Measuring-different-types-of-Cepheid-Variables.pdf

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Measuring different types of Cepheid Variables
Physics Department, University of California, Santa Barbara, CA 93106
(Дата: submitted June 3, 2022)
In this paper, I used a 0.4 meter telescope to observe three Cepheid variables. The three Cepheid
variables each belong to a specific type of Cepheid variables, which are classical Cepheid variables,
type II Cepheid variables, and RR type Cepheid variables. Photos of the three Cepheid variables
are taken using telescope. Photometry techniques are conducted to obtain the luminosity-period
relation of the Cepheid variables
PACS коды:
I. BACKGROUND
A Cepheid variable is a type of star that varies in both diameter and temperature, resulting changes in brightness.
The variation of Cepheid variables is highly predictable. The period of the Cepheid variables is directly related to
their absolute magnitude. They change their brightness in a well-defined stable p eriod and amplitude.Almost every
Cepheid variables follow this relation extremely well. [3] The characteristic of Cepheids was first discovered in 1908
by Henrietta Swan Leavitt after studying thousands of variable stars in the Magellan Clouds.[2] The name, Cepheid,
comes from Delta Cephei in the constellation Cepheus, which is the first star to b e identified as this kind variables.
The strong direct relation between the Cepheid variable’s luminosity and pulsation period established Cepheids as
important indicators in the distance measurement in astronomy. Once the period of a Cepheid variable is measured,
its absolute magnitude can be inferred from the luminosity-period relation. Compare the absolute magnitude and
the apparent magnitude, the distance to the Cepheid variable can be obtain. The distance measurement by Cepheid
variables can be used to measure extremely large distance, such as the distance between the Milky way and another
galaxy. Astronomer Edwin Hubble used Cepheid variable for the first t ime t o m easure t he d istance b etween the
Earth and the Andromeda Galaxy. The result is far greater than the diameter of the Milky way. Thus, he determined
that Andromeda is not located in the Milky way. There are many galaxies existing out of the Milky way.[4]. The
mechanism of the varying nature of the Cepheid variables was understood as star heat engine proposed by Arthur
Stanley Eddington in 1917.[9] The mechanism was further explained by S.A. Zhevakin in 1957, who identified ionized
helium as a likely valve for the engine. [10]
II. APPROACH
A. objects choosing
three Cepheid variables, each belong to a specific type of Cepheid variables, which are classical Cepheid variables,
type II Cepheid variables, and RR type Cepheid variables, were chosen. RR lyrae, NSV 4148, and BI Cas were chosen

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Рис. 1: The transmission data of SDSS u’, SDSS g’, SDSS r’, and SDSS i’ filters.[7]
as observation target. RR lyrae is classified as anomalous Cepheid variables, or more specifically, RR lyrae variables.
Since RR lyrae is the most iconic one among this type. NSV 4148 is classified as classical Cepheid variables while BI
Cas is classified as type II Cepheid variables.[1]. According to the variable data base, NSV 4148 has a mean apparent
magnitude of 11.79, period 0.894 day. BI Cas has a mean apparent magnitude of 12.8, period 1.099 days. And RR
lyrae has a mean apparent magnitude of 8.55, period 0.567 day.
B. observing instruments and parameters
In this paper, a 0.4 meter telescope from the Las Cumbres Observatory was used to take photos of the Cepheid
variables. The telescope has a FOV of 29.2 ∗ 19.5 in arc min. The instrument on the telescope is SBIG STL6303
camaera, which has a pixel size of 0.571 arc sec and 2048 ∗ 3072 pixels in total. Further information about the
instruments used can be found in the reference.[5] Exposure time is calculated using LCO exposure time calculator.[6]
SDSS u’, SDSS g’, SDSS r’, and SDSS i’ filters are used. These four filters combined offer a view from near infrared
to near ultraviolet.
The objects chosen also need to be visible during certain time period for LCO 0.4m telescopes. LCO visibility
calculator was used to determined visibility.[18]
Рис. 2: One example observation window from 2022-12-04 to 2022-12-11 of BI CAS. The observation window for other targets
can be found in the same way.

