# Period Changes of 14,127 Contact Eclipsing Binaries in the Galactic Bulge

Kyeongsoo Hong Institute for Astrophysics, Chungbuk National University,
Chungdae-ro 1, Seowon-Gu, Cheongju 28644, Republic of Korea  Jae Woo Lee Korea
Astronomy and Space Science Institute, Daejeon 34055, Republic of Korea  Jang-
Ho Park Korea Astronomy and Space Science Institute, Daejeon 34055, Republic
of Korea  Hye-Young Kim Department of Astronomy and Space Science, Chungbuk
National University, Cheongju 28644, Republic of Korea  Chung-Uk Lee Korea
Astronomy and Space Science Institute, Daejeon 34055, Republic of Korea
Hyoun-Woo Kim Department of Astronomy and Space Science, Chungbuk National
University, Cheongju 28644, Republic of Korea  Dong-Jin Kim Korea Astronomy
and Space Science Institute, Optical Division, 776 Daedeok-daero, Yuseong-gu,
Daejeon 34055, Republic of Korea  Cheongho Han Department of Physics, Chungbuk
National University, Cheongju 28644, Republic of Korea

###### Abstract

We present the orbital period variations of 14,127 contact eclipsing binaries
(CEBs) based on the OGLE-III&IV observations in the Galactic bulge. New times
of minimum lights for the CEBs were derived by binary modeling for the full
seasonal light curves, which were made from survey observations at an interval
of 1 year. The orbital period changes of the systems were classified based on
the statistical inference, multiple-hypothesis testing error measure, and
visual inspection of the eclipse timing diagrams. As the results, we
identified 13,716 CEBs with a parabola, 307 CEBs with a sinusoid, and 104 CEBs
with the two variations. The period distributions of the inner close binaries
and the outer companions were in the ranges of $0.235-0.990$ days and
$5.0-14.0$ years, respectively. In our sample of 13,820 CEBs showing a
parabolic variation, the highest decreasing and increasing period rates were
determined to be $\dot{P}=-1.38\pm 0.06\times 10^{-5}$ day year-1 for OGLE-
BLG-ECL-169991 and $\dot{P}=+8.99\pm 0.44\times 10^{-6}$ day year-1 for OGLE-
BLG-ECL-189805, respectively. The secular period change rates were distributed
almost symmetrically around zero, and most of them lie within $\dot{P}=\pm
5.0\times 10^{-6}$ day year-1.

binaries: eclipsing – Galaxy: bulge
Supporting material: figure sets, machine-readable tables, tar file

##  1 Introduction

Generally, contact eclipsing binaries (CEBs) consist of two dwarf stars
surrounded by a common convective envelope. Most of these systems have
spectral classes from F to K, and orbital periods between 0.2 and 1.0 days
(Maceroni & van’t Veer 1996; Paczyński et al. 2006). The light curves of these
binaries show similar eclipse depths because they have almost the same
effective temperatures due to the mass and energy transfers between the
components (Lucy 1968; Webbink 2003). A good review of the possible formation
and evolution processes of close binary system have been provided by Webbink
(1976), Stȩpień (2006), and Eggleton (2012).

The investigation of orbital period changes of the CEBs has contributed to the
understanding of binary evolution. In general, the eclipse timing variations
are represented by one or a combination of parabolic or cyclical variations
created by various effects, as follows: (i) the light-travel-time effect
(LTTE) due to additional companions (Irwin 1952, 1959) or period oscillations
because of magnetic activity cycles (Applegate 1992; Lanza et al. 1998) show
cyclical period changes, (ii) the angular momentum loss (AML) due to a
gravitational radiation or magnetic braking (Paczyński 1967; van’t Veer 1979;
Mochnacki 1981; Rappaport et a. 1983; van’t Veer & Maceroni 1989) indicate
downward parabolic changes created by the period decrease, (iii) mass ejection
from the binary system (Prendergast & Taam 1974) demonstrates a upward
parabolic variation induced by the period increase; and (iv) mass transfers
between the binary components (Lucy 1976; Webbink 1976, 2003) can show upward
or downward variations.

Since 1992, the Optical Gravitational Lensing Experiment (OGLE) project has
carried out long-term wide-field survey observations, searching for dark and
unseen matter using the microlensing phenomena (Udalski et al. 1992).
Recently, Soszyński et al. (2016) published a list of 450,598 eclipsing and
ellipsoidal binary systems (86,560 CEBs, 338,633 semi-detached and detached
EBs, and 25,405 non-eclipsing ellipsoidal variables) based on the OGLE-II,
OGLE-III, and OGLE-IV observations in the Galactic Bulge. The long-term
homogeneous data of eclipsing binaries allows us to study variations in their
orbital period. Kubiak et al. (2006) reported secular period variations for
569 CEBs with a short period ($P\leq 1$ day) from the OGLE data between 1992
and 2005. Pietrukowicz et al. (2017) identified 56, 52, and 35 systems with
increasing, decreasing, and cyclic period variations, respectively, by
searching for potential stellar mergers in a sample of 22,462 short period EBs
($P\leq 4$ day) in the Galactic bulge using OGLE observations from 1992-2015.
Hajdu et al. (2019) reported LTTE solutions for 992 hierarchical multiple
candidates, which were determined by analyzing the eclipse timing diagrams of
about 80,000 EBs with a period shorter than 6 days.

