11institutetext:  Observatoire astronomique de l’Université de Genève, Chemin
Pegasi 51, 1290 Versoix, Switzerland 22institutetext:  Instituto de
Astrofisica e Ciencias do Espaco, Universidade do Porto, CAUP, Rua das
Estrelas, 4150-762 Porto, Portugal 33institutetext:  Departamento de Fisica e
Astronomia, Faculdade de Ciencias, Universidade do Porto, Rua do Campo Alegre,
4169-007 Porto, Portugal 44institutetext:  Space Research Institute, Austrian
Academy of Sciences, Schmiedlstrasse 6, A-8042 Graz, Austria 55institutetext:
INAF, Osservatorio Astrofisico di Torino, Via Osservatorio, 20, I-10025 Pino
Torinese To, Italy 66institutetext:  Weltraumforschung und Planetologie,
Physikalisches Institut, University of Bern, Gesellschaftsstrasse 6, 3012
Bern, Switzerland 77institutetext:  Center for Space and Habitability,
University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland
88institutetext:  Department of Astronomy, Stockholm University, AlbaNova
University Center, 10691 Stockholm, Sweden 99institutetext:  Centre for
Exoplanet Science, SUPA School of Physics and Astronomy, University of St
Andrews, North Haugh, St Andrews KY16 9SS, UK 1010institutetext:  IMCCE,
UMR8028 CNRS, Observatoire de Paris, PSL Univ., Sorbonne Univ., 77 av.
Denfert-Rochereau, 75014 Paris, France 1111institutetext:  INAF, Osservatorio
Astrofisico di Catania, Via S. Sofia 78, 95123 Catania, Italy
1212institutetext:  Department of Physics, University of Warwick, Gibbet Hill
Road, Coventry CV4 7AL, United Kingdom 1313institutetext:  Cavendish
Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK 1414institutetext:
Institute of Planetary Research, German Aerospace Center (DLR),
Rutherfordstrasse 2, 12489 Berlin, Germany 1515institutetext:  Instituto de
Astrofisica de Canarias, Via Lactea s/n, 38200 La Laguna, Tenerife, Spain
1616institutetext:  Departamento de Astrofisica, Universidad de La Laguna,
Astrofísico Francisco Sanchez s/n, 38206 La Laguna, Tenerife, Spain
1717institutetext:  European Space Agency (ESA), ESTEC, Keplerlaan 1, 2201 AZ
Noordwijk, The Netherlands 1818institutetext:  Admatis, 5. Kandó Kálmán
Street, 3534 Miskolc, Hungary 1919institutetext:  Depto. de Astrofisica,
Centro de Astrobiologia (CSIC-INTA), ESAC campus, 28692 Villanueva de la
Cañada (Madrid), Spain 2020institutetext:  Université Grenoble Alpes, CNRS,
IPAG, 38000 Grenoble, France 2121institutetext:  INAF, Osservatorio
Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy
2222institutetext:  Institute of Optical Sensor Systems, German Aerospace
Center (DLR), Rutherfordstrasse 2, 12489 Berlin, Germany 2323institutetext:
Université de Paris Cité, Institut de physique du globe de Paris, CNRS, 1 Rue
Jussieu, F-75005 Paris, France 2424institutetext:  Centre for Mathematical
Sciences, Lund University, Box 118, 221 00 Lund, Sweden 2525institutetext:
Aix Marseille Univ, CNRS, CNES, LAM, 38 rue Frédéric Joliot-Curie, 13388
Marseille, France 2626institutetext:  Astrobiology Research Unit, Université
de Liège, Allée du 6 Août 19C, B-4000 Liège, Belgium 2727institutetext:  Space
sciences, Technologies and Astrophysics Research (STAR) Institute, Université
de Liège, Allée du 6 Août 19C, 4000 Liège, Belgium 2828institutetext:  Centre
Vie dans l’Univers, Faculté des sciences, Université de Genève, Quai Ernest-
Ansermet 30, 1211 Genève 4, Switzerland 2929institutetext:  Leiden
Observatory, University of Leiden, PO Box 9513, 2300 RA Leiden, The
Netherlands 3030institutetext:  Department of Space, Earth and Environment,
Chalmers University of Technology, Onsala Space Observatory, 439 92 Onsala,
Sweden 3131institutetext:  Dipartimento di Fisica, Università degli Studi di
Torino, via Pietro Giuria 1, I-10125, Torino, Italy 3232institutetext:
Department of Astrophysics, University of Vienna, Türkenschanzstrasse 17, 1180
Vienna, Austria 3333institutetext:  Institute for Theoretical Physics and
Computational Physics, Graz University of Technology, Petersgasse 16, 8010
Graz, Austria 3434institutetext:  Konkoly Observatory, Research Centre for
Astronomy and Earth Sciences, 1121 Budapest, Konkoly Thege Miklós út 15-17,
Hungary 3535institutetext:  ELTE Eötvös Loránd University, Institute of
Physics, Pázmány Péter sétány 1/A, 1117 Budapest, Hungary 3636institutetext:
Institut d’astrophysique de Paris, UMR7095 CNRS, Université Pierre & Marie
Curie, 98bis blvd. Arago, 75014 Paris, France 3737institutetext:  Astrophysics
Group, Lennard Jones Building, Keele University, Staffordshire, ST5 5BG,
United Kingdom 3838institutetext:  Physikalisches Institut, University of
Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland 3939institutetext:
Dipartimento di Fisica e Astronomia ”Galileo Galilei”, Università degli Studi
di Padova, Vicolo dell’Osservatorio 3, 35122 Padova, Italy 4040institutetext:
ETH Zurich, Department of Physics, Wolfgang-Pauli-Strasse 2, CH-8093 Zurich,
Switzerland 4141institutetext:  Zentrum für Astronomie und Astrophysik,
Technische Universität Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany
4242institutetext:  Institut fuer Geologische Wissenschaften, Freie
Universitaet Berlin, Maltheserstrasse 74-100,12249 Berlin, Germany
4343institutetext:  Institut de Ciencies de l’Espai (ICE, CSIC), Campus UAB,
Can Magrans s/n, 08193 Bellaterra, Spain 4444institutetext:  Institut
d’Estudis Espacials de Catalunya (IEEC), Gran Capità 2-4, 08034 Barcelona,
Spain 4545institutetext:  ELTE Eötvös Loránd University, Gothard Astrophysical
Observatory, 9700 Szombathely, Szent Imre h. u. 112, Hungary
4646institutetext:  HUN-REN–ELTE Exoplanet Research Group, Szent Imre h. u.
112., Szombathely, H-9700, Hungary 4747institutetext:  Institute of Astronomy,
University of Cambridge, Madingley Road, Cambridge, CB3 0HA, United Kingdom
4848institutetext:  CFisUC, Departamento de Física, Universidade de Coimbra,
3004-516 Coimbra, Portugal

# The tidal deformation and atmosphere of WASP-12 b from its phase
curve††thanks: Based on data from CHEOPS guaranteed time observations (GTO)
with Program IDs: CH_PR100013, CH_PR100016, and CH_PR330093, ††thanks: The
CHEOPS photometric time-series data used in this paper are available in
electronic form at the CDS via anonymous ftp to cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u- strasbg.fr/cgi-bin/qcat?J/A+A/

B. Akinsanmi ${}^{\href https://orcid.org/0000-0001-6519-1598}$ 11 S. C. C.
Barros ${}^{\href https://orcid.org/0000-0003-2434-3625}$ 2233 M. Lendl
${}^{\href https://orcid.org/0000-0001-9699-1459}$ 11 L. Carone 44 P. E.
Cubillos 5544 A. Bekkelien 11 A. Fortier ${}^{\href
https://orcid.org/0000-0001-8450-3374}$ 6677 H.-G. Florén 88 A. Collier
Cameron ${}^{\href https://orcid.org/0000-0002-8863-7828}$ 99 G. Boué
${}^{\href https://orcid.org/0000-0002-5057-7743}$ 1010 G. Bruno ${}^{\href
https://orcid.org/0000-0002-3288-0802}$ 1111 B.-O. Demory ${}^{\href
https://orcid.org/0000-0002-9355-5165}$ 7766 A. Brandeker ${}^{\href
https://orcid.org/0000-0002-7201-7536}$ 88 S. G. Sousa ${}^{\href
https://orcid.org/0000-0001-9047-2965}$ 22 T. G. Wilson ${}^{\href
https://orcid.org/0000-0001-8749-1962}$ 1212 A. Deline 11 A. Bonfanti
${}^{\href https://orcid.org/0000-0002-1916-5935}$ 44 G. Scandariato
${}^{\href https://orcid.org/0000-0003-2029-0626}$ 1111 M. J. Hooton
${}^{\href https://orcid.org/0000-0003-0030-332X}$ 1313 A. C. M. Correia
${}^{\href https://orcid.org/0000-0002-8946-8579}$ 4848 O. D. S. Demangeon
${}^{\href https://orcid.org/0000-0001-7918-0355}$ 2233 A. M. S. Smith
${}^{\href https://orcid.org/0000-0002-2386-4341}$ 1414 V. Singh ${}^{\href
https://orcid.org/0000-0002-7485-6309}$ 1111 Y. Alibert ${}^{\href
https://orcid.org/0000-0002-4644-8818}$ 7766 R. Alonso ${}^{\href
https://orcid.org/0000-0001-8462-8126}$ 15151616 J. Asquier 1717 T. Bárczy
${}^{\href https://orcid.org/0000-0002-7822-4413}$ 1818 D. Barrado Navascues
${}^{\href https://orcid.org/0000-0002-5971-9242}$ 1919 W. Baumjohann
${}^{\href https://orcid.org/0000-0001-6271-0110}$ 44 M. Beck ${}^{\href
https://orcid.org/0000-0003-3926-0275}$ 11 T. Beck 66 W. Benz ${}^{\href
https://orcid.org/0000-0001-7896-6479}$ 6677 N. Billot ${}^{\href
https://orcid.org/0000-0003-3429-3836}$ 11 X. Bonfils ${}^{\href
https://orcid.org/0000-0001-9003-8894}$ 2020 L. Borsato ${}^{\href
https://orcid.org/0000-0003-0066-9268}$ 2121 C. Broeg ${}^{\href
https://orcid.org/0000-0001-5132-2614}$ 6677 M. Buder 2222 S. Charnoz
${}^{\href https://orcid.org/0000-0002-7442-491X}$ 2323 Sz. Csizmadia
${}^{\href https://orcid.org/0000-0001-6803-9698}$ 1414 M. B. Davies
${}^{\href https://orcid.org/0000-0001-6080-1190}$ 2424 M. Deleuil ${}^{\href
https://orcid.org/0000-0001-6036-0225}$ 2525 L. Delrez ${}^{\href
https://orcid.org/0000-0001-6108-4808}$ 26262727 D. Ehrenreich ${}^{\href
https://orcid.org/0000-0001-9704-5405}$ 112828 A. Erikson 1414 J. Farinato
2121 L. Fossati ${}^{\href https://orcid.org/0000-0003-4426-9530}$ 44 M.
Fridlund ${}^{\href https://orcid.org/0000-0002-0855-8426}$ 29293030 D.
Gandolfi ${}^{\href https://orcid.org/0000-0001-8627-9628}$ 3131 M. Gillon
${}^{\href https://orcid.org/0000-0003-1462-7739}$ 2626 M. Güdel 3232 M. N.
Günther ${}^{\href https://orcid.org/0000-0002-3164-9086}$ 1717 A. Heitzmann
11 Ch. Helling 443333 S. Hoyer ${}^{\href
https://orcid.org/0000-0003-3477-2466}$ 2525 K. G. Isaak ${}^{\href
https://orcid.org/0000-0001-8585-1717}$ 1717 L. L. Kiss 34343535 K. W. F. Lam
${}^{\href https://orcid.org/0000-0002-9910-6088}$ 1414 J. Laskar ${}^{\href
https://orcid.org/0000-0003-2634-789X}$ 1010 A. Lecavelier des Etangs
${}^{\href https://orcid.org/0000-0002-5637-5253}$ 3636 D. Magrin ${}^{\href
https://orcid.org/0000-0003-0312-313X}$ 2121 P. F. L. Maxted ${}^{\href
https://orcid.org/0000-0003-3794-1317}$ 3737 M. Mecina 3232 C. Mordasini 6677
V. Nascimbeni ${}^{\href https://orcid.org/0000-0001-9770-1214}$ 2121 G.
Olofsson ${}^{\href https://orcid.org/0000-0003-3747-7120}$ 88 R. Ottensamer
3232 I. Pagano ${}^{\href https://orcid.org/0000-0001-9573-4928}$ 1111 E.
Pallé ${}^{\href https://orcid.org/0000-0003-0987-1593}$ 15151616 G. Peter
${}^{\href https://orcid.org/0000-0001-6101-2513}$ 2222 D. Piazza 3838 G.
Piotto ${}^{\href https://orcid.org/0000-0002-9937-6387}$ 21213939 D. Pollacco
1212 D. Queloz ${}^{\href https://orcid.org/0000-0002-3012-0316}$ 40401313 R.
Ragazzoni ${}^{\href https://orcid.org/0000-0002-7697-5555}$ 21213939 N. Rando
1717 H. Rauer ${}^{\href https://orcid.org/0000-0002-6510-1828}$ 141441414242
I. Ribas ${}^{\href https://orcid.org/0000-0002-6689-0312}$ 43434444 N. C.
Santos ${}^{\href https://orcid.org/0000-0003-4422-2919}$ 2233 D. Ségransan
${}^{\href https://orcid.org/0000-0003-2355-8034}$ 11 A. E. Simon ${}^{\href
https://orcid.org/0000-0001-9773-2600}$ 6677 M. Stalport 27272626 Gy. M. Szabó
${}^{\href https://orcid.org/0000-0002-0606-7930}$ 45454646 N. Thomas 66 S.
Udry ${}^{\href https://orcid.org/0000-0001-7576-6236}$ 11 V. Van Grootel
${}^{\href https://orcid.org/0000-0003-2144-4316}$ 2727 J. Venturini
${}^{\href https://orcid.org/0000-0001-9527-2903}$ 11 E. Villaver 15151616 N.
A. Walton ${}^{\href https://orcid.org/0000-0003-3983-8778}$ 4747

(Received 06 October 2023; accepted 05 February 2024 )

###### Abstract

Context. Ultra-hot Jupiters present a unique opportunity to understand the
physics and chemistry of planets, their atmospheres, and interiors at extreme
conditions. WASP-12 b stands out as an archetype of this class of exoplanets,
with a close-in orbit around its star that results in intense stellar
irradiation and tidal effects.

