# Physical properties of the eclipsing binary KIC 9851944 and analysis of its
tidally-perturbed p- and g-mode pulsations

Z. Jennings1, J. Southworth1 K. Pavlovski2 T. Van Reeth3
1 Astrophysics Group, Keele University, Staffordshire, ST5 5BG, UK
2 Department of Physics, Faculty of Science, University of Zagreb, 10000
Zagreb, Croatia
3 Institute of Astronomy, KU Leuven, Celestijnenlaan 200D, B-3001 Leuven,
Belgium
E-mail<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

Stars that are both pulsating and eclipsing offer an important opportunity to
better understand many of the physical phenomena that occur in stars, because
it is possible to measure the pulsation frequencies of stars for which the
masses and radii are known precisely and accurately. KIC 9851944 is a double-
lined detached eclipsing binary containing two F-stars which show both
pressure and gravity mode pulsations. We present an analysis of new high-
resolution spectroscopy of the system and high-quality light curves from the
Kepler and TESS space missions. We determine the masses and effective
temperatures of the stars to 0.6% precision, and their radii to 1.0% and 1.5%
precision. The secondary component is cooler, but larger and more massive than
the primary so is more evolved; both lie inside the $\delta$ Scuti and
$\gamma$ Doradus instability strips. We measure a total of 133 significant
pulsation frequencies in the light curve, including 14 multiplets that each
contain between 3 and 19 frequencies. We find evidence for tidal perturbations
to some of the p- and g-modes, attribute a subset of the frequencies to either
the primary or secondary star, and measure a buoyancy radius and near-core
rotational frequency for the primary component. KIC 9851944 is mildly metal-
rich and MIST isochrones from the MESA evolutionary code agree well with the
observed properties of the system for an age of 1.25 Gyr.

###### keywords:

stars: oscillations — stars: variables: Scuti — binaries: eclipsing —
binaries: spectroscopic — stars: fundamental parameters

††pubyear: 2022††pagerange: Physical properties of the eclipsing binary KIC
9851944 and analysis of its tidally-perturbed p- and g-mode pulsations–C

##  1 Introduction

Binary and multiple systems make up the vast majority of all observed medium-
and high-mass ($\textit{M}>1.15\,{\rm M}_{\odot}$) stellar objects in the
galaxy (Duchêne & Kraus, 2013). For double-lined spectroscopic binary systems
(SB2s), where light emitted from both components is visible in the stellar
spectrum, the radial velocity (RV) variations of each component throughout the
orbit can be measured. Combining this with the analysis of the light
variability due to geometrical effects when the components also eclipse one
another, i.e., eclipsing binaries (EBs), can lead to estimations of the mass
and radius to within $1\%$ (Torres et al., 2010), making these objects our
best source of precise fundamental stellar parameters. Since the methods used
to derive the properties of the components in double-lined eclipsing binaries
(DLEBs) do not rely on theoretical stellar models, these objects are critical
for testing and verifying stellar evolution theory (Pols et al., 1997;
Pourbaix, 2000; Lastennet & Valls-Gabaud, 2002; de Mink et al., 2007).

Such measurements for the masses and radii of stars allow for a precise
determination of evolutionary status and age (e.g., Higl & Weiss 2017) and the
calibration of interior physics. As examples, the mass dependence of
convective core overshooting (Claret & Torres, 2016, 2017, 2018, 2019), and
the amount of core mixing in massive stars has been probed using DLEBs
(Pavlovski et al., 2018; Tkachenko et al., 2020).

It has been known since the 1970s that stellar pulsations occur in the
components of EBs (e.g., Tempesti, 1971, for the system AB Cassiopeiae).
Oscillation signals measured in the light curves of pulsating stars can be
studied using asteroseismology. This is an alternative way by which accurate
stellar parameters can be derived (Aerts et al., 2010) and is a powerful tool
for probing the internal structures of stars (Aerts et al., 2017; Chaplin &
Miglio, 2013). The analysis of space-based photometry of pulsating stars has
revealed information about important phenomena such as internal rotation, core
overshooting and angular momentum transport (e.g., Saio et al. 2015; Lovekin &
Guzik 2017).

It is advantageous to study pulsating stars in EBs because analysing the
intrinsic variability due to oscillations of the stellar interior and
atmosphere as well as the light variability during eclipses leads to
independent constraints on theoretical models (Guo et al., 2019; Liakos, 2021;
Miszuda et al., 2022). The parameters derived from EBs can also be used to
constrain the seismic models (e.g., Sekaran et al., 2020), and this can be
useful, for example, in mode identification for $\delta$ Scuti stars which is
difficult (Streamer et al., 2018; Murphy et al., 2021). The complimentary
nature of these two methods of analysis makes pulsating stars in EBs
(particularly DLEBs) the best objects to use to refine our knowledge of
stellar structure and evolution (Guo et al., 2016). Lampens (2021) and
Southworth (2021) have reviewed the impact of space missions on the study of
EBs with pulsating components, and on binary stars in general, respectively.

$\delta$ Scuti stars are a class of short-period (15 min to 8 h; Uytterhoeven
et al. 2011; Aerts et al. 2010), multiperiodic pressure mode (p-mode)
pulsators. They are dwarfs or subgiants located at the lower end of the
classical instability strip in the Hertzsprung-Russell (HR) diagram (Breger,
2000; Dupret et al., 2005; Murphy et al., 2019). Their spectral types are A2
to F5, corresponding to typical effective temperatures ($T_{\rm eff}$s) of
6500 – 9500 K (Liakos, 2021). Their masses range from 1.5 to 2.5$\,{\rm
M}_{\odot}$ (Aerts et al., 2010; Yang et al., 2021), which places them in the
transition region between lower-mass stars with radiative cores and thick
outer convection zones, and massive stars with convective cores and thin outer
convective zones (Yang et al., 2021). They thus provide an opportunity to
study the structure and evolution of stars in this transition region (Bowman &
Kurtz, 2018). Low-order, non-radial modes are generally observed for these
stars and these modes are driven by the $\kappa$ mechanism acting in the
partial ionization zone of He II (Pamyatnykh, 1999; Breger, 2000; Antoci et
al., 2014; Murphy et al., 2020). Higher-order non-radial pulsations have been
observed in some $\delta$ Scuti stars such as $\tau$ Peg (Kennelly et al.,
1998), where the $\kappa$ mechanism may not be sufficient to explain the
modes, and may instead be attributed to the turbulent pressure in the hydrogen
convective zone (Antoci et al., 2014; Grassitelli et al., 2015).

The $\gamma$ Doradus class of pulsators are also located at the lower end of
the classical instability strip in the HR diagram; they exist near the red
edge of the $\delta$ Scuti instability strip (Yang et al., 2021). Such stars
pulsate with typical periods between 8 and 80 h (Handler, 1999) in high order
gravity modes (g-modes) (Kaye et al., 1999), believed to be excited by the
interaction of convection and pulsations (e.g. Guzik et al., 2000; Grigahcène
et al., 2005; Dupret et al., 2005). Typical masses are between 1.3 and 1.8
$\,{\rm M}_{\odot}$ (Hong et al., 2022; Aerts et al., 2023) with spectral
types of F5 to A7. The high radial order of their pulsations puts them in the
asymptotic regime, meaning their oscillations are equally spaced in period for
non-rotating, non-magnetic, chemically homogeneous stars (Tassoul, 1980).
Deviations from a homogeneous spacing emerge due to chemical gradients and
rotation (Sekaran et al., 2020), where the former are influenced by the
effects of diffusive mixing; mixing reduces the steepness of chemical
gradients. Period spacing diagrams can therefore be used to derive information
on chemical composition gradients, internal rotation rates and diffusive
mixing processes (e.g. Bouabid et al., 2013; Bedding et al., 2015; Saio et
al., 2015; Van Reeth et al., 2016; Ouazzani et al., 2017; Li et al., 2019;
Miglio et al., 2008; Moravveji et al., 2015; Sekaran et al., 2021).

Space missions such as _Kepler_ (Koch et al., 2010; Borucki et al., 2010),
CoRoT (Baglin et al., 2006) and TESS (Ricker et al., 2015), have delivered a
large amount of photometric data with precisions unachievable from the ground.
The unprecedented precision has allowed for the detection of extremely low-
amplitude frequencies (Murphy et al., 2013; Bowman & Kurtz, 2018) while long
sequences of continuous observations (Lehmann et al., 2013) means that longer-
period pulsations, such as those typical of $\gamma$ Doradus stars, can be
studied. The overlap of the $\delta$ Scuti and $\gamma$ Doradus instability
strips supports the existence of $\delta$ Scuti/$\gamma$ Doradus hybrids
(Breger & Beichbuchner, 1996; Handler & Shobbrook, 2002; Yang et al., 2021).
These were expected to be rare, based on early calculations by Dupret et al.
(2005), but the lower detection thresholds provided by space missions has led
to the discovery that such behaviour is indeed common (Uytterhoeven et al.,
2011; Grigahcène et al., 2010; Bradley et al., 2015; Balona et al., 2015; Guo
et al., 2019). Later calculations by Xiong et al. (2016) conform better with
these findings.

Hybrids have great potential for asteroseismology (Schmid & Aerts, 2016). The
p-modes probe the stellar envelope while the g-modes carry information about
the near-core regions (Yang et al., 2021; Grigahcène et al., 2010; Saio et
al., 2015; Kurtz et al., 2014). The behaviour also means that information
about two different driving mechanisms can be obtained (Hong et al., 2022).

KIC 9851944 is an EB showing $\delta$ Scuti/$\gamma$ Doradus hybrid pulsation
signatures. Therefore, the object is an ideal candidate for testing our
understanding of stellar structure and evolution given the large amount of
constraints that can be obtained due to the advantages associated with hybrid
behaviour as well as binarity. KIC 9851944 is included in the _Kepler_
Eclipsing Binary Catalogue (KEBC; Prša et al. 2011; Kirk et al. 2016), as well
as a study by Matson et al. (2017) who presented RVs for 41 _Kepler_ EBs. Gies
et al. (2012, 2015) studied the eclipse times for KIC 9851944 and found no
evidence of apsidal motion or a third body; the object was also included in a
catalogue of precise eclipse times of 1279 _Kepler_ EBs by Conroy et al.
(2014).

Guo et al. (2016) combined the analysis of _Kepler_ photometry with medium-
resolution spectra ($R=\lambda/\Delta\lambda\sim 6000$) to determine the
atmospheric and physical properties of KIC 9851944; we list these results in
Table 8. Evolutionary modelling based on these properties shows the post-main-
sequence secondary to be more evolved than the main-sequence (MS) primary. The
authors concluded that both components show $\delta$ Scuti type pulsations,
which they interpreted as p-modes and p and g mixed modes, and attempted to
identify the modes by comparison with theoretically computed frequencies; the
range of theoretically predicted unstable modes agreed roughly with
observations but the authors conclude that mode identification is still
difficult in $\delta$ Scuti stars, even with constrained mass, radius and
$T_{\rm eff}$.

This work aims to be complementary to the work by Guo et al. (2016) on KIC
9851944; we additionally include observations by TESS in our photometric
analysis and combine this with the analysis of high-resolution ($R=60000$)
spectroscopic observations. Section 2 outlines the photometric and
spectroscopic observations. We determine the orbital ephemeris based on the
photometric observations in Section 3. In Section 4 we analyse RVs derived
from the spectroscopic observations and in Section 5 we present the
spectroscopic analysis. The analysis of the photometric light curves is given
in section 6 and in Section 6 we present the physical properties of the
system. An investigation of the pulsations is given in Section 8. The
discussion and conclusion are given in Sections 9 and 10, respectively.

##  2 Observations

###  2.1 Photometry

Figure 1: Cuts of the _Kepler_ Quarter 0 (top panel) and TESS Sector 15
(middle panel) processed light curves. Also shown in the bottom panels are the
phase-folded light curves observed by WASP between May 2007 and July 2007
(left), and between June 2008 and July 2008 (right).

The _Kepler_ space telescope (Koch et al., 2010) was launched in March 2009
with the primary aim of detecting Earth-like extrasolar planets around solar-
like stars. However, the mission has also provided a huge amount of high-
quality data on stars and stellar systems (Hong et al., 2022). The data were
collected using two modes of observation (Gilliland et al., 2010), long
cadence (29.5 min sampling rate) and short cadence (58.5 s sampling rate). To
keep the solar arrays pointed toward the Sun during its Earth-trailing
heliocentric orbit, the spacecraft was rotated by $90^{\circ}$ every three
months (one quarter). Primary functions of the _Kepler_ photometric analysis
module of the _Kepler_ science data processing pipeline are responsible for
generating the Simple Aperture Photometric (SAP) flux time series of the
observed objects, while secondary photometric analysis functions performed by
the Pre-Search Data Conditioning (PDC) module support systematic error
correction, giving rise to the PDCSAP fluxes (see Jenkins, 2017, for further
details on the SAP and PDCSAP measurements). KIC 9851944 was observed by
_Kepler_ in six quarters (0, 12, 13, 14, 16, 17) in short cadence mode between
May 2009 and May 2013.

The Transiting Exoplanet Satellite Survey (TESS; Ricker et al. 2015) was also
designed to discover extrasolar planets and also produced a large amount of
high-quality data for many other celestial objects. More than $2\times 10^{5}$
main-sequence dwarfs were observed at 2 min cadence during the all-sky survey,
which was divided into overlapping sectors on the sky. KIC 9851944 was
observed by TESS in sectors 14, 15, 41, 54, 55 and 56 between July 2019 and
September 2022. The TESS science pipeline is based on the _Kepler_ science
pipeline so the architecture and algorithms are similar; the photometric
analysis module computes the SAP flux time-series and the PDC module corrects
for systematics to compute the PDCSAP fluxes (Jenkins et al., 2016).