Luminous Red novae (LRNe) are very important targets when investigating the
evolution of binary systems, and have been related to common-envelope
evolution (Soker & Tylenda 2003; Kulkarni et al. 2007; Tylenda et al. 2011;
Ivanova et al. 2013; Nandez et al. 2014; Pastorello et al. 2019; Howitt et al.
2020). Only four of these LRNe have been noted in our Galaxy: V4332 Sgr
(Hayashi et al. 1994), V838 Mon (Brown et al. 2002), V1309 Sco (Nakano et al.
2008), and OGLE-2002-BLG-360 (Tylenda et al. 2013). The LRNe are known to be
the result of the merger of the cores of contact binaries. They have a total
energy in the range of $10^{45}-10^{48}$ ergs (Kashi 2018), which have
intermediate luminosities between classical novae and super-novae, and
relatively long outbursts. Kochanek et al. (2014) estimated that these
transients are produced about once per decade. According to the suggestion by
Ferreira et al. (2019), the magnitude of LRNe V1309 Sco has nearly stabilized
during about 9 years after the outburst, and the system is likely to be in a
blue straggler state. Recently, the Contact Binaries Towards Merging (CoBiToM)
Project was performed to study stellar evolution before the merging processes
by Gazeas et al. (2021).

The aim of this study was to search for potential stellar mergers through the
eclipse timing analysis of a total of 86,560 CEBs classified by Soszyński et
al. (2016). The homogeneous data obtained from the OGLE-III&IV surveys provide
long-term stable and uniform photometry for the objects, over a long period of
15 years. Section 2 provides the information about the OGLE data used in this
study. We briefly describe the determination of the times of minimum lights
for the eclipse timing diagrams in Section 3. Section 4 presents the study of
the orbital period variations of the selected CEBs. Finally, we present the
summary and discussions of our results in Section 5.

##  2 Data collections

For this study, we collected photometric data for 86,560 CEBs from the OGLE-
III&IV archive data from 2001 to 2015 for the Galactic bulge fields. The OGLE
observations were carried out using the 1.3-m Warsaw telescope at the Las
Campanas Observatory in Chile. The telescope is equipped with an eight-CCD
mosaic camera covering a 0.35 deg2 field of view during the OGLE-III and with
a 32 chip mosaic camera which covers a 1.4 deg2 field of view for the OGLE-IV.
The sky coverages were nearly 92 deg2 for the OGLE-III in 2001$-$2009 and 182
deg2 for the OGLE-IV in 2010$-$2015\. In this study, only light curves in $I$
band were used because the surveys were obtained mostly in $I$ band,
occasionally in $V$ band. The overall specification and reductions of the OGLE
observations were provided by Udalski et al. (2008 for OGLE-III; 2015 for
OGLE-IV).

##  3 Times of minimum light

For the orbital period studies of the CEBs, we used the observed minus
computed ($O-C$) diagram, which is a powerful diagnostic tool that can measure
very subtle changes in the orbital period. The eclipse timings for the $O-C$
diagram are usually determined using consecutive observations during primary
or secondary eclipses. However the CEBs were observed only one to about thirty
points per night for lowest and highest cadence fields by the OGLE-III&IV
surveys in the Galactic bulge. The number of data points in the $I$ band
during the OGLE-III observation periods was collected from about 35 to 2,500
points per field, while the OGLE-IV observations were obtained about 150 to
17,000 measurements per field for their observing periods. More detailed
information of the observation cadence for the OGLE surveys was presented in
the papers of Udalski et al. (2003 for OGLE-III; 2015 for OGLE-IV).

In order to determine the times of minimum lights from the OGLE data, we
adopted the method of determining the primary eclipse timing from the
synthetic light curve, as applied in the papers of Hong et al. (2015, 2016,
and 2019). First, full light curves with an interval of 1 year were
constructed from the OGLE observations. Then, we analyzed the light curves
using the binary modeling and obtained the primary eclipse timings from the
binary solutions. For the binary modeling, we used the Wilson-Devinney program
(Wilson & Devinney 1971; Van Hamme & Wilson 2007, hereafter W-D), which is an
excellent tool for solving a binary star light curve. The light curve
solutions by the W-D program can provide times of minimum light without the
impact of spot activity (Maceroni & van’t Veer 1994; Lee et al. 2014).

##  4 Eclipse timing variations

In the eclipse timing analysis, times of minima with a significantly large
error can increase the uncertainty of period changes. Therefore we excluded
the minimum times that had an uncertainty of more than three times the mean
error (0.0025 days), which was calculated using the errors of all minimum
times obtained from the individual light curves of 86,560 CEBs. Then the CEBs
with eclipse timings with more than six points were selected by considering a
variable number of the following equations in this section, to prevent
incorrect results due to insufficient data. Consequently, we selected 50,824
of 86,560 CEBs with an orbital period range of $0.166-2.07$ days.

First of all, the mean light ephemeris for all selected systems was calculated
introducing all minimum times into the following linear ephemeris:

$C_{1}=T_{0}+PE.$ (1)

Then, the selected systems were classified into three types (parabolic,
sinusoidal, and parabolic $plus$ sinusoidal variations) based on the approach
as presented below.

> 1\. In order to classify the parabolic variations in the $O-C$ diagrams, we
> adopted the Bayes factor for a statistical inference (Kass & Raftery 1995;
> Wagenmakers 2007) and the controlling false discovery rate (FDR) for a
> multiple-hypothesis testing error measure (Benjamini & Hochberg 1995) based
> on the R software environment111http://www.rproject.org, which is an open-
> source free statistical environment developed under the GNU GPL (Ihaka &
> Gentleman 1996). The Bayesian information criterion (BIC) is given by
>
> $BIC(H_{i})=-2{\rm log}L_{i}+k_{i}{\rm log}~{}n,$
>
> where $L_{i}$ and $k_{i}$ are the maximum likelihood and the number of free
> parameters for model $H_{i}$, respectively, and $n$ is the number of minimum
> times. The Bayes factor is given by
>
> $BF_{10}=\exp\left(\frac{BIC(H_{0})-BIC(H_{1})}{2}\right).$
>
> In this case, the hypothesis $H_{0}$ and $H_{1}$ represent the linear and
> parabolic regressions for individual CEBs, respectively. We selected the
> models with the Bayes factor of $BF_{10}>10$ (cf. the scales of evidence for
> model selection are proposed in the paper of Kass & Raftery 1995). The
> controlling FDR is given by
>
> $P_{i}\leq\frac{i}{m}\alpha,$
>
> where $P_{i}$ is the ordered p-values and $m$ is the total number of
> hypotheses. We strict set the $\alpha$ to 0.01 for a false positive to be
> less than 1%. In the case of the p-value of $H_{0}$ less than $H_{1}$, the
> hypothesis is rejected. Applying the criterion, we identified 13,716 CEBs
> exhibiting parabolic variations.

