088.339677 0.000281 56088.341888 -0.002211
38.5 56089.24441 0.000098 56089.244012 0.000395
39 56089.422013 0.000259 56089.424436 -0.002423
41.5 56090.325531 0.000225 56090.326560 -0.001029
44 56091.22828 0.000106 56091.228683 -0.000399
47 56092.30802 0.000116 56092.311231 -0.003209
49.5 56093.21398 0.000268 56093.213355 0.000627
50 56093.39403 0.000163 56093.393780 0.000251
52.5 56094.293 0.000242 56094.295903 -0.002906
55 56095.197875 0.000169 56095.198027 -0.000152
55.5 56095.37917 0.000126 56095.378451 0.000714
58 56096.279566 0.000173 56096.280575 -0.001009
60.5 56097.183822 0.000190 56097.182698 0.001124
61 56097.362215 0.000119 56097.363123 -0.000908
63.5 56098.265903 0.000145 56098.265246 0.000657
66.5 56099.349029 0.000121 56099.347795 0.001234
69 56100.249601 0.000126 56100.249918 -0.000317
72 56101.331516 0.000108 56101.332466 -0.000950
74.5 56102.235440 0.000256 56102.234590 0.000850
75 56102.416431 0.000191 56102.415014 0.001417
80 56104.218851 0.000096 56104.219261 -0.000410
80.5 56104.401718 0.000298 56104.399686 0.002032
Figure 22: O-C data for GSC 2083 – 1870
31
Figure 23: O-C values with error bars for GSC 2083 – 1870
The corrected period obtained for GSC 2083–1870 is 0.3608493911 d.
-0.004000
-0.003000
-0.002000
-0.001000
0.000000
0.001000
0.002000
0.003000
0 10 20 30 40 50 60 70 80 90
O-C Values
Cycle Number
O-C Calculations for GSC 2083-1870
32
3.5 GSC 1622-1875 Peranso estimate and Phase Diagram, O-C data and
Calculations
Figure 24: Lomb-Scargle Periodogram for GSC 1622 - 1875.
Notice that this periodogram is much messier than both of the previous stars. This is because the
SuperWASP data, which is what we combined with the data for this star, is very messy. Because
of this, the periodogram does not look as orderly as the previous two. The phase diagram is
messy for this reason as well.
Figure 25: The respective phase diagram is not perfect because of the untidy data from SuperWASP.
33
O-C data for GSC 1622 - 1875
n O Error (+/-) C O-C
0 56076.195750 0.000175 56076.198641 -0.002891
2 56077.251420 0.000384 56077.250868 0.000552
4 56078.305332 0.000130 56078.303096 0.002236
6 56079.35921 0.000484 56079.355323 0.003890
8 56080.414633 0.000247 56080.407550 0.007083
15.5 56084.346303 0.000177 56084.353403 -0.007100
19 56086.191551 0.000210 56086.194800 -0.003249
21 56087.243413 0.000184 56087.247028 -0.003615
23 56088.299039 0.000256 56088.299255 -0.000216
25 56089.348210 0.000188 56089.351482 -0.003272
27 56090.406613 0.000233 56090.403710 0.002903
28.5 56091.197100 0.000245 56091.192880 0.004220
30.5 56092.243924 0.000258 56092.245108 -0.001184
32.5 56093.298713 0.000232 56093.297335 0.001378
34.5 56094.356604 0.000376 56094.349562 0.007042
36.5 56095.398932 0.000174 56095.401790 -0.002858
40 56097.236620 0.000146 56097.243187 -0.006567
42 56098.293638 0.000163 56098.295415 -0.001777
44 56099.344878 0.000373 56099.347642 -0.002764
46 56100.39113 0.000175 56100.399869 -0.008743
49.5 56102.238849 0.000962 56102.241267 -0.002418
51.5 56103.317981 0.000209 56103.293494 0.024487
53.5 56104.338586 0.000189 56104.345722 -0.007136 Figure 26: O-C data for GSC 1622-1875.
34
Figure 27: O-C calculations for GSC 2622-1875.
The large gaps in the cycle number are due to the fact that SuperWASP data was taken long before the
present work.
-0.200000
-0.150000
-0.100000
-0.050000
0.000000
0.050000
0.100000
0.150000
0 500 1000 1500
O-C Values
Cycle Number
O-C Calculations for GSC 2087-0364
35
4.1 Results 1) GSC 2087 – 0364:
Peranso period estimate: 0.526 d
Slope of O-C vs. cycle number graph: 1.64372 E-10
Intercept of O-C vs. cycle number graph: 1.50369 E-8
O-C Period: 0.526113662 d
O-C Epoch: 56076.1986409
Due to the period found, we speculate this star is a W UMa type binary system. It’s period
lies between the 0.25 and 1.0 d that is typical for such a system
2) GSC 2083 – 1870:
Peranso Period estimate: 0.3610 d
Slope of O-C vs. cycle number graph: 9.20777 E-11
Intercept of O-C vs. cycle number graph: 8.79315 E-6
O-C Period: 0.3608493911 d
O-C Epoch: 56075.35131
These results are of particular interest to us because Bob Nelson found the period of this
binary last year to be 0.360846911 days using the same method as us. His O-C graph for the
calculations are shown below. The period and epoch calculated in this work for V1097
yielded a slope and intercept that are both much smaller than Bob Nelson’s, so possibly the
new period and ephemeris are more precise. This binary system is also most likely a W UMa
type binary.
