11institutetext:  INAF, Osservatorio Astrofisico di Catania, Via S. Sofia 78,
95123 Catania, Italy 22institutetext:  Observatoire Astronomique de
l’Université de Genève, Chemin Pegasi 51, CH-1290 Versoix, Switzerland
33institutetext:  ETH Zurich, Department of Physics, Wolfgang-Pauli-Strasse 2,
CH-8093 Zurich, Switzerland 44institutetext:  Cavendish Laboratory, JJ Thomson
Avenue, Cambridge CB3 0HE, UK 55institutetext:  Physikalisches Institut,
University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland
66institutetext:  Instituto de Astrofisica e Ciencias do Espaco, Universidade
do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal 77institutetext:
Department of Astronomy, Stockholm University, AlbaNova University Center,
10691 Stockholm, Sweden 88institutetext:  Centre for Exoplanet Science, SUPA
School of Physics and Astronomy, University of St Andrews, North Haugh, St
Andrews KY16 9SS, UK 99institutetext:  Aix Marseille Univ, CNRS, CNES, LAM, 38
rue Frédéric Joliot-Curie, 13388 Marseille, France 1010institutetext:  Space
sciences, Technologies and Astrophysics Research (STAR) Institute, Université
de Liège, Allée du 6 Août 19C, 4000 Liège, Belgium 1111institutetext:  Center
for Space and Habitability, University of Bern, Gesellschaftsstrasse 6, 3012
Bern, Switzerland 1212institutetext:  Instituto de Astrofisica de Canarias,
Via Lactea s/n, 38200 La Laguna, Tenerife, Spain 1313institutetext:
Departamento de Astrofisica, Universidad de La Laguna, Astrofísico Francisco
Sanchez s/n, 38206 La Laguna, Tenerife, Spain 1414institutetext:  Institut de
Ciencies de l’Espai (ICE, CSIC), Campus UAB, Can Magrans s/n, 08193
Bellaterra, Spain 1515institutetext:  Institut d’Estudis Espacials de
Catalunya (IEEC), Gran Capità 2-4, 08034 Barcelona, Spain 1616institutetext:
Admatis, 5. Kandó Kálmán Street, 3534 Miskolc, Hungary 1717institutetext:
Depto. de Astrofisica, Centro de Astrobiologia (CSIC-INTA), ESAC campus, 28692
Villanueva de la Cañada (Madrid), Spain 1818institutetext:  Departamento de
Fisica e Astronomia, Faculdade de Ciencias, Universidade do Porto, Rua do
Campo Alegre, 4169-007 Porto, Portugal 1919institutetext:  Space Research
Institute, Austrian Academy of Sciences, Schmiedlstrasse 6, A-8042 Graz,
Austria 2020institutetext:  Université Grenoble Alpes, CNRS, IPAG, 38000
Grenoble, France 2121institutetext:  INAF, Osservatorio Astronomico di Padova,
Vicolo dell’Osservatorio 5, 35122 Padova, Italy 2222institutetext:  Université
de Paris Cité, Institut de physique du globe de Paris, CNRS, 1 Rue Jussieu,
F-75005 Paris, France 2323institutetext:  ESTEC, European Space Agency,
Keplerlaan 1, 2201AZ, Noordwijk, NL 2424institutetext:  Institute of Planetary
Research, German Aerospace Center (DLR), Rutherfordstrasse 2, 12489 Berlin,
Germany 2525institutetext:  INAF, Osservatorio Astrofisico di Torino, Via
Osservatorio, 20, I-10025 Pino Torinese To, Italy 2626institutetext:  Centre
for Mathematical Sciences, Lund University, Box 118, 221 00 Lund, Sweden
2727institutetext:  Astrobiology Research Unit, Université de Liège, Allée du
6 Août 19C, B-4000 Liège, Belgium 2828institutetext:  Centre Vie dans
l’Univers, Faculté des sciences, Université de Genève, Quai Ernest-Ansermet
30, CH-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, Universita 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:  Science and Operations Department - Science
Division (SCI-SC), Directorate of Science, European Space Agency (ESA),
European Space Research and Technology Centre (ESTEC), Keplerlaan 1, 2201-AZ
Noordwijk, The Netherlands 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:  German Aerospace Center (DLR), Institute of Optical Sensor
Systems, Rutherfordstraße 2, 12489 Berlin 3737institutetext:  IMCCE, UMR8028
CNRS, Observatoire de Paris, PSL Univ., Sorbonne Univ., 77 av. Denfert-
Rochereau, 75014 Paris, France 3838institutetext:  Institut d’astrophysique de
Paris, UMR7095 CNRS, Université Pierre & Marie Curie, 98bis blvd. Arago, 75014
Paris, France 3939institutetext:  Astrophysics Group, Lennard Jones Building,
Keele University, Staffordshire, ST5 5BG, United Kingdom 4040institutetext:
Institute of Optical Sensor Systems, German Aerospace Center (DLR),
Rutherfordstrasse 2, 12489 Berlin, Germany 4141institutetext:  Dipartimento di
Fisica e Astronomia ”Galileo Galilei”, Universita degli Studi di Padova,
Vicolo dell’Osservatorio 3, 35122 Padova, Italy 4242institutetext:  Department
of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United
Kingdom 4343institutetext:  Zentrum für Astronomie und Astrophysik, Technische
Universität Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany
4444institutetext:  Institut fuer Geologische Wissenschaften, Freie
Universitaet Berlin, Maltheserstrasse 74-100,12249 Berlin, Germany
4545institutetext:  Department of Astrophysics, University of Vienna,
Tuerkenschanzstrasse 17, 1180 Vienna, Austria 4646institutetext:
Physikalisches Institut, University of Bern, Sidlerstrasse 5, 3012 Bern,
Switzerland 4747institutetext:  Université de Liège, Allée du 6 Août 19C, 4000
Liège, Belgium 4848institutetext:  ELTE Eötvös Loránd University, Gothard
Astrophysical Observatory, 9700 Szombathely, Szent Imre h. u. 112, Hungary
4949institutetext:  MTA-ELTE Exoplanet Research Group, 9700 Szombathely, Szent
Imre h. u. 112, Hungary 5050institutetext:  Institute of Astronomy, University
of Cambridge, Madingley Road, Cambridge, CB3 0HA, United Kingdom
5151institutetext:  Institute for Theoretical Physics and Computational
Physics, Graz University of Technology, Petersgasse 16, 8010 Graz, Austria

# Constraining the reflective properties of WASP-178 b using CHEOPS
photometry.††thanks: The CHEOPS program ID is CH_PR100016.††thanks: The CHEOPS
photometric data used in this work are only available in electronic form at
the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or via
https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/

I. Pagano
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11 G. Scandariato
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11 V. Singh
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11 M. Lendl
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22 D. Queloz
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3344 A. E. Simon
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55 S. G. Sousa
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66 A. Brandeker
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77 A. Collier Cameron
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88 S. Sulis
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99 V. Van Grootel
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1010 T. G. Wilson
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88 Y. Alibert
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111155 R. Alonso
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12121313 G. Anglada
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14141515 T. Bárczy
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1616 D. Barrado Navascues
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1717 S. C. C. Barros
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661818 W. Baumjohann
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1919 M. Beck
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22 T. Beck 55 W. Benz
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551111 N. Billot
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22 X. Bonfils
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2020 L. Borsato
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2121 C. Broeg
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551111 G. Bruno
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11 L. Carone 1919 S. Charnoz
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2222 C. Corral van Damme 2323 Sz. Csizmadia
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2424 P. E. Cubillos 25251919 M. B. Davies
${}^{\href{https://orcid.org/0000-0001-6080-1190}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2626 M. Deleuil
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99 A. Deline 22 L. Delrez
${}^{\href{https://orcid.org/0000-0001-6108-4808}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
27271010 O. D. S. Demangeon
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661818 B.-O. Demory
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111155 D. Ehrenreich
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222828 A. Erikson 2424 A. Fortier
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551111 L. Fossati
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1919 M. Fridlund
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29293030 D. Gandolfi
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3131 M. Gillon
${}^{\href{https://orcid.org/0000-0003-1462-7739}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2727 M. Güdel 3232 M. N. Günther 2323 Ch. Helling 19195151 S. Hoyer
${}^{\href{https://orcid.org/0000-0003-3477-2466}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
99 K. G. Isaak
${}^{\href{https://orcid.org/0000-0001-8585-1717}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
3333 L. L. Kiss 34343535 E. Kopp 3636 K. W. F. Lam
${}^{\href{https://orcid.org/0000-0002-9910-6088}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2424 J. Laskar
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3737 A. Lecavelier des Etangs
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3838 D. Magrin
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2121 P. F. L. Maxted
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3939 C. Mordasini 551111 M. Munari
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11 V. Nascimbeni
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2121 G. Olofsson
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77 R. Ottensamer 3232 E. Pallé
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12121313 G. Peter
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4040 G. Piotto
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21214141 D. Pollacco 4242 R. Ragazzoni
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21214141 N. Rando 2323 H. Rauer
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242443434444 C. Reimers 4545 I. Ribas
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14141515 M. Rieder 46461111 N. C. Santos
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661818 D. Ségransan
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22 A. M. S. Smith
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2424 M. Stalport 4747 M. Steller
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1919 Gy. M. Szabó 48484949 N. Thomas 55 S. Udry
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22 J. Venturini
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22 N. A. Walton
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5050