> 2\. The sinusoidal and parabolic $plus$ sinusoidal variations in the $O-C$
> diagrams of 50,824 CEBs were selected by visual inspection because it is
> difficult to solve the nonlinear least-squares fitting for these two models.
> Their fitting process requires good starting values of amplitude, angular
> frequency, and phase. In the procedure, we excluded the systems with the
> period modulation shorter than 5 years by considering the one-year interval
> of eclipse timing data. Then the $O-C$ diagrams of the selected systems were
> fitted according to the procedures described in Section 4.2 and 4.3, and
> verified using the Bayes factor and FDR. As a result, we found the 307 CEBs
> with a sinusoid and 104 CEBs with two variations. Note, all of the
> individual eclipse timings for 14,127 CEBs are listed in the machine-
> readable data table.

###  4.1 Parabolic variations

The parabolic variations of CEBs can be explained by angular momentum loss due
to a magnetic stellar wind, a mass transfer from the more massive to the less
one, or a part of the periodic variation by the Applegate (1992) mechanism or
LTTE. All of the eclipse timings of the 13,716 CEBs showing parabolic
variations and the 104 CEBs with parabolic $plus$ sinusoidal variations were
applied by using a least-squares parabolic fitting as follows,

$C_{2}=T_{0}+PE+AE^{2}.$ (2)

In order to check whether our processes were appropriate, the period change
rates were compared with the results of 45 CEBs overlapped with ours in the
paper by Pietrukowicz et al. (2017), where a total of 23 and 22 CEBs had
decreased and increased period changes, respectively. The resulting period
change rates ($\dot{P}$) for the overlapping CEBs are listed in Table 1, where
the value of $\dot{P}$ is calculated from the relation $\dot{P}=2A/P$. As one
can see in Figure 1, the results are in satisfactory agreement with each other
within their uncertainties.

Among the 13,716 CEBs, 6,263 and 7,453 systems were classified with decreasing
and increasing period variations, respectively. Of the 104 CEBs with two
variations, 69 and 35 systems were identified with decreasing and increasing
period variations, respectively, which will be discussed in the following
Subsection 4.3. The resulting parabolic ephemeris for the 13,716 CEBs are
provided in Table 2, which are sorted from the highest decreasing rate.

In our samples, the systems OGLE-BLG-ECL-169991 and OGLE-BLG-ECL-052921 had
the highest period decreasing rates of $\dot{P}=-1.38\pm 0.06\times 10^{-5}$
day year-1 and $\dot{P}=-9.74\pm 0.54\times 10^{-6}$ day year-1, respectively,
and the systems OGLE-BLG-ECL-189805 and OGLE-BLG-ECL-288708 had the highest
period increasing rates of $\dot{P}=+8.99\pm 0.44\times 10^{-6}$ day year-1
and $\dot{P}=+8.29\pm 0.66\times 10^{-6}$ day year-1, respectively. The
eclipse timing diagrams for the two CEBs with the highest period decreasing
rate and for two CEBs with the highest period increasing rate are displayed in
the upper and lower panels of Figure 2, respectively. The $I$ light curves
phased with the linear and quadratic ephemeris for the four CEBs are plotted
based on the OGLE-III and OGLE-IV observations in the left and right panels of
Figure 3, respectively. In the left panels, the linear ephemeris taken from
the OGLE EBs catalog presented by Soszyński et al. (2016) were used, while the
parabolic ephemeris derived from Equation (2) were applied in the right
panels. As one can see in each panel of this figure, the scatters around the
light curves were remarkably diminished after applying the parabolic
ephemeris.

###  4.2 Sinusoidal variations

The sinusoidal variations in the $O-C$ diagram can be caused by LTTE due to an
additional companion in a binary system (Irwin 1952, 1959) or period
oscillations because of magnetic activity cycles (Applegate 1992; Lanza et al.
1998). To analyze the $O-C$ diagrams of 307 CEBs with a sinusoidal period
variation, all times of minimum of each EB were fitted using a sine curve, as
follows:

$C_{3}=T_{0}+PE+K\sin(\omega E+\omega_{0}).$ (3)

The sinusoidal ephemeris gave an acceptable fit to the eclipse timings for 307
CEBs. In all calculations, the parameters in Equation (3) were solved based on
the Levenberg-Marquart (hereafter LM; Press et al. 1992) technique. The basic
parameters with fitting results for 307 CEBs are listed in Table 3, where the
outer period ($P_{2}$) of the third companion is derived from the relation
$P_{2}=2\pi P/\omega$. The $O-C$ diagrams for 307 CEBs with the sinusoidal
ephemeris are displayed in Figure 4. For comparison, the periods of the triple
system candidates given in the papers of Pietrukowicz et al. (2017) and Hajdu
et al. (2019) are presented in the last column of Table 3. A total of 22 of
our and those systems were overlapped.