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The integration time is calculated using LCO exposure and SNR calculator.[19]. A proper integration time is chosen
so that CCD won’t be saturated
Таблица I: observation parameter. Period data comes from [17]
Target Name RA (J2000) Dec (J2000) Filter # Exposures Integration Time (s) Observational Windows
ASASSN-V J004331.19+623955.9 / BI Cas 10.880 62.666 SDSS u’ 3 120
Period: 1.099 day.
10 observations in 2 days
ASASSN-V J004331.19+623955.9 / BI Cas 10.880 62.666 SDSS g’ 3 120
Period: 1.099 day.
ASASSN-V J004331.19+623955.9 / BI Cas 10.880 62.666 SDSS r’ 3 120
Period: 1.099 day.
ASASSN-V J004331.19+623955.9 / BI Cas 10.880 62.666 SDSS i’ 3 120
Period: 1.099 day.
ASASSN-V J004331.19+623955.9 / BI Cas 128.550 -68.600 SDSS u’ 3 50
Period: 0.894 days.
10 observations in a day
ASASSN-V J083412.06-683601.2 / NSV 4148 128.550 -68.600 SDSS g’ 3 50
Period: 0.894 days.
ASASSN-V J083412.06-683601.2 / NSV 4148 128.550 -68.600 SDSS r’ 3 50
Period: 0.894 days.
ASASSN-V J083412.06-683601.2 / NSV 4148 128.550 -68.600 SDSS i’ 3 50
Period: 0.894 days.
RR Lyrae 19 25 27.913 +42 47 03.694 SDSS u’ 3 1
Period: 0.567 day
RR Lyrae 19 25 27.913 +42 47 03.694 SDSS g’ 3 1
Period: 0.567 day
RR Lyrae 19 25 27.913 +42 47 03.694 SDSS r’ 3 1
Period: 0.567 day
RR Lyrae 19 25 27.913 +42 47 03.694 SDSS i’ 3 1
Period: 0.567 day
C. photometry
The data from the 0.4m telescope is pre-processed by the LCO BANZAI pipeline.[8] The data from the telescope was
then imported into python using astropy package for further analysis. First the pictures taken under the same filter
are sorted out. Calibration techniques is used to measure the apparent magnitude of the variable stars. Calibration is
conducted using nearby stars in the field of view and the background sky. The light curve under different filters is then
plotted. Pictures taken under different filters allow us to observe the varying magnitude under different wavelength.
Compare the observed period-luminosity relation to other result. Period luminosity for classical Cepheid variables is
Mv = (−2.43 ± 0.12)(log10 P − 1) − (4.05 ± 0.02)
given by Hubble Space Telescope trigonometric parallaxes for 10 nearby Cepheids.[15] The following relations can
then be used to calculate the distance d to classical Cepheids
5 log10 d = V + 3.34 log10 P − 2.45(V − I) + 10.52[15]

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or
5 log10 d = V + 3.34 log10 P − 2.55(V − I) + 10.48[16]
I and V represent near infrared and visual apparent mean magnitudes. The distance d is given in parsecs. The
period-luminosity relation for type II Cepheid variables is given by
M = −0.35 − 1.75 log P
Astro art is used to find the calibration star. In order to make more precise photo metric measurement, it is better
to choose brighter reference star to maximize signal-to-noise ratio. Coordinates of the reference stars were then input
into the database to search for their magnitude. The information of the reference stars comes from SIMBAD[11]. Note
that since a multi-band photometry was conducted, it is important to use magnitude data from different bands. Since
the magnitude of a star may vary on different wave band, using a reference magnitude for all bands may result in
discrepancy. Some of the stars in the SIMBAD database have magnitude measured in sdss u g r i filters. The images
taken from the telescope is processed using python. Astropy and photutils packages are used to conduct photometry.
First the images is sorted according their filter information. The Findstar function from photutils packages is used to
locate stars and measure their flux on the image. The functions works based on such mechanism: The function locates
pixel points that deviate from the background signal by a certain amount of standard deviation. This threshold can
be customized. In practice, I set the threshold to be over 150 σ. The function also excludes pixel points that have
a full-width-at-half-maximum lower than some threshold to prevent locating hot pixels. I set the threshold to be 3
standard deviation
Рис. 3: The program finds out 20 brightest stars in the image taken for RR Lyrae.
D. period and frequency analysis
After plotting the calibrated magnitude, a special technique is used to find out the period of the variables. The
Lomb-Scargle periodogram is a commonly used statistical tool designed to detect periodic signals in unevenly spaced
observations.[12] It can be conveniently applied to our data using the LombScargle class in Astrpy package. Since the
data obtained from the observation is unevenly spaced, it becomes the perfect tool that can be applied in the analysis
of the data.

