# Pulsation Phases and Mode Identification of Tidally Excited Oscillations in
Fourteen Kepler Heartbeat Stars

Min-Yu Li Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216,
People’s Republic of China  School of Physics and Astronomy, Yunnan
University, Kunming 650500, People’s Republic of China  University of Chinese
Academy of Sciences, No.1 Yanqihu East Road, Huairou District, Beijing 101408,
People’s Republic of China  Center for Astronomical Mega-Science, Chinese
Academy of Sciences, 20A Datun Road, Chaoyang District, Beĳing, 100012,
People’s Republic of China  Sheng-Bang Qian E-mail<EMAIL_ADDRESS>Yunnan
Observatories, Chinese Academy of Sciences, Kunming 650216, People’s Republic
of China  School of Physics and Astronomy, Yunnan University, Kunming 650500,
People’s Republic of China  University of Chinese Academy of Sciences, No.1
Yanqihu East Road, Huairou District, Beijing 101408, People’s Republic of
China  Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A
Datun Road, Chaoyang District, Beĳing, 100012, People’s Republic of China  Li-
Ying Zhu E-mail<EMAIL_ADDRESS>Yunnan Observatories, Chinese Academy of
Sciences, Kunming 650216, People’s Republic of China  University of Chinese
Academy of Sciences, No.1 Yanqihu East Road, Huairou District, Beijing 101408,
People’s Republic of China  Center for Astronomical Mega-Science, Chinese
Academy of Sciences, 20A Datun Road, Chaoyang District, Beĳing, 100012,
People’s Republic of China  Zhao Guo Department of Applied Mathematics and
Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK  Wen-Ping
Liao Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216,
People’s Republic of China  University of Chinese Academy of Sciences, No.1
Yanqihu East Road, Huairou District, Beijing 101408, People’s Republic of
China  Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A
Datun Road, Chaoyang District, Beĳing, 100012, People’s Republic of China  Er-
Gang Zhao  Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216,
People’s Republic of China  University of Chinese Academy of Sciences, No.1
Yanqihu East Road, Huairou District, Beijing 101408, People’s Republic of
China  Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A
Datun Road, Chaoyang District, Beĳing, 100012, People’s Republic of China
Xiang-Dong Shi Yunnan Observatories, Chinese Academy of Sciences, Kunming
650216, People’s Republic of China  University of Chinese Academy of Sciences,
No.1 Yanqihu East Road, Huairou District, Beijing 101408, People’s Republic of
China  Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A
Datun Road, Chaoyang District, Beĳing, 100012, People’s Republic of China  Fu-
Xing Li Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216,
People’s Republic of China  University of Chinese Academy of Sciences, No.1
Yanqihu East Road, Huairou District, Beijing 101408, People’s Republic of
China  Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A
Datun Road, Chaoyang District, Beĳing, 100012, People’s Republic of China  Qi-
Bin Sun Yunnan Observatories, Chinese Academy of Sciences, Kunming 650216,
People’s Republic of China  University of Chinese Academy of Sciences, No.1
Yanqihu East Road, Huairou District, Beijing 101408, People’s Republic of
China  Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A
Datun Road, Chaoyang District, Beĳing, 100012, People’s Republic of China

###### Abstract

Tidally excited oscillations (TEOs) in Heartbeat Stars (HBSs) are an essential
probe of the internal properties of the systems, but their potential has not
yet been fully exploited. Based on the orbital parameters and TEO candidates
from our previous works, we identify the pulsation phases and mode of TEOs in
fourteen Kepler HBSs. The pulsation phases of most systems can be explained by
the dominant modes spherical harmonic $l=2$, $m=0$, or $\pm 2$ modes, assuming
the spin and orbital axes are aligned and the pulsations are adiabatic, except
for five systems that show large median deviations ($>2\sigma$). The largest
deviation ($>6\sigma$), in KIC 8459354, can be explained by the spin-orbit
misalignment since the high eccentricity and the long orbital period. Besides
the two existing reasons for the large deviations, we additionally suggest
that the apsidal motion could be another reason. In addition, for KIC
11122789, almost half of the harmonics show large deviations, we suggest that
these harmonics may not be considered as TEO candidates. This phases and mode
identification approach can then be used inversely to verify the TEO
candidates derived by the Fourier analysis.

Binary stars (154) – Elliptical orbits (457) – Stellar oscillations (1617) –
Pulsating variable stars (1307)

##  1 Introduction

Tidally excited oscillations (TEOs) are induced by the phase-dependent tides
in binary stars with eccentric orbits (Zahn, 1975; Kumar et al., 1995; Fuller,
2017; Jackiewicz, 2021; Kołaczek-Szymański et al., 2022). They can occur in
some of the Heartbeat Stars (HBSs) and in turn affect the evolution of the
HBSs, including the synchronization and circularization of binary stars.
Therefore, TEOs can be a probe to the binary system parameters and potentially
to the stellar interiors (Guo et al., 2020). Theoretical work was done several
decades ago (Zahn, 1975; Kumar et al., 1995), but it is the Kepler satellite
(Borucki et al., 2010) that provides the opportunity to study such small
variations in photometric light curves. A series of works have reported some
studies of TEOs in the Kepler HBSs.

