# Characteristics of _Kepler_ Eclipsing Binaries Displaying a Significant
O’Connell Effect

Matthew F. Knote Florida Institute of Technology  Saida M. Caballero-Nieves
Florida Institute of Technology  Vayujeet Gokhale Truman State University
Kyle B. Johnston Florida Institute of Technology  Booz Allen Hamilton  Eric S.
Perlman Florida Institute of Technology

###### Abstract

The O’Connell effect – the presence of unequal maxima in eclipsing binaries –
remains an unsolved riddle in the study of close binary systems. The _Kepler_
space telescope produced high precision photometry of nearly 3,000 eclipsing
binary systems, providing a unique opportunity to study the O’Connell effect
in a large sample and in greater detail than in previous studies. We have
characterized the observational properties – including temperature,
luminosity, and eclipse depth – of a set of 212 systems (7.3% of _Kepler_
eclipsing binaries) that display a maxima flux difference of at least 1%,
representing the largest sample of O’Connell effect systems yet studied. We
explored how these characteristics correlate with each other to help
understand the O’Connell effect’s underlying causes. We also describe some
system classes with peculiar light curve features aside from the O’Connell
effect ($\sim$24% of our sample), including temporal variation and asymmetric
minima. We found that the O’Connell effect size’s correlations with period and
temperature are inconsistent with Kouzuma’s starspot study. Up to 20% of
systems display the parabolic eclipse timing variation signal expected for
binaries undergoing mass transfer. Most systems displaying the O’Connell
effect have the brighter maximum following the primary eclipse, suggesting a
fundamental link between which maximum is brighter and the O’Connell effect’s
physical causes. Most importantly, we find that the O’Connell effect occurs
exclusively in systems where the components are close enough to significantly
affect each other, suggesting that the interaction between the components is
ultimately responsible for causing the O’Connell effect.

††facilities: _Gaia_ , _Kepler_ ††software: AstroPy (Astropy Collaboration et
al., 2013, 2018), BinaryMaker3 (Bradstreet & Steelman, 2002), IPython/Jupyter
(Pérez & Granger, 2007), Matplotlib (Hunter, 2007), NumPy (Harris et al.,
2020), Peranso (Paunzen & Vanmunster, 2016), PHOEBE v0.31a (Prša & Zwitter,
2005), SciPy (Virtanen et al., 2020)

##  1 Introduction

The O’Connell effect (O’Connell, 1951; Milone, 1968) is a poorly understood
asymmetry seen in some eclipsing binaries where the maxima between eclipses
are not equal in brightness (see Figure 1). At the quadrature phases
($\phi=0.25,~{}0.75$ for circular orbits), we see the stars side-by-side, so a
difference in brightness implies that one hemisphere of a component emits a
different amount of radiation than the other hemisphere. Roberts (1906) first
discussed the O’Connell effect, where he attributed it to stars in an
eccentric orbit becoming tidally distorted near periapsis. His theory did not
explain the presence of the O’Connell effect in systems with circular orbits,
however, and is no longer considered a likely cause of the O’Connell effect in
most systems. Wilsey & Beaky (2009) outlines a few possible explanations for
the O’Connell effect: chromospheric spots on one or both components, a hotspot
caused by mass transfer, or circumbinary material impacting the stars as they
orbit (Liu & Yang, 2003). Wilsey & Beaky (2009) notes that none of these
explains the O’Connell effect in all cases, however, and Papageorgiou et al.
(2014) states that “the O’Connell effect is still one of the most perplexing
challenges in binary studies.”

###  1.1 Background

O’Connell (1951, hereafter O51) found strong correlations between the
O’Connell effect size (OES, also commonly known as $\Delta m$) and the
following parameters: the difference in OES at different wavelengths, the
ellipticity of the component stars, the ratio of stellar radii, and the ratio
of stellar densities. Most notably, he found that the OES becomes larger at
shorter wavelengths as the OES increases, which is to say OESblue – OESred
increases as OES increases. O51 restricted his study to systems with a
constant OES and excluded W Ursae Majoris systems.

Davidge & Milone (1984, hereafter D84) built upon the work of O51 by
introducing more systems displaying the O’Connell effect and using a second,
non-parametric form of analysis less prone to outliers in addition to the
parametric analysis used by O51. Like O51, D84 restricted their sample to
systems with a constant OES and excluded overcontact systems. The higher
precision photometry available to D84 meant that many systems included in O51
were found to have a variable OES and so were excluded. D84 found essentially
the same strong correlations as O51. The correlation between the OES and the
size at different wavelengths was the strongest correlation in O51 and D84.
However, D84 found that the OES becomes larger at _longer_ wavelengths as the
OES increases, the opposite of O51’s correlation. They attributed this
discrepancy to the sample selection difference between O51 and D84, as there
are only six systems in common between the two studies. Note that both O51 and
D84 used ground-based data taken in the era before CCDs.

Figure 1: Averaged light curve of KIC 5282464 showing the O’Connell effect.
The maximum following the primary eclipse (indicated with the upper dashed
line) is noticeably brighter than the maximum preceding it (indicated by the
lower dashed line).

Wilsey & Beaky (2009, hereafter W09) is the most recent major work on the
O’Connell effect. W09 examines the current theoretical models used to explain
the O’Connell effect: starspots, gas stream impacts, circumstellar material,
and asymmetric circumfluence due to Coriolis forces. They state that starspots
are the most commonly used explanation for the O’Connell effect. These spot
models can be used to explain both chromospheric starspots like those found on
our Sun and starspots arising from the impact of a matter stream. However, W09
states that the spots need to be unrealistically large to explain the observed
O’Connell effect in many systems, and Maceroni & van’t Veer (1993) states that
the photometric light curves used to build these models are often insufficient
to justify the existence of starspots. Furthermore, starspots introduce four
new parameters per spot to the parameter space: spot latitude, longitude,
radius, and temperature factor. These additional parameters increase the
uniqueness problem (Kallrath & Milone, 2009), making finding the parameter
space’s global minimum (and thus the best model of the system) nearly
impossible. Next, W09 discusses how a hotspot created by a matter stream can
create a maximum difference if it is offset from the central axis connecting
the centers of both stars. They also identify two candidate systems where this
matter stream hypothesis has explained the O’Connell effect: V361 Lyrae and GR
Tauri. Finally, W09 discusses two papers (Liu & Yang 2003 and Zhou & Leung
1990) that propose models for the O’Connell effect. Liu & Yang (2003) proposed
that the impact of material surrounding both stars heats the leading
hemispheres, causing a difference in brightness and temperature. However, W09
notes that their model’s assumptions are unrealistic. Zhou & Leung (1990)
proposed that the deflection of circulating material by Coriolis forces can
explain the O’Connell effect in overcontact systems.

Recent surveys such as the _Kepler_ (Borucki et al., 2010) and Transiting
Exoplanet Survey Satellite (TESS; Ricker et al., 2014) missions and future
surveys such as the Legacy Survey of Space and Time (LSST; Ivezić et al.,
2019) conducted by the Vera C. Rubin Observatory provide an enormous amount of
data on the variable universe. These surveys uncover and provide complete
samples of sources that display poorly understood phenomena (including the
O’Connell effect), allowing us to completely characterize these systems for
the first time. The first step to understanding these phenomena is determining
the characteristics of sources exhibiting them. This characterization allows
us to anticipate which newly observed sources might show them and provides
clues to the physical processes underlying the phenomenon in question.

###  1.2 Our Study

Our project aims to characterize the observational properties (including
period, color, OES, and eclipse depth) of systems displaying a significant
O’Connell effect, which serves as a prelude to understanding the O’Connell
effect’s physical causes. We have selected 258 eclipsing binaries observed by
_Kepler_ , of which 212 constitute a core sample showing an average $|$OES$|$
larger than 1% of their normalized flux. Our core sample represents the
largest sample of O’Connell effect binaries studied to date. This paper
describes our characterization of the core sample. In addition to this
characterization, we determined how the characteristics correlate with each
other. These correlations provide important insights into the processes
underlying the O’Connell effect. This paper also details our use of two
statistical measures introduced by McCartney (1999) and refined by W09. Our
project serves as a continuation of the studies presented in O51 and D84 using
modern observational techniques like space-based CCD photometry. However, our
lack of bulk physical characteristics (mass, density, surface gravity, etc.)
due to lacking radial velocity data forces us to probe a different parameter
space than those works.

Our project seeks to answer questions about the O’Connell effect using a
sample selected in a consistent and unbiased manner. We primarily see the
O’Connell effect in short-period eclipsing binaries, where the stars are close
enough to interact with each other. Is this a selection bias, or is there a
genuine link between binary interaction and the O’Connell effect? O51 found
that the brighter maximum almost uniformly followed the primary eclipse rather
than preceding it. D84, by contrast, found a more even distribution of maxima
preference, with about 60% of their sample having the brighter maximum follow
the primary eclipse. Which result better describes the maxima distribution of
the underlying population? Do the characteristics of these systems correlate
in the manner Kouzuma (2019) suggests spotted stars should? Or do we instead
see the eclipse timing variations characteristic of mass transfer in a
majority of systems? Is the OES the best way to characterize the O’Connell
effect, or are the O’Connell Effect Ratio and Light Curve Asymmetry
statistical measures introduced by McCartney (1999) better? Finally, how
common is the O’Connell effect?

We now define the light curve classifications (Algol-, $\beta$ Lyrae-, and W
Ursae Majoris-type systems) and morphological classifications (detached, semi-
detached, and overcontact) we will adopt in this paper. Algol-type systems
have sharp, well-defined eclipses and minimal out-of-eclipse variations,
$\beta$ Lyrae-type systems have continuously variable light curves and minima
of significantly unequal depth, and W Ursae Majoris-type systems have
continuously variable light curves and minima of equal or nearly equal depth.
We quantify our light curve classification in Section 3.3. Our light curve
classification does not consider the binary components’ spectral type or
evolutionary state (e.g. we would classify V382 Cygni as W Ursae Majoris-type
rather than $\beta$ Lyrae-type as Landolt 1975 did). Meanwhile, detached
systems have neither star filling their Roche lobe, semi-detached systems have
one star (either the primary or secondary) exactly filling their Roche lobe,
and overcontact systems have both stars overflowing their Roche lobes. We
define the primary star as the component eclipsed at the deeper minimum
without regard to the component masses.

Section 2 describes how we determined our sample and the biases that are
inherent to it. Section 3 discusses our methodology in analyzing the _Kepler_
data. Section 4 gives the characteristics of our core sample, including their
light curve properties, information obtained from other sources, and
properties derived from available data. Section 4 also describes four system
classes with peculiar features in their light curves aside from the O’Connell
effect, including temporal variation of their light curves and asymmetric
minima. Section 5 discusses the observed trends found in our study and the
implications of these trends. Finally, Section 6 summarizes our major results
and details future work on this project.

##  2 Target Selection

We drew our sample from the _Kepler_ Eclipsing Binary Catalog (KEBC), a
compilation of eclipsing binary systems observed by the _Kepler_ space
telescope. We defined a single criterion to define our sample: $|$OES$|\geq
0.01$, where OES is in units of normalized flux and $|$OES$|$ is the absolute
value of the OES that folds the negative OES values over the positive ones. We
created a code to determine the OES, which we used to select a core sample of
212 system based on their long-cadence _Kepler_ data.

###  2.1 _Kepler_ Space Telescope

The _Kepler_ space telescope was launched in 2009 with the purpose of
observing transiting planets. The spacecraft imaged over 150,000 objects every
thirty minutes for four years, observing numerous variable stars in the
process. _Kepler_ ’s photometry is accurate within 29 parts per million (ppm)
for a star with an apparent magnitude of 12 and 80.7 ppm at an apparent
magnitude of 14.5 (Gilliland et al., 2011). For comparison, Tregloan-Reed &
Southworth (2013) cites a precision of 258 ppm (of a V = 11.45 star using the
4-meter Mayall telescope; Gilliland et al., 1993) as the most precise ground-
based observations known. Therefore, it is clear that _Kepler_ ’s photometry
is much more precise than ground-based data, even for fainter stars, making
features such as kinks or subtle changes over time more apparent. As a result
of this precision, we are more confident that any peculiar features observed
in a light curve are real structures rather than a result of statistical
noise. Some targets have short-cadence data (sampling every minute) in
addition to the long-cadence data (sampling every 30 minutes) that all targets
have, allowing us to study short-timescale variations in these systems.
However, our analysis in this paper is based exclusively on the long-cadence
data.

_Kepler_ observed most systems nearly continuously for over three years,
providing an opportunity to see how each system changes over that time. Such
changes can be attributed to various factors, including starspot evolution,
accretion disk instabilities, or a change in temperature due to thermal
equilibration. The long observation span and high precision also maximizes our
sensitivity to transient effects like flares, allowing us to characterize the
prevalence of these events in our sample. Additionally, variations in the
eclipse timing can indicate the presence of a third body in the system or an
actual change in the period due to mass transfer. Finally, observing targets
for such a long, continuous time minimizes the risk of artifacts due to poor
sampling or transient effects such as flares. The effects of such artifacts
are unpredictable, with one such effect being an oscillatory signal introduced
in systems with a period that is a near-integer multiple of the _Kepler_
cadence.