Aims. The goals are to measure the planet’s tidal deformation, atmospheric
properties, and also to refine its orbital decay rate.

Methods. We performed comprehensive analyses of the transits, occultations,
and phase curves of WASP-12 b by combining new CHEOPS observations with
previous TESS and Spitzer data. The planet was modeled as a triaxial ellipsoid
parameterized by the second-order fluid Love number of the planet, $h_{2}$,
which quantifies its radial deformation and provides insight into the interior
structure.

Results. We measured the tidal deformation of WASP-12 b and estimated a Love
number of $h_{2}$ =$1.55_{-0.49}^{+0.45}$ (at 3.2$\sigma$) from its phase
curve. We measured occultation depths of $333\pm 24$ ppm and $493\pm 29$ ppm
in the CHEOPS and TESS bands, respectively, while the nightside fluxes are
consistent with zero, and also marginal eastward phase offsets. Our modeling
of the dayside emission spectrum indicates that CHEOPS and TESS probe similar
pressure levels in the atmosphere at a temperature of $\sim$2900 K. We also
estimated low geometric albedos of $A_{g}=0.086\pm 0.017$ and $A_{g}=0.01\pm
0.023$ in the CHEOPS and TESS passbands, respectively, suggesting the absence
of reflective clouds in the high-temperature dayside of the planet. The CHEOPS
occultations do not show strong evidence for variability in the dayside
atmosphere of the planet at the median occultation depth precision of 120 ppm
attained. Finally, combining the new CHEOPS timings with previous measurements
refines the precision of the orbital decay rate by 12% to a value of
–30.23$\pm$0.82 ms/yr, resulting in a modified stellar tidal quality factor of
$Q^{\prime}_{\star}=1.70\pm 0.14\times 10^{5}$.

Conclusions. WASP-12 b becomes the second exoplanet, after WASP-103b, for
which the Love number has been measured from the effect of tidal deformation
in the light curve. However, constraining the core mass fraction of the planet
requires measuring $h_{2}$ with a higher precision. This can be achieved with
high signal-to-noise observations with JWST since the phase curve amplitude,
and consequently the induced tidal deformation effect, is higher in the
infrared.

###### Key Words.:

Planetary systems – Planets and satellites: individual: WASP-12 b,
atmospheres, interiors – Techniques: photometric

##  1 Introduction

Ultra-hot Jupiters (UHJs) orbit very close to their host stars and are
subjected to immense tidal forces and irradiation which impact the orbital,
atmospheric, and geometric characteristics of the planets. Depending on the
stellar properties, the strong tidal interaction may cause the orbits of the
planets to become circular and coplanar (zero eccentricity and obliquity),
while their rotation rates and orbital periods may become synchronized (tidal
locking; Hut, 1980). These effects can impact the atmospheric circulation and
also result in tidal deformation of the planet’s shape in response to the
perturbing force (Correia & Rodríguez, 2013; Correia et al., 2014). Another
consequence of the tidal interaction is the shrinkage of the planetary orbit
(tidal decay) due to loss of angular momentum to the star if the planetary
orbital period is not synchronized with the stellar rotation period.

The intense irradiation received by UHJs results in extremely high dayside
atmospheric temperatures such that molecular water (H2O) is thermally unstable
in favor of atomic hydrogen (Woitke et al., 2018). The hot daysides of UHJs
generally favor the atomic form rather than the molecular form (e.g., Si/SiO
or Mg/MgH). Other metal elements may appear in their ionized form, such as
Na+, K+, but also in the less abundant Ti+ and Al+ (Helling et al., 2019,
2021). Therefore, UHJs are unique laboratories to study the physics and
chemistry of planets at extreme conditions, shedding valuable insights into
their orbital evolution, atmospheres, and interiors.

WASP-12 b stands out as one of the most irradiated UHJs, with an orbital
period of only 1.09 days around a G0 star of ${T_{\mathrm{eff}}}$=6300 K (Hebb
et al., 2008). Separated from the star by less than 3 stellar radii, the
planet is exposed to strong tidal forces and irradiation, which makes it an
attractive target for characterization via transit, eclipse, and phase-curve
observations. WASP-12 b is indeed one of the most extensively studied
exoplanets, with numerous observations ranging from ultraviolet to infrared
wavelengths, revealing remarkable properties of the planet. The planet is
inflated with a large radius of $1.9R_{\mathrm{Jup}}$ likely due to high
stellar irradiation. Evidence suggests that it is undergoing atmospheric mass
loss with gas outflowing toward the star (Fossati et al., 2010). The proximity
of its orbit to the Roche limit of its star makes it one of the few exoplanets
where the tidal distortion of the planet can be probed from its light curve
(Correia, 2014; Akinsanmi et al., 2019). It is also the only exoplanet that
has been observationally confirmed to be spiraling into the star due to tidal
orbital decay (Yee et al., 2019). Secondary eclipse observations of WASP-12 b
have resulted in discrepant eclipse depth measurements at various passbands
which could be indicative of variability in the dayside atmosphere (Hooton et
al., 2019). Previous work on WASP-12 b found evidence for water absorption in
its terminator (Stevenson et al., 2014; Kreidberg, 2015), whereas the dayside
spectrum showed no signs of water (Swain et al., 2013), supporting previous
gas-phase modeling results (e.g., Helling et al., 2019).

In this paper, we report on the transit, occultation, and phase curve
observations of WASP-12 b by the CHaracterizing ExOplanet Satellite (CHEOPS;
Benz et al., 2021). Previous CHEOPS observations (e.g., Lendl et al., 2020;
Barros et al., 2022; Deline et al., 2022; Hooton et al., 2022; Ehrenreich et
al., 2023) have shown its remarkable photometric capability in characterizing
exoplanets. Here, we analyze the CHEOPS observations alongside archival data
of WASP-12 to characterize the shape, orbit, and atmosphere of the planet. In
Section 2.1, we described the observations obtained by the different
instruments, and also the pre-processing of the datasets. We also derived the
stellar properties and summarized the theoretical background on planetary
tidal deformation. The analyses of the datasets are detailed in Section 3. In
Section 4, we use the ellipsoidal planet model to probe the tidal deformation
of the planet in the joint phase curve and transit observations. We
characterize the atmosphere of the planet in Section 5: modeling its emission
spectrum and probing for variability in its dayside atmosphere. In Section 6,
we use our CHEOPS transit timing measurements along with published ones to
refine the tidal decay timescale of the planet and the stellar tidal
dissipation. Finally, we summarize the main results of our work in Section 7.

##  2 Observations and system properties

###  2.1 Observations

WASP-12 has been observed by several space- and ground-based instruments,
capturing the transit, eclipse, and also the phase curve of the planet,
WASP-12 b. In this paper, we analyze new observations from CHEOPS in addition
to existing space-based observations from TESS and Spitzer. Although HST/WFC3
observations of WASP-12 b are also available, we excluded them due to poor
transit ingress and/or egress coverage that makes it challenging to probe the
planetary deformation.

####  2.1.1 CHEOPS

We obtained 47 visits of WASP-12 spanning 3 observation seasons between
2022–11–02 and 2022–12–24 using CHEOPS as part of the Guaranteed Time
Observations (See observation log in Table 4). The visits, identified by
unique file keys, consist of 21 transits, 25 occultations, and half of a phase
curve of WASP-12 b, all taken at an exposure time of 60 s. The phase curve
visit lasted for 24 hrs, starting before an occultation and ending after
transit. The visit durations of the transits and occultations range between
7.1 – 12.6 hrs, capturing significant baselines before and after the
transits/occultations (see Figs. 7 and 8). Due to the short orbital period of
the planet, the transit and occultation visits also fortuitously combine to
construct a phase curve for the planet. In some cases, consecutive
observations of transits and occultations result in half or full phase curves.

Due to the low-Earth orbit of CHEOPS, its line of sight is often interrupted
by Earth occultations of the target or spacecraft passages through the South
Atlantic Anomaly resulting in data gaps. For our observations of WASP-12, this
resulted in light curve efficiencies between 49 and 63%. CHEOPS data are
automatically processed by the official Data Reduction Pipeline (DRP version
13; Hoyer et al., 2020) which performs aperture photometry after calibration
of the images and correcting for instrumental and environmental effects. The
DRP provides light curves extracted with different aperture radii. Point-
spread function (PSF) photometry can also be extracted using the
PIPE111https://github.com/alphapsa/PIPE package developed specifically for
CHEOPS data.

In our analyses of WASP-12, we used the PSF-extracted light curves which are
less sensitive to contamination from background stars (see e.g., Morris et
al., 2020; Brandeker et al., 2022; Delrez et al., 2023). The resulting light
curves have a lower scatter than the DRP apertures in all visits. Data points
that were flagged to have poor photometry (e.g., due to cosmic ray hits or bad
pixels) were discarded. We further removed points with high background ($BG>3$
times the median background level) where the correlation with the flux becomes
nonlinear due to scattered light from the moon or the Earth’s limb. Finally, a
15-point moving median filter was used to eliminate points $>$5 times the
median absolute deviation (MAD). In total, 987 points were discarded
corresponding to 6.2% of the data points across all visits.

The Nadir-locked orientation of CHEOPS, as it orbits around the Earth, causes
its field of view to rotate around the target. Combined with the irregular
shape of the CHEOPS PSF, this leads to time-variable flux contamination in the
aperture that is correlated with the spacecraft’s roll-angle. Thus, it is
usually necessary to decorrelate against the roll-angle when analyzing CHEOPS
data (e.g., Lendl et al., 2020; Morris et al., 2021; Barros et al., 2022).
Spacecraft pointing jitter can also result in flux trends that can be
accounted for by decorrelating against the X and Y centroid positions of the
target PSF on the CCD. Furthermore, CHEOPS observations can feature ramp
effects at the beginning of each visit caused by the thermal settling of the
telescope as it adjusts to a new target position. The ramp is accounted for by
decorrelating the flux against the deviation of telescope tube temperature
from the median value $\Delta T_{\mathrm{tube}}$ (see e.g., Morris et al.,
2021; Deline et al., 2022). Data points ($\sim$45) in the first orbit of
visits 44, 45, and 47 were removed as they featured a strong, nonlinear
increase in the telescope temperature.

We model the systematic trends in each CHEOPS visit using spline
decorrelations222implemented with the scipy.interpolate.LSQUnivariateSpline
and scipy.interpolate.LSQBivariateSpline python classes. against the roll-
angle, background flux, telescope tube temperature, and the $X$ and $Y$
centroid positions. The spline fit is performed simultaneously with the fit of
the astrophysical model (§3.1) and involves successively fitting splines to
the residuals of the astrophysical model and then evaluating the likelihood of
the joint model. First, we performed a 2D spline fit of the residual against
$BG$ and $\Delta T_{\mathrm{tube}}$. The resulting residual is then used for
another 2D spline fit against the $X$ and $Y$ centroid positions. Since the
flux trends with these variables are approximately linear, the 2D spline
functions are defined with a single degree and knot in each dimension.
Finally, we model the roll-angle trend with a 1D cubic spline fit with knots
every 18$\degree$.

####  2.1.2 TESS

TESS observed WASP-12 with 2-minute cadence in sectors 20, 43, 44, and 45 with
a span of almost 2 years between December 2019 and December 2021. Across the
four sectors, TESS observed 74 transits and occultations. Details of the TESS
dataset are given in Table 5. We utilized the Pre-search Data Conditioning
Single Aperture Photometry (PDCSAP) light curve data produced by the Science
Processing Operations Center (SPOC) pipeline which has been corrected for
known instrumental systematics and contamination (Stumpe et al., 2012; Smith
et al., 2012). The light curves from these sectors were recently published by
Wong et al. (2022) where the transits, occultations, phase curve, and transit
timings for WASP-12b were analyzed. We also analyzed these light curves to
complement our CHEOPS observations.

The lightcurves were downloaded from the Mikulski Archive for Space Telescope
(MAST) archive using the lightkurve python package (Lightkurve Collaboration
et al., 2018). Data points flagged by the SPOC pipeline were removed after
which a 15-point moving median filter was used to remove points
$>$5$\times$MAD. We separated the light curves into segments by the time of
the momentum dumps and removed any strong flux ramps at the beginning of data
segments. In fitting the astrophysical model, we simultaneously account for
long-term temporal trends in each data segment using a cubic spline with knots
every 3 days so as to preserve the phase variation within an orbital period.

####  2.1.3 Spitzer

We also analyzed archival Spitzer data in the 3.6$\mu$m and 4.5$\mu$m channels
of the InfraRed Array Camera (IRAC) that have already been published (Cowan et
al., 2012; Bell et al., 2019). These observations consist of 2 phase curves in
each channel acquired in 2010 (PID 70060, PI P. Machalek) and 2013 (PID 90186,
PI K. Todorov). We downloaded the data from the Spitzer Heritage
Archive333http://sha.ipac.caltech.edu.

The reduction and analysis of these datasets are similar to Demory et al.
(2016b), where we modeled the IRAC intra-pixel sensitivity (Ingalls et al.,
2012) using a modified implementation of the BLISS (BiLinearly-Interpolated
Sub-pixel Sensitivity) mapping algorithm (Stevenson et al., 2012). In addition
to the BLISS mapping (BM), our baseline model includes a linear function of
the Point Response Function’s (PRF) FWHM along the $x$ and $y$ axes, which
significantly reduces the level of correlated noise as shown in previous
studies (e.g., Demory et al., 2016a, b; Mendonça et al., 2018; Barros et al.,
2022; Jones et al., 2022). This baseline model (BM + PRF FWHM) does not
include time-dependent parameters. We implemented this instrumental model in a
Markov Chain Monte Carlo (MCMC) framework already presented in the literature
(Gillon et al. 2012). We included all data described in the paragraph above in
the same fit. We ran two chains of 200,000 steps each to determine baseline
corrected light curves at 3.6 and 4.5 $\mu$m that were used subsequently.