###  4.3 Parabolic $plus$ sinusoidal variations

As mentioned in Section 4.1, the eclipse timing diagrams of 104 CEBs show the
parabola $plus$ sinusoidal variations. Therefore the $O-C$ diagrams were
fitted using the quadratic ephemeris together with a sine curve, as follows:

$C_{4}=T_{0}+PE+AE^{2}+K\sin(\omega E+\omega_{0}).$ (4)

The basic parameters of 104 systems with parabolic and sinusoidal fitting
results are listed in Table 4. In the panels of Figure 5, the blue dashed and
red solid lines represent the parabolic term and the full ephemeris,
respectively.

###  4.4 Tertiary periods of triple system candidates

Although the period changes of the CEBs with a sinusoidal variation can be
explained by both magnetic activity cycles and LTTE, we only assumed the
period variation of the systems by LTTE in this study. According to our
results, the distributions of the inner ($P_{1}$; the period of close binary)
and outer periods ($P_{2}$; the period of the third companion) for 411 triple
system candidates were in the ranges of $0.235-0.990$ days and $5.0-14.0$
years, respectively. In this study, we first found 379 triple candidates,
excepting the 32 EBs investigated previously in the papers of Pietrukowicz et
al. (2017) and Hajdu et al. (2019). A histogram of their outer periods
($P_{2}$) is presented in Figure 6, where most values are located in the range
of $3,400-3,800$ days. The mean of the outer periods in our results was
determined to be $P_{2}\sim 3200$ days. This is longer than the values of
$P_{2}\sim 2000$ days reported in the paper by Hajdu et al. (2019) because the
observation period used in the study is longer than theirs. In the case of the
real outer period that is significantly longer than the observation length,
the calculated value would be shorter or closer period than the duration of
the observation time span (cf. Hajdu et al. 2019).

Figure 7 shows the locations of the triple candidates (open circles) on the
$P_{1}$ versus $P_{2}$ plane, where the grey circles represent the triple (or
multiple) system candidates identified by Hajdu et al. (2019). In the figure,
the horizontal and sloped blue lines represent the limits of the amplitudes
with 50 seconds of the LTTE and dynamical effects (cf. Borkovits et al. 2016),
respectively. As one can see in the figure, almost all of the triple
candidates in this study are located in the region of higher LTTE
contributions ($A_{\rm LTTE}$) than the dynamical effect ($A_{\rm dyn}$).

##  5 Summary and Discussion

In this study, we present the results of an eclipse timing analysis of 14,127
CEBs based on OGLE-III&IV observations between 2001 and 2015 in the Galactic
bulge. Among the 14,127 CEBs, 13,716 CEBs with parabolic variations were
classified by the Bayes factor for a statistical inference (Kass & Raftery
1995; Wagenmakers 2007) and the controlling false discovery rate (FDR) for a
multiple-hypothesis testing error measure (Benjamini & Hochberg 1995). The 307
CEBs with a sinusoid, and 104 CEBs with the two variations were also
identified by visual inspection and verified based on the Bayes factor and
FDR. Of the CEBs with a parabolic variation, 6,332 and 7,488 systems were
classified as having decreasing and increasing period variations,
respectively. Of a total of 411 CEBs showing periodic variations, their inner
and outer orbital periods are in the ranges of $0.235-0.990$ days and
$5.0-14.0$ years, respectively. These were mainly dominated by the LTTE
contribution rather than by the dynamical effect (cf. Figure 7).

Kubiak et al. (2006) suggested that the period change rates of the CEBs show
around zero and up to $\pm 2.3\times 10^{-7}$ day year-1 based on the
distribution of the period variations for 134 CEBs in the Galactic bulge. They
did not find CEBs with a period change rate greater than $\pm 5\times 10^{-6}$
day year-1. In this study, the number of CEBs with decreasing and increasing
periods were 6,332 and 7,488 objects, respectively. A histogram of the orbital
periods and the diagram of the period change rate versus the orbital periods
are presented in the upper and lower panels of Figure 8, respectively. As
shown in the upper panel of the figure, most CEBs have orbital periods in the
range of $0.2\leq P\leq$ 0.6 days, and the peak of the decreasing period shows
slightly shorter orbital periods than the peak of the increasing period. In
the lower panel of the figure, their period change rates are distributed
almost symmetrically around zero, and they were mostly located within the
range of $\dot{P}=\pm 5.0\times 10^{-6}$ day year-1. In our sample, the
highest decreasing and increasing period rates were determined to be
$\dot{P}=-1.38\pm 0.06\times 10^{-5}$ day year-1 for OGLE-BLG-ECL-169991 and
$\dot{P}=+8.99\pm 0.44\times 10^{-6}$ day year-1 for OGLE-BLG-ECL-189805,
respectively. The period change rates in particular seem to be decreased on
both sides of the peak around the orbital period of 0.4 days.

The variation in the period of the precursor to the LRNe V1309 Sco, with an
orbital period of 1.4 days, decreased exponentially, from $\dot{P}=-9.64\times
10^{-4}$ day year-1 in 2002 to $\dot{P}=-3.93\times 10^{-3}$ day year-1 in
2007 before the eruption, and the shape of their light curves was transformed
from a double to single wave (Tylenda et al. 2011). Molnar et al. (2017)
suggested that the contact binary KIC 9832227 could be considered as a merger
candidate based on a tentative exponential period decrease. However the system
may not be a merger candidate according in a recent study (Socia et al. 2018;
Kovacs et al. 2019). Pietrukowicz et al. (2017) presented three potential
merger candidates, EBs with relatively long orbital periods of more than
$P>1.0$ days and a relatively high decreasing period rate of more than
$\dot{P}>-1.0\times 10^{-5}$ day year-1. According to our results (cf. Table
2), a total of 70 systems have a period decreasing rate higher than
$\dot{P}=-3.65\times 10^{-6}$ day year-1, which is a necessary criterion to be
a serious merger candidate as presented by Molnar et al. (2017). Although the
decreasing period of the systems can be interpreted as a part of the periodic
variation due to the Applegate (1992) mechanism or LTTE, we carefully propose
that the CEBs may be potential merger candidates.