The light curves from the _Kepler_ quarters and TESS sectors mentioned above
were downloaded from the Mikulski Archive for Space Telescopes (MAST) and are
used in Section 3 to determine the ephemeris of the system, as well as Section
6 to obtain the final model of the light curve. Both SAP and PDCSAP
measurements are available; the SAP and PDCSAP fluxes were similar, which we
verified by inspecting the SAP fluxes with the PDCSAP fluxes over-plotted
after dividing them by their median flux values to put them on the same scale.
Thus, we used the SAP measurements to avoid possible biases due to the
additional processing applied to the PDCSAP data. A second-order polynomial
was fitted to fluxes that correspond to positions of quadrature, i.e., the
maximum of the ellipsoidal brightening, to estimate systematic trends present
in the light curves. Subtracting the difference between this polynomial and
the median flux of the light curve yields a model for the local median level
of out-of-eclipse flux. This model was then divided out to remove systematic
trends. The residual value of a smoothed version of the light curve subtracted
from the observed light curve was calculated and observed fluxes that deviated
by more than $1\%$ were rejected. Fluxes were converted to magnitudes and
errors were propagated following Prša (2018). The short cadence _Kepler_ light
curve from Quarter 0 and the two-minute cadence TESS light curve from Sector
15, after performing these operations, are shown in the top and middle panels
of Fig.1, respectively.

KIC 9851944 was also observed by the Wide Angle Search for Planets (WASP)
telescope (Butters et al., 2010) between May 2007 and July 2010. WASP
consisted of two robotic observatories, one in the Nothern Hemisphere at
Observatorio del Roque de los Muchachos on La Palma and the other at the South
African Astronomical Observatory (SAAO), each with eight 20 mm telescopes on a
single mount. Observations of KIC 9851944 collected by WASP between May 2007
and July 2007 as well as between June 2008 and July 2008 are shown in the
bottom panels of Fig.1 phase folded about the orbital period determined in
Section 3. The data collected by WASP for KIC 9851944 were only used to
constrain the times of primary minima when performing preliminary fits to the
_Kepler_ and TESS light curves in Section 3, to determine the orbital
ephemeris.

###  2.2 Spectroscopy

A set of 33 spectroscopic observations were obtained for the target using the
Hamilton échelle spectrograph (Vogt, 1987) on the Shane 3 m telescope at Lick
Observatory during two observing runs, one in July 2012 and the other in June
2013. The data were obtained using CCD chip no. 4, giving a wavelength
coverage of 340 – 900 nm over 89 échelle orders at a resolving power of
$R\approx 60000$.

The data were reduced using the standard pipeline for the spectrograph (e.g.
Fischer et al., 2014). Flat-fields were obtained with a quartz lamp and
divided from the spectra. The wavelength calibration was obtained from
exposures of a thorium-argon emission lamp taken roughly every hour during the
night.

Details for the normalization of the one-dimensional extracted spectra are
given in Sections 4 and 5 because each of those components of the analysis
used different approaches.

Table 1 gives the epochs of each of the 33 spectrosopic observations as well
as the signal-to-noise ratio (S/N), estimated as the square root of the counts
close to the peak of the best-exposed échelle order. Also given are the
corresponding RVs of each component that are derived in Section 4.

Table 1:  RVs and S/N corresponding to the spectroscopic observations taken at times given in the BJD column. An asterisk next to the BJD value means that observation was not used to derive the orbital parameters in Section 4.  BJD | RV Star A | RV Star B | S/N
---|---|---|---
2456133.71296 |  $-$122.47 $\pm$ | 1.33 | 103.45 $\pm$ | 1.14 | 59
2456133.72724 |  $-$119.89 $\pm$ | 1.37 | 101.95 $\pm$ | 1.08 | 59
2456133.74151 |  $-$118.46 $\pm$ | 1.26 | 99.54 $\pm$ | 1.03 | 57
2456133.75578 |  $-$114.54 $\pm$ | 1.21 | 96.79 $\pm$ | 0.93 | 57
2456133.80101 |  $-$104.26 $\pm$ | 1.28 | 87.87 $\pm$ | 1.13 | 60
2456133.83920 |  $-$94.81 $\pm$ | 1.26 | 79.68 $\pm$ | 1.25 | 58
2456133.87945 |  $-$84.36 $\pm$ | 1.02 | 71.67 $\pm$ | 1.67 | 59
2456133.90977 |  $-$77.35 $\pm$ | 1.31 | 63.03 $\pm$ | 1.97 | 58
2456133.94924 |  $-$64.66 $\pm$ | 1.51 | 51.37 $\pm$ | 2.85 | 60
2456133.98702* |  $-$41.04 $\pm$ | 5.71 | 34.50 $\pm$ | 5.52 | 57
2456134.70808 | 117.69 $\pm$ | 1.01 |  $-$121.35 $\pm$ | 1.39 | 57
2456134.72235 | 116.88 $\pm$ | 0.88 |  $-$120.01 $\pm$ | 1.68 | 59
2456134.73662 | 116.27 $\pm$ | 0.92 |  $-$119.12 $\pm$ | 1.65 | 58
2456134.75090 | 114.89 $\pm$ | 0.81 |  $-$118.39 $\pm$ | 1.66 | 55
2456134.78975 | 109.50 $\pm$ | 0.96 |  $-$113.29 $\pm$ | 1.81 | 53
2456134.82917 | 104.240 $\pm$ | 1.02 |  $-$107.26 $\pm$ | 1.72 | 50
2456134.86653 | 97.71 $\pm$ | 0.94 |  $-$99.91 $\pm$ | 1.60 | 56
2456134.91057 | 86.88 $\pm$ | 0.84 |  $-$91.01 $\pm$ | 1.74 | 53
2456134.94964 | 77.43 $\pm$ | 1.15 |  $-$82.09 $\pm$ | 1.50 | 54
2456134.98739 | 65.92 $\pm$ | 1.34 |  $-$72.46 $\pm$ | 1.18 | 52
2456469.93848 | 115.04 $\pm$ | 1.08 |  $-$115.25 $\pm$ | 1.17 | 31
2456469.97977 | 115.86 $\pm$ | 0.98 |  $-$120.41 $\pm$ | 1.65 | 55
2456470.72325* |  $-$41.06 $\pm$ | 2.92 | 29.87 $\pm$ | 5.18 | 41
2456470.79028 |  $-$69.87 $\pm$ | 1.09 | 56.92 $\pm$ | 2.79 | 35
2456470.84339 |  $-$84.53 $\pm$ | 0.94 | 69.97 $\pm$ | 1.69 | 57
2456470.90594 |  $-$101.34 $\pm$ | 1.04 | 83.79 $\pm$ | 1.11 | 56
2456470.96852 |  $-$116.02 $\pm$ | 1.22 | 95.28 $\pm$ | 1.17 | 57
2456470.99830 |  $-$121.03 $\pm$ | 1.23 | 99.82 $\pm$ | 1.12 | 54
2456471.71677* |  $\pm$ |  |  $\pm$ |  | 43
2456471.77503* | 19.52 $\pm$ | 2.84 |  $-$29.63 $\pm$ | 2.40 | 30
2456471.87251 | 58.97 $\pm$ | 1.67 |  $-$68.30 $\pm$ | 1.74 | 50
2456471.90957 | 69.15 $\pm$ | 1.34 |  $-$79.35 $\pm$ | 1.53 | 54
2456471.93874 | 79.31 $\pm$ | 1.00 |  $-$85.78 $\pm$ | 1.15 | 54

##  3 Orbital ephemeris

A first model of the _Kepler_ and TESS light curves was obtained using the
jktebop code (Southworth, 2013), which approximates stellar shapes in detached
eclipsing binaries based on the simple and efficient NDE biaxial ellipsoidal
model, so is computationally fast (Nelson & Davis, 1972; Southworth et al.,
2004). We fitted for the period $P$, epoch $T_{0}$, surface brightness ratio
$J$, sum of the fractional radii $r_{\rm A}+r_{\rm B}$, ratio of the radii
$k=\frac{r_{\rm B}}{r_{\rm A}}$, inclination $i$, the Poincaré elements
$e\cos\omega$ and $e\sin\omega$, and a light scale factor. We define star A to
be the star eclipsed at the primary (deeper) minimum and star B to be its
companion.

We also fitted for the linear coefficients $u$ of the quadratic limb darkening
(LD) law while quadratic terms were taken from Claret & Bloemen (2011) and
Claret (2017) for the _Kepler_ and TESS bands, respectively; we used the
estimated atmospheric parameters reported in the KIC to choose the appropriate
values for the quadratic coefficients. We performed fits to the WASP light
curve but the lower quality of this data compared to the _Kepler_ and TESS
light curves means that including these results in the calculation of the
overall preliminary eclipse model would lead to less well determined
parameters. Thus, we simply include the epochs of primary minimum

estimated from the WASP light curves as additional observational constraints
on $T_{0}$ in the preliminary fits to the _Kepler_ and TESS light curves.

The adopted values for the light curve parameters from this preliminary
analysis were taken as the weighted averages of the results from the fits to
the individual _Kepler_ quarters and TESS sectors, where the reciprocal of the
squared errors from the covriance matrix were used as weights.

Figure 2: jktebop model fit to the phase folded _Kepler_ light curve from
quarter 0.

These values are given in Table 9, and Fig. 2 shows the fit to the _Kepler_
Quarter 0 light curve; high-frequency variability due to pulsations of the
$\delta$ Scuti type is clearly visible at all orbital phases in the residuals
shown in the lower panel, with amplitudes of around $\pm$0.2 mmag.

The value for $r_{\rm A}+r_{\rm B}$ in Table 9 suggests that the components of
KIC 9851944 are deformed beyond the limits of applicability of the biaxial
ellipsoidal approximation (Popper & Etzel, 1981) due to their close proximity
and thus strong mutual deformations; the ellipsoidal variation ($\sim 0.04$
mag) observed in Fig. 1 and Fig. 2 is another indication of this. Furthermore,
plotting the results from individual quarters and sectors against each other
reveals strong degeneracies between the fitted parameters, specifically
$r_{\rm A}+r_{\rm B}$, $k$, $i$, $J$ and the LD coefficients. This is
highlighted by the results giving a wide range of ratios of the light
contributions of the two stars. Therefore, we present a detailed analysis of
the _Kepler_ and TESS light curves using a more sophisticated modelling code,
as well as constraints on the light ratio $\ell_{\rm B}/\ell_{\rm A}$ from our
spectral analysis, to reliably determine the light curve solution in Section
6.

Among the reliably constrained parameters from this preliminary analysis are
the period and epochs of primary minimum from individual quarters and sectors.
Therefore, the analysis is useful for establishing the orbital ephemeris. A
linear ephemeris was fit to the resulting values of $T_{0}$ against orbital
cycle using the reciprocal of the squared errors from the covariance matrix as
weights; these errors were rescaled during the fitting procedure to yield a
reduced chi-squared value of $\chi^{2}_{\nu}=1$. Fig. 3 shows the residuals of
the fit and the red line represents the linear ephemeris.

There seems to be a trend in the O-C diagram compared with the linear
ephemeris, with data points corresponding to TESS observations ($>$1000
cycles) all appearing above the red line. Thus, we also attempted to fit for a
quadratic ephemeris, which is represented by the blue dashed line in Fig. 3.
However, the corresponding quadratic term was a similar size to its errorbar.
We therefore decided to stick with the linear ephemeris which was measured to
be:

Min I (BJDTDB) = 2456308.304072(14) + 2.163901775(98) E.

Figure 3: O-C plot from the linear fit (red line) to the times of primary
minimum (black dots), where the grey shaded region represents 1 $\sigma$
uncertainty associated to the corresponding calculated values.. The blue
dashed line represents an attempt to additionally fit for a quadratic term.

##  4 Radial velocity analysis

The spectral range between 4400–4800 Å is a suitable region to carry out the
RV extraction procedure because there are many well-resolved spectral lines
and the region does not contain any wide Balmer lines. Thirteen échelle orders
within this range were corrected for cosmic ray spikes by first resampling
onto a homogeneous wavelength grid before using a median filter to smooth
them, then resampling the orders back to the original grid and subtracting
them from the observed spectrum (Blanco-Cuaresma et al., 2014a). Pixels where
the residual deviated by a threshold percentage, typically between 5 and
10$\%$, were masked and their values were corrected for using linear
interpolation. The appropriate value for the threshold was investigated for
each order individually based on visual inspections of resulting flagged
pixels. The blaze signature was removed using the method of Xu et al. (2019),
where alpha shape fitting (see, Edelsbrunner et al., 1983) combined with local
polynomial regression is used to estimate the continuum without dividing out
important spectral features.

Small errors in the continuum estimation are magnified where the blaze
function approaches zero. Following Xu et al. (2019), a better estimation at
the edges can be obtained by taking a weighted average in the overlapping
regions and applying weights such that pixels further away from the edge
within their own order are favoured. For this to be applied correctly, the
orders were re-sampled onto grids with a common wavelength step so that the
pixel averages were calculated for corresponding wavelength values between
neighbouring orders. A useful result from the edge correction procedure is
that neighbouring orders share a common edge and can be merged after
truncating the orders in the centre of the overlapping regions. The following
analysis was carried out after merging the 13 spectral orders with wavelengths
between 4400-4800 Å using this process.

Figure 4: Top: orbital fit to the corrected RVs using the merged approach.
Bottom: velocity corrections for the corresponding RVs.

Preliminary estimations for the projected rotational velocity, $v\sin i$, of
the components of KIC 9851944 were obtained by cross correlating the observed
spectra with a template broadened to a range of $v\sin i$ values between 0 and
150 km s-1 in steps of 10 km s-1. Observations taken less than 0.125 orbital
phases from the eclipses were excluded to minimise issues arising due to lines
from each component blending or being eclipsed. Interpolating the peak heights
of the correlation functions and taking the maximum yields $v\sin i_{\rm
A}=46.3\pm 0.4$ km s-1 and $v\sin i_{B}=57.2\pm 0.5$ km s-1 for the primary
and secondary component, respectively. These values for $v\sin i$ were then
used and applied to synthesise primary and secondary templates using the ispec
code (Blanco-Cuaresma et al., 2014b). In the first iteration, the $T_{\rm
eff}$ and $\log(g)$ values were taken from the Kepler Input Catalogue (KIC).
For the final iteration of the RV extraction, the atmospheric parameters of
the templates corresponded to those derived from our atmospheric analysis of
the disentangled spectra (see Section 5.1.)

Our own implementation of the todcor two-dimensional cross-correlation
algorithm (Zucker & Mazeh, 1994) was used to extract the RVs. Systematic
errors arise when using the cross-correlation technique to extract RVs because
neighbouring peaks and side-lobes in the doubled-peaked CCF disturb each
other, and this is related to blending between the spectral lines of the two
components (Andersen, 1975; Latham et al., 1996). The result is that the
estimated RVs may be shifted relative to the true value, where the size and
direction of the shift depends on the phase (see Fig.4), i.e, the relative
Doppler shifts between the components (Latham et al., 1996).