Previous analyses of these datasets by Cowan et al. (2012) and Bell et al.
(2019) reported an anomalous phase modulation (with a periodicity of half the
planet’s orbital period) in the 4.5$\mu$m data but not at 3.6$\mu$m. If the
anomalous modulation is due to the tidal deformation of the planet, Cowan et
al. (2012) estimated that the substellar axis would have to be at least 1.5
times longer than the polar axis. However, the effect of such a large
deformation is not supported by the transit. Bell et al. (2019) instead
proposed that the anomalous signal at 4.5$\mu$m may be due to heated CO
emission from gas outflowing from the planet. Bell et al. (2019) also found
that the measured hotspot offset of the 3.6$\mu$m phase curves significantly
changed from eastward in 2010 to westward in 2013. As the unexpected phase
curve features in these datasets make them difficult to combine, we chose to
use only the transit regions of the Spitzer data in our analysis (0.2 d before
and after mid-transit). Similar to the CHEOPS and TESS datasets, we remove
outlier points $>$5$\times$MAD with a 15-point moving median filter. Details
of the Spitzer datasets are given in Table 6.

###  2.2 The host star

####  2.2.1 Stellar parameters

Table 1: Properties of the star WASP-12 system

Parameter name | Symbol | Value | Source
---|---|---|---
Effective temperature |  $T_{\mathrm{eff}}$ [K] | $6313\pm 52$ | This work
Surface gravity |  $\log{g}$ [dex] | $4.37\pm 0.12$ | This work
Metallicity | [Fe/H] [dex] | $0.21\pm 0.04$ | This work
Stellar radius |  $R_{\star}$ [$R_{\odot}$] | $1.734\pm 0.022$ | This work
Stellar mass |  $M_{\star}$ [$M_{\odot}$] | $1.422_{-0.069}^{+0.077}$ | This work
Stellar age |  $t_{\star}$ [Gyr] | $2.3\pm 0.5$ | This work
Planet Mass |  $M_{p}$ [$M_{J}$] | $1.470\pm 0.073$ | Collins et al. (2017)
RV semi-amplitude |  $K_{\mathrm{RV}}$ [m/s] | $226.4\pm 4.1$ | Collins et al. (2017)

To facilitate our analysis of the observations, we refined the stellar
parameters of WASP-12 (V = 11.5) as shown in Table 1. The spectroscopic
stellar parameters ($T_{\mathrm{eff}}$, $\log g$, microturbulence, [Fe/H])
were originally taken from a previous version of SWEET-Cat (Santos et al.,
2013; Sousa et al., 2018). The spectroscopic parameters for WASP-12 were
estimated with the ARES+MOOG methodology where we used the latest version of
ARES444The latest version, ARES v2, can be downloaded at
https://github.com/sousasag/ARES (Sousa et al., 2007, 2015) to consistently
measure the equivalent widths (EW) of selected iron lines on the spectrum of
WASP-12. The list of iron lines is the same as the one presented in Sousa et
al. (2008). In this analysis, we used a minimization process to find the
ionization and excitation equilibrium to converge on the best set of
spectroscopic parameters. This process makes use of the ATLAS grid of stellar
model atmospheres (Kurucz, 1993) and the radiative transfer code MOOG (Sneden,
1973). More recently the same methodology was applied on a combined HARPS–N
spectrum where we derived completely consistent spectroscopic stellar
parameters ($T_{\mathrm{eff}}\,=\,6301\pm 64$ K, $\log g\,=\,4.26\pm 0.10$
dex, and [Fe/H] = $0.18\pm 0.04$ dex; Sousa et al. 2021). We also derived a
more accurate trigonometric surface gravity using recent GAIA data following
the same procedure as described in Sousa et al. (2021) which provided a
consistent value when compared with the spectroscopic surface gravity (4.23
$\pm$ 0.01 dex).

We determined the radius of WASP-12 using the infrared flux method (IRFM) with
an MCMC approach (Blackwell & Shallis, 1977; Schanche et al., 2020). We
conducted a comparison between observed and synthetic broadband photometry to
determine the stellar effective temperature and angular diameter that is
converted to the stellar radius with knowledge of the target’s parallax. We
constructed the spectral energy distributions (SEDs) of WASP-12 using the
results of our spectral analysis above and the ATLAS stellar atmosphere
models. We then produce synthetic photometry in the Gaia, 2MASS, and WISE
passbands that are compared to Gaia G, GBP, and GRP, 2MASS J, H, and K, and
WISE W1 and W2 broadband fluxes from the most recent data releases (Skrutskie
et al., 2006; Wright et al., 2010; Gaia Collaboration et al., 2022). Using the
offset-corrected Gaia DR3 parallax, we obtained the stellar radius that is
reported in Table 1.

We used $T_{\mathrm{eff}}$, [Fe/H], and $R_{\star}$ along with their
uncertainties to determine the stellar mass $M_{\star}$ and age $t_{\star}$ by
employing two different stellar evolution models. In fact, a first pair of
mass and age values ($M_{\star,1}$, $t_{\star,1}$) were computed by the
isochrone placement algorithm (Bonfanti et al., 2015, 2016) that interpolates
the input values within precomputed grids of PARSEC555PAdova and TRieste
Stellar Evolutionary Code: http://stev.oapd.inaf.it/cgi-bin/cmd v1.2S (Marigo
et al., 2017) isochrones and tracks. A second pair ($M_{\star,2}$,
$t_{\star,2}$), instead, was computed via the CLES (Code Liègeois d’Évolution
Stellaire Scuflaire et al., 2008) code, which generates the best-fit
evolutionary track according to the provided input and the Levenberg-Marquadt
minimization scheme (Salmon et al., 2021). We finally merged the two pairs of
outcomes after successfully checking their mutual consistency through the
$\chi^{2}$-based criterion as described in Bonfanti et al. (2021) and we
obtained $M_{\star}=1.422_{-0.069}^{+0.077}\,M_{\odot}$ and $t_{\star}=2.3\pm
0.5$ Gyr, consistent with literature values.

####  2.2.2 Stellar limb darkening

Accurate modeling of stellar limb darkening (LD) is important when analyzing
exoplanetary transits in order to obtain unbiased transit parameters, and also
to measure higher-order effects such as tidal deformation (Espinoza & Jordán,
2015; Akinsanmi et al., 2019). The theoretical LD profile of different stars
can be obtained from stellar atmosphere models that compute the stellar
intensities as a function of the foreshortening angle $\mu$ measured from the
limb to the center of the stellar disk. The most widely used stellar libraries
for this purpose are the PHOENIX (Husser et al., 2013) and ATLAS (Kurucz,
1993) stellar models. Since they are both theoretical models, they may not
always provide an accurate representation of the actual stellar intensity
profile (see e.g., Espinoza & Jordán, 2015; Patel & Espinoza, 2022). Indeed,
both libraries predict slightly different intensity profiles for the same star
in the same passband, making it difficult to select one over the other.
However, since these libraries represent our current knowledge of stellar
atmospheres, they may still be used to put useful priors on the stellar limb
darkening profiles. Obtaining priors from these models can also be beneficial
to multi-passband observations since the derived priors will ensure that the
LD profile in all passbands relates to the same star.

First, we compute stellar intensity profiles for each passband using the LDCU
python package666https://github.com/delinea/LDCU which queries both stellar
libraries using the stellar parameters ($T_{\mathrm{eff}}$, $\log g$, and
[Fe/H]) given in Table 1. Following Claret & Bloemen (2011), the obtained
intensity profile computed from each library, and in each passband, is
represented by 100 interpolated points (evenly spaced in $\mu$). The
uncertainties in the stellar parameters are accounted for by generating 10000
intensity profile samples using random stellar parameter sets drawn from their
normal distribution. The median and standard deviation of the profiles are
then computed. Thus, for each passband, we have a median intensity profile
from each stellar library with error bars at each $\mu$ point.

Different parametric limb darkening laws can be adopted to approximate the
model intensity profile to derive limb darkening coefficients (LDCs) that can
be used in transit analyses. In this work, we adopted the power-2 LD law
(Hestroffer, 1997) parameterized by the LDCs, $c$ and $\alpha$. The power-2
law has been shown to be superior to other two-parameter laws in modeling
intensity profiles generated by stellar atmosphere models (Morello et al.,
2017; Claret & Southworth, 2022). The power-2 LD law is only surpassed by the
four-parameter law which is difficult to use in transit model fitting due to
the higher number of parameters and the strong correlations between them,
which can lead to nonphysical intensity profiles.

Similar to Barros et al. (2022), we leveraged the computed model intensity
profiles from the two libraries to derive priors on the LDCs. We obtain LDCs
in each passband by fitting the preferred power-2 law to the corresponding
combined PHOENIX and ATLAS model intensity profiles. The $1\sigma$
uncertainties of the obtained LDCs are inflated such that the allowed
parameter space encompasses both intensity profiles and associated $1\sigma$
uncertainties. This approach is illustrated in Fig. 9a and the derived LDC
priors for the passbands are reported in Table 7. Using the derived LDCs as
priors allows the transit fit to determine the best-fit limb darkening profile
without being too restricted to the predictions of either library. Alternative
approaches to modeling limb darkening using the intensity profiles are
presented in Section 4.2.

###  2.3 The planet: Tidal deformation

WASP-12 b orbits so close to its host star that it is predicted to be one of
the most tidally deformed planets (Akinsanmi et al., 2019; Hellard et al.,
2019; Berardo & De Wit, 2022). The deformation of a planet in response to
perturbing forces depends on its interior structure and can be quantified by
the second-degree Love number for radial displacement $h_{2}$ (Love, 1911;
Kellermann et al., 2018). For a fluid planet, $h_{2}$ is related to the tidal
Love number $k_{2}$ (as $h_{2}$ = 1 + $k_{2}$) which measures the distribution
of mass within the planet. Therefore, measurement of $h_{2}$ from the
detection of tidal deformation allows constraining the planet’s interior
structure (Kramm et al., 2011, 2012). The relationship between the tidal
response of a planet and its core mass has been investigated in previous
studies (e.g., Batygin et al., 2009; Ragozzine & Wolf, 2009; Kramm et al.,
2011) showing that an incompressible fluid planet with a homogeneous interior
mass distribution will have the highest $h_{2}$ value of 2.5. However, the
value of $h_{2}$ decreases as a planet becomes more centrally condensed (more
mass at the core), since the presence of a massive core reduces the response
of a planet to the perturbing potential (Leconte et al., 2011b). For instance,
the lower measured Love number of 1.39 for Saturn (Lainey et al., 2017)
reflects a core mass fraction higher than that of Jupiter which has a Love
number of 1.565 (Durante et al., 2020).

The shape of a tidally deformed planet can be described by a triaxial
ellipsoid (with axes $r_{1},\,r_{2},\,r_{3}$) where $r_{1}$ is the planet
radius oriented along the star-planet (substellar) axis, $r_{2}$ is along the
orbital direction (dawn-dusk axis), and $r_{3}$ is along the polar axis. The
volumetric radius of the ellipsoid is given by
$R_{\mathrm{v}}$$\leavevmode\nobreak\ =\leavevmode\nobreak\
(r_{1}\,r_{2}\,r_{3})^{1/3}$. According to the analytical shape model
formulated in Correia (2014), the axes of the ellipsoid follow the
relation777This relation also satisfies the condition in the linear theory of
figures (Zharkov & Trubitsyn, 1978; Dermott, 1979) that a synchronously
rotating body in hydrostatic equilibrium subjected to both rotation and tidal
forces deforms such that $(r_{2}-r_{3})/(r_{1}-r_{3})=1/4$:

$\displaystyle r_{2}=R_{\mathrm{v}}(1-2q/3),\leavevmode\nobreak\
\leavevmode\nobreak\ r_{1}\,=\,r_{2}(1+3q),\leavevmode\nobreak\
\leavevmode\nobreak\ \text{and}\leavevmode\nobreak\ \leavevmode\nobreak\
r_{3}\,=\,r_{2}(1-q)$ (1)

where $q$ is an asymmetry parameter defined as

$q=\frac{h_{2}}{2Q_{\mathrm{M}}}\left(\frac{R_{\mathrm{v}}{}}{a}\right)^{3}\,.$
(2)

Therefore the axes $r_{1},\,r_{2},\,r_{3}$ of the deformed planet depend on
the Love number $h_{2}$, the proximity of the planetary orbit to the star $a$,
the planet-to-star mass ratio $Q_{\mathrm{M}}=M_{p}/M_{\star}$, and the
volumetric radius of the planet $R_{\mathrm{v}}$.

The nonspherical shape of the ellipsoidal planet causes the projected cross-
sectional area to vary as it rotates with orbital phase. The projected area as
a function of orbital phase angle ($\phi$ = 2$\pi$ phase) is given (e.g., by
Leconte et al., 2011a) as

$A(\phi)=\pi\sqrt{r_{1}^{2}r_{2}^{2}\cos^{2}i+r_{3}^{2}\sin^{2}i\,\left(r_{1}^{2}\sin^{2}{\phi}+r_{2}^{2}\cos^{2}{\phi}\right)},$
(3)

where $i$ is the orbital inclination of the planet. The projected area can be
calculated using Eqs. (1) and (2), given the values of $h_{2}$,
$R_{\mathrm{v}}$, and $Q_{M}$. The ellipsoid projects its maximum area at
quadrature, while the minimum area is projected at mid-transit (and mid-
occultation). The effective planetary radius at mid-transit ($\phi=0$) can be
obtained from Eq. (3) as $R_{p}^{\mathrm{eff}}$ = $\sqrt{A_{\phi=0}/\pi}$. The
varying planet projection can lead to anomalies in the transit light curve
(Correia, 2014; Akinsanmi et al., 2019) and also in the shape of the full-
orbit phase curve (Leconte et al., 2011b; Akinsanmi et al., 2023) when
compared to a spherical planet. Measuring the deviation of the planet’s shape
from sphericity in high-precision light curves can thus provide a measurement
of the Love number.

##  3 Light curve analysis

In this section, we describe the analytical light curve model that we use to
model the phase curves, transits, and occultations of WASP-12 b. We further
describe our fitting methodology for each analysis.