Using _Kepler_ as a basis for our study introduces the same biases inherent to
the _Kepler_ mission. The observed targets were drawn from the _Kepler_ Input
Catalog (KIC; Brown et al., 2011), which estimated each star’s spectral type
and luminosity class based on photometrically determined colors. Batalha et
al. (2010) describes the _Kepler_ target prioritization and statistics for the
operational target list. They assigned the highest priority to systems with a
_Kepler_ magnitude $K_{p}<14$, and where an Earth-sized planet in the
habitable zone would produce at least three transits in the 3.5-year mission
with a signal-to-noise ratio greater than $7.1\sigma$. They chose the first
condition to facilitate follow-up high-precision spectroscopy to confirm
planet detections. The second condition, meanwhile, excluded most O- and
B-type stars. As a result of this prioritization, there are more G- and F-type
stars than K- or M-type stars among _Kepler_ targets, and _Kepler_ only
observed $\sim$3,000 M-type stars and fewer than 200 O- and B-type stars. The
_Kepler_ target list also has a bias against giants and subgiants because
planetary transits are harder to detect due to the smaller ratio between the
star and planet radii. Comparing Tables 2 and 3 of Batalha et al. (2010) shows
that the _Kepler_ target list included fewer than 10% of giants located within
the _Kepler_ field of view. Because of these selection biases, our study
undersamples the giants and subgiants as well as the low- and high-mass
stellar population, a fact reinforced by the results we present in Section
4.2. Consequently, our study provides less information on the O’Connell effect
in systems containing these stars.

Bryson et al. (2020) and Wolniewicz et al. (2021) further investigated the
_Kepler_ completeness using _Gaia_ DR2 data (Gaia Collaboration et al., 2018),
the former for exoplanet occurrence rates and the latter for stellar
populations. Wolniewicz et al. (2021) looked specifically at _Kepler_ ’s
selection function, finding that the _Kepler_ sample is nearly unbiased for
stars brighter than $K_{p}=14$. For fainter stars ($K_{p}>14$), they found a
bias toward main-sequence and subgiant stars with late-F to early-M spectral
types and a bias against cool giants. They state that the bias toward
subgiants was because the KIC misidentified them as main-sequence stars. They
also found a bias against binaries by analyzing the re-normalized unit weight
error (RUWE), which is the normalized $\chi^{2}$ obtained from fitting the
point-spread function of _Gaia_ sources re-normalized to correct for color-
dependent biases (Lindegren, 2018). Wolniewicz et al. (2021) considered a
system to be binary when its RUWE was greater than 1.2 (Kraus et al. 2020, in
prep.). For $K_{p}>14$, they found that _Kepler_ ’s completeness was 8% lower
for main-sequence binaries than for solitary main-sequence stars. It is
unclear how significantly this impacts the KEBC, however, because Wolniewicz
et al. (2021) considered systems with stellar separations of order tens of AU
and above. Even the most widely separated systems in the KEBC ($P\sim
1,\\!000$ d) have separations of only a few AU. _Gaia_ EDR3 data (Gaia
Collaboration et al., 2021) shows that some of the KEBC systems (24.5% of the
2,861 systems with _Gaia_ parallaxes) have an RUWE above the 1.2 limit
Wolniewicz et al. (2021) used. Based on their results, the fact that nearly a
quarter of the KEBC shows a RUWE above this limit means that the _Kepler_
target list likely excluded some close eclipsing binaries.

###  2.2 _Kepler_ Eclipsing Binary Catalog

Kirk et al. (2016) compiled _Kepler_ ’s data on eclipsing binaries into the
KEBC, from which we drew our sample. The KEBC contains _Kepler_ observations
on 2,907 unique identified eclipsing binary systems in 2,920 entries. Each
system has an extracted light curve, and the catalog can be searched based on
several parameters, including period and eclipse depth. The KEBC provides an
excellent source for our study for three primary reasons: the large number of
systems observed, the data’s photometric precision, and the observation span.
The 2,907 systems in the catalog represent a large, complete sample of
eclipsing binaries. For comparison, the All-Sky Automated Survey (ASAS) found
11,099 eclipsing binaries south of declination +28∘ (Obu & Okeke, 2013), while
the Optical Gravitational Lensing Experiment (OGLE) survey found over 425,000
eclipsing binaries toward the galactic bulge (Soszyński et al., 2016). The
KEBC also contains eclipsing binaries with many different characteristics and
allows us to study a complete, statistically significant sample. Kirk et al.
(2016) states that the KEBC completeness is 89.1% for eclipsing binaries, with
essentially 100% completeness for systems with periods of order one day or
less. However, Bienias et al. (2021) recently found 547 short-period
($P\lesssim 0.5$ d) eclipsing binaries in the _Kepler_ field not included in
the KEBC. These new eclipsing binaries are fainter than the KEBC systems, with
an average magnitude of 18.2. Nevertheless, the KEBC’s high completeness for
brighter ($K_{p}\lesssim 16$) systems means it is well-representative of the
true eclipsing binary population, increasing its suitability for our study.

We compare our core sample to the entire KEBC throughout this paper. However,
we use three separate subsets of the KEBC in this paper, each more restrictive
than the last. The first subset – the comparative subset – removed four
systems, one due to data corruption and three because they are not in the KEBC
(note that the three non-KEBC systems are not included in the system count
given in the previous paragraph). We use the comparative subset in the
histograms in Sections 4 and 5 and the color-magnitude diagram in Section 4.
The second subset – the trend subset – removed a further 228 systems due to
missing certain parameters or having multiple KEBC entries. We use the trend
subset in the plots in Sections 5.1 and 5.2. Finally, the third subset – the
analysis subset – removed a still further 1,309 systems to select only systems
similar in period and light curve shape to our sample. We use the analysis
subset in our statistical analysis explained in Section 3.4 and discussed in
Section 5.2. We discuss these subsets further in Sections 4, 5.1, and 5.2,
respectively.

###  2.3 Selection Criterion

To ensure that the sample only includes systems with a significant O’Connell
effect, we instituted a cutoff in the absolute value of the normalized flux
difference between the two maxima of 0.01 (1% of the normalized flux). We
applied this cutoff to averaged light curves (defined in Section 2.4) similar
to the one Figure 1 shows. We determined the median, mean, and standard
deviation ($\sigma$) of the KEBC’s OES by applying the definitions of these
measures to the KEBC systems’ OESs. This method shows that the KEBC’s median
OES is 0.002%, the mean value is 0.075%, and its $\sigma$ is 0.837%. The
corresponding values for our core sample are a median of 1.303%, a mean value
of 0.071%, and a $\sigma$ of 2.886%. Figure 2 shows a histogram of the OES
distribution for the KEBC with the 1% cutoff indicated by dashed lines. We
determined that our core sample does not include 45 systems with an $|$OES$|$
larger than the 1$\sigma$ value but below our 1% cutoff (equivalent to
1.19$\sigma$). Our criterion is therefore more restrictive than the 1$\sigma$
value. Unlike O51 and D84, we do not exclude W Ursae Majoris-type or
overcontact systems, respectively, nor do we exclude systems with a variable
OES.

Figure 2: Histogram showing the OES distribution for the KEBC. The central
peak at an OES of 0.00 rises to a value of 1,367 systems and is truncated for
clarity. There are 26 additional systems with $|$OES$|>0.03$ not included in
this figure for clarity. The dashed lines indicate the $|$OES$|=0.01$ cutoff
defining our sample.

###  2.4 Selection Method: O’Connell Effect Size Determination

We tried several methods to produce our sample. We first tried manually
selecting our sample based on visually inspecting KEBC light curves, but this
method is inherently subjective and non-reproducible. We then tried using the
automated detector Johnston et al. (2019) describes, as they made this
detector to create our sample. However, due in part to the light curve scaling
Johnston et al. (2019) used, it found over 70 systems that did not meet our
criterion. Additionally, the detector did not find several other systems that
we knew from visual inspection should be in our sample. Therefore, we
developed a code that directly calculated the OES, which we used to select our
sample.

First, we downloaded the short- and long-cadence _Kepler_ data for each
eclipsing binary from the KEBC website. Our code split each cadence’s data
into 1,001 equally-spaced bins in phase space for $\phi\in[-0.5,0.5]$, where
the bins for $\phi=-0.5$ and 0.5 contain the same data. It then calculated the
weighted average flux $F$ for each bin using inverse-variance weighting
(Hartung et al., 2008):

$F=\frac{\sum_{i}{f_{i}/\sigma_{f_{i}}^{2}}}{\sum_{i}{1/\sigma_{f_{i}}^{2}}}$
(1)

along with the error:

$\sigma_{F}=\sqrt{\frac{1}{\sum_{i}{1/\sigma_{f_{i}}^{2}}}}$ (2)

where $\sigma_{f_{i}}$ is the error of the $i$th data point in the bin. We
estimated the flux value for any phase bins without data by linearly
interpolating between the nearest bins with data. The code introduced a phase
shift setting the phase bin with the lowest average flux to $\phi=0$. The
resultant light curve represents an averaged light curve for the system, an
example of which is shown in Figure 1. Note that, throughout this work, we
removed all data from KIC 9164694 with BJD in the ranges [2455309.2870355,
2455336.8939506] and [2455432.2188883, 2455552.5485149] to correct an apparent
data processing issue wherein the flux values during these intervals were
systematically reduced by $\sim$2%. We also removed a single outlier datum
from KIC 8029708’s data. Finally, 160 KEBC systems have the flag QAM, which
indicates that these systems’ amplitudes differ between _Kepler_ quarters.
Eighteen of these systems are in our sample. The KEBC corrected for this
amplitude mismatch by rescaling the data to match between quarters, and we use
this rescaled flux in our analysis. However, in at least one case (KIC
4474637), this rescaling did not correct the issue, and so this system
exhibits significantly increased scatter in its data and its averaged light
curve.

The code produced a smoothed version of the light curve by convolving each
point in the averaged light curve with the immediately adjacent points. The
convolution reduces the effect of data scatter and oscillations on the light
curve, giving a more accurate measure of the OES. Since convolution in time
space is equivalent to multiplication in frequency space (Press et al., 2007),
we performed the convolution in frequency space after applying a discrete
Fourier transform to the phased data. Our code multiplied the transformed data
by the transformed convolution kernel 50 times in frequency space, equivalent
to convolving the data 50 times in time space. It then transformed the data
back into time space using the inverse discrete Fourier transform and measured
the maximum value of this convolved curve on either side of the primary
minimum. The difference between these two maxima is the OES. The OES is
positive when the maximum after the primary eclipse is brighter than the one
preceding it, and we say that a system with a positive OES has a positive
O’Connell effect. Conversely, we say that a system where the maximum before
the primary eclipse is brighter than the one following it has a negative
O’Connell effect. We found the error of each bin in the convolved curve by
convolving the errors given by Equation 2 in the same fashion as we convolved
the averaged data. The error in the OES is then given by propagating the error
of the bins used to determine the OES. The maxima difference data from this
code provided the basis for the histogram in Figure 2. Because we determine
the OES from the convolved curve, which is derived from the averaged light
curve, this OES represents the average size of the O’Connell effect over the
duration of _Kepler_ ’s observations.

This method for finding O’Connell effect binaries is viable only because we
applied it to a set of known eclipsing binaries. If applied to a general
population of stars, this method would find many non-eclipsing systems, such
as pulsating variables and spotted stars. The KEBC therefore acts as a filter
that allows this method to find O’Connell effect binaries. This method is also
quite sensitive, and several systems near the $|$OES$|$ cutoff of 0.01 would
cross that threshold when we would change our method. For instance, when we
began using Equation 1 instead of the standard definition of the mean, three
systems with $|$OES$|$ above the criterion threshold decreased to under it,
while three systems under it increased to above it. This sensitivity to the
precise method of determining the OES makes it more challenging to accurately
reproduce our sample. It also explains why this method initially added systems
to our sample that ultimately did not meet the criterion. To estimate the
sensitivity of our method, we varied the cutoff by a small amount and
determined the number of systems meeting the new cutoff. We found that varying
the cutoff value by 0.0002 in either direction changed the sample size by
about $\pm 5$ ($\pm 2.3\%$), while varying it by 0.001 changed the sample size
by about $\pm 28$ ($\pm 13.2\%$).

Figure 3: _Kepler_ short-cadence (left) and long-cadence (right) light curves
of KIC 7667885. The system displays a negative O’Connell effect and at least
one significant flare event.

It is also important to keep in mind that our averaging process removes
information about the temporally varying properties of the system. The non-
averaged OES changes in magnitude and even sign in several systems, as does
the overall shape of the light curve. Removing this information simplifies the
analysis of these systems but reduces the utility of the averaged light curve
in studying temporal variation. We will analyze the temporal variation
separately by looking at the eclipse timing variation (discussed further in
Section 4.3).

