# Reanalysis of c-type RR Lyrae Variable BE Dor, Period Modulations and
Possible Mechanism

L. -J. Li,1 S. -B. Qian,1,2 and L. -Y. Zhu1,2
1Yunnan Observatories, Chinese Academy of Sciences, P.O.Box110, Kunming
650011, Yunnan, P.R.China
2University of Chinese Academy of Sciences, No.19(A) Yuquan Road, Shijingshan
District, Beijing 100049, P.R.China
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Using the photometric data of several sky surveys, we reanalyzed the c-type RR
Lyrae star BE Dor (MACHO 5.4644.8, OGLE-LMC-RRLYR-06002), which was discovered
to show cyclic period changes. The $O-C$ diagram and pulsation period obtained
from Fourier analysis show that there are significant period modulations in BE
Dor. However, different from previous viewpoint, the changes are quasi-
periodic and abrupt. Therefore, the light-travel time effect caused by the
companion motion cannot explain the changes. Noting a same subtype star KIC
9453114 with similar phenomena has a high macroturbulent velocity, and the
degree of $O-C$ changes seem to be positively correlated with these
velocities, we consider that the mechanism leading to period modulation should
be caused by the interaction between turbulent convection and magnetic field
activity in the ionization zone, i.e., Stothers model. This model may not
explain the general Blazhko effect, but should explain such phenomena as
period modulations in BE Dor and those other c-type RR Lyrae stars.

###### keywords:

techniques: photometric – stars: fundamental parameters – stars: variables :
RR Lyrae stars: individual : BE Dor

††pubyear: 2016††pagerange: Reanalysis of c-type RR Lyrae Variable BE Dor,
Period Modulations and Possible Mechanism–Reanalysis of c-type RR Lyrae
Variable BE Dor, Period Modulations and Possible Mechanism

##  1 Introduction

RR Lyrae stars (RRLs) are a kind of short pulsation period variable stars,
which are widely distributed in various celestial systems in the Galaxy.
According to the characteristics of the light curves and pulsation modes, RRLs
can be divided into ab-type RRLs (fundamental mode, abbreviated as RRab),
c-type RRLs (first overtone mode, RRc) and d-type RRLs (double modes, RRd).
Because of their characteristics such as period-luminosity relation, RRLs are
usually used as probes to study the dynamics and chemical evolution of old age
low mass stars in the Milky Way (Smith, 2004).

In the field of RRLs, the relevant photometric observations and researches are
mostly carried out around their pulsations. The pulsations are affected by
different factors. Therefore, the researchers can use the pulsations as probes
to explore the mechanism behind. The pulsation variations of long-term scale
(decades or even hundreds of years) are generally considered to be caused by
stellar evolutions and light-travel time effect (Li et al. 2021 and references
therein), while the short-term variations, are usually known as Blazhko
effect, which is defined as the periodic or quasi-periodic variations of the
pulsation parameters (i.e., pulsation period, amplitude and phase) with time
(Blažko, 1907). With the running of multiple sky survey projects, a large
number of photometric data have been accumulated, and hundreds of Blazhko star
candidates have been obtained (Skarka 2013; Netzel et al. 2018, and references
therein). Recently, with the launch of space projects (e.g. $CoRoT$,
$Kepler$), researchers have obtained unprecedented high-precision
uninterrupted data, and great progress has been made in the discoveries and
researches of Blazhko stars (Chadid et al., 2009, 2010; Poretti et al., 2010;
Chadid et al., 2011; Kolenberg, 2013).

Compared with the research on the RRab stars, there are relatively fewer
studies on RRc stars. Recently, Percy & Tan (2013) studied the period changes
of 40 RRc stars by using $O-C$ method, and pointed out that the linear period
changes of most targets are consistent with the prediction of stellar
evolution model. Some RRc stars are also found to have special
characteristics, e.g., existing two close pulsation frequencies (Antipin &
Jurcsik, 2005; Khruslov, 2012), or additional non-radial pulsation (Khruslov,
2015; Moskalik et al., 2015). In addition, it is also noted that the
pulsations of some RRc stars have significant period modulations/jumps, but in
contrast, the pulsation amplitude has no significant corresponding variation
(Oja, 1981; Derekas et al., 2004; Wils et al., 2007; Wils, 2008; Berdnikov et
al., 2017; Le Borgne & Klotz, 2019; Benkő et al., 2021).

