11institutetext: Nicolaus Copernicus Astronomical Centre, Polish Academy of
Sciences, Bartycka 18, 00-716 Warszawa, Poland
11email<EMAIL_ADDRESS>22institutetext: Departamento de Astronomía,
Universidad de Concepción, Casilla 160-C, Concepción, Chile 33institutetext:
Astronomical Observatory, University of Warsaw, Aleje Ujazdowskie 4, Warsaw,
00-478, Poland

# Non-evolutionary effects on period change in Magellanic Cepheids.

I. New binary systems revealed from Light Travel Time Effect

Rajeev Singh Rathour 11 0000-0002-7448-4285 Gergely Hajdu 11
0000-0003-0594-9138 Radosław Smolec 11 0000-0001-7217-4884 Paulina Karczmarek
22 0000-0002-0136-0046 Vincent Hocdé 11 0000-0002-3643-0366 Oliwia Ziółkowska
11 0000-0002-0696-2839 Igor Soszyński 33 0000-0002-7777-0842 Andrzej Udalski
33 0000-0001-5207-5619

(Received ??; accepted ??)

###### Abstract

Context. Period change studies offer a novel way to probe the evolution and
dynamics of Cepheids. While evolutionary period changes have been well studied
both observationally and theoretically, non-evolutionary period changes lack a
systematic and quantitative description. Here, we deal with one such aspect of
non-evolutionary period changes related to a crucial property, namely, the
binarity-based nature of a Cepheid. With the advent of long-term photometry
surveys covering Magellanic fields, the census of classical Cepheids in binary
(or multiple) systems outside the Milky Way is timely. This may have
implications for crucial aspects such as the period-luminosity relationship
calibrations and our understanding of the nature of Cepheid companions.

Aims. The overall objective is to have a quantitative understanding of the
full picture of non-evolutionary period changes in Cepheids to develop a
formalism to disentangle it from the secular evolutionary period change. In
the first paper in the series, we aim to conduct a systematic search for non-
evolutionary period changes to look for Cepheids in likely binary
configurations and quantify their incidence rates in the Magellanic Clouds.

Methods. We collected more than a decade-long time-series photometry from the
publicly available, Optical Gravitational Lensing Experiment (OGLE) survey,
with more than 7200 Cepheids altogether from the Large Magellanic Cloud (LMC)
and Small Magellanic Cloud (SMC). Our sample contains both fundamental-mode
and first-overtone mode Cepheids. Then, we calculate d the observed minus
calculated ($O-C$) diagrams to reveal the light-travel time effect (LTTE).
Finally, we calculated the minimum companion masses of the Cepheids and
compared them with the predictions from Cepheid population synthesis results.

Results. In our search, out of an overall sample of more than 7200 Cepheids,
we found 52 candidate Cepheid binary systems in the LMC (30 fundamental and 22
first-overtone mode) and 145 in the SMC (85 fundamental and 60 first-overtone
mode). The majority of the sample is characterized by orbital periods of
2000-4000 d and eccentricities of 0.2-0.5. Moreover, we report two candidates
in each galaxy with the Cepheid likely existing with a giant companion. The
incidence rate ratio for SMC to LMC calculated from our sample is in agreement
with binary Cepheid population synthesis predictions.

Conclusions. In our attempt to quantify the non-evolutionary period change
connected with the LTTE, our systematic search has enriched the Cepheid binary
sample by a factor of about 2 in both galaxies. The future spectroscopic
follow-up can confirm the binarity nature of our sample and constrain the
orbital parameters.

###### Key Words.:

Techniques : photometric – Methods: data analysis – stars: variables:
Cepheids– stars: binaries: general

##  1 Introduction

Type I or classical Cepheids (hereafter referred to as Cepheids) have long
been one of the benchmark stars for stellar evolution and pulsation studies.
They are a class of variable stars with intermediate and high masses ($\sim
3-13{\,}{\rm M}_{\odot}$) and with pulsation periods of $\sim 1-100$ days.
Cepheids have ages ranging between $10-100$ Myr and are reported to be
pulsating in radial and non-radial modes (e.g., Catelan & Smith, 2015;
Soszyński et al., 2008, 2010, 2011, 2015b).

The pulsations in Cepheids are excited when they cross a region on the
Hertzprung-Russell diagram, known as the instability strip (IS). These
pulsations are driven by the $\kappa$ and $\gamma$ mechanisms, operating in
the partially ionized helium and hydrogen zones (e.g., Cox, 1980). As these
stars move across the IS, they experience overall changes in stellar radius
(alongside effective temperature and luminosity), thereby changing the
pulsation period. As a consequence, from the observational viewpoint, the
measure of these period changes is a direct insight into the Cepheid
evolution. However, Cepheid also exhibits period changes that are not related
to its secular evolution, and thus of much shorter timescales ($\sim$100-10000
days). These period changes are classified as non-evolutionary and come in two
types. The first kind manifests as periodic period changes and is a
consequence of a Cepheid in a binary system. The second type is observed as
short timescale irregularities in the period changes, hence the designation of
irregular period changes. The occurrence of this kind of period change is
still a puzzle with no consolidated mechanism proposed yet. In this work, we
investigate the non-evolutionary period changes of the first kind, namely,
those related to Cepheids in binary systems.

In the Galactic sample, $60-80$ % or more of Cepheids are likely to be part of
a binary system, as observations indicate (Szabados, 2003a; Evans et al.,
2013; Kervella et al., 2019b). The well maintained Galactic binary Cepheids
database111https://konkoly.hu/CEP/intro.html contains $\sim$200 confirmed or
suspected binary targets (Szabados, 2003b). However, the Magellanic Clouds’
photometry and spectroscopic observations are way short of reproducing these
statistics with 9 and 25 reported Cepheid binaries in the Small and Large
Magellanic Clouds (SMC and LMC), respectively (Szabados, 1983; Pilecki et al.,
2021). The significance of binary Cepheids is on two fronts. First, from a
cosmological standpoint, Cepheids follow the well-known period-luminosity
relationship (Leavitt Law, Leavitt, 1908; Leavitt & Pickering, 1912), making
them excellent extragalactic distance indicators (e.g., Freedman et al., 2001;
Riess et al., 2021, and references therein). However, the presence of any
companion can affect the measured brightness and color of the Cepheid (e.g.,
Szabados et al., 2013), as a consequence that may possibly affect the period-
luminosity relationship (Szabados & Klagyivik, 2012; Pilecki et al., 2021;
Karczmarek et al., 2023). Second, binary Cepheids are significant to bridge
the gap between stellar evolution and pulsation theory predictions. Nearly
half a century has passed since the realization that the evolutionary masses
of Cepheids are systematically higher (currently about $10-20$ %) than the
ones predicted by pulsation theory. This has led to a long-standing enigma in
astrophysics, known as the Cepheid mass discrepancy problem (Stobie, 1969;
Fricke et al., 1972; Cox, 1980; Keller, 2008; Bono et al., 2008). The first
purely dynamical mass of Cepheid (Polaris Aa) was presented by Evans et al.
(2008), using Hubble Space Telescope (HST) UV imaging observations.
Thereafter, a landmark study by Pietrzyński et al. (2010) reported the
dynamical mass of a Cepheid in an eclipsing binary system, OGLE-LMC-CEP-0227,
to an accuracy of 1 %, which endorsed the pulsation theory predictions and
inspired revising stellar evolutionary models. Many more studies to measure
dynamical masses of Cepheids in binary systems have been conducted thereafter
(Pietrzyński et al., 2011; Gieren et al., 2014; Pilecki et al., 2015, 2018).
Therefore, Cepheids in binary systems (even better if eclipsing), can provide
crucial and accurate knowledge in boosting the above frontiers, since we can
no longer ignore the companion effects on avenues explored via Cepheids.

Binary Cepheids are increasingly getting due attention, as solving for their
orbital parameters can not only provide dynamical masses (and other physical
properties) of the Cepheids, which has its crucial applications as described
before, but also may contribute to better understanding of orbital evolution
of the binary. Recently, a number of population synthesis studies have tried
to approach Cepheids in binary systems (Neilson et al., 2015; Karczmarek et
al., 2022, 2023). The predicted minimum orbital period of Cepheid in a binary
system is $200-300$ days for the Galactic Cepheid population. This prediction
ensures that the binaries evolve without interacting when one of the
components goes through the red giant branch phase. However, this does not
apply to first-crossing Cepheids that are still in the sub-giant phase and may
have shorter binary orbital periods, as observationally confirmed for OGLE-
LMC-CEP-1347 (Pilecki et al., 2022). It is a double-mode Cepheid (first- and
second-overtone modes) with a very short, 59-day orbit with a less massive
companion (mass ratio $q=0.55$).

Searching Cepheid-bearing binary systems is traditionally done via
spectroscopic studies looking for radial velocity variation. Evans (1992)
compiled several systems via direct imaging also for binaries containing
luminous companions. Recently, a technique of searching Cepheids and RR Lyrae
stars in binary and multiple systems using proper motion anomalies due to
orbiting companions was demonstrated by Kervella et al. (2019b, a). Eclipsing
binaries have also been reported using light curve modeling and radial
velocity measurements (e.g. Pietrzyński et al., 2011; Pilecki et al., 2015).
However, while effective and accurate, the spectroscopic methods are limited
in the volume of binaries that can be searched for. Alternatively, there is a
photometry-based method to significantly improve the number of binaries that
can be searched. In binary systems, while the stars are orbiting around each
other, there are apparent changes in the pulsation period as a consequence of
the distance variation between Cepheid (or any regular variable) and the
observer, known as the light-travel time effect (LTTE, Irwin, 1952, 1959).
This can be noticed from the Cepheid’s pulsational variability manifesting
periodic phase shifts, and is most commonly measured using the technique of
observed minus calculated ($O-C$) diagrams (e.g., Sterken, 2005).

For more than two decades the Optical Gravitational Lensing Experiment (OGLE,
Udalski et al., 1999, 2015b) has been continuing to collect V\- and I-band
photometry of selected Galactic fields and the Magellanic Clouds (MCs). The
Cepheid catalog in both LMC and SMC has reached more than 99 % in terms of
completeness (Soszyński et al., 2008, 2010, 2017) and it covers fundamental-
mode, overtone-mode, and even multi-mode objects. This inventory provides a
valuable dataset to search for Cepheids in binary systems throughout the MCs.
Therefore, our aim is to analyze $O-C$ diagrams of the MCs’ single-mode
Cepheids (fundamental or first-overtone mode) from the OGLE-III and OGLE-IV
survey, to present a systematic investigation of Cepheids in binary system
candidates and determine their incidence rate.

We describe the data in Section 2 and present the analysis methodology in
Section 3. Thereafter we discuss the final sample of binary candidates
separately for SMC and LMC fields in Section 4. We present the distribution of
the binary parameters of the two galaxies in Section 5. Finally, we conclude
our findings in Section 6.

##  2 Data

We analyzed the publicly available photometric
data222http://www.astrouw.edu.pl/ogle/ from OGLE-III and OGLE-IV (Soszyński et
al., 2010, 2015a, 2017) which provide the ideal sample for searching for
binary candidates due to data homogeneity and a long temporal baseline. The
span of observations for OGLE-III, which also contains OGLE-II data, is
1997-2009 and for OGLE-IV is 2010-present, hereby providing more than 20 years
of data. The data are collected using the 1.3-m Warsaw telescope at the Las
Campanas Observatory, Chile.

The database contains photometry in the Johnson-Kron-Cousins V and I bands,
however, we utilized the I-band data only since it is more densely sampled
than the V-band data. The OGLE data is highly suitable for searching binary
candidates with variable stars, as shown in several works (e.g., Soszyński et
al., 2009, 2011; Hajdu et al., 2015, 2021).

For the final list of our candidate binary systems, we obtained non-public
data from the recent OGLE-IV extended survey (starting from August 12, 2022).
Adding the new OGLE observations ($\sim$1.3 year) does improve the credibility
of binary model fits.

##  3 Methodology

###  3.1 Combining photometric data

Our analysis uses photometry from two phases of OGLE, with data obtained with
different cameras between them. Therefore, as a first step, we combine the
time-series of OGLE-III and OGLE-IV as follows. The I-band data is converted
from magnitudes to an arbitrary intensity scale. Then, a Fourier series is fit
individually to both phased data, and the possible mean intensity difference
is corrected. The order of the Fourier series is decided by the
$A_{k}/\sigma(A_{k})>4$ empirical criterion, with ${A_{k}}$ and
$\sigma(A_{k})$ being the amplitude and uncertainty of the $k$-th order term.
This criterion is effective to prevent an overfitting of the light curves and
typically results in higher ($\sim$10) order fits for fundamental-mode stars
and lower ($\sim$6) order fits for first-overtone mode stars. The resulting
combined homogeneous data set is then transformed back to the magnitude scale.
Following a visual inspection, we chose not to use Cepheids with significant
zero-point variations across observing seasons, as these affect the overall
light curve and are not fit for our analysis.

###  3.2 $O-C$ procedure applied to OGLE database

An important tool for measuring and quantifying period changes is the well
known observed-minus-calculated ($O-C$) diagram. A brief review of $O-C$
techniques is given in Zhou (1999) and Sterken (2005). The $O-C$ is estimated
by recording the differences between specific points in the light curve
(typically the maxima or minima) against the corresponding calculated (or
expected) times, assuming a constant period. Such a method requires a good
observing cadence around the specific points. Therefore, sampling of the OGLE
survey is not suitable for utilizing this technique. However, there is an
alternative method proposed by Hertzsprung (1919), which uses a light curve
template to measure phase shifts using the whole light curve to derive $O-C$
points. This technique has been used recently for period change studies in
classical pulsators (e.g., Hajdu et al., 2021; Rodríguez-Segovia et al., 2022)

Here we describe the procedure to compute the $O-C$ for the combined OGLE-III
and OGLE-IV data. We started by performing a Fourier series fitting of
suitable order (criteria same as in Sect. 3.1) on the combined OGLE-III+IV
data. This serves as our template to compute ”O” in the $O-C$. For ”C”, we
used the linear ephemeris with a pulsation period derived for all data using
our back-end scripts for the discrete Fourier transform (DFT) via the Kurtz
algorithm (Kurtz, 1985). When phasing the data, we use this period and the
starting observation timestamp as the phase folding reference epoch. The full
data set for each Cepheid is divided into time bins corresponding to observing
seasons by default. In order to resolve the period change effect due to
binarity, it is pertinent to increase the resolution of the $O-C$ curve.
Therefore, we further divided the seasons into finer time bins if the number
of data points in each season exceeds a threshold of a minimum of 10 data
points or the time span of the bin is larger than 160 d. The individual phase-
folded segments were then compared with the template, and $O-C$ points were
derived as a phase shift by minimizing the scatter between the template and
light curve folded within each time bin, using a non-linear least-squares
technique. This provided us with 7212 $O-C$ diagrams covering both SMC and LMC
samples.

Using linear and parabolic fits we used the Akaike information criterion (AIC)
(Akaike, 1974), Bayesian information criterion (BIC) (Schwarz, 1978) and chi-
square statistics on the data to determine goodness of fit along with
analyzing the resultant residuals (Anderson-Darling and Ljung-Box tests,
Anderson & Darling, 1952; Ljung & Box, 1978, respectively) to filter the
sample into three categories: (1) flat $O-C$, (2) parabolic trend and (3)
variation of a third kind (neither flat nor parabolic). This third category
sample was semi-automatically divided further into two categories (3a)
periodic $O-C$ trend and (3b) irregular shape $O-C$ trend. After visual
confirmation of periodic structures in the $O-C$, we identified relevant
candidates which were later fitted with a binary model to confirm good
candidates (described in the following section). Full details of the
classification method along with a sample of irregular period change
candidates (performed as a parallel analysis) will be presented in a separate
paper (Rathour et al., in prep.).

###  3.3 Modeling the Light-Travel Time Effect

Below, we describe the analysis of the Light-Travel Time Effect (LTTE),
applied to our Cepheid inventory. A phase modulation of the light curve that
is periodic in nature, is a signature of the LTTE. This is also reflected in
the $O-C$ diagram as a repeating structure. Due to secular evolution, the
linear period change effect, which manifests as a parabolic trend, is also
overlaid on top of the LTTE. Hence, we use a composite LTTE and quadratic
model presented below:

$z(t)=\left[a\sin
i\frac{1-e^{2}}{1+e\cos(\nu)}\sin(\nu+\omega)\right]+\left[c_{0}+c_{1}t+c_{2}t^{2}\right]\,.$
(1)

The first term is the LTTE model (Irwin, 1952) with $a\sin i$ as projected
semi-major axis of the orbit (where $i$ is the inclination), $e$ denotes
eccentricity of the orbit, $\nu$ is the true anomaly and $\omega$ denotes the
argument of the periastron. The second term represents the quadratic function
of time to model the linear period change with $c_{0}$, $c_{1}$, and $c_{2}$
being the coefficients. Here, $c_{0}$ and $c_{1}$ are the correction terms in
reference epoch and pulsation period whereas the linear period-change rate
(PCR) is calculated using the $c_{2}$ coefficient and pulsation period
$P_{\mathrm{pulsation}}$ via the equation

$\displaystyle\mathrm{PCR}=2\cdot P_{\mathrm{pulsation}}\cdot c_{2}\,.$ (2)

The true anomaly, $\nu$, and the mean anomaly, $M$, are calculated from the
following relations:

$\displaystyle\cos\nu=\frac{\cos E-e}{1-e\cos E},$ (3)

$\displaystyle M=E-e\sin
E=\frac{2\pi}{P_{\mathrm{orbital}}}\left(t-T_{0}\right),$ (4)

where $E$, $P_{\mathrm{orbital}}$ and $T_{0}$ represent eccentric anomaly,
orbital period and time of periastron passage, respectively.

Our procedure, similarly to Hajdu et al. (2021), is iterative. After we fit
eq. (1) to the obtained $O-C$ diagram, we subtracted the LTTE+quadratic effect
from the data, which refines the phased light curve. This less scattered light
curve was employed again to obtain a new template (typically resulting in a
Fourier series of higher order) as well as the new pulsation period. Then we
repeated the $O-C$ calculation procedure a second time using the new template.
For estimating the uncertainties on the improved $O-C$ points, we used a
bootstrapping method. We drew random samples from individual time bins with
replacement, which were used to obtain $O-C$ points with a total of 500
iterations. The standard deviation of the resulting $O-C$ with respect to the
original $O-C$ value of the respective time bin was treated as the uncertainty
on that $O-C$ value.