###  3.1 Light curve model

We adopt an analytical light curve model composed of the transit
($F_{\mathrm{tra}}$) and occultation ($F_{\mathrm{occ}}$) signals, the phase
variation signal by the planet ($F_{p}$: due to reflection and thermal
emission from the atmosphere), and also the phase variation signal by the star
($F_{\star}$: due to ellipsoidal distortion of the star by the planet and
Doppler beaming of the stellar light). The total phase curve model is given as
a function of the orbital phase angle as:

$F(\phi)=F_{\mathrm{tra}}\,F_{\star}(\phi)\leavevmode\nobreak\
+\leavevmode\nobreak\ F_{\mathrm{occ}}\,F_{p}(\phi).$ (4)

Given the expected deformed shape of WASP-12 b, an adequate light curve model
should account for the deformation in the relevant component signals
(Akinsanmi et al., 2023). We describe the components of Eq. (4) in the
following sections.

####  3.1.1 Stellar phase variation model

The flux from the star $F_{\star}$ varies as a function of phase as

$\displaystyle F_{\star}(\phi)$
$\displaystyle=1+F_{\mathrm{EV}}+F_{\mathrm{DB}}$ (5)
$\displaystyle=1+A_{\mathrm{EV}}(1-\cos{2\phi})+A_{\mathrm{DB}}\sin{\phi},$

such that $F_{\star}$ is unity at mid-transit and mid-eclipse. The value of
$F_{\star}$ at other phases depends on the ellipsoidal variation
$F_{\mathrm{EV}}$ and Doppler beaming $F_{\mathrm{DB}}$ signals which have
semi-amplitudes, $A_{\mathrm{EV}}$ and $A_{\mathrm{DB}}$ respectively given
(e.g., in Loeb & Gaudi, 2003; Esteves et al., 2013; Shporer, 2017) by:

$\displaystyle A_{\mathrm{EV}}$
$\displaystyle=\alpha_{\mathrm{EV}}\,Q_{\mathrm{M}}\left(\frac{a}{R_{\star}}\right)^{-3}\sin^{2}{i},$
(6) $\displaystyle A_{\mathrm{DB}}$
$\displaystyle=\alpha_{\mathrm{DB}}\,\frac{K_{\mathrm{RV}}}{c}$ (7)

where $Q_{\mathrm{M}}$ is a again the planet-to-star mass ratio, $a/R_{\star}$
is the semi-major axis scaled by the stellar radius, $i$ is the orbital
inclination, $K_{\mathrm{RV}}$ is the radial velocity (RV) semi-amplitude, and
$c$ is the speed of light. The coefficient $\alpha_{\mathrm{EV}}$ depends on
the linear limb darkening coefficient $u$, and gravity darkening coefficient
$g$ as

$\alpha_{\mathrm{EV}}=0.15\frac{(15+u)(1+g)}{3-u},$ (8)

while the coefficient $\alpha_{\mathrm{DB}}$ depends on the stellar flux
$\mathcal{F}_{\lambda}$ and passband transmission $\mathcal{T}_{\lambda}$ at
wavelength $\lambda$ as

$\alpha_{\mathrm{DB}}=\frac{\int\left(5+\frac{d\ln\mathcal{F}_{\lambda}}{d\ln\lambda}\right)\lambda\mathcal{F}_{\lambda}\mathcal{T}_{\lambda}d\lambda}{\int\lambda\mathcal{F}_{\lambda}\mathcal{T}_{\lambda}d\lambda}.$
(9)

####  3.1.2 Transit and occultation models

The transit, $F_{\mathrm{tra}}$, and occultation, $F_{\mathrm{occ}}$, signals
are generated using the ellc transit tool (Maxted, 2016) which allows modeling
the planet shape as a sphere or as an ellipsoid parameterized by the Love
number as implemented in Akinsanmi et al. (2019). The ellipsoidal planet model
parameters are the same as the usual spherical planet model except that the
spherical planet radius $R_{p}$ is replaced by the volumetric radius
$R_{\mathrm{v}}$ 888$R_{\mathrm{v}}$ of an ellipsoidal planet is always larger
than the spherical planet transit radius since only a small part of the long
axis is projected during transit. However, for the spherical planet,
$R_{\mathrm{v}}$ is equivalent to the spherical planet transit radius., and
the addition of $h_{2}$ and $Q_{M}$.

####  3.1.3 Planetary phase variation model

Although the flux from the planet ($F_{p}$) is composed of both reflected
light and thermal emission from the atmosphere, the degeneracy between both
components makes it challenging to model them simultaneously (see e.g., Lendl
et al., 2020; Deline et al., 2022; Parviainen et al., 2022). Following recent
optical phase curve studies (e.g., Shporer et al., 2019; Wong et al., 2021;
Daylan et al., 2021), we model the total planetary phase variation using a
sinusoidal function given by:

$F_{p}(\phi)=\left(F_{\mathrm{max}}-F_{\mathrm{min}}\right)\,\frac{1-\cos{(\phi+\delta)}}{2}+F_{\mathrm{min}},$
(10)

where $F_{\mathrm{max}}$ and $F_{\mathrm{min}}$ are the maximum and minimum
planet fluxes respectively, and $\delta$ is the hotspot offset (positive value
means an eastward offset). The planet’s dayside flux $F_{d}$ (i.e.,
occultation depth) and nightside flux $F_{n}$ are derived as the value of
$F_{p}(\phi)$ at $\phi=\pi$ and $\phi=0$ respectively999Our code actually
transforms Eq. (10) as a function of $F_{d}$ and $F_{n}$ instead of
$F_{\mathrm{max}}$ and $F_{\mathrm{min}}$.. The semi-amplitude of the
atmospheric phase variation $A_{\mathrm{atm}}$ is
$(F_{\mathrm{max}}-F_{\mathrm{min}})/2$.

Following the recommendation of Akinsanmi et al. (2023), we account for the
planetary tidal deformation in the out-of-transit phases by multiplying the
planet’s phase variation (Eq. 10) by the normalized phase-dependent projected
area of the ellipsoid (Eq. 3) to have

$F_{p}(\phi)^{\mathrm{def}}=F_{p}(\phi)\,\frac{A(\phi)}{A(0)}.$ (11)

This implies that the phase variation of the deformed planet depends on
$h_{2}$, $R_{\mathrm{v}}$, and $Q_{M}$. Bell et al. (2019) employed a similar
approach to model the tidal deformation of WASP-12 b in an attempt to explain
its anomalous phase curve shape in the Spitzer 4.5$\mu$m band. However, the
planet’s shape was modeled as a biaxial ellipsoid (instead of the triaxial
model used here). They found that even if tidal deformation might contribute
to the phase curve, it is not sufficient to explain the observed anomaly
present only in the 4.5$\mu$m phase curve. Similarly, in the analysis of the
HST and Spitzer phase curves of WASP-103 b, Kreidberg et al. (2018) accounted
for the planetary deformation by including the normalized phase-dependent
projected area of the ellipsoid. However, instead of fitting the shape of the
planet, it was fixed based on the tabulated predictions in Leconte et al.
(2011b).

####  3.1.4 Light travel time

We corrected for light-travel time across the planetary system by converting
the observation times ($t_{\mathrm{obs}}$) into a reference time
($t_{\mathrm{ref}}$). The reference time accounts for the projected distance
between the current position of the planet along the orbit and its position at
inferior conjunction. This is given for a circular orbit as (Deline et al.,
2022)

$t_{\mathrm{ref}}=t_{\mathrm{obs}}-\frac{a}{c}\left(1-\cos{\phi}\right)\sin{i}.$
(12)

###  3.2 Fitting process

To obtain results regarding different planetary properties, we perform
different model fits to the datasets:

  * •

We analyzed all datasets by performing a global fit using the ellipsoidal and
spherical planet models and comparing the results (§3.3). This analysis allows
to constrain the shape and atmospheric properties of the planet.

  * •

We analyzed the CHEOPS transit observations individually (§3.4) to derive
transit timings for orbital decay analysis .

  * •

We analyzed the CHEOPS occultation observations individually (§3.5) to measure
the occultation depth of each visit and probe for potential atmospheric
variability.

In these fits, the astrophysical and systematic trends are modeled
simultaneously. In cases where we perform model comparison, we sample the
parameter space using the nested sampling algorithm, dynesty (Speagle, 2019)
which provides posteriors of the fit and also the Bayesian evidence for each
model. We used 1000 live points to explore the parameter space until the
estimated log-evidence was smaller than 0.1. In other fits, we use the affine-
invariant MCMC ensemble sampler implemented in emcee (Foreman-Mackey et al.,
2013). The results of these analyses are discussed in Sections 4, 5 and 6.

###  3.3 Phase curve analysis

We used dynesty to fit the light curve model (Eq. 4) to all the datasets
simultaneously, considering both a spherical and an ellipsoidal planet. We fit
the CHEOPS and TESS observations using the full phase curve model, and the
Spitzer observations using only the transit component. We fit for the orbital
parameters: planetary period $P$, mid-transit time $T_{0}$, planet-star radius
ratio $R_{p}/R_{\star}$, scaled semi-major axis $a/R_{\star}$ and impact
parameter $b$. We also fit for the phase curve parameters: $F_{d}$, $F_{n}$,
$\delta$, $A_{\mathrm{EV}}$, and $A_{\mathrm{DB}}$. Fitting for the total EV
amplitude, $A_{\mathrm{EV}}$, ensures that we consider all possible
combinations of the component limb and gravity darkening coefficients (in Eqs.
6 and 8) and not just their theoretically estimated values.

As seen in Table 2, we assumed wide uniform priors on all parameters except
for the LDCs which have Gaussian priors as derived in Section 2.2.2, and
$A_{\mathrm{DB}}$ for which we used Gaussian priors centered on theoretical
values calculated from Eq. (9) for each passband with a standard deviation of
0.2. The prior on $F_{n}$ includes negative values to ensure Gaussian
posteriors and avoid overestimating the nightside flux. For the fit using the
ellipsoidal planet model, we adopt a uniform prior for $h_{2}$ and a normal
prior for the log of the mass ratio ($\log Q_{M}$) based on literature values.

We performed the analysis using a two-step fitting process in which an initial
fit of the data from each instrument is performed to estimate the noise
properties of each visit/sector (e.g., Lendl et al., 2017; Wong et al., 2021;
Demory et al., 2023). We estimated the amplitude of additional white noise
$\beta_{w}$, which is calculated as the ratio of the residual RMS to the mean
photometric uncertainty. We also calculated the amplitude of the red noise
$\beta_{r}$ by taking the average of the ratio of the binned residuals at
several timescales to the expected Gaussian $1/\sqrt{n}$ noise scaling at that
timescale (e.g., Winn et al., 2008). We then rescaled the photometric
uncertainties of the light curve by multiplying them by $\beta_{w}\beta_{r}$.
This method allows us to propagate the extra noise contributions to the best-
fit parameters. Fitting instead for a jitter parameter to account for extra
white noise in each visit/sector of the instruments would introduce 54
additional parameters to the global fit of the datasets, making the
convergence time of the fit much longer. We chose not to model the red noise
using Gaussian Processes since this might be capable of absorbing the subtle
signal of tidal deformation in the light curve.

After determining the $\beta_{w}\beta_{r}$ for all datasets, the global
results are obtained from the final joint fit of the datasets. The orbital
parameters ($P$, $T_{0}$, $b$, $a/R_{\star}$) are common between the datasets
while other passband-dependent parameters including
$R_{\mathrm{v}}$/$R_{\star}$ are different between the datasets. We also fit
the models using fixed quadratic ephemeris determined from our orbital decay
result in Section 6, and find that the derived parameters are consistent
within 1$\sigma$. The results of the spherical and ellipsoidal fits are given
in Table 2 while the phase-folded data and best-fit models are shown in Fig. 1
and Fig. 2.

The results of the phase curve fit are discussed in Sections 4.1 and 5.1.

###  3.4 CHEOPS transit analysis

With the aim of deriving precise transit times, we individually analyzed the
22 CHEOPS transit observations using the $F_{\mathrm{tra}}$ model in Eq. (4).
Since the phase variation signals ($F_{p}$ and $F_{\star}$) impact the transit
baseline of each visit, we first subtracted out the best-fit phase variation
signals determined from the phase curve fit (in § 3.3). We propagate the
uncertainties of the $F_{p}$ and $F_{\star}$ models by quadratically adding
their standard deviations to the flux error bars. Each transit was allowed to
have a unique mid-transit time and systematics model, but the shape parameters
(i.e., $R_{p}/R_{\star}$, $b$, $a/R_{\star}$) were constrained using Gaussian
priors based on the parameter posteriors from the spherical planet phase curve
fit. We sample the parameter space using emcee.

Figure 7 shows the best-fit transit+systematics model overplotted on each
transit light curve. It also shows the best-fit transit model overplotted on
each detrended light curve and annotated with the obtained timing precision.
Finally, the residual of each light curve fit is shown with the measured RMS.
The mid-transit times obtained for the transits and their $1\sigma$
uncertainties are given in Table 8 with a mean precision of 27.5 s.

Figure 1: CHEOPS (left) and TESS (right) phase curves of WASP-12 b. Panels
(a): The total phase curve models overplotted on the detrended and phase-
folded data. Panels (b): Zoom of panel (a) showing the different components of
the total phase curve. The solid blue curve is the total phase curve model for
an ellipsoidal planet, while the red dashed curve is for a spherical planet.
The solid and dashed green curves represent the best-fit planetary phase
variation component ($F_{p}$) for the deformed and spherical planets,
respectively. The cyan curve is the computed thermal-only 3D GCM for the
planet (see §5.2.2) which shows for CHEOPS that some reflection from the
atmosphere is required to reach the amplitude of the green curves. The black
curves at the bottom of this panel show the best-fit stellar ellipsoidal
variation ($F_{\mathrm{EV}}$) from the deformed (solid) and spherical (dashed)
planet model fits. The yellow curve represents the contribution of tidal
deformation to the total phase curve, with circles denoting the peaks. Panels
(c) & (d): The residuals of the phase curve fits using the ellipsoidal (blue)
and spherical (red) planet models. The difference between the models
(ellipsoidal–spherical) representing the deformation signature in each
passband is overplotted on the spherical model residuals. Figure 2: Phase-
folded transits of WASP-12 b from the different instruments with the best-fit
models overplotted. For each instrument, the 5-min binned residuals for the
ellipsoidal (blue) and spherical planet (red) model fits are shown. The
difference between both models is again overplotted on the spherical model
residuals.