This systematic error can be minimised by correcting the RVs for this effect
(e.g., Latham et al., 1996; Torres et al., 1997, 2000; Torres & Ribas, 2002;
Southworth & Clausen, 2007). A preliminary orbit was obtained by making an
initial fit to the RVs extracted using todcor. A synthetic SB2 model was then
produced with orbital parameters corresponding to the initial fit by Doppler
shifting and adding the template spectra weighted by their relative light
contributions as estimated by the resulting value for $\ell_{\rm B}/\ell_{\rm
A}$ from the initial todcor run. The todcor algorithm was then used to extract
the known RVs from the synthetic SB2 model at orbital phases corresponding to
the observations. The difference between the calculated RV and the known RV of
the model gives an estimation for the magnitude and direction of the
systematic error at that orbital phase. The correction was then added to the
actual RVs measured from the observed spectra.

This procedure was iterated multiple times using templates with different
atmospheric parameters; the top panel of Fig. 4 shows the resulting orbital
fit to the corrected velocities that were derived in our final iteration,
where we used templates with atmospheric parameters corresponding to those
derived in Section 5.1, and the results are presented in Table 2. The middle
panel of Fig. 4 shows the residuals of the fit and the lower panel, split into
two for the primary and secondary, gives the corrections that were applied to
the extracted RVs. RVs with corrections larger than $2.5$ km s-1 were excluded
and are not shown because these RVs correspond to phases of conjunction, where
line blending effects are most severe, and negligible information is contained
on the velocity semiamplitudes. The fit was constrained to a circular orbit
because attempts to fit for eccentricity yielded values consistent with zero
and the study by Guo et al. (2016) suggests that this system has circularised.
The uncertainties on the RV measurements were rescaled by a constant factor
during the fitting procedure to yield a $\chi^{2}_{r}$ of unity. The root mean
square (RMS) of the residuals of the fit compared to the RVs for the primary
and the secondary are 1.1 km/s and 1.4 km/s, respectively.

The differences between our final results for $K_{\rm A}$ and $K_{\rm B}$
presented in Table 2 and those derived from the initial run (both after
applying corrections), which used templates with atmospheric parameters
corresponding to those from the KIC, was $\sim 0.1\%$. The corresponding
difference in $\ell_{\rm B}/\ell_{\rm A}$ was $\sim 9\%$. This shows that the
todcor light ratio is more sensitive to the atmospheric parameters of the
templates than the derived RVs. We added these differences in quadrature, to
the uncertainties derived from the covariance matrix of the fit for the
orbital parameters, and to the standard error in the mean value of $\ell_{\rm
B}/\ell_{\rm A}$ derived from different observations, in our calculation of
the parameter error bars in Table 2.

Table 2: Orbital Parameters. Parameter | Value
---|---
$K_{\rm A}$ (km/s) | 125.282 $\pm$ | 0.269
$K_{\rm B}$ (km/s) | 117.453 $\pm$ | 0.328
$\gamma$ (km/s) |  $-$5.52 $\pm$ | 0.16
$\ell_{\rm B}/\ell_{\rm A}$ | 1.222 $\pm$ | 0.136

Applying the corrections resulted in an increase in the velocity
semiamplitudes $K_{\rm A}$ and $K_{\rm B}$ by $\mathbf{0.2}\%$ and
$\mathbf{0.3}\%$, respectively. This is a small increase, suggesting that
todcor is less sensitive to blending effects than the one-dimensional cross
correlation technique. However, the $0.2\%$ and $0.3\%$ increase in the
velocity semiamplitudes translates to a $0.6\%$ and $0.9\%$ increase in the
derived masses, which is significant considering that we aim to achieve
precisions smaller than these values. This suggests that the corrections are
necessary. Furthermore, the corrections are clearly systematic, and depend on
the relative Doppler shifts of the components, as shown in the bottom panels
of Fig.4.

##  5 Spectral analysis

###  5.1 Atmospheric parameters

The method of spectral disentangling (SPD) makes it possible to separate the
spectra of individual components from a time-series of binary or multiple star
spectra (Simon & Sturm, 1994). SPD has other advantages: the optimal orbital
parameters are determined at the same time, and the disentangled spectra have
a higher S/N than the original observed spectra. The disentangled spectra
contain information on the atmospheric parameters ($T_{\rm eff}$, surface
gravity, $v\sin i$ and metallicity) of each component. The disentangled
spectra are still in the common continuum of the binary system, but can be
renormalised if the light ratio between the components is known from other
sources, e.g. from the analysis of light curves (Hensberge et al., 2000;
Pavlovski & Hensberge, 2005, 2010). Alternatively, the disentangled spectra
can be fitted with synthetic spectra to determine the light ratio between the
components, and this can be useful for partially-eclipsing systems where there
is a degeneracy between the radius ratio and light ratio in the system (Tamajo
et al., 2011; Tkachenko, 2015; Pavlovski et al., 2023). In well-determined
cases, the precision of the light ratio can approach 1% (Pavlovski et al.,
2022).

Since we planned to apply the method of SPD to extract the individual spectra
of the components, normalisation of the observed spectra was of critical
importance. We used a different approach than that in Section 4, where we
extracted RVs. Here, we used the dedicated code described in Kolbas et al.
(2015). First, the blaze function of échelle orders was fitted with a high-
order polynomial function. Then, the normalised échelle orders were merged.
When overlapping regions of successive échelle orders are sufficiently long,
the very ends were cut off because of their low S/N. Échelle orders containing
broad Balmer lines, in which it is not possible to define the blaze function
with enough precision, were treated in a special way. For these échelle
orders, the blaze function was interpolated from adjacent orders. This
produces more reliable normalisation in spectral regions with broad Balmer
lines than the usual pipeline procedures. For recent applications of this
approach, please see Pavlovski et al. (2018, 2023); Lester et al. (2019,
2022); Wang et al. (2020, 2023).

SPD was performed in the Fourier domain with the prescription by Hadrava
(1995). The FDBinary code, developed in Ilijic et al. (2004), was applied to
the time-series of normalised échelle spectra of KIC 9851944. FDBinary uses
the fast Fourier transform approach which allows flexibility in selection of
spectral segments for SPD whilst still keeping the original spectral
resolution. The orbital parameters, specifically $K_{\rm A}$ and $K_{\rm B}$,
were fixed to the solution reported in Table 2, thus SPD was used in pure
separation mode (Pavlovski & Hensberge, 2010). At this point the reconstructed
individual spectra of the components were still in the common continuum of the
binary system. A portion of separated spectra for both components in the
spectral range 4900–5290 Å is shown in Fig. 5.

Figure 5: Portion of disentangled spectra in the spectral range 4900–5290 Å. The similarity between the spectra of both components is obvious. Figure 6: Optimal fitting of the wings for H$\beta$ lines in disentangled spectra of binary system KIC 9851944. The spectral lines of metals were masked during the fitting, so were not included in calculations of the merit function. The disentangled spectra are shown in black, and the optimal fits in red. Table 3:  Determination of the atmospheric parameters from disentangled spectra for the components of KIC 9851944. The surface gravity for each component was fixed to the values determined from light curve and RVs solution as listed in Table 6. Parameter | H$\beta$ | H$\alpha$ | Metals
---|---|---|---
Wavelength range (Å) | 4750–4890 | 6500–6600 | 5120–5220
Primary |  |  |
$T_{\rm eff}$[K] | 6980$\pm$55 | 6940$\pm$70 | 7205$\pm 90$
$\log g$ [cgs] | 3.873 | 3.873 | 3.873
$v\sin i$ [ km s-1] | 56.2 | 56.22 | 56.5$\pm$1.9
LDF | 0.432$\pm$0.005 | 0.434$\pm 0.007$ | 0.439$0.009$
Secondary |  |  |
$T_{\rm eff}$[K] | 6875$\pm$45 | 6770$\pm$65 | 6990$\pm$80
$\log g$ [cgs] | 3.761 | 3.761 | 3.761
$v\sin i$ [ km s-1] | 67.2 | 67.2 | 67.2$\pm$1.5
LDF | 0.568$\pm$0.004 | 0.566$\pm 0.006$ | 0.561$\pm$0.009

Overall, a slight difference in line depth between the two components is seen.
The more massive component has deeper lines and slightly faster rotation. The
Balmer lines are broadened by Stark broadening, and generally are not
sensitive to the rotational broadening. Thus, we first optimised portions of
the disentangled spectra free of the Balmer lines, primarily to discern the
$v\sin i$ values. We then performed optimal fitting of disentangled spectra
centred on the H$\beta$ and H$\alpha$ lines, with fixed $v\sin i$.

It is well-known that the hydrogen Balmer lines are excellent diagnostic tools
for the determination of the $T_{\rm eff}$s for stars with $T_{\rm eff}<8000$
K because the degeneracy with the surface gravity vanishes (Gray, 2005).
Moreover, we can use $\log(g)$ determined for the components since this
quantity is determined with high precision from the analysis of DLEBs. In the
case of KIC 9851944, the $\log(g)$ values are determined with uncertainties of
about 0.01 dex, for both components. Therefore, preference is given to the
determination of the $T_{\rm eff}$ for the components in KIC 9851944 from line
profile fitting of Balmer lines, with fixed surface gravity. This is
advantageous over the calculation of the $T_{\rm eff}$ from metal lines
because their strength depends on the metallicity of the stellar atmosphere.

Without any appreciable changes in the light ratio in the course of the
orbital cycle, i.e. no spectra were observed in eclipse, ambiguity exists in
the reconstruction of the components’ spectra and only separation of the
spectra still in the common continuum of the binary system is possible. The
components’ spectra are correctly reconstructed but with scaling factors, i.e.
the shapes of the spectral lines are correct, but not their strength. In such
a case, the light ratio can be determined from fitting the separated spectra
with synthetic (theoretical) spectra, in the course of determination of the
atmospheric parameters. The optimal fitting was performed in constrained mode,
as defined in Tamajo et al. (2011), with the condition that the sum of the
light dilution factors is equal to unity. The starfit code (Kolbas et al.,
2015) uses a genetic algorithm to search for the best fit within a grid of
synthetic spectra pre-calculated using the uclsyn code (Smalley et al., 2001).
The uncertainties were calculated using the MCMC approach described in
Pavlovski et al. (2018). This task was straightforward due to the fixed
surface gravities and $v\sin i$ values: only the light ratio and $T_{\rm
eff}$s were optimised.

The results of the optimal fitting for all three spectral segments are given
in Table 3. We adopt the weighted average of the results from H$\alpha$ and
H$\beta$ for $T_{\rm eff}$, which is $6964\pm 43$ K and $6840\pm 37$ K for the
primary and secondary, respectively; these values are used to obtain the light
curve solution in Section 6 and are presented formally in Section 7. Our
$T_{\rm eff}$ values indicate that the KIC $T_{\rm eff}$ (6204 K) for this
object is underestimated, consistent with prior studies (e.g., Molenda-
Żakowicz et al., 2011; Lehmann et al., 2011; Tkachenko et al., 2013; Niemczura
et al., 2015) reporting similar underestimations for stars with $T_{\rm
eff}\sim 7000$ K.

With the $T_{\rm eff}$ determined from the Balmer lines, the depths of the
metal lines cannot be matched – the metal lines for both components are
slightly deeper than the synthetic spectra for given parameters. This could be
explained with the metallicity effect, which affects metal lines more than
broad H$\beta$ and H$\alpha$ lines for which Stark broadening dominates. The
effects of metallicity explain the somewhat higher $T_{\rm eff}$ (by about 200
K) found from the metal lines. From a grid of calculated synthetic spectra for
given atmospheric parameters derived from Balmer lines, we found that the
metallicity for both components is [M/H] $\sim 0.15$ dex. Since the components
of KIC 9851944 are at the cool end of where Am stars are found, this motivated
us to disentangle the spectra between 3853 – 4067 Å and compare the Ca K lines
to synthetic spectra; the findings did not suggest any Am peculiarity although
the Ca K line for the primary appears slightly weakened. Due to the relatively
high $v\sin i$ for both components and thus severe line blending, as well as
the low S/N of the individual spectra, we did not attempt to determine
individual elemental abundances.

The light ratio from the optimal fitting of the wings of H$\beta$ and
H$\gamma$ lines, spectral segments containing only metal lines, and the Mg i
triplet at around 5180 Å, are $1.315\pm 0.018$, $1.304\pm 0.025$ and $1.278\pm
0.033$ (Table 3). All three values for the light ratio are consistent within
their 1$\sigma$ uncertainties, with the light ratio determined from the wings
of H$\beta$ line being the most precise of the three.

Guo et al. (2016) determined atmospheric parameters from tomographically
reconstructed spectra of the components. Their spectra cover the wavelength
range 3930–4610 Å at a medium spectral resolution of $R=6000$. The analysis by
Guo et al. (2016) was similar to ours, but with an important difference that
they fitted complete separated spectra, whilst we concentrate on the wings of
Balmer lines. Our results for the Balmer lines corroborate their findings to
within 1$\sigma$. We also find good agreement for the $v\sin i$ values, which
are also within 1$\sigma$ for both components. Guo et al. (2016) obtained a
mean light ratio $1.34\pm 0.03$ from spectra centred at $4275$ Å – this is
slightly larger than but still consistent with our own results.

###  5.2 Direct fitting for the light ratio

We were initially unable to constrain the light ratio of the system from the
photometric analysis alone (see Section 3), so estimated it using several
independent methods. The first method was the todcor light ratio reported in
Table 2, the second was the optimal fitting of the disentangled spectra (ODS),
and a third method is developed here.

We minimize the sum of the squared residuals between synthetic composite
spectra and the observed spectra, where the synthetic composite spectra are
built by adding Doppler-shifted synthetic spectra weighted by light fractions
according to some value of $\ell_{\rm B}/\ell_{\rm A}$, and with atmospheric
parameters from the ODS analysis. We search in a grid of $\ell_{\rm
B}/\ell_{\rm A}$ values between 0.8 and 2 with ten uniformly spaced samples;
these bounds were determined after an initial trial run with bounds of 0.5 to
3. The minimum of a polynomial fit to the sum of the square residuals against
trial values for $\ell_{\rm B}/\ell_{\rm A}$ then yields a best estimate for
its value. The only free parameters of the minimization for each trial
$\ell_{\rm B}/\ell_{\rm A}$ were the coefficients of a sixth-order polynomial
that was used to normalize the observed spectra against the synthetic
spectrum. The applied Doppler shifts were fixed according to the corresponding
RV values derived in Section 4. The minimization was carried out using the
Nelder-Mead method as implemented in the Scipy python package minimize
(Virtanen et al., 2020).