###  2.5 Target Sample

Our final, complete sample consists of 211 systems found by our final OES
determination code, 1 system (KIC 7667885) found in Ramsay et al. (2014), and
46 systems found using earlier methods, for a total of 258 systems. Our OES
determination code showed that three additional systems met our criterion. Of
these, KIC 7950964 is a duplicate entry of a system already in the sample (KIC
7950962), and KIC 9137819 appears to be an RR Lyrae variable and not an
eclipsing binary. These two systems have since been removed from the KEBC (A.
Prša 2021, private comm.). The third system, KIC 8456774, is a heartbeat star
(Thompson et al., 2012), which is an eccentric binary that becomes tidally
distorted at periapsis. KIC 8456774, like many heartbeat stars, does not
appear to be eclipsing, and we do not consider it to have an O’Connell effect
due to the fundamentally different origin of its asymmetry. Additionally, it
is inappropriate to apply the phase offset we introduced to place the primary
minimum at $\phi=0$ to non-eclipsing systems like KIC 8456774, and the OES
without this phase offset is negligible. We do not include these three systems
in our sample. We also do not include systems like KIC 11560447 that meet our
criterion with their short-cadence data but not with their long-cadence data.

_Kepler_ observed KIC 7667885 during quarters 14-17, but the KEBC does not
include the system. Inspection of KIC 7667885’s short-cadence data from
MAST444Mikulski Archive for Space Telescopes;
http://archive.stsci.edu/kepler indicated that it met our criterion. We
detrended the single-aperture photometry (SAP) flux by fitting Legendre
polynomials up to order 150 to continuous data blocks (defined as a set of
data with no gaps longer than a day) and then dividing the flux by the fitted
trend. We then normalized the data by dividing the flux by the median flux
value of each data block. This method is similar to the one outlined in
Section 4.2 of Slawson et al. (2011), which they used to produce the KEBC
data. We determined a period of $0.314840\pm 0.000004$ d using the analysis of
variance (Schwarzenberg-Czerny, 1989, 1996) method with the software Peranso.
Figure 3 shows the resultant phased short- and long-cadence light curves. We
did not determine the morphology parameter of KIC 7667885, nor did we perform
an analysis of its ETV, but we otherwise fully incorporated the system into
our sample.

One system in our sample, KIC 7879399, lies only 4” from another _Kepler_
system, KIC 7879404. As a result of their proximity, these two systems’ data
are blended, making it difficult to determine which one is the eclipsing
binary. The first two releases of the KEBC (Prša et al. 2011 and Slawson et
al. 2011) identified KIC 7879404 as the binary. The third release (Kirk et al.
2016 and Abdul-Masih et al. 2016), however, identified KIC 7879399 as the
binary. C. Cynamon of the American Association of Variable Star Observers
(AAVSO) observed these stars for us (C. Cynamon 2021, private comm.), and our
analysis of his data conclusively confirms KIC 7879399 as the eclipsing
binary.

Of the 258 systems in our sample, 46 selected before we finalized the
selection method do not meet our criterion. We will describe these 46 marginal
sample systems in a separate paper. The remaining 212 systems (the 211 KEBC
systems found by our OES determination code and KIC 7667885, which our code
also found when we added its processed data to the KEBC’s) constitute our core
sample, which represents 7.3% of the systems in the KEBC. Table 1 lists all
258 systems in our complete sample.

##  3 Methodology

We used several methods to characterize the systems in our sample:
determinations of the eclipse depth, the O’Connell Effect Ratio and Light
Curve Asymmetry, the KEBC’s morphology parameter, and statistical analyses of
the sample characteristics. The majority of these methods involved analyzing
the phased _Kepler_ data taken from the KEBC using our in-house Python codes.

###  3.1 Eclipse Depth Determination

The KEBC provides a value for the eclipse depth determined using the polyfit
method described in detail in Appendix A of Prša et al. (2008). In brief, the
polyfit uses a chain of piecewise smooth polynomials connected at dynamically
determined knots to fit the light curve, therefore providing a smoothed
version of the light curve. This procedure works for the majority of systems
in the KEBC. Unfortunately, it fails to accurately measure the eclipse depth
for a small number of systems in our sample, as shown in Figure 4 with KIC
9777984’s polyfits (note that Kirk et al. 2016 and the KEBC identify this
system as KIC 9777987, while Abdul-Masih et al. 2016 identifies it as KIC
9777984). Furthermore, Kirk et al. (2016) states that the provided eclipse
depths are approximate and only as accurate as the polyfits used to produce
them.

We therefore needed to define our own method for determining eclipse depth.
The convolved curve used to determine the OES consistently underestimates the
depth of the eclipse, so we instead applied a Savitzky–Golay filter (Savitzky
& Golay, 1964) of polynomial order 4 and window size 7 to each averaged light
curve. We applied the filter in the same way we convolved the data in Section
2.4, the sole difference being in the kernel. We used SciPy’s savgol_coeffs
function to determine the kernel values for the Savitzky–Golay curve. This
procedure produced a curve that approximates the eclipse depth much better
than the polyfits for KIC 9777984 – as Figure 4 shows – despite retaining more
of a oscillatory pattern than the convolved curve. We found the Savitzky–Golay
curve’s errors similarly to how we found the convolved curve’s errors. We take
the smallest value of this curve to be the point of maximum eclipse, and the
error of the corresponding bin is the eclipse depth error. A limitation of
this method is that Algol-type systems with narrow eclipses can have their
eclipse depths underestimated by 50% or more. Nevertheless, the Savitzky–Golay
curve measures the eclipse depth more accurately than the Fourier curve
(Section 3.2), particularly for some Algol-type systems like KIC 5700330 that
lie outside of our sample. The Savitzky–Golay curve is also more accurate than
the convolved curve in all but a few edge cases involving totally-eclipsing
Algol-type systems.

Figure 4: _Kepler_ light curve of KIC 9777984 showing three fits to the
system: the KEBC’s original polyfit (green), the KEBC’s reconvolved polyfit
(blue) that corrects for _Kepler_ ’s non-zero integration time, and our fit
using the Savitzky–Golay filter (red). The Savitzky–Golay curve better fits
KIC 9777984’s light curve and provides a more accurate measure of the eclipse
depth for this system.

Establishing a proper reference point from which to measure the depth is a
non-trivial matter. We rejected using either maximum as the reference point as
it is uncertain which maximum (if either) is unaffected by the process causing
the O’Connell effect. Communication with the KEBC team revealed that they face
the same uncertainty about a proper reference point and noted that modeling
the light curve provides the only robust way of determining eclipse depth (A.
Prša 2020, private comm.), a procedure which is beyond the scope of this
paper. Therefore, we used the KEBC’s normalized data and adopted a normalized
flux value of unity to establish each system’s baseline flux. We then
subtracted the flux value at the eclipse minimum from this baseline to find
the eclipse depth. Slawson et al. (2011) details the normalization scheme used
for the KEBC data, and it is “reasonable to assume that, for the most part,
the data are median-normalized” (A. Prša 2020, private comm.).

###  3.2 O’Connell Effect Ratio and Light Curve Asymmetry

The O’Connell Effect Ratio (OER) and Light Curve Asymmetry (LCA) are two
statistical measures introduced by McCartney (1999), who used them with phased
light curves broken into $n$ equally-spaced phase bins. While originally
introduced to characterize the O’Connell effect in W Ursae Majoris-type
systems, OER and LCA are valid measures for all light curve classes. The
original definitions given by McCartney (1999) are:

$\text{OER}=\frac{\sum_{i=1}^{n/2}{I_{i}-I_{0}}}{\sum_{i=(n/2)+1}^{n}{I_{i}-I_{0}}}$
(3)

and:

$\text{LCA}=\sqrt{\sum_{i=1}^{n/2}{\left(\frac{I_{i}-I_{n+1-i}}{I_{i}}\right)}^{2}}$
(4)

where $n$ is the number of phase bins, $I_{i}$ is the average intensity of the
$i$th bin, and $I_{0}$ is the minimum average intensity of the light curve.
Note that the LCA should contain a normalization factor to account for the
number of bins, i.e.:

$\text{LCA}=\sqrt{\frac{1}{n}\sum_{i=1}^{n/2}{\left(\frac{I_{i}-I_{n+1-i}}{I_{i}}\right)}^{2}}$
(5)

as in W09. W09 transformed these summations into integrals (taking
$n\to\infty$) such that:

$\text{OER}=\frac{\int_{0}^{1/2}{[I(\phi)-I(0)]~{}d\phi}}{\int_{1/2}^{1}{[I(\phi)-I(0)]~{}d\phi}}$
(6)

and:

$\text{LCA}=\sqrt{\int_{0}^{1/2}{\left(\frac{I(\phi)-I(1-\phi)}{I(\phi)}\right)^{2}}~{}d\phi}$
(7)

where $I(\phi)$ is the $N$-term Fourier series representation of the light
curve:

$I(\phi)=\frac{a_{0}}{2}+\sum_{i=1}^{N}{[a_{i}\cos{(2\pi
i\phi)}+b_{i}\sin{(2\pi i\phi)}]}$ (8)

We use the integral formulation in this paper. Both OER and LCA assume that
the secondary eclipse occurs at $\phi=0.5$, so systems with eccentric orbits
have inaccurate OER values. The LCA, by contrast, interprets the eccentricity
as an asymmetry in the light curve.

Figure 5: Averaged light curve of KIC 11347875 showing two Fourier series
approximations of the system: the 50-term series used in this paper (solid
red) and a 12-term series (dashed blue) similar to one Akiba et al. (2019)
would use. The 50-term series follows the averaged data more closely than the
12-term series while not capturing the statistical scatter.

We chose to use a 50-term Fourier series based on a $\chi^{2}$ analysis of $N$
for our complete sample of 258 systems. Our code calculated the $\chi^{2}$
statistic by comparing the Fourier curve with $N$ coefficients to the averaged
light curve of the system, where $N\in[2,500]$. This analysis indicated a
median optimal $N$ of 59. We adopted an $N$ of 50 because we found that
varying $N$ in the range $[40,60]$ changes the results little. Figure 5
compares our 50-term Fourier series of the core sample system KIC 11347875 to
a 12-term Fourier series like those used in Akiba et al. (2019) and Hahs et
al. (2020). Figure 5 demonstrates that 50 terms are sufficient to capture most
large-scale features in the light curve without having so many terms that the
series approximates small-scale statistical scatter.

We obtained our Fourier series by applying a discrete Fourier transform to the
averaged light curve described in Section 2.4. Our code converted the
resultant complex-valued coefficients $c_{i}$ to the Fourier series
coefficients $a_{i}$ and $b_{i}$ using the equations:

$\displaystyle a_{i}$ $\displaystyle=\frac{2\text{Re}(c_{i})}{n}$ (9a)
$\displaystyle b_{i}$ $\displaystyle=-\frac{2\text{Im}(c_{i})}{n}$ (9b)

We use the resulting Fourier series to calculate the OER and LCA using
Equations 6 and 7, respectively. We also use the $a_{i}$ coefficients to
determine each system’s light curve class, as we describe in Section 3.3. The
50-term Fourier series is an excellent approximation of $\beta$ Lyrae- and W
Ursae Majoris-type systems. However, it struggles to accurately represent
several Algol-type systems, particularly those with sharp, narrow eclipses.
Such systems dominate the long-period ($P>10$ d) population of the KEBC but
are rare in our sample.

We based our error analysis on the method described in Akiba et al. (2019),
but there were some differences caused by our use of fft, most prominently
that all $a_{i}$ and $b_{i}$ coefficients with $i>0$ have the same error. This
error is given by:

$\sigma_{X}=\frac{\sqrt{2}}{n}\sqrt{\sum_{j=1}^{n}{\displaystyle\sigma_{x_{j}}^{2}}}$
(10)

where $\sigma_{x_{j}}$ are the errors of each bin given by Equation 2. The
error for $a_{0}$ is larger by a factor of $\sqrt{2}$. Our errors for OER and
LCA are:

$\begin{split}&\sigma_{\text{OER}}=\text{OER}\frac{\sigma_{X}}{2}\\\
&~{}~{}~{}\times~{}\sqrt{\frac{N+2\sum_{i=1}^{N}{\left(\frac{1}{\pi
i}\right)^{2}}}{\left[H\left(\frac{1}{2}\right)-H(0)\right]^{2}}+\frac{5N+2\sum_{i=1}^{N}{\left(\frac{1}{\pi
i}\right)^{2}}}{\left[H(1)-H\left(\frac{1}{2}\right)\right]^{2}}}\end{split}$
(11)

and:

$\begin{split}\sigma_{\text{LCA}}&\approx\frac{\sigma_{X}}{\text{LCA}}\int_{0}^{1/2}{}\left[\left(\frac{I(\phi)-I(1-\phi)}{I(\phi)}\right)^{2}\right.\\\
&~{}~{}~{}~{}~{}\left.\times~{}\sqrt{\frac{2\left(N+\frac{1}{2}\right)}{\left[I(\phi)-I(1-\phi)\right]^{2}}+\frac{N+\frac{1}{2}}{\displaystyle
I(\phi)^{2}}}\right]~{}d\phi\end{split}$ (12)

where:

$\begin{split}H(\phi)&=\sum_{i=1}^{N}{\left[\frac{a_{i}}{2\pi i}\sin{(2\pi
i\phi)}-\frac{b_{i}}{2\pi i}\cos{(2\pi i\phi)}\right.}\\\
&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\left.\vphantom{\frac{b_{i}}{2\pi
i}\cos{(2\pi i\phi)}}{}-{}a_{i}\phi\right]\end{split}$ (13)

is the antiderivative of $I(\phi)-I(0)$.