Acknowledgements We are grateful to the anonymous statistics editor, referee,
and Dr. Seung-Lee Kim for very helpful comments and suggestions that have
helped us to improve the quality of the manuscript. We thank the OGLE teams
for all of the observations. This research was supported by Basic Science
Research Program through the NRF funded by the Ministry of Education
(2019R1I1A1A01056776), and by grants of the National Research Foundation of
Korea (2019R1A2C2085965 and 2020R1A4A2002885).

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---
Figure 1: Comparison of the period change rates between ours and those of
Pietrukowicz et al. (2017).
---
Figure 2: $O-C$ diagrams of four CEBs, which have the highest decreasing
(upper panels) and increasing (lower panels) period change rates in our
samples, respectively.

---
Figure 3: $I$ light curves of four CEBs. The blue and red circles represent
the OGLE-III and OGLE-IV observations, respectively. In the left panels, the
light curves were plotted using the linear ephemeris taken from the OGLE EBs
catalog of Soszyński et al. (2016). In the right panels, the light curves were
phased with the quadratic ephemeris determined in this study.

---
Figure 4: Samples for the eclipse timing diagrams of the 307 CEBs with a
sinusoidal variation. In the panels, the red curves represent the sinusoidal
term of the equation (3). (The full figures for the 307 CEBs with a sinusoidal
variation can be obtained in the electronic edition of the paper.)
---
Figure 5: Samples for the eclipse timing diagrams of the 104 CEBs with
parabolic $plus$ sinusoidal variations. In the panels, the blue dashed and red
solid lines represent the parabolic terms and the full contributions,
respectively. (The full figures for the 104 CEBs with parabolic $plus$
sinusoidal variations can be obtained in the electronic edition of the paper.)
---
Figure 6: Histogram of the outer orbital periods ($P_{2}$) of the 411 triple
candidates.
---
Figure 7: Outer triple period ($P_{2}$) versus the inner binary period
($P_{1}$) for the 411 triple star candidates (black circles) found in this
study. The gray circles represent the 992 triple star candidates in the
Galactic Bulge taken from Hajdu et al. (2019). The horizontal black, sloped
blue (a typical outer eccentricity of $e_{2}=0.35$) and green ($e_{2}=0.0$)
lines are boundaries of the regions where the amplitudes of the LTTE and
dynamical effect may exceed $\sim$50 s or greater in amplitude to be
detectable. These limits were calculated using the equations (8) and (12) in
the paper of Borkovits et al. (2016), where the parameters are adopted as
$m_{\rm A}=m_{\rm B}=m_{\rm C}=1$ $M_{\odot}$, $i_{2}=60^{\circ}$, and
$\omega_{2}=90^{\circ}$. The dynamically unstable regions for outer
eccentricities ($e_{2}$) of 0.0 (cyan region) and 0.35 (orange region) were
calculated using the equation (27) given in the paper by Borkovits (2015),
which is rewritten from the expression of Mardling & Aarseth (2001).

---
Figure 8: The histogram of the orbital period (upper panel) and the period
change rate versus the orbital period diagram (lower panel) of the 13,820
CEBs. In the upper panel, the blue and red bars are represent the CEBs with
the decreasing and increasing period change rates, respectively. In the lower
panel, the black and red circles represent the 13,716 CEBs with parabolic
variations and 104 CEBs with parabolic $plus$ sinusoidal variations,
respectively.

Table 1: Period Change Rates of 45 CEBs Overlapped with those of Pietrukowicz
et al. (2017)