BE Dor ($\alpha_{2000}=05^{\rm h}:09^{\rm m}:18^{\rm s}.94$,
$\delta_{2000}=-69^{\circ}:50^{{}^{\prime}}:14^{{}^{\prime\prime}}.43$,
$\langle V\rangle$ = 15.231 mag, $\langle I\rangle$ = 14.861 mag, $\triangle
I$ = 0.302 mag, MACHO 5.4644.8, OGLE LMC-RRLYR-06002) is a foreground RRc star
toward the Large Magellanic Cloud (LMC). It was listed for the first time by
Alcock et al. (1997b) as an RRc star. In that paper, BE Dor, together with
other LMC foreground RRLs, ware used as tracers to find the possible
intervening galaxy located in front of the LMC. At that time, the period
change of BE Dor have be noticed. The further detailed study was made by
Derekas et al. (2004): they carried out $O-C$ analysis, period analysis and
Fourier spectrum analysis on BE Dor by using MACHO and OGLE-III photometric
data; found that the pulsation period has a periodic change with a period of
about 7.95 years, but other pulsation parameters and Fourier coefficients have
no obvious change, indicating that there is only simple period modulation in
BE Dor. Derekas et al. (2004) provided two explanations for the relevant
physical mechanism: one is the light-travel time effect (Irwin, 1952), and the
second is magnetic activity cycles (Stothers, 1980).

In present paper, we collect and organize photometric data from several sky
survey projects, reanalyze the pulsation of BE Dor by using O-C method and
Fourier analysis method, study its period changes on a longer time baseline,
and try to confirm the mechanism behind the phenomenon. Section 2 introduces
the photometric data from different sky surveys, including the properties of
data, the description of data processing and analysis. Section 3 is the
corresponding analyses, and in Section 4 and 5, the discussions and summaries
are presented, respectively.

##  2 Sky Surveys

BE Dor locates in the southern sky region, and we mainly use the data provided
by those photometric sky survey projects. After searching, we find that the
photographic plate digitization project
DASCH111https://projects.iq.harvard.edu/dasch (Digital Access to a sky century
@ Harvard, Grindlay et al. 2009), the space sky survey project
TESS222https://archive.stsci.edu/missions-and-data/tess (Transiting Exoplanet
Survey Saterllite, Ricker et al. 2015), and ASAS-
SN333http://www.astronomy.ohio-state.edu/asassn (All-Sky Automated Survey for
Supernovae, Shappee et al. 2014) provide the relevant data of BE Dor. Of
course, the MACHO444https://wwwmacho.anu.edu.au/ (MAssive Compact Halo
Objects, Alcock et al. 1997a) and OGLE555http://ogle.astrouw.edu.pl/ (The
Optical Gravitational Lensing Experiment, Udalski et al. 1992) data are also
indispensable. We will introduce them respectively below.

###  2.1 DASCH data

DASCH is a project to convert the photographic plate data accumulated by
different telescopes around the world in the early years into digital data
(Grindlay et al., 2009). The characteristics of data is low time resolution
and low data precision (about 0.1 mag), but the advantage is long time
baseline. The span of the light curves of some variable stars can reach
decades or even 100 years. Therefore, selecting appropriate variables star
targets (e.g. period longer than 0.2 days, amplitude larger than 0.2 mag) and
adopting suitable processing and analysis methods, we can obtain credible and
valuable information. Actually, in recent years, researchers have published a
series of works by using DASCH data (Liška et al., 2016b, a; Skarka et al.,
2018; Li et al., 2018a; Li et al., 2018b, 2021).

DASCH provides about 800 data points of BE Dor lasting from 1890 to 1953, of
which there are only 23 data points from 1890 to 1909. Due to the low time
resolution, these 23 points are not used in analysis. The exposure times of
most observations are 45 or 60 minutes, and only a few greater than 60
minutes. Therefore, we finally use 720 data points with exposure times less
than 60 minutes from 1923 to 1953 for analysis. Figure 1 presents the DASCH
light curves of BE Dor in the top panel, and the bottom panel shows the phased
light curves. Due to the low time resolution, the DASCH data are mainly used
for $O-C$ analysis.

We adopt an unconventional method to determine the times of light maximum,
which has been successfully applied in previous works (Li et al., 2018b; Li et
al., 2018a; Li et al., 2021). We first divide the light curves into multiple
segments; fit each segment of data with the following Fourier polynomials,