The Kepler HBS KOI-54 (KIC 8112039) was first reported by Welsh et al. (2011),
with the two dominant TEOs being the 90th and 91st harmonics of the orbital
frequency. KOI-54 was then studied in a series of papers. Fuller & Lai (2012)
studied the nature of the 90th and 91st harmonics and suggested that the
system could naturally evolve to a state where at least one $m=2$ mode is
locked in resonance with the orbit. However, Burkart et al. (2012) found that
chance resonance of $l=2$, $m=0$ modes can also explain the observed large
amplitude of the 90th and 91st harmonics without invoking resonance locking.
O’Leary & Burkart (2014) also found that the oscillation modes are
axisymmetric and thus cannot be responsible for resonance locking. Most of the
high frequency harmonic pulsations are found to be $m=0$ oscillations.
Recently, Guo et al. (2022) reexamined the non-linear TEOs in KOI-54 and found
that the non-orbital-harmonic TEOs show a period spacing pattern, likely
arising from $l=2$, $m=0$ eigen-modes. The two largest-amplitude TEOs at 90th
and 91st harmonics are very close to resonances, and likely from different
stars. Guo et al. (2017, 2019) modeled both the amplitudes and phases of TEOs
in the eclipsing HBSs KIC 3230227 and KIC 4142768, respectively. They found
that most TEO phases in the two systems are consistent with $l=2$, $m=2$
modes.

Guo et al. (2020) studied the pulsation phases of TEOs in eight HBSs and found
that more than half of the systems satisfy the geometric effect of the
dominant $l=2$, $m=0$, or $\pm 2$ modes assuming their spin and orbital axes
are aligned and the pulsations are adiabatic. The phases of TEOs contain
important information about the mode properties and can be used to better
understand the mode physics of HBSs. However, the detailed analysis of
pulsation phases and mode identification in more HBSs with TEOs is lacking due
to the need for better characterization of the orbital and stellar parameters.

In the first paper in the series (Li et al. (2023a), hereafter Paper I), we
derived the orbital parameters of 153 Kepler HBSs in the Kirk et al. (2016)
catalog by fitting a corrected version of the Kumar et al. (1995) model (K95+
model). We then identified the TEOs in 21 HBSs among these systems in the
second paper in the series (Li et al. (2023b), hereafter Paper II). So far, we
have obtained sufficient orbital and stellar parameters of the HBSs with TEOs
based on our previous works. Therefore, we can study the pulsation phases and
mode identification of these TEOs.

This paper examines the pulsation phases of TEOs from fourteen HBSs following
the analysis approach of O’Leary & Burkart (2014) and Guo et al. (2020).
Section 2 describes the equations and the analysis procedure. Section 3
presents the results. Section 4 discusses the deviations of the harmonics and
addresses another potential cause of the deviations. Section 5 summarizes and
concludes our work.

##  2 Analysis

Under the following assumptions: (1) the pulsation, spin, and orbit axes are
all aligned; (2) the pulsations are adiabatic and the observed TEOs are
standing waves; (3) the observed TEOs are not fine-tuned, the pulsation phases
of the TEOs for dominant modes of spherical harmonic degree $l=2$ can be
expressed by the equation (O’Leary & Burkart, 2014; Guo et al., 2020):

$\phi_{{}_{l=2,m}}=0.25+m\phi_{{}_{0}},$ (1)

where azimuthal order $m=0,\pm 2$, $\phi_{{}_{0}}=0.25-\omega/360^{\circ}$ and
$\omega$ is the argument of periastron. In addition, all phases can be
measured with respect to the time of periastron passage $T_{{}_{0}}$ and are
in units of 360∘.

According to Eq. (1), the accurate $\omega$ and $T_{{}_{0}}$ are needed to
identify the pulsation modes from their phases. Since the parameters of these
TEO HBSs have been derived in Paper I, the theoretical phases of the $m=0,\pm
2$ modes of these harmonic TEOs can be calculated. The phases derived from the
Fourier spectrum can then be used to identify the azimuthal order $m$.

However, note that the $\omega$ derived from different approaches may differ
in some systems. We will perform a comprehensive analysis of these
differences. In the cases, including KIC 3547874, KIC 5090937, KIC 5960989,
and KIC 11403032, we further extend the range of theoretical phases $\phi$ for
$m=\pm 2$ modes to account for the different $\omega$ derived by different
approaches.

In addition, Guo et al. (2020) have investigated the amplitude ratio as a
function of orbital inclination $i$ (see their Figure 1). They have concluded
that a low inclination (less than 30∘) would favor the $m=0$ mode, while
medium to high inclinations would favor both the $m=0$ and $m=2$ modes. These
rules can also further constrain the identified modes.

To derive the phases of the harmonic frequencies, we perform a standard
prewhitening procedure using Period04 (Lenz & Breger, 2005), a software
package based on classical Fourier analysis. The flux variation is formulated
as a sinusoidal function $A_{i}{\rm sin}[2\pi(f_{i}\cdot t+\phi_{i})]$, where
$A_{i}$, $f_{i}$, and $\phi_{i}$ are the amplitude, frequency, and phase,
respectively. The uncertainties of the three parameters are obtained using
Period04, which calculates the uncertainties according to Montgomery &
O’Donoghue (1999). In addition, since $t$ is measured relative to the time of
periastron $T_{{}_{0}}$, the time of each data point in the photometric light
curves should subtract $T_{{}_{0}}$ before Fourier analysis. The mean noise
level of each frequency, $N_{i}$, is derived as the mean amplitude in the
frequency range $\pm$0.5 d-1. The signal-to-noise ratio ($S/N$) of the
frequencies is $A_{i}/N_{i}$.