Figure 7: Spectral fit to order 66 using the grid search method with optimal
normalization. Figure 8: Resulting $\ell_{\rm B}/\ell_{\rm A}$ values from the
grid search method averaged over the orders.

We repeated the process for all spectral orders showing sufficient well-
resolved lines in regions unaffected by tellurics from Earth’s atmosphere. To
save computing time, only the two observations closest to positions of
quadrature were used. This approach resulted in 30 spectral orders showing
adequate fits with well defined minima in the sum of the square residuals.
Fig.7 shows the result of this method applied to order 66, which demonstrates
the effectiveness of optimizing the normalization of the observed spectra in
the fitting routine for an order where line blending is significant. The
resulting values for $\ell_{\rm B}/\ell_{\rm A}$ after applying this method to
the selected orders are shown in Fig.8, where each result is the average of
the result from each of the two observations used.

Thus, after taking the average of the results in Fig.8, which corresponds to
the blue line, we have three independent estimations for the light ratio of
the system, i.e., todcor, ODS, and direct fitting for the light ratio. These
estimations are given in Table 4, where we take the average and its standard
deviation from the ODS method. For the rest of this study, we adopt $\ell_{\rm
B}/\ell_{\rm A}=1.300$ as the weighted mean of those values, with an errorbar
of $\pm$ 0.036 which is their standard deviation. This value is used to
ascertain the correct light curve solution is obtained in the next section.

Table 4:  Light ratios measured and adopted for KIC 9851944. Source | Light Ratio $\ell_{\rm B}/\ell_{\rm A}$
---|---
todcor | 1.222 $\pm$ | 0.136
ODS | 1.300 $\pm$ | 0.015
Direct Fit | 1.302 $\pm$ | 0.021
Adopted | 1.300 $\pm$ | 0.036

##  6 Analysis of the light curve

The components of KIC 9841944 are close to each other and thus have a
significant tidal deformation. We therefore sought to model the light curve
using a code that is based on Roche geometry. We selected the Wilson-Devinney
(WD) code (Wilson & Devinney, 1971; Wilson, 1979) for this, and used the 2004
version of the code driven using the jktwd wrapper (Southworth et al., 2011).
The user guide which accompanies the WD code (Wilson & Van Hamme, 2004)
includes a description of all input and output quantities discussed below.

The WD code is computationally expensive and is not suited to the analysis of
the full 500 000 short-cadence datapoints in one step. We therefore used the
orbital ephemeris determined in Section 3 to convert the datapoints to orbital
phase, and then binned them into a much smaller number of points. We chose a
bin size of 0.001 orbital phases during the eclipses and 0.005 outside the
eclipses, resulting in a total of 456 phase-binned datapoints suitable for
analysis with the WD code.

Figure 9:  The best-fitting WD model (green line) to the Kepler phase-binned
light curve of KIC 9851944 (red filled circles). The residuals of the fit are
plotted in the lower panel using a greatly enlarged y-axis to bring out the
detail.

Through a process of trying a large number of different fits with a range of
fixed and fitted parameters, we arrived at a good solution to the light curve.
We adopt this as the default solution, plot it in Fig. 9, and give the fitted
parameters in Table 5. It was obtained in Mode $=$ 0 with the following fitted
parameters: the light contributions of the two stars; their potentials; their
gravity darkening coefficients; the linear coefficients of the logarithmic LD
law; and the orbital inclination. We fixed the mass ratio at the spectroscopic
value, the orbital eccentricity to zero, the albedos to 1.1, the logarithmic
coefficients of LD to values from Van Hamme (1993), third light to zero, and
the rotation rates to synchronous. We adopted the maximum numerical precision
values of N1 $=$ N2 $=60$ and N1L $=$ N2L $=60$, the simple treatment of
reflection, and the Cousins $R$-band as a proxy for the Kepler passband. This
solution gives a light ratio in excellent agreement with the spectroscopic
values in Table 4. The fractional radii in Table 5 are volume-equivalent
values calculated by the lc component of the WD code.

The main confounding factors in fitting the light curve were the albedo and
gravity darkening. Albedo values around 1.0 give a good fit to the data, but
there is a wide local minimum of $\chi^{2}$ at albedos in the region of 0.0
that gives a worse fit but is frequently found by the steepest-descent
minimization method implemented in the WD code. After extensive
experimentation, we found that fixing the albedos to values slightly above 1.0
yielded the best fits and did not risk descending into the local minimum at
lower abedo values. Fitting for gravity darkening turned out to be a crucial
step in obtaining a good model, and the best-fitting values were somewhat
varied but generally around 0.8 for the primary star and 1.1 for the secondary
star. We interpret this as a systematic issue caused by residual pulsations in
the light curve and account for it in the uncertainties.

For the determination of the uncertainties of the fitted parameters we ran a
wide range of solutions with a large number of different possible approaches
to modelling the light curve (see Southworth 2020). We calculated the errorbar
for each parameter by adding in quadrature the contribution from every model
choice, which in turn was taken to be the amount that parameter changed by
versus the default solution. The following different approaches were explored.

  1. 1.

We varied the amount of binning prior to the WD solution, finding that it had
a negligible effect on the fitted parameters.

  2. 2.

We changed the numerical precision to N1 $=$ N2 $=$ N1L $=$ N2L $=50$. This
modified the fractional radii by a maximum of 0.5%.

  3. 3.

We tried the detailed reflection effect and found almost identical results.

  4. 4.

Attempts to fit for mass ratio returned a value very close to the
spectroscopic one and almost no change in the other parameters.

  5. 5.

Fitting for the rotation rates of the stars yielded values close to and
consistent with synchronous rotation and little change in the other
parameters.

  6. 6.

Fitting for albedo gave a poorer fit but again very little change in the other
parameters.

  7. 7.

Allowing for third light gave very similar parameters and a very small amount
of third light consistent with zero.

  8. 8.

Fitting for one of the temperatures of the stars instead of the two light
contributions directly (namely Mode $=$ 2) changed the fractional radii by
0.2%.

  9. 9.

Fixing the limb darkening coefficients to theoretical values changed the
fractional radii by 0.5%.

  10. 10.

Using the square-root instead of logarithmic limb darkening law had a
negligible effect.

The best-fitting parameters were highly robust against all these experiments.
The only significantly discrepant fit (neglecting our original exploratory
ones) was when we used the Cousins $I$-band instead of the $R$-band. However,
we are able to reject this fit as it is not consistent with the spectroscopic
light ratio. The calculated uncertainties are given in Table 5.

Fig. 9 shows that the best fit is extremely good, with an r.m.s. of 0.20 mmag,
but that there are systematics remaining in the residuals. We attribute these
to the WD numerical integration for points during the eclipses, and residual
pulsations for points outside eclipse. Our approach of phase-binning the data
gives a light curve practially without Poisson noise, so makes any
imperfections in the fit easily noticable. We note that the systematics in the
residuals of the fits we found to the data are too small to show up in plots
of unbinned data, so may well be present in previous work on this object.

Table 5: Summary of the parameters for the wd2004 solutions of the light curves of the systems. Detailed descriptions of the control parameters can be found in the WD code user guide (Wilson & Van Hamme, 2004). A and B refer to the primary and secondary stars, respectively. Uncertainties are only quoted when they have been robustly assessed by comparison of a full set of alternative solutions. Parameter |  wd2004 name | Value
---|---|---
Control and fixed parameters: |  |
wd2004 operation mode | mode | 0
Treatment of reflection | mref | 1
Number of reflections | nref | 1
LD law | ld | 2 (logarithmic)
Numerical grid size (normal) | n1, n2 | 60, 60
Numerical grid size (coarse) | n1l, n2l | 60, 60
Fixed parameters: |  |
Mass ratio | rm | 1.06
Phase shift | pshift | 0.0
Orbital eccentricity | e | 0.0
$T_{\rm eff}$ star A (K) | tavh | 6964
$T_{\rm eff}$ star B (K) | tavc | 6840
Bolometric albedos | alb1, alb2 | 1.1, 1.1
Rotation rates | f1, f2 | 1.0, 1.0
Logarithmic LD coefficients | y1a, y2a | 0.294, 0.293
Fitted parameters: |  |
Star A potential | phsv | $5.365\pm 0.044$
Star B potential | phsc | $4.867\pm 0.093$
Orbital inclination (°) | xincl | $73.912\pm 0.044$
Star A gravity darkening | gr1 | $0.75\pm 0.40$
Star B gravity darkening | gr2 | $1.14\pm 0.40$
Star A light contribution | hlum | $5.58\pm 0.12$
Star B light contribution | clum | $7.34\pm 0.12$
Star A linear LD coefficient | x1a | $0.658\pm 0.019$
Star B linear LD coefficient | x2a | $0.662\pm 0.013$
Fractional radius of star A |  | $0.2344\pm 0.0024$
Fractional radius of star B |  | $0.2759\pm 0.0040$

##  7 Physical properties

Table 6: Physical properties measured for the four systems analysed in this work. The units labelled with a ‘N’ are given in terms of the nominal solar quantities defined in IAU 2015 Resolution B3 (Prša et al., 2016). Parameter | Star A | Star B
---|---|---
Mass ratio | $1.0667\pm 0.0038$
Semimajor axis ($\mathcal{R}^{\rm N}_{\odot}$) | $10.805\pm 0.019$
Mass ($\mathcal{M}^{\rm N}_{\odot}$) | 1.749 $\pm$ | 0.010 | 1.866 $\pm$ | 0.010
Radius ($\mathcal{R}^{\rm N}_{\odot}$) | 2.533 $\pm$ | 0.026 | 2.981 $\pm$ | 0.044
Surface gravity ($\log$[cgs]) | 3.874 $\pm$ | 0.009 | 3.760 $\pm$ | 0.013
Synchronous velocity ( km s-1) | 59.2 $\pm$ | 0.6 | 69.7 $\pm$ | 1.0
$T_{\rm eff}$ (K) | 6964 $\pm$ | 43 | 6840 $\pm$ | 37
Luminosity $\log(L/\hbox{$\mathcal{L}^{\rm N}_{\odot}$})$ | 1.133 $\pm$ | 0.014 | 1.244 $\pm$ | 0.016
Absolute bolometric magnitude | 1.907 $\pm$ | 0.035 | 1.631 $\pm$ | 0.039
$E(B-V)$ (mag) | $0.14\pm 0.02$
Distance (pc) | $935\pm 13$

We determined the physical properties of KIC 9851944 from the spectroscopic
and photometric results obtained above. For this we used the jktabsdim code
(Southworth et al., 2005), modified to use the IAU system of nominal solar
values (Prša et al., 2016) plus the NIST 2018 values for the Newtonian
gravitational constant and the Stefan-Boltzmann constant. Errorbars were
propagated via a perturbation analysis. The results are given in Table. 6.

We determined the distance to the system using optical $BV$ magnitudes from
APASS (Henden et al., 2012), near-IR $JHK_{s}$ magnitudes from 2MASS (Cutri et
al., 2003) converted to the Johnson system using the transformations from
Carpenter (2001), and surface brightness relations from Kervella et al.
(2004). The interstellar reddening was determined by requiring the optical and
near-IR distances to match. We found a final distance of $935\pm 12$ pc, which
is significantly shorter than the distance of $999\pm 12$ pc from the Gaia DR3
parallax (Gaia Collaboration, 2016, 2021). We have no explanation for this at
present, but note that the Gaia DR2 and DR3 parallaxes of this object differ
by nearly 2$\sigma$ so might be affected by its binarity.

##  8 Asteroseismic analysis

###  8.1 Frequency analysis

Following the binary modelling, we continue with the asteroseismic analysis of
the target. Because the observed pulsations have much smaller amplitudes than
the binary signal, the quality of the TESS and WASP data are insufficient for
the asteroseismic analysis, and we limit ourselves to using the residual
Kepler light curve. This is the merged light curve of all available Kepler
short-cadence data after subtracting the best-fitting binary model, hereafter
referred to as the pulsation light curve. To minimize the impact of outliers
and instrumental effects on the asteroseismic analysis of small-amplitude
pulsations, we apply additional processing to the data. Firstly, we remove
those parts where coronal mass ejections (CMEs) or thermal and pointing
changes of the spacecraft, such as at the start of a quarter or after a safe-
mode event, have a visible impact on the quality of the light curve. Secondly,
we apply preliminary iterative pre-whitening (as described by, e.g., Van Reeth
et al., 2023) to build a tentative mathematical model of the 20 most dominant
pulsations using a sum of sine waves

$L(t)=\sum_{i=1}^{N=20}a_{i}\sin\left(2\pi\left[f_{i}\left(t-t_{0}\right)+\phi_{i}\right]\right),$
(1)

where $a_{i}$, $f_{i}$ and $\phi_{i}$ are the amplitude, frequency, and phase
of the $i^{th}$ sine wave, respectively, and $t_{0}$ is the average time stamp
of all data points in the pulsation light curve. We then identify individual
outliers in the residuals using 5$\sigma$ clipping, and remove these data
points from the pulsation light curve as well.