$m(t)=A_{0}+\sum^{n}_{k=1}A_{k}\sin[\frac{2\pi kt}{P_{\rm pul}}+\phi^{\rm
s}_{k}],$ (1)

where $m(t)$ is the magnitude observed at time $t$, $A_{0}$ is the mean
magnitude, $A_{k}$ is the amplitude of the $k$ component, and $\phi^{\rm
s}_{k}$ is the phase of the $k$ component in $\sin$ term; then obtained the
times of light maximum according to the fitting results. In the fitting,
taking higher terms cannot get reliable results. Considering that the light
curves of RRc stars are approximately sinusoidal, we set the term $n$ as 1 in
the fitting. Another problem should be considered in the analysis is the
maximum time delay (Liška et al., 2016b; Li et al., 2021), that is, the
obtained times of light maximum will be later than the real times due to long
exposure times and low time resolutions. Similar to previous work (Li et al.,
2021), we took KIC 4064484 as the template (Pulsation period is about 0.337
days), and carried out experiments with $Kepler$ SLC data to correct the delay
effect. We found that, when $n$=1, the times of light maximum obtained is
about 10 minutes later than the real times, and are insensitive to the
exposure times. Table 1 lists the corresponding corrected times of light
maximum. In addition, it can be seen from Equation (1) that we also obtain the
mean magnitude $A_{0}$, pulsation period $P_{\rm pul}$ and other parameters of
each data segment.

###  2.2 MACHO and OGLE

The main scientific objectives of MACHO and OGLE are to find dark matter in
the universe by microlensing technique (Udalski et al., 1992; Alcock et al.,
1997a). They respectively monitor those regions, such as the Galactic Bulge,
the large and small Magellanic Clouds. In addition to many achievements in the
discovery of gravitational lens events, these two projects have also brought
results in the field of variable star (e.g., RRLs in LMC, Alcock et al. 1996;
Soszynski et al. 2003; Soszyński et al. 2009, 2016, 2019). At present MACHO
project has been completed, but OGLE project is still ongoing and has reached
phase IV. We collected these old and new data for the $O-C$ and Fourier
analyses, and compared the results with the work of Derekas et al. (2004).

###  2.3 TESS

TESS is a new space project after the $Kepler$ project, which also uses the
transit method to discover exoplanets in the nearby stars (Ricker et al.,
2015). The observation region of $Kepler$ is relatively fixed, but that of
TESS can basically cover the whole celestial surface. However, the TESS
telescope has a small aperture and its resolution is only 21 arcsecnods/pixel.
Therefore, TESS cannot directly provide the light curves for dark stars in
dense star fields such as BE Dor. Fortunately, TESS also provides the download
of cutouts of images (Brasseur et al., 2019), which enables us to try targeted
processing of data. Until July 24, 2021, TESS provides the image files of 25
sectors for BE Dor (22 files contain valid data), and we download the
corresponding $5\times 5$ pixel cutouts. Through preliminary observation, we
find that the fluxes of the central pixels change periodically, which
indicates that the images do carry the light information of BE Dor. Further,
we plot the light curves of the 25 pixels over time independently, and find
that the flux variations caused by BE Dor are basically concentrated in the
center $3\times 3$ pixels.

Therefore, we assume that there is only one variable star (i.e. BE Dor) in the
whole $5\times 5$ pixel image, the light variations is only concentrated in
the central $3\times 3$ pixel image, and the 16 pixels in the periphery do not
contain the information of BE Dor. In order to eliminate the flux changes
caused by instrument and other factors, the following equation is used to
obtain the light curves:

$\Delta mag(t)=-2.5\lg[\frac{\sum^{9}_{\rm
inner=1}flux\\_inner(t)}{\sum^{16}_{\rm outer=1}flux\\_outer(t)}\frac{16}{9}]$
(2)

In the above formula, other non-pulsation variation factors are well
eliminated by the division. After obtaining the light curves of each sector
(mostly span 30 days), we fit a polynomial to the data, and then remove the
long-term trends by subtracting the polynomial curve. Figure 2 shows the light
curves after treatments. It should be pointed out that due to the low
resolution, BE Dor cannot be distinguished from the background stars, and the
amplitude of light curves in Figure 2 is far less than the real amplitude of
BE Dor.

###  2.4 ASAS-SN

ASAS-SN project uses multiple telescopes located in different countries to
monitor the vast sky with a purpose of discovering bright supernova and other
transients (Shappee et al., 2014). Meanwhile, it also obtains the light curves
of a large number of variable stars, and sorts out the catalogs (Jayasinghe et
al., 2018, 2020). Its variable stars database provides five years of V-band
light curves for BE Dor, which are similar to MACHO and OGLE data in time
resolution and precision. Figure 3 show the corresponding light curves of BE
Dor.

##  3 Analyses

Because BE Dor is located in the Southern Hemisphere, and its apparent
magnitude is only $\langle V\rangle$ $\sim$ 15 mag, most of the data provided
by sky survey projects are obtained by photometric observation. Therefore, our
analyses focus on these photometric data. The corresponding methods are $O-C$
method and Fourier analysis method. The corresponding analyses are presented
in the following subsections.