Finally, the iterative procedure results in an $O-C$ diagram for a Cepheid,
which was then fitted again with the binary model giving the final orbital
parameters [$P_{\mathrm{orbital}}$, $T_{0}$, $e$, $\omega$, $a\sin i$]. Using
these parameters we calculated the derived parameters, namely the semi-
amplitude of the expected radial velocity variation, $K$, and a mass function,
$f(m)$, using the relations:

$\displaystyle K=\frac{2\pi a\sin i}{P_{\mathrm{orbital}}\sqrt{1-e^{2}}},$ (5)

$\displaystyle
f(m)=\frac{a^{3}\sin^{3}i}{P_{\mathrm{orbital}}\sqrt{1-e^{2}}}.$ (6)

We adopted a Bayesian formalism in calculating the binary parameters and their
errors using the EMCEE package (Foreman-Mackey et al., 2013). In cases with
$O-C$ points having large errors resulting in larger errors on eccentricities,
the procedure may return unreasonable $\omega$ values. To avoid such a
situation, in the MCMC procedure, instead of $\omega$ and $e$ fitted as
independent parameters, we used their Cartesian equivalents,
$\sqrt{e}\sin\omega$ and $\sqrt{e}\cos\omega$. After the MCMC calculations, we
recover back the eccentricity and argument of periastron values (for details,
see Hajdu et al., 2021). We used the default ensemble sampler provided within
the EMCEE package and adopted the well tested sampling parameters given in
Hajdu et al. (2021), which were used for a similar study aimed at RR Lyrae
stars using the OGLE data. Relying upon their convergence and autocorrelation
lengths computations, we used 200 walkers, 31000 steps, burn-in of 1000 steps
and thinning by 300 steps. For a few dozen candidates, we had to rely on
manual adjustment, for example (i) where we needed to inspect the seasonal
light curve corresponding to a deviant $O-C$ point, (ii) in cases where we
needed to increase and decrease the adopted period to refine the $O-C$ diagram
in order to make it more symmetric, and (iii) in cases where we needed to
limit priors of some of the binary parameters to improve their posterior
distribution.

Table 1: Sample of MC Cepheids with distribution with respect to pulsation mode and final binary candidates. Field | Pulsation | Stars with data in | Stars with data in | Stars with data in | Binary candidates
---|---|---|---|---|---
| mode | OGLE-III | OGLE-IV | both OGLE-III+IV |
SMC | F | 2626 | 2739 | 2582 | 85
| 1O | 1644 | 1790 | 1617 | 60
LMC | F | 1818 | 2430 | 1801 | 30
| 1O | 1238 | 1769 | 1212 | 22
Total |  | 7326 | 8728 | 7212 | 197

Figure 1: Examples of LMC fundamental-mode Cepheid binary candidate $O-C$
diagrams over-plotted with their MCMC binary fit solutions (left). The blue
dashed line represents the orbital solution obtained from the median parameter
values of the posterior distribution. The purple-shaded regions imply one,
two, and three standard deviations of the resulting MCMC fit in increasing
transparency order. Above each panel the OGLE-ID, adopted pulsation period to
construct the $O-C$ diagram and the orbital period are shown. Same as above
but for SMC fundamental-mode Cepheids (right). The remaining sample is
presented in the appendix.

Figure 2: Examples of LMC first-overtone mode Cepheid binary candidate $O-C$
diagrams over-plotted with their MCMC binary fit solutions (left). The blue
dashed line represents the orbital solution obtained from the median parameter
values of the posterior distribution. The purple-shaded regions imply one,
two, and three standard deviations of the resulting MCMC fit in increasing
transparency order. Above each panel the OGLE-ID, adopted pulsation period to
construct the $O-C$ diagram and the orbital period are shown. Same as above
but for SMC first-overtone mode Cepheids (right). The remaining sample is
presented in the appendix.

##  4 Results

From the possible candidate inventory of periodic variation, we ended up with
197 Cepheids (145 in SMC and 52 in LMC), for which we carefully inspect their
resulting model fits, binary parameters and posterior distributions. To reach
the above-mentioned final sample, many candidates were rejected if any of the
following criteria were met: (i) strong amplitude changes in the raw
photometry time series, (ii) a large gap (2-3 seasons) in the photometric
data, (iii) an $O-C$ diagram that is shorter than one full cycle, thus
preventing the periodic nature from being well established, (iv) significant
$O-C$ point deviation from the resultant fit (more than ten points beyond
3-sigma in the residual), or (v) nonphysical or skewed distribution of fitted
binary parameters in the posterior distribution.

The case of a significant number of $O-C$ points deviating from periodic fit,
as mentioned above, but classified as periodic, was more frequent in overtone-
mode Cepheids in both the LMC and SMC. One likely nuisance factor could be the
presence of non-linear period changes which can distort the original $O-C$
shape leading to misinterpretation. This is supported by earlier studies
(Berdnikov et al., 1997; Poleski, 2008; Evans et al., 2015) that indicated
first-overtone mode pulsation to be less stable and likely to show
irregularities in $O-C$ diagrams.

The final sample consists of both strong candidates and marginal ones, and
they are listed with their binary parameters in Tables 3 and 4 for the LMC and
SMC Cepheids, respectively. Candidates labeled as strong all have less than
two % uncertainty on the derived orbital period, an $O-C$ covering at least
one full orbital cycle, and normal posterior distribution of all the fitted
parameters. Marginal candidates do not satisfy either of these conditions, for
instance: having a higher uncertainty on the orbital period, not enough
coverage of the orbital cycle (mainly in longer orbital period binaries), or
at least one unconstrained binary parameter in the posterior distribution or
multiple $O-C$ points (more than 5) beyond the 3-sigma binary fit. In Figs. 1
and 2 we present the gallery of a few exemplary $O-C$ diagrams from both
galaxies for fundamental-mode and first-overtone candidates, respectively. The
remaining $O-C$ diagrams for the candidates are provided in the appendix Figs.
11, 16, 14 and 24. Each panel in the figure provides an $O-C$ diagram for the
candidate with its OGLE ID, pulsation period (used to construct the $O-C$) and
orbital period.

###  4.1 Distribution of binary parameters

The distribution of the obtained binary parameters for the full sample is
shown in Figs. 3 and 4 for fundamental-mode and first-overtone mode Cepheids,
respectively.

####  4.1.1 Fundamental-mode Cepheids

Firstly we discuss fundamental-mode candidates in both LMC and SMC fields. Due
to constraints on the total data span, our analysis will miss possible long
orbital period ($P_{\mathrm{orbital}}>9500$ d) binary candidates, forming the
upper limit of detection. In the SMC, the orbital periods for fundamental-mode
Cepheid binary candidates range between $500-7500$ d, with a maximum at around
$2000$ d, from where the distribution falls off toward longer periods. In the
LMC, they are distributed in the range of $500-4500$ d, with a wide and flat
maximum at around $3500$ d.

The eccentricity distribution for the SMC candidates shows multiple peaks with
the vast majority of the stars having a value less than 0.7, whereas the LMC
candidates show marginal peaks at around $0.15$ and $0.35$. The statistics are
too low, however, to assign physical significance to these peaks. It is
important to note that the LMC eccentricity distribution is roughly
concentrated between $0.1-0.7$, whereas in the SMC, eccentricities go as high
as $0.9$ with roughly half a dozen candidates with high eccentricities ($\geq
0.8$). We note that the SMC sample is about three times larger, so the LMC
distribution might not be large enough to draw conclusions.

The logarithmic distribution of mass function shows asymmetrical behavior
around the peak value in both SMC and LMC fundamental-mode Cepheids. For the
SMC, the distribution roughly peaks at $-1.2$ whereas for the LMC the
distribution peaks at a slightly higher value of $-0.3$. This may imply that
high-mass binaries in more metal-rich environments result in higher initial
mass ratios $q$. Moving towards higher mass function values, away from the
peak, the SMC distribution decreases gradually, but for the LMC it falls off
sharply, implying a narrower range of total masses for Cepheid binaries for
fundamental-mode candidates.

The lower panels of Fig. 3 show the distribution of three of the binary
parameters: projected semi-major axis, eccentricity and logarithm of the mass
function as a function of the orbital period. The majority of projected semi-
major axis ($a\sin i$) values are within $6$ AU. Only four candidates (three
of them are marginal) extend beyond this limit, and all of them are from the
SMC fundamental-mode sample. A weak relation of increasing $a\sin i$ value
with increasing orbital period can be seen, at least for the LMC sample. The
eccentricity values are uniformly distributed till $3500$ d, and for longer
orbital periods cases with $e>0.5$ are rare in both Magellanic Clouds.
Similarly, the logarithm of mass function is well distributed across the
orbital periods with no particular clustering of points or trends in either
the LMC and the SMC.

####  4.1.2 First-overtone mode Cepheids

The first-overtone Cepheid binary candidate parameters from the SMC and LMC
are shown on Fig. 4. Compared to fundamental-mode candidates, the distribution
of orbital periods extends to longer values ($\sim$8500 d). Looking at the
distribution of the orbital periods, the number of the LMC candidates is too
low to draw conclusions. In the SMC, the maximum is at $\sim$3700 d, from
where the distribution shows an asymmetrical shape, which falls off steeply at
lower orbital periods and is rather broad at the longer orbital periods.
Considering eccentricity, both galaxies have the maximum at a roughly similar
value of $\sim$0.2. A few candidates in the SMC do show much higher
eccentricities (0.8 or more), similar to the SMC fundamental-mode candidates.
The SMC distribution is broader, covering all possible eccentricity bins. In
contrast, the majority of the LMC candidates have eccentricity below 0.3.
However, as with the binary periods, the statistics for the LMC sample is too
low to draw significant conclusions. The distribution of logarithm of the mass
functions for the SMC candidates peaks at two values: $\log f(m)=0.1$ and
$0.9$. Although having fewer first-overtone Cepheids in the LMC, its
distribution also peaks at similar values. At lower values moving from the
first peak ($\log f(m)=0.1$) the distribution is broader in the SMC and quite
narrow in the LMC. However, any metallicity effect is inconclusive due to the
small number of stars in the LMC sample.

For both galaxies, we notice a trend of increasing projected semi-major axis
with increasing orbital period. In comparison to the previously discussed
fundamental-mode candidates, the first-overtone sample extends to much higher
$a\sin i$ values (up to $\sim$30 AU). The eccentricities are uniformly
distributed in the orbital period range of $2000-5500$ d for both Magellanic
Clouds. Below $\sim$2000 d, eccentricities are higher than $0.6$, whereas
above 5500 d they are typically below $0.5$. The logarithm of the mass
function shows a weak positive correlation with the orbital period in both LMC
and SMC Cepheids, implying Cepheids with longer orbits have higher total mass
of the binaries.

Figure 3: Distribution of orbital period, eccentricity and (logarithmic) mass
functions of LMC and SMC fundamental-mode binary Cepheid candidates (shown at
the top, from left to right). The dashed vertical line denotes the median
value of the distribution. Distribution of the projected semimajor axes,
eccentricities and (logarithmic) mass functions as a function of the orbital
periods (bottom, from left to right). Filled blue and red dots denote LMC and
SMC fundamental-mode strong candidates respectively. Empty symbols denote the
marginal candidates. Figure 4: Same as Fig. 3 but for first-overtone binary
Cepheids candidates.

###  4.2 Period change rate

One important property of Cepheids (alongside the binary parameters we
determine in this work) is the period change rate. In our analysis, we fit an
LTTE model along with an additional parabolic term (see Eq. 1). This latter
term results from accounting for evolutionary changes in the pulsation period.
This is calculated using Eq. 2 reported under the PCR column in Tables 3 and 4
in units d/Myr. A positive value of PCR implies a positive period change with
a Cepheid showing secular evolution redwards on the HR diagram, whereas a
negative value of PCR implies the opposite, namely, a blueward evolution
indicating negative period change. In our combined sample we have 116 Cepheids
with positive PCR and 81 Cepheids with negative PCR. The ratio of negative to
positive PCR candidates is 0.7, which is close to the 0.8 value derived from
the sample by Rodríguez-Segovia et al. (2022). This ratio is a proxy of the
slower evolution of Cepheid in the third crossing, compared to the second
crossing. Consequently, our reported PCR values are reasonably indicative of
the long-term Cepheid evolution.

###  4.3 Probing Cepheid companions

####  4.3.1 Estimating mass

We estimated the minimum mass of Cepheid companions under certain assumptions,
using the equation:

$\displaystyle
f(m)=\frac{m_{\text{c}}^{3}\sin^{3}i}{(m_{\text{cep}}+m_{\text{c}})^{2}},$ (7)

where $m_{\text{c}}$ and $m_{\text{cep}}$ are the masses of the companion and
the Cepheid, respectively. If the mass function, inclination angle, and
Cepheid mass are known, it allows us to estimate the companion mass. Since
with the $O-C$ technique, we have no information regarding the inclination
angle, for simplicity, we may assume $i=90^{\circ}$ and then we can determine
the ”minimum” mass of the companion. In a similar study for RR Lyrae stars,
Hajdu et al. (2021) assumed $m_{\text{RR}}=0.65{\,}{\rm M}_{\odot}$ as a
typical mass, since masses of RR Lyrae stars lie in a narrow range. However,
for Cepheids, the range is $3-13{\,}{\rm M}_{\odot}$ and quite large to assume
any typical value. To approximate the Cepheid mass, we use mass estimates
provided by Groenewegen & Lub (2023), who studied fundamental-mode Magellanic
Cloud Cepheids (142 in the LMC and 77 in the SMC) using spectral energy
distributions (SEDs) and several empirical relations: period-luminosity (PL),
period-radius (PR), and mass-luminosity (ML) relations. For each Cepheid, they
calculated Cepheid mass using five different literature relations (see their
appendix, Table B.1, and references therein). Finally, the adopted mass of a
Cepheid was the median of the previous five estimates. From their sample, we
used the Cepheid periods and corresponding mass estimates to construct a
period-mass (PM) relation. Since their data extends for Cepheids with periods
as high as 200 d, and periods in our sample extend only up to 10 d, we
construct the relation using Groenewegen & Lub (2023) data for pulsation
periods less than 50 d (to maintain Cepheids with statistically significant
empirical relation). The period-mass (PM) relation estimated in log–log space
is linear and is given as follows:

$\displaystyle\log(M/{\,}{\rm M}_{\odot})=(0.368\pm 0.022)+(0.352\pm
0.018)\log P.$ (8)

Equation (8) for Cepheid mass is based on an average representation of
literature relations transforming physical parameters to mass. As a
consequence, the resulting estimate of Cepheid mass (and, hence, the minimum
companion mass) is crude. Based on the dispersion of mass values for a given
Cepheid, as given in Groenewegen & Lub (2023), we conclude that Cepheid masses
derived with Eq. (8) are accurate to within $\pm 1{\,}{\rm M}_{\odot}$.

Since Groenewegen & Lub (2023) study was only limited to fundamental-mode
Cepheids, therefore, for mass estimates of our overtone Cepheid sample, the
pulsation periods are fundamentalized using the relation given by Pilecki et
al. (2021):

$\displaystyle P_{\rm F}=P_{\rm 1O}(1.418+0.115\log P_{\rm 1O}),$ (9)

where $P_{\rm F}$ and $P_{\rm 1O}$ are the fundamental-mode and first-overtone
mode periods, respectively.

Figure 5 shows the above relations applied to our Cepheid sample. It shows, in
the logarithmic space, the distribution of the estimated minimum companion
mass, using Eq. (7), and derived Cepheid mass, using the $P-M$ relation, Eq.
(8). For the SMC sample, the minimum companion masses are uniformly
distributed up to the Cepheid mass of $\sim$ 3.2 M⊙, above which they tend to
be systematically higher. For the LMC sample, there is a weak positive
correlation between the minimum companion mass and the binary Cepheid mass.

In Figure 6 we plot the distribution of both derived Cepheid mass and
calculated minimum companion mass for each pulsation mode and both Magellanic
Clouds. For this distribution, we exclude the peculiar mass function Cepheids
(detailed in section 4.6). The majority of the SMC fundamental-mode Cepheids
of our sample have masses in the $2.5-3{\,}{\rm M}_{\odot}$ range whereas in
the LMC masses higher than $3{\,}{\rm M}_{\odot}$ are observed; this is a
consequence of different pulsation period distributions in both Clouds: the
pulsation periods of SMC Cepheids are generally shorter than in the LMC. The
distributions are similar for the first-overtone mode sample. For minimum
companion masses, the majority of the SMC fundamental-mode candidates extend
up to $2{\,}{\rm M}_{\odot}$ with a tail extending up to $4{\,}{\rm
M}_{\odot}$. In the LMC, the distribution favors slightly higher minimum
companion masses ($\sim 1-3{\,}{\rm M}_{\odot}$). In the first-overtone mode
sample, the SMC candidates favor a broader range of companion masses (maximum
at $2{\,}{\rm M}_{\odot}$), extending until $\sim 7.5{\,}{\rm M}_{\odot}$. In
the LMC, the majority of the sample lies in the $\sim 3-5{\,}{\rm M}_{\odot}$
range.

Figure 5: Distribution of estimated minimum companion masses as a function of
derived Cepheid masses (see subsection 4.3). The symbols represent Cepheids
from the LMC F (blue circle), SMC F (brown diamond), LMC 1O (light blue
circle) and SMC 1O (red diamond) samples. Symbols with black edges denote
strong candidates whereas ones without denote marginal candidates. For
comparison, we plot LMC Cepheid in eclipsing binaries (black filled square):
OGLE-LMC-CEP-0227, OGLE-LMC-CEP-1718, OGLE-LMC-CEP-1812, OGLE-LMC-CEP-2532 and
OGLE-LMC-CEP-4506 (Pilecki et al., 2018). The dashed gray line denotes two
times the maximum Cepheid mass in the overall sample. Figure 6: Distribution
of derived Cepheid mass (top) for the fundamental (left) and first-overtone
(right) modes. The corresponding distributions of minimum companion masses
(bottom).

####  4.3.2 Nature of the companions

To investigate further the nature of the companions, we utilized findings from
a recent study by Pilecki et al. (2021), where the authors noted that Cepheids
in binary systems with giant companion manifest as appearing overbright when
compared to ”typical” Cepheids. A typical Cepheid is either a Cepheid with a
main-sequence companion or an isolated Cepheid, where there is negligible to
no additional contribution to actual Cepheid’s brightness, respectively. In
the study by Pilecki et al. (2021), Cepheids with systematic offset from $P-L$
relationship were investigated to be mostly Cepheids with giant companions.
Two fundamental-mode candidates from the LMC (OGLE-LMC-CEP-0837 and OGLE-LMC-
CEP-0889) and two from the SMC (OGLE-SMC-CEP-3061 and OGLE-SMC-CEP-4365), show
excess in their total observed brightness when compared to the respective
Period-Wesenheit index from Soszyński et al. (2008, 2010) as seen in Fig. 7.
These Cepheids (except OGLE-LMC-CEP-0837) appear redder in the color-magnitude
diagram (see Fig. 7 middle panels), in agreement with what was observed by
Pilecki et al. (2021) for their LMC sample. As shown previously, we can
estimate the Cepheid mass and the minimum mass of its companion. Therefore,
for these giant companions, using Eq. (7), we estimate their minimum masses
along with their primary Cepheid represented as $m_{\text{cep}}+m_{\text{c}}$,
to be:

$\displaystyle\text{OGLE-LMC-CEP-0837:}\ 4.3{\,}{\rm M}_{\odot}+3.2{\,}{\rm
M}_{\odot}$ $\displaystyle\text{OGLE-LMC-CEP-0889:}\ 3.5{\,}{\rm
M}_{\odot}+6.8{\,}{\rm M}_{\odot}$ $\displaystyle\text{OGLE-SMC-CEP-3061:}\
3.2{\,}{\rm M}_{\odot}+2.8{\,}{\rm M}_{\odot}$ $\displaystyle\text{OGLE-SMC-
CEP-4365:}\ 2.7{\,}{\rm M}_{\odot}+4.5{\,}{\rm M}_{\odot}$

We note that these are not direct mass measurements and are dependent on the
adopted empirical relation from Eq. (8) and derived parameters from the $O-C$
analysis. Our two LMC candidates were already reported as double-lined
spectroscopic binaries (SB2 type) by Pilecki et al. (2021). We corroborate
their findings regarding their binary nature via our independent, photometry-
based method (LTTE analysis). The other two SMC candidates were not reported
earlier. These targets could be observed spectroscopically to confirm if they
show characteristics of double-lined binaries (SB2 type) in the composite
spectra. This will eventually corroborate the binarity nature and aid in
refining their orbital parameters.