###### Abstract

Context. Multiwavelength photometry of the secondary eclipses of extrasolar
planets is able to disentangle the reflected and thermally emitted light
radiated from the planetary dayside. This leads to the measurement of the
planetary geometric albedo $A_{g}$, which is an indicator of the presence of
clouds in the atmosphere, and the recirculation efficiency $\epsilon$, which
quantifies the energy transport within the atmosphere.

Aims. In this work we aim to measure $A_{g}$ and $\epsilon$ for the planet
WASP-178 b, a highly irradiated giant planet with an estimated equilibrium
temperature of 2450 K.

Methods. We analyzed archival spectra and the light curves collected by CHEOPS
and TESS to characterize the host WASP-178, refine the ephemeris of the system
and measure the eclipse depth in the passbands of the two respective
telescopes.

Results. We measured a marginally significant eclipse depth of 70$\pm$40 ppm
in the TESS passband and statistically significant depth of 70$\pm$20 ppm in
the CHEOPS passband.

Conclusions. Combining the eclipse depth measurement in the CHEOPS
($\lambda_{\rm eff}=6300\leavevmode\nobreak\ \AA$) and TESS ($\lambda_{\rm
eff}=8000\leavevmode\nobreak\ \AA$) passbands we constrained the dayside
brightness temperature of WASP-178 b in the 2250-2800 K interval. The
geometric albedo 0.1¡$\rm A_{g}$¡0.35 is in general agreement with the picture
of poorly reflective giant planets, while the recirculation efficiency
$\epsilon>$0.7 makes WASP-178 b an interesting laboratory to test the current
heat recirculation models.

###### Key Words.:

techniques: photometric – planets and satellites: atmospheres – planets and
satellites: detection – planets and satellites: gaseous planets – planets and
satellites: individual: WASP-178 b

##  1 Introduction

In the last few years we have gained access to a detailed characterization of
exoplanets. Ground-based and space-born instrumentation have progressed such
to allow the analysis of the atmosphere of exoplanets, in terms of
thermodynamic state and chemical composition (Sing et al. 2016; Giacobbe et
al. 2021). In particular, current photometric facilities allow to observe the
secondary eclipse of giant exoplanets in close orbits (Stevenson et al. 2017;
Lendl et al. 2020; Wong et al. 2020; Singh et al. 2022). In this research
area, CHEOPS (Benz et al. 2021) is bringing a valuable contribution given its
ultra-high photometric accuracy capabilities (Lendl et al. 2020; Deline et al.
2022; Hooton et al. 2022; Brandeker et al. 2022; Parviainen et al. 2022;
Scandariato et al. 2022; Demory et al. 2023).

The depth of the eclipse quantifies the brightness of the planetary dayside
with respect to its parent star. Depending on the temperature of the planet
and the photometric band used for the observations, the eclipse depth provides
insight into the reflectivity and energy redistribution of the atmosphere.

WASP-178 b (HD 134004 b) is a hot Jupiter discovered by Hellier et al. (2019)
and independently announced by Rodríguez Martínez et al. (2020) as KELT-26 b.
It orbits an A1 IV-V dwarf star (Table 1) at a distance of 7 stellar radii:
these features place WASP-178 b among the planets that receive the highest
energy budget from their respective host stars. It is thus an interesting
laboratory to test atmospheric models in the presence of extreme irradiation.

In this paper we analyze the eclipse depths measured using the data collected
by the CHEOPS and TESS space telescopes. It is organized as follows. Sect. 2
describes the data acquisition and reduction, while in Sect. 3 we describe how
we derive the stellar radius, mass and age. In Sect. 4 we update the orbital
solution of WASP-178 b, put an upper limit on the eclipse depth using TESS
data and get a significant detection using CHEOPS photometry. Finally, in
Sect. 5 we discuss the implication of the extracted eclipse signal in terms of
geometric albedo and atmospheric recirculation efficiency.

Table 1: Stellar and system parameters. Parameter | Symbol | Units | Value | Ref.
---|---|---|---|---
V mag |  |  | 9.95 | Hellier et al. (2019)
Spectral Type |  |  | A1 IV-V | Hellier et al. (2019)
Effective temperature | $\rm T_{eff}$ | K | $9350\pm 150$ | Hellier et al. (2019)
Surface gravity | $\log g$ | — | $4.35\pm 0.15$ | Hellier et al. (2019)
Metallicity | [Fe/H] | — | $0.21\pm 0.16$ | Hellier et al. (2019)
Projected rotational velocity | $v\sin{i}$ | km/s | $8.2\pm 0.6$ | Hellier et al. (2019)
Stellar radius | $\rm R_{\star}$ | $\rm R_{\sun}$ | $1.722\pm 0.020$ | this work
Stellar mass | $\rm M_{\star}$ | $\rm M_{\sun}$ | $2.169_{-0.089}^{+0.083}$ | this work
Stellar age | $t_{\star}$ | Gyr | $0.05_{-0.05}^{+0.06}$ | this work
Radial velocity semi-amplitude | $\rm K_{RV}$ | m/s | $139\pm 9$ | Hellier et al. (2019)

##  2 Observations and data reduction

###  2.1 TESS observations

TESS (Ricker et al. 2014) observed the WASP-178 system in sector 11 (from 2019
April 23 to 2019 May 20) with a 30 min cadence and in sector 38 (from 2020
April 29 to 2020 May 26) with a 2 min cadence. Rodríguez Martínez et al.
(2020) claimed a modulation with period of 0.369526 days in the photometry of
sector 11, interpreting it as a $\delta$ Scuti pulsation mode. Later,
Lothringer et al. (2022) found out that the TESS photometry is heavily
contaminated by the background eclipsing binary ASASSN-V J150908.07-424253.6,
located at a projected distance of 50.4″ from the target, whose orbital period
matches the periodicity in the photometry of WASP-178. In order to optimize
the photometric extraction and avoid the background variable contamination, we
extracted the Light Curve (LC) corresponding to each pixel in the aperture
mask defined by the TESS pipeline, and we excluded the pixels for which the
periodogram shows a peak at the same orbital period of the binary system. We
thus re-extracted the photometry by retrieving the calibrated Full Frame
Imagess and Target Pixel Files for sector 11 and 38 respectively. We used a
custom extraction pipeline combined with the default quality bitmask. The
extracted LCs were then background-corrected after determining the sky level
using custom background masks on the FFIs and TPFs. A principal component
analysis was then conducted on the pixels in these background masks across all
frames in order to measure the flux contribution of scattered light in the
TESS cameras. We detrended the data using these principal components as a
linear model. Lastly, we further corrected for any photometric trends due to
spacecraft pointing jitter by retrieving the Co-trending Basis Vectorss and
two-second cadence engineering quaternion measurements for the specific
cameras WASP-178 was observed in. For each sector we computed the mean of the
quaternions over the length of a science observation, i.e. 30 min for the FFIs
and 2 min for the TPFs. We subsequently used these vectors along with the CBVs
to detrend the TESS photometry in a similar manner as was done in Delrez et
al. (2021).