Next, we use iterative pre-whitening to measure the pulsation frequencies from
the resulting cleaned pulsation light curve. Hereby we iteratively fit
additional sine waves to the time series, with frequencies that correspond to
the dominant amplitudes in the Lomb-Scargle periodogram (Scargle, 1982) of the
residual pulsation light curve. However, in the particular case of KIC
9851944, there is a significant amount of red noise present in the data, most
likely caused by residual instrumental effects, even after cleaning the
pulsation light curve. To ensure that we detect all significant pulsation
frequencies (with a signal-to-noise ratio $S/N\geq 5.6$; see e.g. Baran et al.
2015), we customise our approach. Firstly, we split the frequency range
between $0\,\rm d^{-1}$ and the Nyquist frequency ($734.21\,\rm d^{-1}$) in
overlapping parts and apply the iterative pre-whitening to these individually:
from $0\,\rm d^{-1}$ to $2\,\rm d^{-1}$, from $1\,\rm d^{-1}$ to $6\,\rm
d^{-1}$, from $4\,\rm d^{-1}$ to $11\,\rm d^{-1}$, from $9\,\rm d^{-1}$ to
$21\,\rm d^{-1}$, from $19\,\rm d^{-1}$ to $51\,\rm d^{-1}$, from $49\,\rm
d^{-1}$ to $201\,\rm d^{-1}$, and from $199\,\rm d^{-1}$ to the Nyquist
frequency. In this stage, we measure all frequencies with $S/N\geq 4.0$.
Secondly, we merge the different frequency lists, keeping only those
frequencies that are dominant in a $2.5\,f_{\rm res}$-window, where $f_{\rm
res}$ is the frequency resolution of the light curve (Loumos & Deeming, 1978)
with a value of $0.00208\,\rm d^{-1}$, and that have $S/N\geq 5.6$. Thirdly,
we non-linearly optimize this filtered frequency list using the least-squares
minimization with the trust region reflective method from the lmfit python
package (Newville et al., 2019). After the optimization, we redetermined the
$S/N$ that are associated with the different frequencies, again only keeping
those with $S/N\geq 5.6$. This leaves us with a final list of 133 measured
frequencies in both the p- and g-mode regimes, considerably more than the 89
frequencies reported by Guo et al. (2016). Notably, we formally detect one
frequency at $391.5095\pm 0.0002\,\rm d^{-1}$ with an amplitude of $6.3\pm
1.2\,\mu\rm mag$. However, a detailed analysis of this sine wave reveals that
it originates from the noise properties of the short-cadence light curve. The
signal has a maximal amplitude during the first parts of quarters 13 and 14,
but is not detectable in most other parts of the light curve. Hence, we
discard this frequency and limit ourselves to the remaining frequencies, with
values below $25\,\rm d^{-1}$. These are illustrated in Fig. 10 and listed in
Table 11 in Appendix B.

Figure 10:  Lomb-Scargle periodogram of the short-cadence Kepler light curve
of KIC 9851944 (black) with the iteratively prewhitened frequencies (full red
lines).

###  8.2 Tidal perturbation analysis

As already demonstrated by Guo et al. (2016) and illustrated in Fig. 11, many
of the detected frequencies form orbital-frequency spaced multiplets, also
called “tidally split multiplets”. In our work, we identify each multiplet by
looping over all measured frequencies $f_{i}$ in order of decreasing
amplitude, and consider frequencies $f_{j}$ to be part of a multiplet around
it if

$|f_{i}-f_{j}-nf_{\rm orb}|<\Delta f,$ (2)

where $n$ is an integer chosen to minimise the left-hand side of the Equation,
and $\Delta f$ is a chosen frequency tolerance of $0.002\,\rm d^{-1}$, that is
$\approx f_{\rm res}$. In each iteration, the considered frequencies $f_{j}$
have amplitudes $a_{j}$ that are smaller than the amplitude $a_{i}$,
associated with $f_{i}$, and are not yet associated with a different
multiplet.

Figure 11:  Échelle diagram of the prewhitened frequencies of KIC 9851944,
folded with the orbital frequency $f_{\rm orb}$. Identified multiplets are
indicated in different colours ranging from purple to dark yellow, with a
white star marking the dominant frequency of each multiplet. Single
frequencies are shown in black. The marker sizes indicate the associated
amplitudes.

We identify 13 multiplets that consist of three or more components, as listed
in Table 11 in Appendix B. One of these, with the dominant frequency at
$1.386342(4)\,\rm d^{-1}$, consists of 19 frequencies that match integer
multiples of the binary orbital frequency $f_{\rm orb}$ within $\Delta f$, and
is discussed in detail below in Sect. 8.3. In their work, Guo et al. (2016)
explained the other multiplets as consequences of (i) rotational frequency
splitting, and (ii) partial occultations of the pulsation mode geometries
during the eclipses (e.g., Reed et al., 2001, 2005; Gamarova et al., 2003;
Rodríguez et al., 2004; Gamarova et al., 2005). However, it is now known that
pulsation modes are often tidally perturbed (e.g., Samadi Ghadim et al., 2018;
Steindl et al., 2021; Van Reeth et al., 2023) or tilted (e.g., Fuller et al.,
2020; Handler et al., 2020; Kurtz et al., 2020; Van Reeth et al., 2022) in
close binaries.

Hence, to determine the true origin of the detected multiplets, we evaluate
their dependence on binary orbital phase in detail. Each multiplet with three
or more components is studied individually by removing all measured
variability that is not associated with that multiplet from the pulsation
light curve, that is, we subtract all sine waves with frequencies that are not
in the multiplet from the pulsation light curve. We then fold the residual
light curve with the orbital period, split the data in 50 bins, and fit the
dominant sine wave of the multiplet to the data, optimising the amplitude and
phase for each orbital-phase bin. The reason for this approach is twofold: (i)
the frequency spacings within the detected multiplets are not always exact
multiples of $f_{\rm orb}$, and (ii) there can be frequencies missing within
the detected multiplets. As a result, the analytical reconstruction from
Jayaraman et al. (2022) is not well suited for this star.

The results are illustrated in Fig. 12 for the dominant pulsation, and shown
in Appendix C in Figs. 18 to 29 for all pulsations. In each figure, the
relevant frequency multiplet is plotted in the left-hand panel, with a white
star marking the dominant frequency. On the right-hand side, the middle and
bottom panels show the orbit-phase dependence of the pulsation amplitude and
phase, respectively. For reference, the top right panel shows the orbital-
phase-folded light curve of the binary. From these figures we can draw several
conclusions. Firstly, because most pulsation amplitudes and phases vary
significantly at all orbital phases, and the scales of the observed pulsation
phase modulations are of the order of 0.5 to 1.5 rad, we can conclude that the
observed pulsations are tidally perturbed. While tidally tilted pulsations are
expected to have pulsation phase modulations of 0 rad, $\pi$ rad or $2\pi$
rad, which can be smeared out (Fuller et al., 2020), tidal perturbations can
result in much smaller pulsation phase modulations (e.g., Van Reeth et al.,
2023). Moreover, the observed tidally split multiplets are not perfectly
equidistant. This is in contradiction with current theoretical predictions
made for tidally perturbed (e.g., Smeyers, 2005) and tidally tilted pulsations
(Fuller et al., 2020), indicating that aspects that are currently not included
in these theoretical frameworks, such as the Coriolis force, also play a role.
Secondly, based on the different morphology of the curves for the different
pulsations, we can conclude that the pulsations have different mode geometries
(Van Reeth et al., 2023). For example, while most observed pulsations are
modulated twice per orbit, the amplitude of the pulsation with frequency
$f=5.097165(3)\,\rm d^{-1}$ only reaches one maximum per orbital cycle.
Moreover, for some pulsations the pulsation phase decreases as a function of
the orbital phase when the observed amplitudes are maximal, while for others
the pulsation phase increases. There is no detectable correlation between
these effects and the pulsation frequency, but the tidal modulations are most
easily observed for pulsations with higher (average) amplitudes. This suggests
that most if not all pulsations in this target are tidally perturbed, but that
the S/N of the lower-amplitude pulsations is too low for a detection.

Finally, we can confirm that some pulsations belong to the primary, while
others could belong to either the primary or the secondary component, in
agreement with the inferences made by Guo et al. (2016) based on asteroseismic
models. As seen in Fig. 12, the observable amplitude of the p mode with
frequency $10.39971\,\rm d^{-1}$ drops during the primary eclipse, indicating
that it belongs to the primary. The higher observed amplitudes just before and
after the primary eclipse signifies that the pulsation has a higher amplitude
on the side of the primary that is facing the secondary component, similar to
what has been observed for $g$-mode pulsations in V456 Cyg (Van Reeth et al.,
2022), and KIC 3228863 and KIC 12785282 (Van Reeth et al., 2023). By contrast,
the observable amplitudes of the p modes with frequencies $11.52234\,\rm
d^{-1}$ and $11.89048\,\rm d^{-1}$ peak during the primary eclipse, as shown
in Figs. 23 and 24. This can either indicate that these pulsations belong to
the secondary component, or it can be caused by reduced geometric mode
cancellation during the eclipse, depending on the geometry of these pulsation
modes. The origin of the observed pulsations can be investigated further using
detailed asteroseismic modelling, which would allow us to calculate the
probability that specific pulsations belong to one or the other component.
However, such a modelling study lies outside the scope of the current work.

Figure 12:  Tidally perturbed pulsation with frequency $f=10.399706(2)\,\rm
d^{-1}$. Left: Associated frequency multiplet, as shown in Fig. 11. The white
star marks the dominant frequency of the multiplet. Top right: Orbital-phase
folded light curve. Middle right: Orbital-phase dependent modulations of the
pulsation amplitude, calculated in 50 data bins, with the 1$\sigma$
uncertainty range indicated by the dashed lines. Bottom right: Orbital-phase
dependent modulations of the pulsation phase, calculated in 50 data bins, with
the 1$\sigma$ uncertainty range indicated by the dashed lines.

###  8.3 Orbital harmonic frequencies

In addition to the tidal perturbation of self-excited pulsations, and in
agreement with Guo et al. (2016), we also detect an orbital harmonic frequency
comb, illustrated in Fig. 13, which can indicate tidally excited oscillations.
However, we do not recover the same orbital harmonics as Guo et al. (2016).
They reported the detection of $8f_{\rm orb}$, $22f_{\rm orb}$, $46f_{\rm
orb}$, and $50f_{\rm orb}$. While our detected orbital harmonic frequency comb
consists of 19 frequencies, we only recover $22f_{\rm orb}$. This discrepancy
is likely caused by differences between the reduced short- and long-cadence
Kepler light curves, and between the photometric binary models. Thus, while we
confirm the presence of orbital-phase dependent variability in the pulsation
light curve, as seen in the bottom right panel of Fig. 13, its exact observed
characteristics have to be treated with caution.

As pointed out by Guo et al. (2016), the detection of an orbital harmonic
frequency comb is somewhat unexpected for synchronised binaries with circular
orbits, though it has also been detected for other such systems (da Silva et
al., 2014). Because the orbital eccentricity of the binary is zero, the
equilibrium tides that are responsible for deforming the star and perturbing
the pulsations are considerably larger than the dynamical tides that excite
oscillations. Hence, this can indicate that this system has a slightly
eccentric orbit or that one or both of the components is asynchronously
rotating, within the uncertainty margins of our measurements.

Figure 13:  Tidally excited oscillations in KIC 9851944. Left: Associated
frequency multiplet, as shown in Fig. 11. The white star marks the dominant
frequency of the multiplet. Top right: Orbital-phase folded light curve.
Bottom right: Orbital-phase folded residuals of the light curve after
subtraction of the binary model. Individual data points are shown in black.
The overplotted purple line shows the average variability as a function of
orbital phase, evaluated in 50 equal bins.

###  8.4 Gravity-mode period-spacing pattern

Finally, we analyse the observed g-mode pulsations in detail. Li et al. (2020)
reported the detection of a short period-spacing pattern of prograde sectoral
quadrupole modes, that is with $(k,m)=(0,2)$, and assigned them to the primary
component of the system, based on the stellar masses determined by Guo et al.
(2016). In this work, we confirm the pattern detection by Li et al. (2020)
using the methodology from Van Reeth et al. (2015), as shown in Fig. 14. Only
the g mode with frequency $\sim 0.461\,\rm d^{-1}$ was not recovered because
of the higher signal-to-noise ratio cutoff that we used, that is, $S/N\geq
5.6$ instead of 4.0. Moreover, the dominant g mode in the pattern, with a
frequency $f$ of $2.239718(6)\,d^{-1}$, was found to be tidally perturbed and
exhibit spatial filtering during the primary eclipse, as shown in Fig. 18 and
with the corresponding multiplet listed in Table 11. These observations
confirm that the g modes belong to the primary component of the system: the
observed tidal perturbations exhibit a dip in the amplitude and a saw-tooth-
like “glitch” in the pulsation phase during the primary eclipse, which can
only be explained if the pulsation belongs to the primary (Van Reeth et al.,
2022; Van Reeth et al., 2023).

To investigate the potentially asynchronous rotation of the pulsating star, we
modelled our detected period-spacing pattern by fitting an asymptotic period-
spacing series, following the method described by Van Reeth et al. (2016). We
confirmed the pulsation mode identification found by Li et al. (2020),
$(k,m)=(0,2)$, and found that the star has a buoyancy radius $\Pi_{0}$ of
$4370^{+690}_{-660}$ s and a near-core rotation frequency $f_{\rm rot}$ of
$0.49^{+0.05}_{-0.06}\,\rm d^{-1}$. These values agree within 1$\sigma$ with
the results from Li et al. (2020), who found $\Pi_{0}$ and $f_{\rm rot}$
values of $3500\pm 500$ s and $0.41\pm 0.05\,\rm d^{-1}$, respectively.
Moreover, both sets of values are consistent with synchronous rotation.
However, because the detected pattern is so short, the uncertainties on these
$f_{\rm rot}$ values are relatively large. Hence, there is still a possibility
that the primary is asynchronously rotating, but insufficiently strongly to be
detected with the available data.

Figure 14:  Detected period-spacing pattern of g modes with $(k,m)=(0,2)$ that
belong to the primary component of KIC 9851944. Top: part of the Lomb-Scargle
periodogram of the pulsation light curve (black) with the pulsation periods of
the modes that form the pattern (red dashed lines). Bottom: the period spacing
between consecutive modes in the detected pattern, as a function of the
pulsation period. Because there is an undetected mode between the fourth and
fifth detected pulsation modes, the fourth period spacing in the pattern is
not shown.

The error margins are smaller than the symbol sizes.