###  3.5 CHEOPS occultation analysis

With the aim of measuring the individual occultation depths, we individually
analyzed the 26 CHEOPS occultation observations using the $F_{\mathrm{occ}}$
model in Eq. (4). Similar to the transit analyses, we also subtracted out the
best-fit phase variation signals ($F_{p}$ and $F_{\star}$) determined from the
phase curve fit and propagated their uncertainties to the flux error bars.
Here, we fit for the occultation depth but fixed the orbital and shape
parameters (i.e., $P$, $R_{p}/R_{\star}$, $b$, $a/R_{\star}$) to the best-fit
CHEOPS values obtained from the phase curve fit. However, since precise timing
measurements cannot be obtained from the individual occultations due to the
low signal-to-noise, we use Gaussian prior on $T_{0}$ based on the posterior
from the phase curve fit. Figure 8 shows the best-fit occultation+systematics
model overplotted on each occultation light curve. It also shows the best-fit
occultation model overplotted on each detrended light curve and lastly, the
residuals of each light curve fit and its RMS.

We further jointly analyzed the occultation observations within each of the
three seasons, freely fitting for the mid-occultation time and occultation
depth. The epoch of the occultation time for each season was set to that of
the occultation closest to the center of the time series. The resulting
timings for each of the three seasons (labeled S1–S3) are listed in Table 8.

Section 5.4 discusses the measured occultation depths with respect to
previously reported hints of atmospheric variability in dayside of WASP-12 b.

Table 2: Result of the fit to the CHEOPS and TESS phase curves. We note that $\mathcal{N}(\mu,\sigma)$ represents a Gaussian prior with mean $\mu$ and standard deviation $\sigma$ while $\mathcal{U}(a,b)$ is a uniform prior between $a$ and $b$. Parameter | Symbol | Prior | Posterior
---|---|---|---
|  |  |  | Spherical planet
---
| Ellipsoidal planet
---
Orbital Period [d] | $P$ |  $\mathcal{N}$(1.0914,0.0001) |  ${1.0914185}\pm 2$E-7 |  ${1.0914185}\pm 2$E-7
Transit time [BJDTBD–2459000] | $T_{0}\,^{\diamond}$ |  $\mathcal{U}$(196,197) |  $196.624122\pm 6.6$E-4 |  $196.624121\pm 6.6$E-4
Impact parameter | $b$ |  $\mathcal{U}$(0,1) | $0.392\pm{0.011}$ | $0.369\pm{0.016}$
Scaled semi-major axis | $a/R_{\star}$ |  $\mathcal{U}$(2.5, 3.5) | ${3.006}\pm{0.013}$ | ${3.036}\pm{0.018}$
Love number | $h_{2}$ |  $\mathcal{U}$(0,2.5) | – | $1.55_{-0.49}^{+0.45}$
Log planet-to-star mass ratio | $\log Q_{M}$ |  $\mathcal{N}$(-6.935, 0.025) | – | $-6.931\pm{0.022}$
| Planet–star radius ratio∗
---
|  $R_{\mathrm{v}}$/$R_{\star_{{}_{\mathrm{CHEOPS}}}}$
---
$R_{\mathrm{v}}$/$R_{\star_{{}_{\mathrm{TESS}}}}$
$R_{\mathrm{v}}$/$R_{\star_{{}_{\mathrm{Spitzer3.6}}}}$
$R_{\mathrm{v}}$/$R_{\star_{{}_{\mathrm{Spitzer4.5}}}}$
|  $\mathcal{U}$(0, 0.25)
---
| $0.1177\pm 0.0002$
---
$0.1164\pm 0.0002$
$0.1133\pm 0.0004$
$0.1071\pm 0.0006$
| $0.1242\pm 0.0023$
---
$0.1226\pm 0.0023$
$0.1188\pm 0.0020$
$0.1115\pm 0.0017$
EV semi-amplitude [ppm] |  | $A_{\mathrm{EV}_{{}_{\mathrm{CHEOPS}}}}$
---
$A_{\mathrm{EV}_{{}_{\mathrm{TESS}}}}$
$\mathcal{U}$(0,200) |  | $69\pm 14$
---
$58\pm 14$
| $54\pm 15$
---
$40\pm 15$
DB semi-amplitude [ppm] |  | $A_{\mathrm{DB}_{{}_{\mathrm{CHEOPS}}}}$
---
$A_{\mathrm{DB}_{{}_{\mathrm{TESS}}}}$
|  $\mathcal{N}$(2.92, 0.2)
---
$\mathcal{U}$(2.32, 0.2)
| $2.92\pm 0.17$
---
$2.31\pm 0.17$
| $2.91\pm 0.17$
---
$2.31\pm 0.17$
| Dayside flux [ppm]
---
(occultation depth)
| $F_{d}/F_{\star_{{}_{\mathrm{CHEOPS}}}}$
---
$F_{d}/F_{\star_{{}_{\mathrm{TESS}}}}$
$\mathcal{U}$(0, 1000) |  | $340\pm 24$
---
$513\pm 29$
| $333\pm 24$
---
$493\pm 29$
Nightside flux [ppm] |  | $F_{n}/F_{\star_{{}_{\mathrm{CHEOPS}}}}$
---
$F_{n}/F_{\star_{{}_{\mathrm{TESS}}}}$
$\mathcal{U}$(-100, 100) |  | $6.2\pm 23$
---
$6.1\pm 21$
| $7.0\pm 23$
---
$11.8\pm 21$
Hotspot offset $[\degree]$ |  | $\delta_{{}_{\mathrm{CHEOPS}}}$
---
$\delta_{{}_{\mathrm{TESS}}}$
$\mathcal{U}$(-90,90) |  | $9.8\pm 4.5$
---
$6.3\pm 2.7$
| $9.3\pm 4.6$
---
$5.8\pm 2.6$
Derived parameters
|  Effective planetary radius ratio
---
mid-occultation
| $R_{p}^{\mathrm{eff}}/R_{\star_{{}_{\mathrm{CHEOPS}}}}$
---
$R_{p}^{\mathrm{eff}}/R_{\star_{{}_{\mathrm{TESS}}}}$
– |  | $0.11770\pm{0.00020}$
---
$0.11640\pm{0.00020}$
| $0.11730\pm 0.00027$
---
$0.11600\pm 0.00025$
| Ellipsoid axis ratios
---
|  $r_{1}$ : $r_{2}$ : $r_{3}\,{}_{{}_{\mathrm{CHEOPS}}}$
---
$r_{1}$ : $r_{2}$ : $r_{3}\,{}_{{}_{\mathrm{TESS}}}$
$r_{1}$ : $r_{2}$ : $r_{3}\,{}_{{}_{\mathrm{Spitzer3.6}}}$
$r_{1}$ : $r_{2}$ : $r_{3}\,{}_{{}_{\mathrm{Spitzer4.5}}}$
| –
---
–
–
–
| –
---
–
–
–
| 1.23 : 1.06 : 1.0
---
1.22 : 1.05 : 1.0
1.20 : 1.05 : 1.0
1.16 : 1.04 : 1.0
| Planet phase variation
---
semi-amplitude [ppm]
| $A_{\mathrm{atm}_{{}_{\mathrm{CHEOPS}}}}$
---
$A_{\mathrm{atm}_{{}_{\mathrm{TESS}}}}$
– |  | $170\pm 16$
---
$254\pm 14$
| $165\pm 16$
---
$241\pm 14$
| Planetary mass
---
from EV [$M_{\mathrm{Jup}}$]
| $M_{p_{{}_{\mathrm{CHEOPS}}}}$
---
$M_{p_{{}_{\mathrm{TESS}}}}$
– |  | $2.34\pm 0.48$
---
$2.09\pm 0.49$
| $1.89\pm 0.53$
---
$1.49\pm 0.58$
Geometric albedo |  | $A_{g_{{}_{\mathrm{CHEOPS}}}}$
---
$A_{g_{{}_{\mathrm{TESS}}}}$
– |  | $0.089\pm 0.017$
---
$0.022\pm 0.023$
| $0.086\pm 0.017$
---
$0.010\pm 0.023$
Dayside temperature [K] |  | $T_{\mathrm{day}_{{}_{\mathrm{CHEOPS}}}}$
---
$T_{\mathrm{day}_{{}_{\mathrm{TESS}}}}$
– |  | $2821\pm 25$
---
$2915\pm 25$
| $2821\pm 25$
---
$2915\pm 25$
Nightside temperature [K] |  | $T_{\mathrm{night}_{{}_{\mathrm{CHEOPS}}}}$
---
$T_{\mathrm{night}_{{}_{\mathrm{TESS}}}}$
– |  | $<2025\leavevmode\nobreak\ (2\sigma)$
---
$<1890\leavevmode\nobreak\ (2\sigma)$
| $<2126\leavevmode\nobreak\ (2\sigma)$
---
$<1921\leavevmode\nobreak\ (2\sigma)$
∗ For the spherical planet, this is equal to the transit radius.

##  4 Tidal deformation

###  4.1 Measuring deformation from phase curve

From the joint fit of the CHEOPS, TESS, and Spitzer datasets, we compare the
results of the ellipsoidal and spherical planet models. Table 2 reports the
median posterior and 1$\sigma$ uncertainties of the parameters of both models.
The posterior probability distributions of some relevant parameters are also
shown in Fig. 11 where we see greater uncertainties in the determination of
the planet-star radius ratios for the ellipsoidal model due to strong
correlations with the Love number.

From the ellipsoidal planet phase curve fit, we measured a Love number of
$1.55^{+0.45}_{-0.49}$ corresponding to a 3.16$\sigma$ detection. This is the
first measurement of the Love number of a planet from the analysis of its
full-orbit phase curve. Previous work by Barros et al. (2022) obtained a
3$\sigma$ measurement of the Love number of WASP-103 b from the analysis of
transit-only observations from different space telescopes. Since planetary
tidal deformation signal in the phase curve is correlated with the stellar
ellipsoidal variation signal, we additionally follow the strategy of Barros et
al. (2022) to measure the deformation in the transit-only regions where the
stellar ellipsoidal variation is insignificant. We measured $h_{2}$
=$1.56^{+0.47}_{-0.52}$ in agreement with the phase curve derived value, but
at a slightly reduced significance of $\sim$3$\sigma$. This indicates that the
detection of deformation is robust against the modeling of ellipsoidal
variation and that the inclusion of out-of-transit data in our phase curve fit
slightly enhances the detection of deformation (see Akinsanmi et al., 2023).
Both detections of tidal deformation from its induced subtle effects on light
curves have been facilitated by CHEOPS, allowing to reach the 3$\sigma$
measurement significance. We further assess the significance of our detection
by computing the Bayes factor from the log-evidence of each model obtained
from the dynesty fit. The Bayes factor, $B_{\mathrm{ES}}$ is computed as the
exponent of the difference in log-evidence between the ellipsoidal and
spherical models. We obtained a Bayes factor of 6.7 in positive favor of the
ellipsoidal planet model (Kass & Raftery, 1995).

Figure 1 shows the best-fit phase curve models to the CHEOPS and TESS light
curves and the contribution of the component signals. We see that the major
impact of tidal deformation on the total phase curve is the modification of
the planet’s atmospheric phase variation $F_{p}$ (green curves). Compared to
the spherical planet case, $F_{p}$ for the deformed planet features additional
flux contribution between transit and eclipse due to the larger and varying
projected size of the ellipsoid at these phases. The contribution of tidal
deformation to the total ellipsoidal planet phase curve (yellow curve at the
bottom of panel b) peaks between quadrature and eclipse (at phases 0.3 and
0.69). The spherical planet model fit attempts to account for the deformation
contribution by increasing the amplitude of the stellar ellipsoidal variation
$F_{\mathrm{EV}}^{sph}$ (dashed black curve). However, since the peaks of
$F_{\mathrm{EV}}$ occur at quadratures and are always spaced by 0.5 phases, it
is unable to completely absorb the deformation signal (which has shorter peak
spacing) even if it is shifted in phase. The bottom two panels of Fig. 1 show
the residuals from the fits. In each passband, the difference between the
ellipsoidal and spherical planet models is overplotted on the spherical planet
fit residuals, highlighting the deformation signature – that is, the remaining
signal due to deformation that cannot be accounted for by a spherical planet
model. The deformation signature is concentrated within the in-transit phases
while the out-of-transit phases show only slight variations. This is due to
the low amplitude of the planetary phase variation in these optical bands (a
few hundred ppm) which influences the out-of-transit deformation signature.
Phase curve observations in the infrared, where the amplitude of the planet’s
atmospheric phase variation can reach thousands of ppm, will lead to a
significant increase in the deformation signature outside transit, thereby
allowing a more precise detection of the deformation and Love number
measurement (Akinsanmi et al., 2023).

Figure 2 shows the transit phases of the fit for all the datasets. We see that
the amplitude and shape of the deformation signature are passband dependent,
respectively, due to the varying size of the planet and also different limb
darkening compensation at each passband. The ratios of the ellipsoidal planet
axes are also passband-dependent since the asymmetry between the axes depends
on the radius at that passband (Eq. 2). Using Eqs. (1) and (2) with the
derived values from the ellipsoidal planet model fit, we calculate the axis
ratios, $r_{1}$:$r_{2}$:$r_{3}$, in the different passbands and report them in
Table 2. As opposed to the large $r_{1}$/$r_{3}$ ratio of 1.5 estimated in
(Cowan et al., 2012) to explain the Spitzer 4.5$\mu$m phase curve anomaly, our
fit to the datasets results in a more physical ratio of 1.16 in this passband.
Therefore, the source of the Spitzer 4.5$\mu$m remains unexplained and
requires further investigation.

Figure 3: Physical radius and density for WASP-12 b derived from the spherical
and ellipsoidal planet model fits.

Comparing the posterior parameters between the ellipsoidal and spherical
planet model fits, we find that most parameters differ only by
$\lesssim$2$\sigma$ due to the limited precision of $h_{2}$ obtained from
these datasets. Figure 3 compares the physical radius and mean density of the
planet in the different passbands, calculated using the posteriors from the
fits and stellar parameters in Table 1. The results show that the assumption
of sphericity leads to an underestimation of the volumetric radius of WASP-12
b by 4.0–5.2% and an overestimation of the planetary density by $\gtrsim$12%,
in line with theoretical expectations (Leconte et al., 2011b; Burton et al.,
2014; Correia, 2014). The biases in radius and density are respectively only
$\sim$2$\sigma$ and $\sim$1$\sigma$ significant due to the limited precision
of the derived $h_{2}$ from these datasets and the planetary mass from radial
velocity (RV). Therefore, an increase in photometric and RV precisions will
lead to more significant parameter biases and the derived properties of the
planet if sphericity is assumed (Berardo & De Wit, 2022).