Figure 7: Cepheid Period-Luminosity ($P-L$) relation (top-left) using the
reddening-free Wesenheit index for fundamental-mode (blue diamond symbol) and
first-overtone mode (light blue diamond symbol) Cepheids from the LMC sample.
The black solid line depicts the adopted Period-Wesenheit relation for the LMC
whereas the black dashed lines indicate if Cepheids are 50% and 100% brighter
than the P-L relation respectively. Color-magnitude diagram for the LMC
(middle), where the solid red line denotes the color of a typical Cepheid,
while the dashed red line denotes the total brightness of a Cepheid if it had
a companion of similar color and luminosity. Distribution of I-band amplitude
as a function of the logarithm of the pulsation period (bottom-left). The
right three panels are the same as above but for the SMC. For both LMC and SMC
fields, the candidate overbright Cepheids are marked with an arrow. Note:
Filled symbols denote strong candidates, whereas empty symbols denote marginal
candidates. The background green and gray points denote fundamental and first-
overtone mode Cepheids from the full OGLE database respectively for LMC
Soszyński et al. (2008) and SMC Soszyński et al. (2010). The first-overtone
candidates are ”fundamentalized” using Eq. 9.

###  4.4 Low-amplitude variability or additional radial mode

Cepheids may show low-amplitude variability such as the presence of non-radial
modes (mainly in first-overtone stars, Moskalik & Kolaczkowski, 2008; Moskalik
& Kołaczkowski, 2009; Soszyński et al., 2008, 2010; Rathour et al., 2021;
Plachy et al., 2021; Smolec et al., 2023) and periodic modulation of the
pulsation amplitude and/or phase (Moskalik & Kołaczkowski, 2009; Soszyński et
al., 2015b; Smolec, 2017; Rathour et al., 2021). The latter is very common in
another type of classical pulsators, RR Lyrae stars, and is known as the
Blazhko effect (Blazko, 1907), as also discussed, for example, in Kovacs
(2016); Smolec (2016); Netzel (2023). Such phenomena may be present in our
sample and, particularly the possible periodic modulation of pulsation, may
alter the light curve and, eventually, the LTTE-determined orbital parameters
for our binary candidates.

####  4.4.1 Modulation and non-radial mode verification

Smolec (2017) searched through the OGLE Magellanic Cloud sample of
fundamental-mode Cepheids to look for periodic modulation of the pulsation and
compiled a list of 51 modulated Cepheids, 29 from the SMC and 22 from the LMC.
Another recent study by Smolec et al. (2023) reported periodic modulation in
the LMC and SMC first-overtone OGLE samples. They reported 24 such candidates
in the LMC and only 3 in the SMC. We cross-matched both our fundamental-mode
and first-overtone samples and found no overlap with the above two reported
studies, which was expected since the modulation periods ($\sim$10-100 d) are
much shorter than the ones captured by LTTE analysis. Hence, our Cepheid
binary candidates are free from any likely low amplitude periodic modulation
of pulsation.

We also check if our sample contains Cepheids with non-radial modes reported.
We find an overlap in five SMC Cepheids (OGLE-SMC-CEP-1611, OGLE-SMC-CEP-1615,
OGLE-SMC-CEP-2033, OGLE-SMC-CEP-2293 and OGLE-SMC-CEP-3097) and two LMC
Cepheids (OGLE-LMC-CEP-0764 and OGLE-LMC-CEP-0839) which show periodicities
with period ratios of $P_{\rm x}/P_{\rm 1O}\in(0.60,\,0.65)$ co-existing with
the radial first-overtone mode. Here, $P_{\rm x}$ denotes the period of
additional variability. In certain cases, instead of the above-mentioned
period-ratio range, there can be additional periodicity at the sub-harmonic of
$\nu_{x}$ (i.e. 1/$P_{\rm x}$), which is the direct detection of the non-
radial mode according to the theoretical framework proposed by Dziembowski
(2016). After cross-matching our first-overtone sample with a list of such
candidates (Smolec et al., 2023), we find an overlap of one SMC candidate
(OGLE-SMC-CEP-3097) and two LMC Cepheids (OGLE-LMC-CEP-0764 and OGLE-LMC-
CEP-0839) from our list. As a sanity check, namely, to inspect whether the low
amplitude periodicities may affect the binary fits, we took the marginal LTTE
candidate OGLE-LMC-CEP-0839 and performed a frequency analysis. Thus, we could
identify the low-amplitude variability to eventually pre-whiten it from the
frequency spectra. Then, we used the prewhitened data, with the additional
variability filtered out, to obtain the $O-C$ diagram and re-estimate the
binary fit and orbital parameters. Our analysis resulted in marginal
differences in the final orbital parameters (well within the errors) implying
that the low-amplitude variability indeed does not affect the LTTE signature
in our sample.

####  4.4.2 Additional radial mode verification

Apart from the above-mentioned low-amplitude variability, the light curves can
be distorted in the presence of another radial mode of pulsation. Therefore,
as a check and to scrutinize our candidate binary sample, we check whether
these are genuine single-mode Cepheids. Smolec et al. (2023) provide a list of
first-overtone Cepheids that likely have a presence of another radial mode of
pulsation. We find an overlap with two Cepheids from the LMC only (OGLE-LMC-
CEP-0839 and OGLE-LMC-CEP-1134). Both stars are candidate F+1O double-mode
pulsators. This concludes that the majority of our samples are genuine single-
mode Cepheids. OGLE-LMC-CEP-0839, as mentioned previously, also contains
additional non-radial variability, whereas OGLE-LMC-CEP-1134 is classified as
strong LTTE candidate in our first-overtone sample, so unlikely to be affected
by the presence of the additional radial mode. We tested this hypothesis by
pre-whitening the fundamental-mode signal in the frequency spectra and
performing the $O-C$ analysis again to observe only a marginal difference in
the fitted orbital parameters (within the errors). For OGLE-LMC-CEP-0839, we
have already checked the negligible contribution of both the additional low-
amplitude variability and the radial mode in LTTE analysis. In Tables 3 and 4,
Cepheids are listed with their binary parameters, whereas the candidates with
additional low amplitude variability are marked with ${\ddagger}$ symbol,
candidates with a sub-harmonic signal are marked with $\dagger$ symbol and
candidates with additional radial mode are marked with $\ast$.

###  4.5 Previously detected Cepheid binary candidates

We compared the list of candidates in our study with earlier detections
reported in the literature. In the LMC, OGLE discovered six Cepheids in
eclipsing binary systems (OGLE-LMC-CEP-0227, OGLE-LMC-CEP-1718, OGLE-LMC-
CEP-1812, OGLE-LMC-CEP-2532, OGLE-LMC-CEP-3782, and OGLE-LMC-CEP-4506 (Udalski
et al., 1999; Soszyński et al., 2008; Udalski et al., 2015a; Soszyński et al.,
2012). Then, OGLE-LMC-CEP-3782 and OGLE-LMC-CEP-4506 were not included in our
original sample, since they lacked OGLE-III data (see our sample selection
criteria in Sect. 3.1), but we also went on to analyse the available data for
these stars.

In the automatic analysis of eclipsing binary stars, the $O-C$ reveals no
strong periodic feature. This is due to the template getting distorted due to
the eclipse feature getting incorporated into the phased light curve.
Therefore, we manually re-analyzed $O-C$ diagrams of these candidates and
detected LTTE in three candidates, OGLE-LMC-CEP-1812, OGLE-LMC-CEP-2532 and
OGLE-LMC-CEP-4506. We report the orbital periods for these candidates:
554$\pm$1 d, 788$\pm$4 d, and 1671$\pm$176 d, which are consistent with the
ones derived from the eclipse timing: 551.776$\pm$0.003 d, 800.419$\pm$0.009
d, and 1550.354$\pm$0.009 d, respectively (Pilecki et al., 2018). For the
remaining three candidates: OGLE-LMC-CEP-0227, OGLE-LMC-CEP-1718 and OGLE-LMC-
CEP-3782, we still do not see any LTTE. The reason will be explained at the
end of this section, along with similar SMC candidates.

From the SB2 candidate sample compiled by Pilecki et al. (2021), we could
recover only two targets in our sample, OGLE-LMC-CEP-0837 and OGLE-LMC-
CEP-0889. OGLE-LMC-CEP-0889 is not a spectroscopically confirmed SB2-type
binary, but it does show an LTTE signature in the $O-C$. The orbital period we
derived is $2569\pm 12$ d which is close to the value of $2533$ d derived by
Pilecki et al. (2021). However, their mass function estimate
($f(m)=1.83{\,}{\rm M}_{\odot}$) is lower than ours ($f(m)=2.95\pm
0.33{\,}{\rm M}_{\odot}$). Another fundamental-mode candidate from Pilecki et
al. (2021), OGLE-LMC-CEP-0837, is a spectroscopically confirmed SB2-type
binary (see their tab. 3) and is also among our strong $O-C$ candidates. We
derived an orbital period of $3637\pm 62$ d with an eccentricity of $0.2\pm
0.1$. From our empirical relations in Sect. 4.3, we estimated the mass of the
Cepheid and its companion to be $4.3{\,}{\rm M}_{\odot}$ and $3.2{\,}{\rm
M}_{\odot}$, respectively.

In the SMC, the OGLE catalog contains five detections of classical Cepheids in
eclipsing binary systems (OGLE-SMC-CEP-0411, OGLE-SMC-CEP-1996, OGLE-SMC-
CEP-2199, OGLE-SMC-CEP-3235, and OGLE-SMC-CEP-4795; Soszyński et al., 2010;
Udalski et al., 2015a). It is important to note that none of the above SMC
eclipsing binary Cepheids have been studied yet with radial velocity
observations to obtain precise orbital parameters. We analyzed these stars
manually, including OGLE-SMC-CEP-4795, for which only OGLE-IV data are
available. After the manual analysis, we still did not see any LTTE in OGLE-
SMC-CEP-1996 and OGLE-SMC-CEP-4795 that could corroborate the orbital periods
reported from the analysis of the light curves. In OGLE-SMC-CEP-2199, we
observed a LTTE signature corresponding to 1204$\pm$31 d, which is in good
agreement with the orbital period of 1210 d estimated from photometric data.
Lastly, OGLE-SMC-CEP-3235 is particularly interesting, as it not only shows a
clear signature of LTTE in the $O-C$ diagram but also a single eclipse in
$\sim$20-year OGLE baseline data, further validating its binarity nature. Our
analysis recovered this first-overtone candidate with an orbital period of
$7528\pm 429$ d, with other binary parameters presented in Table 4. Udalski et
al. (2015a) estimated the orbital period from the photometric data to be
$\sim$4200 d, which is discrepant from our value. This difference in the
orbital period parameter has two reasons. First, our analysis uses data
extended beyond 2015. Second, in Udalski et al. (2015a), their analysis was
limited to fitting a sinusoidal fit to the $O-C$ data assuming a circular
orbit ($e=0$) whereas we fit full binary model (including a period change term
in Eq.1) within a Bayesian framework.

In principle, all eclipsing binaries should show LTTE. However, to reveal LTTE
in the $O-C$ diagram, the orbital period should be relatively long (and
correspondingly the orbit should be sufficiently large) so that the amplitude
of the $O-C$ variation is above the detection limit and the relatively coarse
photometric data allow us to resolve the orbital variation. A total of six
reported eclipsing binaries (three in each MC) from the literature did not
show LTTEs when subject to analysis. A closer inspection reveals that their
orbital periods are short, from 43 d for OGLE-SMC-CEP-0411 to 412 d for OGLE-
LMC-CEP-1718. For such orbital periods, the amplitude of the $O-C$ produced by
the LTTE clearly falls below the detection limit.

It is important to note that the majority of known Magellanic Cloud Cepheid
eclipsing binaries have orbital periods less than 800 d (8 out of 11 Cepheid).
While these Cepheid binaries are valuable, searching for them and obtaining
precise orbital parameters via radial velocity follow-ups and long-term
monitoring is time-consuming and costly. Therefore, our sample is
complementary to the known Magellanic Cloud Cepheid eclipsing binaries, as it
extends from orbital periods starting above 600 days up to $\sim$9000 days.
This shows that due to the longer temporal baseline of photometric
observations, the $O-C$ technique is valuable for searching Cepheid binaries
in the parameter space that is unattainable by other methods.

###  4.6 Peculiar high-mass-function candidates

Figure 8: Examples of $O-C$ diagram for peculiar high-mass-function
candidates. The remaining sample is presented in the appendix.

Within our binary sample, an inspection of Tables 3 and 4 reveals that for
several candidates, the mass function is significantly larger than it is for
the rest of the sample.

In Sect. 4.3, we describe how we estimated Cepheid mass from empirical period-
mass relation for all candidates. Candidates for which mass function is twice
as large as Cepheid’s mass, namely, $f(m)\geq 2\times m_{\mathrm{cep}}$, are
classified as high-mass-function candidates. Using this criterion, there are
21 such Cepheids (LMC F mode: 1, LMC 1O mode: 6, SMC F mode: 2 and SMC 1O
mode: 12). These candidates are marked with $\diamondsuit$ symbol in Tables 3
and 4. Their $O-C$ diagrams are compiled in Fig. 8 and Fig. 29.

Out of the 21 candidates, when we calculate the minimum companion masses, the
lowest mass is $8.7{\,}{\rm M}_{\odot}$. Eleven targets have minimum companion
masses in the $10-20{\,}{\rm M}_{\odot}$ range, six in the $20-30{\,}{\rm
M}_{\odot}$ range, one in the $50-100{\,}{\rm M}_{\odot}$ range, and two have
even higher masses. These two extreme cases are OGLE-SMC-CEP-1267 and OGLE-
LMC-CEP-3095 with derived minimum companion masses of $133.5{\,}{\rm
M}_{\odot}$ and $226.8{\,}{\rm M}_{\odot}$, respectively. We recall that these
are estimates and depend on an uncertain estimate of Cepheid’s mass. These
high values are difficult to explain in terms of a single Cepheid companion.
Interestingly, the well-confirmed first-overtone mode Cepheid binary candidate
OGLE-SMC-CEP-3235, discussed in the previous section, which shows eclipse as
well as periodic $O-C$ variation (Udalski et al., 2015a), is also one of the
candidates in the peculiar high-mass-function category ($f(m)=12.32\pm
1.43{\,}{\rm M}_{\odot}$). This is one of the arguments in favor of not
discarding all of these candidates straight away. We discuss possible
explanations for these candidates below.

Firstly, stars can exist in more complex configurations than just a binary,
for example, a hierarchical triple system with a tight massive inner binary
and the third component being the Cepheid on a wide orbit. Theoretical
population study by Dinnbier et al. (2022) validates that the fraction of
Cepheids in multiple systems in significant. In principle, a companion may
also be a black hole. (e.g. El-Badry et al., 2023). However, with the
currently known mass distribution of black hole binaries (e.g. Bavera et al.,
2023) it is not possible to reconcile the most extreme cases with the largest
estimated minimum companion mass.

Secondly, the periodic $O-C$ may also arise due to a periodic modulation of
pulsation phase intrinsic to Cepheid, such as the Blazhko effect in RR Lyrae
stars, but with pure phase modulation.

Thirdly, the $O-C$ diagrams may also be affected by the other category of non-
evolutionary period changes. These are typically irregular, but we cannot
exclude that on a shorter timescale, these changes may mimic periodic
variations. Such changes may be the sole explanation for the observed $O-C$,
and, consequently, a peculiar orbital solution, but may also be superimposed
on a genuinely periodic $O-C$ due to LTTE in a binary system, distorting the
determined orbital solutions and affecting the mass function. Some examples of
candidates from our sample that may be affected by these non-linear period
changes are OGLE-LMC-CEP-0889, OGLE-LMC-CEP-1454, OGLE-LMC-CEP-3095, OGLE-SMC-
CEP-0340, and OGLE-SMC-CEP-3955 (see their $O-C$ diagrams). The proposed
mechanisms for non-evolutionary period changes in classical pulsators are
magnetic field (Stothers, 1980), short-term variations in the atmospheric
structure (Deasy & Wayman, 1985), changes in helium abundance gradients (Cox,
1998), and semiconvection (Sweigart & Renzini, 1979). For Cepheids with longer
pulsation periods, convection could play a significant role in the unstable
behavior of the period (e.g. Neilson & Ignace, 2014).

The third explanation seems most probable. This is because first-overtone
Cepheids are more prone to non-evolutionary period changes (e.g. Berdnikov et
al., 1997; Poleski, 2008; Evans et al., 2015). In our high-mass-function
sample, most of the candidates (18 out of 21), the Cepheid is indeed the
first-overtone pulsator. The incidence rate and characteristics of irregular
period changes (in particular, the dependence on pulsation mode) will be
discussed in the next paper of the series (Rathour et al., in prep.). In that
work, we will also systematically search for irregular period changes in
Magellanic Cloud Cepheids.

Radial velocity observations will be the most straightforward way to confirm
whether the observed variation in the $O-C$ is indeed due to LTTE in binary
system. The longer time base of observations may allow us to more firmly
conclude about the nature of $O-C$ variation in high-mass-function candidates.

##  5 Discussion

###  5.1 Incidence rates

We calculate the incidence rates of binary Cepheids, with respect to the
working sample of Cepheids (mentioned in Table 1 column under stars with data
in both OGLE-III+IV). Starting with the LMC, our binary candidate incidence
rate for fundamental-mode Cepheids is $1.7\pm 0.3$ % and nearly the same for
first-overtone Cepheids, namely, $1.8\pm 0.4$ %. In the SMC the incidence rate
is higher, $3.3\pm 0.4$ % for fundamental-mode Cepheids and $3.7\pm 0.5$ % for
first-overtone Cepheids. In a given galaxy, first-overtone Cepheids have a
similar binarity detection fraction within the errors as compared to
fundamental-mode Cepheids. However, when compared between the two galaxies,
SMC incidence rates are about twice as high as the LMC ones. These incidence
rates are clearly much lower than estimates for Galactic Cepheids, which are
well above 50 percent (Evans, 1992; Szabados, 2003a; Evans et al., 2013;
Kervella et al., 2019b).