36
Figure 28: Bob Nelson's O-C graph.
3) GSC 1622-1875:
Peranso Period estimate: 1.7552 d
Slope of O-C vs. cycle number graph: 8.60106 E-11
Intercept of O-C vs. cycle number graph: 8.79315 E-6
O-C Period: 1.7539078 d
From the period found, this binary system is an eclipsing-Algol type (EA).
y = -3E-08x + 0.0015 R² = 1
-0.004
-0.002
0
0.002
0.004
0.006
0.008
0.01
0.012
0.014
-4000 -2000 0 2000 4000 6000 8000 10000 12000
O-C
(d
ays)
Cycle
V1097 Her - O-C Diagr.
GCVS 4 IBVS Nelson S4
37
5 Discussion The research discussed in this paper was executed by a small team of three undergraduate
physics majors and two professors. The limited funds definitely limited the depth of research in
which we could delve, but it did not greatly affect the quality of the research. The project was
able to obtain very precise, clean data with a high signal to noise ratio of 22 – 60, depending on
the magnitude of the star. A few things could be done to improve this project in the future.
Having two or more telescopes would be very beneficial because the likelihood to find binaries
and exoplanets would be increased by the number of fields observed. Due to the significant
amount of work it takes to convert the raw data into the information we use for data analysis of
this nature, there was not enough time to process the data from the second telescope. So faster
data analysis is needed. Another factor that would improve the quality of the results is extending
the integration time of the data acquisition phase. The O-C results from this work are being used
by a colleague who is using PHOEBE software to model the binary systems and determine
characteristics such as mass, luminosity, and radii of the component stars. Through the research
methods and calculations discussed in this work, it has been shown that new binaries can be
identified examined to properly classify them.
38
6 Appendix A
O-C for Minima of GSC 2083-1870 0.360841 56075.171596
n O Error (+/-
) Mag C Epoch Phase O-C
0 0 56075.171596 0.000138 leave blank 56075.171596 0.000000 0.000000 0.000000
0.5 0.49488833 56075.350172 0.000310 56075.352017 0.494888 0.494888 -0.001845
2 56075.893278 0.000000
3 3.002161617 56076.254899 0.000148 56076.254119 3.002162 0.002162 0.000780
4 56076.614960 0.000000
5 56076.975801 0.000000
6 5.9980878 56077.335952 0.000135 56077.336642 5.998088 0.998088 -0.000690
7 56077.697483 0.000000
8.5 8.497717831 56078.237921 0.000117 56078.238745 8.497718 0.497718 -0.000823
9 9.001895572 56078.419849 0.000228 56078.419165 9.001896 0.001896 0.000684
10 56078.780006 0.000000
11.5 11.49948869 56079.321083 0.000222 56079.321268 11.499489 0.499489 -0.000185 Figure 29: Example of O-C calculations spreadsheet in Excel with lines in the cycle number skipped when not obtaining data
i.e. during the day or during cloud cover at night.
39
7 References
Vanmunster, Tony. "Peranso 2.0 User Manual." Peranso Light Curve and Period Analysis
Software. N.p., 2004. Web.
"W Ursae Majoris." American Association of Variable Star Observers. Ed. BSJ. Web. 26 Apr.
2013. Web.
Hopkins, Jeff. "Eclipsing Binaries - Algol." 14 Dec. 2004.
"The Algol System." Astronomy 162 Stars, Galaxies, and Cosmology. Web Santa Barbara
Instrument Group. "Operating Manual CCD Camera Models." Sbig.
Scargle, Jeffrey D. "Statistical Aspects of Spectral Analysis of Unevenly Spaced Data." The
Astrophysics Journal 263 (1982): 835-38.
10 Richard Berry & James Burnell, The handbook of Astronomical Image Processing, 2nd
Ed.
(Willman-Bell, Inc, VA, 2005), pp89,165,170,186,191
Model ST-10XE/XME CCD Imaging Camera. PlaneWave.
"Astronomy Filters." Lot-oriel.
"Orion StarShoot AutoGuider." Orion Telescopes & Binoculars.
Archer, Russell. "TheSkyX Pro Review." Russell Archer's Astronomy Blog. N.p., 15 Jan. 2013.
Richard Berry & James Burnell, The handbook of Astronomical Image Processing, 2nd
Ed.
(Willman-Bel, Inc, VA, 2005), pp89,165,170,186,191