11institutetext:  Konkoly Observatory, Research Centre for Astronomy and Earth
Sciences, HUN-REN, MTA Centre of Excellence, Konkoly-Thege Miklós út 15–17.,
H-1121, Hungary 22institutetext: HUN-REN-ELTE Exoplanet Research Group,
Szombathely, Szent Imre h. u. 112., H-9700, Hungary 33institutetext: ELTE
Eötvös Loránd University, Doctoral School of Physics, Budapest, Pázmány Péter
sétány 1/A, H-1117, Hungary 44institutetext: ELTE Eötvös Loránd University,
Gothard Astrophysical Observatory, Szombathely, Szent Imre h. u. 112., H-9700,
Hungary 55institutetext: MTA-ELTE Lendület ”Momentum” Milky Way Research
Group, Szombathely, Szent Imre h. u. 112., H-9700, Hungary 66institutetext:
Deutsches Zentrum für Luft- und Raumfahrt, Institute of Planetary Research,
Rutherfordstrasse 2, D-12489 Berlin, Germany 77institutetext: HUN-REN-SZTE
Stellar Astrophysics Research Group, H-6500 Baja, Szegedi út, Kt. 766, Hungary
88institutetext: ELTE Eötvös Loránd University, Institute of Physics, Pázmány
Péter sétány 1/A, H-1117 Budapest, Hungary 99institutetext: University of
Szeged, Doctoral School of Physics, Dóm tér 9, 6720 Szeged, Hungary
1010institutetext: Baja Astronomical Observatory of University of Szeged,
H-6500 Baja, Szegedi út, Kt. 766, Hungary

# The phase curve of the ultra-hot Jupiter WASP-167b as seen by TESS

Sz. Kálmán  112233 A. Derekas 445577 Sz. Csizmadia 66 A. Pál 11 R. Szabó 1188
A. M. S. Smith 66 K. Nagy 99 V. Hegedűs  33 5 5 T. Mitnyan  771010 L. Szigeti
44 Gy. M. Szabó 224455

(Received …; accepted …)

###### Abstract

Context. Ultra-hot Jupiters (UHJs) orbiting pulsating A/F stars represent an
important subset of the exoplanetary demographic, as they are excellent
candidates for the study of exoplanetary atmospheres, as well as being
astrophysical laboratories for the investigation of planet-to-star
interactions.

Aims. We analyse the TESS (Transiting Exoplanet Survey Satellite) light curve
of the WASP-167 system, consisting of an F1V star and a substellar companion
on a $\sim 2.02$ day orbit.

Methods. We model the combination of the ellipsoidal variability and the
Doppler beaming to measure the mass of WASP-167b, and the reflection effect to
obtain constraints on the geometric albedo, while placing a special emphasis
on noise separation. We implement a basic model to determine the dayside
($T_{\rm Day}$), nightside ($T_{\rm Night}$) and intrinsic ($T_{\rm
Internal}$) temperatures of WASP-167b and put a constraint on its Bond albedo.

Results. We confirm the transit parameters of the planet seen in the
literature. We find that a resonant $\sim 2P^{-1}$ stellar signal (which may
originate from planet-to-star interactions) interferes with the phase curve
analysis. After considerate treatment of this signal, we find $M_{p}=0.34\pm
0.22$ $M_{J}$. We measure a dayside temperature of $2790\pm 100$ K,
classifying WASP-167b as an UHJ. We find a $2\sigma$ upper limit of $0.51$ on
its Bond albedo, and determine the geometric albedo at $0.34\pm 0.11$
($1\sigma$ uncertainty).

Conclusions. With an occultation depth of $106.8\pm 27.3$ ppm in the TESS
passband, the UHJ WASP-167b will be an excellent target for atmospheric
studies, especially those at thermal wavelength ranges, where the stellar
pulsations are expected to be be less influential.

###### Key Words.:

Planets and satellites: individual: WASP-167b – Planets and satellites:
atmospheres – Planets and satellites: interior – Planets and satellites:
general – Techniques: photometric

##  1 Introduction

Close-in, substellar mass companions orbiting pulsating host stars of A/F
spectral types, such as WASP-33b (Christian et al., 2006; Collier Cameron et
al., 2010; Herrero et al., 2011), HAT-P-2b(Bakos et al., 2007; de Wit et al.,
2017), HD 31221b (Kálmán et al., 2023a), HAT-P-57 (Hartman et al., 2015) or
KOI-976 (Ahlers et al., 2019), represent a small fraction of the entire
exoplanet population (see e.g. Hey et al., 2021). On the one hand, these
systems are analogous to those binary stars, where the pulsation of one of the
stars is influenced by the tidal forces of its companion, such as V453 Cyg
(Southworth et al., 2020) or RS Cha (Steindl et al., 2021). Planet-to-star
interactions of this kind have been observed in several of these systems,
including HAT-P-2 (de Wit et al., 2017), WASP-33 (Kálmán et al., 2022) and HD
31221 (Kálmán et al., 2023a). Recently, Bryan et al. (2024) explored the
potential of these resonances in revealing the evolution of exoplanets,
primarily the highly eccentric HAT-P-2b. The emerging class of interacting
pulsators and their planetary companions is expected to yield new
opportunities in stellar characterisation as well (Bryan et al., 2024). On the
other hand, due to the strong irradiation, these systems are prime candidates
for atmospheric studies via phase curve analyses. The ultra-hot Jupiter (UHJ)
WASP-33b, being the epitome of these systems, was the subject of many such
studies, including analyses based on the occultation detections (e.g. Smith et
al., 2011; Deming et al., 2012; von Essen et al., 2015; Haynes et al., 2015)
and the full phase curve (e.g. Zhang et al., 2018). Both of these require
careful treatment of the stellar pulsations, which is enabled by the
continuous photometry done by spaceborne observatories such as Kepler (Borucki
et al., 2010) or Transiting Exoplanet Survey Satellite (TESS; Ricker et al.,
2015), as seen in von Essen et al. (2020). Steindl et al. (2021) explored the
known $\delta$ Scuti population in the Kepler field and found no UHJ
companions of these stars. This suggests that these objects are rare and all-
sky surveys, such as TESS or PLATO (PLAnetary Transits and Oscillations of
stars Rauer et al., 2014) might provide a more in-depth insight into these
objects.

The host stars in question typically present either $\delta$ Scuti, $\gamma$
Doradus, or hybrid $\delta$ Scuti – $\gamma$ Doradus pulsations. These objects
often rotate rapidly (Ahlers et al., 2022), leading to spin-orbit misalignment
detectable by Doppler tomography – a technique also pioneered on the WASP-33
system (Collier Cameron et al., 2010; Johnson et al., 2015). A continuous
monitoring of the stellar obliquity $\lambda$ also allows for the detection of
nodal precession (e.g. Johnson et al., 2015; Szabó et al., 2020; Stephan et
al., 2022; Watanabe et al., 2022).

Figure 1: TESS Simple Aperture Photometry light curve of WASP-167 (top row,
grey dots) from Sectors 10 (left), 37 (middle), and 64 (right). The masked,
detrended and centroid-corrected LC is shown with black. The pulsation model,
constructed from the 16 highest amplitude sinusoidal curves, is shown in the
second row with a solid magenta line. Blue dots in the third row represent the
LC when these 16 highest amplitude pulsations have been removed, while the
best-fit combination of time-correlated noise and phase curve models are shown
with red. The residuals, along with their Root Mean Square Error (RMSE) are
shown in the bottom row.

In the WASP-167 system (also known as KELT-13, TOI-748,TIC 104024556, Gaia DR3
6154982877300947840), Temple et al. (2017) discovered a hot Jupiter with an
orbital period of $2.0219596\pm 0.0000006$ days and a radius of $1.58\pm 0.05$
RJ. The mass measurement of WASP-167b through radial velocities (RVs) was
inconclusive, as the star is both ‘line-poor’ and ‘broad-lined’, owing to its
spectral type ($T_{\rm eff}=6900\pm 150$ K) and rapid rotation ($v\sin
i_{\star}=52\pm 8$ km s-1; Temple et al., 2017). Based on Doppler tomography,
Temple et al. (2017) established that WASP-167b/KELT-13b has a misaligned,
retrograde orbit, determined by the spin-orbit angle $\lambda=-165\pm
5^{\circ}$. The line-profile tomography also led the authors to conclude that
the host star presents pulsations of either $\delta$ Scuti or $\gamma$ Doradus
type, which presents further difficulties in the mass measurements to the ones
discussed above (see e.g. Lehmann et al., 2015).

We undertake an analysis of WASP-167b, based solely on its TESS light curve
(LC). This paper is structured as follows. In Sect. 2, we describe the LC pre-
processing and fitting. In Sect. 3, we present the best-fit parameters,
describing both the transits and the out-of-transit variations, as well as
give constraints on the mass of WASP-167b. We also explore how the pulsations
of the host affect the determination of these parameters. We characterise the
pulsations of the host star, discuss the possibilities of planet-to-star
interactions and the possible effects of the pulsations on the precision and
accuracy of the retrieved parameters in Sect. 4. We also estimate the dayside
and nightside temperatures of WASP-167b in Sect. 4.

##  2 Methods

###  2.1 Light curve preparation

We downloaded the short-cadence Simple Aperture Photometry (SAP) light curves
obtained in Sectors 10, 37 and 64 by TESS using the lightkurve software
(Lightkurve Collaboration et al., 2018), utilising the astroquery and astropy
packages (Ginsburg et al., 2019; Astropy Collaboration et al., 2022). We
removed all data points with a non-zero quality flag, and corrected for the
third light contamination via the CRWDSAP keyword. The obtained light curves
are shown in Fig. 1. We then proceeded to mask apparent anomalies and remove a
linear trend from the data of all three sectors. We also corrected for the
jitter in the pointing of the telescope. This was done by subtracting the
best-fit linear model of the centroid position of the PSF (Point Spread
Function) of WASP-167 in terms of row and column pixels.

After the initial preparatory steps, we examined the Fourier spectra of the
light curves using Period04 (Lenz & Breger, 2005). We identified a ‘critical
frequency’ ($\sim 0.98$ d-1, labelled F2 in Table 3, see Sect 4 for more
details) that is close to being in resonance with the second orbital harmonic
of WASP-167b. To explore the effect of the stellar pulsations on the stability
of the phase curve parameters, we prepared 17 different light curves, 16 of
which have been pre-whitened by the 16 highest-amplitude sinusoidal signals
from the LCs of the three sectors and the original one (i.e. without
whitening). To do this, we first masked the transits. We then computed the
Fourier spectrum of the LCs. Breger et al. (1993) suggest that in order to
consider a peak in the power spectrum significant, an S/N $\geq$ 4 is
required. Relying on this criterion is common practise in the analysis of
pulsating stars (e.g. Derekas et al., 2009, 2017; Aerts et al., 2018; Bowman
et al., 2020; Samadi-Ghadim et al., 2022; Sowicka et al., 2023). We identified
$16$ peaks with S/N $\geq$ 4 using Period04, separately for all three sectors.
The whitened light curves are then given as

$\Phi_{i}(t)=\Phi(t)-\sum_{i=0}^{16}A_{i}\sin\left(2\pi\left(\nu_{i}\cdot
t+\varphi_{i}\right)\right),$ (1)

where $\Phi(t)$ is the original detrended SAP light curve, and $A_{i}$,
$\nu_{i}$ and $\varphi_{i}$ are the amplitudes, frequencies and phases of the
individual Fourier-terms and their values are taken from the Fourier-analysis.
We emphasise that the pre-whitening process is done for each Sector
individually. The light curve corresponding to $\Phi_{16}$ is shown in Fig. 1,
and it contains $32$ transits and $32$ full occultations as we fitted the data
from the three sectors jointly. We note that the sole purpose of this initial
Fourier analysis is to enable a consistent phase curve modelling. The Fourier
spectrum after the subtraction of the first $16$ highest-amplitude sinusoidal
curves is shown on Fig. 2. For the more in-depth discussion of the
astrophysical aspect of the pulsations (Sect. 4), we need a stable LC model
that can take advantage of a dataset that does not need masked transits.

Figure 2: Fourier spectrum after subtracting the 16 highest-amplitude
sinusoidal curves for Sectors 10 (left), 37 (middle), and 64 (left). The solid
red line corresponds to the noise level as estimated by Period04. The window
function are shown in the top right corner of each panel.

###  2.2 Analysis of the light curves

Figure 3: Best-fit $K_{\rm phot}$ and $q_{\rm ell}$ as a function of the
number of sinusoidal curves subtracted from the light curve. Note the change
in the scale of the y axis after subtracting one sinusoidal curve (i.e. before
and after the break). The dashed lines correspond to the uncertainty of the
parameters obtained from the $\Phi_{16}(t)$ LC.