We clip out the photometry taken before BJD 2459334.7 and in the windows
2458610–2458614.5 and 2459346–2459348: these data present artificial trends
due to the momentum dumps of the telescope. We also visually identified and
excluded a short bump in the LC between BJD 2459337.6 and 2459337.9, most
likely an instrumental artifact or some short term photometric variability
feature. The final LCs cover 7 and 8 transits for sector 11 and 38
respectively.

###  2.2 CHEOPS observations

CHEOPS (Benz et al. 2021) observed the WASP-178 system during six secondary
eclipses of the planet WASP-178 b with a cadence of 60 s. The aim is the
measurement of the eclipse depth and derivation of the brightness of the
planet in the CHEOPS passband (3500–11000 Å). Each visit is $\sim$12 hr long,
scheduled in order to bracket the eclipse and equally long pre- and post-
eclipse photometry. The logbook of the observations, which are part of the
CHEOPS Guaranteed Time Observation (GTO) program, is summarized in Table 2.

Table 2: Logbook of the CHEOPS observations of WASP-178. The filekey is the unique identifier associated with each dataset processed by the CHEOPS DRP. Filekey | Start time ]UT] | Visit duration [hr] | Exposure time [s] | N. frames | Efficiency [%]
---|---|---|---|---|---
PR100016_TG014201_V0200 | 2021-04-05 12:49:30 | 11.54 | 60.0 | 427 | 61.7
PR100016_TG014202_V0200 | 2021-04-15 13:54:09 | 12.14 | 60.0 | 473 | 64.9
PR100016_TG014203_V0200 | 2021-05-02 06:56:09 | 11.54 | 60.0 | 465 | 67.2
PR100016_TG014204_V0200 | 2021-05-22 08:52:09 | 11.54 | 60.0 | 431 | 62.3
PR100016_TG014205_V0200 | 2021-05-28 23:49:09 | 11.44 | 60.0 | 415 | 60.5
PR100016_TG014206_V0200 | 2022-05-21 22:33:49 | 11.54 | 60.0 | 445 | 64.3

The data were reduced using version 13 of the CHEOPS Data Reduction Pipeline
(DRP) (Hoyer et al. 2020). This pipeline performs the standard calibration
steps (bias, gain, non-linearity, dark current and flat fielding) and corrects
for environmental effects (cosmic rays, smearing trails from nearby stars, and
background) before the photometric extraction.

As for the case of TESS LCs, the aperture photometry is significantly
contaminated by ASASSN-V J150908.07-424253. To decontaminate the LC of
WASP-178 we performed the photometric extraction using a modified version of
the package PIPE111https://github.com/alphapsa/PIPE (Brandeker et al. 2022;
Morris et al. 2021; Szabó et al. 2021), upgraded in order to compute the
simultaneous Point Spread Function (PSF) photometry of the target and the
background contaminant.

The extracted LCs present gaps due to Earth occultations which cover $\sim$40%
of the visits. The exposures close to the gaps are characterized by high value
of the background flux, due to stray-light from Earth. The corresponding flux
measurements are thus affected by a larger photometric scatter. To avoid these
low quality data, we applied a 5$\sigma$ clipping to the background
measurements: this selection criterion removes less than 20% of the data with
the highest background counts. Finally, for a better outlier rejection, we
smoothed the data with a Savitzky-Golay filter, computed the residuals with
respect to the filtered LC and 5$\sigma$-clipped the outliers. This last
rejection criterion excludes a handful of data points in each LC.

Finally, we normalize the unsmoothed LCs by the median value of the
photometry. These normalized LCs are publicly available at CDS PUT LINK.

##  3 Stellar radius, mass and age

We use a Infra-Red Flux Method (IRFM) in a Markov Chain Monte Carlo (MCMC)
approach to determine the stellar radius of WASP-178 (Blackwell & Shallis
1977; Schanche et al. 2020). We downloaded the broadband fluxes and
uncertainties from the most recent data releases for the following bandpasses:
Gaia G, GBP, and GRP, 2MASS J, H, and K, and WISE W1 and W2 (Skrutskie et al.
2006; Wright et al. 2010; Gaia Collaboration et al. 2021). Then we matched the
observed photometry with synthetic photometry computed in the same bandpasses
by using the theoretical stellar Spectral Energy Distributions corresponding
to stellar atmospheric parameters (Table 1). The fit is performed in a
Bayesian framework and, to account for uncertainties in stellar atmospheric
modeling, we averaged the atlas (Kurucz 1993; Castelli & Kurucz 2003) and
phoenix (Allard 2014) catalogs to produce weighted averaged posterior
distributions. This process yields a $R_{\star}=1.722\pm 0.020\,R_{\odot}$.

Assuming the $T_{\mathrm{eff}}$, [Fe/H], and $R_{\star}$ listed in Table 1 as
input parameters, we also computed the stellar mass $M_{\star}$ and age
$t_{\star}$ by using two different sets of stellar evolutionary models. In
detail, we employed the isochrone placement algorithm (Bonfanti et al. 2015,
2016) and its capability of interpolating within pre-computed grids of
PARSEC222PAdova and TRieste Stellar Evolutionary Code:
http://stev.oapd.inaf.it/cgi-bin/cmd v1.2S (Marigo et al. 2017) for retrieving
a first pair of mass and age estimates. A second pair of mass and age values,
instead, was computed by CLES (Code Liègeois d’Évolution Stellaire; Scuflaire
et al. 2008), which generates the best-fit evolutionary track of the star by
entering the input parameters into the Levenberg-Marquadt minimisation scheme
as described in Salmon et al. (2021). After carefully checking the mutual
consistency of the two respective pairs of estimates through the
$\chi^{2}$-based criterion broadly discussed in Bonfanti et al. (2021), we
finally merged the outcome distributions and we obtained
$M_{\star}=2.169_{-0.089}^{+0.083}\,M_{\odot}$ and $t_{\star}=50_{-50}^{+60}$
Myr.

##  4 Light curve analysis

###  4.1 TESS photometry

To compare in a homogeneous way the LCs of sectors 11 and 38, we rebinned the
photometry of sector 38 to 30 min. The standard deviation of the TESS LCs,
after transits and secondary eclipses are clipped, is of $\sim$300 ppm for
both sectors, and in both cases the photometric uncertainty can account for
only $\sim$70% of the variance. This indicates that there is some noise in the
LCs due to astrophysical signals and/or instrumental leftovers. To investigate
if the unexplained variance is related to periodic signals, for both sectors
we compute the Generalized Lomb-Scargle (GLS) periodogram (Zechmeister &
Kürster 2009) of the out-of-transit and out-of-eclipse photometry (Fig. 1).
For sector 11 we did not find any significant periodic signal, while for
sector 38 there is a strong peak at frequency $\rm\nu=0.304\pm
0.003\leavevmode\nobreak\ d^{-1}$ with False Alarm Probability (FAP) lower
than 0.1%. The amplitude of the corresponding sinusoidal signal is $90\pm 10$
ppm.

Figure 1: Out-of-transit and out-of-eclipse 30 min cadence TESS photometry of
WASP-178 during sector 11 (left panel) and sector 38 (central panel). In each
panel, the dashed vertical lines mark the planetary transit, while the solid
blue line is a smoothing of the data points to emphasize the correlated noise.
The right panel shows the GLS periodogram of sector 11 and 38 (top and bottom
box respectively). In each box, we report the bootstrap-computed 0.1% and 1%
FAP levels (horizontal dashes) and the planetary orbital period (vertical red
dotted line).