###  3.1 $O-C$ analysis

For pulsating variable stars, $O-C$ analysis usually selects the light maximum
as the reference phase to calculate the $O-C$ values (Sterken, 2005). We have
briefly introduced the method of determining the times of light maximum of BE
Dor in Section 2.1. Based on the different characteristics of photometric
data, we set different Fourier orders when calculating the times of light
maximum: for DASCH data, $n$ = 1; for MACHO, OGLE and ASAS-SN data, $n$ = 3;
for TESS data, $n$ = 5. Table 1 lists the corresponding times of light
maximum. Using the following corrected linear ephemeris,

$\rm HJD_{\rm max}=2449197.4241+0.^{d}328038\cdot E,$ (3)

we calculate the $O-C$ values and plot the $O-C$ diagram (see Figure 4). In
Figure 4, the $O-C$ values provided by Derekas et al. (2004) are also plotted
(black pentagram). It can be seen that the change trends of $O-C$ points
obtained by us and Derekas et al. (2004) according to MACHO and OGLE-III data
are consistent, and there is no obvious systematic deviation between them.

It is easy to see that the $O-C$ diagram of BE Dor shows more complex changes
than the previous results. Firstly, when epoch number $<$ 15000 cycle, the
$O-C$ curve shows a relatively periodic change, but the light-travel time
effect model cannot well describe the change trend, which means that the
change is not strictly periodic. It should be noted that there is a long gap
between DASCH and MACHO data in $O-C$ diagram (range from -50000 to -5000
epoch cycles). Since the $O-C$ curve of BE Dor fluctuates greatly, it cannot
be ruled out the epoch cycles of DASCH data are incorrect. Therefore, we
abandoned the DASCH data, and only fitted the $O-C$ data of MACHO and OGLE-
III. It is found that the $O-C$ diagram can be well described by the
sinusoidal component with a period of about 8.1 year (red dotted line).
According to the above result, it can be seen that the $O-C$ values suddenly
changed the trend near 15000 cycle, and the $O-C$ values that should be
reduced, becomes increased. In other words, it seems that there is a $\pi$
phase jump around 15000 cycle (black solid line refers the $\pi$ phase jump).
After that, the $O-C$ changes obtained by OGLE-IV, ASAS-SN and TESS are also
different from the previous change trend. Considering that the light-travel
time effect model will generally cause the $O-C$ diagram to show strict
periodic changes, it can not well explain the $O-C$ diagram of BE Dor anymore.

###  3.2 Fourier analysis

The changes in the $O-C$ diagram means that the pulsation of BE Dor occurs
period modulation. In order to explore whether there are corresponding changes
in other pulsation parameters, we adopted Fourier analysis method to analyze
the MACHO, OGLE and TESS data (due to the large errors, the results of ASAS-SN
data are not listed), and the software used here is Period04 (Lenz & Breger,
2004, 2005). Due to the strong period modulation, we cannot simply Fourier
decompose the light curves as a whole, but divide the light curves into many
segments, and then Fourier analyze each segment. The Fourier formula is the
same as Equation (1). Table 2 lists the corresponding Fourier coefficients
obtained from each segments, including pulsation period $P_{\rm pul}$, mean
magnitude $A_{0}$, $R_{21}$, $R_{31}$, $\phi_{21}$, $\phi_{31}$ and
corresponding errors.

Figure 5 describes the trend chart of pulsation period with time, in which the
black and green hollow points represent the pulsation period obtained from
DASCH and ASAS-SN data, respectively. These values are obtained during the
process of determined the times of light maximum. It can be seen that at
first, the pulsation period changes quasi-periodically with time. The
corresponding period change is about 12 s, which is consistent with the
previous results (Derekas et al., 2004). The red dotted line indicates the
corresponding sinusoidal fitting (only those $P_{\rm pul}$ determined from
DASCH, MACHO, OGLE-III data are used), and the sinusoidal period is about 8.2
yr. However, between OGLE-III and IV, the pulsation period changes suddenly.
These results are consistent with those in the $O-C$ diagram.