We made use of the Transit and Light Curve Modeller (TLCM; Csizmadia, 2020) to
model the light variations induced by the presence of WASP-167b. TLCM uses a
model similar to the one of Mandel & Agol (2002) to describe the transits. It
is characterised by the relative planetary radius ($R_{\rm p}/R_{\star}$); the
impact parameter which is related to the orbital inclination via
$b=a/R_{\star}\cos i_{\rm p}$ ($i_{\rm p}$ being the orbital inclination of
the planet relative to the line of sight); the time of mid-transit ($t_{C}$)
and the scaled semi-major axis ($a/R_{\star}$). The stellar parameters listed
in Table 1 were taken into account via the isochrones of Hurley et al. (2000).
We calculated the theoretical values for the $u_{1}$ and $u_{2}$ quadratic
limb-darkening coefficients in the TESS passband with PyLDTK (Parviainen &
Aigrain, 2015), which uses the PHOENIX library of stellar atmospheres (Husser
et al., 2013). The stellar parameters used in these calculations were adopted
from Temple et al. (2017) and are listed in Table 1. We implemented a
reparametrization of quadratic limb-darkening law (based on Principal
Component Analysis), by fitting the two coefficients $A$ and $B$, defined as

$\displaystyle A$
$\displaystyle=\frac{1}{4}\left(u_{+}\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)+u_{-}\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)\right),$
(2) $\displaystyle B$
$\displaystyle=\frac{1}{4}\left(u_{+}\left(\frac{1}{\alpha}+\frac{1}{\beta}\right)+u_{-}\left(\frac{1}{\alpha}-\frac{1}{\beta}\right)\right),$
(3)

where $u_{\pm}=u_{1}\pm u_{2}$,
$\alpha=\frac{1}{2}\cos\left(77^{\circ}\right)$ and
$\beta=\frac{1}{2}\sin\left(77^{\circ}\right)$. Note that TLCM uses this
definition of limb darkening coefficients in its latest version. We also
applied Gaussian priors on $A$ and $B$, around their theoretical values, in
the form of $\mathcal{N}(0.91,0.10)$ and $\mathcal{N}(1.40,0.10)$.

Table 1: Stellar parameters of WASP-167, adopted from Temple et al. (2017). $T_{\rm eff}$ [K] | $\log g$ | [Fe/H] |  $v\sin I_{\star}$ [km s-1] |  $R_{\star}$ [$R_{\odot}$]
---|---|---|---|---
$6900\pm 150$ | $4.13\pm 0.02$ | $0.1\pm 0.1$ | $52\pm 8$ | $1.79\pm 0.05$

We modelled the out-of-transit variations including the Doppler beaming, the
ellipsoidal variability (Zucker et al., 2007; Faigler & Mazeh, 2011, and
references therein), and the reflection effect, as described in Csizmadia
(2020) and Csizmadia et al. (2023b). Both the Doppler beaming and the
ellipsoidal variations can be used to estimate the mass of WASP-167b, as the
former is characterised by the (photometric) RV semi-amplitude $K_{\rm phot}$,
while the latter introduces the ellipsoidal mass ratio, $q_{\rm ell}$, as a
fitting parameter. The contribution of these two effects to the photometric
baseline variations depend on the stellar parameters and are taken into
account according to the description in Csizmadia (2020). The reflection of
the stellar light from the planetary atmosphere was taken into account via a
Lambertian phase function:

$\frac{F_{\rm p}}{F_{\star}}=\frac{I_{\rm p}}{I_{\star}}\left(\frac{R_{\rm
p}}{R_{\star}}\right)^{2}+A_{g}\left(\frac{R_{\rm
p}}{R_{\star}}\frac{R_{\star}}{a}\right)^{2}\frac{\sin\alpha-\alpha\cos\alpha}{\pi},$
(4)

where $I_{\rm p}/I_{\star}$ is the surface brightness ratio of the planet and
the star, $A_{g}$ is the geometric albedo, and $F_{\rm p}$ and $F_{\star}$ are
the planetary and stellar fluxes, respectively. The phase angle $\alpha$ is
related to the true anomaly $v$ by

$\cos\left(\alpha+\varepsilon\right)=\cos\left(\omega+v\right)\sin i_{\rm p},$
(5)

where $\omega$ is the argument of periastron and $\varepsilon$ is the offset
of the brightest point on the planetary surface from the substellar point. We
assumed that WASP-167b has a circular orbit, as expected for a hot Jupiter.
Additional free parameter was the orbital period $P$. Furthermore, we applied
the so-called height correction, denoted by the fitting parameter $h$ (see Eq.
(47) of Csizmadia, 2020). This parameter describes an offset in flux that is
designed to perform renormalisation in a similar way to third light
contamination.

###  2.3 Time-correlated noise

In their systematic phase curve study of the TESS data, Wong et al. (2020)
excluded WASP-167 specifically due to the stellar variability, which
introduces difficulties to the analysis of both the transits and the out-of-
transit variations. One possible way to account for this was presented in the
case of WASP-33 by von Essen et al. (2020), where the $\delta$ Scuti/$\gamma$
Doradus type stellar oscillations had been removed from the light curve as
simple sinusoidal signals, computed via a periodogram. TLCM incorporates the
wavelet-based routines of Carter & Winn (2009) to account for time-correlated
noise (Csizmadia, 2020; Csizmadia et al., 2023b). By treating the residual
stellar oscillations and the remaining systematic effects (Fig. 1) in this
way, we are able to improve the precision and accuracy of the fitted
parameters, as was demonstrated in Csizmadia et al. (2023b); Kálmán et al.
(2023b); Bókon et al. (2023). Stellar pulsations in the case of WASP-33 were
also accounted for successfully via this noise filtering approach in Kálmán et
al. (2022). These routines introduce two additional parameters to describe the
noise: $\sigma_{r}$ and $\sigma_{w}$ for the red and white components,
respectively.

We used these wavelets to account for any systematic effects and residual
stellar pulsations for the 16 pre-whitened light curves and the original one.
In every case, we ran the MCMC analysis for a maximal number of 100,000 steps,
with 10 chains (also known as walkers) and a thinning factor of 10.

##  3 Results

###  3.1 Influence of the stellar pulsations on the out-of-transit variability

Figure 4: Best-fit $A_{g}$, $I_{\rm p}/I_{\star}$, $\varepsilon$, occultation
depth and $R_{\rm p}/R_{\star}$ as a function of the number of sinusoidal
curves subtracted from the light curve. The dashed lines correspond to the
$1\sigma$ uncertainty of the parameters obtained from the $\Phi_{16}(t)$ LC.

In order to explore the effect that the stellar pulsation has on the precision
and accuracy of the out-of-transit variations, we show the behaviour of
$A_{g}$, $I_{\rm p}/I_{\star}$, $\varepsilon$, the occultation depth, $K_{\rm
phot}$, and $q_{\rm ell}$ as a function of the number of sinusoidal curves
removed as described in Eq. (1). As a check for consistency, we also show the
estimated $R_{\rm p}/R_{\star}$ in every case. The parameters are shown in
Figs. 4. The occultation depth can be calculated as follows. First, we divide
the best-fit model (including the transits, occultations and the out-of-
transit variations) with the sum of Eq. (4), the ellipsoidal distortion and
the Doppler boosting. The median flux level during the (total) occultation can
then be compared to the median of the flat baseline that is out-of-transit and
out-of-occultation. The uncertainty of the occultation depth can be estimated
as the standard deviation divided by the square root of the number of LC
points during the occultation and the standard deviation divided by the square
root of the number of points out-of-transit and out-of-occultation, added in
quadrature.

In TLCM, the Doppler beaming and ellipsoidal variations are fitted
simultaneously to constrain each other Csizmadia (2020). As discussed in Sect.
2.1, one component of the stellar variability is close to second orbital
harmonic of WASP-167b – the same time-dependence as the ellipsoidal effect.
The wavelet-based noise filtering of Carter & Winn (2009), as implemented in
TLCM Csizmadia (2020); Csizmadia et al. (2023b), ‘force’ the LC into the most
likely shape. Given the additional signal around the $2/P$ frequency region,
this component of the stellar variability can yield an inaccurate (either
underestimated or overestimated) mass for WASP-167b. Theoretically, it would
be possible to rely on the Doppler boosting alone for the mass estimation
(Mazeh et al., 2012). However, as Csizmadia (2020) pointed out, that effect is
often degenerate with the model for the light reflected from the planetary
atmosphere. Although Csizmadia (2020) suggests the usage of radial velocity
data to break this degeneracy between the out-of-transit parameters, the
available measurements from Temple et al. (2017) imply that for this system,
the mass of the companion can not be determined with adequate precision from
the present data. The two parameters representing these two effects (beaming
and ellipsoidal variations), $K_{\rm phot}$ and $q_{\rm ell}$ do not change
significantly after subtracting the sinusoidal curve corresponding to the
frequency near $2/P$, labelled F2 in Table 3 (Fig. 3, Table 4), however, they
are consistent with each other when modelling $\Phi_{2}(t)$ – $\Phi_{16}(t)$.

When we model $\Phi_{2}(t)$, the offset between the brightest point on the
stellar surface and the substellar point ($\varepsilon$) is $<-90^{\circ}$
i.e. on the nightside of the planet, assuming tidal locking. This case also
corresponds to the most shallow occultation (Fig. 4, Table 4). This value of
$\varepsilon$ is nonphysical. All of the best-fit parameters are consistent
with each other when extracted from $\Phi_{3}(t)$ – $\Phi_{16}(t)$ (Figs. 3
and 4). The marginal detection of an eastward offset111According to Csizmadia
et al. (2023b), positive $\varepsilon$ values correspond to eastward shift,
while negative values imply a westward shift, under the assumption that the
planet has a prograde rotation. between the brightest point on the planetary
surface and the substellar point observed in $\Phi_{15}(t)$ and $\Phi_{16}(t)$
are within $1.8\sigma$ to the respective values from $\Phi_{3}(t)$ and within
$2\sigma$ to $0^{\circ}$.

To decide which set of parameters to accept, we calculated the Bayesian
Information Criteria (BIC) for all 17 light curve solutions (Table 4). When
calculating the BIC values, we used the best-fit model without the correlated
noise identified via the wavelets in comparison to $\Phi_{0}(t)$ –
$\Phi_{16}(t)$. The total number of parameters was therefore $13+3i$, where
$i$ is the number of sinusoidal curves subtracted – that are described by
three independent parameters: their frequency, amplitude and phase – (without
$\sigma_{w}$ and $\sigma_{r}$). The BIC analysis suggests that the
$\Phi_{16}(t)$ light curve produces the optimal set of parameters, we
therefore adopt this one. We note that $R_{\rm p}/R_{\star}$ does not change
significantly across the $17$ tested cases. This is expected, since the
transits have higher S/N than the out-of-transit variability, and the transit
depths can therefore be estimated with higher precision and accuracy in
comparison to other parameters (Csizmadia et al., 2023b). We further emphasise
that the uncertainties of the analysed parameters are also consistent with
each other (Table 4).

The uniform priors (also known as searchboxes) used in the analysis for each
individual parameter were the same throughout the analysis of the 17 light
curves. These are shown in Table 2. The combination of the best-fit model for
the planetary transits, occultations, out-of-transit variability and the red
noise (for $\Phi_{16}(t)$) is plotted in Fig. 1.

###  3.2 System parameters

Table 2: Fitted and derived parameters from the $\Phi_{16}(t)$ light curve.