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The Lomb-Scargle technique utilizes Fourier Transform, which is a useful tool in signal analysis and processing.
Fourier transform of a continuous signal f(t) is given by the integral [14]
F(f) =
Z ∞
−∞
g(t)e−2πift
dt
This transformation transforms a signal in time domain to frequency domain. For Convenience the Fourier transform
is usually denoted by operator F. Such that the Fourier transform and the inverse Fourier transform can be denoted
as
F(f) = F(g(t))
and for the inverse Fourier transform
g(t) = F−1
However, in real world, there is only discrete data points but not continuous function. In this case, discrete Fourier
transform, usually denoted as DFT, is used.
F(f) =
∞
X
−∞
g(n∆t)e−2πifn∆t
In this case, a continuous variable is substituted by a small block n∆t The Fourier Transform gives a power spectrum
density, known as PSD for a set of data. From the PSD of the data, the periodic behaviour and the frequency can be
read out.
Lomb-Scargle method then use least squre method to fit the data to a periodic model with three parameters
y(t, f, θ) = θ0 +
nterms
X
n=1
[θ2n−1sin(2πnft) + θ2ncos(2πnft)]
E. error analysis and caliberation
The magnitude measured for the reference stars chosen may not be the same for each images. However, the reference
stars chosen are not variable stars, The reasons that may explain this phenomenon are probably the background noise
of the night sky and the CCD may be in different configuration each time. Thus a normalizing constant has to be
introduced to normalized the magnitude of the three reference star to the same magnitude as they are in the first
image. The normalizing constant is given by the equation
n =
M1
Mi
The calibrated magnitude is normalized by the mormalizing constant
Mcalibrated = n ∗ (Mraw − Mzp)
The error in measuring is given by the equation
δMcalibrated = Mcalibrated ∗
v
u
u
t
(
δn
n
)2 + (
q
δM2
raw + δM2
zp
Mraw − Mzp
)2
Mraw stands for the magnitude before calibration, which is adding the zero point offset. The uncertainty δn and
δMraw can be calculated using the standard deviation of these data.

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III. ANALYSIS
A. RR Lyrae
Plot out the raw data measured from the aperture sum of the images of RR Lyrae
Рис. 4: The raw data, measuring the aperture sum, from the observation of RR Lyrae under different filters. A significant
variation can be seen in the graph.
The magnitude of a star is calculated using
m = −2.5log10(Naperture sum)
before calibration The magnitude after calibration is calculated using
Рис. 5: Plot the data using magnitude before calibration
mcalibrated = m + ZP
Where ZP is the zero point difference. When doing photometry, we note that the photos taken under sdss u filter has
very low quality. On the photos taken under the u filer, only RR lyrae can be barely recognized. Other stars which can
be used as reference star can be hardly recognized. As a result, calibration can not be done for images taken under u
filter. It seems like the stars themselves are too dark on ultraviolet band. For the calibration process below, we use
only images taken under g r and i filters, neglecting the images taken under the ultraviolet band. From the picture it
can be seen that the image taken under sdss U filter is too bad to conduct analysis.