In addition, Figure 6 plots the variation of mean magnitudes with time. It can
be seen that the mean magnitudes obtained from OGLE I-band data seem to
increase with time (bottom panel), indicating that the luminosity in I-band is
darkened. And accordingly, the values from MACHO b-band are reduced with time
(upper panel), but those from MACHO r-band data do not change significantly
(middle panel). The effective wavelength range of I-band used in OGLE is 700 -
1100 nm, the ranges of r and b filter used in MACHO are 590 - 780 nm and 440 -
590 nm, respectively. Assuming that the mean magnitudes do change as shown in
Figure 6, it means that the luminosity in short wave band becomes brighter and
in long wave band becomes darker. That is, the color index (b - I) decreases,
and the effective temperature increases with time. Of course, this change is
still very small, which is equivalent to the photometric error of the two
projects. The long-term change of the magnitudes cannot be ruled out to be
caused by instruments or other factors. In fact, for the photometric data
accumulated for many years, how to retain the information of the long-term
changes as much as possible in data processing and analysis is still a
difficult problem. Therefore, we doubt the authenticity of the long-term
changes of BE Dor mean magnitudes, and just make a presentation in Figure 6.

Figure 7 presents the diagrams of Fourier coefficients with time. It can be
seen that these parameters fluctuate with time, but the changes are not so
significant relative to the pulsation period (see Figure 5).

##  4 Discussion

The above analyses show that the pulsation modulation of the pulsating
variable BE Dor is mainly manifested in its period modulation, while the
variations of the other pulsation parameters and Fourier coefficients, such as
the $R_{21}$, $R_{31}$, $\phi_{21}$ and $\phi_{31}$, are relatively weaker. In
addition, the $O-C$ diagram (Figure 4) and the $P_{\rm pul}$ \- Time diagram
(Figure 5) also indicate that the pulsation period shows quasi-periodic
variation and abrupt change. Therefore, the light-travel time effect caused by
the presence of a companion object cannot explain these variations. Moreover,
the variation characteristics (mainly pulsation period modulation, quasi-
periodic variation with long period up to several years) are also different
from normal Blazhko RRc stars (Netzel et al., 2018). In Section 1 of this
paper, we mentioned some RRc stars showing the similar phenomena. It can be
seen that the variations of BE Dor is not a rare case. Maybe we should look
for some more similar samples, get more clues by studying their
characteristics to find the mechanism leading the period modulation.

###  4.1 Four RRc stars in $Kepler$ field

Nemec et al. (2013) used the large aperture grounded telescopes to observe the
RRLs in $Kepler$ field, obtained the high-resolution spectra and determined
the corresponding physical characteristics. In their paper, four RRc stars are
included. Using the uninterrupted photometric data provided by $Kepler$, we
determine their times of light maximum and plot their $O-C$ diagrams (see
Figure 8, refer to Li & Qian 2014 for the corresponding data processing and
analysis methods), in which the $O-C$ diagram for KIC 9453114 shows extremely
large variation, similar to the changes of BE Dor, those of KIC 5520878 and
8832417 have little changes and are almost straight lines, and that of KIC
4064484 has weak fluctuation. Table 3 lists their physical parameters provided
by Nemec et al. (2013) (see Table 9 in their paper), in which the four stars
are arranged from left to right according to the degree of $O-C$ changes. It
can be found that the parameters of KIC 9453114 showing the largest $O-C$
variation are obviously different from other 3 RRc stars: its effective
temperature $T_{\rm eff}$, surface gravity acceleration $\log g$ and metal
abundance [M/H] are the smallest, while the macroturbulent velocity $v_{\rm
mac}$ and microturbulent velocity $\xi_{t}$ are the largest. Especially, the
$v_{\rm mac}$ reaches 29.3 km s-1, which is the largest among RRLs in $Kepler$
field, about 2-3 times that of other stars. The two RRc stars with almost no
$O-C$ changes have the highest $T_{\rm eff}$, [M/H] and the smaller $v_{\rm
mac}$. Therefore, our direct idea is that there is a positive correlation
between the change severity of $O-C$ diagram and the $v_{\rm mac}$, and the
reason for the large $v_{\rm mac}$ may be related to the $T_{\rm eff}$, [M/H]
and the implied evolution history.

###  4.2 the Stothers model

The strong period modulation shown by BE Dor may be explained by Stothers
model according to its quasi-periodic variation. Of course, Derekas et al.
(2004) have put forward corresponding views. Stothers (1980) pointed out that
the hydrogen and helium ionization zones of RRLs are convective unstable, and
it can be reasonably assumed that the local turbulent motions can directly
interfere with the pulsation of the envelope. However, Stothers expected that
the periodic variation of pulsation caused by convection is random and short-
term (hours to days), would not accumulate on a long time scale, and the
observed pulsation period variations are related to the activity of solar like
magnetic field. Stothers inferred that the sudden change of some pulsation
periods may correspond to the sudden generation or destruction of the magnetic
field. If so, the quasi-periodic and abrupt changes in BE Dor $O-C$ diagram
can be explained by the viewpoint of Stothers (1980). It is worth noting that
the energy equipartition and the Equation (1) mentioned in that paper:

$H^{2}/8\pi\approx 1/2\rho\upsilon_{\rm turb}^{2}.$ (4)

It can be seen that the magnetic field strength $H$ is positively correlated
with the turbulence velocity $\upsilon_{\rm turb}$, and they are interrelated.