##  9 Discussion

We compared the observed properties of the components of KIC 9851944 to
isochrones calculated by MIST (Mesa Isochrones and Stellar Tracks) using the
mesa code (Paxton et al., 2011; Paxton et al., 2013, 2015; Dotter, 2016; Choi
et al., 2016; Paxton et al., 2019). We searched for the best MIST isochrone
using two methods. For method 1, we interpolated radius, $T_{\rm eff}$ and
luminosity as functions of mass. We then estimated these parameters using the
interpolants at our measured masses. The objective function to minimize is
then the sum of the quadrature distances between the interpolated and observed
locations of the components with age and metallicity as free parameters. For
method 2, we included mass in the calculation of the objective function, i.e.,
masses are not constrained to the observed values. For the minimization, we
used SciPy’s implementation of the stochastic differential evolution algorithm
(Storn & Price, 1997; Virtanen et al., 2020). Our grid of isochrones spanned
from $-$4.0 to 0.5 dex in [Fe/H] and 0.5 to 10.3 dex in $\log_{10}(\rm age)$.
The grid spacing in [Fe/H] was 0.05 dex between $-$0.5 and 0.5 dex and 0.25
dex outside that range up to $\pm 2$ dex, beyond which the grid spacing was
0.5 dex. A grid spacing of $0.05$ dex in $\log_{10}(\rm age)$ was used through
the full grid.

After the best matching coeval isochrone was found, we removed the
corresponding metallicity value from our isochrone grid and repeated the
procedure to explore the effect of varying [Fe/H] on the predicted age of the
system. We did this twice, leading to first, second, and third best matching
coeval isochrones, each corresponding to a different metallicity. For method
1, the metallicity values of the first, second and third best matching
isochrones were 0.05, 0.1 and 0.0 dex, respectively. For method 2, these
values correspond to 0.05, 0.0 and 0.1 dex, i.e., the same set, except the
second and third best matches are swapped. In all cases, the predicted age of
the system is $1.259\pm 0.073$ Gyr. The error on this age estimate is taken to
be half the average grid spacing either side of the best fitting isochrone. We
note that this age estimation is in excellent agreement with the estimation by
Guo et al. (2016) of 1.25 Gyr.

Figure 15: The three best fitting isochrones with the observed locations of
the objects over-plotted in the $M-\log(T_{\rm eff})$ (top-left), $M-\log(R)$
(top-right), $M-\log(L)$ (bottom-left) and HR (bottom-right) planes. Also
plotted are the best fitting isochrone neighbours, i.e., $\pm$ the grid
spacing in age. The locations of the isochrones within those planes
corresponding to the model with the smallest quadrature distance to the
measured quantities that were included in the objective function are indicated
for those from both model comparison methods.

The three best-fitting isochrones are shown in Fig. 15 with the observed
locations of the objects over-plotted in the $M-\log(T_{\rm eff})$,
$M-\log(R)$, $M-\log(L)$ and HR diagrams; also shown are their isochrone
neighbours, i.e., $\pm$ the grid spacing in age. For both model comparison
methods, the isochrone with [Fe/H] = 0.05 dex yielded the model with the
smallest quadrature distance to the observed quantities that were included in
the objective functions so we adopt the corresponding model predictions from
now on. Those values and their differences relative to the observed values are
given in Table 7 for both model comparison methods. The locations of these
values in the planes of Fig.15 are also shown; note the relative scales of the
axes, particularly for $T_{\rm eff}$ where the scale of the axis is much
smaller than the scales of the other axes.

Table 7 confirms that the luminosity predictions from our best matching
isochrone resulting from both comparison methods are accurate for both
components and this is also clear in the lower, left panel of Fig.15. The
other parameters are also in good agreement for star A, particularly regarding
those resulting from method 2 where the agreement is excellent. However, for
star B, the radius is underestimated while the $T_{\rm eff}$ is overestimated,
and these discrepancies compensate for each other to yield the accurate
luminosity prediction.

A possible explanation is that the secondary star is approaching the terminal
age main sequence (TAMS), i.e., an evolved stage where the sensitivity of the
models increases. Thus, a full evolutionary modelling analysis, such as that
carried out by Guo et al. (2016), might yield better model predictions because
metallicity and other parameters relating to, e.g., overshooting, are included
as free parameters so the model is more flexible. This would also provide the
means for a more detailed discussion of model comparison with observations but
is beyond the scope of this paper. Here, we simply searched in grids of pre-
computed models, which are limited by the size of the grid steps in
metallicity and age, as well as fixed input physics, to estimate the age of
the system. The parameter ranges that the isochrone neighbours span in the
planes shown in Fig.15 (i.e., about twice the uncertainty), and the agreement
with the value determined by Guo et al. (2016), is evidence that this age
estimation is accurate.

Table 7:  Model parameters of the best fitting isochrone with [Fe/H] = 0.05 dex and an age of $1.259\pm 0.073$ Gyr. The results obtained from both comparison methods are given. Also given are the differences $\Delta$ between the observations and model for each parameter; we quote this difference in units of the uncertainty associated to the observations in brackets. Parameter | Method 1 |  | Method 2
---|---|---|---
| Value |  $\Delta$[%] |  | Value |  $\Delta$[%]
Mass ratio | 1.0667 | 0.0 ( | 0.0 $\sigma$) |  | 1.0530 | -1.2 ( | 3.6 $\sigma$)
$M_{\rm A}$ [$\,{\rm M}_{\odot}$] | 1.749 | 0.0 ( | 0.0 $\sigma$) |  | 1.758 | 0.5 ( | 0.9 $\sigma$)
$M_{\rm B}$ [$\,{\rm M}_{\odot}$] | 1.866 | 0.0 ( | 0.0 $\sigma$) |  | 1.851 | -0.8 ( | 1.5 $\sigma$)
$R_{\rm A}$ [$\,{\rm R}_{\odot}$] | 2.469 | -2.5 ( | 2.5 $\sigma$) |  | 2.512 | -0.8 ( | 0.8 $\sigma$)
$R_{\rm B}$ [$\,{\rm R}_{\odot}$] | 3.214 | 7.8 ( | 5.3 $\sigma$) |  | 3.096 | 3.8 ( | 2.6 $\sigma$)
$T_{\rm eff,A}$ [K] | 6987 | 0.33 ( | 0.5 $\sigma$) |  | 6971 | 0.1 ( | 0.2 $\sigma$)
$T_{\rm eff,B}$ [K] | 6636 | -3.0 ( | 5.5 $\sigma$) |  | 6694 | -2.1 ( | 3.9 $\sigma$)
$\log(L_{\rm A}/\,{\rm L}_{\odot})$ | 1.117 | -1.4 ( | 1.0 $\sigma$) |  | 1.128 | -0.4 ( | 0.4 $\sigma$)
$\log(L_{\rm B}/\,{\rm L}_{\odot})$ | 1.256 | 1.0 ( | 0.8 $\sigma$) |  | 1.239 | -0.4 ( | 0.3 $\sigma$)

The evolutionary tracks corresponding to the masses of the models in Table 7
are shown in the $T_{\rm eff}$-$R$, and HR diagrams in Figs. 16 and 17,
respectively, as well as the observed locations of the components. While Guo
et al. (2016) find the secondary to be in the hydrogen-shell burning phase for
the same age estimate as ours, we find that the secondary has not yet
exhausted the hydrogen in its core, and is approaching the TAMS. The scenario
found here is more likely to be observed on a statistical basis since the
evolution up the HR diagram after the blue loop occurs on a very short
timescale. In any case, the secondary is more evolved and thus larger and more
massive than the primary, but cooler.

Figure 16: The $T_{\rm eff}$-$R$ plane showing the evolutionary tracks
corresponding to the models presented in Table 7. The evolutionary tracks are
shown in green and purple for star A and star B, respectively, and their
observed locations are indicated by the blue and black markers. The blue (blue
lines) and red (red lines) edges of the instability domains for low-order p-
and g-modes calculated by Xiong et al. (2016) for $\delta$ Scuti (dashed
lines) and $\gamma$ Doradus (dotted lines) stars are indicated. The thin,
black line is the ZAMS corresponding to the models in Table 7 ([Fe/H] = 0.05
dex) and the dotted black line is the ZAMS for a metallicity of [Fe/H]
$=-0.25$ dex. The thick, black line represents the best fitting isochrone
which is shown as a solid grey line in Fig.15. Transparent grey dotted lines
show the evolutionary tracks of [Fe/H] = 0.05 dex stars with other labelled
masses. Figure 17: Same as Fig.16 but in the Hertzsprung-Russell diagram.

Also shown in those figures are is the blue and red edges of the instability
strips for low-order p- (dashed lines), and g-modes (dotted lines) in $\delta$
Scuti and $\gamma$ Doradus stars, respectively, from Xiong et al. (2016). Both
stars are well within both instability domains which is in agreement with the
findings in Section 8 that both components might be pulsating. Guo et al.
(2016) find that the secondary is slightly hotter than the blue edge of the
$\gamma$ Doradus instability domain calculated by Dupret et al. (2005). This
reflects the fact that the calculations by Xiong et al. (2016) (used here)
predict a much broader overlap between the $\delta$ Scuti and $\gamma$ Doradus
instability domains so stars with a larger range in $T_{\rm eff}$ are expected
to pulsate simultaneously in p- and g-modes, i.e., hybrids are expected to be
more common. We also plotted a sub-solar ZAMS ([Fe/H] = -0.25 dex) as well as
the ZAMS corresponding to our best matching isochrone ([Fe/H = 0.05 dex] in
Fig.16 and Fig.17; this shows how the ZAMS is raised to higher luminosities
and radii for higher metallicity.

We present the results previously derived by Guo et al. (2016) in Table 8. The
13 spectroscopic observations that the previous authors used to derive their
RVs and atmospheric parameters have resolutions of $R=6000$. While it was
already discussed in section 5.1 that our atmospheric parameters agree with
those derived by Guo et al. (2016), the higher resolution ($R=60000$) of the
spectra, and larger number of observations (33) used in this work yielded a
much higher quality RV curve. This is reflected by comparing the reported
masses between the two studies; the precision in the mass estimates by Guo et
al. (2016) are $4.0\%$ and $3.9\%$ for the primary and secondary,
respectively, while we attain precisions of $0.57\%$ and $0.59\%$. The two
sets of mass measurements agree to within $0.2\sigma$ for $M_{\rm A}$ and
1.1$\sigma$ for $M_{\rm B}$. In contrast, our radius measurements differ from
those of Guo et al. (2016) by 6.6$\sigma$ for star A and 3.5$\sigma$ for star
B. However, Guo et al. (2016) note the discrepancy between their values for
the radius ratio derived from two spectroscopic methods ($k=1.22\pm 0.05$ and
$k=1.27\pm 0.29$) and the value derived from their light curve modelling
($1.41\pm 0.018$), where they _tentatively_ adopt the radii associated to the
latter. The radius ratio derived here agrees with the values derived by Guo et
al. (2016) using their spectroscopic methods within 0.8 $\sigma$ and 0.3
$\sigma$, but shows the same discrepancy compared to the value derived from
their light curve modelling, for which Guo et al. (2016) note there exists a
family of comparable solutions due to the partial nature of the eclipses.

We have assumed a circularised and synchronized orbit for KIC 9851944 because
our RV- and light-curve solutions were consistent with a circular orbit, and
the values for the component of the synchronous velocity along our line of
site are in excellent agreement with the values for $v\sin i$ derived in
section 5.1 (see Table 3). Further justification was provided by Guo et al.
(2016) where they examined the eclipse times by Conroy et al. (2014) and Gies
et al. (2015), finding that the median deviation of the phase difference
between the primary and secondary eclipses from that of a circular orbit
suggests $e\geq 0.0001$. Furthermore, the circularization time-scale for a
binary system like KIC 9851944 is 600 Myr, and the synchronization timescale
is an order of magnitude shorter (Zahn, 1977; Khaliullin & Khaliullina, 2010;
Guo et al., 2016); these timescales are shorter than the age of the system
reported by both studies.

Our estimations of the near-core and surface rotations reported in section 5.1
and 8 suggest KIC 9851944 is rotating rigidly and synchronously. This is
similar to the findings by e.g. Guo et al. (2017); Guo & Li (2019), that the
short-period EBs KIC 9592855 and KIC 7385478 both contain a $\gamma$ Doradus
pulsator that is tidally synchronised at the surface as well as at the near-
core regions. Such measurements allow for a calibration of the time-scales for
synchronization at the surface compared to the core, and thus the time-scales
associated to angular momentum transport, which further aids in the
discrimination among different theories.

In contrast to the above, our detection of the orbital harmonic frequency comb
for KIC 9851944 either suggests asynchronous rotation or a non-zero
eccentricity that has not been detected due to observational error because
these can be indicative of tidally excited oscillations. It is unclear whether
the very small value for the lower limit on the eccentricity, i.e.,
$e>0.0001$, reported above for this system would be enough to induce tidal
excitation of modes at the amplitudes observed here.

The pulsation analysis here complements the study by Guo et al. (2016) who
report splittings to some of the pulsation modes. We confirm the detection of
tidally split multiplets, and explain their origin; we present evidence to
suggest that these are due to perturbations to the pulsation mode cavaties,
i.e., tidally perturbed pulsations, by investigating their phase and amplitude
dependencies with orbital phase. Guo et al. (2016) attempted to interpret the
modes by comparing the observations with theoretically computed frequencies,
from which they concluded that the observations can be explained by low-order
p modes in the primary and the secondary, or the g mode and mixed modes of the
secondary (Guo et al., 2016). We confirm that some of the p modes belong to
the primary, and others could belong to either the primary or the secondary
from modulation of the amplitudes during eclipses. Our evidence suggests the
primary is the hybrid, and this is because the saw-tooth-like "glitch" in
pulsation phase of the tidally perturbed g mode (see Fig.18) can only be
explained if the pulsation belongs to the primary (Van Reeth et al., 2022; Van
Reeth et al., 2023). Guo et al. (2016) report that mode identification was
inconclusive, reflecting the difficulty in identifying the p-modes in $\delta$
Scuti stars.