Consider KIC 3547874 as an example, shown in Figure 1. The top panel shows the
K95+ model fitted to the phase-folded light curve, and all the orbital
parameters are derived in Paper I. The middle panel shows the Fourier spectrum
of the TEOs analysis results derived in Paper II. The harmonic numbers $n$ of
the TEOs are plotted as blue dashed lines. The bottom panel shows the
pulsation phases of the TEOs. The theoretical pulsation phases $\phi_{{}_{m}}$
for $m=0,\pm 2$ strips are derived by Eq. (1). In addition, since the phases
alone cannot determine whether the modes are retrograde ($m=+2$) or prograde
($m=-2$) due to the 180∘ phase ambiguity (Burkart et al., 2012), a phase
offset of 0.5 (180∘) can be added or subtracted from Eq. (1) when comparing
with observations (Guo et al., 2020). Therefore, we obtain two strips for each
$m=0,\pm 2$ modes. The width of the strips result from the uncertainties of
$T_{{}_{0}}$ and $\omega$ (O’Leary & Burkart, 2014) that are obtained from
Paper I. However, for this system, the $\omega$=40∘.7 derived by Dimitrov et
al. (2017) has a more significant deviation from ours. We then plot the dashed
lines to represent the theoretical phase calculated by the more significant
deviation $\omega$. The lighter-colored areas show the total uncertainties.
Each TEO is represented by a blue circle. Its position corresponds to its
phase and harmonic number $n$; the size of each circle corresponds to the
amplitude of the frequency; the error bar corresponds to the uncertainty of
the phase. In Table 1, the orbital parameters are from Paper I, and the
frequency, amplitude, and phase of each harmonic are derived in this work.

Among the HBSs reported in Paper II, we exclude the systems with $\omega$
close to 90∘, since the $m=0,\pm 2$ strips will be too close to distinguish.
We also exclude the systems with lower $S/N$ TEOs, since the error bars of the
phases will be too large to match the modes. Finally, we select the fourteen
samples discussed in Section 3.

##  3 Results

###  3.1 KIC 3547874 (Figure 1; Table 1)

Thompson et al. (2012) reported that KIC 3547874 is a double-lined HBS with
$e$=0.648, $i$=46∘.4, and $\omega$=40∘.40 based on the K95+ model. They also
reported that the TEOs are present in this HBS. Dimitrov et al. (2017)
obtained $e$=0.655, a higher inclination $i$=67∘.7, and a similar argument of
periastron $\omega$=40∘.68 using PHOEBE. We obtained an eccentricity of 0.648
and an inclination of 53∘.7 and $\omega$=32∘.424 from Paper I. Note that the
$\omega$ has a deviation of about 8∘ from that of Thompson et al. (2012) and
Dimitrov et al. (2017). The dashed lines in the bottom panel represent the
theoretical pulsation phases of $\omega$=40∘.68, and the lighter-colored
strips represent the corresponding deviations. The $n$ = 37 prominent TEO
stands out clearly in the middle panel of Figure 1. The bottom panel shows
that its pulsation phase is close to an $m=0$ mode. Although there are some
variations in the inclinations from previous works, the $m=0$ mode can be
visible in such a system with an intermediate inclination (Guo et al., 2020).

Figure 1: Top panel: The K95+ model (solid red line) fitted to the phase-folded light curve (black dots). Middle panel: The Fourier spectrum of the TEOs analysis results. The red and blue vertical dashed lines represent the orbital harmonics n; the blue ones indicate that they are TEOs. The solid red line represents the amplitudes at $S/N$ = 4.0 as a function of frequency. Bottom panel: The pulsation phases of the TEOs. The gray, light blue, and green strips indicate the $m=0,+2$, and $-2$ modes, respectively. The phases of the $m=+2,-2$ modes are shown under the center of the strips. The dashed lines represent the theoretical phase calculated from the $\omega$ with a larger deviation; the lighter-colored areas indicate the total uncertainties. The blue circle represents a TEO with its harmonic number $n$; the size corresponds to its amplitude; the error bar corresponds to the uncertainty of its phase.  Table 1: Orbital and TEO parameters of KIC 3547874. Orbital Parameter |  |  |  |
---|---|---|---|---
$P$(d) | 19.6919178(70) |  |  |
$e$ | 0.64755(10) |  |  |
$i$(∘) | 53.663(18) |  |  |
$\omega$(∘) | 32.424(42) |  |  |
$T_{{}_{0}}$(BJD$-$2,400,000) | 54959.43861(47) |  |  |
Harmonic number | Frequency | Amplitude | Phase | S/N
n | (d-1) | (mmag) |  |
37 | 1.878924(10) | 0.03600(98) | 0.798(4) | 32.44

###  3.2 KIC 4377638 (Figure 2; Table 2)

This is a short-period 2.8 day HBS system with a low eccentricity of 0.22 and
an intermediate inclination of 54∘.6. Eight TEOs are reported in Paper II, and
among those, $n$ = 7, 3, 6 are prominent. The bottom panel of Figure 2 shows
that the phases of $n$ = 7, 3, 8, and 9 harmonics are consistent with or close
to the $m=-2$ mode; the $n$ = 5 harmonics are consistent with the $m=0$ mode;
the $n$ = 4 and 10 harmonics are close to the $m=2$ mode. However, the $n$ = 6
harmonic shows a large deviation ($>2\sigma$) from the adiabatic expectations.
This system has eight consecutive harmonic TEOs, and some harmonics may be
dominated by the imperfect fit. Given the large deviation of the $n$ = 6
harmonic, we suggest that imperfect removal may explain it, and it cannot be a
TEO candidate.