There are multiple factors affecting the number of detected binary candidates.
(a) The sample we analyze contains Cepheids with both OGLE-III and OGLE-IV
data. In total, we discarded 1630 Cepheids (SMC F mode: 201; SMC 1O mode: 200;
LMC F mode: 646; LMC 1O mode: 583) which had data in only one of the OGLE
phases. It is likely we may miss some binary candidates just based on the
sample quality cut. (b) The total time base including OGLE-III, OGLE-IV and
OGLE-IV extended data is $\sim$10,000 d. Any Cepheid binary candidate with an
orbital period nearby or greater than this limit cannot be detected. Further
observations of the OGLE survey in the future are necessary to recover those
candidates. (c) Similarly, there is a detection limit on the lower end of
orbital periods (below $\sim$600 d), as we discussed in Section 4.5 while
searching for LTTE in eclipsing binary candidates. We could only establish
LTTE for Cepheids with orbital periods longer than $\sim$600 d. Hence, we
would miss an important fraction of Cepheid binaries with orbital period less
than this limit. (d) Irregular, non-evolutionary period changes may affect the
$O-C$ diagrams (see Sect. 4.6). This is also supported by Pilecki et al.
(2021), with some of their Cepheid $O-C$ diagrams showing quasiperiodic cyclic
variation (marked as QPOC in that work). In extreme cases, real binary
Cepheids will have a distorted $O-C$ diagram appearing semi-periodic or not
periodic at all. (e) Finally, detectability is influenced by the inclination
angle. The amplitude of LTTE signal measured from $O-C$ diagram is dependent
on the inclination of the binary. In an ideal scenario at $90^{\circ}$ (edge-
on configuration), the $O-C$ will have maximum amplitude and on the other hand
at $0^{\circ}$ angle (face-on configuration) of the binary with respect to the
observer, the $O-C$ will have no LTTE. Hence, assuming all possible
inclination angles, the incidence rate of detected binaries will definitely be
affected due to the ones missed in low inclination configuration binaries.

###  5.2 Comparison with binary population synthesis models

In order to understand the two-times higher incidence rate of binary Cepheid
candidates in the SMC as compared to the LMC, we resorted to the synthetic
populations of binary Cepheids (Karczmarek et al., 2022). These populations
consist of only fundamental-mode Cepheids in binary systems (i.e., the
incidence rate of binary Cepheids is 100 percent) and show systematic
differences in the distributions of luminosities, masses, pulsation periods,
and so on, between the variants of different metallicities. We used these
differences in an attempt to interpret the two-times higher incidence rate of
binary candidates with fundamental-mode Cepheids in the SMC than in the LMC.

Firstly, the theory of stellar evolution predicts that the Cepheid blue loops
are more pronounced in metal-poorer stars of a given mass than in their metal-
richer counterparts (see, e.g., Fig. 1 from Anderson et al., 2016). This
results in the SMC Cepheids crossing the instability strip (IS) during the
blue loop at systematically lower masses, and therefore lower luminosities,
than the LMC Cepheids. Indeed, synthetic SMC Cepheids (i.e. coming from the
synthetic population, which reflects the average metallicity of the SMC
Cepheids, $Z=0.004$) have masses starting from $3.3{\,}{\rm M}_{\odot}$ and a
median mass of $3.7{\,}{\rm M}_{\odot}$, while synthetic LMC Cepheids
($Z=0.008$) have minimum and median masses of $3.8{\,}{\rm M}_{\odot}$ and
$5.0{\,}{\rm M}_{\odot}$, respectively. The initial mass function (Kroupa &
Weidner, 2003), which was used to create both synthetic populations of LMC and
SMC Cepheids, explains that low-mass stars are more abundant that massive
ones; therefore, we can conclude that the SMC Cepheids should indeed be more
numerous than the LMC Cepheids because they have systematically lower masses.

We notice that the aforementioned synthetic masses are systematically higher
than the ones calculated for our sample. This discrepancy is clearly shown in
Figs. 9 and 10 (bottom panels). Models of stellar evolution used in the binary
population synthesis calculations fail to render Cepheid blue loops for lower-
mass stars (below 3.3 and 3.8 ${\,}{\rm M}_{\odot}$ for the SMC and LMC,
respectively), even though Cepheids of even lower masses (derived from Eq. 8)
are evident in the observed data. The relative difference between the median
mass of the synthetic and observed samples from the bottom panels of Figs. 9
and 10 is 26% and 31% for the SMC and LMC, respectively, which is consistent
with the mass discrepancy phenomenon, existing between the mass derived from
evolutionary models and other methods. Interestingly enough, the synthetic
Cepheids from SMC and LMC alike follow the Period-Mass relation in Eq. (8),
only starting at larger masses and pulsation periods, as shown in Figs. 9 and
10 (middle panels).

We also compared the time that synthetic Cepheids spend inside the instability
strip (IS) in order to check the hypothesis that low-mass Cepheids spend more
time inside the IS, because they evolve slower than the more massive ones. The
median time that synthetic SMC Cepheids spend inside the IS is 4 and 6.3 Myr
for the second and third IS crossing, respectively, whereas for the synthetic
LMC Cepheids these times are 3.9 and 6 Myr. We regard these differences
between crossing times for the SMC and LMC Cepheids as insignificant and
insufficient to explain the twice higher incidence rate of SMC to LMC binary
Cepheid candidates.

Another possibility might be different star formation rates (SFR) in the LMC
and SMC. For example, a significantly larger SFR in the SMC about 200 Myr ago
would result in a surplus of the 200 Myr-old SMC Cepheids today.
Unfortunately, the star formation history used in the synthetic population
approach was constructed from the pulsation period-age relation (Bono et al.,
2005); the number of Cepheids in a given pulsation period bin was translated
to the SFR at a given age bin. This was, in turn, fed to the population
synthesis code to induce the occurrence of Cepheids of a given age and
pulsation period. This circular logic prevents us from using the synthetic
population to establish the relevance of the SFR on the incident rates of SMC
and LMC Cepheids in our sample.

As a next test, we compared the masses of Cepheids and their companions from
our sample with the synthetic populations, as shown in Figs. 9 and 10. The
masses of our companions generally fall into the range of synthetic companion
masses, with some exceptions, which are located above the upper limit for the
synthetic companion mass. These exceptions are mostly overtone Cepheids
(light-blue circle symbols in Fig. 9 and bright red diamond symbols in Fig.
10), which endorses our hypothesis from Section 4.6 that such massive objects
cannot exist within a framework of non-interacting stars. These more massive
companions can be close binaries in hierarchical triples with the more distant
Cepheids, mergers from a hierarchical triple, or an outcome of the mass gain,
neither of which was explored in the population synthesis method (Karczmarek
et al., 2022). However, the recent theoretical studies on evolution of stars
in stellar clusters hint at the possibility that significant percentage of
Cepheids might be in multiple systems (Dinnbier et al., 2022), which sets an
interesting course for future exploration.

Figure 9: Distribution of estimated minimum companion masses as a function of
derived Cepheid masses in the LMC (top), represented by dark blue circles
(fundamental-mode Cepheids) and light blue circles (first-overtone Cepheids).
Black line indicates the median companion mass from the binary synthetic
population (Karczmarek et al., 2022), with half of the companions (1st to 3rd
quartile) lying inside the dark-shaded area, and the entire sample enclosed by
the light-shaded area. Relation between the Cepheid mass and pulsation period
(middle) for observed fundamental-mode Cepheids (dark blue circles, derived
from the inverse of Eq. 8) and for synthetic ones (black dots). Mass
distribution of synthetic LMC Cepheids (bottom), shown as a black histogram,
which offers additional information about the number of systems that were used
to calculate the median companion mass (in the top panel). For comparison the
mass distribution of observed LMC fundamental-mode Cepheids is given in dark
blue. Figure 10: Same as Fig 9, but for the SMC Cepheid binary candidates:
Dark red diamonds (fundamental-mode) and bright red diamonds (first-overtone),
with the corresponding synthetic binary population.

Finally, we calculated the percentage of systems that could be recovered from
the entire population of binary Cepheids using the LTTE method. We considered
the following limitations, namely, the minimum and maximum values of our
observed distributions of strong candidates among fundamental-mode Cepheids:
(i) orbital period, (ii) logarithm of the mass function, and (iii) $a\sin(i)$
assuming isotropic inclination angle. We imposed them on the synthetic
populations of binary Cepheids. Table 2 presents the upper and lower limits,
and the recovery rate of SMC and LMC fundamental-mode Cepheids for each
limitation separately, and all combined. We were able to recover about 6.1 and
2.6 % of the initial sample of SMC and LMC synthetic Cepheids, respectively.
We notice that the percentages of individual limitations added together exceed
100 %, which means that the recovery of some Cepheid binaries is affected by
more than one limiting factor. The recovered incidence rates from the
synthetic populations are about twice as large as the incidence rates from the
empirical data, which can be explained on the basis of the Cepheid binarity
fractions in the Magellanic Clouds. The binarity fractions in the synthetic
populations are 100% (all Cepheids are in binary systems), whereas the
empirical binarity fractions of the SMC and LMC Cepheids are unknown. If the
real binarity fractions were about 50%, then the synthetic incidence rates
should be halved, bringing them to a closer agreement with the empirical ones.
The synthetic populations of binary Cepheids in the LMC and SMC were rendered
using identical initial distributions; therefore, any differences between the
SMC and LMC synthetic incidence rates arise from the limitations imposed on
the orbital parameters (see Table. 2). This does not, however, explain why the
limiting parameters have different ranges in the first place.

Table 2: Limitations imposed on the synthetic population of binary Cepheids and the recovery rates. | SMC | LMC
---|---|---
Parameter | Limits | Recov. | Limits | Recov.
$P_{\mathrm{orbital}}$ [days] | [863, 5353] | 11.2% | [945, 3897] | 8.1%
$\log f(m)$ [${\,}{\rm M}_{\odot}$] | [-2.29, 0.25] | 66.4% | [-1.74, 0.33] | 45.6%
$a\sin(i)$ [AU] | [0.66, 6.80] | 22.8% | [1.18, 5.80] | 15.2%
Combined | — | 6.1% | — | 2.6%

Galactic field studies estimated binarity fractions to be as high as 80 %
(Evans, 1992; Szabados, 2003a; Evans et al., 2013; Kervella et al., 2019b),
whereas population synthesis of LMC Cepheids implied the binarity fraction to
be at least 55 %, and likely much more (Karczmarek et al., 2022). That study
also predicts most of this fraction (up to 90 percent) to be dominated by hot
main sequence companions, leaving only 3-5 percent for evolved giant star
companions. Given the small sample statistic for our LMC candidates (52 out of
roughly more than 3000 stars in the survey), it is clear why we did not detect
any new overbright Cepheids (apart from previously established OGLE-LMC-
CEP-0837 and OGLE-LMC-CEP-0889), which are likely tracers of Cepheids with
giant companion origins. The SMC sample is three times as large as the LMC
one, and does show new Cepheid candidates (OGLE-SMC-CEP-3061 and OGLE-SMC-
CEP-4365) with giant companions (see Section 4.3).

##  6 Conclusions

We have applied the modified Hertzsprung method to analyze period changes of
classical Cepheids in the Magellanic Clouds based on OGLE data. This resulted
in $O-C$ diagrams of more than 7200 Cepheids. Our systematic search for
periodic $O-C$ features resulted in a sample of 197 candidates for binary
Cepheids. The key findings are as follows:

  * •

In the LMC, we found 52 candidates for Cepheid binary systems (30 fundamental
and 22 first-overtone mode) and a roughly three times larger sample of 145
candidates in the SMC (85 fundamental and 60 first-overtone mode). The
detection fraction of binary Cepheid candidates is as follows: LMC F 1.7 %;
LMC 1O 1.8 %; SMC F 3.3%; and SMC 1O 3.7%.

  * •

The Cepheid binary population synthesis results, when subject to the limits of
our detected binary sample, provided a similar incidence rate ratio for SMC to
LMC, as calculated from our observational sample, showing the agreement
between theoretical predictions and observations.

  * •

Given that only a few spectroscopic detections of Cepheids in a binary system
were reported, 25 in the LMC (Szabados & Nehéz, 2012; Pilecki et al., 2021)
and 9 in the SMC (Szabados & Nehéz, 2012), our sample of Cepheid binary
candidates is twice as large for the LMC and sixteen times as large for the
SMC. We note that the sample compiled in this paper is a list of probable
binary candidates and further follow-up is needed for verification.

  * •

From the SMC binary candidate sample, two fundamental-mode Cepheids, OGLE-SMC-
CEP-3061 and OGLE-SMC-CEP-4365, are found to be in the class of overbright
Cepheids (see Pilecki et al., 2021), which is likely a consequence of them
being in a binary system with a giant companion. These two candidates are the
very first detections of LTTE in overbright Cepheids in the SMC. Using
empirical relations we estimate the Cepheid mass and minimum companion mass
($m_{\text{Cep}}+m_{\text{c}}$) to be: ($3.2{\,}{\rm M}_{\odot}+2.8{\,}{\rm
M}_{\odot}$) for OGLE-SMC-CEP-3061 and ($2.7{\,}{\rm M}_{\odot}+4.5{\,}{\rm
M}_{\odot}$) for OGLE-SMC-CEP-4365. In the LMC we report LTTE variation for
two overbright Cepheids, OGLE-LMC-CEP-0837 and OGLE-LMC-CEP-0889, where the
former is also a spectroscopically confirmed SB2-type binary system. For the
remaining SB2-type systems in the LMC compiled by Pilecki et al. (2021), we do
not see any significant LTTE variation.

  * •

Considering known literature Cepheids in eclipsing binary systems, from our
$O-C$ analysis we confirm the LTTE signature for three candidates in the LMC
(OGLE-LMC-CEP-1812, OGLE-LMC-CEP-2532 and OGLE-LMC-CEP-4506) and two in the
SMC (OGLE-SMC-CEP-2199 and OGLE-SMC-CEP-3235). The reasons for no such
signature for remaining eclipsing binary candidates are described in Sect.
4.5.

  * •

We found 21 candidate binary systems with very high-mass-function values. From
these, 85 % come from the first-overtone mode sample, likely indicating a non-
evolutionary period change phenomenon playing a role in observed the $O-C$
variation. Other explanations, however, including triple systems with a tight
inner binary, or even black hole companions, cannot be excluded. To establish
a better understanding of their nature, long-term monitoring and/or
spectroscopic observations are needed.

Detailed spectroscopic follow-ups on at least the strong candidates may be
able to accurately establish the binarity nature and lead to improved orbital
parameters. A longer photometric baseline in the future will not only refine
the proposed sample but also uncover more candidates in the longer orbital
period regime. In addition, for a better characterization of non-evolutionary
period changes in classical Cepheids, we will tackle the more complicated,
second type of the non-evolutionary period change related to irregular $O-C$
diagrams (Rathour et al. in prep). The subject of our follow-up paper will be
crucial for better understanding the biases and limitations in the study of
binary candidates via the $O-C$ technique as well.

###### Acknowledgements.

In tribute to late Krzysztof Belczyński, whose expert discussions on high-
mass-function candidates have directly contributed to this article. We thank
all the OGLE observers for their contribution to the collection of the
photometric data over the decades. RSR, RS, VH and OZ are supported by the
National Science Center, Poland, Sonata BIS project 2018/30/E/ST9/00598. GH
and PK acknowledge grant support from the European Research Council (ERC)
under the European Union’s Horizon 2020 research and innovation program (grant
agreement No. 695099). IS acknowledge support from the National Science
Centre, Poland, grant no. 2022/45/B/ST9/00243. For the purpose of Open Access,
the author has applied a CC-BY public copyright license to any Author Accepted
Manuscript (AAM) version arising from this submission. This research made use
of the SIMBAD and VIZIER databases at CDS, Strasbourg (France) and the
electronic bibliography maintained by the NASA/ADS system.

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##  Appendix A Table with binary parameters

Table 3: List of LMC binary Cepheid candidates with their orbital parameters.