| This study |  | Pietrukowicz et al. (2017)
---|---|---|---
Object ID | R.A. | Decl. | $I$ | $V-I$ | $P$ | $T_{0}$ | $\dot{P}$ |  | $\dot{P}$
(OGLE-BLG-) | (J2000) | (J2000) | (mag) | (mag) | (day) | (HJD+2450000) | (day year-1) |  | (day year-1)
ECL-106230 | 17:47:21.97 |  $-$34:40:49.3 | 16.056 | 1.330 | 0.4890080(1) | 7000.4740(2) |  $-1.15\pm 0.02$ E$-$6 |  |  $-8.88\pm 0.45$ E$-$7
ECL-114280 | 17:48:28.67 |  $-$35:05:09.9 | 16.314 | 1.292 | 0.4530149(3) | 7000.2223(6) |  $-1.78\pm 0.05$ E$-$6 |  |  $-9.99\pm 1.64$ E$-$7
ECL-122558 | 17:49:35.14 |  $-$30:48:09.5 | 16.769 | 1.927 | 0.4109427(1) | 7000.3873(1) |  $-3.10\pm 0.02$ E$-$7 |  |  $-6.79\pm 1.20$ E$-$7
ECL-125481 | 17:49:57.46 |  $-$29:16:03.0 | 15.070 | 1.611 | 0.3585634(1) | 7000.0308(1) |  $6.51\pm 0.01$ E$-$7 |  |  $5.88\pm 0.27$ E$-$7
ECL-149279 | 17:52:26.29 |  $-$32:13:12.7 | 17.688 | 2.043 | 0.3147148(2) | 7000.2520(5) |  $-1.39\pm 0.04$ E$-$6 |  |  $-2.03\pm 0.32$ E$-$6
ECL-153698 | 17:52:48.61 |  $-$29:06:42.9 | 16.678 | 1.672 | 0.3866363(1) | 7000.2255(2) |  $1.13\pm 0.02$ E$-$6 |  |  $1.08\pm 0.14$ E$-$6
ECL-157410 | 17:53:07.74 |  $-$30:27:52.4 | 15.436 | 1.307 | 0.3634261(1) | 7000.0597(1) |  $1.49\pm 0.01$ E$-$6 |  |  $1.69\pm 0.08$ E$-$6
ECL-159116 | 17:53:16.11 |  $-$29:48:59.7 | 15.842 | 1.261 | 0.5873745(3) | 7000.2863(5) |  $-1.87\pm 0.04$ E$-$6 |  |  $-1.72\pm 0.06$ E$-$6
ECL-172659 | 17:54:22.13 |  $-$30:01:42.0 | 18.196 | 1.630 | 0.6261682(11) | 7000.5112(16) |  $-5.39\pm 0.17$ E$-$6 |  |  $-5.85\pm 0.21$ E$-$6
ECL-174839 | 17:54:34.06 |  $-$29:24:38.3 | 16.385 | 1.431 | 0.4612320(1) | 7000.2364(2) |  $7.66\pm 0.02$ E$-$7 |  |  $9.62\pm 0.24$ E$-$7
ECL-181083 | 17:55:07.21 |  $-$29:58:17.8 | 16.138 | 1.619 | 0.3493719(1) | 7000.1897(2) |  $-1.78\pm 0.03$ E$-$6 |  |  $-1.02\pm 0.04$ E$-$6
ECL-181311 | 17:55:08.47 |  $-$29:33:46.8 | 16.602 | 1.341 | 0.4226041(2) | 7000.1641(5) |  $-1.41\pm 0.04$ E$-$6 |  |  $-1.43\pm 0.05$ E$-$6
ECL-181955 | 17:55:11.75 |  $-$30:02:12.1 | 16.212 | 1.419 | 0.4143063(1) | 7000.0863(1) |  $4.45\pm 0.01$ E$-$7 |  |  $2.62\pm 0.14$ E$-$7
ECL-183497 | 17:55:19.37 |  $-$32:55:40.9 | 17.803 | 1.105 | 0.5232081(6) | 7000.4452(10) |  $1.38\pm 0.10$ E$-$6 |  |  $3.36\pm 0.79$ E$-$6
ECL-184767 | 17:55:25.23 |  $-$29:56:15.9 | 17.363 | 1.821 | 0.4732967(4) | 7000.2416(6) |  $-1.01\pm 0.06$ E$-$6 |  |  $-8.57\pm 1.18$ E$-$7
ECL-187430 | 17:55:38.54 |  $-$29:17:24.1 | 17.464 | 2.089 | 0.2753081(1) | 7000.0050(2) |  $-3.99\pm 0.01$ E$-$7 |  |  $-3.18\pm 0.54$ E$-$7
ECL-192324 | 17:56:03.39 |  $-$29:25:10.6 | 17.785 | 2.144 | 0.4369188(3) | 7000.3381(6) |  $1.71\pm 0.04$ E$-$6 |  |  $2.05\pm 0.24$ E$-$6
ECL-192437 | 17:56:03.96 |  $-$29:59:12.6 | 16.808 | 1.627 | 0.4022991(2) | 7000.0568(3) |  $-1.67\pm 0.04$ E$-$6 |  |  $-1.60\pm 0.07$ E$-$6
ECL-192939 | 17:56:06.40 |  $-$29:29:21.4 | 18.206 | 2.032 | 0.5318869(8) | 7000.4773(15) |  $-4.68\pm 0.12$ E$-$6 |  |  $-4.40\pm 0.20$ E$-$6
ECL-194517 | 17:56:14.16 |  $-$29:43:37.7 | 17.159 | 1.580 | 0.4216062(2) | 7000.2498(5) |  $1.35\pm 0.03$ E$-$6 |  |  $1.58\pm 0.20$ E$-$6
ECL-197927 | 17:56:32.09 |  $-$29:15:43.0 | 17.211 | 1.761 | 0.5455864(3) | 7000.4951(5) |  $2.13\pm 0.04$ E$-$6 |  |  $2.26\pm 0.27$ E$-$6
ECL-198303 | 17:56:33.75 |  $-$30:14:33.7 | 14.734 | 1.197 | 0.4387309(1) | 7000.1706(1) |  $-3.44\pm 0.01$ E$-$7 |  |  $-3.81\pm 0.72$ E$-$7
ECL-199259 | 17:56:38.43 |  $-$29:48:45.8 | 13.812 | 0.923 | 0.3892928(1) | 7000.3713(3) |  $8.86\pm 0.02$ E$-$7 |  |  $1.37\pm 0.09$ E$-$6
ECL-200313 | 17:56:43.78 |  $-$30:49:34.2 | 17.265 | 1.492 | 0.4791391(1) | 7000.1185(2) |  $9.30\pm 0.03$ E$-$7 |  |  $1.03\pm 0.05$ E$-$6
ECL-200942 | 17:56:46.93 |  $-$30:00:57.7 | 18.390 | 1.915 | 0.3583932(3) | 7000.0824(6) |  $1.81\pm 0.06$ E$-$6 |  |  $2.86\pm 0.52$ E$-$6
ECL-201682 | 17:56:50.80 |  $-$29:44:49.4 | 17.355 | 1.837 | 0.4973019(3) | 7000.2404(6) |  $-1.21\pm 0.05$ E$-$6 |  |  $-1.89\pm 0.29$ E$-$6
ECL-202706 | 17:56:55.80 |  $-$28:39:16.6 | 15.889 | 1.656 | 0.4497383(1) | 7000.3855(1) |  $4.11\pm 0.01$ E$-$7 |  |  $4.08\pm 0.71$ E$-$7
ECL-203756 | 17:57:00.79 |  $-$29:39:38.4 | 17.980 | 1.917 | 0.4183556(4) | 7000.3487(10) |  $2.10\pm 0.06$ E$-$6 |  |  $2.14\pm 0.41$ E$-$6
ECL-205743 | 17:57:12.10 |  $-$29:56:21.2 | 15.745 | 1.394 | 0.3004912(1) | 7000.0351(1) |  $3.05\pm 0.01$ E$-$7 |  |  $8.24\pm 0.97$ E$-$7
ECL-207879 | 17:57:22.78 |  $-$28:56:59.0 | 15.064 | 1.474 | 0.3930792(1) | 7000.0772(1) |  $-2.88\pm 0.01$ E$-$7 |  |  $-5.51\pm 0.33$ E$-$7
ECL-212252 | 17:57:44.48 |  $-$31:03:40.9 | 18.765 | … | 0.4132157(4) | 7000.2266(7) |  $-3.27\pm 0.09$ E$-$6 |  |  $-3.88\pm 0.17$ E$-$6
ECL-215121 | 17:58:00.27 |  $-$29:57:49.4 | 17.125 | 1.867 | 0.3078372(1) | 7000.0778(1) |  $-6.10\pm 0.01$ E$-$7 |  |  $-5.51\pm 1.13$ E$-$7
ECL-220028 | 17:58:24.67 |  $-$29:57:42.0 | 16.705 | 1.492 | 0.9207411(6) | 7000.7851(4) |  $-5.91\pm 0.12$ E$-$6 |  |  $-6.02\pm 1.10$ E$-$6
ECL-233754 | 17:59:37.56 |  $-$29:12:27.8 | 16.919 | 1.298 | 0.5702720(4) | 7000.5608(5) |  $3.51\pm 0.07$ E$-$6 |  |  $1.84\pm 0.09$ E$-$6
ECL-233821 | 17:59:37.95 |  $-$28:50:22.4 | 15.276 | 1.220 | 0.3599062(2) | 7000.3556(4) |  $-1.19\pm 0.03$ E$-$6 |  |  $-1.59\pm 0.04$ E$-$6
ECL-279326 | 18:03:46.02 |  $-$29:47:40.5 | 15.765 | 1.519 | 0.2277294(1) | 7000.1718(1) |  $-3.49\pm 0.01$ E$-$7 |  |  $-3.29\pm 0.24$ E$-$7
ECL-280295 | 18:03:51.13 |  $-$27:54:12.3 | 16.145 | 1.356 | 0.4842285(1) | 7000.4589(1) |  $-1.02\pm 0.01$ E$-$6 |  |  $-1.10\pm 0.10$ E$-$6
ECL-282055 | 18:04:00.72 |  $-$28:39:33.7 | 17.620 | 1.560 | 0.4828396(4) | 7000.2603(5) |  $2.93\pm 0.07$ E$-$6 |  |  $2.04\pm 0.14$ E$-$6
ECL-284531 | 18:04:14.75 |  $-$29:52:19.9 | 13.598 | 0.883 | 0.4355595(1) | 7000.2722(1) |  $5.58\pm 0.01$ E$-$7 |  |  $5.17\pm 0.21$ E$-$7
ECL-294795 | 18:05:10.60 |  $-$29:21:03.9 | 13.019 | 1.000 | 0.2936554(1) | 7000.1750(2) |  $-3.35\pm 0.01$ E$-$7 |  |  $-4.84\pm 0.41$ E$-$7
ECL-306303 | 18:06:16.05 |  $-$29:00:46.8 | 17.850 | 1.099 | 0.4155388(5) | 7000.0956(9) |  $2.36\pm 0.08$ E$-$6 |  |  $3.31\pm 0.59$ E$-$6
ECL-311367 | 18:06:47.01 |  $-$30:47:49.1 | 17.965 | 1.497 | 0.4202339(6) | 7000.3097(14) |  $3.57\pm 0.11$ E$-$6 |  |  $4.44\pm 0.80$ E$-$6
ECL-317072 | 18:07:22.35 |  $-$29:42:13.1 | 15.582 | 1.173 | 0.3537176(1) | 7000.2037(2) |  $6.53\pm 0.02$ E$-$7 |  |  $7.28\pm 0.82$ E$-$7
ECL-317088 | 18:07:22.46 |  $-$29:18:39.2 | 17.678 | 1.297 | 0.4874658(5) | 7000.1490(8) |  $2.56\pm 0.09$ E$-$6 |  |  $3.47\pm 0.69$ E$-$6
ECL-351804 | 18:11:16.73 |  $-$25:42:55.3 | 16.684 | 1.426 | 0.3853975(2) | 7000.3813(4) |  $-2.33\pm 0.03$ E$-$6 |  |  $-2.12\pm 0.26$ E$-$6