Table 8:  Previously reported results for KIC 9851944 by Guo et al. (2016). Also given are the discrepancies $\Delta$ of our results compared to those results, given as a percentage of the previous result as well as units of sigma $\sigma$. Parameter | Star A |  | Star B
---|---|---|---
| Value |  $\Delta$ [%] |  | Value |  $\Delta$[%]
Mass ($\,{\rm M}_{\odot}$) | 1.76 $\pm$ | 0.07 | -0.6 ( | 0.2 $\sigma$) |  | 1.79 $\pm$ | 0.07 | 4.24 ( | 1.1 $\sigma$)
Radius ($\,{\rm R}_{\odot}$) | 2.27 $\pm$ | 0.03 | 11.6 ( | 6.6 $\sigma$) |  | 3.19 $\pm$ | 0.04 | -6.55 ( | 3.5 $\sigma$)
$\log(g\,[\rm cgs])$ | 3.96 $\pm$ | 0.03 | -2.2 ( | 2.7 $\sigma$) |  | 3.69 $\pm$ | 0.03 | 1.9 ( | 2.1 $\sigma$)
$v_{\rm sync}$ ( km s-1) | 51.4 $\pm$ | 0.7 | 15.2 ( | 8.4 $\sigma$) |  | 72.1 $\pm$ | 0.09 | -3.3 ( | 2.4 $\sigma$)
$T_{\rm eff}$ (K) | 7026 $\pm$ | 100 | -0.9 ( | 0.6 $\sigma$) |  | 6902 $\pm$ | 100 | -0.9 ( | 0.6 $\sigma$)
$q$ | $1.01\pm 0.03$ |  | 5.6 (1.9$\sigma$)
$a$ ($\,{\rm R}_{\odot}$) | $10.74\pm 0.014$ |  | 0.6 ($2.8\sigma$)

##  10 Conclusions

We have determined the physical properties of KIC 9851944, a short-period
detached eclipsing binary containing two F-type stars, both of which pulsate.
Our analysis is based on 33 échelle spectra plus light curves from the
_Kepler_ and TESS missions. We measure masses and $T_{\rm eff}$s to 0.6%,
radii to 1.0% and 1.5%, and 133 frequencies due to p- and g-mode pulsations.
We find no evidence of a third component, apsidal motion, or eccentricity. We
estimate the age of the system to be $\sim$1.26 Gyr by comparison of the
measured properties to the MIST model isochrones.

We investigated the systemic errors associated to using the cross correlation
technique to extract RVs, which arise due to blending between the spectral
lines of the components in a binary system. We find that the effect is small
when using todcor but correcting for the issue is still necessary because the
resulting shifts are clearly systematic in nature and have a non-negligible
impact on the results. We used three independent spectroscopic methods to
determine the light ratio for the system. The results are in agreement with
each other and consistent with the value obtained from modelling the light
curve which supports the reliability of our light curve solution. We compared
our results to those reported by Guo et al. (2016) and find that we have
improved the precision of the measured masses significantly, but the precision
in the radius estimates are comparable. Both these outcomes are expected since
we use much higher-resolution spectra but the same photometry (_Kepler_).

By analysing the residuals of the light curve model, we confirm the detection
of tidally perturbed p-mode pulsations, possibly in both components of KIC
9851944. A short period spacing pattern was detected among the g modes and was
assigned to the primary component, where perturbations to one of the g-modes
was detected. Thus, the primary component is a $\delta$ Scuti/$\gamma$ Doradus
hybrid. If pulsation mode identification can be performed (as in, e.g.,
Bedding et al., 2020) on the p modes, KIC 9851944 will become a well-equipped
laboratory for stellar physics.

## Data Availability

This paper includes data collected by the _Kepler_ and TESS missions which is
publicly available at the Mikilski Archive for Space Telescopes (MAST) at the
Space Telescope Science institute (STScl) (https://mast.stsci.edu).

## Acknowledgements

We gratefully acknowledge Barry Smalley for his insightful comments regarding
the properties of Am stars, comparing the Ca K lines with synthetics, and
sharing his thoughts on the possibility that the components of KIC 9851944 are
Am stars.

We are grateful to Kelsey Clubb for assisting in obtaining and reducing the
spectroscopic observations from the Shane telescope. TVR gratefully
acknowledges funding from the KU Leuven Research Council (grant C16/18/005:
PARADISE). We gratefully acknowledge financial support from the Science and
Technology Facilities Council. This research has made use of the SIMBAD and
CDS databases operated by the Centre de Données astronomiques de Strasbourg,
France. This paper also made use of data from the _Kepler_ and TESS missions
obtained from the MAST data archive at the Space Telescope Science Institue
(STScI). Funding for the _Kepler_ and TESS missions is provided by the NASA
Science Mission Directorate and NASA Explorer Program, respectively. STScI is
operated by the Association of Universities for Research in Astronomy, Inc.,
under NASA contract NAS 5–26555.

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##  Appendix A Ephemeris Determination

This appendix presents the results from the preliminary jktebop analaysis as
well as times of primary minimum and fitted values calculated in section 3.

Table 9:  Prelimimary light curve results from jktebop fits to the _Kepler_ and TESS light curves. We do not present the error bars associated to each parameter in Table 9 because 1; uncertainties from the covariance matrix of a light curve fit are notoriously underestimated and the results from these preliminary fits are not reliable (see below) so we did not attempt to derive better uncertainties, and 2; excluding the error bar from the result makes it clear that these are not our final values for the light curve parameters. Paramter | Value
---|---
$J$ | 0.7956
$r_{A}+r_{A}$ | 0.5091
$k$ | 1.591
$i(^{\circ})$ | 74.294
$e\cos\omega$ | -0.000120
$e\sin\omega$ | -0.012329
$u_{A}$ | 0.684
$u_{B}$ | 0.310
Table 10:  Measured and fitted times of primary minimum derived in section 3. Cycle | Observed (BJD) | Calculated (BJD) | O-C (days)
---|---|---|---
-624 | 2454958.029393 (85) | 2454958.0293642 | 0.0000293
-168 | 2455944.768607 (47) | 2455944.7685736 | 0.0000337
-154 | 2455975.063169 (69) | 2455975.0631985 | -0.0000287
-140 | 2456005.357832 (82) | 2456005.3578233 | 0.0000091
-128 | 2456031.324618 (54) | 2456031.3246446 | -0.0000261
-113 | 2456063.783268 (56) | 2456063.7831712 | 0.0000973
-100 | 2456091.913868 (59) | 2456091.9138943 | -0.0000260
-88 | 2456117.880630 (67) | 2456117.8807156 | -0.0000855
-72 | 2456152.503214 (57) | 2456152.5031440 | 0.0000708
-56 | 2456187.125554 (54) | 2456187.1255724 | -0.0000181
0 | 2456308.30403 (14) | 2456308.3040718 | -0.0000409
14 | 2456338.598621 (52) | 2456338.5986967 | -0.0000748
31 | 2456375.385071 (58) | 2456375.3850268 | 0.0000442
44 | 2456403.515741 (61) | 2456403.5157499 | -0.0000083
53 | 2456422.99070 (15) | 2456422.9908659 | -0.0001687
1106 | 2458701.57955 (17) | 2458701.5794350 | 0.0001219
1119 | 2458729.71047 (17) | 2458729.7101581 | 0.0003167
1439 | 2459422.15886 (22) | 2459422.1587261 | 0.0001436
1600 | 2459770.54729 (25) | 2459770.5469118 | 0.0003871
1613 | 2459798.67785 (22) | 2459798.6776349 | 0.0002222
1626 | 2459826.80849 (21) | 2459826.8083580 | 0.0001361