Figure 2: Same as Fig. 1 for KIC 4377638.  Table 2: Orbital and TEO parameters of KIC 4377638. Orbital Parameter |  |  |  |
---|---|---|---|---
$P$(d) | 2.82141983(21) |  |  |
$e$ | 0.22004(4) |  |  |
$i$(∘) | 54.609(11) |  |  |
$\omega$(∘) | 150.643(16) |  |  |
$T_{{}_{0}}$(BJD$-$2,400,000) | 54956.28394(10) |  |  |
Harmonic number | Frequency | Amplitude | Phase | S/N
n | (d-1) | (mmag) |  |
7 | 2.480888(16) | 0.0735(25) | 0.038(5) | 47.83
3 | 1.063304(37) | 0.0320(25) | 0.580(12) | 12.40
6 | 2.126581(49) | 0.0243(25) | 0.662(16) | 14.29
5 | 1.77220(11) | 0.0107(25) | 0.259(37) | 6.79
4 | 1.41789(11) | 0.0106(25) | 0.960(37) | 5.40
8 | 2.83540(12) | 0.0098(25) | 0.514(40) | 7.62
9 | 3.18978(19) | 0.0063(25) | 0.559(63) | 6.66
10 | 3.54428(29) | 0.0041(25) | 0.460(96) | 4.60

###  3.3 KIC 5090937 (Figure 3; Table 3)

Shporer et al. (2016) reported an orbital period of 8.8 days, an eccentricity
$e$=0.241, and an argument of periastron $\omega$ of 153∘.071. We obtained
eccentricity $e$=0.25, $\omega$=163∘.059 from Paper I. However, the
inclination of this HBS may be overestimated (Paper I). Saio & Kurtz (2022)
reported the Rossby mode (r-mode) features of this system. In addition, clear
TEOs can also be seen in the top panel of Figure 3. The $n$ = 6 and 5
harmonics are prominent TEOs, shown in the middle panel. In the bottom panel,
the dashed lines represent the theoretical pulsation phases of
$\omega$=153∘.071, and the lighter-colored strips represent the corresponding
deviations. The $n$ = 5 and 7 harmonics are consistent with the $m=2$ mode.
However, the $n$ = 6 harmonic shows a large deviation ($>2\sigma$) from the
adiabatic expectation. It cannot be explained by imperfect removal because of
its largest amplitude. It can be expected to be a travelling wave rather than
a standing wave. The similar scenarios also exist in KOI-54 (O’Leary &
Burkart, 2014).

Figure 3: Same as Fig. 1 for KIC 5090937.  Table 3: Orbital and TEO parameters of KIC 5090937. Orbital Parameter |  |  |  |
---|---|---|---|---
$P$(d) | 8.7998816(35) |  |  |
$e$ | 0.24862(8) |  |  |
$i$(∘) | 90.000(63) |  |  |
$\omega$(∘) | 163.059(28) |  |  |
$T_{{}_{0}}$(BJD$-$2,400,000) | 54958.06945(58) |  |  |
Harmonic number | Frequency | Amplitude | Phase | S/N
n | (d-1) | (mmag) |  |
6 | 0.6818365(45) | 0.1240(15) | 0.965(2) | 38.79
5 | 0.5681886(95) | 0.0590(15) | 0.320(4) | 12.45
7 | 0.795427(53) | 0.0105(15) | 0.897(23) | 4.43

###  3.4 KIC 5129777 (Figure 4; Table 4)

This is a 26 day binary system, with an eccentricity of 0.71, an inclination
of 60∘, and an argument of periastron $\omega$ of 20∘, as shown in Figure 4.
The $n$ = 102 harmonic is clearly visible in the middle panel. The bottom
panel shows that its pulsation phase is close to the $m=0$ mode. However,
since the predicted TEO phases of the $m=0$ and $m=2$ modes are very close,
the $m=2$ mode cannot be excluded.

Figure 4: Same as Fig. 1 for KIC 5129777.  Table 4: Orbital and TEO parameters of KIC 5129777. Orbital Parameter |  |  |  |
---|---|---|---|---
$P$(d) | 26.15918(18) |  |  |
$e$ | 0.7080(15) |  |  |
$i$(∘) | 60.34(46) |  |  |
$\omega$(∘) | 19.87(72) |  |  |
$T_{{}_{0}}$(BJD$-$2,400,000) | 54975.2984(86) |  |  |
Harmonic number | Frequency | Amplitude | Phase | S/N
n | (d-1) | (mmag) |  |
102 | 3.899293(21) | 0.0188(10) | 0.200(19) | 15.53

###  3.5 KIC 5877364 (Figure 5; Table 5)

Shporer et al. (2016) reported a long orbital period of 89 days, a high
eccentricity $e$=0.8875, and an argument of periastron $\omega$ of 97∘. We
obtained a similar eccentricity $e$=0.719 and a different argument of
periastron $\omega$=15∘.6 from Paper I. We suggest that a lower $\omega$ is
more appropriate in this system (see section 5.1.6 in Paper I), so there is no
need to extend the uncertainties of the strips. Saio & Kurtz (2022) reported
the r-mode features and a rotation frequency of 0.30d-1 in this system. In
addition, Paper II reported eight TEOs in this system. The $n$ = 194, 202, and
232 harmonics are prominent TEOs shown in the middle panel of Figure 5. The
bottom panel shows that the $n$ = 194, 202, 180 and 211 harmonics are close to
the $m=-2$ mode according to the $\omega$ from Paper I. However, the $n$ = 232
and 245 harmonics show large deviations ($>3\sigma$) from the adiabatic
expectations, and they cannot be the TEO candidates.