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Рис. 6: The image taken under filter u. Only RR lyrae can be barely recognized
Рис. 7: Locate the reference stars using astroart. This image is taken under the sdss r filter.
Choosing these three stars as the reference star. The data comes from SIMBAD database[11]. They are chosen
because they are relatively bright and have a complete magnitude data measured under sdss g r and i filter.
Таблица II: reference star parameters for RR Lyrae[11]
Reference stars name RA coordinate DEC coordinate g filter magnitude r filter magnitude i filter magnitude
UCAC4 663-075895 19 25 31.6321974576 +42 34 49.029074832 12.991±0.05 12.170±0.04 11.861±0.02
UCAC4 664-074538 19 25 17.1222077784 +42 37 01.642304496 13.099±0.04 12.432±0.03 12.153±0.03
UCAC4 663-076028 19 26 09.7910026752 +42 34 24.371817276 13.624±0.03 13.270±0.04 13.157±0.03

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Plotting the magnitude of RR lyrae after caliberation through time.
Рис. 8: The magnitude of RR Lyrae after using the three calibration star.
Separate the data point under different filter band can plot them into different graphs.
Рис. 9: The magnitude of RR Lyrae under G, R, and I filter band, respectively
Plotting the magnitude of the calibration stars calculated directly from the images.
Рис. 10: the magnitude of the calibration stars calculated directly from the images.
We can see from the figure that the magnitudes of the calibration stars are generally stable from image to image.
However there are still some outliers
Next step is calculating the normalizing constant and its uncertainties. In case of there are three reference stars
here, we calculate the normalizing constant of one particular image for each of the stars and take the average of the
three numbers as the normalizing constant for this image. To calculate the uncertainty of the mormalizing constant

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δn, the standard deviation of the normalizing constants for all of the three stars is taken. Thus, the normalizing
constant and the uncertainty of the normalizing constant for each image is determined.
The uncertainty provided in the data base for the magnitude of the reference stars is used asδMzp. The uncertainty
can be looked up in table II. Since three reference stars were chosen, the average of the uncertainty of the magnitude
of the three reference stars is taken as δMzp
To calculate δMraw, the error propagating feature in photutils.aperture photometry is used. To use this feature, we
need to first input an array of error for each of the pixel points in the image. After that, the function will propagate
the error value use
∆F =
sX
i∈A
σ2
tot,i
Where A are the non-masked pixels in the aperture and σtot,i is the input error array.[13] In this case we take error
to be 0.015 times the pixel value. The 0.015 time value corresponds to a signal to noise ratio of approximately 70. This
value is the upper bond of the uncertainty. Measuring the signal to noise ratio of the three reference stars using astro
art, I found that they both have a signal to noise ratio higher than 70. The uncertainty value given by this process
is the uncertainty in the flux measured from the image. This uncertainty needed to be converted into uncertainty in
magnitude. However, logrithm is not linear, so the average of both the change in magnitude in the positive direction
and in the negative direction is taken to be δMraw. It is expressed by the equation
δMraw = −2.5 ∗
1
2
(lg(
Naperatruesum + δN
Naperatruesum
) + lg(
Naperatruesum
Naperatruesum − δN
))
Calculating the uncertainty using the above equations and plot out error bars.
Рис. 11: The calibrated magnitude for RR Lyrae with error bars plotted
Note that in these figures, the outlier values has longer error bars, indicating that systematic errors may occur.
Using the Lomb-Scargle method to analyze the frequency of the RR Lyrae variable star and then draw out the power
spectrum density diagram.

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Рис. 12: The power spectrum density after using Lomb-Scargle method to analyze the data with uncertainty under g r and i
band for RR Lyrae
However, due to some build-in error or limitations of the function lomb-Scargle.autopower(). The power spectrum
density for the data under G band is not successfully drawn out. After checking the program, we found out that the
frequency data points returned by this function when inputting G filter data are all none type variables. I failed to
solve this problem so only analyses for r filter data and i filter data are done below. Find out the maximum and the
standard deviation of the frequency data
Таблица III: results of the r and i filter power spectrum density for RR Lyrae
R filter I filter
Most probable frequency 149.882 day−1
43.3032 day−1
Standard deviation 6.243 1.8038
Using the best fit parameters calculated by the LombScargle function. Plot out the model which the data is fitted
to and adjust the position of the data points to be in phase of the model
Рис. 13: Plot the model togather with the data for RR Lyrae