The periodicity detected in sector 38 is consistent within uncertainties with
the planetary orbital period. Nonetheless, we exclude that this signal is of
planetary origin because it remained undetected in sector 11 and is not
coherent from orbit to orbit in sector 38. This signal might be due to stellar
rotation, and is consistent with the typical amplitude reported by Balona
(2011) for A-type stars. Its periodicity would correspond to the $v\sin
i_{\star}$ in Table 1 if the inclination of the stellar rotation axis is $\sim
15^{\circ}$. This speculation supports the hypothesis of Rodríguez Martínez et
al. (2020) that the star is seen nearly pole-on. A significant misalignment
between the stellar rotation axis and the planetary orbit axis is not
surprising, both because of its young age (no time for realignment to occur),
and the stellar temperature. As a matter of fact, hot stars (Teff¿6200 K) are
observed to often have large misalignments, which is thought to be due to
their lack of a convective zone (Winn et al. 2010), needed to tidally align
the planetary orbit with the stellar spin axis. Furthermore, Albrecht et al.
(2021) found that misaligned orbits are most often polar, or close to polar,
as seems to be the case of WASP-178 b. We expand the discussion on the
photometric variability and the orientation of the stellar rotation axis in
Appendix B.

The fact that the period of the correlated noise is similar to the orbital
period of WASP-178 b makes it difficult to extract the complete planetary
Phase Curve (PC) signal. We thus first tried a simpler and more robust
approach to analyze the planetary transits and eclipses (Sect. 4.1.1), then we
attempted a more complex analysis framework aimed at retrieving the full PC of
WASP-178 b (Sect. 4.1.2).

####  4.1.1 Fit of transits and eclipses

We computed the ephemeris of the planet by trimming segments of the LCs
centered on the transit events (7 transits in sector 11 and 8 transits in
sector 8) and as wide as 3 times the transit duration. To further constrain
the ephemeris of WASP-178 b, we included in our analysis the WASP-South
photometry (2006 May – 2014 Aug) and EulerCAM I-band photometry (2018 Mar 26)
presented in Hellier et al. (2019). Also for the case of the WASP-South
photometry, we trimmed the LC around the transits and kept the 24 intervals
containing more than 20 data points.

We fit the data using the same Bayesian approach described in Scandariato et
al. (2022). In summary, it consists in the fit of a model - in a likelihood-
maximization framework - which includes the transits, a linear term for each
transit to detrend against stellar/instrumental systematics and a jitter term
to fit the white noise not included in the photometric uncertainties. The
transit profile is formalized using the quadratic limb darkening (LD) law
indicated by Mandel & Agol (2002) with the reparametrization of the LD
coefficients suggested by Kipping (2013). For TESS sector 11 the model is
rebinned to the same 30 min cadence of the data. The likelihood maximization
is performed with MCMC using the python emcee package version 3.1.3 (Foreman-
Mackey et al. 2013), using a number of samplings long enough to ensure
convergence. We used flat priors for all the fitting parameters but the
stellar density, for which we used the Gaussian prior $N(0.43,0.02)$ given by
the stellar mass and radius in Table 1. We also used the Gaussian priors for
the Limb Darkening (LD) coefficients given by the LDTk package
333https://github.com/hpparvi/ldtk. For simplicity, we use the same LD
coefficients for the three datasets. This is motivated by the fact that the
WASP-South photometry is not accurate enough to constrain the LD profile, and
that the TESS passband is basically centered on the standard I band, thus the
same LD profile is expected for the TESS and EulerCAM LCs. Since the model
fitting is computationally demanding, we ran the code in the HOTCAT computing
infrastructure (Bertocco et al. 2020; Taffoni et al. 2020).

The result of the model fitting is listed in Table 3. The orbital solution we
derived is consistent with previous studies (Hellier et al. 2019; Rodríguez
Martínez et al. 2020) within 5$\sigma$. The best fit model, corresponding to
the Maximum A Posteriori (MAP) parameters, is over-plotted to the phase-folded
data in Fig 2, where we rebinned the model and the LC of sector 38 to the same
30 min cadence of sector 11 (we do not show the WASP-South and EulerCAM
photometry not to clutter the plot). In Fig. 9 we show the corner plot of the
system’s parameters from the fit of the transit LCs.

Table 3: Model parameters for the fit of the TESS WASP-South and EulerCAM
data.

Jump parameters | Symbol | Units | MAP | C.I.aa$a$Uncertainties expressed in parentheses refer to the last digit(s). | Prior
---|---|---|---|---|---
Time of transit | $T_{0}$ | BJDTDB-2400000 | 56612.6581 | 56612.6581(3) |  $U$(56612.6, 56612.7)
Orbital frequency | $\nu_{\rm orb}$ | days-1 | 0.29896856 | 0.29896855(3) |  $U$(0.2989,0.2990)
Stellar density | $\rho_{\star}$ | $\rho_{\sun}$ | 0.45 | 0.44(1) |  $N$(0.43,0.02)
Radii ratiobb$b$Fitting WASP-South, EulerCAM and TESS data altogether. | $R_{p}/R_{\star}$ | — | 0.1124 | 0.1125(2) |  $U$(0.05,0.12)
Radii ratiocc$c$Fitting TESS sector 11 only. | $R_{p}/R_{\star}$ | — | 0.1109 | 0.1108(4) |  $U$(0.05,0.12)
Radii ratiodd$d$Fitting TESS sector 38 only. | $R_{p}/R_{\star}$ | — | 0.1141 | 0.1141(4) |  $U$(0.05,0.12)
Impact parameter | b | — | 0.52 | 0.51(1) |  $U$(0.01,0.9)
First LD coef.bb$b$Fitting WASP-South, EulerCAM and TESS data altogether. | q1 | — | 0.147 | 0.147(8) |  $N$(0.133,0.014)
Second LD coef.bb$b$Fitting WASP-South, EulerCAM and TESS data altogether. | q2 | — | 0.344 | 0.34(2) |  $N$(0.333,0.023)
Secondary eclipse depth | $\delta_{\rm ecl}$ | ppm | 70 | 70(40) |  $U$(0,400)
Derived parameters | Symbol | Units | MAP | C.I. |
Planetary radiusbb$b$Fitting WASP-South, EulerCAM and TESS data altogether. | Rp | RJ | 1.88 | 1.88(2) | including the stellar radius uncertainty
Planetary radiuscc$c$Fitting TESS sector 11 only. | Rp | RJ | 1.86 | 1.85(2) | including the stellar radius uncertainty
Planetary radiusdd$d$Fitting TESS sector 38 only. | Rp | RJ | 1.91 | 1.91(2) | including the stellar radius uncertainty
Orbital period | Porb | day | 3.3448332 | 3.3448333(4) |
Transit duration | T14 | hr | 3.488 | 3.488(8) |
Scaled semi-major axis | $a/R_{\star}$ | — | 7.20 | 7.19(6) |
Orbital inclination | $i$ | degrees | 85.8 | 85.8(1) |

444

Since the TESS LCs of sector 11 and 38 show different levels of variability,
we investigated any seasonal dependence of the apparent planet-to-star radius
ratio. We used the same Bayesian framework as above, where we fix the
ephemeris of WASP-178. The planet-to-star radius ratio we derived is
0.1108$\pm$0.0004 for sector 11 and 0.1141$\pm$0.0004 for sector 38. The
difference is thus 0.0033$\pm$0.0006, that is we found different transit
depths with a 5.5$\sigma$ significance. In particular, we remark that the
planet looks larger in sector 38, where the rotation signal is stronger. We
thus speculate that WASP-178 was in a low variability state during sector 11,
while one year later ($\sim$80 stellar rotations) during sector 38 the stellar
surface hosted dark spots in co-rotation with the star. This hypothesis is
consistent with Hümmerich et al. (2018), according to which magnetic
chemically peculiar stars may show complex photometric patterns due to surface
inhomogeneities.

We used a similar framework to extract the secondary eclipse signal of
WASP-178: we simultaneously fit segments of the TESS LCs centered on the
planetary eclipses keeping fixed the ephemeris of WASP-178 to the values
listed in Table 3. We did not attempt the disjoint analysis of the two sectors
as the expected eclipse depth is of the order of $\sim$100 ppm (see below),
and the photometric precision of the LCs is not good enough to appreciate
differences in the eclipse depth with enough statistical evidence. The only
free planetary parameter of the model is thus the eclipse depth, which turned
out to be $\delta_{\rm ecl}=70\pm 40$ ppm (MAP: 70 ppm). The detrended phase-
folded eclipses are shown in Fig 2 together with the MAP model.