###  3.3 Morphology Parameter and Light Curve Classification

Matijevič et al. (2012) introduced the morphology parameter (which we
represent with the symbol $\mu$) to determine the morphology class of KEBC
systems. They used local linear embedding (Roweis & Saul, 2000) to reduce the
light curve’s polyfit representation (Prša et al., 2008) to a single, one-
dimensional parameter. This parameter proved to correlate very well with their
manually identified morphology class for most KEBC systems. They determined
that $\mu\leq 0.5$ implied detached systems, $0.5<\mu\leq 0.7$ implied semi-
detached systems, $0.7<\mu\leq 0.8$ implied overcontact systems, and $\mu>0.8$
implied ellipsoidal variables (non-eclipsing systems where the variation is
due to the changing aspect of tidally distorted stars). They note, however,
that the morphology parameter is a “best guess” at the morphology class and
that system modeling can more accurately determine the morphology class.
Nevertheless, we use the morphology parameter to estimate a system’s
morphology class throughout this paper, and Table 1 lists each system’s
morphology parameter.

Based on our own visual classification of the systems in our sample, we
estimated that Algol-type systems have $\mu<0.6$, $\beta$ Lyrae-type systems
have $0.6\leq\mu<0.75$, and W Ursae Majoris-type systems have $\mu\geq 0.75$.
These boundaries agree with those in Matijevič et al. (2012), as Algol-type
systems are generally detached or semi-detached, $\beta$ Lyrae-type systems
can be of any morphological class, and W Ursae Majoris-type systems are
predominantly overcontact (although detached and semi-detached eclipsing
binaries, as well as ellipsoidal variables, may also appear as W Ursae
Majoris-type systems).

In addition, we quantified the light curve classification using the Fourier
coefficients obtained from Equation 9a. Our procedure is a modification of the
one described in Section 3.2 of Akiba et al. (2019) and uses the following
criteria to classify systems:

  1. 1.

If $a_{4}<a_{2}(0.125-a_{2})$, the system is classified as an Algol-type
system.

  2. 2.

If $a_{4}\geq a_{2}(0.125-a_{2})$ and $|a_{1}/a_{2}|>0.25$, the system is
classified as a $\beta$ Lyrae-type system.

  3. 3.

If $a_{4}\geq a_{2}(0.125-a_{2})$ and $|a_{1}/a_{2}|\leq 0.25$, the system is
classified as a W Ursae Majoris-type system.

Criterion 1 comes from Rucinski (1997) and is unchanged from Akiba et al.
(2019), as are the first parts of Criteria 2 and 3. However, the second parts
of Criteria 2 and 3 differ from Akiba et al. (2019), who instead used
$|a_{1}|>0.05$ to define $\beta$ Lyrae-type systems and $|a_{1}|<0.05$ to
define W Ursae Majoris-type systems. These criteria arise from W09’s
observation that $a_{1}$ is proportional to the difference in eclipse depth.
Unfortunately, $a_{1}$ does not contain any information about the eclipse
depth _ratio_ , which is the parameter related to the component temperature
ratio (Kallrath & Milone, 2009). We introduced $a_{2}$ (which W09 notes is
proportional to the light curve amplitude) to quantify the eclipse depth
ratio.

We note that our sample likely contains several ellipsoidal variables. It is
difficult to distinguish an ellipsoidal variable from an overcontact system
due to their similar light curves. Preliminary modeling of four systems in our
sample with PHOEBE v0.31a (Prša & Zwitter, 2005) shows that the morphology
parameter is not wholly effective at identifying ellipsoidal variables. As an
example, KIC 8285349 has $\mu=0.90$ but clearly shows an eclipse. Meanwhile,
KICs 10815379 and 10979669 have $\mu=0.79$ and 0.83, respectively, but are
non-eclipsing according to our models. Identifying and removing ellipsoidal
variables would therefore require large-scale modeling of the sample systems,
which is beyond the scope of this paper.

###  3.4 Statistical Analysis

To determine if our sample population was similar to the KEBC population as a
whole, we performed the Kolmogorov–Smirnov test (K–S test; Kolmogorov, 1933;
Smirnov, 1948) on the observational characteristics we studied. If the test
statistic differs for a given characteristic, it suggests a potential link
between that characteristic and a significant O’Connell effect. We expect the
OES and $|$OES$|$ populations to differ due to how we defined our sample. We
do not expect the distance populations to differ because the O’Connell effect
should be independent of distance.

We determined correlations between characteristics using Spearman’s $\rho$
coefficient (Spearman, 1904), as in D84. We also used Kendall’s $\tau$
coefficient (Kendall, 1938) as a second test of correlation. Our decision to
use both coefficients stemmed from research showing that the two measures
complement each other (Xu et al., 2013). Unlike O51 and D84, we did not use
Pearson’s r coefficient due to its sensitivity to outliers and assumption of a
linear correlation. We tested six functional forms for each correlation using
SciPy’s implementation of orthogonal distance regression (ODR; Boggs et al.,
1987) to determine the best fit to the correlation. The functional forms we
tested were linear, quadratic, exponential, logarithmic, power law, and
inverse.

In order to test the robustness of our correlation analysis results, we
performed a bootstrapping procedure on our sample. Our code randomly selected
20 subsets of 40 systems from our core sample (and 400 systems from the KEBC)
and computed the Spearman’s $\rho$ coefficients for each subset. We considered
a correlation robust if at least 19 of the 20 subsets recovered the
correlation.

##  4 Results

We now present a detailed summary of various characteristics of the core
sample, specifically the light curve characteristics (distributions of eclipse
depth, OES, and positive and negative OESs) and physical characteristics
(spatial, luminosity, temperature, and period distributions). Determining the
characteristic distributions of the underlying population allows us to make
inferences about the physical processes underlying the O’Connell effect. We
also discuss the results of our analysis of the ETV, OER, and LCA. Finally, we
present several system classes in our sample that show peculiar features in
their light curves aside from the O’Connell effect. These features include
changes in the light curve over time, asymmetric minima, and concave-up
regions. In the figures in Sections 4 and 5, the distribution or data set
labeled “Core Targets” includes all 212 core sample systems, while the
distribution or data set labeled “KEBC Systems” includes the core sample
(except KIC 7667885) and the rest of the KEBC. In this section, we use the
comparative subset of the KEBC discussed in Section 2.2, which excluded four
systems: KIC 5217781, a long-period system with severe data corruption, KIC
7667885, due to it not being in the KEBC, and KICs 7950964 and 9137819, due to
their removal from the KEBC (as explained in Section 2.5).

###  4.1 Light Curve Characteristics

We determined several characteristics from analyzing the _Kepler_ photometry
of the KEBC systems: light curve class, OES, the proportion of positive and
negative O’Connell effects (defined in Section 2.4), primary eclipse depth,
and the ratio between $|$OES$|$ and the primary eclipse depth. This section
discusses the range and distribution of these characteristics. Using the
method described in Section 3.3, we classified 54 systems (25% of our sample)
as Algol-type, 40 (19%) as $\beta$ Lyrae-type, and 118 (56%) as W Ursae
Majoris-type. This distribution strongly differs from the KEBC’s, wherein
1,990 (68%) are Algol-type, 250 (9%) are $\beta$ Lyrae-type, and 679 (23%) are
W Ursae Majoris-type.

The largest OES in our sample is KIC 11347875, which has an OES of $-0.265$ in
units of normalized flux. However, KIC 1134785 is an unusual system in that
one of its inter-eclipse maxima is instead a minimum. Therefore, it lacks an
O’Connell effect in the traditional sense described in Section 1. We discuss
KIC 11347875 further in Section 4.5.4. The largest OES among systems with a
more traditional O’Connell effect is KIC 9935311, with an OES of 0.115. A
clear majority of systems (147, or 69% of the sample) show a positive
O’Connell effect, where the brighter maximum is the one following the primary
eclipse. We discuss the implications of this preference for a positive
O’Connell effect in Section 5.2.1.

The primary eclipse depth ranges from 0.025 (KIC 8190491) to 0.916 (KIC
9101279) in units of normalized flux. Figure 6 shows the ratio between
$|$OES$|$ and the primary eclipse depth for the core sample and the KEBC. We
have determined that the overwhelming majority of systems with a ratio above
0.5 are heartbeat stars, a type of binary discussed in Section 2.5. The only
system in our sample among such high ratio systems is the aforementioned non-
heartbeat star KIC 11347875, which has a ratio of 0.658.

Figure 6: Histogram comparing the ratio $|$OES$|$/primary eclipse depth for
our core sample (solid red) and all entries in the KEBC (dashed grey). Systems
with a ratio above 1 are not displayed. Core sample systems have a larger
ratio on average than other KEBC systems.

###  4.2 Physical Characteristics

We determined several physical characteristics from the published information
on the KEBC: distance, luminosity, period, temperature, and spectral type.
This section focuses on these characteristics’ distributions for our core
sample and the KEBC. We also discuss our sample’s color-magnitude diagram and
references that indicate the presence of flares, spots, or mass transfer in
our systems. Figure 7 shows a histogram of the distances derived from _Gaia_
parallaxes (Bailer-Jones et al., 2021) for our sample, along with the distance
distribution for the KEBC. These two distributions appear similar. Figure 8
shows a histogram of the luminosity distribution in our sample and the KEBC,
and we note that our sample’s systems are less luminous on average than those
in the KEBC. Figure 9 shows a histogram of the period distribution for our
sample and the KEBC. We obtained these periods directly from the KEBC, and
Figure 9 has a logarithmic scale to capture the KEBC’s wide range of periods.
Figure 9 shows that our sample has a similar distribution for $P\leq 0.5$ d,
but the proportion of systems with longer periods drops precipitously.

Figure 7: Histogram comparing the implied distances for the 212 core sample
targets (solid red) and 2,860 of 2,862 entries in the KEBC (dashed grey) with
_Gaia_ EDR3 parallaxes (excluding two systems with $\text{d}>9,\\!000$ pc).
Our core sample’s distances range from 210 pc (KIC 7671594) to 7,743 pc (KIC
4474637). The two distributions are similar.

Both the KEBC and _Gaia_ DR2 (Gaia Collaboration et al., 2018) provide
temperatures for most of our targets based on their colors. The KEBC
temperatures range from 3,717 K (KIC 7671594) to 8,540 K (KIC 10857342), while
the _Gaia_ temperatures range from 3,808 K (KIC 7671594) to 8,540 K (KIC
10857342). The _Kepler_ and _Gaia_ temperatures are similar for the systems in
our sample. Nine target systems do not have a _Kepler_ temperature, while
seven do not have a _Gaia_ temperature. We chose to use _Gaia_ as our primary
source of temperature in light of the more complete coverage _Gaia_ provides.

Frasca et al. (2016) provides spectral types for a handful of target systems
determined using optical spectra (although they do not state if they assumed
solitary stars during their data processing), and the main-sequence spectral
types range from K5 (KIC 12109575) to A2 (KIC 8904448). There are eleven
systems with an explicit non-main-sequence spectral type listed in Table 1.
Ramsay et al. (2014) spectrally classified KIC 7667885 and KIC 9786165 as “mid
G” systems. We used the online
table555http://www.pas.rochester.edu/~emamajek/EEM_dwarf_UBVIJHK_colors_Teff.txt
of photometric colors periodically updated by Dr. Eric Mamajek (Pecaut &
Mamajek, 2013) in combination with the _Gaia_ colors provided by EDR3 to
expand the spectral classification to the entire sample. While this does not
account for interstellar reddening, evolutionary stage, or color blending of
the binary components, it provides a first approximation estimate of the
spectral class for all systems in our sample, which Table 1 also lists. By
this estimate, the spectral types of our sample range from M2.5 V (KIC
7671594) to A7 V (KIC 10857342).