In fact, Stothers model has been continuously improved and attracted attention
since then (Stothers, 2006, 2010; Smolec et al., 2011; Molnár et al., 2012).
Stothers (2006) considered the interaction between turbulent convection and
magnetic field activity, and pointed out that the cyclically strengthening and
weakening of turbulent convection in the hydrogen and helium ionization zones
is the mechanism causing the Blazhko effect. However, through research, Smolec
et al. (2011) and Molnár et al. (2012) found that Stothers model cannot
explain Blazhko modulations with short period (less than 100 days), but it may
be effective for long-term modulations. Noting the period modulation
characteristics of RRc stars such as BE Dor and KIC 9453114, we suppose that
they may be explained by Stothers model, that is, the strong modulations are
caused by the interaction of turbulent convection and magnetic field
activities, which affects the pulsation in the coupled state.

Stothers (2010) considers that his theoretical model can explain those RRLs
with significant amplitude modulation but no periodic modulation. However, the
samples are RRab stars, and the Blazhko periods are less than 150 days. As
Smolec et al. (2011) pointed out, for Blazhko modulation on a relatively short
time scale, huge modulation of the mixing length is required. In our opinion,
when there is a large change in the pulsation amplitude on the short scale, it
means that the amplitudes of the radial velocity caused by pulsation also
change greatly (Sesar, 2012). Turbulent convection and magnetic field
activities may affect the macroturbulent and microturbulent velocities, but
would not have a great impact on the overall radial velocities of RR Lyrae
variables.

The problem following is that what factors is related to the magnitude of
macroturbulent velocity? According to Table 3, it can be found that KIC
9453114 has the longest pulsation period and the lowest $T_{\rm eff}$, $\log
g$ and [M/H]. These parameters seem to imply that it is an old and evolved RR
Lyrae star, at the late stage of horizontal branch. The variable stars in this
evolution stage move to the upper right in the HR diagram, the $T_{\rm eff}$
decrease and the radii increase (resulting in the decrease of $\log g$). At
relatively low temperature, the turbulence develops more fully, which affects
the pulsation, resulting the pulsation period variation on the order of
$\delta P/P$ $\sim$ 10-4. Back to BE Dor, due to its faint nature, there is no
relevant spectral observation so far, therefore we cannot know its atmospheric
physical parameters. Gaia Dr2 (Gaia Collaboration et al., 2018) provided its
temperature of 7688 K, which is higher than that of normal RRc stars (Li et
al., 2020). Therefore, we are cautious about this result, and suggest that
relevant spectral observations are needed in the future.

Other macroturbulent velocity measurements and research works on RRLs are also
being carried out. Preston et al. (2019) derived the macroturbulent velocity
variations of RRLs, and found that the RRab stars exhibit an upper limit on
$\upsilon_{\rm mac}$ of $5\pm 1$ km s-1, and the RRc minima range from 2 to 12
km s-1. Compared with their results, those obtained by Nemec et al. (2013) are
generally and systematically larger. The reason for this may be related to the
use of different analysis roadmaps and methods.

###  4.3 More discussions

Admittedly, due to the lack of spectral data of BE Dor, our view that the
pulsation period modulation is related to the macroturbulent velocity is
obtained indirectly, which is a preliminary conclusion. More works need to be
done in the future: on the one hand, BE Dor should be studied more deeply. The
most urgent task is to carry out spectral observation and obtain its
atmospheric physical parameters; on the other hand, find more RRc stars which
show the similar modulation phenomena, discover their nature features, and
provide clues for theoretical work. For example, Netzel et al. (2018)
presented the analyses of Blazhko RRc stars in the Galactic bulge by using
OGLE-IV data. Figure 6 in their paper plotted the histogram of Blazhko
periods. We note that there is a weak centralized distribution of Blazhko RR
stars in the period of 1500 - 3000 days. Perhaps similar period modulations
exist in these targets and they can be used as research objects of special
concern.

Our work on BE Dor draws on the observation results of several sky survey
projects. Their main scientific objects are not pulsating variables (but the
baryonic dark matters, supernovae, and exoplanets). However, they have
actually greatly helped the development of this research field. It is worth
mentioning that TESS project, due to its design characteristics (small
aperture and low spatial resolution), it seems that the dark source in crowded
star field such as BE Dor cannot obtain any pulsation information.
Nevertheless, our work shows that as long as targeted data processing is
adopted, valuable data and information can be obtained from the original image
data.