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B. BI Cas
For the star BI Cas, which is a classical type Cepheid variable, similar process is conducted. This time, choosing
the reference stars to be Note that this time the object star is not the brightest star in the field. As a result, first
Таблица IV: reference star parameter for BI Cas[11]
Reference stars name RA coordinate DEC coordinate g filter magnitude r filter magnitude i filter magnitude
HD 3949 00 42 48.5115495240 +62 45 57.097433448 7.763291±0.00277 7.79±0.03 7.79±0.42
UCAC4 764-009281 00 43 54.3609383483 +62 44 29.58664229 12.260±0.02 11.807±0.03 11.639±0.04
UCAC4 764-009265 00 43 41.9666592251 +62 45 53.367092577 14.122±0.03 13.0050±02 12.490±0.04
I need to modify the python functions that were previously defined to do the photometry for RR Lyrae. Since RR
Lyrae is the brightest star in the field, to locate RR Lyrae, Locating RR Lyrae only needs to locate the brightest pixel
point. However, that is not the case for BI Cas. The stars that is closest to the center of the field need to be located.
Also Note that for HD 3949, there is no Sloan Digital Sky Survey data for this star. As a result, its magnitude is not
measured under sdss u’ g’ r’ i’ filter. Instead, GAIA data is used. GAIA data only provides magnitude measured under
B, V,G,J,H,K filters. Among then, filter G can be consider approximately the same as sdss g’ filter. The magnitude
measured under V filter is used as r filter magnitude and i filter magnitude. The difference between G and V magnitude
is used as the uncertainty in g filter magnitude. Similarly, the difference between J and V filter is used the uncertainty
in i filter magnitude
Рис. 14: Locate the reference stars and the object stars using astroart. This image is taken under the sdss g filter.

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Again we neglect the images taken under filter u’ because of their low signal to noise ratio. It is very hard to
recognize our reference stars and object star on such images.
Рис. 15: The image taken under filter u. Only the brightest star, HD 3949 can be barely recognized.
plotting out the light curve of the calibration stars for BI Cas
Рис. 16: the magnitude of the calibration stars calculated directly from the images.
Note that from the image it can be concluded that the measured flux of the reference stars is not always constant.
There are also some outliers. As a result, it is necessary to do normalization.
Plot out the normalized light curve of the reference stars.
Рис. 17: the normalized magnitude of the calibration stars .

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Plot out the caliberated magnitude through time of BI Cas as below
Рис. 18: The ca liberated but unnormalized magnitude of BI Cas
Normalized the magnitude data and add in error bars. Plot different filter data respectively
Рис. 19: The normalized magnitude-time data of BI Cas under g, r and i filters, with error bars plotted.
Utilizing Lomb-Scargle method, obtain the power density spectrum for the data
Рис. 20: The PSD for the magnitude-time data under g, r and i filters for Bi Cas.

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Plotting out the best fit model that Lomb-Scargle method fits to, and adjust the x location of the data points to
plot it in phase with the model
Рис. 21: The phase data of Bi Cas.
Таблица V: Frequency and uncertainty of BI Cas
G filter R filter I filter
Most probable frequency 8.801682 day1
10.6128 day1 13.3734 day−1
Standard deviation 3.8648 3.8649 3.865
This is the frequency given by the Lomb-Scargle method.
IV. RESULTS
A. conclusion
In this paper, we measured the magnitude to light relations for two Cepheid variable stars, RR Lyrae and BI Cas.
RR Lyrae is classified as anomalous Cepheid variable and BI Cas is classified as classical Cepheid variable. Calibration
stars from the field of view are chosen to caliberate the magnitude. Normalization technique is also used to offset the
difference from image to image. Finally, Lomb-Scargle method is used to obtain the power spectrum density and fit
the data to a periodic model. Lomb-Scargle method gives the most probable frequency for the data sampled with an
uneven time spacing from a periodic source.
By observing the caliberated and normalized magnitude figures that were drawn, some conclusions can be drawn.
Many of the magnitude of our measurements are close to the mean magnitude that is given by the database. However,
there are also some outliers, but these outliers all have very long error bars, which suggest that the outlying values
may not be very far from the mean magnitude. We can conclude that a good job of error analysis is done. The power
spectrum density figures that were drawn suggest that there is not noticeably outstanding frequency. The reason why
is explained below