Figure 8: Histogram comparing the luminosities for the 212 core sample targets
(solid red) and all 2,862 entries in the KEBC (dashed grey) with _Gaia_ EDR3
parallaxes. Note the logarithmic scale of the _x_ -axis. Our core sample’s
calculated luminosities range from 0.02 L$\odot$ (KIC 7671594) to 50.25
L$\odot$ (KIC 5820209). Systems in our sample are less luminous on average
than other KEBC systems.  Figure 9: Histogram comparing the periods for all
212 core sample targets (solid red) and all 2,915 entries in the KEBC (dashed
grey) with a period below 1,000 d. Note the logarithmic scale of the _x_
-axis. Our core sample’s orbital periods range from 0.234 d (KIC 6050116) to
9.752 d (KIC 6197038). The distributions are similar under 0.5 d, but our
sample’s long-period population decreases much more quickly than the KEBC’s.

Figure 10 shows a color-magnitude diagram using _Gaia_ EDR3 data. The marker
color and shape show the light curve classification for each system determined
using the criteria given in Section 3.3. Figure 10 shows the _Kepler_
selection bias discussed in Batalha et al. (2010) and Section 2.1, as there
are few systems with a _Gaia_ color index $\text{BP}-\text{RP}\geq 1.5$ or
$\text{BP}-\text{RP}\leq 0.5$. Our sample’s distribution in Figure 10 is
similar to the KEBC’s along most of the main sequence, suggesting that our
sample is not strongly biased beyond _Kepler_ ’s selection function.
Additionally, only two clear giants exist in our sample (KICs 5820209 and
9489411, although the former’s 0.656-day period is incongruent with containing
a giant because such a period with two 1 M$\odot$ stars implies a maximum star
size of only $\sim$2.5 R$\odot$).

Numerous systems in our sample are thought to have observed flares (57% of the
sample; Balona, 2015; Gao et al., 2016; Davenport, 2016) or be spotted (42%;
Tran et al., 2013; Balaji et al., 2015). Table 1 lists these systems with
reference numbers 1-3 and 4-5, respectively, and the flags F and S,
respectively. Kouzuma (2018) found evidence for mass transfer in 22 (10%)
systems in our sample. Table 1 lists these systems with reference number 6 and
the flag M. Therefore, our sample consists mainly of solar-type main-sequence
stars with evidence for flares, spots, or mass transfer in 148 (70%) sample
systems.

###  4.3 Eclipse Timing Variations

Eclipse timing variation (ETV) measures how much the primary and secondary
eclipses times vary from their theoretical values calculated assuming a
constant period. ETV patterns reveal apparent period changes due to light
travel time effects (LTTEs) caused by a third body, along with actual period
changes due to effects such as mass transfer. Mass transfer creates a
parabolic signal (as Figure 11’s left panel shows), while an LTTE creates a
sinusoidal signal with a period equal to the third body’s orbital period (as
Figure 11’s right panel shows). For this paper, we use the ETV data published
by Conroy et al. (2014), a paper describing the ETVs of KEBC systems with
morphology parameters between 0.5 and 1.0. Conroy et al. (2014) distinguished
a parabolic signal from a long-period sinusoidal signal using the Bayesian
Information Criterion (Schwarz, 1978). Only three core sample systems (KICs
5020034, 6044064, and 7696778) show a parabolic ETV signal, while none from
our marginal sample do. By contrast, forty core sample systems and seven
marginal sample systems show a sinusoidal signal. Furthermore, forty-seven
core sample systems and seven marginal sample systems show an “interesting”
ETV, meaning it has a pattern that does not clearly fit into the other two
categories.

Figure 10: Color-magnitude diagram showing the 212 systems in the core sample
overplotting the rest of the KEBC systems. The _Gaia_ color is plotted against
the absolute magnitude in _Gaia_ ’s G band. The systems are color coded
according to their light curve class. Most of our sample are solar-type main-
sequence stars.

Figure 11: Eclipse timing variation of KIC 7696778 (left) and KIC 8904448
(right). KIC 7696778’s ETV has a parabolic shape, consistent with a linear
change in period caused by mass transfer between components, while KIC
8904448’s ETV has a sinusoidal shape, consistent with LTTEs caused by a third
body orbiting the binary.

The twenty-two core sample systems Kouzuma (2018) cited as undergoing mass
transfer do not include the three systems Conroy et al. (2014) flagged with a
parabolic ETV. However, the KEBC flagged sixteen of these twenty-two systems
with a sinusoidal or interesting ETV, and visual inspection of their ETVs
suggests that, aside from KIC 2437038, their signals are roughly parabolic.
Figure 12 shows the ETV of two systems Kouzuma (2018) references: KIC 6791604
(flagged as interesting) and KIC 11924311 (flagged as sinusoidal). To further
investigate our core sample’s ETVs, we applied a digital Butterworth filter
(Butterworth, 1930) of order one to the ETV data using SciPy’s butter function
to reduce noise from effects such as starspots. The sampling frequency is the
orbital frequency, and we set the cutoff frequency to five cycles over the
system’s observation span, thus removing signals such as starspot modulation
that recur more than five times during _Kepler_ ’s observations. Due to the
need for continuous sampling, we removed any system with an observation gap
longer than 20 d, leaving a sample of 122 core sample systems. We fit a
parabola to each system’s filtered data using SciPy’s curve_fit function and
calculated the $R^{2}$ value of the fit. Thirteen systems (11% of the 122
investigated systems) have $R^{2}\geq 0.9$, while 25 systems (20%) have a
less-stringent $R^{2}\geq 0.8$ (for reference, KIC 6791604’s ETV – shown in
Figure 12’s left panel – has $R^{2}=0.895$). We regard these 25 systems as
having an ETV that is not incompatible with being parabolic. Because these 122
systems form an unbiased subset of our core sample, we expect the same
percentage of core sample systems to show a possibly parabolic ETV. Therefore,
we estimate that up to 43 core sample systems show evidence of mass transfer
in their ETV, making mass transfer a plausible, if rare, cause of the
O’Connell effect.

Figure 12: Eclipse timing variation of KIC 6791604 (left) and KIC 11924311
(right). KIC 6791604 was flagged with an interesting ETV while KIC 11924311
was flagged as a sinusoidal ETV, but both are consistent with a roughly
parabolic ETV as well.

###  4.4 O’Connell Effect Ratio and Light Curve Asymmetry

Figures 13 and 14 compare the OES to the OER (Equation 6) and the LCA
(Equation 7), respectively. The data labeled “Non-Core Systems” include all
KEBC systems not in the core sample, excluding KIC 5217781. Our analysis found
many systems with an OES of $\sim$0 but OER $\neq 1$ or LCA $\neq 0$,
producing significant vertical scatter in Figures 13 and 14. Systems in this
scatter region of Figure 13 are low-amplitude (total change in flux $\Delta
F\lesssim 0.03$) binaries with an OES that is large relative to their eclipse
depth but small on an absolute scale. In Figure 14, the scatter region also
includes long-period eccentric binaries. We removed a single data point from
KICs 9701423 and 10614158 to avoid a nonphysical negative OER, although the
low-amplitude system KIC 6948480 retains a negative OER. We excluded the two
systems with $|$OES$|>0.1$ (KICs 9935311 and 11347875) from Figures 13 and 14
for clarity. The figures do not plot errors because they are too small to see
at this scale for most systems.

Figure 13 shows a strong correlation between the OER and OES. This correlation
is unsurprising because both measures account for the different amounts of
light under both maxima, albeit in different ways. McCartney (1999) notes that
the presence of a constant, third source of light contaminating the data will
reduce the OES. However, the OER is not affected by third light due to
subtracting the minimum flux value of the light curve. Therefore, the OER
provides a more consistent measure of the O’Connell effect in this respect. On
the other hand, the OER calculation method also means that the OER will be
significant for systems like KIC 8912911 and KIC 10905824 with O’Connell
effects that are large relative to their eclipse depth but small on an
absolute scale. Based on this, we believe that the OER is ill-suited to
describe the O’Connell effect in a general population of eclipsing binaries
but is applicable to a subset of binaries with amplitudes larger than
$\sim$0.01 in normalized flux.

Figure 14 shows that most of our sample lies along lines with slopes of
$\pm$0.5 that meet at the origin. Many systems lie above these lines, but none
lie significantly below them, implying that the O’Connell effect produces an
LCA at least half as large as the OES. The LCA is also clearly sensitive to
asymmetries aside from the O’Connell effect, as seen by the vertical scatter
mentioned earlier. Many systems in this scatter (like KIC 10909274) are long-
period systems with eccentric orbits, implying that the LCA is sensitive to
unevenly-spaced eclipses. Therefore, the LCA serves as a good measure of a
light curve’s asymmetry, but its sensitivity to other asymmetries means that
it should not be relied on to detect the O’Connell effect.

###  4.5 Systems Showing Peculiar Features

Figure 13: Plot comparing the OER to the OES. Core sample targets are shown in
red diamonds and non-core sample systems in black circles. KICs 9935311 and
11347875 are excluded for clarity. Figure 14: Plot comparing the LCA to the
OES. Core sample targets are shown in red diamonds and non-core sample systems
in black circles. KICs 9935311 and 11347875 are excluded for clarity.

Four system classes showing peculiar features are present in our sample: those
displaying significant temporal variation, those with an asymmetric minimum,
those with a concave-up region in their light curves, and a white dwarf.
Systems can belong to multiple classes. We have chosen an exemplar system to
serve as an identifier for each class for the purpose of categorization. We
discuss each class and identify characteristics systems in each class share.
Future work will discuss each of these classes more in-depth. In Section 4.5,
our discussion also includes systems from the marginal sample defined in
Section 2.5.

####  4.5.1 KIC 7433513: Temporally Varying Systems

KIC 7433513 and systems like it (labeled with the flag TV in Table 1) display
strong temporal variation in their light curves. Figure 15 shows KIC 7433513’s
phased light curve using all _Kepler_ data, clearly showing the significant
data scatter caused by the temporal variation. Figure 16 shows 10-day time
slices of KIC 7433513’s _Kepler_ data separated by a few months, along with
the Barycentric Julian Date (BJD) for the midpoint of each time slice. The
system changes drastically even over these short timescales. It is difficult
to rigorously quantify the number of these systems in our sample because there
is a continuum between systems with stable light curves like KIC 5282464
(Figure 1) and systems like KIC 7433513. However, we estimate that about 20%
of our core sample and about 65% of our marginal sample exhibits strong
temporal variation. Three further examples from our sample are KICs 2569494,
3659940, and 9137992. Systems displaying this temporal variation are among our
sample’s cooler systems, with the hottest (KIC 8294484) estimated at 5,760 K.
Kunt & Dal (2017) described KIC 7885570 – a system from our marginal sample
showing strong temporal variation – as an RS Canum Venaticorum variable. The
stars in these systems have active chromospheres that produce large spots, and
the evolution of these spots is a plausible source for the observed temporal
variation.

####  4.5.2 KIC 9164694: Asymmetric Minima Systems

KIC 9164694 and systems like it (labeled with the flag AM in Table 1) contain
an asymmetric minimum, as seen in Figure 17’s left panel showing KIC 9164694’s
light curve. This asymmetry is rather subtle, and in KIC 9164694’s case, is
best described as looking like someone has taken their finger and “pushed” the
light curve upward and leftward in the region just to the right of minimum
light. Asymmetric minima are a rarely discussed feature seen in some eclipsing
binary systems, including RY Scuti (Djurašević et al., 2008), GR Tauri (Zhang
et al., 2002), AG Virginis (Pribulla et al., 2011), and NSVS 7322420 (Knote et
al., 2019). Asymmetries can appear in the primary minimum (as in KIC 9164694)
or the secondary minimum (as in KIC 9717924; see Figure 17’s right panel).
About 13% of our sample displays an asymmetric minimum in either eclipse.
There appear to be two sets of asymmetric minima systems: those showing strong
temporal variation (like KIC 7433513), and those showing comparatively little
(like KICs 9164694 and 9717924). We focus on the latter set for this class,
which consists of the marginal sample system KIC 5283839 and 15 core sample
systems: KICs 2159783, 2449084, 6205460, 8248967 (according to a 20-term
Fourier series), 8696327, 8822555, 8842170, 9164694, 9283826, 9717924, 9786165
(with short-cadence data only), 10528299, 10861842, 11395645, and 11924311.
These 16 are among our sample’s hotter systems, with nearly half having a
temperature over 6,000 K, and most show minimal temporal variation. We
distinguish this asymmetry from what older literature like Brownlee (1957) and
van’t Veer (1973) call an asymmetric minimum, as the latter asymmetry occurs
closer to the eclipse’s beginning and end. By contrast, what we call an
asymmetric minimum occurs around the eclipse’s center.

Figure 15: _Kepler_ light curve of KIC 7433513 showing significant scatter
caused by temporal variation.  Figure 16: Ten-day slices of phased _Kepler_
data for KIC 7433513 separated by several months, showing the significant
changes in the light curve over time. The number in each time slice’s subplot
is the BJD for the midpoint of that time slice.

Figure 17: Averaged light curves of KIC 9164694 (left) and KIC 9717924 (right)
showing an asymmetric primary and secondary minimum, respectively.