Figure 5: Same as Fig. 1 for KIC 5877364.  Table 5: Orbital and TEO parameters of KIC 5877364. Orbital Parameter |  |  |  |
---|---|---|---|---
$P$(d) | 89.650258(77) |  |  |
$e$ | 0.71930(9) |  |  |
$i$(∘) | 48.615(13) |  |  |
$\omega$(∘) | 15.632(31) |  |  |
$T_{{}_{0}}$(BJD$-$2,400,000) | 54996.8240(10) |  |  |
Harmonic number | Frequency | Amplitude | Phase | S/N
n | (d-1) | (mmag) |  |
194 | 2.163987(52) | 0.00583(81) | 0.903(22) | 12.70
202 | 2.253208(63) | 0.00484(81) | 0.922(27) | 10.76
232 | 2.587824(74) | 0.00409(81) | 0.981(31) | 10.56
245 | 2.73282(14) | 0.00222(81) | 0.028(58) | 6.04
180 | 2.00306(14) | 0.00224(81) | 0.162(57) | 4.05
211 | 2.35363(16) | 0.00190(81) | 0.906(68) | 4.09

###  3.6 KIC 5960989 (Figure 6; Table 6)

Shporer et al. (2016) reported a long orbital period of 50 days, a high
eccentricity $e$=0.813, and an argument of periastron $\omega$ of 37∘.873.
Dimitrov et al. (2017) obtained $e$=0.806, an inclination $i$=51∘, and an
argument of periastron $\omega$ of 29∘.82. We obtained an eccentricity of
0.811, an inclination of 50∘.8, and $\omega$=25∘.7 from Paper I. The bottom
panel shows that the $n$ = 77, 123, and 162 harmonics are close to the
$m=2,-2$, and $0$ modes, respectively. The dashed lines represent the
theoretical pulsation phases of $\omega$=37∘.873, and the lighter-colored
strips represent the corresponding deviations.

Figure 6: Same as Fig. 1 for KIC 5960989.  Table 6: Orbital and TEO parameters of KIC 5960989. Orbital Parameter |  |  |  |
---|---|---|---|---
$P$(d) | 50.708376(41) |  |  |
$e$ | 0.81126(15) |  |  |
$i$(∘) | 50.792(40) |  |  |
$\omega$(∘) | 25.70(11) |  |  |
$T_{{}_{0}}$(BJD$-$2,400,000) | 54983.6124(10) |  |  |
Harmonic number | Frequency | Amplitude | Phase | S/N
n | (d-1) | (mmag) |  |
77 | 1.51838(12) | 0.0102(32) | 0.669(50) | 5.02
123 | 2.42553(18) | 0.0068(32) | 0.413(75) | 4.74
162 | 3.19463(23) | 0.0054(32) | 0.751(96) | 4.30

###  3.7 KIC 6290740 (Figure 7; Table 7)

This is a 15 day HBS system, with a lower eccentricity of 0.21, an inclination
of 45∘.8, and an argument of periastron $\omega$ of 121∘ shown in the top
panel of Figure 7. The bottom panel shows that the $n$=5 harmonic is in agrees
with the $m=-2$ mode. However, the $n$ = 3 and 4 harmonics show large
deviations ($>2\sigma$) from the adiabatic expectations. We suggest that these
two harmonics may be dominated by the imperfect fit and cannot be real TEOs.

Figure 7: Same as Fig. 1 for KIC 6290740.  Table 7: Orbital and TEO parameters of KIC 6290740. Orbital Parameter |  |  |  |
---|---|---|---|---
$P$(d) | 15.151353(99) |  |  |
$e$ | 0.21454(99) |  |  |
$i$(∘) | 45.80(13) |  |  |
$\omega$(∘) | 121.25(29) |  |  |
$T_{{}_{0}}$(BJD$-$2,400,000) | 54965.131(11) |  |  |
Harmonic number | Frequency | Amplitude | Phase | S/N
n | (d-1) | (mmag) |  |
5 | 0.330129(13) | 0.0317(11) | 0.405(5) | 7.46
3 | 0.197976(17) | 0.0245(11) | 0.848(7) | 5.32
4 | 0.264164(18) | 0.0232(11) | 0.148(7) | 5.16

###  3.8 KIC 8264510 (Figure 8; Table 8)

Thompson et al. (2012) reported the eccentricity $e$=0.338, the inclination
$i$=29∘, and the argument of periastron $\omega$ of 120∘. They also reported
the presence of TEOs in this HBS. We obtained an eccentricity of 0.34, an
inclination of 29∘.2, and an argument of periastron $\omega$ of 119∘.2 from
Paper I. The $n$=4, 8, 9, 10, and 12 harmonics are consistent with or close to
the $m=0$ mode; the $n$=6, 7, and 11 harmonics are close to the $|m|=2$ modes.

