11institutetext:  Department of Astronomy, University of Geneva, Chemin Pegasi
51, 1290 Versoix, Switzerland 22institutetext: Physikalisches Institut,
University of Bern, Gesellsschaftstrasse 6, 3012 Bern, Switzerland
33institutetext: Center for Space and Habitability, University of Bern,
Gesellsschaftstrasse 6, 3012 Bern, Switzerland 44institutetext: Department of
Astronomy, Stockholm University, AlbaNova University Center, 10691 Stockholm,
Sweden 55institutetext: Aix-Marseille Université, CNRS, CNES, Laboratoire
d’Astrophysique de Marseille, 38 rue Frédéric Joliot-Curie, 13388 Marseille,
France 66institutetext: Space sciences, Technologies and Astrophysics Research
(STAR) Institute, Université de Liège, Allée du 6 Août 19C, 4000 Liège,
Belgium 77institutetext: Instituto de Astrofísica e Ciências do Espaço,
Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal
88institutetext: Departamento de Física e Astronomia, Faculdade de Ciências,
Universidade do Porto, Rua do Campo Alegre 687, 4169-007 Porto, Portugal
99institutetext: Department of Physics, University of Warwick, Gibbet Hill
Road, Coventry CV4 7AL, United Kingdom 1010institutetext: University
Observatory Munich, Ludwig Maximilian University, Scheinerstraße 1, Munich
81679, Germany 1111institutetext: Instituto de Astrofísica de Canarias, 38200
La Laguna, Tenerife, Spain 1212institutetext: Departamento de Astrofísica,
Universidad de La Laguna, 38206 La Laguna, Tenerife, Spain 1313institutetext:
Dipartimento di Fisica, Università degli Studi di Torino, Via Pietro Giuria 1,
10125, Torino, Italy 1414institutetext: INAF, Osservatorio Astronomico di
Padova, Vicolo Osservatorio 5, 35122 Padova, Italy 1515institutetext: Space
Research Institute, Austrian Academy of Sciences, Schmiedlstraße 6, 8042 Graz,
Austria 1616institutetext: Centre for Exoplanet Science, SUPA School of
Physics and Astronomy, University of St Andrews, North Haugh, St Andrews KY16
9SS, United Kingdom 1717institutetext: Institut de Ciències de l’Espai (ICE,
CSIC), Campus UAB, Carrer de Can Magrans, s/n, 08193 Barcelona, Spain
1818institutetext: Admatis, Kandó Kálmán út 5, 3534 Miskolc, Hungary
1919institutetext: Departamento de Astrofísica, Centro de Astrobiología (CSIC-
INTA), ESAC campus, 28692 Villanueva de la Cañada, Spain 2020institutetext:
Université Grenoble Alpes, CNRS, Institut de Planétologie et d’Astrophysique
de Grenoble, 38000 Grenoble, France 2121institutetext: Institute of Planetary
Research, German Aerospace Center (DLR), Rutherfordstraße 2, 12489 Berlin,
Germany 2222institutetext: Université de Paris, Institut de Physique du Globe
de Paris, CNRS, 75005 Paris, France 2323institutetext: European Space Research
and Technology Centre (ESTEC), European Space Agency (ESA), Keplerlaan 1,
2201-AZ Noordwijk, The Netherlands 2424institutetext: Centre for Mathematical
Sciences, Lund University, Box 118, 22100 Lund, Sweden 2525institutetext:
Astrobiology Research Unit, Université de Liège, Allée du 6 Août 19C, 4000
Liège, Belgium 2626institutetext: Leiden Observatory, University of Leiden, PO
Box 9513, 2300 RA Leiden, The Netherlands 2727institutetext: Department of
Space, Earth and Environment, Chalmers University of Technology, Onsala Space
Observatory, 43992 Onsala, Sweden 2828institutetext: Department of
Astrophysics, University of Vienna, Türkenschanzstraße 17, 1180 Vienna,
Austria 2929institutetext: Division Technique, Institut National des Sciences
de l’Univers (INSU), CS 20330, 83507 La Seyne-sur-Mer, France
3030institutetext: Konkoly Observatory, Research Centre for Astronomy and
Earth Sciences, Konkoly-Thege Miklós út 15-17, 1121 Budapest, Hungary
3131institutetext: IMCCE, UMR8028 CNRS, Observatoire de Paris, PSL Université,
Sorbonne Université, 77 avenue Denfert-Rochereau, 75014 Paris, France

3232institutetext: Institut d’astrophysique de Paris, UMR7095 CNRS, Université
Pierre & Marie Curie, 98 bis boulevard Arago, 75014 Paris, France
3333institutetext: Astrophysics Group, Keele University, Staffordshire, ST5
5BG, United Kingdom 3434institutetext: INAF, Osservatorio Astrofisico di
Catania, Via Santa Sofia 78, 95123 Catania, Italy 3535institutetext: Institute
of Optical Sensor Systems, German Aerospace Center (DLR), Rutherfordstraße 2,
12489 Berlin, Germany 3636institutetext: Dipartimento di Fisica e Astronomia
“Galileo Galilei”, Università degli Studi di Padova, Vicolo Osservatorio 3,
35122 Padova, Italy 3737institutetext: Cavendish Laboratory, JJ Thomson
Avenue, Cambridge CB3 0HE, United Kingdom 3838institutetext: Department of
Physics, ETH Zürich, Wolfgang-Pauli-Strasse 27, 8093 Zürich, Switzerland
3939institutetext: Center for Astronomy and Astrophysics, Technical University
Berlin, Hardenberstraße 36, 10623 Berlin, Germany 4040institutetext: Institut
für Geologische Wissenschaften, Freie Universität Berlin, 12249 Berlin,
Germany 4141institutetext: Institut d’Estudis Espacials de Catalunya (IEEC),
08034 Barcelona, Spain 4242institutetext: ELTE Eötvös Loránd University,
Gothard Astrophysical Observatory, Szent Imre herceg utca 112, 9700
Szombathely, Hungary 4343institutetext: MTA-ELTE Exoplanet Research Group,
Szent Imre herceg utca 112, 9700 Szombathely, Hungary 4444institutetext:
Institute of Astronomy, University of Cambridge, Madingley Road, Cambridge,
CB3 0HA, United Kingdom