The most striking similarity between these systems is that all known
asymmetric minima systems exhibit total eclipses. Total eclipses occur when
one star is fully occluded during an eclipse and are distinguished by flat-
bottomed or nearly flat-bottomed minima. It is unclear why asymmetric minima
are only found in totally eclipsing systems and not in partially eclipsing
systems. One hypothesis is that spots near one component’s poles cause the
asymmetry. Another is that the change in flux causing the asymmetry is small
enough that it is only detectable during totality, when the flux is nearly
constant. We describe these two possibilities in the next two paragraphs. Some
systems with total eclipses and a significant O’Connell effect do not show an
asymmetric minimum, such as KIC 8386048. Additionally, some totally eclipsing
systems with no O’Connell effect show an asymmetric minimum, such as the non-
sample system KIC 8265951.

One difference between partially and totally eclipsing binaries is that parts
of both stars are always visible for partially eclipsing systems. These
regions of persistent visibility occur near each star’s visible pole. By
contrast, one star is fully occluded in totally eclipsing systems, making it
impossible for regions on that star to be persistently visible. Therefore,
features near the poles (like spots) can be occluded in totally eclipsing
systems but not partially eclipsing systems. BinaryMaker3 (BM3) tests showed
that both polar and equatorial spots could produce an asymmetric minimum,
however.

Our BM3 testing also indicated that the effect is strongly dependent on the
presence of total eclipses. Figure 18 shows that a one-degree change in
inclination is enough to transform a largely symmetrical eclipse into a
significantly asymmetric one. Therefore, another hypothesis is that total
eclipses are a prerequisite for a significant minimum asymmetry. We posit that
a feature on one component, which we will call a hot starspot or spot, causes
the asymmetry via its changing aspect. The spot must be azimuthally offset
from the plane both 1.) perpendicular to the orbital plane and 2.) containing
both stellar centers to produce an asymmetric minimum. The change in flux
caused by the spot’s changing aspect is small compared to the change caused by
the eclipsed star being covered (or uncovered) during the eclipse’s partial
phase. Therefore, the large change in flux during the partial phase prevents
the spot from creating a significant asymmetry. During the almost flux
constant total phase, however, the small change in flux caused by the spot’s
changing aspect has no other change in flux to compete with and can produce a
significant asymmetry. Therefore, under our starspot model, asymmetric minima
imply total eclipses.

Figure 18: Light curves of four eclipsing binary models detailing the area
around the primary minimum. Parameters for all four models are identical
except for inclination. The asymmetry grows significantly more pronounced as
the system changes from partially eclipsing ($i=74,~{}75^{\circ}$) to totally
eclipsing ($i=76,~{}77^{\circ}$), demonstrating the asymmetry’s strong
dependence on total eclipses.

####  4.5.3 KIC 10544976: The White Dwarf

Figure 19: Averaged (left) and _Kepler_ (right) light curves of KIC 10544976
using short-cadence data showing important features, including the sharp
primary eclipse and numerous flares. The right panel also plots the averaged
light curve in grey for comparison.

KIC 10544976 (labeled with the flag WD in Table 1) is unique within our sample
as it is the only system containing a degenerate component. The lack of white
dwarfs in our sample is unsurprising because white dwarfs were not priority
targets for _Kepler_. Almenara et al. (2012) identifies the system as
consisting of a DA white dwarf primary and an M4 V secondary in an 8.4-hour
orbit. The left panel of Figure 19 displays KIC 10544976’s short-cadence
average light curve, showing that the system has a very sharp, well-defined
primary eclipse and no secondary eclipse. The red dwarf companion produces
many flares, as Figure 19’s right panel shows. We consider this system to have
a non-traditional O’Connell effect because the observed flux increases
monotonically after the primary eclipse until phase $-0.35$. Therefore, the
light curve lacks two inter-eclipse maxima to measure a difference between.
KIC 10544976 has an OES of $-0.0096$, placing it within our marginal sample
near the core sample cutoff.

####  4.5.4 KIC 11347875: Concave-Up Systems

KIC 11347875 has the largest measured O’Connell effect in our sample, but its
light curve has an unusual appearance shared with six systems in the sample
(labeled with the flag CU in Table 1). Figure 20 shows that KIC 11347875’s
light curve is concave-up after the primary minimum, giving the system the
appearance of an eclipsing signal superimposed on a rough sinusoid. Figure 5
shows an enlarged view of KIC 11347875’s concave-up region. As with KIC
10544976, the lack of inter-eclipse maxima indicates that the system has a
non-traditional O’Connell effect. Gao et al. (2016) identifies KIC 11347875 as
containing two late-type red giants based on effective temperature and surface
gravity estimates, although our luminosity estimate from _Gaia_ EDR3 data
suggests a total luminosity of only 1.306 L$\odot$. Six systems (KICs 5300878,
6044064, 6197038, 6697716, 7671594, and 9119652) show a similar concave-up
region, as do some temporally varying systems like KIC 8479107 during some
time intervals. However, KIC 11347875 is by far the most extreme example. All
systems exhibiting a concave-up region are quite cool ($T<5,\\!100$ K), have a
longer orbital period ($P>1.2$ d), and show flares.

##  5 Analysis & Discussion

We now present a graphical representation of the correlations between the
characteristics we studied (except distance and luminosity). We further
discuss several correlations and their implications on the study of the
O’Connell effect, with a focus on correlations involving OES, eclipse depth,
and the morphology parameter.

###  5.1 Characteristic Trends

Figures 21, 22, and 23 show corner plots comparing period, OES, _Gaia_
temperature, _Gaia_ BP – RP color index, morphology parameter, absolute _Gaia_
G magnitude, and primary eclipse depth. We chose the plot limits for clarity
and, as such, the OES plots exclude KICs 9777984, 9935311, and 11347875. The
banding seen in the temperature plots (most prominent in Figure 23’s right
column) is a known issue with _Gaia_ DR2 data (Andrae et al., 2018). The
banding seen in the morphology plots (most prominent in the KEBC data of
Figure 23’s top-center panel) is a quantization effect due to the morphology
parameter being given to only two decimal places.

Figure 20: Averaged light curve of KIC 11347875 the system’s concave-up region
following the primary eclipse.

Figures 21-24 used the trend subset of the KEBC discussed in Section 2.2,
which excluded systems for two reasons: to analyze our sample without raising
Python errors and not have results be affected by invalid parameter values. We
excluded a given system because either it had no _Gaia_ parallax data (43
systems, plus 1 system without a known _Gaia_ EDR3 identifier), multiple
entries in the KEBC (11 systems, 24 total entries), or had $\mu=-1$,
indicating that the KEBC team could not assign a valid morphology parameter to
the system (174 systems). We also explicitly excluded KICs 5217781, 7667885,
7950964, and 9137819, as in Section 4. These removals reduced the KEBC’s size
to 2,678 systems. While we removed these 228 systems (241 entries) from our
KEBC analysis in Section 5.2, we did not remove any from our core sample
analysis.

###  5.2 Statistical Analysis

Table 2 gives the results of the K–S test described in Section 3.4. The K–S
statistic describes the degree of difference between the populations the two
samples were drawn from. The _p_ -value indicates how compatible the K–S test
results are with the null hypothesis that the two samples were drawn from the
same population. We expected the significantly different OES distributions as
we are looking at the wings of the distribution shown in Figure 2. This
difference is intended by design as it was how we defined our sample. We also
expected the distance distribution similarity, reinforced by Figure 7, because
a phenomenon that should not depend on distance differentiates the samples.
There is no reason to expect spatial differences between the two samples, and
any biases in spatial distribution are due to biases in the _Kepler_ sample.
The remaining characteristics in Table 2 are of greater interest as the
results point to possible underlying connections between those characteristics
and the O’Connell effect.

Table 2: Kolomgorov–Smirnov Test Results Characteristic | K–S Statistic |  _p_ -Value
---|---|---
O’Connell Effect Size | 0.604 | ¡0.001
$|$O’Connell Effect Size$|$ | 0.871 | ¡0.001
Primary Eclipse Depth | 0.398 | ¡0.001
Morphology Parameter | 0.242 | ¡0.001
Temperature | 0.277 | ¡0.001
Distance | 0.092 | 0.080
Absolute Magnitude | 0.285 | ¡0.001
Period | 0.274 | ¡0.001

Table 3 gives the results of our correlation analysis described in Section
3.4. Kendall’s $\tau$ coefficients are uniformly smaller than Spearman’s
$\rho$ coefficients and indicate the same correlations, so Table 3 presents
only Spearman’s $\rho$ coefficients for the sake of brevity. The coefficient
describes the correlation’s strength, while the _p_ -value indicates how
compatible the results of the Spearman test are with the null hypothesis that
the characteristics are uncorrelated. We consider two characteristics
correlated if $|\rho|\geq 0.1$ – consistent with D84 – and strongly correlated
if $|\rho|\geq 0.2$. Characteristic pairs have their values bolded whenever
they are correlated for a given sample. Table 3 also lists how many of the 20
random subsets produced by our bootstrapping procedure (described in Section
3.4) showed a correlation for each characteristic pair. Several correlations,
such as the one between distance and absolute magnitude, are expected and
serve as validations of our correlation analysis. We do not discuss such
expected correlations despite their robustness as they are present for reasons
unrelated to the O’Connell effect.

Table 3: Spearman’s $\rho$ Test Results Characteristic | Characteristic |  Core SampleaaEntries in bold indicate correlated characteristics (i.e. having $|\rho|\geq 0.1$) |  KEBCaaEntries in bold indicate correlated characteristics (i.e. having $|\rho|\geq 0.1$)
---|---|---|---
One | Two | Coeff. |  _p_ -Value | Surv.bbNumber of “surviving” (i.e. having $|\rho|\geq 0.1$) random subsets out of 20; see Section 3.4 | Coeff. |  _p_ -Value | Surv.bbNumber of “surviving” (i.e. having $|\rho|\geq 0.1$) random subsets out of 20; see Section 3.4
Informative O’Connell Effect Results
OES | Temperature |  0.385 |  ¡0.001 | 20 | 0.020 | 0.423 | 0 |
OES | Morphology |  -0.148 |  0.032 | 16 | 0.024 | 0.326 | 0 |
OES | Absolute Mag. |  -0.286 |  ¡0.001 | 19 | 0.033 | 0.183 | 2 |
OES | Distance |  0.218 |  0.001 | 17 | -0.012 | 0.641 | 0 |
OES | Period |  0.212 |  0.002 | 17 | -0.069 | 0.005 | 4 |
Eclipse Depth | Morphology |  -0.489 |  ¡0.001 | 20 |  -0.158 |  ¡0.001 | 18 |
Absolute Mag. | Period |  -0.609 |  ¡0.001 | 20 |  -0.370 |  ¡0.001 | 20 |
Other Results
OES | Eclipse Depth | 0.033 | 0.628 | 13  |  0.124 |  ¡0.001 | 12 |
$|$OES$|$ | Eclipse Depth |  0.136 |  0.048 | 10 |  0.573 |  ¡0.001 | 20 |
$|$OES$|$ | Morphology | -0.055 | 0.428 | 10  |  0.210 |  ¡0.001 | 20 |
$|$OES$|$ | Temperature | -0.025 | 0.726 | 9  |  -0.213 |  ¡0.001 | 19 |
$|$OES$|$ | Distance | 0.044 | 0.524 | 9  | 0.010 | 0.687 | 0 |
$|$OES$|$ | Absolute Mag. | -0.027 | 0.696 | 11  |  0.236 |  ¡0.001 | 20 |
$|$OES$|$ | Period | 0.046 | 0.506 | 8  |  -0.366 |  ¡0.001 | 20 |
Eclipse Depth | Temperature | -0.039 | 0.575 | 9  | -0.091 | ¡0.001 | 10 |
Eclipse Depth | Distance | 0.057 | 0.405 | 10  | 0.071 | 0.004 | 6 |
Eclipse Depth | Absolute Mag. | -0.004 | 0.955 | 12  |  0.110 |  ¡0.001 | 13 |
Eclipse Depth | Period | 0.054 | 0.432 | 12  | -0.084 | ¡0.001 | 10 |
Morphology | Temperature |  0.113 |  0.109 | 12 |  0.196 |  ¡0.001 | 20 |
Morphology | Distance |  -0.148 |  0.032 | 13 | 0.025 | 0.317 | 1 |
Morphology | Absolute Mag. | 0.065 | 0.350 | 12  |  -0.156 |  ¡0.001 | 19 |
Morphology | Period |  -0.591 |  ¡0.001 | 20 |  -0.562 |  ¡0.001 | 20 |
Temperature | Distance |  0.245 |  ¡0.001 | 17 |  0.139 |  ¡0.001 | 19 |
Temperature | Absolute Mag. |  -0.705 |  ¡0.001 | 20 |  -0.695 |  ¡0.001 | 20 |
Temperature | Period |  0.248 |  ¡0.001 | 16 |  0.236 |  ¡0.001 | 20 |
Distance | Absolute Mag. |  -0.512 |  ¡0.001 | 20 |  -0.364 |  ¡0.001 | 20 |
Distance | Period |  0.348 |  ¡0.001 | 20 |  0.100 |  ¡0.001 | 8 |

We found that seven of the characteristic pairs we studied provided more
insight into the O’Connell effect than the remaining ones. Table 3 highlights
these seven pairs (OES and temperature, OES and morphology parameter, OES and
absolute magnitude, OES and distance, OES and period, eclipse depth and the
morphology parameter, and absolute magnitude and period). The bootstrapping
procedure discussed in Section 3.4 guided our choice in which characteristic
pairs were important: five of these pairs are robust and the other two are
nearly so. These levels of robustness further raises our confidence in the
results we present. Our discussion in Sections 5.2.1-5.2.4 focuses primarily
on these characteristic pairs, along with several trends visible in Figures
21, 22, and 23. These trends include the lack of core systems with morphology
parameters between 0.6 and 0.7 in the morphology panels, the clustering of
systems along the right edge of Figure 22’s center panel, and the stark
contrast between the positive and negative O’Connell effect systems’
temperature distributions in Figure 21’s center panel.