Target | Fit order | P${}_{\textbf{pulsation}}$ | P${}_{\textbf{orbital}}$ | T${}_{\textbf{0}}$ | e | a sini | $\omega$ | PCR | K | f(m) | M${}_{\textbf{c}}$ (M${}_{\textbf{cep}}$)
---|---|---|---|---|---|---|---|---|---|---|---
|  | (d) | (d) | (d) |  | AU | (deg) | (d Myr-1) | (km s-1) | (M⊙) | (M⊙)
F mode |  |  |  |  |  |  |  |  |  |  |
Gold sample |  |  |  |  |  |  |  |  |  |  |
0543 | 10 | 3.16340973 | 3649 $\pm$ 60 | 9115 $\pm$ 106 | 0.512 $\pm$ 0.085 | 3.556 $\pm$ 0.296 | -151 $\pm$ 9 | -0.046 $\pm$ 0.182 | 12.34 $\pm$ 1.79 | 0.450 $\pm$ 0.130 | 2.5(3.5)
0631 | 10 | 3.58766239 | 2304 $\pm$ 28 | 5370 $\pm$ 89 | 0.588 $\pm$ 0.134 | 1.505 $\pm$ 0.107 | 129 $\pm$ 17 | +0.809 $\pm$ 0.163 | 8.77 $\pm$ 2.69 | 0.086 $\pm$ 0.019 | 1.3(3.7)
0637 | 6 | 2.60685284 | 945 $\pm$ 10 | 2882 $\pm$ 180 | 0.25 $\pm$ 0.224 | 1.18 $\pm$ 0.169 | 148 $\pm$ 69 | +0.426 $\pm$ 0.242 | 14.217 $\pm$ 9.33 | 0.245 $\pm$ 0.122 | 1.9(3.3)
0662 | 10 | 2.95839254 | 3861 $\pm$ 69 | 8801 $\pm$ 257 | 0.468 $\pm$ 0.161 | 1.645 $\pm$ 0.307 | -13 $\pm$ 18 | +0.019 $\pm$ 0.094 | 5.23 $\pm$ 2.19 | 0.040 $\pm$ 0.038 | 0.9(3.4)
0685 | 9 | 3.18198049 | 1413 $\pm$ 10 | 6471 $\pm$ 124 | 0.295 $\pm$ 0.135 | 2.62 $\pm$ 0.243 | 58 $\pm$ 32 | -1.355 $\pm$ 0.282 | 21.103 $\pm$ 3.23 | 1.201 $\pm$ 0.384 | 4.1(3.5)
0802 | 8 | 2.91257982 | 3385 $\pm$ 34 | 7044 $\pm$ 148 | 0.252 $\pm$ 0.067 | 3.769 $\pm$ 0.143 | -50 $\pm$ 15 | +0.06 $\pm$ 0.166 | 12.535 $\pm$ 0.59 | 0.623 $\pm$ 0.070 | 2.9(3.4)
0837 | 10 | 5.77731047 | 3692 $\pm$ 55 | 5408 $\pm$ 338 | 0.281 $\pm$ 0.143 | 3.932 $\pm$ 0.379 | -3 $\pm$ 33 | -0.130 $\pm$ 0.590 | 12.09 $\pm$ 1.74 | 0.595 $\pm$ 0.186 | 3.2(4.3)
0952 | 6 | 3.12319326 | 3495 $\pm$ 65 | 6312 $\pm$ 264 | 0.398 $\pm$ 0.177 | 2.692 $\pm$ 0.293 | -38 $\pm$ 28 | +1.011 $\pm$ 0.437 | 9.086 $\pm$ 2.12 | 0.213 $\pm$ 0.083 | 1.8(3.5)
1047 | 7 | 3.59110858 | 2498 $\pm$ 54 | 4107 $\pm$ 187 | 0.600 $\pm$ 0.261 | 3.139 $\pm$ 1.156 | -177 $\pm$ 25 | 0.112 $\pm$ 0.504 | 17.26 $\pm$ 16.66 | 0.672 $\pm$ 1.771 | 3.2(3.7)
1265 | 10 | 2.74481384 | 1860 $\pm$ 19 | 5599 $\pm$ 271 | 0.101 $\pm$ 0.074 | 1.801 $\pm$ 0.082 | 72 $\pm$ 53 | 2.564 $\pm$ 0.251 | 10.62 $\pm$ 0.52 | 0.225 $\pm$ 0.031 | 1.8(3.3)
1856 | 10 | 2.07536078 | 2525 $\pm$ 24 | 7335 $\pm$ 164 | 0.373 $\pm$ 0.13 | 1.244 $\pm$ 0.108 | 44 $\pm$ 24 | -0.041 $\pm$ 0.059 | 5.794 $\pm$ 0.8 | 0.040 $\pm$ 0.011 | 0.8(3.0)
2036 | 10 | 4.46453151 | 3473 $\pm$ 25 | 7618 $\pm$ 399 | 0.135 $\pm$ 0.082 | 2.399 $\pm$ 0.144 | 68 $\pm$ 41 | -3.117 $\pm$ 0.138 | 7.593 $\pm$ 0.56 | 0.153 $\pm$ 0.029 | 1.7(4.0)
2039 | 9 | 1.58174298 | 3849 $\pm$ 49 | 5628 $\pm$ 516 | 0.136 $\pm$ 0.1 | 1.268 $\pm$ 0.055 | 57 $\pm$ 48 | -0.06 $\pm$ 0.034 | 3.639 $\pm$ 0.19 | 0.018 $\pm$ 0.003 | 0.6(2.7)
2197 | 8 | 3.42832749 | 2573 $\pm$ 22 | 6037 $\pm$ 269 | 0.164 $\pm$ 0.086 | 2.467 $\pm$ 0.146 | -14 $\pm$ 38 | +0.013 $\pm$ 0.241 | 10.601 $\pm$ 0.67 | 0.302 $\pm$ 0.056 | 2.2(3.6)
2295 | 10 | 2.91158082 | 1555 $\pm$ 8 | 6856 $\pm$ 25 | 0.691 $\pm$ 0.052 | 2.412 $\pm$ 0.113 | 33 $\pm$ 7 | +0.063 $\pm$ 0.188 | 23.317 $\pm$ 2.38 | 0.772 $\pm$ 0.111 | 3.2(3.4)
2491 | 10 | 1.0659162 | 1810 $\pm$ 20 | 5339 $\pm$ 101 | 0.32 $\pm$ 0.107 | 1.183 $\pm$ 0.111 | 15 $\pm$ 20 | +0.051 $\pm$ 0.047 | 7.484 $\pm$ 1.09 | 0.067 $\pm$ 0.023 | 0.9(2.4)
2537 | 10 | 3.6185962 | 1110 $\pm$ 4 | 5357 $\pm$ 160 | 0.146 $\pm$ 0.135 | 1.68 $\pm$ 0.113 | 92 $\pm$ 53 | +0.462 $\pm$ 0.16 | 16.729 $\pm$ 1.71 | 0.513 $\pm$ 0.107 | 2.8(3.7)
2553 | 10 | 2.91710686 | 1513 $\pm$ 7 | 6478 $\pm$ 202 | 0.121 $\pm$ 0.095 | 1.61 $\pm$ 0.105 | -74 $\pm$ 47 | -0.736 $\pm$ 0.086 | 11.685 $\pm$ 0.96 | 0.243 $\pm$ 0.051 | 1.9(3.4)
2634 | 6 | 3.35721485 | 2431 $\pm$ 59 | 7436 $\pm$ 424 | 0.237 $\pm$ 0.18 | 1.411 $\pm$ 0.197 | 36 $\pm$ 61 | +1.816 $\pm$ 0.357 | 6.541 $\pm$ 4.71 | 0.063 $\pm$ 0.031 | 1.1(3.6)
2742 | 10 | 3.84564344 | 3721 $\pm$ 65 | 4563 $\pm$ 563 | 0.133 $\pm$ 0.109 | 1.798 $\pm$ 0.144 | -89 $\pm$ 54 | -1.627 $\pm$ 0.263 | 5.333 $\pm$ 0.5 | 0.056 $\pm$ 0.014 | 1.1(3.8)
2829 | 9 | 2.87070846 | 2628 $\pm$ 18 | 7564 $\pm$ 95 | 0.377 $\pm$ 0.086 | 3.163 $\pm$ 0.185 | 10 $\pm$ 12 | -0.844 $\pm$ 0.17 | 14.121 $\pm$ 1.34 | 0.611 $\pm$ 0.111 | 2.9(3.4)
3032 | 7 | 2.71904257 | 2486 $\pm$ 43 | 5598 $\pm$ 366 | 0.159 $\pm$ 0.117 | 2.487 $\pm$ 0.193 | 169 $\pm$ 54 | +0.423 $\pm$ 0.346 | 11.107 $\pm$ 0.97 | 0.332 $\pm$ 0.079 | 2.2(3.3)
3133 | 9 | 4.83273655 | 3145 $\pm$ 38 | 7082 $\pm$ 62 | 0.584 $\pm$ 0.117 | 5.396 $\pm$ 0.621 | -42 $\pm$ 7 | +5.401 $\pm$ 0.548 | 22.813 $\pm$ 7.72 | 2.112 $\pm$ 1.008 | 6.0(4.1)
3213 | 6 | 2.6542454 | 3613 $\pm$ 56 | 7432 $\pm$ 141 | 0.52 $\pm$ 0.115 | 3.272 $\pm$ 0.267 | 124 $\pm$ 15 | +0.652 $\pm$ 0.269 | 11.557 $\pm$ 5.66 | 0.358 $\pm$ 0.096 | 2.2(3.3)
3298 | 10 | 2.89343793 | 3727 $\pm$ 66 | 6260 $\pm$ 206 | 0.367 $\pm$ 0.075 | 3.793 $\pm$ 0.137 | -60 $\pm$ 17 | +5.915 $\pm$ 0.236 | 11.865 $\pm$ 0.82 | 0.520 $\pm$ 0.061 | 2.7(3.4)
F mode |  |  |  |  |  |  |  |  |  |  |
Bronze sample |  |  |  |  |  |  |  |  |  |  |
0687 | 9 | 3.36220963 | 3444 $\pm$ 36 | 5806 $\pm$ 36 | 0.700 $\pm$ 0.070 | 3.196 $\pm$ 0.317 | -34 $\pm$ 5 | 1.221 $\pm$ 0.158 | 14.11 $\pm$ 3.16 | 0.367 $\pm$ 0.125 | 2.3(3.6)
0889 | 10 | 3.16841733 | 2549 $\pm$ 8 | 8061 $\pm$ 25 | 0.552 $\pm$ 0.079 | 5.118 $\pm$ 0.292 | 164 $\pm$ 3 | +3.789 $\pm$ 0.149 | 26.17 $\pm$ 3.14 | 2.750 $\pm$ 0.491 | 6.5(3.5)
2772 | 10 | 2.94871874 | 3958 $\pm$ 74 | 6398 $\pm$ 76 | 0.7 $\pm$ 0.136 | 3.684 $\pm$ 1.275 | 27 $\pm$ 7 | -2.091 $\pm$ 0.44 | 14.166 $\pm$ 14.81 | 0.426 $\pm$ 1.138 | 2.5(3.4)
2952 | 10 | 3.20001924 | 6564 $\pm$ 347 | 3108 $\pm$ 315 | 0.196 $\pm$ 0.078 | 5.406 $\pm$ 1.041 | 92 $\pm$ 14 | +2.911 $\pm$ 2.521 | 9.134 $\pm$ 1.35 | 0.487 $\pm$ 0.269 | 2.6(3.5)
3267 $\diamondsuit$ | 10 | 2.28513112 | 1400 $\pm$ 10 | 4873 $\pm$ 16 | 0.886 $\pm$ 0.077 | 5.803 $\pm$ 2.19 | -8 $\pm$ 3 | +4.782 $\pm$ 0.151 | 96.989 $\pm$ 95.42 | 13.296 $\pm$ 28.260 | 18.2(3.1)
1O mode |  |  |  |  |  |  |  |  |  |  |
Gold sample |  |  |  |  |  |  |  |  |  |  |
0030 | 5 | 0.603548 | 2155 $\pm$ 30 | 6874 $\pm$ 115 | 0.576 $\pm$ 0.173 | 1.23 $\pm$ 0.278 | 22 $\pm$ 18 | +0.332 $\pm$ 0.042 | 7.536 $\pm$ 4.72 | 0.053 $\pm$ 0.059 | 0.8(2.2)
0088 | 2 | 2.04553516 | 3215 $\pm$ 48 | 6877 $\pm$ 431 | 0.134 $\pm$ 0.091 | 4.471 $\pm$ 0.297 | -29 $\pm$ 47 | +3.553 $\pm$ 0.31 | 15.327 $\pm$ 1.15 | 1.155 $\pm$ 0.237 | 4.0(3.4)
0472 $\diamondsuit$ | 4 | 3.28359809 | 4064 $\pm$ 73 | 6466 $\pm$ 362 | 0.164 $\pm$ 0.084 | 11.026 $\pm$ 0.465 | -111 $\pm$ 32 | +8.127 $\pm$ 1.201 | 29.935 $\pm$ 1.66 | 10.790 $\pm$ 1.444 | 16.7(4.1)
0816 | 4 | 2.8070886 | 2155 $\pm$ 37 | 5619 $\pm$ 135 | 0.425 $\pm$ 0.149 | 3.301 $\pm$ 0.287 | 101 $\pm$ 26 | +4.652 $\pm$ 0.629 | 18.449 $\pm$ 3.37 | 1.033 $\pm$ 0.283 | 4.0(3.8)
0900 $\diamondsuit$ | 2 | 2.18609105 | 5271 $\pm$ 80 | 5447 $\pm$ 131 | 0.171 $\pm$ 0.027 | 25.476 $\pm$ 0.468 | 159 $\pm$ 9 | +4.771 $\pm$ 0.849 | 53.338 $\pm$ 0.93 | 79.301 $\pm$ 3.584 | 85.9(3.5)
1046 $\diamondsuit$ | 3 | 1.96984052 | 1211 $\pm$ 5 | 5147 $\pm$ 26 | 0.885 $\pm$ 0.111 | 4.253 $\pm$ 2.061 | 29 $\pm$ 11 | +1.347 $\pm$ 0.285 | 82.487 $\pm$ 140.27 | 6.999 $\pm$ 27.987 | 11.7(3.4)
1134 $\diamondsuit$ $\ast$ | 3 | 3.09020297 | 4719 $\pm$ 64 | 5017 $\pm$ 126 | 0.441 $\pm$ 0.134 | 15.031 $\pm$ 2.047 | -56 $\pm$ 10 | -5.971 $\pm$ 1.177 | 38.541 $\pm$ 12.85 | 20.344 $\pm$ 12.133 | 26.8(4.0)
2793 | 4 | 2.91121489 | 3165 $\pm$ 123 | 6082 $\pm$ 380 | 0.364 $\pm$ 0.214 | 3.753 $\pm$ 0.533 | -19 $\pm$ 44 | +6.34 $\pm$ 0.858 | 13.964 $\pm$ 5.14 | 0.708 $\pm$ 0.345 | 3.3(3.9)
1O mode |  |  |  |  |  |  |  |  |  |  |
Bronze sample |  |  |  |  |  |  |  |  |  |  |
0134 | 4 | 3.24235812 | 4797 $\pm$ 159 | 5436 $\pm$ 174 | 0.594 $\pm$ 0.101 | 9.125 $\pm$ 0.998 | -31 $\pm$ 15 | -14.237 $\pm$ 1.46 | 25.659 $\pm$ 5.71 | 4.387 $\pm$ 1.622 | 9.1(4.1)
0315 | 3 | 1.73136384 | 4477 $\pm$ 124 | 4283 $\pm$ 438 | 0.179 $\pm$ 0.09 | 4.81 $\pm$ 0.234 | -79 $\pm$ 35 | -3.291 $\pm$ 0.382 | 11.907 $\pm$ 0.66 | 0.740 $\pm$ 0.103 | 3.1(3.2)
0764 $\diamondsuit$ ${\ddagger}$ $\dagger$ | 5 | 2.31569813 | 5587 $\pm$ 37 | 9807 $\pm$ 31 | 0.444 $\pm$ 0.022 | 20.391 $\pm$ 0.318 | -20 $\pm$ 2 | -1.494 $\pm$ 0.317 | 44.29 $\pm$ 1.14 | 36.209 $\pm$ 1.692 | 42.6(3.6)
0839 ${\ddagger}$ $\dagger$ $\ast$ | 4 | 2.19325821 | 5710 $\pm$ 62 | 6987 $\pm$ 106 | 0.370 $\pm$ 0.041 | 10.701 $\pm$ 0.270 | -30 $\pm$ 7 | +3.669 $\pm$ 0.339 | 21.92 $\pm$ 0.91 | 5.007 $\pm$ 0.395 | 9.4(3.5)
0924 | 6 | 2.94963963 | 2849 $\pm$ 25 | 3756 $\pm$ 316 | 0.217 $\pm$ 0.138 | 3.772 $\pm$ 0.302 | -32 $\pm$ 41 | -0.450 $\pm$ 0.263 | 14.82 $\pm$ 1.66 | 0.883 $\pm$ 0.220 | 3.7(3.9)
0950 | 5 | 1.51765486 | 9673 $\pm$ 286 | 4349 $\pm$ 136 | 0.255 $\pm$ 0.024 | 16.094 $\pm$ 0.600 | 154 $\pm$ 6 | +1.323 $\pm$ 0.225 | 18.72 $\pm$ 0.28 | 5.928 $\pm$ 0.347 | 10.1(3.1)
1147 | 3 | 1.56527096 | 6783 $\pm$ 183 | 4244 $\pm$ 117 | 0.484 $\pm$ 0.056 | 7.568 $\pm$ 0.317 | 131 $\pm$ 6 | +4.963 $\pm$ 0.124 | 13.82 $\pm$ 1.11 | 1.250 $\pm$ 0.164 | 4.0(3.1)
1303 | 3 | 1.10871849 | 7026 $\pm$ 368 | 1086 $\pm$ 379 | 0.288 $\pm$ 0.036 | 8.659 $\pm$ 1.26 | -8 $\pm$ 10 | -0.063 $\pm$ 0.973 | 13.979 $\pm$ 1.41 | 1.745 $\pm$ 0.641 | 4.5(2.7)
1454 | 4 | 2.86627508 | 4824 $\pm$ 38 | 9061 $\pm$ 270 | 0.26 $\pm$ 0.086 | 9.87 $\pm$ 0.459 | -85 $\pm$ 20 | -12.68 $\pm$ 0.344 | 23.072 $\pm$ 1.54 | 5.512 $\pm$ 0.781 | 10.4(3.9)
1666 | 5 | 0.81963911 | 2868 $\pm$ 73 | 6281 $\pm$ 389 | 0.292 $\pm$ 0.213 | 1.098 $\pm$ 0.169 | 42 $\pm$ 50 | -0.471 $\pm$ 0.061 | 4.413 $\pm$ 1.92 | 0.022 $\pm$ 0.012 | 0.6(2.5)
2172 | 4 | 1.955718 | 4269 $\pm$ 104 | 5071 $\pm$ 318 | 0.468 $\pm$ 0.265 | 4.559 $\pm$ 1.322 | 102 $\pm$ 28 | +3.369 $\pm$ 0.339 | 13.046 $\pm$ 66.73 | 0.701 $\pm$ 1.008 | 3.1(3.4)
2967 | 2 | 1.91301 | 7371 $\pm$ 291 | 2707 $\pm$ 375 | 0.268 $\pm$ 0.097 | 7.628 $\pm$ 0.415 | -122 $\pm$ 19 | -1.599 $\pm$ 0.479 | 11.745 $\pm$ 0.66 | 1.098 $\pm$ 0.138 | 3.8(3.3)
3007 | 3 | 1.31733058 | 5770 $\pm$ 280 | 10923 $\pm$ 522 | 0.227 $\pm$ 0.064 | 7.178 $\pm$ 0.461 | 156 $\pm$ 21 | +3.439 $\pm$ 0.419 | 13.876 $\pm$ 0.66 | 1.480 $\pm$ 0.192 | 4.2(2.9)
3095 $\diamondsuit$ | 2 | 2.04440386 | 4619 $\pm$ 13 | 6280 $\pm$ 84 | 0.185 $\pm$ 0.016 | 32.78 $\pm$ 0.263 | -54 $\pm$ 6 | +12.949 $\pm$ 0.251 | 78.557 $\pm$ 0.68 | 220.128 $\pm$ 5.439 | 226.8(3.4)

333Notes: Strong candidates are marked in bold and constitute the gold sample.
The remaining ones are the marginal candidates referred to as the bronze
sample. Symbols denote: [$\diamondsuit$] peculiar mass function candidates;
[${\ddagger}$] additional low amplitude variability; [$\dagger$] sub-harmonic
signal; [$\ast$] additional radial mode.

Table 4: Same as Table 3 but for SMC binary Cepheid candidates.