Parameter | Searchbox | This work | This work (zero mass) | Temple et al. (2017)
---|---|---|---|---
$a/R_{\star}$ | [$0.1$, $9.9$] | $4.126\pm 0.054$ | $4.122\pm 0.053$ | $4.38\pm 0.36$
$R_{\rm p}/R_{\star}$ | [$0.0$, $0.2$] | $0.08945\pm 0.00051$ | $0.08953\pm 0.00053$ | $0.0906\pm 0.0043$
$b$ | [$0.0$, $1.0$] | $0.7627\pm 0.0089$ | $0.7635\pm 0.0086$ | $0.77\pm 0.01$
$P$ [days] | [$2.017$, $2.027$] | $2.02195830\pm 0.00000053$ | $2.02195831\pm 0.00000056$ | $2.0219596\pm 0.0000006$
$t_{C}$ [BTJD] | [$8592.13$, $8592.23$] | $8592.18253\pm 0.00027$ | $8592.18254\pm 0.00028$ | $6592.4643\pm 0.0002$
$A$aafootnotemark: $a$ | [$-1.5$, $1.5$] | $0.86\pm 0.10$ | $0.85\pm 0.10$ | –
$B$bbfootnotemark: $b$ | [$-1.5$, $1.5$] | $1.291\pm 0.083$ | $1.290\pm 0.082$ | –
$\sigma_{r}$ [100 ppm] | [$0.0$, $10000.0$] | $703.1\pm 7.1$ | $703.1\pm 7.2$ | –
$\sigma_{w}$ [100 ppm] | [$0.0$, $6000.0$] | $10.527\pm 0.034$ | $10.526\pm 0.034$ | –
$A_{g}$ | [$0.0$, $1.0$] | $0.34\pm 0.11$ | $0.32\pm 0.12$ | –
$I_{p}/I_{\star}$ | [$0.0$, $1.0$] | $0.0010\pm 0.0013$ | $0.0025\pm 0.0027$ | –
$\varepsilon$ [∘] | [$-90.0$, $90.0$] | $17.7\pm 11.2$ | $18.2\pm 12.0$ | –
$K_{\rm phot}$ [m s-1] | [$0.0$, $200000.0$] | $45\pm 34$ | – | –
$K$ [m s-1] | – | – | – |  $<897$ccfootnotemark: $c$
$q_{\rm ell}$ | [$0.0$, $0.4$] | $0.00022\pm 0.00014$ | – | –
$h$ | [$-0.5$, $0.5$] | $-0.00055\pm 0.00015$ | $-0.00055\pm 0.00015$ | –
Derived parameters |
$R_{\star}$ [$R_{\odot}$] | $1.861\pm 0.057$ | $1.84\pm 0.17$ | $1.79\pm 0.05$
$M_{\star}$ [$R_{\odot}$] | $1.49\pm 0.13$ | $1.44\pm 0.32$ | $1.59\pm 0.08$
$R_{\rm p}$ [$R_{\rm J}$] | $1.621\pm 0.059$ | $1.60\pm 0.16$ | $1.58\pm 0.05$
$i_{p}$ [∘] | $79.3\pm 0.2$ | $79.3\pm 0.2$ | $79.9\pm 0.3$
$u_{1}$ | $0.453\pm 0.062$ | $0.453\pm 0.062$ | –
$u_{2}$ | $0.030\pm 0.067$ | $0.029\pm 0.069$ | –
$M_{\rm P,Beaming}$ [$M_{\rm J}$] | $0.37\pm 0.28$ | – |  $<8$ccfootnotemark: $c$
$M_{\rm P,Ellipsoidal}$ [$M_{\rm J}$] | $0.34\pm 0.22$ | – | –
Occultation depth [ppm] | $106.8\pm 27.3$ | $108.8\pm 27.3$ |  $<1100$ ppm

222aafootnotemark: $a$Gaussian prior applied: $\mathcal{N}(0.91,0.10)$.
bbfootnotemark: $b$Gaussian prior applied:
$\mathcal{N}(1.40,0.10)$.ccfootnotemark: $c$Estimated from radial velocities,
a method similar in principle to Doppler beaming.

We applied the same uniform priors for the free parameters (Table 2) on all 17
light curves, with the exception of the two limb darkening coefficients, where
the same Gaussian priors were used. The best-fitting light curve model,
obtained from $\Phi_{16}(t)$ is shown in Fig. 5 and the corresponding
parameters are shown in Table 2. The basic transit parameters, including
$a/R_{\star}$, $R_{\rm p}/R_{\star}$ and $b$, agree within $1\sigma$ with the
respective values presented in the discovery paper of (Temple et al., 2017) –
after converting those to the formalism used in our analysis. A slight
($\lesssim 1.7\sigma$) discrepancy can be observed in the measurements of
$i_{\rm p}$, which may be attributed to the different treatment of limb
darkening. The results of Temple et al. (2017) are based on the four parameter
law (Claret, 2000, 2004) with theoretical coefficients, the applicability of
which is disputable for moderately fast rotating, pulsating stars, such as
WASP-167. It is also possible that the transit chord is moving due to nodal
precession, as in the cases of WASP-33b(e.g. Johnson et al., 2015; Stephan et
al., 2022), KELT-9b (Stephan et al., 2022) or Kepler-13Ab (e.g. Szabó et al.,
2014; Herman et al., 2018; Szabó et al., 2020), The planetary radius of
$1.621\pm 0.059$ RJ is also in good agreement with the size reported in Temple
et al. (2017). We also present slight improvement in the precision of these
parameters from the TESS data alone.

The well-known degeneracies between the scaled semi-major axis, the relative
planetary radius and the impact parameter are observable in Fig. 15. These can
be quantified via the correlation coefficients from the posteriors: $r=-0.49$
(between $a/R_{\star}$ and $R_{\rm p}/R_{\star}$), $-0.95$ (between
$a/R_{\star}$ and $b$), and $0.51$ (between $R_{\rm p}/R_{\star}$ and $b$).
The reparametrization of the quadratic limb-darkening coefficients, expressed
in Eqs. (2) and (3) yield no significant degeneracies between these and the
transit parameters discussed above.

Temple et al. (2017) provide an upper limit of $8$ MJ on the mass of WASP-167
by analysing the radial velocity curve. Our adopted set of solutions suggest
that the mass is considerably lower at $0.37\pm 0.28$ MJ or $0.34\pm 0.22$ MJ,
as obtained by modelling the Doppler beaming effect and the ellipsoidal
variability, respectively, placing the planet in the mass regime of Saturn. We
note however that due to the stellar signal near the second orbital harmonic
(Fig. 6), these masses, obtained from the marginal detection of the two
effects, should not be taken at face value. Furthermore, the correlation
between the posteriors (Fig. 15) of $K_{\rm phot}$ and $q_{\rm ell}$
($r=0.83$) also implies that caution must be taken when considering the mass
of this object. The true mass of WASP-167b may be estimated by modeling these
two effects from infrared light curve observations, where the stellar activity
is known to be less significant.

The observed occultation depth of $106.8\pm 27.3$ ppm is well below the upper
limit of $1100$ ppm in $z^{\prime}$ band, as obtained from TRAPPIST
observations (Temple et al., 2017) (see Ricker et al., 2015, for comparison of
the passbands). Incidentally, this result also emphasises the necessity of
space-based observations in the study of exoplanets. This occultation depth is
detected with $p=1.12\cdot 10^{-6}$, based on the two-sample t-test in which
the out-of-transit and out-of-occultation flat baseline is compared to the
signal level during the occultation. We also measure a slight eastward offset
in the brightest point of the planetary surface at $17.7^{\circ}\pm
11.2^{\circ}$, a value that is similar to the observed by von Essen et al.
(2020) for WASP-33b. The posteriors of the parameters describing the observed
light variation of WASP-167b are not correlated with each other, or the mass
measuring parameters (Fig. 15).

The wavelet-based noise fitting in TLCM is constrained by prescribing that the
standard deviation of the residuals (of the LC and red noise models) must be
equal to the average photometric uncertainty (Csizmadia et al., 2023b). This
implies that the efficiency of the noise treatment is dependent on, among
other things, the brightness of the star (in this case, $V=10.52$ mag).
Accordingly, some autocorrelated effects may be present in the residuals, as
seen in Fig. 5. To rule out the possibility that the residuals seen in Fig. 5
are caused by an overestimation of the ellipsoidal variability (and
consequently, the mass of WASP-167b),we ran another test with the beaming and
ellipsoidal effects turned off (i.e. a mass-less case) on the $\Phi_{16}$ LC.
The resultant parameters are also shown in Table 2. All of the best-fit
parameters are in excellent agreement with the case when we also modelled the
two mass-related effects. We measure an increase in $I_{p}/I_{\star}$, which
is a consequence of the lack of a (near) constant signal that would have been
the sum of the beaming and ellipsoidal effects. The surface brightness ratio
is therefore enlarged (yet still consistent with the uncertainty range from
the original analysis) to account for the underlying occultation depth (which
is observed to be $108.8\pm 27.3$ ppm). The resultant (‘zero mass’) LC model
is shown in Fig. 12. The same auto-correlated effects are present in the
residuals of this modelling as in the case of Fig. 5, implying that the mass
of WASP-167b is not overestimated in the original calculations. Consequently,
we adopt the more thorough ‘massive’ case for the rest of the analysis.

As an additional check for consistency, we also modelled the light curves from
Sectors 10, 37 and 64 separately, for all 17 stages of pre-whitening. We
fitted the same parameters, over the same intervals as in the global analysis
(Table 2). The resultant parameters (Table 5) are shown on Figs. 13 and 14.
None of the parameters extracted in this way differ by more than $3\sigma$
from the adopted ones. The mass-related parameters (Fig. 13) from Sector 10
are always consistent with a zero-mass object. The effect of the ‘critical
frequency’ F2 is, however, present in the $K_{\rm phot}$ and $q_{\rm ell}$
values from Sectors 37 and 64. As in the case of the global analysis, the
relevant values from $\Phi_{2}$ to $\Phi_{16}$ are also consistent with zero
in all three Sectors. Certain $\varepsilon$ values are non-physical (14),
yielding negative occultation depths (as in the case of $\Phi_{15}$ and
$\Phi_{16}$ from Sector 37). These are, however, consistent with $0$ within
the estimated uncertainty range (Table 5), implying that the out-of-transit
variability can not be distinguished from the time-correlated noise in these
cases. The pre-whitening process is not perfectly efficient at the removal of
all quasi-periodic noise features (Fig. 1). Consequently, its residuals (third
row of 1) show variability on timescales comparable to the orbital period, but
much shorter than the duration of the light curve of the three combined
sectors. As a result, in our global fit, these quasi-periodic events get
‘averaged out’, yielding parameters that are fundamentally more precise and
accurate. This phenomenon can also be observed in the case of WASP-33b, where
Lehmann et al. (2015) used a large number of radial velocity observations to
deal with the effects of stellar pulsations (in contrast with Kovács et al.
(2013), who only had access to $12$ data points in total). We note that
because of the deep transits (yielding high S/N for the transit depth), the
$R_{\rm p}/R_{\star}$ are always consistent with each other within $1\sigma$.
We also note that in general, Sector 10 yield the highest uncertainties for
any given parameter, corresponding to the highest RMSE seen in the data from
these three Sectors (Fig. 1).

Figure 5: Full phase curve model of WASP-167b (solid red line). The top panel
shows the $\Phi_{16}(t)$ light curve with the fitted red noise subtracted via
the wavelet-based filtering algorithm. The occultation is clearly visible on
the zoomed-in middle panel, with the reflection effect, ellipsoidal
variability and Doppler beaming separated (magenta, blue, and orange lines,
respectively). The residuals are shown on the bottom panel. White circles and
the respective error bars represent bins of $500$ data points each. The less
opaque red lines correspond to $1000$ model LCs are randomly selected from the
($1\sigma$) uncertainty ranges of each of the parameters.

We further note that the height correction parameter $h$ is not degenerated
with any other fitting parameter (Fig. 15). This implies that its use is
effective in treating any possible light contamination that is not corrected
for via the CROWDSAP keyword during the photometry.