For both deformed and spherical model phase curve fits, the measured stellar
ellipsoidal variation semi-amplitude $A_{\mathrm{EV}}$ is larger in the CHEOPS
band compared to TESS. This is due to the passband-dependent limb and gravity
darkening parameters in $\alpha_{\mathrm{EV}}$ in Eq. (8). The measured
$A_{\mathrm{EV}}$ in both bands can be used to independently estimate the mass
of WASP-12 b using Eqs. (6) and (8), and we expect to obtain the same mass
estimate in both passbands. First, the linear limb-darkening parameter for
each passband was estimated using LDCU, while the gravity darkening
coefficient was estimated from the tables in Claret (2021) and Claret (2017)
for the CHEOPS and TESS bands, respectively. We use the stellar mass from
Table 1 and orbital parameters from the joint fit in Table 2. For the
spherical planet model, we derive a planetary mass of $2.34\pm 0.48$
$M_{\mathrm{Jup}}$ from CHEOPS parameters and $2.09\pm 0.49$
$M_{\mathrm{Jup}}$ from TESS which are consistent with one another within
$1\sigma$. These masses are higher but consistent with the value of $1.47\pm
0.073$ $M_{\mathrm{Jup}}$ derived from radial velocity (RV; Collins et al.,
2017) at 1.2–1.8$\sigma$. For the ellipsoidal planet model, we derive $1.89\pm
0.53$ $M_{\mathrm{Jup}}$ for CHEOPS and $1.49\pm 0.58$ $M_{\mathrm{Jup}}$ for
TESS which are more consistent with the RV value at $<$0.8$\sigma$. Although
the discrepancy between the masses from EV and RV in the spherical model case
is not significant due to the large uncertainties on EV masses, we see a
slight indication that accounting for deformation reconciles both estimates
better.

The derived $h_{2}$ for WASP-12 b is unexpectedly close to the value of
Jupiter despite its higher insolation and mass. This was also the case for
WASP-103 b with the measured $h_{2}$ of 1.585 (Barros et al., 2022).
Theoretical models predict a decrease in the Love number with mass above
$\sim$1 $M_{\mathrm{Jup}}$ since more massive objects are more compressible
and therefore tend to become more centrally condensed (Leconte et al., 2011b).
Lower $h_{2}$ values are also expected at higher equilibrium temperatures due
to the lower density of the planetary envelope compared to the core (Kramm et
al., 2012; Wahl et al., 2021). Nonetheless, our derived $h_{2}$ value is
consistent with the theoretical models (e.g., Leconte et al., 2011b; Wahl et
al., 2021) predicting a maximum $h_{2}$ of 1.6 for tidally locked hot
Jupiters. It has been suggested that the ellipsoidal planet shape model might
not sufficiently account for nonlinear tidal response of the planet causing it
to systematically overestimate the Love number (Wahl et al., 2021). However,
transit models that account for these are not currently available and will
require more parameters to model the planetary shape which will be strongly
correlated.

Following the procedure of Buhler et al. (2016), we attempt to estimate the
core mass fraction of the WASP-12 b using the measured $h_{2}$ value. We found
that the 3$\sigma$ measurement of the Love number does not provide valuable
constraints as the result remain consistent with a core mass fraction of 0 and
1. Indeed, Akinsanmi et al. (2023) showed that valuable constraints on the
core mass fraction require $h_{2}$ precisions higher than 4$\sigma$. They also
showed that a single JWST phase curve of a target such as WASP-12 b is capable
of attaining 17$\sigma$ measurement of $h_{2}$ due to the large phase curve
amplitude in the infrared, the reduced effect of limb darkening, and the
unparalled precision of the instrument.

###  4.2 Impact of limb darkening

As most of the optical band deformation signature is concentrated within the
transit phases, the detection is sensitive to the modeling of the limb
darkening profile. In the fit for tidal deformation, we used the method
described in Section 2.2.2 to derive priors on the LDCs based on model
intensity profiles from the two spectral synthesis libraries (we call this our
fiducial analysis). We also explore two alternative methods to leveraging the
generated model intensity profiles, which involve simultaneously fitting the
model intensity profiles and the transit observations in order to find the
best-fit LD profile.

• Alternative-1: This approach merges the PHOENIX and ATLAS model intensity
profiles to create a new joint intensity profile whose 1$\sigma$ uncertainty
at each $\mu$ encompasses the 1$\sigma$ uncertainty of the individual model
profiles (see Fig. 9b). The joint intensity profile is fitted with the power-2
LD law simultaneously with the transit observations so that a joint likelihood
of the suggested limb darkening profile and transit model is obtained (for
each passband) at each iteration in our sampling process. This approach allows
the joint model intensity profile and transit observations to determine the
best-fit limb darkening profile. The parameters of the LD law here have
uninformative wide priors since their values are constrained during the joint
fit.

• Alternative-2: In this approach, a uniform bound is defined that spans the
median of both model intensity profiles (see Fig. 9c). During the likelihood
estimation, each $\mu$ point in the suggested limb darkening profile that is
anywhere within the defined bounds is assigned zero likelihood. Profile points
outside the bound are still acceptable but have lower likelihood values
(Gaussian) depending on their distance from the bounds. This approach is
agnostic to the eventual choice of limb darkening profile as long as its
points are within or close to the bounds defined by both theoretical stellar
intensity profiles.

We perform the ellipsoidal planet model fits to the data again using the
aforementioned limb darkening alternative approaches and compare the results
with our fiducial analysis. The result is given in Table 3 where a consistent
value of $h_{2}$ is derived with the 3 methods, indicating that our fit is
robust against the modeling of limb darkening.

Table 3: Sensitivity of derived Love number on different approaches of modeling limb darkening. LD method | $h_{f}$ | Significance
---|---|---
Fiducial | $1.55^{+0.45}_{-0.49}$ | 3.16$\sigma$
Alternative–1 | $1.53^{+0.45}_{-0.51}$ | 3.00$\sigma$
Alternative–2 | $1.54^{+0.47}_{-0.52}$ | 2.96$\sigma$

##  5 Atmospheric characterization

Phase curves provide a wealth of information to characterize the atmosphere of
a planet such as the day and nightside temperatures, the longitudinal
temperature and reflectivity map, and also the efficiency of heat transport
among others (see e.g., Cowan & Agol, 2011; Shporer, 2017; Parmentier &
Crossfield, 2018). In this section, we infer the atmospheric properties of
WASP-12 b from our CHEOPS and TESS phase curve analyses (§3.3). Unless
otherwise stated, the discussion below is based mostly on the phase curve fit
using the ellipsoidal planet model. However, we still report the derived
parameters for both planet models in Table 2 and we found them to be in
agreement within 1$\sigma$.

###  5.1 Phase curve constraints

The results of the phase curve fit for the ellipsoidal and spherical planet
model fits (Table 2) reveal a larger planet-to-star radius ratio in the CHEOPS
band compared to the TESS band, indicating stronger atmospheric opacity in the
bluer CHEOPS band. We also measure a significantly lower dayside flux
(occultation depth) in CHEOPS compared to TESS. The nightside fluxes in both
passbands are consistent with zero at $<$1$\sigma$ ($2\sigma$ upper limits of
$\sim$50 ppm). The best-fit phase curves show only marginally significant
eastward phase offsets of $9\pm 5\degree$ and $6\pm 3\degree$ in the CHEOPS
and TESS bands, respectively. The low nightside flux and small phase offset in
both bands indicate low-efficiency day-night heat redistribution at the
atmospheric layers probed by the instruments. This is consistent with the
theoretical and observed trend of decreasing phase offset with increasing
temperature for ultra-hot Jupiters (Parmentier et al., 2016; Komacek &
Showman, 2016; Komacek et al., 2017) possibly due to their short radiative
timescales (Perna et al., 2012).

###  5.2 Atmospheric modeling

We model the atmosphere of WASP-12 b by performing 1D retrievals on the
emission spectrum and computing the forward global circulation model. This
will facilitate the proper interpretation of the phase curve, thereby enabling
the determination of the relative contribution of both reflection and thermal
emission to the observed occultation depths.

Figure 4: Eclipse spectrum of WASP-12 b. Left: Observed occultation depths of
WASP-12 b (gray points) with the CHEOPS and TESS occultation depths are shown
as colored diamonds. The best-fit thermal retrieval model and its 1$\sigma$
uncertainties are overplotted in green while the forward thermal 3D GCM
spectrum is plotted in violet. The gray dashed curves show the blackbody model
spectra for brightness temperatures of 2600 K, 3000 K, and 3400 K. Right: T-P
profile for the retrieval showing a gradual temperature decrease with
decreasing pressure depth between 10-1 – $10^{-4}$ bar indicative of no
thermal inversion in the atmosphere of WASP-12 b. The contribution function at
each passband is also shown indicating that TESS and CHEOPS essentially probe
the same pressure levels within the atmosphere.

####  5.2.1 1D retrieval

We model the emission spectra of WASP-12 b using the open-source Pyrat Bay
framework (Cubillos & Blecic, 2021) to constrain its atmospheric properties
based on the tabulated occultation depth measurements in Hooton et al. (2019,
and references therein). This dataset consists of several ground-based,
HST/WFC3, and Spitzer measurements spanning 0.5–8$\mu$m. Since reflection is
not accounted for in Pyrat Bay, our model only considered the infrared
observation ($\lambda>1.0$ $\mu$m) to avoid interference from the reflected
flux at shorter wavelengths. The retrieved thermal emission spectrum can then
be computed including shorter wavelengths to estimate the thermal contribution
in the CHEOPS and TESS passbands.

We model the atmosphere of WASP-12 b between 102–10-9 bar adopting the
parametric temperature–pressure (T-P) prescription of Guillot (2010). The
parametric model depends on the irradiation temperature $T_{\mathrm{irr}}$,
the mean thermal opacity $\log\kappa^{\prime}$, and the ratio of the visible
to thermal opacities $\log\gamma$. For this analysis, we modeled the
composition under thermochemical equilibrium, considering the most relevant
neutral and ionic species expected for hot-Jupiter atmospheres. The chemistry
was parameterized by the abundance of carbon [C/H], oxygen [O/H], and all
other metals [M/H] relative to solar-abundance values. The radiative transfer
calculation considered HITEMP and ExoMol opacity line lists for H2O, CO, CO2,
CH4, HCN, NH3, TiO, VO (Rothman et al., 2010; Li et al., 2015; Hargreaves et
al., 2020; Harris et al., 2006, 2008; Polyansky et al., 2018; Coles et al.,
2019; Yurchenko, 2015; McKemmish et al., 2016, 2019), which were preprocessed
using the repack algorithm to extract the dominant line transitions (Cubillos,
2017). Additional opacities include the Na and K resonant lines (Burrows et
al., 2000), collision-induced absorption from H2–H2 (Borysow et al., 2001;
Borysow, 2002), Rayleigh opacity from H2, H, and He (Kurucz, 1970), and H-
free-free and bound-free opacity (John, 1988). For the stellar spectrum, we
used a synthetic PHOENIX spectrum (Husser et al., 2013) according to the
stellar properties (Table 1). Finally, the atmospheric Bayesian retrieval
employed a differential-evolution MCMC algorithm implemented in Cubillos et
al. (2017) to construct posterior distributions for the atmospheric
parameters.

We tested four retrieval scenarios: with and without the Spitzer 5.8 and
8.0$\mu$m data points; and also with and without TiO/VO absorption (since
these species may or may not be present in the atmosphere). We found
statistically consistent results between all scenarios, thus in the following,
we report the results of the retrieval including TiO/VO and all Spitzer
observations.

Figure 4 shows the retrieved T-P profile and thermal spectrum, which we
extended over the CHEOPS and TESS passbands. The retrieval spectral fit is
mostly driven by the HST/WFC3 and the Spitzer 3.6 and 4.5 $\mu$m observations.
The model fits most of the occultation depths relatively well, although
several mutually inconsistent measurements lead to a reduced $\chi^{2}$ of
2.9. The observations at wavelengths $\lambda<2\,\mu$m are well fit by a
$\sim$3000 K blackbody model, indicating that these bands probe a nearly
isothermal region of the atmosphere. The depths at the Spitzer 3.6 $\mu$m and
4.5 $\mu$m bands imply lower brightness temperatures ($\sim$2600 K) than in
the HST/WFC3 band, possibly due to absorption by carbon-bearing species. These
measurements drive the retrieved model towards a non-inverted T-P profile
since these bands probe the upper atmosphere (see contribution function and
T–P profile in the right panel of Fig. 4). Previous work by Stevenson et al.
(2014a) and Oreshenko et al. (2017) similarly finds a noninverted T-P profile.
The retrieval model underestimates the Spitzer 5.8 $\mu$m and 8.0 $\mu$m which
would require higher brightness temperatures of $\sim$3400 K, and thus are
inconsistent with the rest of the observations. These Spitzer points have
larger uncertainties and are therefore not very constraining in the fit.

In all scenarios, we find insignificant contributions from N-bearing species,
which are not expected to be prominent at high-temperature atmospheres in
thermochemical equilibrium. Likewise, TiO and VO do not seem to have an impact
on the retrieved spectrum, probably because the bluer end of the spectrum
(where these molecules absorb the most) is nearly isothermal, which decreases
the amplitude of spectral features. On the other hand, C- and O-bearing
species (CO, CO2, CH4, H2O) are more prominent. Figure 10 shows the retrieval
posterior distribution, we find super-solar abundances for both carbon and
oxygen while keeping mostly C/O ¿ 1 ratios. For other metals, we find sub-
solar abundances. These results are consistent with previous retrieval
analyses of the WASP-12 b emission spectrum (e.g., Oreshenko et al., 2017;
Himes & Harrington, 2022).