Target | Fit order | P${}_{\textbf{pulsation}}$ | P${}_{\textbf{orbital}}$ | T${}_{\textbf{0}}$ | e | a sini | $\omega$ | PCR | K | f(m) | M${}_{\textbf{c}}$ (M${}_{\textbf{cep}}$)
---|---|---|---|---|---|---|---|---|---|---|---
|  | (d) | (d) | (d) |  | AU | (deg) | (d Myr-1) | (km s-1) | (M⊙) | (M⊙)
F mode |  |  |  |  |  |  |  |  |  |  |
Gold sample |  |  |  |  |  |  |  |  |  |  |
0024 | 10 | 1.22561463 | 2940 $\pm$ 51 | 5035 $\pm$ 189 | 0.414 $\pm$ 0.158 | 1.62 $\pm$ 0.205 | 146 $\pm$ 23 | +0.08 $\pm$ 0.081 | 6.594 $\pm$ 1.8 | 0.066 $\pm$ 0.031 | 0.9(2.5)
0092 | 10 | 0.97886425 | 2989 $\pm$ 27 | 6831 $\pm$ 265 | 0.184 $\pm$ 0.075 | 1.86 $\pm$ 0.086 | 167 $\pm$ 31 | +0.047 $\pm$ 0.05 | 6.893 $\pm$ 0.39 | 0.096 $\pm$ 0.014 | 1.0(2.3)
0268 | 10 | 1.20302354 | 1006 $\pm$ 15 | 6251 $\pm$ 141 | 0.531 $\pm$ 0.264 | 0.66 $\pm$ 0.223 | -139 $\pm$ 51 | +0.071 $\pm$ 0.078 | 8.357 $\pm$ 23.51 | 0.038 $\pm$ 0.086 | 0.7(2.5)
0349 | 10 | 1.47939779 | 2606 $\pm$ 44 | 4389 $\pm$ 233 | 0.511 $\pm$ 0.254 | 0.702 $\pm$ 0.147 | 78 $\pm$ 34 | +0.029 $\pm$ 0.044 | 3.411 $\pm$ 10.31 | 0.007 $\pm$ 0.006 | 0.4(2.7)
0381 | 10 | 1.69160087 | 873 $\pm$ 5 | 1473 $\pm$ 87 | 0.398 $\pm$ 0.188 | 1.044 $\pm$ 0.173 | -158 $\pm$ 35 | -0.665 $\pm$ 0.063 | 14.199 $\pm$ 5.61 | 0.199 $\pm$ 0.129 | 1.6(2.8)
0450 | 6 | 1.27595376 | 2169 $\pm$ 36 | 7189 $\pm$ 137 | 0.665 $\pm$ 0.211 | 1.819 $\pm$ 0.734 | 177 $\pm$ 19 | -0.283 $\pm$ 0.116 | 12.198 $\pm$ 15.24 | 0.171 $\pm$ 0.554 | 1.4(2.5)
0586 | 10 | 1.90344226 | 1031 $\pm$ 9 | 6324 $\pm$ 69 | 0.592 $\pm$ 0.202 | 1.102 $\pm$ 0.234 | 89 $\pm$ 30 | +0.015 $\pm$ 0.064 | 14.378 $\pm$ 32.72 | 0.168 $\pm$ 0.195 | 1.5(2.9)
0643 | 10 | 1.13272049 | 1734 $\pm$ 27 | 5965 $\pm$ 294 | 0.286 $\pm$ 0.194 | 0.708 $\pm$ 0.099 | -188 $\pm$ 63 | +0.063 $\pm$ 0.049 | 4.68 $\pm$ 1.9 | 0.016 $\pm$ 0.008 | 0.5(2.4)
0647 | 10 | 2.57753659 | 2743 $\pm$ 16 | 5378 $\pm$ 58 | 0.474 $\pm$ 0.079 | 2.287 $\pm$ 0.129 | 145 $\pm$ 8 | -0.1 $\pm$ 0.076 | 10.293 $\pm$ 1.02 | 0.212 $\pm$ 0.037 | 1.7(3.3)
0709 | 10 | 1.73208107 | 3311 $\pm$ 14 | 5318 $\pm$ 67 | 0.407 $\pm$ 0.043 | 2.982 $\pm$ 0.099 | 166 $\pm$ 7 | +0.111 $\pm$ 0.039 | 10.723 $\pm$ 0.57 | 0.323 $\pm$ 0.033 | 2.0(2.8)
0771 | 10 | 1.84251862 | 1078 $\pm$ 7 | 6235 $\pm$ 80 | 0.485 $\pm$ 0.183 | 1.17 $\pm$ 0.153 | 145 $\pm$ 26 | +0.042 $\pm$ 0.052 | 13.388 $\pm$ 4.09 | 0.184 $\pm$ 0.088 | 1.5(2.9)
0777 | 6 | 1.50491854 | 3867 $\pm$ 64 | 5204 $\pm$ 223 | 0.368 $\pm$ 0.131 | 2.711 $\pm$ 0.213 | -179 $\pm$ 20 | +0.003 $\pm$ 0.084 | 8.173 $\pm$ 1.17 | 0.177 $\pm$ 0.047 | 1.5(2.7)
0849 | 10 | 1.74006829 | 3389 $\pm$ 19 | 8597 $\pm$ 142 | 0.214 $\pm$ 0.057 | 2.297 $\pm$ 0.068 | -40 $\pm$ 16 | +0.019 $\pm$ 0.04 | 7.548 $\pm$ 0.3 | 0.141 $\pm$ 0.013 | 1.4(2.8)
0918 | 9 | 1.2558028 | 2385 $\pm$ 35 | 5671 $\pm$ 319 | 0.23 $\pm$ 0.178 | 1.067 $\pm$ 0.099 | -64 $\pm$ 49 | +0.067 $\pm$ 0.052 | 5.056 $\pm$ 1.07 | 0.029 $\pm$ 0.008 | 0.7(2.5)
1146 | 9 | 2.02200462 | 3451 $\pm$ 33 | 5958 $\pm$ 91 | 0.535 $\pm$ 0.12 | 2.115 $\pm$ 0.151 | -87 $\pm$ 11 | -0.03 $\pm$ 0.084 | 7.897 $\pm$ 1.68 | 0.106 $\pm$ 0.024 | 1.2(3.0)
1215 | 10 | 1.51245282 | 1725 $\pm$ 18 | 6070 $\pm$ 206 | 0.256 $\pm$ 0.164 | 0.873 $\pm$ 0.087 | -102 $\pm$ 45 | -0.149 $\pm$ 0.055 | 5.733 $\pm$ 1.12 | 0.030 $\pm$ 0.010 | 0.7(2.7)
1245 | 6 | 2.11006384 | 5260 $\pm$ 96 | 2811 $\pm$ 138 | 0.568 $\pm$ 0.089 | 3.221 $\pm$ 0.162 | -94 $\pm$ 10 | +0.112 $\pm$ 0.118 | 8.091 $\pm$ 1.2 | 0.161 $\pm$ 0.026 | 1.5(3.0)
1283 | 10 | 2.13681841 | 3106 $\pm$ 20 | 7020 $\pm$ 82 | 0.377 $\pm$ 0.076 | 2.649 $\pm$ 0.094 | 106 $\pm$ 10 | +0.125 $\pm$ 0.095 | 10.021 $\pm$ 0.63 | 0.257 $\pm$ 0.027 | 1.8(3.1)
1348 | 10 | 1.60295544 | 1086 $\pm$ 3 | 4779 $\pm$ 38 | 0.423 $\pm$ 0.083 | 1.637 $\pm$ 0.103 | 154 $\pm$ 14 | +0.025 $\pm$ 0.037 | 18.082 $\pm$ 1.86 | 0.495 $\pm$ 0.096 | 0.4(2.4)
1581 | 10 | 1.25120461 | 3165 $\pm$ 10 | 7481 $\pm$ 65 | 0.206 $\pm$ 0.032 | 3.763 $\pm$ 0.063 | 12 $\pm$ 7 | +0.501 $\pm$ 0.033 | 13.218 $\pm$ 0.28 | 0.710 $\pm$ 0.035 | 2.7(2.5)
1645 | 10 | 1.69847602 | 3080 $\pm$ 46 | 6336 $\pm$ 181 | 0.431 $\pm$ 0.161 | 1.124 $\pm$ 0.132 | -38 $\pm$ 21 | -0.062 $\pm$ 0.076 | 4.393 $\pm$ 1.08 | 0.020 $\pm$ 0.008 | 0.6(2.8)
1719 | 10 | 1.97438818 | 2482 $\pm$ 19 | 5312 $\pm$ 287 | 0.143 $\pm$ 0.085 | 1.636 $\pm$ 0.065 | 148 $\pm$ 42 | +0.541 $\pm$ 0.058 | 7.268 $\pm$ 0.32 | 0.095 $\pm$ 0.011 | 1.2(3.0)
1795 | 10 | 2.05372202 | 1430 $\pm$ 9 | 7606 $\pm$ 34 | 0.823 $\pm$ 0.083 | 2.406 $\pm$ 0.686 | -171 $\pm$ 5 | -0.622 $\pm$ 0.053 | 32.18 $\pm$ 23.82 | 0.909 $\pm$ 1.414 | 3.3(3.0)
1818 | 10 | 3.03115737 | 1558 $\pm$ 9 | 5967 $\pm$ 101 | 0.261 $\pm$ 0.1 | 1.969 $\pm$ 0.115 | -21 $\pm$ 24 | -1.064 $\pm$ 0.098 | 14.231 $\pm$ 1.2 | 0.419 $\pm$ 0.077 | 2.4(3.5)
1868 | 10 | 1.52833063 | 4619 $\pm$ 118 | 6097 $\pm$ 703 | 0.14 $\pm$ 0.104 | 1.39 $\pm$ 0.073 | -48 $\pm$ 53 | -0.091 $\pm$ 0.061 | 3.322 $\pm$ 0.23 | 0.017 $\pm$ 0.003 | 0.6(2.7)
1883 | 8 | 1.48845328 | 1231 $\pm$ 11 | 6840 $\pm$ 91 | 0.528 $\pm$ 0.251 | 0.937 $\pm$ 0.196 | 127 $\pm$ 29 | +0.159 $\pm$ 0.041 | 9.712 $\pm$ 13.75 | 0.073 $\pm$ 0.066 | 1.0(2.7)
1921 | 10 | 1.93682713 | 2832 $\pm$ 39 | 7603 $\pm$ 160 | 0.439 $\pm$ 0.134 | 1.427 $\pm$ 0.157 | -29 $\pm$ 22 | +0.137 $\pm$ 0.066 | 6.092 $\pm$ 1.29 | 0.048 $\pm$ 0.018 | 0.9(2.9)
1942 | 10 | 1.84741798 | 4852 $\pm$ 92 | 6150 $\pm$ 682 | 0.185 $\pm$ 0.129 | 0.97 $\pm$ 0.081 | 72 $\pm$ 51 | +0.028 $\pm$ 0.048 | 2.23 $\pm$ 0.26 | 0.005 $\pm$ 0.001 | 0.4(2.9)
1960 | 10 | 4.16810092 | 3483 $\pm$ 27 | 6349 $\pm$ 238 | 0.222 $\pm$ 0.07 | 2.637 $\pm$ 0.124 | 37 $\pm$ 25 | -0.563 $\pm$ 0.189 | 8.454 $\pm$ 0.49 | 0.201 $\pm$ 0.029 | 1.9(3.9)
2166 | 10 | 2.70734099 | 863 $\pm$ 4 | 4115 $\pm$ 31 | 0.643 $\pm$ 0.166 | 1.155 $\pm$ 0.213 | 127 $\pm$ 15 | -0.814 $\pm$ 0.093 | 18.925 $\pm$ 12.96 | 0.275 $\pm$ 0.225 | 2.0(3.3)
2206 | 10 | 1.12805011 | 2298 $\pm$ 24 | 5938 $\pm$ 53 | 0.68 $\pm$ 0.12 | 1.513 $\pm$ 0.301 | -141 $\pm$ 9 | +0.039 $\pm$ 0.044 | 9.731 $\pm$ 6.53 | 0.087 $\pm$ 0.093 | 1.0(2.4)
2358 | 10 | 1.72183113 | 4849 $\pm$ 32 | 3942 $\pm$ 202 | 0.197 $\pm$ 0.059 | 3.049 $\pm$ 0.085 | 52 $\pm$ 14 | +0.08 $\pm$ 0.043 | 6.983 $\pm$ 0.25 | 0.161 $\pm$ 0.014 | 1.4(2.8)
2457 | 10 | 1.52821245 | 1241 $\pm$ 5 | 5758 $\pm$ 175 | 0.091 $\pm$ 0.065 | 1.739 $\pm$ 0.06 | 15 $\pm$ 53 | -0.031 $\pm$ 0.063 | 15.351 $\pm$ 0.54 | 0.456 $\pm$ 0.047 | 2.2(2.7)
2486 | 10 | 1.46259811 | 2355 $\pm$ 7 | 7773 $\pm$ 34 | 0.384 $\pm$ 0.037 | 3.505 $\pm$ 0.066 | 122 $\pm$ 5 | +0.156 $\pm$ 0.049 | 17.535 $\pm$ 0.48 | 1.035 $\pm$ 0.058 | 3.4(2.7)
2747 | 10 | 4.56171961 | 3107 $\pm$ 15 | 7251 $\pm$ 55 | 0.555 $\pm$ 0.083 | 4.374 $\pm$ 0.326 | -43 $\pm$ 7 | +2.126 $\pm$ 0.267 | 18.381 $\pm$ 3.02 | 1.156 $\pm$ 0.291 | 4.3(4.0)
2833 | 8 | 1.75469245 | 3079 $\pm$ 43 | 7081 $\pm$ 431 | 0.117 $\pm$ 0.089 | 1.821 $\pm$ 0.083 | -55 $\pm$ 49 | -0.06 $\pm$ 0.086 | 6.503 $\pm$ 0.35 | 0.085 $\pm$ 0.012 | 1.1(2.8)
2844 | 10 | 1.72678651 | 1336 $\pm$ 6 | 8630 $\pm$ 54 | 0.359 $\pm$ 0.116 | 1.477 $\pm$ 0.082 | 49 $\pm$ 17 | -0.035 $\pm$ 0.061 | 12.89 $\pm$ 1.35 | 0.241 $\pm$ 0.042 | 1.7(2.8)
2849 | 8 | 1.58862022 | 4822 $\pm$ 56 | 6968 $\pm$ 193 | 0.23 $\pm$ 0.054 | 3.358 $\pm$ 0.11 | 4 $\pm$ 13 | +0.24 $\pm$ 0.056 | 7.789 $\pm$ 0.35 | 0.217 $\pm$ 0.022 | 1.6(2.8)
2929 | 10 | 1.58662693 | 951 $\pm$ 6 | 8456 $\pm$ 34 | 0.647 $\pm$ 0.121 | 1.438 $\pm$ 0.298 | -18 $\pm$ 10 | +0.031 $\pm$ 0.053 | 21.5 $\pm$ 10.41 | 0.438 $\pm$ 0.441 | 2.2(2.8)