##  Appendix B Iteratively prewhitened frequencies

Table 11: Parameter values associated with the iteratively prewhitened frequencies, which were obtained as described in Sec. 8.1. The frequencies are grouped, with the first set consisting of those labelled as independent frequencies, and the following groups consisting of combination frequencies, sorted according to the dominant parent frequency. In the last column, we list the identified combinations. Combination frequencies that are found within the frequency resolution $f_{\rm res}$ but not within $3\sigma$, are indicated with ∗ in the first column and with $\approx$ in the final column. | frequency $f$ | amplitude $a$ | phase $\phi$ | signal-to-noise $S/N$ | comments
---|---|---|---|---|---
| ($\rm d^{-1}$) | (mmag) | ($2\pi$ rad) |  |
$f_{1}$ | 0.029864(4) | 0.3292(12) | 0.4796(6) | 20.42 |
$f_{2}$ | 0.04104(12) | 0.1145(12) | 0.246(2) | 7.15 |
$f_{3}$ | 0.050798(11) | 0.1231(12) | -0.482(2) | 7.72 |
$f_{4}$ | 0.429539(12) | 0.1126(12) | -0.412(2) | 8.25 |
$f_{6}$ | 1.13642(2) | 0.0559(12) | 0.196(3) | 5.68 |
$f_{7}$ | 1.26821(2) | 0.0855(12) | 0.151(2) | 9.41 |
$f_{8}$ | 1.297029(9) | 0.1389(12) | 0.044(14) | 15.59 |
$f_{14}$ | 2.1344(2) | 0.0598(12) | 0.345(3) | 10.12 |
$f_{15}$ | 2.20293(2) | 0.0554(12) | -0.028(3) | 9.51 |
$f_{16}$ | 2.239718(6) | 0.23(12) | -0.0492(8) | 39.16 |
$f_{17}$ | 2.27839(4) | 0.0363(12) | -0.258(5) | 6.12 |
$f_{22}$ | 4.14574(3) | 0.04(12) | -0.092(5) | 5.63 |
$f_{27}$ | 5.097165(3) | 0.5158(12) | -0.301(4) | 55.8 |
$f_{28}$ | 5.53879(2) | 0.0597(13) | -0.271(3) | 5.91 |
$f_{30}$ | 5.99419(2) | 0.0644(12) | 0.442(3) | 5.87 |
$f_{34}$ | 6.59017(2) | 0.0672(12) | 0.469(3) | 6.12 |
$f_{35}$ | 7.22671(2) | 0.0824(12) | -0.288(2) | 8.08 |
$f_{45}$ | 9.59223(5) | 0.0284(12) | -0.486(7) | 7.4 |
$f_{53}$ | 10.176017(2) | 0.6515(12) | 0.429(3) | 172.1 |
$f_{54}$ | 10.399706(2) | 0.8015(12) | -0.2904(2) | 221.19 |
$f_{58}$ | 10.90789(6) | 0.0211(12) | 0.045(9) | 5.79 |
$f_{60}$ | 11.00533(2) | 0.0579(12) | 0.059(3) | 16.02 |
$f_{61}$ | 11.018536(5) | 0.2785(12) | 0.3112(7) | 76.68 |
$f_{63}$ | 11.08192(6) | 0.0237(12) | 0.408(8) | 6.61 |
$f_{66}$ | 11.27246(3) | 0.0487(12) | 0.117(4) | 13.83 |
$f_{67}$ | 11.3651(5) | 0.0275(12) | -0.247(7) | 7.76 |
$f_{68}$ | 11.41982(7) | 0.1941(12) | -0.3938(10) | 54.46 |
$f_{70}$ | 11.43819(3) | 0.0376(12) | -0.15(5) | 10.57 |
$f_{71}$ | 11.52234(5) | 0.2606(12) | -0.4854(7) | 73.33 |
$f_{74}$ | 11.890477(2) | 0.5704(12) | 0.1807(3) | 164.93 |
$f_{90}$ | 13.61013(6) | 0.0213(12) | 0.222(9) | 7.7 |
$f_{93}$ | 13.76588(7) | 0.019(12) | 0.161(10) | 7.05 |
$f_{98}$ | 14.01097(2) | 0.0589(12) | -0.244(3) | 23.17 |
$f_{100}$ | 14.21089(3) | 0.0417(12) | -0.354(5) | 16.79 |
$f_{102}$ | 14.315077(5) | 0.2819(12) | 0.4401(7) | 115.05 |
$f_{103}$ | 14.39805(2) | 0.0527(12) | -0.117(4) | 21.96 |
$f_{104}$ | 14.448108(7) | 0.1824(12) | -0.3085(11) | 77.2 |
$f_{105}$ | 14.4932(3) | 0.0396(12) | 0.333(5) | 16.83 |
$f_{107}$ | 14.69426(7) | 0.0175(12) | 0.023(11) | 7.6 |
$f_{111}$ | 14.83253(6) | 0.022(12) | 0.083(9) | 9.79 |
$f_{125}$ | 19.126701(8) | 0.1584(12) | -0.4735(12) | 91.81 |
$f_{126}$ | 19.42779(2) | 0.0847(12) | -0.419(2) | 47.64 |
$f_{127}$ | 20.15601(9) | 0.0139(12) | -0.124(14) | 7.51 |
$f_{129}$ | 20.82366(3) | 0.0432(12) | 0.429(4) | 21.19 |
$f_{5}^{*}$ | 0.462195(5) | 0.2625(12) | -0.3711(7) | 19.39 | $\approx 1f_{\rm orb}$
$f_{10}^{*}$ | 1.386342(4) | 0.3628(12) | -0.3605(5) | 44.54 | $\approx 3f_{\rm orb}$
$f_{13}^{*}$ | 1.848604(10) | 0.1336(12) | 0.2485(14) | 20.41 | $\approx 4f_{\rm orb}$
$f_{18}$ | 2.310647(4) | 0.3462(12) | 0.3839(6) | 58.23 | 5$f_{\rm orb}$
$f_{20}$ | 2.77278(7) | 0.1827(12) | 0.0065(11) | 32.88 | 6$f_{\rm orb}$
$f_{21}$ | 3.234899(8) | 0.1651(12) | -0.3581(12) | 29.22 | 7$f_{\rm orb}$
$f_{24}^{*}$ | 4.62137(3) | 0.051(12) | 0.015(4) | 6.14 | $\approx 10f_{\rm orb}$
$f_{26}^{*}$ | 5.08347(2) | 0.0676(12) | -0.349(3) | 7.34 | $\approx 11f_{\rm orb}$
$f_{31}^{*}$ | 6.007556(11) | 0.1215(12) | -0.13(2) | 11.09 | $\approx 13f_{\rm orb}$
$f_{36}^{*}$ | 7.85624(2) | 0.0794(12) | 0.441(2) | 9.71 | $\approx 17f_{\rm orb}$
$f_{39}$ | 8.7805(4) | 0.0331(12) | -0.314(6) | 7.19 | 19$f_{\rm orb}$
$f_{52}^{*}$ | 10.16693(3) | 0.0453(12) | -0.448(4) | 11.96 | $\approx 22f_{\rm orb}$
$f_{64}$ | 11.09106(2) | 0.0642(12) | -0.195(3) | 17.87 | 24$f_{\rm orb}$
$f_{77}$ | 12.01533(3) | 0.0517(12) | 0.058(4) | 15.06 | 26$f_{\rm orb}$
$f_{95}$ | 13.86382(6) | 0.021(12) | 0.064(9) | 7.98 | 30$f_{\rm orb}$
$f_{109}$ | 14.7881(4) | 0.0326(12) | 0.314(6) | 14.44 | 32$f_{\rm orb}$
Table 12: continued | frequency $f$ | amplitude $a$ | phase $\phi$ | signal-to-noise $S/N$ | comments
---|---|---|---|---|---
| ($\rm d^{-1}$) | (mmag) | ($2\pi$ rad) |  |
$f_{118}$ | 15.71249(6) | 0.0201(12) | -0.435(10) | 9.87 | 34$f_{\rm orb}$
$f_{122}$ | 17.56078(6) | 0.0202(12) | -0.407(10) | 10.72 | 38$f_{\rm orb}$
$f_{124}$ | 18.48512(5) | 0.0242(12) | -0.172(8) | 13.89 | 40$f_{\rm orb}$
$f_{9}$ | 1.315491(12) | 0.1099(12) | 0.252(2) | 12.54 |  $f_{16}$ \- 2$f_{\rm orb}$
$f_{12}^{*}$ | 1.77778(2) | 0.0577(12) | -0.476(3) | 8.7 |  $\approx f_{16}$ \- $f_{\rm orb}$
$f_{23}^{*}$ | 4.17278(2) | 0.062(12) | -0.128(3) | 8.61 |  $\approx f_{27}$ \- 2$f_{\rm orb}$
$f_{25}$ | 4.635033(12) | 0.1126(12) | 0.394(2) | 13.54 |  $f_{27}$ \- $f_{\rm orb}$
$f_{29}$ | 5.559303(5) | 0.2567(12) | -0.1643(8) | 25.3 |  $f_{27}$ \+ $f_{\rm orb}$
$f_{32}^{*}$ | 6.021343(13) | 0.1002(12) | -0.298(2) | 9.15 |  $\approx f_{27}$ \+ 2$f_{\rm orb}$
$f_{33}^{*}$ | 6.48364(2) | 0.0743(12) | 0.31(3) | 6.72 |  $\approx f_{27}$ \+ 3$f_{\rm orb}$
$f_{131}$ | 22.06635(5) | 0.0237(12) | -0.077(8) | 12.1 |  $f_{52}$ \+ $f_{73}$
$f_{132}$ | 24.70539(5) | 0.0264(12) | 0.45(7) | 14.83 |  $f_{73}$ \+ $f_{81}$
$f_{11}$ | 1.39562(2) | 0.0572(12) | 0.312(3) | 7.06 |  $f_{53}$ \- 19$f_{\rm orb}$
$f_{19}$ | 2.319817(12) | 0.1116(12) | -0.005(2) | 18.75 |  $f_{53}$ \- 17$f_{\rm orb}$
$f_{37}$ | 8.32746(2) | 0.0582(12) | -0.077(3) | 9.42 |  $f_{53}$ \- 4$f_{\rm orb}$
$f_{40}$ | 8.78961(3) | 0.0418(12) | 0.348(5) | 9.09 |  $f_{53}$ \- 3$f_{\rm orb}$
$f_{42}$ | 9.25174(2) | 0.0625(12) | 0.162(3) | 15.84 |  $f_{53}$ \- 2$f_{\rm orb}$
$f_{47}$ | 9.71391(2) | 0.0599(12) | -0.347(3) | 15.77 |  $f_{53}$ \- $f_{\rm orb}$
$f_{57}$ | 10.63816(2) | 0.058(12) | -0.107(3) | 15.93 |  $f_{53}$ \+ $f_{\rm orb}$
$f_{65}^{*}$ | 11.10018(3) | 0.0449(12) | -0.321(4) | 12.47 |  $\approx f_{53}$ \+ 2$f_{\rm orb}$
$f_{72}^{*}$ | 11.56252(4) | 0.0362(12) | 0.098(5) | 10.34 |  $\approx f_{53}$ \+ 3$f_{\rm orb}$
$f_{78}$ | 12.02459(4) | 0.0327(12) | -0.062(6) | 9.52 |  $f_{53}$ \+ 4$f_{\rm orb}$
$f_{84}$ | 12.94893(4) | 0.0356(12) | 0.191(5) | 11.25 |  $f_{53}$ \+ 6$f_{\rm orb}$
$f_{96}$ | 13.87319(5) | 0.0283(12) | 0.428(7) | 10.81 |  $f_{53}$ \+ 8$f_{\rm orb}$
$f_{110}^{*}$ | 14.79735(8) | 0.0163(12) | -0.299(12) | 7.22 |  $\approx f_{53}$ \+ 10$f_{\rm orb}$
$f_{38}^{*}$ | 8.55106(4) | 0.0327(12) | -0.308(6) | 6.3 |  $\approx f_{54}$ \- 4$f_{\rm orb}$
$f_{43}^{*}$ | 9.47551(2) | 0.067(12) | 0.496(3) | 17.51 |  $\approx f_{54}$ \- 2$f_{\rm orb}$
$f_{48}$ | 9.93758(14) | 0.0905(12) | 0.007(2) | 23.92 |  $f_{54}$ \- $f_{\rm orb}$
$f_{73}^{*}$ | 11.78622(4) | 0.0348(12) | 0.096(6) | 10.11 |  $\approx f_{54}$ \+ 3$f_{\rm orb}$
$f_{81}^{*}$ | 12.71017(5) | 0.0264(12) | 0.386(7) | 8.21 |  $\approx f_{54}$ \+ 5$f_{\rm orb}$
$f_{91}^{*}$ | 13.63457(8) | 0.0158(12) | -0.362(12) | 5.72 |  $\approx f_{54}$ \+ 7$f_{\rm orb}$
$f_{49}$ | 10.08101(3) | 0.0377(12) | -0.22(5) | 9.93 |  $f_{60}$ \- 2$f_{\rm orb}$
$f_{50}$ | 10.094274(11) | 0.1133(12) | 0.134(2) | 29.92 |  $f_{61}$ \- 2$f_{\rm orb}$
$f_{75}$ | 11.942814(7) | 0.1987(12) | 0.13(10) | 57.43 |  $f_{61}$ \+ 2$f_{\rm orb}$
$f_{41}$ | 9.21174(5) | 0.0262(12) | 0.358(7) | 6.61 |  $f_{71}$ \- 5$f_{\rm orb}$
$f_{46}$ | 9.67396(5) | 0.0273(12) | -0.086(7) | 7.21 |  $f_{71}$ \- 4$f_{\rm orb}$
$f_{51}$ | 10.13587(4) | 0.0313(12) | -0.405(6) | 8.28 |  $f_{71}$ \- 3$f_{\rm orb}$
$f_{56}$ | 10.598064(10) | 0.1353(12) | 0.2578(14) | 37.3 |  $f_{71}$ \- 2$f_{\rm orb}$
$f_{62}$ | 11.06027(3) | 0.0476(12) | -0.186(4) | 13.31 |  $f_{71}$ \- $f_{\rm orb}$
$f_{76}$ | 11.9845(5) | 0.0274(12) | 0.103(7) | 8.01 |  $f_{71}$ \+ $f_{\rm orb}$
$f_{80}$ | 12.44658(5) | 0.0237(12) | -0.383(8) | 7.04 |  $f_{71}$ \+ 2$f_{\rm orb}$
$f_{83}$ | 12.90873(7) | 0.0189(12) | 0.327(10) | 5.96 |  $f_{71}$ \+ 3$f_{\rm orb}$
$f_{87}$ | 13.37075(5) | 0.0246(12) | -0.004(8) | 8.46 |  $f_{71}$ \+ 4$f_{\rm orb}$
$f_{101}$ | 14.29517(9) | 0.0146(12) | 0.283(13) | 5.95 |  $f_{71}$ \+ 6$f_{\rm orb}$
$f_{44}$ | 9.58038(4) | 0.0297(12) | -0.181(6) | 7.74 |  $f_{74}$ \- 5$f_{\rm orb}$
$f_{55}^{*}$ | 10.50457(2) | 0.0551(12) | 0.106(3) | 15.18 |  $\approx f_{74}$ \- 3$f_{\rm orb}$
$f_{59}$ | 10.96637(3) | 0.042(12) | -0.292(5) | 11.57 |  $f_{74}$ \- 2$f_{\rm orb}$
$f_{69}$ | 11.42854(2) | 0.0649(12) | 0.357(3) | 18.21 |  $f_{74}$ \- $f_{\rm orb}$
$f_{79}$ | 12.35285(2) | 0.08(12) | -0.375(2) | 23.84 |  $f_{74}$ \+ $f_{\rm orb}$
$f_{82}^{*}$ | 12.814957(5) | 0.2491(12) | 0.0752(8) | 77.37 |  $\approx f_{74}$ \+ 2$f_{\rm orb}$
$f_{85}$ | 13.27707(3) | 0.0514(12) | -0.101(4) | 17.07 |  $f_{74}$ \+ 3$f_{\rm orb}$
$f_{92}$ | 13.73916(3) | 0.0399(12) | -0.488(5) | 14.76 |  $f_{74}$ \+ 4$f_{\rm orb}$
Table 13: continued | frequency $f$ | amplitude $a$ | phase $\phi$ | signal-to-noise $S/N$ | comments
---|---|---|---|---|---
| ($\rm d^{-1}$) | (mmag) | ($2\pi$ rad) |  |
$f_{99}$ | 14.2014(3) | 0.0509(12) | 0.167(4) | 20.4 |  $f_{74}$ \+ 5$f_{\rm orb}$
$f_{106}$ | 14.66346(4) | 0.031(12) | -0.274(6) | 13.37 |  $f_{74}$ \+ 6$f_{\rm orb}$
$f_{113}$ | 15.12563(4) | 0.0365(12) | 0.415(5) | 17.04 |  $f_{74}$ \+ 7$f_{\rm orb}$
$f_{117}$ | 15.5876(6) | 0.0227(12) | -0.035(8) | 11.38 |  $f_{74}$ \+ 8$f_{\rm orb}$
$f_{119}$ | 16.04978(7) | 0.0198(12) | -0.329(10) | 10.23 |  $f_{74}$ \+ 9$f_{\rm orb}$
$f_{112}$ | 14.93524(5) | 0.0278(12) | 0.498(7) | 12.66 |  $f_{98}$ \+ 2$f_{\rm orb}$
$f_{86}$ | 13.28676(5) | 0.0259(12) | 0.304(7) | 8.65 |  $f_{100}$ \- 2$f_{\rm orb}$
$f_{88}$ | 13.39085(4) | 0.0341(12) | 0.204(6) | 11.76 |  $f_{102}$ \- 2$f_{\rm orb}$
$f_{94}^{*}$ | 13.85328(4) | 0.0326(12) | -0.282(6) | 12.38 |  $\approx f_{102}$ \- $f_{\rm orb}$
$f_{108}$ | 14.77747(4) | 0.031(12) | -0.005(6) | 13.62 |  $f_{102}$ \+ $f_{\rm orb}$
$f_{114}$ | 15.2393(3) | 0.0473(12) | -0.302(4) | 22.42 |  $f_{102}$ \+ 2$f_{\rm orb}$
$f_{120}$ | 16.16355(5) | 0.0262(12) | -0.061(7) | 13.58 |  $f_{102}$ \+ 4$f_{\rm orb}$
$f_{115}$ | 15.32214(8) | 0.0173(12) | -0.351(11) | 8.23 |  $f_{103}$ \+ 2$f_{\rm orb}$
$f_{89}$ | 13.52388(2) | 0.0822(12) | 0.432(2) | 29.63 |  $f_{104}$ \- 2$f_{\rm orb}$
$f_{116}$ | 15.37227(8) | 0.0156(12) | -0.108(12) | 7.54 |  $f_{104}$ \+ 2$f_{\rm orb}$
$f_{97}$ | 13.90834(6) | 0.02(12) | 0.279(10) | 7.68 |  $f_{111}$ \- 2$f_{\rm orb}$
$f_{121}$ | 17.27819(3) | 0.0423(12) | -0.464(5) | 22.4 |  $f_{125}$ \- 4$f_{\rm orb}$
$f_{123}$ | 17.74024(7) | 0.019(12) | 0.113(10) | 10.38 |  $f_{125}$ \- 3$f_{\rm orb}$
$f_{128}$ | 20.3521(2) | 0.0686(12) | -0.152(3) | 35.16 |  $f_{126}$ \+ 2$f_{\rm orb}$
$f_{130}$ | 21.27623(2) | 0.0601(12) | -0.429(3) | 29.62 |  $f_{126}$ \+ 4$f_{\rm orb}$

##  Appendix C Detected tidally perturbed pulsations

In this Appendix we illustrate all detected tidally perturbed pulsations of
KIC 9851944. In each figure, the left panel shows a zoom-in of the associated
frequency multiplet, as plotted in Fig. 11. In the top right panel, we see the
orbital-phase folded light curve of the binary. In the middle and bottom right
panels, we see the orbital-phase dependent modulations of the pulsation
amplitudes and phases, respectively. These are calculated in 50 data bins,
with the 1$\sigma$ uncertainty range indicated by the dashed lines.

Figure 18:  Tidally perturbed pulsation with frequency $f=2.239718(6)\,\rm
d^{-1}$. Figure 19:  Tidally perturbed pulsation with frequency
$f=5.097165(3)\,\rm d^{-1}$. Figure 20:  Tidally perturbed pulsation with
frequency $f=10.176017(2)\,\rm d^{-1}$. Figure 21:  Tidally perturbed
pulsation with frequency $f=10.399706(2)\,\rm d^{-1}$. Figure 22:  Tidally
perturbed pulsation with frequency $f=11.018536(5)\,\rm d^{-1}$. Figure 23:
Tidally perturbed pulsation with frequency $f=11.522340(5)\,\rm d^{-1}$.
Figure 24:  Tidally perturbed pulsation with frequency $f=11.890477(2)\,\rm
d^{-1}$. Figure 25:  Tidally perturbed pulsation with frequency
$f=14.210888(3)\,\rm d^{-1}$. Figure 26:  Tidally perturbed pulsation with
frequency $f=14.315077(5)\,\rm d^{-1}$. Figure 27:  Tidally perturbed
pulsation with frequency $f=14.448108(7)\,\rm d^{-1}$. Figure 28:  Tidally
perturbed pulsation with frequency $f=19.126701(8)\,\rm d^{-1}$. Figure 29:
Tidally perturbed pulsation with frequency $f=19.427792(15)\,\rm d^{-1}$.