##  4 Discussion

###  4.1 Stellar signal

Table 3: Frequencies, amplitudes, phases and the respective S/N values extracted from the Fourier transform. |  Frequency [$d^{-1}$] | Amplitude | Phase | S/N
---|---|---|---|---
Sector 10
F1 | $0.0831\pm 0.0005$ | $810.0\pm 17.2$ | $0.5519\pm 0.0034$ | $44.6$
F2 | $0.9874\pm 0.0006$ | $650.0\pm 17.2$ | $0.2938\pm 0.0042$ | $27.8$
F3 | $0.6235\pm 0.0009$ | $435.9\pm 17.2$ | $0.6381\pm 0.0063$ | $20.7$
F4 | $0.3009\pm 0.0006$ | $618.5\pm 17.2$ | $0.3291\pm 0.0044$ | $30.4$
F5 | $0.8305\pm 0.0008$ | $493.5\pm 17.2$ | $0.3118\pm 0.0055$ | $22.7$
F6 | $1.7426\pm 0.0013$ | $307.6\pm 17.2$ | $0.9985\pm 0.0089$ | $11.7$
F7 | $0.3591\pm 0.0008$ | $503.4\pm 17.2$ | $0.0406\pm 0.0054$ | $25.1$
F8 | $0.8781\pm 0.0008$ | $476.3\pm 17.2$ | $0.7507\pm 0.0057$ | $21.9$
F9 | $0.7350\pm 0.0011$ | $358.2\pm 17.2$ | $0.5646\pm 0.0076$ | $16.8$
F10 | $1.6400\pm 0.0012$ | $317.0\pm 17.2$ | $0.6849\pm 0.0086$ | $12.4$
F11 | $2.1232\pm 0.0017$ | $227.8\pm 17.2$ | $0.6201\pm 0.0120$ | $7.6$
F12 | $0.5293\pm 0.0014$ | $274.5\pm 17.2$ | $0.4919\pm 0.0100$ | $12.9$
F13 | $1.2725\pm 0.0017$ | $228.2\pm 17.2$ | $0.2254\pm 0.0120$ | $8.7$
F14 | $0.2112\pm 0.0018$ | $216.4\pm 17.2$ | $0.3051\pm 0.0126$ | $11.1$
F15 | $1.8348\pm 0.0019$ | $205.8\pm 17.2$ | $0.4257\pm 0.0133$ | $7.5$
F16 | $1.5909\pm 0.0019$ | $210.5\pm 17.2$ | $0.2256\pm 0.0130$ | $8.2$
F17 | $1.0819\pm 0.0031$ | $124.8\pm 17.2$ | $0.6277\pm 0.0219$ | $5.1$
F18 | $2.0317\pm 0.0032$ | $120.5\pm 17.2$ | $0.3283\pm 0.0227$ | $4.1$
F19 | $2.5274\pm 0.0034$ | $113.9\pm 17.2$ | $0.9292\pm 0.0240$ | $3.7$
F20 | $0.7661\pm 0.0020$ | $191.9\pm 17.2$ | $0.5987\pm 0.0143$ | $8.9$
F21 | $0.0083\pm 0.0010$ | $376.4\pm 17.2$ | $0.7295\pm 0.0073$ | $22.1$
F22 | $1.3414\pm 0.0033$ | $118.9\pm 17.2$ | $0.2652\pm 0.0230$ | $4.5$
Sector 37
F1 | $0.8537\pm 0.0001$ | $438.4\pm 14.5$ | $0.2611\pm 0.0005$ | $244.8$
F2 | $0.9818\pm 0.0005$ | $664.9\pm 14.5$ | $0.8651\pm 0.0035$ | $37.0$
F3 | $0.2145\pm 0.0008$ | $438.8\pm 14.5$ | $0.8865\pm 0.0053$ | $25.0$
F4 | $0.1748\pm 0.0009$ | $360.4\pm 14.5$ | $0.2597\pm 0.0064$ | $20.8$
F5 | $1.7489\pm 0.0006$ | $571.8\pm 14.5$ | $0.8042\pm 0.0040$ | $26.0$
F6 | $1.8180\pm 0.0007$ | $512.4\pm 14.5$ | $0.2650\pm 0.0045$ | $23.1$
F7 | $0.4225\pm 0.0012$ | $274.5\pm 14.5$ | $0.5237\pm 0.0084$ | $14.8$
F8 | $0.0463\pm 0.0014$ | $235.9\pm 14.5$ | $0.8155\pm 0.0098$ | $17.4$
F9 | $2.1126\pm 0.0016$ | $215.3\pm 14.5$ | $0.9823\pm 0.0107$ | $9.6$
F10 | $1.0427\pm 0.0013$ | $251.6\pm 14.5$ | $0.6853\pm 0.0092$ | $13.4$
F11 | $1.6955\pm 0.0011$ | $308.3\pm 14.5$ | $0.7244\pm 0.0075$ | $14.0$
F12 | $1.4170\pm 0.0025$ | $134.2\pm 14.5$ | $0.4598\pm 0.0172$ | $6.5$
F13 | $0.7817\pm 0.0017$ | $197.1\pm 14.5$ | $0.5396\pm 0.0117$ | $10.6$
F14 | $0.2704\pm 0.0017$ | $202.0\pm 14.5$ | $0.1712\pm 0.0114$ | $10.7$
F15 | $0.3733\pm 0.0025$ | $135.3\pm 14.5$ | $0.4879\pm 0.0170$ | $7.2$
F16 | $2.2255\pm 0.0032$ | $104.5\pm 14.5$ | $0.7059\pm 0.0221$ | $4.9$
F17 | $3.0125\pm 0.0034$ | $98.9\pm 14.5$ | $0.3743\pm 0.0233$ | $4.4$
F18 | $0.8548\pm 0.0001$ | $381.8\pm 14.5$ | $0.2452\pm 0.0006$ | $213.2$
F19 | $1.8874\pm 0.0025$ | $135.7\pm 14.5$ | $0.1838\pm 0.0170$ | $6.0$
F20 | $1.5438\pm 0.0035$ | $96.1\pm 14.5$ | $0.6740\pm 0.0240$ | $4.5$
F21 | $3.4828\pm 0.0042$ | $80.8\pm 14.5$ | $0.2763\pm 0.0285$ | $3.8$
F22 | $0.5737\pm 0.0040$ | $84.6\pm 14.5$ | $0.2580\pm 0.0273$ | $4.7$
Sector 64
F1 | $0.1114\pm 0.0004$ | $843.0\pm 13.9$ | $0.7294\pm 0.0026$ | $34.1$
F2 | $0.9874\pm 0.0005$ | $654.4\pm 13.9$ | $0.2529\pm 0.0034$ | $20.7$
F3 | $0.8437\pm 0.0004$ | $691.2\pm 13.9$ | $0.6012\pm 0.0032$ | $23.2$
F4 | $0.1589\pm 0.0005$ | $584.4\pm 13.9$ | $0.4518\pm 0.0038$ | $22.3$
F5 | $0.0690\pm 0.0004$ | $732.1\pm 13.9$ | $0.4386\pm 0.0030$ | $33.7$
F6 | $0.3003\pm 0.0006$ | $513.9\pm 13.9$ | $0.4419\pm 0.0043$ | $19.5$
F7 | $0.4490\pm 0.0008$ | $406.3\pm 13.9$ | $0.0158\pm 0.0054$ | $15.4$
F8 | $1.8261\pm 0.0008$ | $390.5\pm 13.9$ | $0.0408\pm 0.0057$ | $9.8$
F9 | $1.7409\pm 0.0009$ | $325.6\pm 13.9$ | $0.8936\pm 0.0068$ | $8.5$
F10 | $0.2676\pm 0.0007$ | $434.2\pm 13.9$ | $0.3894\pm 0.0051$ | $16.2$
F11 | $0.9031\pm 0.0009$ | $344.6\pm 13.9$ | $0.4086\pm 0.0064$ | $11.3$
F12 | $0.7871\pm 0.0010$ | $317.7\pm 13.9$ | $0.5489\pm 0.0069$ | $10.6$
F13 | $1.6481\pm 0.0012$ | $250.0\pm 13.9$ | $0.0324\pm 0.0088$ | $6.8$
F14 | $0.5599\pm 0.0012$ | $264.0\pm 13.9$ | $0.6380\pm 0.0084$ | $9.3$
F15 | $0.3661\pm 0.0015$ | $207.4\pm 13.9$ | $0.4260\pm 0.0106$ | $8.1$
F16 | $0.7305\pm 0.0014$ | $225.3\pm 13.9$ | $0.3250\pm 0.0098$ | $7.5$
F17 | $1.3476\pm 0.0017$ | $185.0\pm 13.9$ | $0.3958\pm 0.0119$ | $5.5$
F18 | $2.1325\pm 0.0016$ | $197.5\pm 13.9$ | $0.1294\pm 0.0112$ | $5.0$
F19 | $2.2308\pm 0.0022$ | $141.8\pm 13.9$ | $0.2955\pm 0.0156$ | $3.7$
F20 | $0.6110\pm 0.0017$ | $180.7\pm 13.9$ | $0.4660\pm 0.0122$ | $6.3$
F21 | $0.6792\pm 0.0019$ | $160.2\pm 13.9$ | $0.1076\pm 0.0138$ | $5.5$
F22 | $1.2744\pm 0.0021$ | $144.0\pm 13.9$ | $0.5849\pm 0.0153$ | $4.4$

The analysis of the stellar pulsations generally requires uninterrupted light
curves. To this end, we subtracted the phase curve model (including the
transits, occultations and out-of-transit effects) obtained by the analysis of
$\Phi_{16}(t)$ from $\Phi_{0}(t)$. We then proceeded with calculating the
Fourier spectrum of these residuals using Period04, separately for the LCs of
the three Sectors. These spectra are shown in Fig. 6. The frequencies,
amplitudes and phases, that are the result of the Fourier analysis, are
presented in Table 3, for peaks with S/N $\gtrsim 4$. In all three Sectors, we
extracted data from $22$ peaks with the highest amplitudes. While a detailed
analysis of the stellar signal is beyond the scope of this paper, several
observations can be made from Fourier spectra. ($i$) There are no significant
peaks in the data beyond $\sim 3.5\leavevmode\nobreak\ {\rm d^{-1}}$. ($ii$)
There is a peak in the Fourier spectra – F2 from Table 3 – that almost exactly
coincides with the second orbital harmonic of WASP-167b in all three Sectors
(Fig. 6). When comparing to the best-fit orbital frequency ($f_{\rm
orb}=(0.49547000\pm 1.3\cdot 10^{-7})$ d-1), we find that F2 can be expressed
as $1.996$, $1.958$, and $1.996$ times $f_{\rm orb}$ in the three Sectors,
respectively. Similar commensurability was noted by von Essen et al. (2014)
and later confirmed by Kálmán et al. (2022) in the case of the WASP-33 system.
($iii$) The frequencies and amplitudes of the spectral peaks vary considerably
from Sector to Sector, with the exception of F2, the ‘critical frequency’.
Given the incompatible nature of the Fourier spectra in the three Sectors,
searching for a phase shift in the individual sinusoidal curves(e.g. von Essen
et al., 2014) is not feasible.

We find that $2f_{\rm orb}$ matches F2 within $\approx 3\sigma$ (in Sectors 10
and 64), and within $15\sigma$ in Sector 37, when using the formal
uncertainties of the pulsational frequencies, calculated via Period04. These
calculations are based on the idealistic case where the pulsations can be
expressed as simple sinusoidal signals with constant phases, amplitudes and
frequencies. As we demonstrate in this paper, this is not the case for
WASP-167, and, as a consequence, these uncertainties may be considerably
underestimated. We constructed a simple test to investigate the possibility
that the ‘critical frequency’ only appears close to $2f_{\rm orb}$ without
underlying tidal interactions. To that end, similarly to von Essen et al.
(2014) and Kálmán et al. (2023a), we constructed $10^{5}$ synthetic ‘frequency
spectra’, each consisting of 22 frequencies (corresponding to the number of
peaks identified and presented in Table 3), drawn from a uniform distribution
between $0$ d-1 and $3.5$ d-1. We then proceeded to match these to $2f_{\rm
orb}$ to within $3\cdot 0.0015$ d-1, or roughly the average uncertainty in the
frequencies from Table 3, thus providing a conservative estimate for the
synthetic F2. We found that commensurability defined in this way only arises
in $5.6\%$ of the cases. This statistic further serves as evidence for tidal
interactions between WASP-167b and its host.

The frequency range in which the stellar activity presents itself means that
the observing windows might not be sufficient for a detailed characterisation
of its nature. If it originates from pulsations, as stated in Temple et al.
(2017), then WASP-167b is a $\gamma$ Doradus pulsator. Indeed, instability for
this type of stellar pulsation may be expected on the time scales available
through the TESS photometry (e.g. Bravo et al., 2014; Kurtz et al., 2015).
Although they are generally not expected for $T_{\rm eff}\approx 7000$ K,
stellar spots can also produce variability compatible with our observations
(Fig. 6) and they might offer a better explanation for the apparent changes in
the frequencies and amplitudes that are seen in the Fourier decomposition. It
is also possible that the star presents both $\gamma$ Dor type pulsations and
spots. Future studies focusing on the Zeeman effect (Babcock, 1947), or more
precise photometric observations with longer baselines (with e.g. PLATO Rauer
et al., 2014) may help understanding the origins of the stellar variability of
WASP-167. We note, however, that the accurate interpretation of the stellar
signal depends strongly on the phase curve model of the planet (Fig. 5), as
the latter is also comprised of sinusoidal terms with comparable amplitudes to
the ones from stellar oscillational frequencies (Table 3).

Figure 6: Fourier spectrum of the residuals of the LC model of WASP-167 from
Sectors 10, 37, and 64 (left, middle and right panel). The inset plots show
the spectral windows. The dashed red line marks the second orbital harmonic of
WASP-167b, demonstrating the ‘critical frequency’ that influences primarily
the mass measurements of the planet. The noise level, as calculated by
Period04, is shown with a solid red line.

The peak in the Fourier spectra near the second orbital harmonic (Fig. 6) is
likely composed of both stellar variability (regardless of its nature) and the
ellipsoidal effect. As discussed in Sect. 3.2, this implies that the mass
measurements of WASP-167bmight be inaccurate. This issue might be solved by
obtaining more, higher quality radial velocity data than that presented in
Temple et al. (2017), as in the case of WASP-33 (Lehmann et al., 2015). It is
also possible that, as stated in Sect. 3.2, IR observations by e.g. Ariel
(Tinetti et al., 2021) may be able to provide more reliable mass constraints,
since the pulsational amplitudes are known to be lower at higher wavelengths
(e.g. Derekas et al., 2009).