  * Note: The coordinates, $I$, and $V-I$ are taken from the OGLE-IV catalogue by Soszyński et al. (2016).

Table 2: The Secular Period Change Rates of 13,716 CEBs

Object ID | R.A. | Decl. | $I$ | $V-I$ | $P$ | $T_{0}$ | $\dot{P}$
---|---|---|---|---|---|---|---
(OGLE-BLG-) | (J2000) | (J2000) | (mag) | (mag) | (day) | (HJD+2450000) | (day year-1)
ECL-169991 | 17:54:09.00 |  $-$32:05:55.0 | 17.104 | 1.965 | 1.1757013(27) | 2457000.8180(15) |  $-1.38\pm 0.06$ E$-$5
ECL-052921 | 17:38:43.52 |  $-$22:36:37.2 | 16.425 | 1.672 | 1.3190508(36) | 2457000.3542(22) |  $-9.74\pm 0.54$ E$-$6
ECL-218952 | 17:58:19.54 |  $-$28:08:21.9 | 16.662 | 2.211 | 0.9275454(7) | 2457000.9101(5) |  $-9.57\pm 0.18$ E$-$6
ECL-272254 | 18:03:07.25 |  $-$28:15:19.0 | 15.549 | 1.168 | 0.8535252(4) | 2457000.8072(2) |  $-8.46\pm 0.08$ E$-$6
ECL-151250 | 17:52:36.28 |  $-$30:42:53.2 | 17.127 | 2.175 | 0.9717356(9) | 2457000.7534(6) |  $-8.22\pm 0.19$ E$-$6
ECL-122763 | 17:49:36.96 |  $-$34:40:44.4 | 17.252 | 1.560 | 0.9023381(25) | 2457000.1415(25) |  $-8.07\pm 0.43$ E$-$6
ECL-339477 | 18:09:50.66 |  $-$25:58:29.2 | 16.812 | 1.535 | 1.1381204(25) | 2457001.1215(15) |  $-7.67\pm 0.44$ E$-$6
… |  |  |  |  |  |  |