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B. Comparison
Look up the result given in other data base and compare those result with our data. From the ASAS-SN data
base,[17] the period of Bi Cas is given by 1.099287day, which correspond to a frequency of 0.91day−1
. The result
given by ASAS-SN is very far from the measurement and analysis that was done above. The frequency obtained
ranging from 8.801682day−1
to 13.3734day−1
For magnitude measured under G filter and I filter. The period for Bi
Cas given from the database is over 3σ over from my measurement and analysis.
The period of RR Lyrae is not provided by asas-sn database. It is given by Kolenberg, K.; et al. (February 2011)
[20] as 0.56686776 days (13 hours, 36 minutes), which in terms of period, is 1.764day−1
. Obviously my analysis result
for the period of RR Lyrae is also far from this value. The discrepancy is larger then 3σ There must be some problems
associate with the measurement or analysis above. Many reasons can be attributed to this phenomenon.
Рис. 22: The phase diagram of BI Cas given by ASAS-SN.
In the asas-sn database, the phase diagram and the period of RR Lyrae are both not provided, so I download the
magnitude data from asas-sn database and try to do some analysis myself.[17]
The magnitude data from asas-sn[17] is provided in three columns, which are time, magnitude, and error in
magnitude. There are also two types of data, magnitude measured using bb camera and magnitude measured using
ba camera. Only bb data is used in this case. First plot out the magnitude versus time figure.
Рис. 23: The magnitude-time figure plotted using data from asas-sn
This set of data contains 198 samples. Note that the samples have larger amount and are distributed more evenly
through time then our measurement. It can be noted that though the data is also made up of several sampling period

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seperated by time gaps, the gaps are relatively shorter then the overall sampling time. Error bars at a length of 0.02
magnitude are also plotted, but they are too small to be viewed on the figure
Similarly, Lomb-Scargle method is used to analyze the data Note that the PSD of this set of data contains several
Рис. 24: The power spectrum density figure of the data
spikes. It is necessary to adjust the frequency range of the Lomb-Scargle function in order to convey the interested
frequency band.
It can be concluded from the PSD that the most probable frequency is
1.7643273297635784day−1
with a standard deviation
σ = 1.4145106233476583
The result is very close to the cited period of RR Lyrae above, within one standard deviation. It can be concluded
that Lomb-Scargle method can work well if more sampling points with more even distribution through time can be
provided.
Plot out the phase diagram of RR Lyrae data with the best fit model provided by Lomb-Scargle method
Рис. 25: The data and the best fit model provided by Lomb-Scargle method

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Note that the sampling points are not distributed evenly in phase. This may explain why the most probable
frequency only has a power 0.64796. If the sampling points are distributed more evenly in phase, a better light curve
may be drawn.
Compare this model to the light curve of RR Lyrae drawn by others
Рис. 26: B band light curve of RR Lyrae, with the sidelobes of the frequency triplets removed. Cited from[22]
Рис. 27: V band light curve of RR Lyrae, with the sidelobes of the frequency triplets removed. Cited from[22]
Note that RR Lyrae exhibits different light curve when it is measured under different filter band. Also note that The
light curves were plotted with the sidelobes of the frequency triplets removed. Only in this way the light curves can
have such a neat shape. Usually the light curves can be a combination of two or more frequencies. This information
can be read out in the PSD. The dominant frequency is chosen to be the frequency of the variable stars. There are
also Cepheid variables that have been observed to pulsate in two modes at the same time.[23]