We also see that the F2 frequency is present in the Fourier spectra from all
three Sectors (Fig. 6), with similar amplitudes. This suggest that WASP-167b
is influencing the activity of its host. To examine the intra-sector
stability, we used a time-frequency method known as the discrete wavelet
transform (Foster, 1996) after filling in the data downlink gaps (seen in Fig.
1) with zeros. The wavelet maps of Fig. 7 show the $0.25$ – $2$ d-1 frequency
regime (thus incorporating F2). There is clear amplitude modulation around F2,
with a period of $\sim 4$ days, or $\sim 2P$. A similar behaviour has also
been observed in the case of WASP-33band it’s host star Kálmán et al. (2022).
This timescale is well above the estimated orbital period $\leq 1.81$ d-1,
suggesting that the origin of this particular signal is not related to stellar
rotation. We also observe that in the $\leq 0.8$ d-1 frequency regime there
are considerable amplitude and phase curve modulations both intra-sector, and
from Sector to Sector. However, the long-term effects, such as detector
systematics, would also be observable in this range, and they can not be
distinguished from the stellar activity from the light curves used in this
analysis. A deeper exploration of the stellar signal would need to carefully
remove all of the known instrumental effects from the light curves, however,
such a study is beyond the scope of this paper. Instrumental noise sources
could therefore be a possible explanation for the inter-Sector variability in
the Fourier spectrum (Fig. 6 and Table 3). Another possibility might be the
instability of the pulsational or spot patterns of the star, at least on the
time scales at which the three observed Sectors sample the LC.

Figure 7: Wavelet maps of the stellar activity from Sectors 10 (left), 37
(middle) and 64 (right).

###  4.2 Dayside, nightside and intrinsic temperature of WASP-167b

In the phase curve parametrization of Eq. (4), the second term, characterised
by the geometric albedo, is made up of light that is reflected from the
planetary atmosphere (determined by the Bond albedo, $A_{B}$) as well as light
that is emitted from the dayside. To constrain the dayside and nightside
temperatures, we use the toy model presented in Csizmadia et al. (2023a) and
used in Kálmán et al. (2024) for the energy budget of WASP-167b, to
distinguish between the two phenomena. In a thermodynamical equilibrium, we
can state that the incoming and outgoing luminosities are related to each
other by:

$L_{\rm income}+L_{\rm intrinsic}=L_{\rm day}+L_{\rm night}+L_{\rm absorbed}.$
(6)

In this formulation, $L_{\rm intrinsic}$ is the result of the intrinsic
planetary heat. The incident luminosity is related to the bolometric stellar
luminosity via

$L_{\rm income}=L_{\star}\left(\frac{R_{p}}{2a}\right)^{2}.$ (7)

By definition, the reflected light from the planetary surface can be
characterised as

$L_{\rm reflected}=(1-A_{B})L_{\rm absorbed}.$ (8)

Assuming blackbody radiation for the dayside and nightside hemispheres of the
planet, we get:

$\displaystyle T_{\rm day}$ $\displaystyle=\sqrt[4]{T_{\rm
intrinsic}^{4}+\frac{1}{2}f(1-A_{B})T_{\rm
eff}^{4}\left(\frac{R_{\star}}{a}\right)^{2}},$ (9) $\displaystyle T_{\rm
night}$ $\displaystyle=\sqrt[4]{T_{\rm
intrinsic}^{4}+\frac{1}{2}(1-f)(1-A_{B})T_{\rm
eff}^{4}\left(\frac{R_{\star}}{a}\right)^{2}}.$ (10)

Here $T_{\rm intrinsic}$, $T_{\rm day}$ and $T_{\rm night}$ are the internal,
dayside and nightside temperature of WASP-167b, respectively. The heat
redistribution efficiency, $f$, determines the amount of incident stellar
luminosity that passes through to the nightside of the planet.

The amplitude of the phase curve in Eq. (4) is the combination of emitted and
reflected light. By neglecting the marginal eastward offset, the theoretical
geometric albedo can be characterised as

$A_{\rm g,\leavevmode\nobreak\ theo}(A_{\rm B},f,T_{\rm
int})=\frac{1}{2}\left(\frac{a}{R_{\star}}\right)^{2}\frac{I_{\rm day}-I_{\rm
night}}{I_{\star}}+\frac{1}{4}A_{B}\left(\frac{R_{\star}}{a}\right)^{2},$ (11)

where $I_{\rm day}$ are the passband-specific luminosities of a unit-sized
surface area. The surface brightness ($I_{\rm p}/I_{\star}$) ratio of Eq. (4),
taken from $\Phi_{16}(t)$ (Table 2) can be matched to $I_{\rm
night}/I_{\star}$.

There are therefore two observable quantities and three parameters ($A_{B}$,
$f$ and $T_{\rm intrinsic}$) that need to be determined from them. Thus, no
unique solution can be found for these via observations in a single passband.
We therefore carried out a Monte Carlo integration by generating a sample of
$2\cdot 10^{6}$ independent random values for $A_{B}$, $f$ and $T_{\rm
intrinsic}$, within intervals of $[0,1]$, $[0.5,1]$ and $[0\ \text{K},5000\
\text{K}]$, respectively, with a uniform distribution. The uncertainties of
$T_{\rm eff}$ and $a/R_{\star}$ (Tables 1 and 2) were taken into account via
an additional $2\cdot 10^{6}$ independent random values, with Gaussian
distributions (with means and standard deviations taken from Tables 1 and 2).
We then proceeded by matching $A_{g,\text{theo}}$ and $I_{\rm
night}/I_{\star}$ to the $2\sigma$ uncertainty ranges for $A_{g}$ and $I_{\rm
p}/I_{\star}$ from Table 2.

The resultant distributions of the Bond albedo, intrinsic temperature and the
heat redistribution efficiency are shown in Fig. 8. From these, we can draw
the following conclusions. WASP-167b/KELT-13b has a moderately low Bond
albedo, with $A_{B}<0.51$ at the $2\sigma$ level. This is generally in line
with what is expected from a hot Jupiter based on empirical studies (e.g. Rowe
et al., 2008; Knutson et al., 2009; Cowan & Agol, 2011; Deming & Knutson,
2020; Singh et al., 2023; Kálmán et al., 2024) and theoretical predictions
(e.g. Sudarsky et al., 2000). We also place a lower limit on the heat
redistribution efficiency of $f>0.74$ (at $2\sigma$), implying that the
atmosphere of the planet is structured in a way that it retains the majority
of the incident (and absorbed) stellar flux on the dayside, similarly to
WASP-18Ab (Nymeyer et al., 2011). We find that the intrinsic temperature of
WASP-167b is $\leq 2340$ K, with $1470\pm 640$ K representing the median and
standard deviation of the distribution shown in Fig. 8. Combining these facts
with the $5.6$ L⊙ stellar luminosity (in the TESS passband) implies that the
dayside temperature of the planet (Fig. 9) is $2790\pm 100$ K, implying that
WASP-167b is a so-called ultra hot Jupiter. We also place an upper limit on
the nightside temperature (Fig. 9) of $2360$ K, with the uncertainty range
estimated as $1780\pm 380$ K. In fact, given that our $I_{\rm p}/I_{\star}$
estimates are consistent with $0$ in the majority of the cases (Fig 4), it is
safe to assume that all of the atmospheric parameters provided here can be
regarded as upper limits. We note that $A_{B}$, $f$ and $T_{\rm intrinsic}$
are heavily degenerate, which is the consequence of analysing data in a single
passband. Similarly to the nature of the stellar signal, future observations
with e.g. `PLATO` or `Ariel` will have the possibility of a more in-depth
exploration of the atmospheric temperature regimes of this planet.

We emphasise that all of the solutions represented on Fig. 8 satisfy the
requirements for $A_{g}$ and $I_{p}/I_{\star}$ equally well, and that no
solutions were found beyond the limits for $A_{B}$ and $f$ presented above.
The $2\sigma$ uncertainty range stems from taking the $2\sigma$ uncertainties
for $A_{g}$ and $I_{p}/I_{\star}$.

Figure 8: Distribution of the Bond albedo, the heat redistribution parameter
and the intrinsic temperature, the combination of which reproduces the
observed phase curve of WASP-167b. Figure 9: Distribution of the dayside and
nightside temperatures that reproduce the observed phase curve of WASP-167b.

To put our results into context, we adopted the dayside, nightside and
equilibrium temperatures, defined as ${T_{0}=T_{\rm
eff}\sqrt{\frac{R_{\star}}{2a}}}$, of several well studied exoplanets, from
Bell et al. (2021). We compare the derived $T_{\rm day}$ and $T_{\rm night}$
to these values on Figs. 10 and 11. From these, we can conclude that our
results are clear outliers from the planet population. Slight discrepancies
might occur due to differences in the surface gravity on the planets or
deviations from blackbody radiation. The latter could be enhanced as Bell et
al. (2021) present results from infra red phase curves. We also note that the
data adopted from Bell et al. (2021) is not compiled in a homogeneous way,
and, as was presented in Kálmán et al. (2023b), untreated time-correlated
noise can induce biases in the modelling parameters. This would imply that
such a direct comparison between different studies is not ideal. A further
possibility for the observed deviation of our results from the overall
population would be that we overestimate $T_{\rm day}$ and $T_{\rm night}$, by
overestimating $T_{\rm intrinsic}$, which is related to $I_{\rm p}/I_{\star}$.
The posterior distribution (Fig. 15) suggest that $I_{\rm p}/I_{\star}$ is
compatible with $0$, and, by using the median of this distribution in our
analysis, we might overestimate the amount of light emitted by the planet. A
deeper exploration of the population statistics is beyond the scope of this
paper.

Figure 10: Dayside temperature as a function of equilibrium temperature from
Bell et al. (2021) (black dots and error bars). Our result forWASP-167 is
shown with red. Figure 11: Nightside temperature as a function of equilibrium
temperature from Bell et al. (2021) (black dots and error bars). Our result
forWASP-167 is shown with red.

Although the $T_{\rm intrinsic}$ of WASP-167b is poorly constrained, we
explore the possible sources for the intrinsic heat. We list several
possibilities in Csizmadia et al. (2023a) and Kálmán et al. (2024), however,
at the $1.29^{+0.36}_{-0.27}$ Gyr age of the system (Temple et al., 2017),
with a planet on a circular orbit with a $\lambda=-167\pm 5^{\circ}$ stellar
obliquity, we assume that it originates from the so-called obliquity tides.
Levrard et al. (2007) and Millholland (2019) expressed the thermal
contribution of this heating process as

$L_{\rm tide}=\frac{6\pi}{P}\frac{k_{2}}{Q_{p}}\frac{R_{\rm
p}^{5}}{a^{6}}GM_{\star}^{2},$ (12)

where $k_{2}$ and $Q_{p}$ are the Love number and tidal dissipation factor of
the planet. The Love number can be measured by accounting for the slight
changes in the shape of the transit (e.g. Hellard et al., 2019, 2020; Barros
et al., 2022a, b), or, for eccentric orbits, via the modelling of the apsidal
motion (Csizmadia et al., 2019). A direct $k_{2}$-measurement for WASP-167b is
beyond the scope of this paper, we therefore adopt $k_{2}=0.61$ from Csizmadia
et al. (2019), a value which is also in line with $k_{2}$ measurements of
Jupiter (e.g. Durante et al., 2020). We find that the $\sim 10^{-4}$ L⊙
intrinsic luminosity of the planet yield $Q_{p}=(7.2\pm 1.3)\cdot 10^{5}$,
which is in line with Csizmadia et al. (2023a) and the direct measurements for
the giant planets of the Solar System (Goldreich & Soter, 1966).

##  5 Summary and conclusions

We present the phase curve study of the WASP-167/KELT-13 system, discovered by
Temple et al. (2017), based on its TESS photometry from Sectors 10, 37 and 64.
We confirm that the system hosts a hot Jupiter on a $\sim 2.022$ day orbit,
and with a radius $\sim 1.6$ RJ (Table 3.2). We also present refined transit
parameters. Previous attempts at constraining its mass from radial velocity
data were unsuccessful due to the moderately rapid rotation ($52\pm 8$ km
s-1), spectral type and activity of the host star. We found a peak in the
Fourier spectrum of WASP-167 that is in near-resonance with the second orbital
harmonic $\sim 0.989$ d-1 (Fig. 6) of its companion – the same time dependence
as the ellipsoidal variability caused by WASP-167b.