Figure 8: Same as Fig. 1 for KIC 8264510.  Table 8: Orbital and TEO parameters of KIC 8264510. Orbital Parameter |  |  |  |
---|---|---|---|---
$P$(d) | 5.68674956(31) |  |  |
$e$ | 0.34111(3) |  |  |
$i$(∘) | 29.211(1) |  |  |
$\omega$(∘) | 119.179(4) |  |  |
$T_{{}_{0}}$(BJD$-$2,400,000) | 54957.401056(55) |  |  |
Harmonic number | Frequency | Amplitude | Phase | S/N
n | (d-1) | (mmag) |  |
7 | 1.2309341(99) | 0.1967(52) | 0.356(4) | 35.44
4 | 0.703461(14) | 0.1364(52) | 0.792(6) | 13.84
8 | 1.406766(25) | 0.0794(52) | 0.247(10) | 15.75
9 | 1.582618(29) | 0.0669(52) | 0.249(12) | 15.13
11 | 1.934332(49) | 0.0401(52) | 0.910(21) | 12.17
6 | 1.055140(55) | 0.0356(52) | 0.609(23) | 5.37
10 | 1.758472(60) | 0.0324(52) | 0.178(25) | 8.76
12 | 2.110059(86) | 0.0227(52) | 0.259(36) | 7.90

###  3.9 KIC 8456774 (Figure 9; Table 9)

This system has a short-period of near 3 days, with a low eccentricity of 0.27
and a low inclination of 23∘.9. Paper II reported the $n$ = 8, 3, 5, and 6
harmonics are prominent TEOs shown in the middle panel of Figure 9. The bottom
panel shows that the 3, 4, 5, and 6 harmonics agree with the $m=0$ mode, i.e.,
they meet the expectation for such a low inclination (Guo et al., 2020). The
8, 7, and 9 harmonics are close to the $m=-2$ mode. However, the uncertainty
of $n$ = 10 harmonic is too significant to match the modes.

Figure 9: Same as Fig. 1 for KIC 8456774.  Table 9: Orbital and TEO parameters of KIC 8456774. Orbital Parameter |  |  |  |
---|---|---|---|---
$P$(d) | 2.88624552(21) |  |  |
$e$ | 0.27480(10) |  |  |
$i$(∘) | 23.897(4) |  |  |
$\omega$(∘) | 19.347(19) |  |  |
$T_{{}_{0}}$(BJD$-$2,400,000) | 54955.416037(79) |  |  |
Harmonic number | Frequency | Amplitude | Phase | S/N
n | (d-1) | (mmag) |  |
8 | 2.771776(13) | 0.358(12) | 0.325(5) | 53.32
3 | 1.039410(15) | 0.306(12) | 0.780(6) | 21.62
5 | 1.732333(23) | 0.198(12) | 0.248(10) | 22.60
6 | 2.078805(40) | 0.115(12) | 0.744(17) | 15.69
4 | 1.385810(48) | 0.094(12) | 0.301(20) | 8.57
7 | 2.425258(75) | 0.061(12) | 0.869(32) | 8.23
10 | 3.46467(14) | 0.033(12) | 0.806(59) | 7.57
9 | 3.11817(16) | 0.029(12) | 0.886(66) | 4.97

###  3.10 KIC 8459354 (Figure 10; Table 10)

The orbital period is 53.56 days, the eccentricity $e$=0.94 and the
inclination $i$=27∘.31. Such a low inclination will make the $m=0$ mode more
visible according to Guo et al. (2020). However, the bottom panel shows that
its pulsation phases have large deviations ($>6\sigma$) from the adiabatic
expectations. Given the highest eccentricity and long orbital period, we
propose a strong spin-orbit misalignment in this HBS. Therefore, the TEOs in
this system cannot be modeled with Eq. (1). A further study of the $|m|=1$
modes for the misaligned cases (Fuller, 2017) is left for future work.

Figure 10: Same as Fig. 1 for KIC 8459354.  Table 10: Orbital and TEO parameters of KIC 8459354. Orbital Parameter |  |  |  |
---|---|---|---|---
$P$(d) | 53.555278(22) |  |  |
$e$ | 0.94122(4) |  |  |
$i$(∘) | 27.311(17) |  |  |
$\omega$(∘) | 100.624(44) |  |  |
$T_{{}_{0}}$(BJD$-$2,400,000) | 54963.41426(20) |  |  |
Harmonic number | Frequency | Amplitude | Phase | S/N
n | (d-1) | (mmag) |  |
99 | 1.84854(13) | 0.150(28) | 0.578(30) | 15.93
79 | 1.47503(20) | 0.097(28) | 0.448(46) | 8.74
103 | 1.92319(23) | 0.085(28) | 0.551(52) | 9.31
91 | 1.69919(26) | 0.077(28) | 0.173(58) | 7.75
115 | 2.14729(28) | 0.072(28) | 0.432(62) | 8.7
98 | 1.82986(34) | 0.059(28) | 0.562(76) | 6.08

###  3.11 KIC 9535080 (Figure 11; Table 11)

This is a HB system with a nearly 50-day period, with a high eccentricity of
0.816 and an inclination of 66∘ shown in Figure 11. The $n$ = 10 harmonic is
present in the middle panel. The bottom panel shows that its pulsation phase
is close to the $m=2$ mode.