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V. DISCUSSION
A. Limitation and possible improvement
Some reasons can explain the large discrepancy between our measurement and the data provided form the database.
First, the sampling rate is too low and the samples are too far from each other. For more optimal use of Lomb-Scargle
method, longer sampling time and larger amount of samples is needed. The amount of samples is limited by the
observational window. The telescope time that was provided is very limited, we can only take some samples within
a relatively short period of time compared to the period of the Cepheid Variables. The time difference between two
period of time that we can take samples is large. As a result, this limitation prevented us from obtaining the data for
a complete cycle. We are always observing the same phase in the period of the Cepheid variable stars. The large time
gap between two sampling period results in a bad fit to the model when conducting Lomb-Scargle analysis. If we can
obtain larger samples over a longer sampling period and if the samples are distributed relatively more even than over
data, Lomb-Scargle method may gives a closer frequency and a better model.
Second, the sdss u, g, r, and i filters are not perfectly same with V and G filters, resulting in some error in the
magnitude of the reference stars that were chosen. However, five of the six reference stars that were chosen have
magnitude data that is measured under sdss u’g’r’ and i filters. The only one reference star that does not have
sdss data is HD3949. This error may only have minor contribution to the final result since the discrepancy itself is
not significant and magnitude data is averaged with other more precised magnitude data, significantly reducing the
contribution. If possible, images under more filters but not just the sdss filters can be taken in order to provide broader
insight to the research problem
Third, The method I used to locating stars is by running a python program that involves using functions in astropy
and photuils packages. However, the star finding function is somehow too sensitive even if I set the sigma to over 1500
and can not distinguished stars from other noise or hot pixels very well. When designing the program, I tuned the
program so that it located stars based on the following mechanism: The program will first find the position of all the
stars on a image and compare the distance from each stars to the star coordinates I input. It will then choose out all
the stars within a certain distance to the coordinates and use the largest flux among these stars as the flux of the star
I am finding. In practice, I set the threshold to be 100 pixels. This mechanism is supposed to work well since the stars
I chosen are all relatively bright and any star is brighter then them which is within 100-pixel-distance can be hardly
found. However, In practice I also noticed that sometimes the star coordinate that was located using the program
was over 60 to 80 pixels away from the coordinate I input. As a result, I am not entirely sure that the star which the
program locates is the same as what I suppose it to be. This reason may results in the outliers in the magnitude of
the reference stars and thus abnormal normalizing constant. It can explain why some outlier points have very long
error bars. Since a wrong star is located, the normalizing constant for this star is greatly different from those of the
other two stars, resulting in big standard deviation and thus large δn Better star finding functions which are able to
eliminate noise can give more reliable star finding results. Additionally, if possible, package from astrometry.net[21]
can also be utilized. This package can locate stars precisely using their relative positions. API ports to star database
may also be used to look up the data for reference stars atomically and can effectively reduces workload
Forth, in this experiment, the images taken under sdss u’ filters are all discarded because of low quality, failing to

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provide any insight for the light variation of variable stars on ultraviolet band, since most of the objects on the image
can not be effectively identified. The reason for this result may be attributed to the low magnitude on ultraviolet band
of most of the stars when observing on earth. This is because the Earth’s atmosphere blocks most of the ultraviolet
light. The images taken under ultraviolet band requires longer exposure time. To improve this, we may have to
customize observing parameters for different filters
VI. ACKNOWLEDGEMENTS
This work makes use of observations from the Las Cumbres Observatory global telescope network. This paper is based
on observations made with the SBIG 6303 0.4-meter telescope, McDonald Observatory, Fort Davis, Texas, operated
by the Las Cumbres Observatory
This work makes use of data from ASAS-SN and simbad database
Thank you for helping provided by PHYS 134L Professor Lubin and TA Kim.
Thank you for cooperation by group mates Shihao Zhou and Vincent Wei.

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VII. REFERENCE
[1] ASAS-SN Variable Stars Database https://asas-sn.osu.edu/variables
[2] Leavitt, Henrietta S. (1908). "1777 variables in the Magellanic Clouds". Annals of the Astronomical Observatory of Harvard
College. 60 (4): 87–108.
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