To analyse how the stellar pulsations affect the precision and accuracy of the
transit parameters, we prepared 17 LCs, with an increasing number of
sinusoidal components subtracted from them, according to Eq. 1. The parameters
of these harmonics were obtained via the Fourier transform using Period04. The
sets of examined out-of-transit parameters are shown in Figs. 3 and 4. Using
the Bayesian Information Criterion, we find that subtracting 16 sinusoidal
curves yields the optimal solution. This is coincidentally the total number of
harmonics that can be determined from the Fourier transform, with the commonly
used ${\rm S/N}\geq 4$ criteria. This process of whitening the data yields
self-consistent results, with the caveat that we may be overcompensating for
the ellipsoidal variability. Thus, the obtained mass of WASP-167b at $0.34\pm
0.22$ MJ may be considerably underestimated. We also raise the possibility
that the ‘critical frequency’ influencing the mass estimates of WASP-167b may
arise from planet-to-star interactions. If the stellar signal originates from
pulsations, then the WASP-167 system shows similarities with HAT-P-2 (de Wit
et al., 2017), or WASP-33 (Kálmán et al., 2022) and HD 31221b (Kálmán et al.,
2023a), depending on whether its orbit has a small (but non-zero) eccentricity
or not. On the other hand, if the variability of WASP-167 originates from
spots, then it is similar to AU Mic (Szabó et al., 2021) in this regard.

We also present the occultation of WASP-167b (Fig. 5), and its dependence on
the stellar pulsations (Fig. 4), via modeling the phase curve of the planet.
The resultant occultation depth of $106.8\pm 27.3$ ppm, and the corresponding
dayside temperature of $2790\pm 100$ K is well below the limits provided by
Temple et al. (2017) from ground-based data. It also raises the possibility
that WASP-167b is an ultra-hot Jupiter, similar to WASP-33 (Smith et al.,
2011). We measure a geometric albedo of $0.34\pm 0.11$, and an eastward offset
of the brightest point on the planetary surface of $17.2^{\circ}\pm
11.2^{\circ}$ – similarly to WASP-33 in the same passband (von Essen et al.,
2020). Furthermore, we obtain ($2\sigma$) upper limits on the nightside and
intrinsic temperature ($2360$ K and $2340$ K, respectively) and the Bond
albedo ($0.51$), and discuss the possible combinations of Bond albedos and
heat redistribution efficiencies (Fig. 8) that can reproduce the observed
phase curve (Fig. 5).

###### Acknowledgements.

This work was supported by the PRODEX Experiment Agreement No. 4000137122
between the ELTE Eötvös Loránd University and the European Space Agency
(ESA-D/SCI-LE-2021-0025). Support of the KKP-137523 ‘SeismoLab’ Élvonal grant
as well as the grant K-138962 of the Hungarian Research, Development and
Innovation Office (NKFIH) are acknowledged. This project has been supported by
the LP2021-9 Lendület grant of the Hungarian Academy of Sciences. Project no.
C1746651 has been implemented with the support provided by the Ministry of
Culture and Innovation of Hungary from the National Research, Development and
Innovation Fund, financed under the NVKDP-2021 funding scheme. This project
has received funding from the HUN-REN Hungarian Research Network under grant
number KMP-2023/79. Supported by the DKOP-23 Doctoral Excellence Program of
the Ministry for Culture and Innovation from the source of the National
Research, Development and Innovation Fund.

## References

  * Aerts et al. (2018) Aerts, C., Molenberghs, G., Michielsen, M., et al. 2018, ApJS, 237, 15
  * Ahlers et al. (2019) Ahlers, J. P., Barnes, J. W., & Myers, S. A. 2019, AJ, 158, 88
  * Ahlers et al. (2022) Ahlers, J. P., Fromont, E. F., Kopparappu, R., Cauley, P. W., & Haqq-Misra, J. 2022, ApJ, 928, 35
  * Astropy Collaboration et al. (2022) Astropy Collaboration, Price-Whelan, A. M., Lim, P. L., et al. 2022, ApJ, 935, 167
  * Babcock (1947) Babcock, H. W. 1947, ApJ, 105, 105
  * Bakos et al. (2007) Bakos, G. Á., Kovács, G., Torres, G., et al. 2007, ApJ, 670, 826
  * Barros et al. (2022a) Barros, S. C. C., Akinsanmi, B., Boué, G., et al. 2022a, A&A, 657, A52
  * Barros et al. (2022b) Barros, S. C. C., Akinsanmi, B., Boué, G., et al. 2022b, A&A, 658, C1
  * Bell et al. (2021) Bell, T. J., Dang, L., Cowan, N. B., et al. 2021, MNRAS, 504, 3316
  * Bókon et al. (2023) Bókon, A., Kálmán, S., Bíró, I. B., & Szabó, M. G. 2023, A&A, 674, A186
  * Borucki et al. (2010) Borucki, W. J., Koch, D., Basri, G., et al. 2010, Science, 327, 977
  * Bowman et al. (2020) Bowman, D. M., Burssens, S., Simón-Díaz, S., et al. 2020, A&A, 640, A36
  * Bravo et al. (2014) Bravo, J. P., Roque, S., Estrela, R., Leão, I. C., & De Medeiros, J. R. 2014, A&A, 568, A34
  * Breger et al. (1993) Breger, M., Stich, J., Garrido, R., et al. 1993, A&A, 271, 482
  * Bryan et al. (2024) Bryan, J., de Wit, J., Sun, M., de Beurs, Z. L., & Townsend, R. H. D. 2024, arXiv e-prints, arXiv:2403.08014
  * Carter & Winn (2009) Carter, J. A. & Winn, J. N. 2009, ApJ, 704, 51
  * Christian et al. (2006) Christian, D. J., Pollacco, D. L., Skillen, I., et al. 2006, MNRAS, 372, 1117
  * Claret (2000) Claret, A. 2000, A&A, 363, 1081
  * Claret (2004) Claret, A. 2004, A&A, 428, 1001
  * Collier Cameron et al. (2010) Collier Cameron, A., Guenther, E., Smalley, B., et al. 2010, MNRAS, 407, 507
  * Cowan & Agol (2011) Cowan, N. B. & Agol, E. 2011, ApJ, 729, 54
  * Csizmadia (2020) Csizmadia, S. 2020, MNRAS, 496, 4442
  * Csizmadia et al. (2019) Csizmadia, S., Hellard, H., & Smith, A. M. S. 2019, A&A, 623, A45
  * Csizmadia et al. (2023a) Csizmadia, S., Smith, A. M. S., Cabrera, J., et al. 2023a, A&A, submitted
  * Csizmadia et al. (2023b) Csizmadia, S., Smith, A. M. S., Kálmán, S., et al. 2023b, A&A, 675, A106
  * de Wit et al. (2017) de Wit, J., Lewis, N. K., Knutson, H. A., et al. 2017, ApJ, 836, L17
  * Deming et al. (2012) Deming, D., Fraine, J. D., Sada, P. V., et al. 2012, ApJ, 754, 106
  * Deming & Knutson (2020) Deming, D. & Knutson, H. A. 2020, Nature Astronomy, 4, 453
  * Derekas et al. (2009) Derekas, A., Kiss, L. L., Bedding, T. R., et al. 2009, MNRAS, 394, 995
  * Derekas et al. (2017) Derekas, A., Plachy, E., Molnár, L., et al. 2017, MNRAS, 464, 1553
  * Durante et al. (2020) Durante, D., Parisi, M., Serra, D., et al. 2020, Geochim. Res. Lett., 47, e86572
  * Faigler & Mazeh (2011) Faigler, S. & Mazeh, T. 2011, MNRAS, 415, 3921
  * Foreman-Mackey et al. (2017) Foreman-Mackey, D., Agol, E., Ambikasaran, S., & Angus, R. 2017, AJ, 154, 220
  * Foster (1996) Foster, G. 1996, AJ, 112, 1709
  * Ginsburg et al. (2019) Ginsburg, A., Sipőcz, B. M., Brasseur, C. E., et al. 2019, AJ, 157, 98
  * Goldreich & Soter (1966) Goldreich, P. & Soter, S. 1966, Icarus, 5, 375
  * Hartman et al. (2015) Hartman, J. D., Bakos, G. Á., Buchhave, L. A., et al. 2015, AJ, 150, 197
  * Haynes et al. (2015) Haynes, K., Mandell, A. M., Madhusudhan, N., Deming, D., & Knutson, H. 2015, ApJ, 806, 146
  * Hellard et al. (2019) Hellard, H., Csizmadia, S., Padovan, S., et al. 2019, ApJ, 878, 119
  * Hellard et al. (2020) Hellard, H., Csizmadia, S., Padovan, S., Sohl, F., & Rauer, H. 2020, ApJ, 889, 66
  * Herman et al. (2018) Herman, M. K., de Mooij, E. J. W., Huang, C. X., & Jayawardhana, R. 2018, AJ, 155, 13
  * Herrero et al. (2011) Herrero, E., Morales, J. C., Ribas, I., & Naves, R. 2011, A&A, 526, L10
  * Hey et al. (2021) Hey, D. R., Montet, B. T., Pope, B. J. S., Murphy, S. J., & Bedding, T. R. 2021, AJ, 162, 204
  * Hurley et al. (2000) Hurley, J. R., Pols, O. R., & Tout, C. A. 2000, MNRAS, 315, 543
  * Husser et al. (2013) Husser, T.-O., Wende-von Berg, S., Dreizler, S., et al. 2013, A&A, 553, A6
  * Johnson et al. (2015) Johnson, M. C., Cochran, W. D., Collier Cameron, A., & Bayliss, D. 2015, ApJ, 810, L23
  * Kálmán et al. (2022) Kálmán, S., Bókon, A., Derekas, A., et al. 2022, A&A, 660, L2
  * Kálmán et al. (2024) Kálmán, S., Csizmadia, S., Derekas, A., et al. 2024, AJ, submitted
  * Kálmán et al. (2023a) Kálmán, S., Derekas, A., Csizmadia, S., et al. 2023a, A&A, 673, L14
  * Kálmán et al. (2023b) Kálmán, S., Szabó, G. M., & Csizmadia, S. 2023b, A&A, 675, A107
  * Knutson et al. (2009) Knutson, H. A., Charbonneau, D., Cowan, N. B., et al. 2009, ApJ, 703, 769
  * Kovács et al. (2013) Kovács, G., Kovács, T., Hartman, J. D., et al. 2013, A&A, 553, A44
  * Kurtz et al. (2015) Kurtz, D. W., Shibahashi, H., Murphy, S. J., Bedding, T. R., & Bowman, D. M. 2015, MNRAS, 450, 3015
  * Lehmann et al. (2015) Lehmann, H., Guenther, E., Sebastian, D., et al. 2015, A&A, 578, L4
  * Lenz & Breger (2005) Lenz, P. & Breger, M. 2005, Communications in Asteroseismology, 146, 53
  * Levrard et al. (2007) Levrard, B., Correia, A. C. M., Chabrier, G., et al. 2007, A&A, 462, L5
  * Lightkurve Collaboration et al. (2018) Lightkurve Collaboration, Cardoso, J. V. d. M., Hedges, C., et al. 2018, Lightkurve: Kepler and TESS time series analysis in Python, Astrophysics Source Code Library
  * Mandel & Agol (2002) Mandel, K. & Agol, E. 2002, ApJ, 580, L171
  * Mazeh et al. (2012) Mazeh, T., Nachmani, G., Sokol, G., Faigler, S., & Zucker, S. 2012, A&A, 541, A56
  * Millholland (2019) Millholland, S. 2019, ApJ, 886, 72
  * Nymeyer et al. (2011) Nymeyer, S., Harrington, J., Hardy, R. A., et al. 2011, ApJ, 742, 35
  * Parviainen & Aigrain (2015) Parviainen, H. & Aigrain, S. 2015, MNRAS, 453, 3821
  * Rauer et al. (2014) Rauer, H., Catala, C., Aerts, C., et al. 2014, Experimental Astronomy, 38, 249
  * Ricker et al. (2015) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003
  * Rowe et al. (2008) Rowe, J. F., Matthews, J. M., Seager, S., et al. 2008, ApJ, 689, 1345
  * Samadi-Ghadim et al. (2022) Samadi-Ghadim, A., Lampens, P., & Gizon, L. 2022, A&A, 667, A60
  * Singh et al. (2023) Singh, V., Scandariato, G., Smith, A. M. S., et al. 2023, arXiv e-prints, arXiv:2311.03264
  * Smith et al. (2011) Smith, A. M. S., Anderson, D. R., Skillen, I., Collier Cameron, A., & Smalley, B. 2011, MNRAS, 416, 2096
  * Southworth et al. (2020) Southworth, J., Bowman, D. M., Tkachenko, A., & Pavlovski, K. 2020, MNRAS, 497, L19
  * Sowicka et al. (2023) Sowicka, P., Handler, G., Jones, D., et al. 2023, ApJS, 269, 32
  * Steindl et al. (2021) Steindl, T., Zwintz, K., & Bowman, D. M. 2021, A&A, 645, A119
  * Stephan et al. (2022) Stephan, A. P., Wang, J., Cauley, P. W., et al. 2022, ApJ, 931, 111
  * Sudarsky et al. (2000) Sudarsky, D., Burrows, A., & Pinto, P. 2000, ApJ, 538, 885
  * Szabó et al. (2021) Szabó, G. M., Gandolfi, D., Brandeker, A., et al. 2021, A&A, 654, A159
  * Szabó et al. (2020) Szabó, G. M., Pribulla, T., Pál, A., et al. 2020, MNRAS, 492, L17
  * Szabó et al. (2014) Szabó, G. M., Simon, A., & Kiss, L. L. 2014, MNRAS, 437, 1045
  * Temple et al. (2017) Temple, L. Y., Hellier, C., Albrow, M. D., et al. 2017, MNRAS, 471, 2743
  * Tinetti et al. (2021) Tinetti, G., Eccleston, P., Haswell, C., et al. 2021, arXiv e-prints, arXiv:2104.04824
  * von Essen et al. (2014) von Essen, C., Czesla, S., Wolter, U., et al. 2014, A&A, 561, A48
  * von Essen et al. (2015) von Essen, C., Mallonn, M., Albrecht, S., et al. 2015, A&A, 584, A75
  * von Essen et al. (2020) von Essen, C., Mallonn, M., Borre, C. C., et al. 2020, A&A, 639, A34
  * Watanabe et al. (2022) Watanabe, N., Narita, N., Palle, E., et al. 2022, MNRAS, 512, 4404
  * Wong et al. (2020) Wong, I., Shporer, A., Daylan, T., et al. 2020, AJ, 160, 155
  * Zhang et al. (2018) Zhang, M., Knutson, H. A., Kataria, T., et al. 2018, AJ, 155, 83
  * Zucker et al. (2007) Zucker, S., Mazeh, T., & Alexander, T. 2007, ApJ, 670, 1326