Figure 11: Same as Fig. 1 for KIC 9535080.  Table 11: Orbital and TEO parameters of KIC 9535080. Orbital Parameter |  |  |  |
---|---|---|---|---
$P$(d) | 49.64577(13) |  |  |
$e$ | 0.81643(50) |  |  |
$i$(∘) | 66.28(37) |  |  |
$\omega$(∘) | 66.97(36) |  |  |
$T_{{}_{0}}$(BJD$-$2,400,000) | 54985.8926(35) |  |  |
Harmonic number | Frequency | Amplitude | Phase | S/N
n | (d-1) | (mmag) |  |
10 | 0.201627(27) | 0.00879(63) | 0.849(11) | 4.45

###  3.12 KIC 11122789 (Figure 12; Table 12)

We obtained a short orbital period of 3 days, a low eccentricity $e$=0.327, an
inclination of 41∘ and an argument of periastron $\omega$ of 14∘ in Paper I.
The bottom panel of Figure 12 shows that the $n$ = 3, 6, and 8 harmonics are
consistent with the $m=2$ mode; the $n$ = 7 and 12 harmonics are consistent
with or close to the $m=-2$ mode; the $n$ = 9 and 11 harmonics are close to
the $m=0$ mode. In addition, the $n$ = 4, 5, 10, and 14 harmonics show large
deviations ($>5\sigma$) from the adiabatic expectations. However, the spin-
orbit misalignment cannot explain these deviations, since other expected modes
exist in the same system. Another explanation is that these harmonics cannot
be the real TEOs, since there may be an imperfect fit in Paper II. Considering
the top and middle panels, we cannot identify all the TEOs using only the
Fourier transform analysis. This mode identification approach can be used to
verify the TEOs identified by the Fourier spectrum. We suggest that if a
harmonic does not fit the modes well, it may not be the TEO candidate and
further investigation is needed.

Figure 12: Same as Fig. 1 for KIC 11122789.  Table 12: Orbital and TEO parameters of KIC 11122789. Orbital Parameter |  |  |  |
---|---|---|---|---
$P$(d) | 3.2382374(22) |  |  |
$e$ | 0.32664(47) |  |  |
$i$(∘) | 40.684(44) |  |  |
$\omega$(∘) | 13.93(19) |  |  |
$T_{{}_{0}}$(BJD$-$2,400,000) | 54955.3471(12) |  |  |
Harmonic number | Frequency | Amplitude | Phase | S/N
n | (d-1) | (mmag) |  |
6 | 1.8524487(63) | 0.03109(52) | 0.184(3) | 55.4
4 | 1.2350466(76) | 0.02583(52) | 0.996(3) | 28.71
5 | 1.5440340(86) | 0.02256(52) | 0.428(4) | 27.05
3 | 0.9255531(94) | 0.02083(52) | 0.670(4) | 18.88
8 | 2.470191(17) | 0.01125(52) | 0.657(7) | 19.81
7 | 2.161618(19) | 0.01043(52) | 0.331(8) | 15.01
10 | 3.087879(23) | 0.00866(52) | 0.463(10) | 17.87
9 | 2.779435(34) | 0.00635(52) | 0.220(14) | 12.19
12 | 3.705339(71) | 0.00276(52) | 0.403(30) | 6.18
14 | 4.322875(83) | 0.00234(52) | 0.033(35) | 5.6
11 | 3.396299(86) | 0.00227(52) | 0.221(36) | 4.81

###  3.13 KIC 11403032 (Figure 13; Table 13)

Shporer et al. (2016) reported a short orbital period of 7.6 days, an
eccentricity $e$=0.288, and an argument of periastron $\omega$ of 148∘.144. We
obtained the eccentricity $e$=0.294, $i$=76∘.14, and $\omega$=153∘.757 from
Paper I. Saio & Kurtz (2022) reported the r-mode features and the rotation
period of 10.5 d in this system. In addition, clear TEOs can also be seen in
the top panel of Figure 13. The dashed lines in the bottom panel represent the
theoretical pulsation phases of $\omega$=148∘.144, and the lighter-colored
strips represent the corresponding deviations. It shows that the $n$ = 6, 5,
and 4 harmonics are consistent with or close to the $m=2$ mode. However, the
$n$ = 7 harmonic shows a large deviation ($>2\sigma$) from the adiabatic
expectations. It cannot be a real TEO because of the large deviation.

Figure 13: Same as Fig. 1 for KIC 11403032.  Table 13: Orbital and TEO parameters of KIC 11403032. Orbital Parameter |  |  |  |
---|---|---|---|---
$P$(d) | 7.6312041(26) |  |  |
$e$ | 0.29355(10) |  |  |
$i$(∘) | 76.144(86) |  |  |
$\omega$(∘) | 153.757(30) |  |  |
$T_{{}_{0}}$(BJD$-$2,400,000) | 54955.86722(48) |  |  |
Harmonic number | Frequency | Amplitude | Phase | S/N
n | (d-1) | (mmag) |  |
5 | 0.6551794(82) | 0.0907(20) | 0.480(3) | 22.66
4 | 0.524152(12) | 0.0597(20) | 0.966(5) | 9.23
6 | 0.786248(17) | 0.0443(20) | 0.890(7) | 14.13
7 | 0.917282(26) | 0.0290(20) | 0.002(11) | 10.79

###  3.14 KIC 11572363 (Figure 14; Table 14)

This is a nearly 20 day binary system, with an eccentricity of 0.7, an
inclination of 44∘, and an argument of periastron $\omega$ of 164∘, as shown
in Figure 14. The $n$ = 50 harmonic is clearly visible in the middle panel.
The bottom panel shows that its pulsation phase is close to the $m=0$ or
$m=-2$ modes.