Figure 2: Best fit of the transits and eclipses observed by TESS. Left -
Detrended and phase-folded planetary transits (top panel). The photometry of
sector 11 and 38 are shown with different colors to emphasize the difference
in transit depth (the photometry of sectors 11 and 38 is systematically offset
respectively upwards and downwards with respect to the bestfit transit
profile). The solid blue line is the best fit model, while the cyan lines
represent 100 models corresponding to random samples of the MCMC fit. For the
sake of comparison, the photometry of sector 38 and the theoretical models are
rebinned to the same 30 min cadence of sector 11. The black dots represent the
rebinned photometry. The corresponding O-C diagram is shown in the bottom
panel. Right - Same as in the left panel, but centered on the eclipse. For
clarity, we do not mark the two sectors with different colors.

####  4.1.2 Fit of the phase curve

As a more advanced data analysis, we jointly fit the two TESS sectors to
extract the full planetary PC (for a homogeneous analysis, we rebinned sector
38 to the same 30 min cadence of sector 11). The fitting model now includes
the transits, secondary eclipses, the planetary PC, a Gaussian Process (GP) to
fit the correlated noise in the data and, for each TESS sector, a long-term
linear trend and a jitter term. Given the system parameters listed in Table 1,
the expected amplitude of ellipsoidal variations (Morris 1985) and Doppler
boosting (Barclay et al. 2012) in the planetary PC are 2.5 ppm and 1 ppm
respectively, out of reach for the TESS photometry. Hence, for the sake of
simplicity, we do not include them in the model fitting.

We jointly fit the two TESS sectors, starting with the simplest model where
the out of transit PC is flat (that is, we assumed that the planetary PC is
not detectable) and the putative stellar rotation is modeled as GP with the
SHO kernel555Provided by the celerite2 python package version 0.2.1 (Foreman-
Mackey et al. 2017; Foreman-Mackey 2018), indicated for quasi-periodic
signals. Checking the residuals of the fit we notice that some aperiodic
correlated noise is left. We thus increased the complexity of the GP model by
adding a Matérn3/2 kernel to capture the remaining correlated noise adding
only two free parameters to the model. This composite GP model turned out to
ensure better convergence to the fit of the model.

Finally, we also included the planetary PC. Assuming that the planet is
basically composed of two homogeneous dayside and nightside, the PC is in
principle the combination of three components: the PC due to reflected light
(with amplitude $A_{\rm refl}$), the PC of the planetary night side (with
amplitude $A_{\rm n}$) and the PC of the planetary day side, whose amplitude
$A_{\rm d}$ is parameterized such to be larger than $A_{\rm n}$ by an
increment $\delta_{\rm d}$ (this parametrization avoids the nonphysical case
of a planetary night side brighter than the dayside). The three terms are
modeled assuming Lambert’s cosine law.

For the reflected PC, the amplitude can be expressed as:

$A_{\rm refl}=A_{\rm g}\left(\frac{R_{\rm p}}{a}\right)^{2},$ (1)

where $A_{\rm g}$ is the planetary geometric albedo (Seager 2010). Assuming a
Lambertian reflective planetary surface, the geometric albedo is fixed by the
Bond albedo $A_{\rm B}$ by $A_{\rm g}$=$\frac{2}{3}A_{\rm B}$. In the
optimistic case of a perfectly reflecting body ($A_{\rm B}$=1), the maximum
amplitude for the reflected PC is thus $A_{\rm refl}=167\pm 5$ ppm, where we
have used the system parameters in Table 3.

With a slight re-adaptation of the formalism in Seager (2010), the amplitude
thermal component $A_{\rm therm}$ can be expressed as:

$A_{\rm therm}=\left(\frac{R_{\rm
p}}{R_{\star}}\right)^{2}\frac{\int\eta(\lambda)B(\lambda,T)d\lambda}{\int\eta(\lambda)I_{\star}(\lambda,T_{\rm
eff})d\lambda},$ (2)

where $\eta(\lambda)$ is the optical throughput of the telescope and
$I_{\star}(\lambda,T_{\rm eff})$ is the expected stellar intensity computed
with the NextGen model (Hauschildt et al. 1999) corresponding to the effective
temperature $T_{\rm eff}$. For the planetary thermal emission we lack the
infrared information which can constrain the emission spectrum. We thus assume
a black body spectrum $B(\lambda,T)$ at temperature $T$.

The amplitude of the thermal emission from the dayside and the nightside
depends on the respective temperatures, which we estimated following Cowan &
Agol (2011). The expected temperature of the substellar point of WASP-178 b
is:

$T_{\rm 0}=T_{\rm eff}\sqrt{\frac{R_{\star}}{a}}=3530\pm
60\leavevmode\nobreak\ {\rm K},$ (3)

where we used the stellar effective temperature in Table 1 and the scaled
semi-major axis in Table 3.

The nightside temperature $T_{\rm n}$ and the dayside temperature $T_{\rm d}$
depend on the Bond albedo $A_{B}$ and heat recirculation coefficient
$\epsilon$:

$\displaystyle T_{\rm n}$ $\displaystyle=T_{\rm 0}\left(1-A_{\rm
B}\right)^{1/4}\left(\frac{\epsilon}{4}\right)^{1/4}$ (4) $\displaystyle
T_{\rm d}$ $\displaystyle=T_{\rm 0}\left(1-A_{\rm
B}\right)^{1/4}\left(\frac{2}{3}-\frac{5}{12}\epsilon\right)^{1/4}$ (5)

The highest dayside temperature, corresponding to $A_{\rm B}$=0 and no heat
recirculation ($\epsilon=0$), is:

$T_{\rm d}=T_{\rm 0}\left(\frac{2}{3}\right)^{1/4}=3190\pm
60\leavevmode\nobreak\ {\rm K},$ (6)

while the maximum nightside temperature, obtained in the case of full energy
recirculation ($\epsilon=1$), is:

$T_{\rm n}=T_{\rm 0}\left(\frac{1}{4}\right)^{1/4}=2500\pm
40\leavevmode\nobreak\ {\rm K}.$ (7)

These maximum temperatures lead to the upper limit on the dayside ($A_{\rm
d}$) and nightside ($A_{\rm n}$) thermal emission amplitude of:

$A_{\rm d}=225\pm 27\leavevmode\nobreak\ {\rm ppm};A_{\rm n}=48\pm
7\leavevmode\nobreak\ {\rm ppm}.$ (8)

In Fig. 3 we plot the upper limits for the three PCs discussed so far. The
figure shows that the three components may reach amplitudes of comparable
orders of magnitude, meaning that none of them can in principle be neglected
in the extraction of the planetary PC. We note that the two components
belonging to the dayside (reflection and thermal emission) have the same
shape, thus it is not possible to disentangle them. To simplify the model and
avoid the degeneracy between dayside reflection and emission, we thus
artificially set $A_{\rm refl}=0$ and let $A_{\rm d}$ absorb the whole signal
belonging to the planetary dayside.

Figure 3: Upper limits on the reflected PC, dayside thermal PC and nightside
thermal PC for WASP-178 b.

The Akaike Information Criterion (AIC, Burnham & Anderson 2002) favors the
model that includes the planetary PCs. The simpler model with flat out-of-
transit PC has a relative likelihood of 55% and cannot be rejected. The
posterior distributions of the common parameters (system ephemeris and planet-
to-star radii ratio) between the two models do not differ significantly.

For both models, the SHO GP has a frequency $\nu\sim$0.30 d-1, an amplitude
$\sim$70 ppm and a timescale $\sim$11 d. Frequency $\nu$ and the amplitude of
the quasi-periodic GP are consistent with the frequency and amplitude of the
periodic signal found in the periodogram (Sect. 4.1). While the correspondence
between the orbital period of WASP-178 b and the periodicity of the GP
suggests that the red noise in the data is of planetary origin, we do not have
other strong evidence that support this hypothesis. On the contrary, the
planetary origin seems odd for the reasons discussed in Sect. 4.1. The
aperiodic red noise fitted as a Matérn3/2 GP has an amplitude $\sim$150 ppm
and a timescale of $\sim$50 min. Despite the aperiodic red noise evolves on
timescales shorter than the transit duration ($T_{\rm 14}$=3.53 hr), the
retrieved posterior distributions of the orbital parameters are similar to the
ones obtained in Sect. 4.1.1.