2979 | 10 | 2.0186305 | 2221 $\pm$ 20 | 7833 $\pm$ 246 | 0.194 $\pm$ 0.131 | 1.49 $\pm$ 0.1 | 35 $\pm$ 40 | -0.543 $\pm$ 0.119 | 7.457 $\pm$ 0.76 | 0.089 $\pm$ 0.019 | 1.2(3.0)
3061 | 10 | 2.54070246 | 1197 $\pm$ 7 | 8733 $\pm$ 64 | 0.454 $\pm$ 0.156 | 1.857 $\pm$ 0.253 | -171 $\pm$ 21 | +0.162 $\pm$ 0.161 | 18.886 $\pm$ 5.19 | 0.595 $\pm$ 0.307 | 2.8(3.2)
3075 | 10 | 2.04232342 | 1032 $\pm$ 6 | 2951 $\pm$ 88 | 0.403 $\pm$ 0.192 | 1.081 $\pm$ 0.134 | 65 $\pm$ 32 | -0.381 $\pm$ 0.075 | 12.458 $\pm$ 7.09 | 0.158 $\pm$ 0.067 | 1.5(3.0)
3087 | 10 | 1.55714085 | 1550 $\pm$ 13 | 7913 $\pm$ 82 | 0.579 $\pm$ 0.173 | 1.239 $\pm$ 0.253 | -130 $\pm$ 18 | +0.103 $\pm$ 0.05 | 10.593 $\pm$ 6.93 | 0.105 $\pm$ 0.103 | 1.2(2.7)
3229 | 8 | 1.0361886 | 2771 $\pm$ 41 | 5607 $\pm$ 61 | 0.867 $\pm$ 0.147 | 1.81 $\pm$ 0.638 | 51 $\pm$ 15 | +0.072 $\pm$ 0.084 | 14.387 $\pm$ 27 | 0.103 $\pm$ 0.251 | 1.1(2.4)
3308 | 10 | 1.36063995 | 1697 $\pm$ 21 | 5793 $\pm$ 106 | 0.654 $\pm$ 0.23 | 0.884 $\pm$ 0.182 | -89 $\pm$ 30 | +0.07 $\pm$ 0.067 | 7.405 $\pm$ 23.88 | 0.032 $\pm$ 0.035 | 0.7(2.6)
3463 | 10 | 2.56622462 | 1828 $\pm$ 8 | 8622 $\pm$ 86 | 0.343 $\pm$ 0.081 | 3.204 $\pm$ 0.189 | -136 $\pm$ 17 | -0.583 $\pm$ 0.122 | 20.281 $\pm$ 1.84 | 1.313 $\pm$ 0.245 | 4.2(3.3)
3466 | 10 | 1.85843731 | 1262 $\pm$ 14 | 6069 $\pm$ 173 | 0.379 $\pm$ 0.269 | 0.752 $\pm$ 0.141 | -73 $\pm$ 52 | -0.079 $\pm$ 0.094 | 7.024 $\pm$ 7.4 | 0.035 $\pm$ 0.028 | 0.8(2.9)
3503 | 10 | 3.04494906 | 1802 $\pm$ 32 | 6279 $\pm$ 140 | 0.495 $\pm$ 0.237 | 1.049 $\pm$ 0.17 | 131 $\pm$ 28 | -0.197 $\pm$ 0.201 | 7.246 $\pm$ 14.26 | 0.047 $\pm$ 0.031 | 1.0(3.5)
3556 | 9 | 3.69854923 | 4305 $\pm$ 30 | 6375 $\pm$ 65 | 0.531 $\pm$ 0.064 | 4.676 $\pm$ 0.219 | -59 $\pm$ 6 | +5.489 $\pm$ 0.182 | 13.932 $\pm$ 1.34 | 0.735 $\pm$ 0.107 | 3.3(3.7)
3693 | 8 | 1.32095132 | 3341 $\pm$ 16 | 6852 $\pm$ 95 | 0.297 $\pm$ 0.05 | 3.903 $\pm$ 0.138 | -167 $\pm$ 10 | -0.073 $\pm$ 0.047 | 13.306 $\pm$ 0.68 | 0.710 $\pm$ 0.079 | 2.7(2.6)
3838 | 10 | 1.61179355 | 2214 $\pm$ 50 | 7392 $\pm$ 180 | 0.635 $\pm$ 0.248 | 0.787 $\pm$ 0.185 | 63 $\pm$ 29 | -0.01 $\pm$ 0.076 | 5.015 $\pm$ 18.86 | 0.013 $\pm$ 0.016 | 0.5(2.8)
3869 | 10 | 2.59092842 | 1478 $\pm$ 7 | 6995 $\pm$ 34 | 0.732 $\pm$ 0.151 | 1.805 $\pm$ 0.349 | 125 $\pm$ 12 | -0.354 $\pm$ 0.088 | 19.352 $\pm$ 15.27 | 0.359 $\pm$ 0.346 | 2.2(3.3)
3881 | 10 | 1.44373436 | 3799 $\pm$ 54 | 6932 $\pm$ 179 | 0.368 $\pm$ 0.124 | 1.361 $\pm$ 0.068 | 96 $\pm$ 17 | -0.124 $\pm$ 0.053 | 4.192 $\pm$ 0.42 | 0.023 $\pm$ 0.004 | 0.6(2.7)
3914 | 9 | 1.60525727 | 2553 $\pm$ 18 | 6440 $\pm$ 38 | 0.719 $\pm$ 0.153 | 2.795 $\pm$ 0.713 | -58 $\pm$ 9 | +0.14 $\pm$ 0.058 | 17.1 $\pm$ 16.88 | 0.448 $\pm$ 0.575 | 2.2(2.8)
3929 | 10 | 2.57717076 | 2218 $\pm$ 33 | 5891 $\pm$ 262 | 0.279 $\pm$ 0.161 | 1.498 $\pm$ 0.131 | 138 $\pm$ 45 | -0.207 $\pm$ 0.186 | 7.677 $\pm$ 1.35 | 0.091 $\pm$ 0.025 | 1.2(3.3)
4016 | 10 | 1.8631558 | 2259 $\pm$ 9 | 6144 $\pm$ 81 | 0.288 $\pm$ 0.059 | 3.597 $\pm$ 0.125 | 73 $\pm$ 13 | -0.12 $\pm$ 0.072 | 18.096 $\pm$ 0.87 | 1.216 $\pm$ 0.130 | 3.8(2.9)
4111 | 10 | 2.77874082 | 3154 $\pm$ 37 | 4978 $\pm$ 134 | 0.493 $\pm$ 0.129 | 2.016 $\pm$ 0.267 | 173 $\pm$ 14 | +0.086 $\pm$ 0.121 | 7.97 $\pm$ 2.4 | 0.110 $\pm$ 0.060 | 1.3(3.3)
4202 | 10 | 1.59026817 | 2035 $\pm$ 14 | 6526 $\pm$ 102 | 0.319 $\pm$ 0.104 | 1.373 $\pm$ 0.072 | 83 $\pm$ 18 | -0.015 $\pm$ 0.044 | 7.757 $\pm$ 0.64 | 0.083 $\pm$ 0.013 | 1.1(2.8)
4218 | 10 | 2.03454707 | 3401 $\pm$ 25 | 5601 $\pm$ 50 | 0.528 $\pm$ 0.097 | 4.336 $\pm$ 0.178 | -102 $\pm$ 6 | +0.506 $\pm$ 0.109 | 16.279 $\pm$ 2.14 | 0.937 $\pm$ 0.128 | 3.4(3.0)
4266 | 10 | 2.54817018 | 5353 $\pm$ 76 | 2539 $\pm$ 59 | 0.48 $\pm$ 0.043 | 6.8 $\pm$ 0.196 | 169 $\pm$ 3 | +1.106 $\pm$ 0.652 | 15.717 $\pm$ 0.75 | 1.460 $\pm$ 0.117 | 4.4(3.2)
4289 | 8 | 3.31679034 | 2272 $\pm$ 41 | 5884 $\pm$ 263 | 0.23 $\pm$ 0.148 | 1.973 $\pm$ 0.164 | 36 $\pm$ 41 | +1.298 $\pm$ 0.516 | 9.772 $\pm$ 1.24 | 0.199 $\pm$ 0.051 | 1.8(3.6)
4365 | 10 | 1.4343708 | 2084 $\pm$ 17 | 5962 $\pm$ 19 | 0.869 $\pm$ 0.066 | 3.865 $\pm$ 1.411 | -165 $\pm$ 4 | +0.287 $\pm$ 0.088 | 40.766 $\pm$ 43.12 | 1.775 $\pm$ 4.516 | 4.5(2.7)
4400 | 9 | 2.57392249 | 1017 $\pm$ 11 | 5135 $\pm$ 35 | 0.852 $\pm$ 0.131 | 2.222 $\pm$ 1.062 | -167 $\pm$ 8 | -0.459 $\pm$ 0.17 | 45.306 $\pm$ 61.53 | 1.412 $\pm$ 4.826 | 4.3(3.3)
4436 | 10 | 2.74656267 | 982 $\pm$ 7 | 6326 $\pm$ 39 | 0.814 $\pm$ 0.161 | 1.809 $\pm$ 0.742 | 133 $\pm$ 18 | -0.321 $\pm$ 0.222 | 35.063 $\pm$ 73.84 | 0.820 $\pm$ 2.510 | 3.3(3.3)
4512 | 10 | 1.20054953 | 3753 $\pm$ 68 | 6958 $\pm$ 253 | 0.284 $\pm$ 0.108 | 1.472 $\pm$ 0.09 | 128 $\pm$ 24 | +0.045 $\pm$ 0.077 | 4.455 $\pm$ 0.43 | 0.030 $\pm$ 0.006 | 0.7(2.5)
4571 | 10 | 2.1145914 | 2213 $\pm$ 35 | 6867 $\pm$ 114 | 0.614 $\pm$ 0.151 | 1.588 $\pm$ 0.363 | 11 $\pm$ 15 | -0.431 $\pm$ 0.146 | 9.861 $\pm$ 5.3 | 0.109 $\pm$ 0.119 | 1.3(3.0)
F mode |  |  |  |  |  |  |  |  |  |  |
Bronze sample |  |  |  |  |  |  |  |  |  |  |
0031 | 10 | 1.52691807 | 1014 $\pm$ 9 | 5125 $\pm$ 149 | 0.313 $\pm$ 0.208 | 0.807 $\pm$ 0.139 | -10 $\pm$ 50 | +0.171 $\pm$ 0.084 | 9.118 $\pm$ 3.82 | 0.068 $\pm$ 0.052 | 1.0(2.7)
0127 | 10 | 1.29405349 | 3236 $\pm$ 72 | 6802 $\pm$ 466 | 0.187 $\pm$ 0.123 | 1.06 $\pm$ 0.086 | 144 $\pm$ 51 | +0.156 $\pm$ 0.07 | 3.642 $\pm$ 0.41 | 0.015 $\pm$ 0.004 | 0.5(2.6)
0660 | 10 | 1.43903929 | 1856 $\pm$ 46 | 5258 $\pm$ 66 | 0.908 $\pm$ 0.072 | 2.792 $\pm$ 1.242 | -10 $\pm$ 6 | +0.184 $\pm$ 0.045 | 39.1 $\pm$ 47.56 | 0.845 $\pm$ 2.424 | 3.0(2.7)
1115 | 10 | 3.90037931 | 1784 $\pm$ 21 | 6412 $\pm$ 143 | 0.66 $\pm$ 0.255 | 1.003 $\pm$ 0.272 | 62 $\pm$ 31 | -1.242 $\pm$ 0.173 | 8.069 $\pm$ 20.85 | 0.042 $\pm$ 0.061 | 1.0(3.8)
1318 | 10 | 1.13801287 | 3893 $\pm$ 103 | 5983 $\pm$ 107 | 0.884 $\pm$ 0.142 | 0.962 $\pm$ 0.187 | 106 $\pm$ 20 | -0.099 $\pm$ 0.066 | 5.882 $\pm$ 17.45 | 0.008 $\pm$ 0.007 | 0.4(2.4)
1707 | 10 | 1.9624896 | 2649 $\pm$ 18 | 7513 $\pm$ 67 | 0.624 $\pm$ 0.133 | 1.724 $\pm$ 0.266 | 51 $\pm$ 10 | -0.666 $\pm$ 0.056 | 9.016 $\pm$ 4.56 | 0.097 $\pm$ 0.066 | 1.2(3.0)
2229 | 10 | 1.54470493 | 970 $\pm$ 4 | 6747 $\pm$ 204 | 0.135 $\pm$ 0.135 | 0.867 $\pm$ 0.09 | -3 $\pm$ 108 | +0.013 $\pm$ 0.046 | 9.893 $\pm$ 1.36 | 0.092 $\pm$ 0.030 | 1.1(2.7)
2504 | 10 | 1.27996851 | 2615 $\pm$ 59 | 6072 $\pm$ 308 | 0.384 $\pm$ 0.236 | 0.59 $\pm$ 0.122 | -45 $\pm$ 44 | +0.001 $\pm$ 0.036 | 2.661 $\pm$ 3.05 | 0.004 $\pm$ 0.004 | 0.3(2.6)
2989 | 10 | 1.96058204 | 5248 $\pm$ 228 | 3975 $\pm$ 176 | 0.277 $\pm$ 0.079 | 1.859 $\pm$ 0.132 | -107 $\pm$ 12 | +0.807 $\pm$ 0.302 | 4.032 $\pm$ 0.21 | 0.031 $\pm$ 0.005 | 0.8(3.0)
3053 | 10 | 1.54027597 | 5048 $\pm$ 146 | 9785 $\pm$ 724 | 0.126 $\pm$ 0.103 | 0.96 $\pm$ 0.059 | 14 $\pm$ 52 | +0.09 $\pm$ 0.058 | 2.095 $\pm$ 0.15 | 0.005 $\pm$ 0.001 | 0.4(2.7)
3247 | 10 | 1.46096693 | 1811 $\pm$ 25 | 5972 $\pm$ 282 | 0.216 $\pm$ 0.176 | 1.045 $\pm$ 0.14 | -134 $\pm$ 57 | +0.108 $\pm$ 0.112 | 6.483 $\pm$ 2.3 | 0.046 $\pm$ 0.026 | 0.8(2.7)
3618 | 10 | 1.61380046 | 2165 $\pm$ 29 | 6384 $\pm$ 86 | 0.704 $\pm$ 0.214 | 0.896 $\pm$ 0.188 | 95 $\pm$ 19 | +0.131 $\pm$ 0.037 | 6.314 $\pm$ 22.91 | 0.021 $\pm$ 0.017 | 0.6(2.8)
3664 | 10 | 2.09275118 | 2110 $\pm$ 63 | 5566 $\pm$ 347 | 0.267 $\pm$ 0.245 | 1.036 $\pm$ 0.191 | 98 $\pm$ 60 | -0.384 $\pm$ 0.173 | 5.595 $\pm$ 5.95 | 0.033 $\pm$ 0.025 | 0.8(3.0)