Figure 21: Corner plot showing the correlations between three characteristics
of interest (period, OES, and temperature) and OES, temperature, and _Gaia_
color. Core sample targets are shown in black while non-core KEBC systems are
shown in grey. A few outlier systems have been removed from the OES plots for
clarity (see discussion in text). Note the logarithmic _x_ -axis of the period
plots.  Figure 22: Corner plot showing the correlations between three
characteristics of interest (_Gaia_ color, morphology parameter, and primary
eclipse depth) and morphology parameter, primary eclipse depth, and absolute
_Gaia_ G magnitude. Core sample targets are shown in black while non-core KEBC
systems are shown in grey.  Figure 23: Corner plot showing the correlations
between three characteristics of interest (period, OES, and temperature) and
morphology parameter, primary eclipse depth, and absolute _Gaia_ G magnitude.
Core sample targets are shown in black while non-core KEBC systems are shown
in grey. A few outlier systems have been removed from the OES plots for
clarity (see discussion in text). Note the logarithmic _x_ -axis of the period
plots.

Our initial analysis of the KEBC found strong correlations between the OES and
both period and the morphology parameter. Our inclusion of many long-period
and well-detached systems that are fundamentally different from our sample’s
systems strengthened these correlations. As a result of these fundamental
differences, we considered these initial correlations less relevant to our
study. Therefore, our final KEBC analysis used the analysis subset of the KEBC
discussed in Section 2.2, which, in addition to the systems excluded in the
earlier subsets, excluded systems with $P\geq 10$ d (690 systems) or $\mu\leq
0.3$ (1,018 systems), leaving only 1,639 KEBC systems. As in Sections 4 and
5.1, we removed no systems from our core sample analysis, and the KEBC
population includes the core sample (except KIC 7667885). Our analysis
considers both OES and $|$OES$|$.

####  5.2.1 O’Connell Effect Size

The OES (as defined in Section 2.4) reflects the difference in brightness at
the two maxima, which in turn depends on the brightness difference between the
leading and trailing hemispheres of each star. We gave particular focus to
correlations and trends involving the OES because it defined our sample and is
the characteristic most directly related to the O’Connell effect. This section
discusses OES and $|$OES$|$ correlations with the other characteristics we
studied. It also discusses the differences between the OES and $|$OES$|$
correlations.

Figure 21’s center panel comparing OES and temperature shows that a negative
O’Connell effect, wherein the brighter maximum occurs before the primary
minimum, mainly occurs in systems with $T<6,\\!000$ K. Thirty-four (23% of 143
systems with a _Gaia_ temperature) positive O’Connell effect systems have a
temperature above 6,000 K, while only two (3% of 62 systems) negative
O’Connell effect systems do. Furthermore, one of the two hot negative
O’Connell effect systems, KIC 7773380, has a _Kepler_ temperature and _Gaia_
color implying that it is likely significantly cooler than its _Gaia_
temperature indicates. The other system, KIC 7950962, has primary and
secondary eclipse depths that differ by only $\sim$0.001 in normalized flux.
Furthermore, our light curves of the system have a phase offset of 0.5 from
the KEBC light curve. These two facts indicate an ambiguity regarding which of
KIC 7950962’s eclipses is the primary, and thus if its O’Connell effect is
positive or negative. The dearth of hot systems displaying a negative
O’Connell effect was unknown before now and is consistent with the idea that
starspots are the predominant cause of a negative O’Connell effect. Starspots
are expected to exist in the convective envelopes of cooler stars but not in
the radiative envelopes of hotter stars.

Figure 24: Plot comparing the OES to the period. Core sample targets are shown
in black while non-target KEBC systems are shown in grey. The _y_ -axis is in
units of normalized flux. KICs 9777984, 9935311, and 11347875 are excluded for
clarity. Note the positive correlation in core sample systems with $P<0.5$ d.

Figure 21’s top-left panel comparing OES and period shows that the OES tends
towards zero at longer periods. This trend is consistent with the idea that
binary interaction ultimately causes the O’Connell effect. The lack of any
systems in our sample with a period greater than ten days strengthens this
idea. Table 3 shows that this correlation is not robust. Of greater concern is
that Table 3 indicates that the correlation between OES and period is positive
in our sample, contradicting the trend observed in Figure 21. Our explanation
for this contradiction is a positive trend between OES and period for the
shortest period systems in our sample.

To better display this short-period system trend, Figure 24 shows a rescaled
view of Figure 21’s top-left panel focusing on systems with periods under 1 d.
Systems with $P\leq 0.5$ d show this positive correlation clearly. Since
Figure 9 shows that most systems in our sample have $P\leq 0.5$ d, this
positive correlation dominates the $\rho$ coefficient. We found that the
$\rho$ coefficient for OES and period more than doubled ($\rho=0.425$) when
analyzing only core sample systems with $P\leq 0.5$ d rather than the entire
core sample, supporting our explanation. Furthermore, the short-period system
correlation is robust (20 of 20 subsets show a correlation). When we analyzed
the core sample systems with $P>0.5$ d, we found no correlation
($\rho=-0.069$). We interpret the short-period system trend as a result of the
following: hotter systems ($T\geq 6,\\!000$ K) almost always have OES $>0$ (as
discussed in the previous paragraph), temperature is positively correlated
with luminosity, and luminosity is positively correlated with orbital period
(as discussed in Section 5.2.4). Therefore, the shortest period systems are
more likely to have OES $<0$ than longer period systems, producing the
observed trend.

Figure 25: Plots comparing the primary eclipse depth to $|$OES$|$. Core sample
targets are shown in black while non-target KEBC systems are shown in grey.
The red dashed line shows the ODR correlation for the KEBC. The left panel’s
axes are scaled linearly, while the right panel’s axes are scaled
logarithmically. Both axes are in units of normalized flux. KICs 9777984,
9935311, and 11347875 are excluded from the left panel for clarity. No
correlation is apparent in the left panel, but a weak correlation can be seen
in the right panel.

We expected strong correlations between $|$OES$|$ and other characteristics
under the premise that the positive and negative O’Connell effects are
fundamentally similar. This premise implies that the positive and negative OES
correlations should strengthen when neglecting the OES’s sign. Therefore, we
were surprised by the _lack_ of such correlations in our core sample. The only
$|$OES$|$ correlation we found was between $|$OES$|$ and the primary eclipse
depth, and the correlation is both weak and not robust. Furthermore, Figure
25’s left panel shows no visible correlation between these two
characteristics. The lack of $|$OES$|$ correlations implies that our premise
that the positive and negative O’Connell effects are fundamentally similar is
incorrect.

Note that every characteristic is correlated with $|$OES$|$ in the KEBC, the
strongest of which is the correlation with primary eclipse depth. We
determined a quadratic fit of $D=-298(19)|\text{OES}|^{2}+81(4)|\text{OES}|$
for this correlation using ODR, where $D$ is the primary eclipse depth and the
parentheticals are the uncertainties. While we do not see this correlation in
Figure 25’s left panel, either, Figure 25’s right panel showing a log-log plot
of these characteristics displays a visible, if weak, correlation between
them. We suspect that the $\rho$ coefficient detected this trend, resulting in
the large coefficient in Table 3. Figure 25’s right panel indicates that this
trend extends from our sample down to much smaller values of $|$OES$|$ and
eclipse depth. Therefore, some systems that are not in our sample may be
fundamentally similar to our sample’s systems, only with smaller values for
$|$OES$|$ and eclipse depth. As such, this correlation may be more related to
the O’Connell effect than its strength in our sample would indicate.

Both panels of Figure 25 also show a lack of systems displaying a significant
O’Connell effect and a small primary eclipse depth. Our selection criterion
($|\text{OES}|\geq 0.01$) does not exclude such systems, and their absence is
conspicuous since most KEBC systems have small eclipse depths. A bias in the
_Kepler_ selection function against these systems may explain their absence,
which would imply a fundamental difference between such systems and the
systems _Kepler_ observed. Their absence may also be a true representation of
O’Connell effect binaries, although we cannot identify a plausible reason why
systems with a significant O’Connell effect cannot have a small primary
eclipse depth. As a final note, Figure 25’s left panel shows that all four
KEBC systems with an eclipse depth above 0.8 have a non-negligible $|$OES$|$.

We found different correlations for the OES (the morphology parameter,
temperature, distance, absolute magnitude, and period) as compared to
$|$OES$|$ (eclipse depth). Because the OES distinguishes between the positive
and negative O’Connell effects but $|$OES$|$ does not, this suggests a more
fundamental difference between the positive and negative O’Connell effects
than previously thought. Assuming that spots are the O’Connell effect’s
primary cause, a larger OES implies that spots cover a greater area, have a
more extreme temperature factor, or are further offset from the line
connecting the stellar centers. Kouzuma (2019) found a weak positive
correlation between stellar temperature and spot temperature factor for cool
starspots in W-type overcontact systems, wherein the smaller star is hotter
(in contrast to A-type overcontact systems; McCartney 1999). The same
correlation is much stronger in semi-detached systems. Kouzuma (2019) found
weak positive correlations in overcontact systems between stellar temperature
and spot size and between orbital period and spot size. The correlations for
cool spots are similar but generally stronger than for hot spots. His results
imply positive correlations between temperature and $|$OES$|$ and between
orbital period and $|$OES$|$. Table 3 shows both correlations for OES but
neither for $|$OES$|$.

To further test this correlation, we found the $\rho$ coefficient for the
positive O’Connell effect systems and for the negative O’Connell effect
systems. We found that the correlations for the positive O’Connell effect
systems ($\rho_{\text{temp}}=0.122$ and $\rho_{\text{per}}=0.083$) are weak.
Furthermore, while the correlations are much stronger for the negative
O’Connell effect systems ($\rho_{\text{temp}}=0.402$ and
$\rho_{\text{per}}=0.142$), the former has the wrong sign (i.e. the OES gets
closer to zero as the temperature increases) while the latter is still weak.
These correlations are therefore inconsistent with the results of Kouzuma2019
starspot study. However, such inconsistencies may not indicate that starspots
do not cause the O’Connell effect. Instead, they may result from our sample’s
mixture of systems with different O’Connell effect causes, or perhaps from
differences between our sample and his. Determining the cause of these
inconsistencies is beyond the scope of this paper.

The three other robust correlations that we have not discussed are between OES
and the morphology parameter, distance, and absolute magnitude. We interpret
these three correlations as arising from unrelated correlations with other
characteristics. For instance, the correlation between the OES and absolute
magnitude arises from the discussed correlation between OES and temperature
combined with the strong correlation between temperature and absolute
magnitude that is a well-known feature of main-sequence stars. We do not
discuss these secondary correlations due to their dependence on other
correlations that have a more fundamental explanation.

The different correlations we find among positive and negative O’Connell
effect systems, combined with the fact that a much larger number of systems
display a positive O’Connell effect, leads us to conclude that the preference
for a brighter maximum following the primary eclipse is not an observational
bias. It is instead fundamental to the O’Connell effect. Additionally, we
conclude that the positive and negative O’Connell effects have different
causes, or that one cause is common in one case and rare in the other. We
consider these findings important results of our study.

####  5.2.2 Eclipse Depth

The primary eclipse depth (as defined in Section 3.1) is influenced by four
parameters: temperature ratio, relative radii, component shapes, and orbital
inclination. This section discusses the correlation between the eclipse depth
and the morphology parameter. It also discusses a couple of trends between
these two characteristics seen in Figure 22’s center panel.

The eclipse depth strongly correlates with the morphology parameter in our
sample, while the KEBC shows a weaker correlation. We expect such a
correlation because, as Wilson (2006) states, the eclipse depth is a monotonic
function of the stellar radii ratio for overcontact systems (i.e. systems with
a large morphology parameter). Meanwhile, Matijevič et al. (2012) notes that
the primary eclipse width increases with the morphology parameter, and the
eclipse widths measure the sum of the relative stellar radii. Thus, in
overcontact systems, the primary eclipse depth decreases as the morphology
parameter increases. The best fit to this correlation is the linear function
$\mu=-2.13(13)D+1.26(9)$. The KEBC’s correlation is weaker as it includes more
small $\mu$ systems, for which the eclipse depth and stellar radii ratio
relation does not hold.