##  Appendix A Influence of the stellar pulsations on the out-of-transit
parameters

Table 4: Behaviour of the monitored out-of-transit parameters and $R_{\rm p}/R_{S}$ with respect to the number of subsequent sinusoidal curves (with decreasing amplitudes) used in the whitening process. |  $K_{\rm phot}$ [m/s] | $q_{\rm{ell}}$ | $A_{g}$ | $I_{\rm p}/I_{\star}$ |  $\varepsilon$ [∘] | Occ. depth [ppm] | $R_{\rm p}/R_{\star}$ | BIC
---|---|---|---|---|---|---|---|---
$\Phi_{0}$ | $2205.5\pm 603.8$ | $0.01187\pm 0.00138$ | $0.72\pm 0.16$ | $0.0126\pm 0.0079$ | $-83.1\pm 11.8$ | $191.1\pm 27.0$ | $0.09054\pm 0.00053$ | $-594783.8$
$\Phi_{1}$ | $2576.1\pm 662.5$ | $0.01362\pm 0.00162$ | $0.91\pm 0.18$ | $0.0101\pm 0.0076$ | $-73.2\pm 9.2$ | $251.1\pm 26.0$ | $0.09064\pm 0.00061$ | $-578129.8$
$\Phi_{2}$ | $82.1\pm 56.7$ | $0.00040\pm 0.00024$ | $0.88\pm 0.19$ | $0.0042\pm 0.0045$ | $-146.6\pm 7.1$ | $18.4\pm 25.9$ | $0.09036\pm 0.00061$ | $-579458.4$
$\Phi_{3}$ | $77.2\pm 60.0$ | $0.00040\pm 0.00026$ | $0.42\pm 0.14$ | $0.0022\pm 0.0030$ | $-12.8\pm 15.7$ | $167.5\pm 25.9$ | $0.09013\pm 0.00059$ | $-579477.8$
$\Phi_{4}$ | $75.2\pm 56.8$ | $0.00037\pm 0.00024$ | $0.43\pm 0.13$ | $0.0022\pm 0.0027$ | $-7.7\pm 14.8$ | $171.4\pm 25.9$ | $0.09007\pm 0.00055$ | $-579858.6$
$\Phi_{5}$ | $83.7\pm 64.1$ | $0.00043\pm 0.00027$ | $0.39\pm 0.15$ | $0.0024\pm 0.0031$ | $3.3\pm 15.3$ | $157.1\pm 25.9$ | $0.08999\pm 0.00064$ | $-579584.7$
$\Phi_{6}$ | $64.2\pm 49.2$ | $0.00032\pm 0.00022$ | $0.40\pm 0.13$ | $0.0014\pm 0.0018$ | $7.1\pm 12.6$ | $135.5\pm 25.9$ | $0.08951\pm 0.00059$ | $-579963.9$
$\Phi_{7}$ | $65.7\pm 50.9$ | $0.00033\pm 0.00020$ | $0.40\pm 0.13$ | $0.0013\pm 0.0019$ | $-6.6\pm 16.0$ | $132.9\pm 25.9$ | $0.08952\pm 0.00059$ | $-581147.2$
$\Phi_{8}$ | $66.9\pm 48.1$ | $0.00033\pm 0.00021$ | $0.37\pm 0.14$ | $0.0015\pm 0.0018$ | $-7.7\pm 15.8$ | $130.5\pm 26.0$ | $0.08971\pm 0.00061$ | $-581641.9$
$\Phi_{9}$ | $65.5\pm 49.3$ | $0.00032\pm 0.00020$ | $0.34\pm 0.14$ | $0.0015\pm 0.0018$ | $-4.5\pm 19.6$ | $119.1\pm 26.0$ | $0.08967\pm 0.00059$ | $-581367.6$
$\Phi_{10}$ | $65.8\pm 48.9$ | $0.00032\pm 0.00020$ | $0.33\pm 0.15$ | $0.0016\pm 0.0020$ | $-4.5\pm 18.3$ | $119.6\pm 26.0$ | $0.08977\pm 0.00059$ | $-582065.9$
$\Phi_{11}$ | $67.3\pm 48.3$ | $0.00033\pm 0.00021$ | $0.39\pm 0.14$ | $0.0019\pm 0.0024$ | $-13.4\pm 19.1$ | $143.0\pm 26.1$ | $0.08989\pm 0.00058$ | $-582802.5$
$\Phi_{12}$ | $64.5\pm 47.5$ | $0.00032\pm 0.00020$ | $0.38\pm 0.13$ | $0.0016\pm 0.0021$ | $2.9\pm 13.5$ | $147.8\pm 26.2$ | $0.08992\pm 0.00059$ | $-586007.5$
$\Phi_{13}$ | $68.3\pm 53.2$ | $0.00034\pm 0.00020$ | $0.30\pm 0.14$ | $0.0019\pm 0.0024$ | $7.1\pm 19.1$ | $105.0\pm 26.2$ | $0.08963\pm 0.00058$ | $-586096.1$
$\Phi_{14}$ | $69.5\pm 50.9$ | $0.00034\pm 0.00022$ | $0.30\pm 0.14$ | $0.0022\pm 0.0026$ | $3.1\pm 21.4$ | $116.2\pm 26.3$ | $0.08965\pm 0.00058$ | $-587206.1$
$\Phi_{15}$ | $46.9\pm 35.1$ | $0.00023\pm 0.00014$ | $0.39\pm 0.11$ | $0.0009\pm 0.0012$ | $20.7\pm 10.3$ | $122.8\pm 27.3$ | $0.08934\pm 0.00053$ | $-600450.9$
$\Phi_{16}$ | $45.3\pm 33.6$ | $0.00022\pm 0.00014$ | $0.34\pm 0.11$ | $0.0010\pm 0.0013$ | $17.7\pm 11.2$ | $106.8\pm 27.3$ | $0.08945\pm 0.00051$ | $-600913.6$

In order to test how the stellar pulsations affect the precision and accuracy
of out-of-transit parameters, we prepared 17 LCs according to Eq. (1). The
monitored parameters include the amplitude of the Doppler beaming ($K_{\rm
phot}$), the ellipsoidal mass ratio ($q_{\rm ell}$), the geometric albedo
($A_{g}$), the surface brightness ratio ($I_{\rm p}/I_{\star}$), the offset
between the brightest point on the dayside hemisphere and the substellar point
($\varepsilon$) and the occultation depth. The behaviour of the relative
planetary radius is also observed for consistency. These are listed in Table
4. In order to distinguish between the 17 sets of solutions, we utilize the
Bayesian Information Criterion (BIC), defined as:

$\text{BIC}=n\ln\left(\frac{\text{RSS}}{n}\right)+k\ln n,$ (13)

where $n$ is the number of LC points and $k$ is number of parameters ($13$
plus three times the number of sinusiodial curves subtracted). We establish
that the LC solutions computed from $\Phi_{16}(t)$ have the lowest BIC value,
and accept this set of parameters.

##  Appendix B Consistency check

We performed a check for consistency for the phase curve analysis by excluding
the Doppler beaming and the ellipsoidal variability from the modelling. The
resultant LC model is shown in Fig. 12.

Figure 12: Same as Fig. 5 but with $0$ mass assumed for WASP-167b.

We then performed a second consistency check, where we modelled the data from
the three sectors independently from each other, from $\Phi_{0}$ to
$\Phi_{16}$. We monitored the behaviour of the same parameters as in Figs. 3
and 4.