Figure 14: Same as Fig. 1 for KIC 11572363.  Table 14: Orbital and TEO parameters of KIC 11572363. Orbital Parameter |  |  |  |
---|---|---|---|---
$P$(d) | 19.028898(25) |  |  |
$e$ | 0.70500(56) |  |  |
$i$(∘) | 44.386(75) |  |  |
$\omega$(∘) | 163.70(26) |  |  |
$T_{{}_{0}}$(BJD$-$2,400,000) | 54960.3827(19) |  |  |
Harmonic number | Frequency | Amplitude | Phase | S/N
n | (d-1) | (mmag) |  |
50 | 2.627626(27) | 0.0425(31) | 0.293(12) | 28.32

##  4 Discussions

In section 3, we study the phases and amplitudes of TEOs in fourteen HBSs. We
find that the median deviations range of the modes are less than $2\sigma$ in
most systems. However, in some systems there are some larger deviations from
the expected adiabatic phases. Guo et al. (2020) have addressed two major
reasons for the deviations. One is the non-adiabaticity of pulsations, and the
other is spin-orbit misalignment. The largest deviations ($>6\sigma$) in KIC
8459354 can be explained by the major reason of spin-orbit misalignment. Since
the high eccentricity ($e$=0.941) and the long orbital period ($P$=53.5553 d)
of this system, the system is not yet tidally synchronized, nor are the spin
and orbit axes aligned. Therefore, the large deviation in this system is not
surprising.

In addition, KIC 11122789 is an interesting sample in this work. Some modes
have large deviations, while others agree well with the theoretical phases.
However, the deviations cannot be explained by the non-adiabaticity of
pulsations or the spin-orbit misalignment. Another explanation is that these
harmonics may be excluded from the TEO candidates due to an imperfect fit.
Given the top and middle panels in Figure 12, the TEOs may not be identified
by Fourier transform analysis alone. We suggest that the mode identification
methodology can be a complementary approach to verify the TEOs identified by
the Fourier spectrum. When the TEO candidates are identified by the Fourier
analysis, further phase and mode identification is required. The harmonic can
be considered a TEO candidate if the modes fit well. The large deviation
harmonics in KIC 4377638, KIC 5877364, KIC 6290740, and KIC 11403032 can also
be explained by this reason. However, for KIC 5090937, the large deviation
harmonic also has the largest amplitude; it cannot be explained by the
imperfect removal. We suggest that it can be expected to be a travelling wave
that does not satisfy the assumption in Section 2.

Furthermore, both the $m=0$ and $m=2$ modes can exist in the same system with
a medium to high inclination. Such systems include KIC 4377638, KIC 5960989,
KIC 8264510, and KIC 11122789.

Besides the two reasons addressed by Guo et al. (2020), we propose another
reason here: the apsidal motion. Note that KIC 3749404 is one of the large
deviation systems in their paper. An interesting feature of this HBS is the
rapid apsidal motion reported by Hambleton et al. (2016). Since the apsidal
motion is present, the argument of periastron $\omega$ varies with time. On
the other hand, there is a positive correlation between $\omega$ and
$T_{{}_{0}}$ in most of the HBSs (see Fig. 2 in Paper I and their online
material). Then $T_{{}_{0}}$ also varies with time in the HBS with apsidal
motion. However, the phases are sensitive to $T_{{}_{0}}$ in the Fourier
spectrum analysis. Therefore, the derived phases may not be correct in an
apsidal motion system. This could be a reason for the large deviations in KIC
3749404. For our samples, further investigation of the apsidal motion is
needed in future work.

##  5 Summary and Conclusions

Based on our previous work, Paper I and Paper II, we identify the pulsation
phases of TEOs in fourteen Kepler HBSs using the approach of O’Leary & Burkart
(2014) and Guo et al. (2020). The pulsation phases of most systems can be
explained by the dominant being $l=2$, $m=0$, or $\pm 2$ spherical harmonic.
Among these samples, KIC 8459354 shows the largest deviations ($>6\sigma$)
from the adiabatic expectations. We suggest that this deviation can be
explained by the spin-orbit misalignment due to its high eccentricity and long
orbital period. For KIC 11122789, the TEO phases with large deviations may not
be considered real TEOs. In four other systems, including KIC 4377638, KIC
5877364, KIC 6290740, and KIC 11403032, there are a small number of harmonics
with larger deviations ($>2\sigma$). These harmonics may also not be
considered TEO candidates. Furthermore, this approach can be used to inversely
verify the TEO candidates derived by the Fourier analysis. For KIC 5090937,
the large deviation harmonic with the largest amplitude is expected to be a
travelling wave. In addition, besides the two reasons mentioned in Guo et al.
(2020), the apsidal motion could be another reason for the large deviations.
For other spin-orbit misalignment systems, a further study of the $|m|=1$
modes approach (Fuller, 2017) is left for future work.

## Acknowledgements

This work was supported in part by the National Natural Science Foundation of
China (Nos. 11933008 and 12103084), the Basic Research Project of Yunnan
Province (Grant No. 202301AT070352), and the Yunnan Revitalization Talent
Support Program. We would like to thank Kirk et al. (2016) for their excellent
work and for sharing the results of their work. The Kepler mission is funded
by the NASA Science Mission Directorate. We are grateful to the Kepler teams
for their support and hard work. We are grateful to the anonymous reviewer for
constructive comments that have been a source of improvement for this
manuscript.

## Data Availability

The data underlying this article are available in the Kirk et al. (2016)
catalog, at http://keplerEBs.villanova.edu

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