Table 5: continued.

Target | Fit order | P${}_{\textbf{pulsation}}$ | P${}_{\textbf{orbital}}$ | T${}_{\textbf{0}}$ | e | a sini | $\omega$ | PCR | K | f(m) | M${}_{\textbf{c}}$ (M${}_{\textbf{cep}}$)
---|---|---|---|---|---|---|---|---|---|---|---
|  | (d) | (d) | (d) |  | AU | (deg) | (d Myr-1) | (km s-1) | (M⊙) | (M⊙)
F mode |  |  |  |  |  |  |  |  |  |  |
Bronze sample |  |  |  |  |  |  |  |  |  |  |
3799 | 10 | 2.85774356 | 3781 $\pm$ 81 | 7254 $\pm$ 90 | 0.696 $\pm$ 0.14 | 2.162 $\pm$ 0.671 | 11 $\pm$ 9 | +0.247 $\pm$ 0.158 | 8.665 $\pm$ 7.08 | 0.095 $\pm$ 0.182 | 1.3(3.4)
3935 $\diamondsuit$ | 6 | 6.58742326 | 1797 $\pm$ 9 | 5721 $\pm$ 120 | 0.279 $\pm$ 0.086 | 7.894 $\pm$ 0.407 | -93 $\pm$ 24 | -5.736 $\pm$ 0.96 | 49.812 $\pm$ 3.7 | 20.312 $\pm$ 3.232 | 27.5(4.5)
3994 $\diamondsuit$ | 6 | 10.18392471 | 6312 $\pm$ 127 | 8123 $\pm$ 365 | 0.2 $\pm$ 0.07 | 16.922 $\pm$ 0.524 | -156 $\pm$ 20 | -8.069 $\pm$ 5.302 | 29.712 $\pm$ 1.26 | 16.175 $\pm$ 1.485 | 24.1(5.3)
4064 | 7 | 1.23047345 | 2586 $\pm$ 55 | 5220 $\pm$ 244 | 0.466 $\pm$ 0.271 | 1.106 $\pm$ 0.199 | -112 $\pm$ 35 | -0.219 $\pm$ 0.141 | 5.265 $\pm$ 11.38 | 0.027 $\pm$ 0.020 | 0.7(2.5)
4355 | 10 | 1.41788388 | 3024 $\pm$ 71 | 7103 $\pm$ 243 | 0.493 $\pm$ 0.225 | 0.973 $\pm$ 0.205 | -52 $\pm$ 31 | +0.053 $\pm$ 0.093 | 4.023 $\pm$ 28.09 | 0.014 $\pm$ 0.013 | 0.5(2.6)
4456 | 4 | 9.47825523 | 3380 $\pm$ 112 | 5087 $\pm$ 539 | 0.219 $\pm$ 0.221 | 8.297 $\pm$ 1.253 | 132 $\pm$ 58 | +41.56 $\pm$ 7.423 | 27.711 $\pm$ 11.09 | 6.656 $\pm$ 3.653 | 13.0(5.2)
1O mode |  |  |  |  |  |  |  |  |  |  |
Gold sample |  |  |  |  |  |  |  |  |  |  |
0093 $\diamondsuit$ | 4 | 0.68543359 | 4634 $\pm$ 33 | 5595 $\pm$ 270 | 0.075 $\pm$ 0.024 | 10.881 $\pm$ 0.146 | -67 $\pm$ 20 | -2.132 $\pm$ 0.103 | 25.619 $\pm$ 0.36 | 8.001 $\pm$ 0.319 | 11.5(2.3)
0197 | 3 | 0.8843626 | 1884 $\pm$ 18 | 5078 $\pm$ 168 | 0.27 $\pm$ 0.134 | 2.412 $\pm$ 0.195 | -124 $\pm$ 32 | -0.18 $\pm$ 0.083 | 14.492 $\pm$ 1.79 | 0.527 $\pm$ 0.131 | 2.3(2.5)
0239 | 3 | 1.37099521 | 2962 $\pm$ 50 | 6035 $\pm$ 101 | 0.797 $\pm$ 0.163 | 2.68 $\pm$ 0.632 | 106 $\pm$ 20 | -1.436 $\pm$ 0.145 | 16.322 $\pm$ 33.51 | 0.291 $\pm$ 0.314 | 1.9(3.0)
0391 | 5 | 0.76872759 | 4053 $\pm$ 56 | 6795 $\pm$ 300 | 0.143 $\pm$ 0.059 | 3.128 $\pm$ 0.093 | -6 $\pm$ 25 | +2.903 $\pm$ 0.073 | 8.488 $\pm$ 0.29 | 0.249 $\pm$ 0.022 | 1.6(2.4)
0600 $\diamondsuit$ | 4 | 0.95505234 | 5706 $\pm$ 35 | 8708 $\pm$ 57 | 0.22 $\pm$ 0.021 | 12.026 $\pm$ 0.115 | -172 $\pm$ 3 | -2.152 $\pm$ 0.115 | 23.49 $\pm$ 0.3 | 7.120 $\pm$ 0.198 | 10.9(2.6)
0656 | 4 | 0.79445363 | 4193 $\pm$ 54 | 7055 $\pm$ 252 | 0.285 $\pm$ 0.106 | 2.316 $\pm$ 0.129 | 57 $\pm$ 22 | +0.281 $\pm$ 0.034 | 6.268 $\pm$ 0.51 | 0.094 $\pm$ 0.017 | 1.0(2.4)
0744 $\diamondsuit$ | 3 | 0.85968448 | 3949 $\pm$ 10 | 6588 $\pm$ 130 | 0.142 $\pm$ 0.02 | 9.236 $\pm$ 0.134 | 59 $\pm$ 11 | -4.436 $\pm$ 0.058 | 25.709 $\pm$ 0.42 | 6.739 $\pm$ 0.301 | 10.4(2.5)
1267 $\diamondsuit$ | 4 | 0.90520386 | 6068 $\pm$ 62 | 9652 $\pm$ 72 | 0.151 $\pm$ 0.014 | 32.855 $\pm$ 0.827 | -73 $\pm$ 3 | +2.713 $\pm$ 0.691 | 59.591 $\pm$ 0.84 | 128.508 $\pm$ 7.221 | 133.5(2.5)
1468 | 3 | 0.91322521 | 4903 $\pm$ 30 | 6775 $\pm$ 86 | 0.436 $\pm$ 0.047 | 6.372 $\pm$ 0.174 | -77 $\pm$ 6 | +2.237 $\pm$ 0.042 | 15.711 $\pm$ 0.73 | 1.436 $\pm$ 0.118 | 3.9(2.6)
1597 | 4 | 1.11620594 | 5373 $\pm$ 19 | 8180 $\pm$ 659 | 0.027 $\pm$ 0.018 | 9.778 $\pm$ 0.095 | -24 $\pm$ 44 | +2.844 $\pm$ 0.042 | 19.807 $\pm$ 0.22 | 4.319 $\pm$ 0.135 | 7.9(2.8)
1611 ${\ddagger}$ | 4 | 1.7322348 | 3051 $\pm$ 51 | 5805 $\pm$ 254 | 0.501 $\pm$ 0.269 | 1.297 $\pm$ 0.381 | 157 $\pm$ 30 | +0.052 $\pm$ 0.071 | 5.315 $\pm$ 5.15 | 0.031 $\pm$ 0.051 | 0.8(3.2)
1615 ${\ddagger}$ | 4 | 1.35334862 | 4294 $\pm$ 78 | 3834 $\pm$ 568 | 0.169 $\pm$ 0.121 | 2.092 $\pm$ 0.152 | 27 $\pm$ 47 | +0.052 $\pm$ 0.068 | 5.408 $\pm$ 0.49 | 0.066 $\pm$ 0.014 | 1.0(3.0)
1871 | 5 | 1.03165542 | 2766 $\pm$ 20 | 4443 $\pm$ 54 | 0.691 $\pm$ 0.121 | 1.518 $\pm$ 0.264 | 157 $\pm$ 7 | +0.049 $\pm$ 0.03 | 8.225 $\pm$ 3.66 | 0.061 $\pm$ 0.044 | 0.9(2.7)
1928 | 6 | 0.80609668 | 6026 $\pm$ 22 | 3165 $\pm$ 64 | 0.249 $\pm$ 0.021 | 8.019 $\pm$ 0.08 | -99 $\pm$ 4 | +1.06 $\pm$ 0.035 | 14.946 $\pm$ 0.21 | 1.894 $\pm$ 0.058 | 4.5(2.4)
1985 | 8 | 1.14034791 | 3497 $\pm$ 15 | 5018 $\pm$ 65 | 0.274 $\pm$ 0.054 | 4.817 $\pm$ 0.113 | -108 $\pm$ 6 | +5.698 $\pm$ 0.054 | 15.586 $\pm$ 0.53 | 1.219 $\pm$ 0.084 | 3.7(2.8)
2015 $\diamondsuit$ | 5 | 0.93782717 | 6531 $\pm$ 23 | 2986 $\pm$ 192 | 0.052 $\pm$ 0.009 | 18.156 $\pm$ 0.086 | 83 $\pm$ 10 | +0.911 $\pm$ 0.068 | 30.285 $\pm$ 0.13 | 18.718 $\pm$ 0.233 | 23.1(2.6)
2127 $\diamondsuit$ | 4 | 0.90222227 | 5352 $\pm$ 56 | 5165 $\pm$ 107 | 0.184 $\pm$ 0.023 | 10.365 $\pm$ 0.139 | 141 $\pm$ 7 | -0.484 $\pm$ 0.097 | 21.432 $\pm$ 0.3 | 5.184 $\pm$ 0.184 | 8.7(2.5)
2250 $\diamondsuit$ | 5 | 2.33089694 | 6182 $\pm$ 59 | 4232 $\pm$ 114 | 0.248 $\pm$ 0.026 | 13.299 $\pm$ 0.162 | -57 $\pm$ 6 | +0.014 $\pm$ 0.169 | 24.15 $\pm$ 0.44 | 8.204 $\pm$ 0.339 | 13.3(3.6)
2357 | 5 | 0.81908458 | 3868 $\pm$ 26 | 7090 $\pm$ 78 | 0.444 $\pm$ 0.066 | 3.955 $\pm$ 0.166 | -75 $\pm$ 7 | -0.127 $\pm$ 0.031 | 12.408 $\pm$ 0.94 | 0.551 $\pm$ 0.072 | 2.3(2.5)
2518 | 5 | 1.08540032 | 2999 $\pm$ 43 | 5115 $\pm$ 94 | 0.578 $\pm$ 0.154 | 1.878 $\pm$ 0.204 | -67 $\pm$ 14 | -0.257 $\pm$ 0.065 | 8.35 $\pm$ 6.79 | 0.099 $\pm$ 0.039 | 1.1(2.7)
2961 | 4 | 0.89066439 | 3108 $\pm$ 64 | 5294 $\pm$ 206 | 0.39 $\pm$ 0.167 | 1.593 $\pm$ 0.12 | -120 $\pm$ 25 | +0.077 $\pm$ 0.043 | 6.005 $\pm$ 1.12 | 0.055 $\pm$ 0.015 | 0.9(2.5)
2973 | 5 | 1.56733549 | 3825 $\pm$ 22 | 7436 $\pm$ 96 | 0.333 $\pm$ 0.049 | 5.22 $\pm$ 0.145 | 82 $\pm$ 8 | +5.14 $\pm$ 0.075 | 15.752 $\pm$ 0.65 | 1.297 $\pm$ 0.110 | 4.1(3.1)
3097 ${\ddagger}$ $\dagger$ | 2 | 2.0168275 | 3544 $\pm$ 65 | 7447 $\pm$ 396 | 0.267 $\pm$ 0.128 | 6.038 $\pm$ 0.426 | -93 $\pm$ 39 | -5.169 $\pm$ 0.418 | 19.294 $\pm$ 2 | 2.337 $\pm$ 0.500 | 5.9(3.4)
3863 | 6 | 0.9672118 | 3938 $\pm$ 59 | 6277 $\pm$ 109 | 0.852 $\pm$ 0.119 | 1.077 $\pm$ 0.176 | 68 $\pm$ 17 | +0.132 $\pm$ 0.022 | 5.714 $\pm$ 10.92 | 0.011 $\pm$ 0.006 | 0.5(2.6)
4075 $\diamondsuit$ | 5 | 0.8063606 | 3524 $\pm$ 10 | 6742 $\pm$ 74 | 0.195 $\pm$ 0.026 | 9.359 $\pm$ 0.126 | -132 $\pm$ 7 | -0.235 $\pm$ 0.059 | 29.46 $\pm$ 0.47 | 8.803 $\pm$ 0.353 | 12.6(2.4)
4430 $\diamondsuit$ | 5 | 1.54368245 | 4185 $\pm$ 22 | 6435 $\pm$ 73 | 0.185 $\pm$ 0.02 | 13.034 $\pm$ 0.148 | -114 $\pm$ 6 | +1.392 $\pm$ 0.22 | 34.479 $\pm$ 0.47 | 16.866 $\pm$ 0.582 | 22.0(3.1)
4496 | 5 | 1.36000613 | 2698 $\pm$ 54 | 6815 $\pm$ 403 | 0.209 $\pm$ 0.158 | 1.338 $\pm$ 0.14 | 100 $\pm$ 54 | +1.622 $\pm$ 0.125 | 5.559 $\pm$ 1.48 | 0.044 $\pm$ 0.015 | 0.9(3.0)
4513 | 2 | 1.28093303 | 2277 $\pm$ 33 | 11289 $\pm$ 141 | 0.634 $\pm$ 0.142 | 4.96 $\pm$ 0.995 | -153 $\pm$ 12 | +0.575 $\pm$ 0.282 | 30.582 $\pm$ 15.42 | 3.122 $\pm$ 2.829 | 6.5(2.9)
4524 | 5 | 0.92472577 | 3391 $\pm$ 65 | 9390 $\pm$ 193 | 0.527 $\pm$ 0.164 | 1.116 $\pm$ 0.12 | 104 $\pm$ 24 | -0.181 $\pm$ 0.058 | 4.218 $\pm$ 5.01 | 0.016 $\pm$ 0.006 | 0.5(2.6)
1O mode |  |  |  |  |  |  |  |  |  |  |
Bronze sample |  |  |  |  |  |  |  |  |  |  |
0180 | 4 | 0.97331376 | 5721 $\pm$ 162 | 4163 $\pm$ 138 | 0.395 $\pm$ 0.076 | 4.057 $\pm$ 0.274 | -44 $\pm$ 10 | -0.059 $\pm$ 0.261 | 8.392 $\pm$ 0.62 | 0.271 $\pm$ 0.047 | 1.7(2.6)
0337 | 4 | 1.13067702 | 3923 $\pm$ 36 | 6969 $\pm$ 584 | 0.064 $\pm$ 0.051 | 5.351 $\pm$ 0.181 | 99 $\pm$ 53 | +3.717 $\pm$ 0.125 | 14.89 $\pm$ 0.54 | 1.327 $\pm$ 0.138 | 3.9(2.8)
0340 | 4 | 0.80186187 | 3292 $\pm$ 15 | 6600 $\pm$ 236 | 0.096 $\pm$ 0.045 | 7.073 $\pm$ 0.141 | -77 $\pm$ 25 | +2.52 $\pm$ 0.08 | 23.504 $\pm$ 0.48 | 4.356 $\pm$ 0.258 | 7.6(2.4)
0536 | 4 | 1.03324391 | 2863 $\pm$ 16 | 6304 $\pm$ 105 | 0.437 $\pm$ 0.08 | 3.755 $\pm$ 0.21 | 62 $\pm$ 14 | -1.484 $\pm$ 0.06 | 15.85 $\pm$ 1.6 | 0.861 $\pm$ 0.150 | 3.0(2.7)
0775 | 4 | 1.59291308 | 5960 $\pm$ 177 | 5003 $\pm$ 202 | 0.351 $\pm$ 0.08 | 4.122 $\pm$ 0.184 | -27 $\pm$ 11 | -1.525 $\pm$ 0.1 | 8.036 $\pm$ 0.5 | 0.263 $\pm$ 0.035 | 1.9(3.1)
0792 | 4 | 1.25033041 | 3479 $\pm$ 66 | 4879 $\pm$ 194 | 0.439 $\pm$ 0.125 | 2.063 $\pm$ 0.248 | -3 $\pm$ 19 | +0.456 $\pm$ 0.084 | 7.175 $\pm$ 1.53 | 0.097 $\pm$ 0.041 | 1.2(2.9)
0859 | 4 | 0.87780105 | 4603 $\pm$ 151 | 5616 $\pm$ 209 | 0.195 $\pm$ 0.045 | 4.311 $\pm$ 0.172 | -106 $\pm$ 16 | -1.14 $\pm$ 0.282 | 10.366 $\pm$ 0.24 | 0.501 $\pm$ 0.035 | 2.3(2.5)
0885 | 5 | 0.83943679 | 4121 $\pm$ 35 | 6469 $\pm$ 69 | 0.523 $\pm$ 0.056 | 3.532 $\pm$ 0.146 | 22 $\pm$ 5 | -0.328 $\pm$ 0.025 | 10.94 $\pm$ 0.86 | 0.346 $\pm$ 0.044 | 1.9(2.5)
1103 | 5 | 1.08102444 | 3697 $\pm$ 52 | 2338 $\pm$ 101 | 0.861 $\pm$ 0.121 | 1.469 $\pm$ 0.623 | -23 $\pm$ 9 | +0.629 $\pm$ 0.042 | 8.469 $\pm$ 11.4 | 0.031 $\pm$ 0.089 | 0.7(2.7)
1176 | 5 | 0.9696587 | 7762 $\pm$ 216 | 3710 $\pm$ 141 | 0.2 $\pm$ 0.02 | 12.326 $\pm$ 0.418 | 141 $\pm$ 8 | +0.142 $\pm$ 0.118 | 17.641 $\pm$ 0.2 | 4.150 $\pm$ 0.205 | 7.5(2.6)
1498 | 5 | 1.27470408 | 2642 $\pm$ 26 | 6664 $\pm$ 182 | 0.255 $\pm$ 0.101 | 2.041 $\pm$ 0.085 | 41 $\pm$ 24 | -0.131 $\pm$ 0.05 | 8.705 $\pm$ 0.53 | 0.163 $\pm$ 0.020 | 1.5(2.9)
1626 | 6 | 0.84449431 | 4620 $\pm$ 51 | 8472 $\pm$ 94 | 0.332 $\pm$ 0.054 | 3.036 $\pm$ 0.08 | -123 $\pm$ 7 | -1.524 $\pm$ 0.047 | 7.58 $\pm$ 0.28 | 0.175 $\pm$ 0.013 | 1.4(2.5)
1951 | 6 | 1.30785364 | 3819 $\pm$ 70 | 5645 $\pm$ 471 | 0.356 $\pm$ 0.202 | 1.099 $\pm$ 0.133 | -87 $\pm$ 49 | +0.313 $\pm$ 0.061 | 3.356 $\pm$ 1.11 | 0.012 $\pm$ 0.005 | 0.5(2.9)
2027 | 5 | 0.79555168 | 5773 $\pm$ 262 | 4265 $\pm$ 299 | 0.329 $\pm$ 0.13 | 1.28 $\pm$ 0.088 | -67 $\pm$ 21 | -0.072 $\pm$ 0.067 | 2.558 $\pm$ 0.29 | 0.008 $\pm$ 0.002 | 0.4(2.4)
2033 ${\ddagger}$ | 5 | 1.40873535 | 1820 $\pm$ 16 | 5652 $\pm$ 55 | 0.815 $\pm$ 0.183 | 1.86 $\pm$ 0.57 | -115 $\pm$ 18 | +0.421 $\pm$ 0.061 | 19.343 $\pm$ 46.99 | 0.259 $\pm$ 0.451 | 1.8(3.0)
2035 | 5 | 0.84843745 | 6045 $\pm$ 64 | 6823 $\pm$ 81 | 0.555 $\pm$ 0.054 | 5.546 $\pm$ 0.322 | -160 $\pm$ 4 | +2.785 $\pm$ 0.067 | 11.985 $\pm$ 1.2 | 0.622 $\pm$ 0.111 | 2.5(2.5)
2117 | 5 | 2.9034422 | 3915 $\pm$ 78 | 6845 $\pm$ 124 | 0.851 $\pm$ 0.11 | 5.205 $\pm$ 0.723 | -78 $\pm$ 16 | +0.572 $\pm$ 0.3 | 27.589 $\pm$ 73.2 | 1.221 $\pm$ 0.659 | 4.4(3.9)
2125 | 6 | 2.16627716 | 7329 $\pm$ 242 | 7399 $\pm$ 162 | 0.307 $\pm$ 0.044 | 7.186 $\pm$ 0.47 | 20 $\pm$ 8 | -4.874 $\pm$ 0.6 | 11.238 $\pm$ 0.44 | 0.923 $\pm$ 0.130 | 3.6(3.5)
2293 ${\ddagger}$ | 5 | 1.28879424 | 6873 $\pm$ 95 | 2986 $\pm$ 88 | 0.955 $\pm$ 0.044 | 3.134 $\pm$ 1.102 | 35 $\pm$ 12 | +0.092 $\pm$ 0.082 | 16.796 $\pm$ 20.2 | 0.087 $\pm$ 0.182 | 1.1(2.9)
2328 | 5 | 0.59565295 | 5286 $\pm$ 109 | 4734 $\pm$ 112 | 0.473 $\pm$ 0.039 | 7.497 $\pm$ 0.218 | -67 $\pm$ 8 | -0.373 $\pm$ 0.076 | 17.46 $\pm$ 0.79 | 2.003 $\pm$ 0.170 | 4.5(2.2)
2491 | 3 | 2.07736033 | 2975 $\pm$ 74 | 6379 $\pm$ 182 | 0.63 $\pm$ 0.154 | 3.461 $\pm$ 0.693 | 39 $\pm$ 22 | +1.903 $\pm$ 0.387 | 16.287 $\pm$ 16 | 0.626 $\pm$ 0.558 | 2.9(3.4)
2499 | 5 | 1.78113087 | 1855 $\pm$ 11 | 7980 $\pm$ 49 | 0.896 $\pm$ 0.147 | 2.742 $\pm$ 0.921 | -58 $\pm$ 21 | +0.928 $\pm$ 0.125 | 36.859 $\pm$ 148.97 | 0.799 $\pm$ 1.636 | 3.2(3.3)
2980 $\diamondsuit$ | 4 | 1.07331802 | 5074 $\pm$ 34 | 3478 $\pm$ 101 | 0.152 $\pm$ 0.019 | 12.021 $\pm$ 0.14 | 130 $\pm$ 7 | +6.592 $\pm$ 0.176 | 26.076 $\pm$ 0.28 | 8.998 $\pm$ 0.268 | 13.1(2.7)
3023 | 5 | 0.87912243 | 4076 $\pm$ 44 | 5489 $\pm$ 447 | 0.19 $\pm$ 0.105 | 1.488 $\pm$ 0.076 | -97 $\pm$ 40 | -0.391 $\pm$ 0.025 | 4.067 $\pm$ 0.26 | 0.026 $\pm$ 0.004 | 0.6(2.5)
3159 | 3 | 1.10072567 | 5996 $\pm$ 150 | 7596 $\pm$ 180 | 0.33 $\pm$ 0.088 | 6.551 $\pm$ 0.323 | 23 $\pm$ 10 | +1.885 $\pm$ 0.278 | 12.545 $\pm$ 0.84 | 1.038 $\pm$ 0.132 | 3.4(2.7)
3235 $\diamondsuit$ | 4 | 0.86408921 | 6721 $\pm$ 224 | 5153 $\pm$ 85 | 0.234 $\pm$ 0.016 | 16.106 $\pm$ 0.965 | 45 $\pm$ 5 | +1.367 $\pm$ 0.643 | 26.795 $\pm$ 0.81 | 12.322 $\pm$ 1.425 | 16.4(2.5)
3955 | 5 | 1.07831811 | 1490 $\pm$ 8 | 6053 $\pm$ 49 | 0.69 $\pm$ 0.182 | 1.98 $\pm$ 0.784 | -139 $\pm$ 13 | -0.146 $\pm$ 0.028 | 19.985 $\pm$ 26.34 | 0.466 $\pm$ 1.318 | 2.3(2.7)
3991 $\diamondsuit$ | 5 | 1.32878771 | 1377 $\pm$ 4 | 4332 $\pm$ 16 | 0.96 $\pm$ 0.052 | 4.793 $\pm$ 2.13 | 24 $\pm$ 9 | -1.252 $\pm$ 0.093 | 134.705 $\pm$ 177.92 | 7.744 $\pm$ 21.732 | 12.0(2.9)
4013 | 4 | 1.35226337 | 3629 $\pm$ 108 | 3771 $\pm$ 165 | 0.521 $\pm$ 0.173 | 2.462 $\pm$ 0.338 | -121 $\pm$ 19 | +2.142 $\pm$ 0.203 | 8.666 $\pm$ 5.53 | 0.152 $\pm$ 0.081 | 1.4(3.0)
4245 | 5 | 0.66073172 | 2921 $\pm$ 76 | 7671 $\pm$ 130 | 0.764 $\pm$ 0.18 | 1.175 $\pm$ 0.37 | -131 $\pm$ 18 | -0.837 $\pm$ 0.048 | 6.778 $\pm$ 9.98 | 0.025 $\pm$ 0.049 | 0.6(2.3)
4277 | 3 | 1.91138938 | 4636 $\pm$ 172 | 2327 $\pm$ 332 | 0.284 $\pm$ 0.114 | 4.284 $\pm$ 0.322 | 157 $\pm$ 23 | -4.136 $\pm$ 0.658 | 10.476 $\pm$ 1.07 | 0.487 $\pm$ 0.106 | 2.6(3.3)

##  Appendix B $O-C$ curves for LMC F mode

Figure 11: Remaining $O-C$ diagrams for LMC fundamental-mode Cepheid binary
candidate sample

Figure 12: continued.

Figure 13: continued.

##  Appendix C $O-C$ curves for LMC 1O mode

Figure 14: Remaining $O-C$ diagrams for LMC first-overtone mode Cepheid binary
candidate sample

Figure 15: continued.

##  Appendix D $O-C$ curves for SMC F mode

Figure 16: Remaining $O-C$ diagrams for SMC fundamental-mode Cepheid binary
candidate sample

Figure 17: continued.

Figure 18: continued.

Figure 19: continued.

Figure 20: continued.

Figure 21: continued.

Figure 22: continued.

Figure 23: continued.

##  Appendix E $O-C$ curves for SMC 1O mode

Figure 24: Remaining $O-C$ diagrams for SMC first-overtone mode Cepheid binary
candidate sample

Figure 25: continued.

Figure 26: continued.

Figure 27: continued.

Figure 28: continued.

##  Appendix F $O-C$ curves for high-mass-function candidates

Figure 29: Remaining $O-C$ diagram for peculiar high mass-function candidates
as described in section 4.6

Figure 30: continued.