##  5 Summary

Using the photometric data of several sky surveys, we carried out $O-C$
analysis and Fourier analysis for RRc star BE Dor. Through analyses, we have
the following discoveries and viewpoints:

1\. The change in the $O-C$ diagram of BE Dor is no longer the periodic change
pointed out by Derekas et al. (2004), but quasi-periodic change (period is
about 8.1 years) and sudden jump (occurring during the observation period of
OGLE project). Therefore, the light-travel time effect caused by the motion of
companion object can no longer explain the change in $O-C$ diagram.

2\. The light curves provided by MACHO, OGLE and TESS are Fourier decomposed
to obtain the pulsation period, mean magnitude and Fourier coefficients of BE
Dor. The change of the pulsation period with time supports the results of
$O-C$ diagram. Other parameters also fluctuate with time, but weaker than the
period modulation. It indicates that BE Dor is an RRc star with significant
period modulation.

3\. Such modulations are not rare in RRc stars. Through the comparison of four
RRc stars in $Kepler$ field, we found that one of RRc stars, KIC 9453114,
showed strong $O-C$ variations similar to BE Dor, and its physical parameters
($T_{\rm err}$, $\log g$, and $v_{\rm mac}$) are different from other three
stars. The most notably is its high $v_{\rm mac}$, up to 29.3 km s-1.
Moreover, the variation degree of $O-C$ diagram for the four RRc stars also
seems to be related to the $v_{\rm mac}$, i.e., the $O-C$ diagrams basically
do not change for two stars with relative low $v_{\rm mac}$; but changes rapid
and dramatic for KIC 9453114.

4\. The period modulations in BE Dor, KIC 9453114 and other RRc stars showing
the similar phenomena (mentioned in Section 1) can be explained by the
Stothers model, that is, the interaction between turbulent convection and
magnetic field activities in the ionization region leads to the instability of
pulsation period (Stothers, 1980, 2006). Stothers’ viewpoint may not explain
the general Blazhko effect, but it may explain the modulation phenomenon of
some RRLs.

Our work mainly used photometric data, and the conclusion has some
limitations. The spectral observations of pulsating variables which can
provide more information are very necessary. In fact, investigators have made
many systematic achievements by using the spectral data from different sources
(Duan et al. 2021 and references therein). It is believed that in the future,
with the help of observation and data accumulation, we can more
comprehensively understand the properties of RRLs and finally solve the
various problems left over from history.

##  6 acknowledgments

This paper utilizes public domain data obtained by the MACHO Project, jointly
funded by the US Department of Energy through the University of California,
Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48, by
the National Science Foundation through the Center for Particle Astrophysics
of the University of California under cooperative agreement AST-8809616, and
by the Mount Stromlo and Siding Spring Observatory, part of the Australian
National University. The DASCH project at Harvard is grateful for partial
support from NSF grants AST-0407380, AST-0909073, and AST-1313370. This work
is supported by the National Natural Science Foundation of China (No.11803084,
No.11933008, and No.11922306).

##  7 Date availability statements

The data used in this article are from five sky survey projects, and the
original data or images can be obtained through the following links query:

https://projects.iq.harvard.edu/dasch (DASCH)

https://mast.stsci.edu/tesscut/(TESS)

http://www.astronomy.ohio-state.edu/asassn (ASAS-SN)

https://wwwmacho.anu.edu.au/ (MACHO)

http://ogle.astrouw.edu.pl/ (OGLE)

The data generated by our analyses are available in the article and in its
online supplementary material.

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Table 1: Times of light maximum and corresponding errors for BE Dor provided
by present paper. (the first 10 lines of the whole table)

Max. | Error | Method/Filter | Ref.
---|---|---|---
HJD | (days) |  |
2425921.9326 | 0.0043 | pg | DASCH
2426001.9809 | 0.0043 | pg | DASCH
2426141.4127 | 0.0044 | pg | DASCH
2426256.8961 | 0.0044 | pg | DASCH
2426564.3070 | 0.0044 | pg | DASCH
2426601.0520 | 0.0044 | pg | DASCH
2426820.5241 | 0.0046 | pg | DASCH
2426939.9301 | 0.0046 | pg | DASCH
2427071.1420 | 0.0046 | pg | DASCH
2427358.1483 | 0.0045 | pg | DASCH
… | … | … | …

Notes. pg - photographic; RI - the wavelength coverage of TESS is 6000 - 10000
angstrom, covering most of the Johnson/Bessell R and I filters. The TESS times
are provided in BJD_TDB. For consistency, we have converted the times in
BJD_TDB to in HJD_UTC (Eastman et al., 2010).