Figure 22’s center panel comparing the primary eclipse depth to the morphology
parameter shows a sharp edge toward the panel’s right side in both our sample
and the KEBC. This trend indicates that eclipses get shallower as $\mu$
increases above 0.7, a consequence of the relation discussed in the previous
paragraph. The systems in our sample appear to be clustered along this edge in
Figure 22’s center panel. This clustering indicates that O’Connell effect
systems tend to have the deepest eclipses of systems with similar light curves
(as measured by the morphology parameter), hinting at a connection between the
OES and the eclipse depth. The clustering around the right edge is also seen
with the non-core systems, although it is not as pronounced.

Figure 22’s center panel shows another trend: all KEBC systems with $D\geq
0.6$ have $\mu\approx 0.6$. Söderhjelm & Dischler (2005) states that the
maximum eclipse depth for two main-sequence stars is 0.75 magnitudes
(equivalent to a relative flux of 0.5), implying that systems with deeper
eclipses must have an evolved component. Matijevič et al. (2012) says that a
$\mu$ of 0.6 indicates a semi-detached system, which occurs when an evolving
star fills its Roche lobe. This trend therefore make sense because close
binaries with evolved components would be expected to have $\mu\approx 0.6$ by
Matijevič et al. (2012). However, we cannot discount the possibility that this
trend is a statistical artifact caused by the rarity of systems with such deep
eclipses in the KEBC. Our eclipse depth determination method discussed in
Section 3.1 may also influence this trend, as it can underestimate the eclipse
depth of Algol-type systems.

####  5.2.3 Morphology Parameter

The morphology parameter (described in Section 3.3) is primarily a measure of
eclipse widths (Matijevič et al., 2012), but it correlates well with the
morphology class of a given system. We wish to reemphasize the point from
Matijevič et al. (2012) that the morphology parameter only provides a “best-
guess” estimate of the morphology class. This section focuses on the
morphology parameter distribution shown in Figure 26. It also discusses the
initially surprising correlation between the morphology parameter and
temperature.

Figure 26: Histogram comparing the morphology parameters for 211 of the 212
core sample targets (solid red) and all 2,745 entries in the KEBC (dashed
grey) that have $\mu\neq-1$. KIC 7667885 is not included because it lacks a
value for $\mu$ (see discussion in Section 2.5). The two distributions differ
significantly, particularly for $\mu<0.5$. The core sample also has few
systems with $0.6\leq\mu\leq 0.7$.

Figure 26 shows a lower abundance of systems with $0.6\leq\mu\leq 0.7$. The
same region is depleted in the KEBC, but not to the extent of our sample,
suggesting that this underabundance is not wholly due to the parent
population. Matijevič et al. (2012) states that systems with $\mu$ in this
range are semi-detached. Our interpretation of systems in this range is that
the accreting star grows larger with increasing morphology parameter, with the
system becoming contact around $\mu=0.7$. The larger the accretor is, the less
deeply a matter stream will penetrate into its potential well. The matter
stream will therefore impact the surface with less energy and cause less
dramatic heating. Assuming the accretor is on the main sequence, a larger star
will be hotter, further reducing the degree of heating. These factors could
cause the paucity of systems in this region of parameter space showing a
significant O’Connell effect.

Figure 26 also shows that systems with $\mu\lesssim 0.5$ rarely show a
significant O’Connell effect, and none with $\mu\leq 0.3$ do. This is
significant because our selection criterion should not be biased with respect
to $\mu$. Stars in systems with a small $\mu$ are generally farther apart and
are less likely to significantly affect each other, so this result reinforces
the link between binary interaction and the O’Connell effect discussed in
Section 5.2.1.

We were surprised by the morphology parameter and temperature correlation
because we assumed they would be uncorrelated. We now believe that this
correlation results from a selection effect. Two processes work in tandem to
cause this selection effect: the influence of geometry on the morphology
parameter and the influence of the initial mass function (IMF) on temperature.
Regarding the first process, eclipses only occur if $\Delta<R_{1}+R_{2}$ at
$\phi=0$ (conjunction), where $\Delta=a\cos{i}$ (Prša, 2006, Equations 3.37
and 3.38) and the $R_{n}$ are the component radii. Therefore, closer binaries
(smaller $a$) exhibit eclipses over a broader range of inclinations. Since the
components of large morphology parameter systems are generally closer than in
small morphology parameter systems, we are more likely to observe a given
system eclipse in the former case. The IMF, meanwhile, makes hot stars rarer
than cool stars, a fact compounded by _Kepler_ ’s selection biases discussed
in Section 2.1. Acting together, these processes mean that most hot eclipsing
binaries observed by _Kepler_ should have large morphology parameters. The
resulting dearth of hot, small morphology parameter systems creates an
apparent positive correlation between the characteristics.

####  5.2.4 Period

The period is a function of component masses and orbital separation due to
Kepler’s third law. This section discusses the correlation between the period
and the absolute magnitude/luminosity in the core sample. Figure 23’s lower-
left panel demonstrates this correlation in the dense clustering seen near the
panel’s left side.

A correlation between period and absolute magnitude initially seems odd. Low
luminosity stars can have very wide orbits, after all, while very luminous
stars can be short-period contact binaries. However, there is a lower period
limit for stars of a given luminosity. For example, on the main sequence, a
star’s size and surface temperature (and thus absolute magnitude) are a
function of its mass. If binary components are too close, however, they will
merge into a single star. Kepler’s third law states that $P\propto a^{1/3}$,
while the Stefan-Boltzmann law states that $L_{*}\propto R^{2}$. Therefore, a
lower limit on component separation based on stellar radius implies a lower
limit on the period for a given absolute magnitude for main-sequence stars.
This limit increases for evolved stars since they become cooler and larger as
they evolve. The correlation between period and absolute magnitude is weaker
in the KEBC because of the KEBC’s numerous long-period systems, which do not
have a relationship between component separation and stellar radius.

Figure 27: Plot comparing the luminosity to the period. Core sample targets
are shown in black while non-target KEBC systems are shown in grey. KICs
3662635 and 5820209 are excluded for clarity. The sharp edge near the lower-
left corner denotes the short-period limit for a given luminosity. Most of the
core sample is clustered near or along this edge.

Both Figure 23’s lower-left panel and Figure 27 show a tight clustering of
systems along their left edge. Our interpretation of this clustering is that
it represents the lower limit on the period for a given absolute magnitude or
luminosity. Systems to the left of this edge are likely subdwarf systems or
have poorly determined distances resulting in large luminosity errors. The
fact that such a large percentage of our sample lies on or near this edge –
particularly systems with $L_{*}\geq 1~{}\text{L}_{\odot}$ in Figure 27 –
suggests that the O’Connell effect is more common in contact or near-contact
binaries. This trend is therefore strong evidence that binary interaction is a
critical factor in causing the O’Connell effect.

##  6 Conclusion

Our study of the characteristics of systems displaying a significant O’Connell
effect takes advantage of the unprecedented amount of data available from the
_Kepler_ and _Gaia_ missions. It represents a significant step forward in our
understanding of this poorly understood phenomenon. We studied a significantly
different parameter space than the earlier works of O51 and D84 out of
necessity: we lack detailed information on the bulk characteristics of the
component stars that were the basis of those works. We instead focused on
those observational properties we could directly determine from the data
currently available.

We found a significant O’Connell effect in 212 eclipsing binaries, with the
211 KEBC systems representing 7.3% of the 2,907 systems cataloged in the KEBC.
This shows that a significant O’Connell effect is present in a non-negligible
fraction of eclipsing binaries. Our core sample systems range in period from a
quarter of a day to nearly ten days, temperature from around 3,800 K to 8,500
K, and luminosity from two-hundredths of a solar luminosity to over fifty
solar luminosities. Clearly, a wide range of system types shows this
phenomenon.

The O’Connell effect also manifests itself quite differently in different
systems. Many systems have light curves that vary wildly with time, and the
O’Connell effect varies along with it. Other systems have extremely stable
light curves and a stable value of the OES. The existence of the latter type
of system would indicate a cause of the O’Connell effect that is more stable
in time than starspots, such as a hotspot caused by mass transfer.
Furthermore, up to 20% of our sample displays the ETV signal associated with
mass transfer. We conclude that there are multiple causes of the O’Connell
effect.

The dearth of hot ($T\geq 6,\\!000$ K) systems displaying a negative O’Connell
effect is a new result and can be interpreted as evidence that starspots cause
the O’Connell effect in negative O’Connell effect systems. Additionally, we
found a moderate correlation between OES and temperature and a weaker
correlation between OES and period, both predicted by Kouzuma2019 study on
starspots, which also suggests that starspots predominately cause the
O’Connell effect. However, looking at positive and negative O’Connell effects
separately does not yield the same correlations. Instead, we found
correlations that are inconsistent with Kouzuma2019 results. Therefore, we
draw no conclusion as to whether starspots are a predominant cause of the
O’Connell effect, but note that the idea is plausible. At the same time, it
should not automatically be assumed that starspots cause the O’Connell effect
in a given system without further supporting evidence for starspots. Many more
systems display a positive O’Connell effect than a negative one, suggesting
that which maximum is brighter is not chance but fundamentally tied to the
O’Connell effect. We conclude that, whatever the physical causes of the
O’Connell effect are, they preferentially cause the maximum following the
primary eclipse to be brighter than the one preceding it.

The overarching trend found in this study is that the O’Connell effect and
binary interaction are strongly correlated. We do not see this phenomenon in
binaries with widely separated stars, and we almost always see evidence
(ellipticity effects in light curves) of non-spherical stars in our sample.
Looking beyond our sample, W Crucis (Zoła, 1996; Pavlovski et al., 2006)
exhibits the O’Connell effect, and its light curve indicates an interacting,
semi-detached system similar to other systems in our sample. However, W
Crucis’ 198-day orbital period – the longest of an O’Connell effect binary
that we are aware of – is over 20 times longer than the longest period in our
sample. Nevertheless, W Crucis’ ellipsoidal variations shows that it has
interacting components despite the long period, reinforcing the correlation
between binary interaction and the O’Connell effect. We conclude that this
correlation and the physical causes of the O’Connell effect are related: the
interaction (whether gravitational, magnetic, or physical) between close
binary stars causes the O’Connell effect instead of a phenomenon an isolated
star (including non-interacting components in wider binaries) would display.
If the phenomenon occurred on isolated stars, we would expect to see the
O’Connell effect in _all_ types of binaries, not just those with interacting
components.

There are several ways to expand upon our work in the future. The recently
published TESS Eclipsing Binary Catalog (TEBC; Prša et al., 2022) provides an
opportunity to expand our analysis to a much larger data set. _Gaia_ Data
Release 3 (DR3) – currently slated for release in 2022 – will include _Gaia_
time-series photometry for eclipsing binaries (Gaia Collaboration et al.,
2019). _Gaia_ ’s tricolor photometry allows temperature measurements for both
stars and an exploration of the O’Connell effect’s temperature dependence.
This data greatly increases the parameter space available for study. We will
further discuss the system classes Section 4.5 described in future work. This
discussion will include searching for periodicities and quasi-periodicities in
temporally varying systems, finding the second and third derivatives of
concave-up systems to quantify their properties, and determining the degree of
asymmetry in asymmetric minima systems. We will also discuss the subset of our
sample brighter than $K_{p}=14$ in future work due to Wolniewicz et al. (2021)
finding such systems to be unbiased. The new _Kepler_ eclipsing binaries
Bienias et al. (2021) found require further study to ensure our sample’s
completeness. Future work will also discuss our observations of ten sample
systems and the PHOEBE (Prša & Zwitter, 2005; Prša et al., 2016) models we
create using our data. Analysis of the sub-groupings seen in Figures 21-23 is
left for a future study. Finally, we intend to further study the differences
between the positive and negative O’Connell effects our study found.

This material is based upon work supported by the National Aeronautics and
Space Administration under Grant No. 80NSSC19K1021 issued through the
NNH18ZDA001N Astrophysics Data Analysis Program (ADAP). This paper includes
data collected by the _Kepler_ mission. Funding for the _Kepler_ mission is
provided by the NASA Science Mission directorate. This research has made use
of the SIMBAD database, operated at CDS, Strasbourg, France. This research has
made use of the VizieR catalogue access tool, CDS, Strasbourg, France (DOI :
10.26093/cds/vizier). The original description of the VizieR service was
published in 2000, A&AS 143, 23. We would like to thank Dr. Ron Kaitchuck for
his helpful feedback during the writing of this paper. We would also like to
thank Drs. Andrej Prša and Kyle Conroy for their help in understanding the
creation and contents of the KEBC. Finally, we would like to thank AAVSO
astronomer Chuck Cynamon for observing several systems in our sample.

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