####  5.2.2 3D global circulation model (GCM)

We also compare the measured occultation depths with the output of a forward
3D GCM that self-consistently calculates the thermal emission from the planet.
Here, we use expeRT/MITgcm as introduced by Carone et al. (2020); Schneider et
al. (2022). The expeRT/MITgcm uses a pre-calculated grid of correlated-k
opacities, where we employ here the S1 spectral resolution as described in
Schneider et al. (2022) and opacity sources for Na, K, CH4, H2O, CO2, CO, H2S,
HCN, SiO, PH3 and FeH with collision-induced absorption from H2, He and H- and
Rayleigh scattering of H2 and He. We omit VO and TiO because our retrievals
showed no evidence of a thermal inversion in the upper atmosphere, which would
indicate the presence of these molecules. In post-processing, we calculate the
thermal emission of the planet for different orbital phases (including the
dayside emission) using a wavelength resolution R=100, where we reduce the
spatial resolution of the dayside to 15 degrees in longitude and latitude.
During the GCM simulation as well as during post-processing, the abundance of
molecules is computed using equilibrium chemistry assuming solar ([Fe/H]=0)
metallicity. Thus, we include the effect of thermal ionization of H2 and H2O
on our predicted phase curves and dayside emission spectrum.

The dayside emission of the GCM is overplotted in Fig. 4 where we see a good
agreement with the retrieval model. In particular, the GCM model correctly
predicts that water features are muted in the HST/WFC3 wavelength range
(between 1.1 and 1.6$\mu$m) due to a combination of thermal ionization and
thus dissociation of H2O over large parts of the dayside and electron
opacities that suppress molecular features in this wavelength range. For
larger wavelengths, where Spitzer (and now JWST) is sensitive, molecular
features are present again. Although the GCM dayside emission yields smaller
amplitude features compared to the best retrieval model, it is still
qualitatively in agreement with the available data.

Using the phase-resolved thermal emission spectrum from the GCM, we integrate
the flux with the CHEOPS and TESS response functions to obtain the thermal
phase variation of the planet in both bands. We overplot the thermal GCM along
with the fitted planetary models in Fig. 1. For CHEOPS, the lower amplitude of
the thermal GCM compared to the fitted $F_{p}$ models indicates the need for
additional flux from reflection. For TESS, the computed thermal GCM is a good
representation of the observed atmospheric phase variation, although the
maximum flux and hotspot offset are slightly overestimated; as often seen when
comparing GCMs to observations (Parmentier & Crossfield, 2018; Zhang et al.,
2018). For ultra-hot Jupiters like WASP-12b, it also has been recognized that
their daysides are so hot that the gas is dominated by atoms and ions and no
longer by molecules like H2O, TiO, and VO. The locally high gas temperatures,
therefore, lead to a moderate ionization of the atmosphere (Arcangeli et al.,
2018; Parmentier et al., 2018). While expeRT/MITgcm takes the H2O thermal
stability into account by calculating a chemical equilibrium model and
adjusting abundances accordingly, we did not take into account that this
partially ionized flow may couple to the exoplanet’s magnetic field. Which
processes dominate a magnetic coupling will depend on the planet’s magnetic
field and the local degree of ionization. A partially ionized gas may couple
to the magnetic field and some species may, hence, be transported with the
flow (Helling et al., 2021). Such interactions may modify the wind flow to
dampen the hot spot offset and change the maximum flux (Rogers & Komacek,
2014; Beltz et al., 2022), in particular in the infrared (May et al., 2022).
In this work, we opted to not add drag to the GCM to mimic the unknown
magnetic field in WASP-12 b and minimize the number of uncertainties.

###  5.3 Albedos and dayside/nightside temperatures

The observed dayside flux in each passband is composed of the thermal emission
from the planet and the reflected light by the atmosphere in that passband.
The observed flux is thus given (e.g., Esteves et al., 2013; Shporer, 2017)
as:

$\frac{F_{d}}{F_{\star}}=A_{g}\left(\frac{R_{p}}{a}\right)^{2}+\frac{\int\mathcal{F}_{\lambda}(T_{\mathrm{p}})\mathcal{T}_{\lambda}d\lambda}{\int\mathcal{F}_{\lambda}(T_{\mathrm{eff}})\mathcal{T}_{\lambda}d\lambda}\left(\frac{R_{p}}{R_{\star}}\right)^{2}.$
(13)

For a planet with radius $R_{p}$ and semi-major axis $a$, the reflective
contribution is determined by the geometric albedo $A_{g}$, while the thermal
contribution is determined by the emission spectra of the planet and the star
$\mathcal{F}_{\lambda}(T_{\mathrm{p}})$ and
$\mathcal{F}_{\lambda}(T_{\mathrm{eff}})$, respectively. Therefore,
interpreting the measured dayside fluxes requires determining the relative
contributions of reflection and thermal emission in the CHEOPS and TESS bands.

As seen in the T–P profile in Fig. 4, CHEOPS and TESS probe similar pressure
levels in the atmosphere consistent with black-body dayside temperatures of
2915$\pm$25 K and 2821$\pm$25 K, respectively. The average measured dayside
temperature across both bands is $2868\pm 17$ K, in agreement with the
effective dayside temperature of $2864\pm 15$ K derived in Schwartz et al.
(2017). Using the retrieved thermal emission spectra, we calculated the
thermal contribution in the CHEOPS and TESS passbands as 205$\pm$10 and
480$\pm$19 ppm, respectively. Subtracting the thermal contribution from the
observed occultation depths, we used Eq. (13) to estimate the geometric albedo
in the passbands. We find $A_{g}$ = 0.083$\pm$0.015 in the CHEOPS band and
$A_{g}$ = 0.010$\pm$0.023 in the TESS band, indicating that the atmosphere has
non-negligible reflectivity in the CHEOPS band but much lower reflectivity in
the TESS band where $A_{g}$ is consistent with zero with $2\sigma$ upper limit
of 0.06. The derived $A_{g}$ values are consistent with the results from the
shorter wavelengths of HST/STIS where Bell et al. (2017) found $A_{g}<0.064$.

The derived low geometric albedo of WASP-12 b follows the trend of low
reflectivity ($Ag\lesssim 0.2$) of UHJs in the optical to near-infrared
transition bands (Mallonn et al., 2019), which is supported by the difficulty
in forming condensates at such high temperatures (Parmentier et al., 2018;
Wakeford et al., 2017). Perhaps unlikely, but the higher albedo in the CHEOPS
band may be due to high-temperature condensates (e.g., silicates, Al2O3,
CaTiO3) on the western terminator that have been transported from the
nightside. Indeed, Sing et al. (2013) found that the best-fit model to the
HST/STIS, HST/WFC3, and Spitzer transmission spectrum of WASP-12 b was Mie
scattering by Al2O3 haze but Bell et al. (2017) ruled out this scenario based
on low dayside reflectivity measured in the HST/STIS band. They instead
favored thermal emission and Rayleigh scattering from atomic hydrogen and
helium.

Figure 5: Occultation depth measurements of individual CHEOPS visits in the
three observation seasons. The horizontal lines and shaded regions represent
the measured depth from jointly fitting all the visits in each season. The
depths measured between seasons are consistent within 1$\sigma$. The dotted
sinusoidal signal is the best-fit periodic signal (P=8.3)̣ to the depth
measurements across all seasons.

We used Eq. (13) to convert the measured nightside fluxes to nightside
brightness temperatures and obtained $2\sigma$ upper limits of
$T_{\mathrm{night}}\,\simeq\,2000$ K in both passbands (Table 2). This implies
a large day-night temperature contrast (¿45%) as expected for UHJs due to
relatively inefficient heat redistribution (Perna et al., 2012; Komacek et
al., 2017). The derived high nightside temperature limit is consistent with
the expectation of increasing values as a function of stellar irradiation
(Keating et al., 2019) as night clouds disperse for highly irradiated planets.

###  5.4 Dayside atmospheric variability

Several authors have reported hints of possible time variability in the
dayside atmospheric brightness of WASP-12 b due to discrepant secondary
eclipse depth measurements obtained at various wavelength bands (see Fig. 4).
For example, analyses of two i′–band observations from different ground-based
telescopes resulted in an eclipse depth difference of more than 2$\sigma$
(Hooton et al., 2019) while a pair of V–band observations taken within a month
of each other revealed eclipse depths that are discrepant with 4.5$\sigma$
significance (von Essen et al., 2019). There is also a 2.5$\sigma$ discrepancy
between published z′–band depth measurements (López-Morales et al., 2010;
Föhring et al., 2013) and similar level of disagreement in measurements taken
around $\sim$2$\mu$m (see Hooton et al., 2019), and also at Spitzer 3.6$\mu$m
and 4.5$\mu$m (e.g., Cowan et al., 2012; Stevenson et al., 2014b). Given that
the reported eclipse depths have come from different instruments and authors,
it is possible that some of the observed disagreements could be due to
instrument systematics, observing conditions, telluric contamination in
ground-based observations, or even in the analysis of the data. However,
Komacek & Showman (2019) showed that the hydrodynamic instabilities in hot
Jupiter atmospheres can impact the thermal emission leading to variability at
the 2% level in eclipse depth measurements. WASP-12 b can particularly show
variability due to magnetohydrodynamic effects in the partially ionized
atmosphere (Rogers & Komacek, 2014). It remains unclear whether the observed
disagreements are astrophysical or due to systematics.

Contrary to these results, the analysis of 4 sectors of TESS data by Wong et
al. (2022) found no strong evidence of variability between the individual
eclipse measurements, although the largest discrepancy between any two
measurements is 3.1$\sigma$. Given the low signal-to-noise of the individual
TESS eclipses, they were only able to place 2$\sigma$ upper limits of 450 ppm
and 80 ppm on orbit-to-orbit and month-long variability, respectively.

We further investigate this potential variability with the CHEOPS occultation
observations spanning 3 seasons. The analyses of the 26 occultations have been
described in Section 3.5. Figure 5 shows the derived occultation depths and
their $1\sigma$ uncertainties ordered chronologically. The depths derived from
the joint fit of occultations with each season of observation are also shown.
The individual occultation depths agree with each other within 1$\sigma$ and
also with the joint fit of each season better than 1.2$\sigma$. The largest
discrepancy between any two depth measurements is 1.8$\sigma$ while the joint
fits for the seasons are in agreement within 1$\sigma$. We search for
periodicity in the occultation depths using a Lomb-Scargle periodogram (Lomb,
1976; Scargle, 1982) and found the highest peak periodicity of 8.3 d to be
nonsignificant with a false alarm probability of $\sim$20%. Therefore, we do
not find strong evidence for variability in the dayside atmosphere of WASP-12
b at the median depth precision of 120 ppm attained by CHEOPS.

##  6 Tidal decay

Previous analyses of the transit times of WASP-12 b have clearly shown that
the orbit of the planet is decaying due to tidal interaction with the star,
causing the planet to lose angular momentum to the star (Maciejewski et al.,
2016; Yee et al., 2019; Turner et al., 2021; Wong et al., 2022). The estimate
of the planet’s decay rate was possible due to the long baseline of available
timing measurements. Since our CHEOPS observations further extend the time
baseline, we perform a fit to refine the ephemeris and decay rate of the
WASP-12 b by combining our transit and occultation timing measurements given
in Table 8. We combined these CHEOPS transit timings with prior transit and
occultation timings compiled by Yee et al. (2019) from various authors and
TESS timings derived in Wong et al. (2022).

We model the orbital decay of the planet using a quadratic ephemeris model
which gives the transit and occultation times (e.g., Turner et al., 2021) as:

$\displaystyle T_{\mathrm{tra}}(E)$
$\displaystyle=T_{0}+PE+\frac{1}{2}\frac{dP}{dE}E^{2}\,,$ (14) $\displaystyle
T_{\mathrm{occ}}(E)$
$\displaystyle=\left(T_{0}+\frac{P}{2}\right)+PE+\frac{1}{2}\frac{dP}{dE}E^{2}\,$

where $T_{0}$ is the reference transit time closest to the middle of the
entire time series, $E$ is the transit epoch, and $dP/dE$ is the orbital decay
rate from which the time decay rate $\dot{P}=\frac{1}{P}\frac{dP}{dE}$ and the
decay timescale $\tau=P/\dot{P}$ can be derived. We used emcee to
simultaneously fit Eq. (14) to the transit and occultation times with $T_{0}$,
$P$, and $dP/dE$ as free parameters. The result from the model fit is given
as:

$\displaystyle T_{0}$ $\displaystyle=2457103.283661\pm 3.1\times
10^{-05}\mathrm{BJD_{TBD}},$ (15) $\displaystyle P$
$\displaystyle=1.091419366\pm 2\times 10^{-08}\mathrm{days},$
$\displaystyle\frac{dP}{dE}$ $\displaystyle=(-1.043\pm 0.029)\times
10^{-09}\mathrm{days/orbit},$

from which we derived:

$\displaystyle\dot{P}$ $\displaystyle=-30.13\pm 0.82\,\mathrm{ms/yr}$ (16)
$\displaystyle\tau$ $\displaystyle=3.13\pm 0.087\,\mathrm{Myr}.$

Figure 6: Deviation in the transit (top) and occultation (bottom) timings of
WASP-12b compared to the best-fit linear ephemeris model. The black and green
points are published measurements taken from Yee et al. (2019) and Wong et al.
(2022), respectively, while the red points are the new CHEOPS timing
measurements.

The decay rate $\dot{P}$ derived from our fit including new timing
measurements by CHEOPS remains consistent with the estimate of $-29.81\pm
0.94$ ms/yr obtained in Wong et al. (2022) but improves the precision by 12%.
Our revised orbital decay timescale $\tau$ is slightly shorter but 13% more
precise than the value of 3.16 $\pm$ 0.1 Myr derived in Wong et al. (2022).
Figure 6 shows the transit and occultation timing deviations of WASP-12 b and
the orbital decay model fit after subtraction of the best-fit linear ephemeris
model.