Regarding the retrieval of the planetary PCs, we derived an upper limit of
$A_{\rm n}<$174 ppm at the 99.9% confidence level, while for the dayside we
derived a 1$\sigma$ confidence band of 110$\pm$40 ppm which includes both the
reflection and thermal emission components. We remark that the $A_{\rm d}$
parameter corresponds in our model to the eclipse depth $\delta_{\rm ecl}$ and
is consistent within uncertainties with the estimate reported in Table 3.

The fit of the full TESS LC thus confirms the results previously obtained in
Sect. 4.1.1. Given the similarities between the results obtained with the two
approaches, we preferred the former one as it is less prone to interference
between the data detrending and the extraction of the orbital parameters and
the eclipse depth.

###  4.2 CHEOPS photometry

To analyze the CHEOPS LCs we use the same approach as for the fit of the
eclipses observed by TESS (Sect. 4.1.1), with an additional module in the
fitting model that takes into account the systematics in the CHEOPS data. As a
matter of fact, CHEOPS photometry is affected by variable contamination from
background stars in the field of view (e.g. Lendl et al. 2020; Deline et al.
2022; Hooton et al. 2022; Wilson et al. 2022; Scandariato et al. 2022). This
variability is due to the interplay of the asymmetric PSF with the rotation of
the field of view (Benz et al. 2021). PIPE by design uses nominal magnitudes
and coordinates of the stars in the field to fit the stellar PSFs, but
residual correlated noise is present in the LCs due to inaccuracies in the
assumptions. This signal is phased with the roll angle of the CHEOPS satellite
and, in the case of WASP-178, is clearly visible in the GLS periodogram of the
data together with its harmonics. To remove this signal, we included in our
algorithm a module which fits independently for each visit the harmonic
expansions of the telecope’s orbital period and its harmonics. A posteriori,
we found that the roll angle phased modulation is adequately suppressed (that
is, there is no peak in the periodograms at the frequencies in the harmonic
series) if we include in the model the fundamental harmonic and its first two
harmonics.

We also noticed that the photometry is significantly correlated with the
coordinates of the centroid of the PSF on the detector. To decorrelate against
this instrumental jitter we thus included in the model a bi-linear function of
the centroid coordinates.

Finally, to take into account a white noise not included in the formal
photometric uncertainties, we add to our model a diagonal GP kernel of the
form:

$k(t_{i},t_{j})=j_{v}^{2}\delta_{i,j},$ (9)

with an independent jitter term $j_{v}$ for each CHEOPS visit.

As for the fit of the TESS data (Sect. 4.1.1), we search the bestfit
parameters through likelihood-maximization in a MCMC framework, and we find
$\rm\delta_{ecl}=70\pm 20$ ppm. The ephemeris of the planet were locked to the
MAP values listed in Table 3, which guarantee an uncertainty on the eclipse
time less than 23 s for each CHEOPS visit. The phase-folded data detrended
against stellar and instrumental correlated noise are shown in Fig. 4.

Figure 4: Same as the right panel in Fig. 2 for the eclipses of WASP-178
observed by CHEOPS.

##  5 Discussion

In the previous sections we have analyzed the TESS and CHEOPS PCs of WASP-178
b in order to extract its planetary eclipse depth $\delta_{\rm ecl}$,
obtaining respectively $70\pm 40$ ppm and $70\pm 20$ ppm. These two
measurements take into account both the reflection from the planetary dayside
and its thermal emission. With the assumptions discussed below, the combined
information on the eclipse depth from two different instruments allows to
disentangle these two contributions. For a given telescope with passband
$\eta(\lambda)$ we can combine Eq. 1 and Eq. 2 obtaining:

$\delta_{\rm ecl}=A_{\rm g}\left(\frac{R_{\rm
p}}{a}\right)^{2}+\left(\frac{R_{\rm
p}}{R_{\star}}\right)^{2}\frac{\int\eta(\lambda)B(\lambda,T_{\rm
d})d\lambda}{\int\eta(\lambda)I_{\star}(\lambda,T_{\rm eff})d\lambda},$ (10)

where $T_{\rm d}$ is the planetary dayside temperature.

Solving for $A_{\rm g}$, and indicating with the subscripts $C$ (for CHEOPS)
and $T$ (for TESS) the wavelength-dependent quantities, we obtain:

$\begin{cases}A_{\rm
g}^{C}=\left(\frac{a}{R_{\star}}\right)^{2}\left[\frac{\delta_{\rm
ecl}^{C}}{\left(\frac{R_{\rm
p}}{R_{\star}}\right)^{2}}-\frac{\int\eta^{C}(\lambda)B(\lambda,T_{\rm
d})d\lambda}{\int\eta^{C}(\lambda)I_{\star}(\lambda,T_{\rm
eff})d\lambda}\right],\\\ A_{\rm
g}^{T}=\left(\frac{a}{R_{\star}}\right)^{2}\left[\frac{\delta_{\rm
ecl}^{T}}{\left(\frac{R_{\rm
p}}{R_{\star}}\right)^{2}}-\frac{\int\eta^{T}(\lambda)B(\lambda,T_{\rm
d})d\lambda}{\int\eta^{T}(\lambda)I_{\star}(\lambda,T_{\rm
eff})d\lambda}\right].\end{cases}$ (11)

In the most general case, $A_{g}^{C}$ and $A_{g}^{T}$ differ according to the
planetary reflection spectrum. Since we do not have any spectroscopic analysis
of the dayside of WASP-178 b, we define the proportionality coefficient
$\alpha=A_{g}^{T}/A_{g}^{C}$ to account for differences between the two
passbands. Solving Eq. 11 for $T_{\rm d}$ we thus obtain the implicit
function:

$\frac{\alpha\delta_{\rm ecl}^{C}-\delta_{\rm ecl}^{T}}{\left(\frac{R_{\rm
p}}{R_{\star}}\right)^{2}}+\frac{\int\eta^{T}(\lambda)B(\lambda,T_{\rm
d})d\lambda}{\int\eta^{T}(\lambda)I_{\star}(\lambda)d\lambda}-\alpha\frac{\int\eta^{C}(\lambda)B(\lambda,T_{\rm
d})d\lambda}{\int\eta^{C}(\lambda)I_{\star}(\lambda)d\lambda}=0.$ (12)

We initially assumed the simplest scenario of a gray albedo spectrum
($\alpha$=1) in the spectral range covered by CHEOPS and TESS (the respective
bandpasses are shown in Fig. 6). In this scenario, we expect that the
contribution of reflection to the eclipse depth is the same in the CHEOPS and
TESS passbands. We also expect that the thermal emission from a $T\simeq$3000
K black body has a larger contribution in the TESS passband, because it favors
redder wavelengths compared with CHEOPS (see Fig. 6).

To estimate the dayside temperature $T_{\rm d}$ that best explains our best
observations together with its corresponding uncertainty, we randomly
extracted 10 000 steps from the Monte Carlo chains obtained in the fit of the
transit and eclipse LCs (Sect. 4.1.1 and 4.2). We also generated 10 000
samples of the stellar effective temperature using a normal distribution with
mean and standard deviations as reported in Table 1. For each of the 10 000
samples we thus numerically solved Eq. 12 using the fsolve method of the scipy
Python package, thus obtaining 10 000 samples of $T_{\rm d}$. Then, we plugged
all the samples back into Eq. 11 to derive $A_{g}^{C}$(=$A_{g}^{T}$).

The root finding algorithm failed for $\sim$50% of the samples. These failures
correspond to the combination of parameters that prevent Eq. 12 from having a
$T_{\rm d}$ root in the domain of real numbers. This happens in particular for
the samples where $\delta_{\rm ecl}^{T}<\delta_{\rm ecl}^{C}$ that, as
discussed above, are not consistent with the gray albedo hypothesis. We thus
reject these samples and show in Fig. 7 the $A_{g}^{C}$–$T_{\rm d}$ density
map of the remaining ones. Our results indicate $A_{g}^{C}=0.2\pm 0.1$ and
$T_{\rm d}=2400\pm 300$ K, consistent with the temperature-pressure profile
derived by (Lothringer et al. 2022) at pressure higher that 1$\rm\mu$bar by
means of ultraviolet transmission spectroscopy. Our result also follows the
general trend of $A_{\rm g}$ increasing with $T_{\rm d}$ indicated by Wong et
al. (2021) (Fig. 5).