Figure 13: Behaviour of the best-fit $K_{\rm phot}$ and $q_{\rm ell}$ parameters with respect to the pre-whitening. The dark blue dots and error bars, as well as the red dashed lines, are the same as on Fig. 3. The respective parameters from Sector 10, 37 and 64 are shown with grey, light blue and pink, respectively. The latter points and error bars are also slightly offset on the x axis for better visibility. Figure 14: Behaviour of the best-fit $A_{g}$, $I_{p}/I_{\star}$, occultation depth and $R_{\rm p}/R_{\star}$ parameters with respect to the pre-whitening. The dark blue dots and error bars, as well as the red dashed lines, are the same as on Fig. 4. The respective parameters from Sector 10, 37 and 64 are shown with grey, light blue and pink, respectively. The latter points and error bars are also slightly offset on the x axis for better visibility. Table 5: System parameters from Sectors 10, 37 and 64. |  $K$ [m/s] | $q_{\rm{ell}}$ | $A_{g}$ | $I_{\rm p}/I_{\star}$ |  $\varepsilon$ [∘] | Occ. depth [ppm] | $R_{\rm p}/R_{\star}$
---|---|---|---|---|---|---|---
Sector 10
$\Phi_{0}$ | $691.2\pm 489.7$ | $0.00344\pm 0.00209$ | $1.32\pm 0.40$ | $0.0263\pm 0.0166$ | $-127.4\pm 12.3$ | $211.3\pm 49.5$ | $0.08942\pm 0.00111$
$\Phi_{1}$ | $749.9\pm 511.7$ | $0.00381\pm 0.00214$ | $1.41\pm 0.38$ | $0.0272\pm 0.0174$ | $-129.9\pm 11.5$ | $214.6\pm 49.5$ | $0.08945\pm 0.00110$
$\Phi_{2}$ | $688.6\pm 507.4$ | $0.00349\pm 0.00213$ | $1.45\pm 0.38$ | $0.0287\pm 0.0170$ | $-133.7\pm 11.1$ | $221.0\pm 49.4$ | $0.08938\pm 0.00114$
$\Phi_{3}$ | $697.6\pm 500.3$ | $0.00348\pm 0.00209$ | $0.41\pm 0.35$ | $0.0305\pm 0.0188$ | $-64.8\pm 92.1$ | $299.0\pm 49.8$ | $0.08903\pm 0.00110$
$\Phi_{4}$ | $597.3\pm 456.9$ | $0.00309\pm 0.00189$ | $0.46\pm 0.34$ | $0.0328\pm 0.0186$ | $-105.6\pm 134.6$ | $260.4\pm 49.9$ | $0.08895\pm 0.00110$
$\Phi_{5}$ | $662.9\pm 456.8$ | $0.00335\pm 0.00204$ | $0.55\pm 0.42$ | $0.0360\pm 0.0185$ | $-114.7\pm 139.9$ | $277.8\pm 50.0$ | $0.08891\pm 0.00103$
$\Phi_{6}$ | $718.6\pm 525.2$ | $0.00356\pm 0.00212$ | $0.59\pm 0.40$ | $0.0378\pm 0.0179$ | $-133.3\pm 135.1$ | $269.7\pm 50.1$ | $0.08896\pm 0.00104$
$\Phi_{7}$ | $649.8\pm 472.0$ | $0.00326\pm 0.00198$ | $0.69\pm 0.43$ | $0.0325\pm 0.0178$ | $-116.1\pm 44.1$ | $260.4\pm 50.0$ | $0.08907\pm 0.00115$
$\Phi_{8}$ | $696.1\pm 491.5$ | $0.00348\pm 0.00211$ | $0.72\pm 0.46$ | $0.0333\pm 0.0179$ | $-121.3\pm 42.2$ | $257.0\pm 50.1$ | $0.08913\pm 0.00113$
$\Phi_{9}$ | $652.4\pm 489.0$ | $0.00337\pm 0.00208$ | $0.71\pm 0.40$ | $0.0318\pm 0.0170$ | $-119.8\pm 44.3$ | $249.2\pm 50.3$ | $0.08918\pm 0.00111$
$\Phi_{10}$ | $665.6\pm 476.5$ | $0.00332\pm 0.00199$ | $0.83\pm 0.42$ | $0.0318\pm 0.0177$ | $-121.7\pm 31.5$ | $253.1\pm 50.4$ | $0.08919\pm 0.00112$
$\Phi_{11}$ | $803.8\pm 554.8$ | $0.00411\pm 0.00232$ | $0.87\pm 0.43$ | $0.0244\pm 0.0168$ | $-96.8\pm 29.1$ | $247.3\pm 50.6$ | $0.08924\pm 0.00111$
$\Phi_{12}$ | $885.9\pm 575.8$ | $0.00434\pm 0.00230$ | $0.35\pm 0.28$ | $0.0229\pm 0.0165$ | $12.5\pm 68.0$ | $285.3\pm 50.7$ | $0.08920\pm 0.00106$
$\Phi_{13}$ | $816.2\pm 566.9$ | $0.00411\pm 0.00226$ | $0.32\pm 0.26$ | $0.0234\pm 0.0174$ | $9.4\pm 81.8$ | $274.5\pm 50.9$ | $0.08917\pm 0.00107$
$\Phi_{14}$ | $831.8\pm 570.1$ | $0.00424\pm 0.00233$ | $0.29\pm 0.24$ | $0.0274\pm 0.0170$ | $-15.3\pm 108.4$ | $292.5\pm 51.0$ | $0.08907\pm 0.00107$
$\Phi_{15}$ | $708.6\pm 500.8$ | $0.00345\pm 0.00205$ | $0.29\pm 0.25$ | $0.0279\pm 0.0175$ | $-9.1\pm 117.3$ | $299.5\pm 51.1$ | $0.08883\pm 0.00105$
$\Phi_{16}$ | $638.0\pm 452.4$ | $0.00321\pm 0.00197$ | $0.26\pm 0.23$ | $0.0320\pm 0.0170$ | $-6.7\pm 128.0$ | $327.5\pm 51.3$ | $0.08920\pm 0.00104$
Sector 37
$\Phi_{0}$ | $5310.9\pm 1970.7$ | $0.02396\pm 0.00265$ | $0.90\pm 0.28$ | $0.0367\pm 0.0124$ | $-143.4\pm 13.5$ | $259.3\pm 46.7$ | $0.09098\pm 0.00085$
$\Phi_{1}$ | $7180.8\pm 2857.6$ | $0.03372\pm 0.00409$ | $1.14\pm 0.43$ | $0.0160\pm 0.0133$ | $95.2\pm 18.7$ | $228.5\pm 41.9$ | $0.09115\pm 0.00105$
$\Phi_{2}$ | $183.7\pm 157.7$ | $0.00091\pm 0.00067$ | $0.75\pm 0.38$ | $0.0055\pm 0.0060$ | $34.4\pm 56.8$ | $243.8\pm 41.7$ | $0.08964\pm 0.00337$
$\Phi_{3}$ | $181.4\pm 140.7$ | $0.00090\pm 0.00061$ | $0.67\pm 0.30$ | $0.0049\pm 0.0056$ | $20.6\pm 41.1$ | $243.1\pm 41.7$ | $0.09010\pm 0.00100$
$\Phi_{4}$ | $163.3\pm 144.9$ | $0.00084\pm 0.00062$ | $0.81\pm 0.27$ | $0.0037\pm 0.0043$ | $9.4\pm 18.6$ | $315.4\pm 41.8$ | $0.09018\pm 0.00107$
$\Phi_{5}$ | $219.4\pm 189.4$ | $0.00111\pm 0.00073$ | $0.76\pm 0.30$ | $0.0034\pm 0.0042$ | $17.7\pm 22.7$ | $286.3\pm 41.5$ | $0.08993\pm 0.00107$
$\Phi_{6}$ | $133.6\pm 103.4$ | $0.00069\pm 0.00045$ | $0.50\pm 0.30$ | $0.0027\pm 0.0032$ | $19.4\pm 55.7$ | $141.4\pm 41.5$ | $0.08915\pm 0.00102$
$\Phi_{7}$ | $162.3\pm 134.4$ | $0.00079\pm 0.00056$ | $0.57\pm 0.30$ | $0.0029\pm 0.0035$ | $5.8\pm 28.3$ | $192.3\pm 41.5$ | $0.08929\pm 0.00103$
$\Phi_{8}$ | $167.9\pm 134.4$ | $0.00081\pm 0.00057$ | $0.46\pm 0.28$ | $0.0028\pm 0.0036$ | $10.2\pm 43.1$ | $142.5\pm 41.5$ | $0.08982\pm 0.00104$
$\Phi_{9}$ | $158.6\pm 132.7$ | $0.00077\pm 0.00055$ | $0.47\pm 0.29$ | $0.0029\pm 0.0035$ | $9.9\pm 38.5$ | $147.8\pm 41.5$ | $0.08987\pm 0.00103$
$\Phi_{10}$ | $226.1\pm 193.4$ | $0.00112\pm 0.00073$ | $0.54\pm 0.30$ | $0.0031\pm 0.0035$ | $5.0\pm 37.8$ | $202.3\pm 41.6$ | $0.09041\pm 0.00106$
$\Phi_{11}$ | $212.6\pm 174.4$ | $0.00107\pm 0.00076$ | $0.57\pm 0.30$ | $0.0027\pm 0.0031$ | $1.8\pm 27.7$ | $210.7\pm 41.9$ | $0.09016\pm 0.00103$
$\Phi_{12}$ | $228.8\pm 198.3$ | $0.00111\pm 0.00086$ | $0.52\pm 0.30$ | $0.0031\pm 0.0036$ | $9.3\pm 51.1$ | $190.3\pm 42.0$ | $0.09020\pm 0.00100$
$\Phi_{13}$ | $224.2\pm 188.0$ | $0.00107\pm 0.00077$ | $0.46\pm 0.30$ | $0.0031\pm 0.0038$ | $35.5\pm 64.4$ | $140.5\pm 42.1$ | $0.09014\pm 0.00103$
$\Phi_{14}$ | $237.2\pm 202.7$ | $0.00118\pm 0.00086$ | $0.50\pm 0.30$ | $0.0033\pm 0.0037$ | $21.6\pm 62.1$ | $169.8\pm 42.0$ | $0.08996\pm 0.00103$
$\Phi_{15}$ | $86.5\pm 67.6$ | $0.00043\pm 0.00030$ | $0.18\pm 0.17$ | $0.0017\pm 0.0021$ | $-3.9\pm 63.4$ | $-31.9\pm 46.1$ | $0.08916\pm 0.00082$
$\Phi_{16}$ | $85.3\pm 62.0$ | $0.00041\pm 0.00027$ | $0.19\pm 0.17$ | $0.0017\pm 0.0021$ | $-8.6\pm 44.5$ | $-29.1\pm 46.1$ | $0.08904\pm 0.00082$
Sector 64
$\Phi_{0}$ | $3821.1\pm 1332.4$ | $0.01684\pm 0.00210$ | $1.23\pm 0.20$ | $0.0048\pm 0.0052$ | $-51.2\pm 6.0$ | $396.6\pm 42.7$ | $0.09050\pm 0.00064$
$\Phi_{1}$ | $4002.0\pm 1426.3$ | $0.01670\pm 0.00211$ | $1.08\pm 0.21$ | $0.0053\pm 0.0055$ | $-47.5\pm 6.8$ | $370.5\pm 42.6$ | $0.09047\pm 0.00064$
$\Phi_{2}$ | $79.6\pm 62.0$ | $0.00041\pm 0.00025$ | $0.56\pm 0.18$ | $0.0024\pm 0.0026$ | $24.7\pm 10.2$ | $202.4\pm 42.5$ | $0.08991\pm 0.00063$
$\Phi_{3}$ | $91.4\pm 74.3$ | $0.00044\pm 0.00031$ | $1.25\pm 0.18$ | $0.0014\pm 0.0017$ | $38.1\pm 5.9$ | $397.1\pm 42.2$ | $0.09032\pm 0.00064$
$\Phi_{4}$ | $89.4\pm 68.9$ | $0.00044\pm 0.00030$ | $1.19\pm 0.18$ | $0.0014\pm 0.0016$ | $36.0\pm 6.4$ | $380.8\pm 42.1$ | $0.09033\pm 0.00065$
$\Phi_{5}$ | $83.0\pm 66.1$ | $0.00042\pm 0.00028$ | $1.23\pm 0.19$ | $0.0011\pm 0.0015$ | $36.8\pm 5.8$ | $394.1\pm 42.1$ | $0.09027\pm 0.00062$
$\Phi_{6}$ | $90.7\pm 73.6$ | $0.00044\pm 0.00030$ | $1.17\pm 0.18$ | $0.0012\pm 0.0016$ | $35.4\pm 6.2$ | $374.4\pm 42.1$ | $0.09034\pm 0.00066$
$\Phi_{7}$ | $95.0\pm 79.2$ | $0.00048\pm 0.00033$ | $0.84\pm 0.18$ | $0.0017\pm 0.0020$ | $30.2\pm 7.4$ | $283.6\pm 42.1$ | $0.09022\pm 0.00063$
$\Phi_{8}$ | $90.2\pm 68.4$ | $0.00044\pm 0.00029$ | $0.76\pm 0.19$ | $0.0016\pm 0.0020$ | $29.0\pm 8.3$ | $278.0\pm 42.1$ | $0.09013\pm 0.00065$
$\Phi_{9}$ | $83.5\pm 60.9$ | $0.00042\pm 0.00027$ | $0.82\pm 0.18$ | $0.0018\pm 0.0020$ | $30.0\pm 7.5$ | $272.2\pm 42.0$ | $0.09003\pm 0.00060$
$\Phi_{10}$ | $77.9\pm 60.7$ | $0.00038\pm 0.00026$ | $0.90\pm 0.19$ | $0.0015\pm 0.0017$ | $32.2\pm 7.0$ | $294.5\pm 42.1$ | $0.08986\pm 0.00065$
$\Phi_{11}$ | $78.7\pm 60.8$ | $0.00040\pm 0.00026$ | $0.89\pm 0.18$ | $0.0017\pm 0.0020$ | $29.7\pm 7.3$ | $302.6\pm 42.2$ | $0.08963\pm 0.00064$
$\Phi_{12}$ | $71.5\pm 57.0$ | $0.00036\pm 0.00023$ | $0.88\pm 0.19$ | $0.0015\pm 0.0021$ | $30.6\pm 7.2$ | $296.8\pm 42.4$ | $0.08969\pm 0.00063$
$\Phi_{13}$ | $80.4\pm 64.0$ | $0.00039\pm 0.00026$ | $0.81\pm 0.20$ | $0.0015\pm 0.0020$ | $30.7\pm 7.8$ | $266.7\pm 42.4$ | $0.08928\pm 0.00065$
$\Phi_{14}$ | $76.8\pm 61.7$ | $0.00039\pm 0.00026$ | $0.81\pm 0.18$ | $0.0015\pm 0.0018$ | $29.9\pm 7.3$ | $264.1\pm 42.5$ | $0.08949\pm 0.00061$
$\Phi_{15}$ | $73.9\pm 57.4$ | $0.00037\pm 0.00026$ | $0.74\pm 0.18$ | $0.0015\pm 0.0019$ | $26.0\pm 8.2$ | $238.2\pm 42.4$ | $0.08950\pm 0.00060$
$\Phi_{16}$ | $89.8\pm 66.1$ | $0.00045\pm 0.00028$ | $0.77\pm 0.25$ | $0.0038\pm 0.0041$ | $-145.4\pm 8.0$ | $13.4\pm 42.5$ | $0.08989\pm 0.00061$

##  Appendix C Posterior distribution

The posterior distribution from the Markov chain Monte Carlo light curve
sampling, prepared via the routines of Foreman-Mackey et al. (2017), are shown
in Fig. 15.

Figure 15: Corner plot showing the posterior distribution from the $\Phi_{16}$
light curve modelling.