The rate at which the planet’s orbital energy ($E_{p}$) and angular momentum
($L_{p}$) are being lost to the star can be calculated (e.g., Yee et al.,
2019) as:

$\displaystyle\frac{dE_{p}}{dt}$
$\displaystyle=\frac{M_{p}}{3}\left(\frac{2\pi
GM_{\star}}{P}\right)^{2/3}\frac{\dot{P}}{P}=(-5.1\pm 0.3)\times
10^{30}\,\mathrm{erg/s}$ (17) $\displaystyle\frac{dL_{p}}{dt}$
$\displaystyle=\frac{M_{p}\dot{P}}{3(2\pi)^{1/3}}\left(\frac{GM_{\star}}{P}\right)^{2/3}=(-7.64\pm
0.45)\times 10^{27}\,\mathrm{kg\,m^{2}\,s^{-2}}.$

The energy is then dissipated inside the star as the tidal oscillations are
converted into heat. The efficiency of tidal dissipation is quantified by the
modified tidal quality factor of the star $Q^{\prime}_{\star}$ which can we
derive from $\dot{P}$ using the constant lag model of Goldreich & Soter (1966)
as:

$Q^{\prime}_{\star}=-\frac{27\pi}{2}Q_{M}\left(\frac{a}{R_{\star}}\right)^{-5}\frac{1}{\dot{P}}=(1.70\pm
0.14)\times 10^{5}.$ (18)

Population studies of stars show that the value of $Q^{\prime}_{\star}$ ranges
between $10^{5}$–$10^{6.5}$ for hot Jupiter systems and may extend up to
$10^{7}$ for binary star systems (Jackson et al., 2008; Ogilvie, 2014). Lower
values of $Q^{\prime}_{\star}$ imply more efficient tidal dissipation. Our
derived value in agreement with the value of $Q^{\prime}_{\star}$ =
(1.75$\pm$0.12)$\,\times\,10^{5}$ derived in Yee et al. (2019). The derived
value is on the low end of the range of $Q^{\prime}_{\star}$ values and
implies a much higher dissipation rate for WASP-12 than is expected for a
main-sequence star (Turner et al., 2021; Yee et al., 2019). Since the
efficiency of tidal dissipation depends also on the structure and evolutionary
state of the star, one explanation would be that WASP-12 is actually a
subgiant star capable of such efficient dissipation due to nonlinear breaking
of gravity waves close to the center of the star (Weinberg et al., 2017).
However, further stellar modeling reports that the observed characteristics of
WASP-12 are consistent with a main-sequence star rather than a subgiant
(Bailey & Goodman, 2019). Our modeling also confirms a young stellar age of
2.3 Gyr (§ 2.2.1).

Tidal decay has not been confirmed for any other exoplanet despite several
ultra-hot Jupiter systems having similar planetary and orbital characteristics
as WASP-12. Recently, Vissapragada et al. (2022) reported evidence for the
tidal decay of Kepler-1658 b orbiting an evolved star and derived a decay rate
of $131_{-22}^{+20}$ ms/yr. However, further observation of this system will
be needed to improve the precision of the orbital decay rate. As this is the
first reported case of planetary inspiral around an evolved star, its
confirmation will give support to the theoretical expectation of planetary
engulfment with stellar evolution, which results in the observed dearth of hot
Jupiters around evolved stars (Grunblatt et al., 2022). Harre et al. (2023)
also reported a $5\sigma$ significant measurement of orbital decay for WASP-4
b but found that more observations are required to differentiate between the
tidal decay and apsidal precession scenarios.

##  7 Conclusion

In this work, we have presented a detailed analysis of WASP-12 b using
observations by the CHEOPS spacecraft alongside publicly available data from
TESS and Spitzer. We leverage these datasets to put constraints on the shape,
atmosphere, and orbital characteristics of the planet. We summarize the main
results below:

  * •

The large number of CHEOPS visits allowed us to construct a phase curve which
we analyzed together with TESS phase curve and Spitzer transits to constrain
the tidal deformation and atmospheric properties of the planet. From our
global fit to the datasets, we measured a Love number of $h_{2}$ =
1.55${}^{+0.45}_{-0.49}$ corresponding to a 3.16$\sigma$ detection. This
measurement makes WASP-12 b the second planet, after WASP-103 b, where tidal
deformation has been significantly detected from the light curve. The
3$\sigma$ Love number measurements for these planets are still consistent with
those of Jupiter despite the strong irradiation of these planets. There is a
need to improve the precision of Love number measurements in order to perform
comparative interior structure analysis between these highly irradiated,
tidally locked planets and the cooler Jupiter. Phase curve observations of
such planets with JWST will provide the precisions needed to measure the Love
number at $\sim$12$\sigma$ significance, which will in turn better constrain
their core mass fractions and interior structures (Akinsanmi et al., 2023).

  * •

We significantly measure occultation depths of $333\pm 24$ ppm and $493\pm 29$
ppm in the CHEOPS and TESS bands, respectively. The nightside fluxes are
consistent with zero at both bands. We compared our phase curve with the
output of 3D-GCM and found close agreement between them. We measured marginal
eastward phase offset of in both bands. We also detected stellar ellipsoidal
variation in both passbands, which leads to mass estimates that marginally
differ from the RV-derived mass at the 1.2–1.8$\sigma$ level if a spherical
planet shape is assumed. However, when we account for tidal deformation, the
mass estimates agree with the RV value at $<1\sigma$ indicating a preference
for the ellipsoidal model.

  * •

We model the emission spectrum of WASP-12 b using published occultation depth
measurements spanning 0.5-8$\mu$m to derive the thermal profile of the planet.
Our best-fit model indicates that CHEOPS and TESS are probing the same
pressure level in the atmosphere of WASP-12 b with an average brightness
temperature of $\sim$2868 K. We found no evidence of temperature inversion in
the atmosphere in agreement with the conclusion from Madhusudhan et al.
(2011). We additionally estimate the geometric albedo of the planet and find
the planet to be more reflective in the CHEOPS band with $A_{g}=0.086\pm
0.017$ than in TESS with $A_{g}=0.01\pm 0.023$.

  * •

Our analysis of the CHEOPS occultations did not show strong evidence of
variability in the dayside atmosphere of the planet at the median occultation
depth precision of 120 ppm attained by CHEOPS.

  * •

Our analysis of the tidal decay of the planet using the new CHEOPS
observations refines the orbital decay rate of the planet to $-30.13\pm 0.82$
ms/yr corresponding to a precision improvement of 12% compared to the latest
estimates from Wong et al. (2021).

###### Acknowledgements.

CHEOPS is an ESA mission in partnership with Switzerland with important
contributions to the payload and the ground segment from Austria, Belgium,
France, Germany, Hungary, Italy, Portugal, Spain, Sweden, and the United
Kingdom. The CHEOPS Consortium would like to gratefully acknowledge the
support received by all the agencies, offices, universities, and industries
involved. Their flexibility and willingness to explore new approaches were
essential to the success of this mission. BA acknowledges the financial
support of the Swiss National Science Foundation under grant number
PCEFP2_194576. ML acknowledges support of the Swiss National Science
Foundation under grant number PCEFP2_194576. S.C.C.B. acknowledges support
from FCT through FCT contracts nr. IF/01312/2014/CP1215/CT0004. LCa and CHe
acknowledge support from the European Union H2020-MSCA-ITN-2019 under Grant
Agreement no. 860470 (CHAMELEON). P.E.C. is funded by the Austrian Science
Fund (FWF) Erwin Schroedinger Fellowship, program J4595-N. ACCa and TWi
acknowledge support from STFC consolidated grant numbers ST/R000824/1 and
ST/V000861/1, and UKSA grant number ST/R003203/1. GBr, GSc, VSi, LBo, VNa,
IPa, GPi, and RRa acknowledge support from CHEOPS ASI-INAF agreement n.
2019-29-HH.0. B.-O. D. acknowledges support from the Swiss State Secretariat
for Education, Research and Innovation (SERI) under contract number
MB22.00046. ABr was supported by the SNSA. S.G.S. acknowledge support from FCT
through FCT contract nr. CEECIND/00826/2018 and POPH/FSE (EC). This work was
supported by FCT - Fundação para a Ciência e a Tecnologia through national
funds and by FEDER through COMPETE2020 - Programa Operacional Competitividade
e Internacionalizacão by these grants: UID/FIS/04434/2019, UIDB/04434/2020,
UIDP/04434/2020, PTDC/FIS-AST/32113/2017 & POCI-01-0145-FEDER- 032113,
PTDC/FIS-AST/28953/2017 & POCI-01-0145-FEDER-028953, PTDC/FIS-AST/28987/2017 &
POCI-01-0145-FEDER-028987, O.D.S.D. is supported in the form of work contract
(DL 57/2016/CP1364/CT0004) funded by national funds through FCT. YAl
acknowledges support from the Swiss National Science Foundation (SNSF) under
grant 200020_192038. RAl, DBa, EPa, and IRi acknowledge financial support from
the Agencia Estatal de Investigación of the Ministerio de Ciencia e Innovación
MCIN/AEI/10.13039/501100011033 and the ERDF “A way of making Europe” through
projects PID2019-107061GB-C61, PID2019-107061GB-C66, PID2021-125627OB-C31, and
PID2021-125627OB-C32, from the Centre of Excellence “Severo Ochoa” award to
the Instituto de Astrofísica de Canarias (CEX2019-000920-S), from the Centre
of Excellence “María de Maeztu” award to the Institut de Ciències de l’Espai
(CEX2020-001058-M), and from the Generalitat de Catalunya/CERCA programme. XB,
SC, DG, MF and JL acknowledge their role as ESA-appointed CHEOPS science team
members. This work has been carried out within the framework of the NCCR
PlanetS supported by the Swiss National Science Foundation under grants
51NF40_182901 and 51NF40_205606. This project was supported by the CNES. The
Belgian participation to CHEOPS has been supported by the Belgian Federal
Science Policy Office (BELSPO) in the framework of the PRODEX Program, and by
the University of Liège through an ARC grant for Concerted Research Actions
financed by the Wallonia-Brussels Federation. L.D. is an F.R.S.-FNRS
Postdoctoral Researcher. This project has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (project Four Aces. grant agreement No 724427). It has
also been carried out in the frame of the National Centre for Competence in
Research PlanetS supported by the Swiss National Science Foundation (SNSF). DE
acknowledges financial support from the Swiss National Science Foundation for
project 200021_200726. MF and CMP gratefully acknowledge the support of the
Swedish National Space Agency (DNR 65/19, 174/18). DG gratefully acknowledges
financial support from the CRT foundation under Grant No. 2018.2323 “Gaseousor
rocky? Unveiling the nature of small worlds”. M.G. is an F.R.S.-FNRS Senior
Research Associate. MNG is the ESA CHEOPS Project Scientist and Mission
Representative, and as such also responsible for the Guest Observers (GO)
Programme. MNG does not relay proprietary information between the GO and
Guaranteed Time Observation (GTO) Programmes, and does not decide on the
definition and target selection of the GTO Programme. SH gratefully
acknowledges CNES funding through the grant 837319. KGI is the ESA CHEOPS
Project Scientist and is responsible for the ESA CHEOPS Guest Observers
Programme. She does not participate in, or contribute to, the definition of
the Guaranteed Time Programme of the CHEOPS mission through which observations
described in this paper have been taken, nor to any aspect of target selection
for the programme. K.W.F.L. was supported by Deutsche Forschungsgemeinschaft
grants RA714/14-1 within the DFG Schwerpunkt SPP 1992, Exploring the Diversity
of Extrasolar Planets. This work was granted access to the HPC resources of
MesoPSL financed by the Region Ile de France and the project Equip@Meso
(reference ANR-10-EQPX-29-01) of the programme Investissements d’Avenir
supervised by the Agence Nationale pour la Recherche. AC acknowledges support
from PTDC/FIS-AST/7002/2020. PM acknowledges support from STFC research grant
number ST/M001040/1. This work was also partially supported by a grant from
the Simons Foundation (PI Queloz, grant number 327127). NCSa acknowledges
funding by the European Union (ERC, FIERCE,101052347). Views and opinions
expressed are however those of the author(s) only and do not necessarily
reflect those of the European Union or the European Research Council. Neither
the European Union nor the granting authority can be held responsible for
them. GyMSz acknowledges the support of the Hungarian National Research,
Development and Innovation Office (NKFIH) grant K-125015, a PRODEX Experiment
Agreement No. 4000137122, the Lendület LP2018-7/2021 grant of the Hungarian
Academy of Science and the support of the city of Szombathely. V.V.G. is an
F.R.S-FNRS Research Associate. NAW acknowledges UKSA grant ST/R004838/1.

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##  Appendix A Tables

Table 4: CHEOPS observation log of WASP-12. The visit types are either
occultation (occ), transit (tra), or phase-curve (PC). $\beta_{w}\beta_{r}$
gives the white and red noise correction factor to the flux uncertainties of
each visit.

| Visit
---
[#]
File key | Type |  | Start time
---
[UTC]
| Duration
---
[hr]
| Data
---
points [#]
| Eff.
---
[%]
Epoch | $\beta_{w}\beta_{r}$
1 | PR100016_TG010201_V0200 | occ | 2020-11-02T00:45:45 | 7.1 | 223 | 52 | 1880 | 1.54
2 | PR100016_TG010202_V0200 | occ | 2020-11-09T16:06:45 | 7.8 | 249 | 52 | 1887 | 1.02
3 | PR100016_TG010203_V0200 | occ | 2020-11-10T19:34:19 | 5.8 | 212 | 61 | 1888 | 1.01
4 | PR100016_TG010204_V0200 | occ | 2020-11-12T22:41:45 | 6.8 | 226 | 55 | 1890 | 1.13
5 | PR100016_TG010205_V0200 | occ | 2020-11-20T14:02:20 | 7.8 | 253 | 53 | 1897 | 1.01
6 | PR100016_TG010206_V0200 | occ | 2020-11-21T16:14:20 | 6.8 | 236 | 57 | 1898 | 1.10
7 | PR100016_TG010207_V0200 | occ | 2020-11-29T07:35:20 | 8.2 | 244 | 49 | 1905 | 1.10
8 | PR100016_TG010208_V0200 | occ | 2020-11-30T10:12:20 | 6.7 | 220 | 54 | 1906 | 1.16
9 | PR100016_TG010209_V0200 | occ | 2020-12-04T19:03:20 | 6.8 | 237 | 57 | 1910 | 1.05
10 | PR100016_TG010210_V0200 | occ | 2020-12-05T21:16:20 | 5.9 | 223 | 63 | 1911 | 0.96
11 | PR100016_TG010211_V0200 | occ | 2020-12-06T23:37:22 | 7.1 | 251 | 58 | 1912 | 1.05
12 | PR100016_TG010212_V0200 | occ | 2020-12-09T03:21:20 | 6.8 | 223 | 54 | 1914 | 1.41
13 | PR100013_TG001201_V0200 | tra | 2021-01-12T11:10:20 | 8.9 | 331 | 62 | 1946 | 1.43
14 | PR100013_TG001701_V0200 | tra | 2021-11-01T00:53:20 | 9.1 | 283 | 51 | 2214 | 1.11
15 | PR100013_TG001702_V0200 | tra | 2021-11-10T19:40:22 | 9.8 | 303 | 51 | 2223 | 1.01