Figure 5: Adaptation of Fig. 10 in Wong et al. (2021) including our analysis
of WASP-178 b (in blue). The green squares indicate the systems from the first
and second year of the TESS primary mission. The black circles indicate the
Kepler-/CoRoT-band geometric albedos for the targets that were observed by
those missions.

By definition, the geometric albedo $A_{\rm g}$ refers to the incident light
reflected back to the star at a given wavelength (or bandpass). Integrating at
all angles, the spherical albedo $A_{\rm S}$ is related to $A_{\rm g}$ by the
phase integral $q$: $A_{\rm S}$=$q$$A_{\rm g}$ (see for example Seager 2010).
Depending on the scattering law, exoplanetary atmospheres have 1¡$q$¡1.5
(Pollack et al. 1986; Burrows & Orton 2010). Unfortunately, for the reasons
explained in Sect. 4.1.2, we could not extract a robust phase curve for
WASP-178 b, hence we could not place any better constraint on $q$. In the
following we thus consider the two limiting scenarios $A_{S}^{\rm min}=A_{g}$
and $A_{S}^{\rm max}=1.5A_{g}$.

The Bond albedo $A_{\rm B}$ is computed as the average of $A_{\rm S}$ weighted
over the incident stellar spectrum:

$A_{\rm B}=\frac{\int_{0}^{\infty}A_{\rm
S}(\lambda)I_{\star}(\lambda)d\lambda}{\int_{0}^{\infty}I_{\star}(\lambda)d\lambda}.$
(13)

The conversion into $A_{\rm B}$ thus relies on the measurement of $A_{\rm S}$
across the stellar spectrum. In this study we only covered the optical and
near-infrared part of the spectrum, as shown in Fig. 6, and we lack the
necessary information in the UV, mid, and far infrared domains. Following the
approach of Schwartz & Cowan (2015), we explore the scenario of minimum Bond
albedo $A_{\rm B}^{\rm min}$ obtained through Eq. 13 assuming $A_{\rm
S}$=$A_{\rm S}^{\rm min}$ in the spectral range covered by CHEOPS and TESS and
$A_{\rm S}$=0 otherwise. The opposite limiting case assumes $A_{\rm
S}$$(\lambda)=A_{\rm S}^{\rm max}$ at all wavelength, which leads to $A_{\rm
B}^{\rm max}=A_{\rm S}^{\rm max}$. To compute the integrals in Eq. 13, we used
the synthetic spectrum in the BT-Settl library corresponding to the parameters
of WASP-178 (Allard et al. 2012).

Figure 6: BT-Settl synthetic stellar spectrum for WASP-178 together with the
CHEOPS and TESS passbands.

Under the hypothesis that $\alpha=1$, we obtained samples of $A_{\rm
g}^{C}=A_{\rm g}^{T}=$$A_{\rm g}$, which now translate into samples of $A_{\rm
B}^{\rm min}$ and $A_{\rm B}^{\rm max}$ with the assumptions discussed above.
Moreover, inverting Eq. 5 yields to:

$\epsilon=\frac{1}{5}\left[8-\left(\frac{T_{\rm d}}{T_{\rm
eff}}\right)^{4}\left(\frac{a}{R_{\star}}\right)^{2}\frac{12}{1-A_{\rm
B}}\right].$ (14)

Equation 14 indicates that $\epsilon$ is a decreasing function of $A_{\rm B}$.
Plugging in the samples of $T_{\rm d}$, Teff, $a/R_{\star}$ and alternatively
$A_{\rm B}^{\rm min}$ and $A_{\rm B}^{\rm max}$, we obtained the corresponding
samples $\epsilon^{\rm max}$ and $\epsilon^{\rm min}$.

The albedo–recirculation density maps of the two scenarios are shown in the
right panel of Fig. 7. Unsurprisingly, we find that the upper limit on $A_{\rm
B}$ is larger ($\sim$0.6) in the scenario where all the factors concur to push
up the reflectivity of the atmosphere (high $q$ and maximum $A_{\rm S}$), and
it decreases to $\sim$0.3 in the opposite scenario of minimum reflectivity.

The recirculation coefficient $\epsilon$ tends towards large values,
eventually exceeding 1 due to measurement uncertainties, in both scenarios.
While $\epsilon>1$ is physically meaningless, still the posterior distribution
indicates a high level of atmospheric energy recirculation in both cases of
minimum and maximum albedo ($\epsilon^{\rm max}=1.0\pm 0.3$ and $\epsilon^{\rm
min}=0.8\pm 0.3$ respectively). Following different atmospheric models (e.g.,
Perez-Becker & Showman 2013; Komacek & Showman 2016; Schwartz et al. 2017;
Parmentier & Crossfield 2018), WASP-178 b is in a temperature regime where
heat recirculation of zonal winds is suppressed and recirculation efficiency
is expected to be low. Nonetheless, Zhang et al. (2018) collected
observational evidence that Hot Jupiters with irradiation temperatures similar
to the one of WASP-178 b are characterized by efficient day-to-night
recirculation. Using the global circulation models of Kataria et al. (2016),
they ascribed this efficiency to the presence of zonal winds, that also
explain the eastward offset of the PC of the same planets. While the
theoretical predictions on the offset (and the recirculation efficiency
correspondingly) likely overestimate the truth (see Fig. 15 in Zhang et al.
2018), still there is an indication that zonal winds might indeed explain the
energy transfer from the dayside to the nightside of WASP-178 b.

Unfortunately, as discussed in Sect.4.1.2, it is difficult to disentangle the
planetary phase curve and the stellar variability. By consequence, any
measurement of the phase curve offset is precluded. Another possibility is
that ionized winds flow from the dayside, where temperatures higher that 2500
K leads to complete dissociation of H2, to the nightside, where lower
temperatures allow the molecular recombination and the consequent energy
release. This scenario is supported by several recent studies (Bell & Cowan
2018; Tan & Komacek 2019; Mansfield et al. 2020; Helling et al. 2021, 2023).

Figure 7: Left panel -  $A_{g}^{C}$–$T_{\rm d}$ density maps in the case of
$\alpha=1$. Right panel -  Albedo–recirculation density maps assuming maximum
$A_{\rm B}$ (red contours) and minimum $A_{\rm B}$ (blue contours).

According to the synthetic models computed by Sudarsky et al. (2000), the
albedo spectrum of highly irradiated giant planets is expected to show
molecular absorption bands by H2O at around 1 $\mu m$ and longer wavelength.
Since the passband of TESS is more sensitive in the near-infrared than CHEOPS,
it is plausible to assume that the geometric albedo in the TESS passband is
lower than for CHEOPS. In order to assess how the assumption on $\alpha$
affects our results, we re-run our analysis assuming an extreme $\alpha=0.5$.
The results are shown in Fig. 8: we find almost the same posterior
distribution for dayside temperature, albedo and recirculation, indicating
that the most important source of uncertainties are not the assumptions we
make but the measurement uncertainties on the eclipse depth extracted from the
CHEOPS and TESS LCs.

Figure 8: Same as in Fig. 7 in the case $\alpha=0.5$.

##  6 Summary and conclusion

In this work we have analyzed the space-borne photometry of the WASP-178
system obtained with TESS and CHEOPS. For both telescopes we have tailored the
photometric extraction in order to avoid the strong contamination by a
background eclipsing binary. We have found evidence that the stellar host is
rotating with a period of $\sim$3 days, consistent with the orbital period of
its HJ, and that it is seen in a pole-on geometry.

The similarity between the stellar rotation period and the orbital period of
the planet, coupled with the quality of the TESS data, does not allow a robust
analysis of the full PC of the planet, nor we could put strong constraints on
the planetary nightside emission. Nonetheless, focusing on the transit and
eclipse events, we could update the ephemeris of the planet and measure an
eclipse depth of 70$\pm$40 ppm and 70$\pm$20 ppm in the TESS and CHEOPS
passband respectively.