Table 2: Fourier parameters and coefficients for BE Dor from different data sources. (the first 10 lines of the whole table) Data Set | Filter | Mean Time | $A_{0}$ | $P_{\rm pul}$ | $R_{21}$ | $\phi_{21}$ | $R_{31}$ | $\phi_{31}$
---|---|---|---|---|---|---|---|---
|  | HJD-2400000 | (mag.) | (days) |  | (rad) |  | (rad)
MACHO_01 | b | 48906.1891 | -9.172 | 0.327981(06) | 0.177(13) | 3.149(.077) | 0.064(12) | 0.218(.228)
MACHO_02 | b | 49020.4754 | -9.166 | 0.327993(10) | 0.164(11) | 3.268(.082) | 0.094(13) | 0.078(.120)
MACHO_03 | b | 49081.0099 | -9.161 | 0.327992(10) | 0.191(09) | 3.261(.061) | 0.081(11) | -0.289(.122)
MACHO_04 | b | 49144.4675 | -9.157 | 0.328007(08) | 0.185(13) | 3.157(.069) | 0.082(13) | -0.189(.153)
MACHO_05 | b | 49233.6808 | -9.166 | 0.327992(05) | 0.198(14) | 3.188(.069) | 0.091(12) | -0.206(.161)
MACHO_06 | b | 49356.5275 | -9.168 | 0.328006(04) | 0.173(09) | 3.233(.063) | 0.080(10) | -0.171(.139)
MACHO_07 | b | 49460.4636 | -9.156 | 0.328018(07) | 0.171(12) | 3.116(.075) | 0.082(13) | 0.099(.143)
MACHO_08 | b | 49525.1692 | -9.156 | 0.328039(11) | 0.149(11) | 3.237(.071) | 0.087(10) | -0.088(.123)
MACHO_09 | b | 49582.6118 | -9.171 | 0.328017(05) | 0.198(11) | 3.093(.041) | 0.102(09) | -0.664(.095)
MACHO_10 | b | 49641.1920 | -9.169 | 0.328039(05) | 0.173(10) | 3.251(.058) | 0.099(09) | -0.019(.103)
… | … | … | … | … | … | … | … | …

Notes. The mean magnitude $A_{0}$ for TESS data are not real, therefore, are
not listed.

Table 3: Physical characteristics of the four RRc stars in $Kepler$ provided by Nemec et al. (2013). KIC Number | 9453114 | 4064484 | 5520878 | 8832417
---|---|---|---|---
$P_{\rm pul}$ (days) | 0.366 | 0.337 | 0.269 | 0.248
$T_{\rm eff}$ (K) | 6204 | 6473 | 7266 | 6999
$\log g$ (cm s-2) | 2.04 | 2.39 | 3.66 | 3.30
$[\rm M/H]$ (dex) | -2.15 | -1.51 | -0.18 | -0.25
$v_{\rm mac}$ (km s-1) | 29.3 | 18.4 | 15.2 | 15.3
$\xi_{t}$ (km s-1) | 6.4 | 4.0 | 3.0 | 3.5
$v\sin i$ (km s-1) | 4.3 | 4.9 | 5.1 | 4.8
Figure 1: Light curves of BE Dor provided by DASCH. The photograph magnitude
is close to B-band magnitude. In the top panel, empty blue circles denote
those data not used in the analyses. The phased light curves are plotted in
the bottom panel, and the linear ephemeris HJDMAX = 2430118.2192 +
0d.3279975$\cdot E$ is used to calculate the phases. Figure 2: Part of light
curves of BE Dor obtained from TESS (Sector 1, Camera 4, CCD1). The phases in
the bottom panel are calculated by the linear ephemeris HJDMAX = 2458335.9211
+ 0d.3280228$\cdot E$. It should be noted that the vertical axis is not the
real magnitude. Figure 3: V-band light curves of BE Dor provided by ASAS-SN.
The phases in the bottom panel are calculated by the linear ephemeris HJDMAX =
2457732.35095 + 0d.32802602$\cdot E$. Figure 4: $O-C$ diagram of BE Dor. The
$O-C$ points show quasi-periodic variation and sudden jump change. Figure 5:
$P_{\rm pul}$ vs Time diagram. The range of the period change is about
$1.4\times 10^{-4}$ days, i.e., 12.1 s. Figure 6: Mean magnitude $A_{0}$ vs
Time diagrams. The mean magnitudes of different bands seem to increase or
decrease with time. Figure 7: Variations of the Fourier coefficients for BE
Dor. Figure 8: $O-C$ diagrams of four RRc stars in $Kepler$ field.