  * Note: The coordinates, $I$, and $V-I$ are taken from the OGLE-IV catalogue by Soszyński et al. (2016).

  * (This table is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content.)

Table 3: Results of Sine-curve fit for the 307 CEBs with a Sinusodal Period
Change

| This study |  | The others
---|---|---|---
OGLE-BLG- | R.A. | Decl. | $I$ | $V-I$ | $P$ | $T_{0}$ | $P_{2}$ |  | $P_{2}$
| (J2000) | (J2000) | (mag) | (mag) | (day) | (HJD+2450000) | (year) |  | (year)
… |  |  |  |  |  |  |  |  |
ECL-124673 | 17:49:51.64 |  $-$30:06:25.3 | 16.771 | 1.650 | 0.3540138(1) | 2457000.0105(1) | 8.3(4) |  | 4.5(1)b
ECL-125353 | 17:49:56.56 |  $-$34:58:02.1 | 17.953 | 1.607 | 0.5121666(3) | 2457000.1523(12) | 10.2(5) |  |
ECL-125525 | 17:49:57.66 |  $-$34:39:54.4 | 14.800 | 1.212 | 0.4120029(1) | 2457000.2652(3) | 10.9(4) |  | 15a
ECL-127170 | 17:50:09.95 |  $-$29:44:30.3 | 17.778 | 1.902 | 0.2915032(1) | 2457000.2528(2) | 11.0(3) |  |
ECL-127614 | 17:50:13.22 |  $-$34:57:38.6 | 17.532 | 1.470 | 0.4789311(2) | 2457000.0820(8) | 5.5(1) |  |
ECL-127683 | 17:50:13.66 |  $-$34:42:15.3 | 18.882 | 1.462 | 0.3518262(1) | 2457000.2028(10) | 5.8(1) |  |
ECL-127744 | 17:50:14.07 |  $-$30:06:22.4 | 15.017 | 1.741 | 0.8920668(1) | 2457000.6614(2) | 5.4(3) |  | 6.5a, 4.0(2)b
… |  |  |  |  |  |  |  |  |

  * Note: The coordinates, $I$, and $V-I$ are taken from the OGLE-IV catalogue by Soszyński et al. (2016).

  * a Value taken from Pietrukowicz et al. (2017).

  * b Value taken from Hajdu et al. (2019).

  * (This table is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content.)

Table 4: The fitted parameters for the parabolic $plus$ sinusoidal ephemeris
of 104 Contact EBs

| This study |  |
---|---|---|---
OGLE-BLG- | R.A. | Decl. | $I$ | $V-I$ | $P$ | $T_{0}$ | $\dot{P}$ | $P_{2}$ |  | $P_{2}^{*}$
| (J2000) | (J2000) | (mag) | (mag) | (day) | (HJD+2450000) | (day year-1) | (year) |  | (year)
… |  |  |  |  |  |  |  |  |  |
ECL-147315 | 17:52:15.66 |  $-$33:20:21.7 | 15.744 | 1.341 | 0.3739789(1) | 7000.1115(2) |  $-6.69\pm 1.94$ E$-$7 | 9.9(2) |  |
ECL-148291 | 17:52:21.01 |  $-$29:46:59.3 | 16.952 | 1.612 | 0.3388654(2) | 7000.2649(4) |  $1.46\pm 0.21$ E$-$6 | 9.8(1) |  | 21a, 10.5(1.1)b
ECL-151538 | 17:52:37.82 |  $-$31:20:26.7 | 18.368 | 2.395 | 0.5786571(4) | 7000.3272(4) |  $-1.05\pm 0.31$ E$-$6 | 9.8(2) |  |
ECL-157106 | 17:53:06.12 |  $-$29:58:36.5 | 17.794 | 1.826 | 0.3224295(2) | 7000.1954(6) |  $7.57\pm 2.04$ E$-$7 | 9.8(1) |  |
ECL-159957 | 17:53:20.30 |  $-$29:24:57.3 | 16.696 | 1.507 | 0.3455664(2) | 7000.1874(5) |  $1.36\pm 0.22$ E$-$6 | 10.0(2) |  |
ECL-163882 | 17:53:39.93 |  $-$33:53:45.1 | 17.241 | 1.481 | 0.4790803(3) | 7000.2601(5) |  $-1.32\pm 0.25$ E$-$6 | 9.8(2) |  |
ECL-166853 | 17:53:54.55 |  $-$33:11:01.9 | 15.796 | 1.535 | 0.3405647(2) | 7000.2570(4) |  $-1.22\pm 0.17$ E$-$6 | 9.6(1) |  |
… |  |  |  |  |  |  |  |  |  |

  * Note: The coordinates, $I$, and $V-I$ are taken from the OGLE-IV catalogue by Soszyński et al. (2016).

  * a Value taken from Pietrukowicz et al. (2017).

  * b Value taken from Hajdu et al. (2019).

  * (This table is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content.)