The joint analysis of the eclipse depth measured in the two passbands allowed
us to constrain the temperature in the 2250–2750 K range, consistent with the
temperature-pressure profile derived by (Lothringer et al. 2022) at pressure
higher that 1$\rm\mu$bar. Moreover, we could also constrain the optical
geometric albedo $A_{\rm g}$¡0.4, which fits the general increasing albedo
with stellar irradiation indicated by Wong et al. (2021).

Finally, we found indication of an efficient atmospheric heat recirculation of
WASP-178 b. If confirmed, this evidence challenges the models that predict a
decreasing heat transport by zonal winds as the equilibrium temperature
increases (e.g., Perez-Becker & Showman 2013; Komacek & Showman 2016; Schwartz
et al. 2017; Parmentier & Crossfield 2018). Conversely, WASP-178 b is an
interesting target to test atmospheric models where heat transport is granted
by ionized winds from the dayside to the nightside, where the recombination of
H2 takes place (e.g. Mansfield et al. 2020; Helling et al. 2021, 2023).
Additional observations of WASP-178 b are needed to better constrain its
atmospheric recirculation and rank current competing models. The WASP-178
system is going to be observed again by TESS in May 2023, but this is not
expected to add much to the two sectors analyzed in this work. Conversely,
dedicated observations from larger space observatories might provide helpful
insights of the physical and chemical equilibrium of the atmosphere of
WASP-178 b.

###### Acknowledgements.

We thank the referee, N. B. Cowan, for his valuable comments and suggestions.
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. IPa, GSc, VSi, LBo, GBr, VNa, GPi,
and RRa acknowledge support from CHEOPS ASI-INAF agreement n. 2019-29-HH.0. ML
acknowledges support of the Swiss National Science Foundation under grant
number PCEFP2_194576. This work was also partially supported by a grant from
the Simons Foundation (PI Queloz, grant number 327127). S.G.S. acknowledge
support from FCT through FCT contract nr. CEECIND/00826/2018 and POPH/FSE
(EC). ABr was supported by the SNSA. ACCa and TWi acknowledge support from
STFC consolidated grant numbers ST/R000824/1 and ST/V000861/1, and UKSA grant
number ST/R003203/1. V.V.G. is an F.R.S-FNRS Research Associate. YAl
acknowledges support from the Swiss National Science Foundation (SNSF) under
grant 200020_192038. We acknowledge support from the Spanish Ministry of
Science and Innovation and the European Regional Development Fund through
grants ESP2016-80435-C2-1-R, ESP2016-80435-C2-2-R, PGC2018-098153-B-C33,
PGC2018-098153-B-C31, ESP2017-87676-C5-1-R, MDM-2017-0737 Unidad de Excelencia
Maria de Maeztu-Centro de Astrobiología (INTA-CSIC), as well as the support
of the Generalitat de Catalunya/CERCA programme. The MOC activities have been
supported by the ESA contract No. 4000124370. S.C.C.B. acknowledges support
from FCT through FCT contracts nr. IF/01312/2014/CP1215/CT0004. XB, SC, DG, MF
and JL acknowledge their role as ESA-appointed CHEOPS science team members.
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 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. B.-O. D.
acknowledges support from the Swiss State Secretariat for Education, Research
and Innovation (SERI) under contract number MB22.00046. This project has
received funding from the European Research Council (ERC) under the European
Union’s Horizon 2020 research 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. and innovation programme (project Four Aces. 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. 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.
PM acknowledges support from STFC research grant number ST/M001040/1. IRI
acknowledges support from the Spanish Ministry of Science and Innovation and
the European Regional Development Fund through grant PGC2018-098153-B- C33, as
well as the support of the Generalitat de Catalunya/CERCA programme. GyMSz
acknowledges the support of the Hungarian National Research, Development and
Innovation Office (NKFIH) grant K-125015, a 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. NAW acknowledges UKSA
grant ST/R004838/1. NCS 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. KWFL was supported by Deutsche
Forschungsgemeinschaft grants RA714/14-1 within the DFG Schwerpunkt SPP 1992,
Exploring the Diversity of Extrasolar Planets. In this work we use the python
package PyDE available at https://github.com/hpparvi/PyDE. This research has
made use of the SVO Filter Profile Service (http://svo2.cab.inta-
csic.es/theory/fps/) supported from the Spanish MINECO through grant
AYA2017-84089.

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##  Appendix A Posterior distributions of the model parameters from the fit of
the TESS transit light curves

Figure 9: Corner plot of the MCMC chains of planetary parameters from the fit
of the TESS transits (see Sect. 4.1.1). In each plot, the solid blue lines
mark the MAP values.

##  Appendix B Analysis of the TESS transits in high cadence

In Sect. 4.1 we analyzed the TESS photometry at low cadence (30 min), and we
found a significant difference between the transit depth between sectors 11
and 38. We speculated that the difference in transit depth is due to
photospheric variability, which manifests also as a quasi-periodic signal in
the LC of sector 38. This hypothesis is supported by the growing evidence that
A-type stars show rotational signals due to photospheric inhomogeneities
(e.g., Balona 2011; Böhm et al. 2015; Sikora et al. 2020). This scenario is
further supported by the fact that the WASP-178 is classified as an Am star by
Hellier et al. (2019).

To test if the visible hemisphere of the star hosts dark spots, we searched
for transit anomalies in the TESS LCs, that are localized bumps in the
residuals of the transit fits. If present, they indicate that the planet’s
projection on the stellar surface crosses a darker area compared to the
quiescent photosphere (see for example Béky et al. 2014; Scandariato et al.
2017). To this purpose, the low cadence photometry analyzed in Sect. 4.1 is of
little help, as a finer time sampling is needed. We thus compared the high
cadence (2 min) TESS photometry of sector 38 with the bestfit model discussed
in Sect. 4.1.2, computed using $R_{\rm p}/R*$=0.1141 (see Sect. 4.1.1). This
comparison does not show any bump inside the transits (Figs. 10–13), and we
concluded that in the eight transits observed in sector 38 there is no spot-
crossing event.

We also remark that, in contrast with Rodríguez Martínez et al. (2020), we did
not detect any systematic asymmetry in the transit profile. The stellar flux
distribution along the transit path is symmetric with respect to the transit
center. This either indicates that the star does not show any gravity
darkening, which is unlikely if the stellar rotation period of $\sim$3.2 days
is confirmed, or supports the hypothesis that the star is in a pole-on
geometry, which leads to a radially symmetric flux distribution over the
visible stellar hemisphere.

Figure 10: Detrended short cadence LC of the first (left panel) and second
(right panel) transit observed in TESS sector 38. In each panel, the top plot
shows the model computed with the MAP parameters in Table 3 as a blue solid
line. The bottom plots show the residuals of the short cadence photometry with
respect to the planetary model shown in the top panel. As a guide, we plot
with a blue line the smoothing of the residuals obtained with a Savitzky-Golay
filter. In all plots, the vertical dashed lines mark the first and fourth
contacts.

Figure 11: Detrended short cadence LC of the third (left panel) and fourth
(right panel) transit observed in TESS sector 38. Details are the same as in
Fig. 10.

Figure 12: Detrended short cadence LC of the fifth (left panel) and sixth
(right panel) transit observed in TESS sector 38. Details are the same as in
Fig. 10.

Figure 13: Detrended short cadence LC of the seventh (left panel) and eighth
(right panel) transit observed in TESS sector 38. Details are the same as in
Fig. 10.

  *[LC]: Light Curve
  *[LCs]: Light Curve
  *[FFIs]: Full Frame Images
  *[TPFs]: Target Pixel File
  *[CBVs]: Co-trending Basis Vectors
  *[DRP]: Data Reduction Pipeline
  *[PSF]: Point Spread Function
  *[IRFM]: Infra-Red Flux Method
  *[MCMC]: Markov Chain Monte Carlo
  *[GLS]: Generalized Lomb-Scargle
  *[FAP]: False Alarm Probability
  *[PC]: Phase Curve
  *[LD]: Limb Darkening
  *[MAP]: Maximum A Posteriori
  *[GP]: Gaussian Process
  *[PCs]: Phase Curve
  *[PSFs]: Point Spread Function
  *[HJ]: Hot Jupiter