# The atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curve

A. Deline The atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curveThe atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curve<EMAIL_ADDRESS>M. J. Hooton The atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curve M. Lendl The
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curve B.
Morris The atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curveThe atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curve S. Salmon The atmosphere and architecture of WASP-189 b probed by
its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by
its CHEOPS phase curve G. Olofsson The atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curve C. Broeg The atmosphere and architecture of
WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture of
WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture of
WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture of
WASP-189 b probed by its CHEOPS phase curve D. Ehrenreich The atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curve M. Beck The
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curve A.
Brandeker The atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curveThe atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curve S. Hoyer The atmosphere and architecture of WASP-189 b probed by
its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by
its CHEOPS phase curve S. Sulis The atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curve V. Van Grootel The atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curve V. Bourrier The
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curve O.
Demangeon The atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curveThe atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curveThe atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curveThe atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curve B.-O. Demory The atmosphere and architecture of WASP-189 b probed
by its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed
by its CHEOPS phase curve K. Heng The atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curve H. Parviainen The atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curve L. M. Serrano The
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curve V.
Singh The atmosphere and architecture of WASP-189 b probed by its CHEOPS phase
curveThe atmosphere and architecture of WASP-189 b probed by its CHEOPS phase
curve A. Bonfanti The atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curve L. Fossati The atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curve D. Kitzmann The atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curve S. G. Sousa The atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curve T. G. Wilson The
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curve Y.
Alibert The atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curveThe atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curve R. Alonso The atmosphere and architecture of WASP-189 b probed by
its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by
its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by
its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by
its CHEOPS phase curve G. Anglada The atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curve T. Bárczy The atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curve D. Barrado Navascues The
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curve S.
C. C. Barros The atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curve W. Baumjohann The atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curve T. Beck The atmosphere and architecture of
WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture of
WASP-189 b probed by its CHEOPS phase curve A. Bekkelien The atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curve W. Benz The
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curve N.
Billot The atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curveThe atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curve X. Bonfils The atmosphere and architecture of WASP-189 b probed by
its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by
its CHEOPS phase curve J. Cabrera The atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curve S. Charnoz The atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curve A. Collier Cameron The
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curve C.
Corral van Damme The atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curve Sz. Csizmadia The atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curve M. B. Davies The atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curve M. Deleuil The atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curve L. Delrez The
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curve T.
de Roche The atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curveThe atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curve A. Erikson The atmosphere and architecture of WASP-189 b probed by
its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by
its CHEOPS phase curve A. Fortier The atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curve M. Fridlund The atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curve D. Futyan The atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curve D. Gandolfi The
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curve M.
Gillon The atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curveThe atmosphere and architecture of WASP-189 b probed by its CHEOPS
phase curve M. Güdel The atmosphere and architecture of WASP-189 b probed by
its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by
its CHEOPS phase curve P. Gutermann The atmosphere and architecture of
WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture of
WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture of
WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture of
WASP-189 b probed by its CHEOPS phase curve J. Hasiba The atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curve K. G. Isaak The
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curve L.
Kiss The atmosphere and architecture of WASP-189 b probed by its CHEOPS phase
curveThe atmosphere and architecture of WASP-189 b probed by its CHEOPS phase
curve J. Laskar The atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curve A. Lecavelier des Etangs The atmosphere and architecture of
WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture of
WASP-189 b probed by its CHEOPS phase curve C. Lovis The atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curve D. Magrin The
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curve P.
F. L. Maxted The atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curve M. Munari The atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curve V. Nascimbeni The atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curve R. Ottensamer The atmosphere
and architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere
and architecture of WASP-189 b probed by its CHEOPS phase curve I. Pagano The
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curve E.
Pallé The atmosphere and architecture of WASP-189 b probed by its CHEOPS phase
curveThe atmosphere and architecture of WASP-189 b probed by its CHEOPS phase
curve G. Peter The atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curve G. Piotto The atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curve D. Pollacco The atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curve D. Queloz The atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curve R. Ragazzoni The
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curve N.
Rando The atmosphere and architecture of WASP-189 b probed by its CHEOPS phase
curveThe atmosphere and architecture of WASP-189 b probed by its CHEOPS phase
curve H. Rauer The atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curve I. Ribas The atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curve N. C. Santos The atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curve G. Scandariato The atmosphere
and architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere
and architecture of WASP-189 b probed by its CHEOPS phase curve D. Ségransan
The atmosphere and architecture of WASP-189 b probed by its CHEOPS phase
curveThe atmosphere and architecture of WASP-189 b probed by its CHEOPS phase
curve A. E. Simon The atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curve A. M. S. Smith The atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189
b probed by its CHEOPS phase curve M. Steller The atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and architecture
of WASP-189 b probed by its CHEOPS phase curve Gy. M. Szabó The atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curveThe atmosphere and
architecture of WASP-189 b probed by its CHEOPS phase curve N. Thomas The
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curveThe
atmosphere and architecture of WASP-189 b probed by its CHEOPS phase curve S.
Udry The atmosphere and architecture of WASP-189 b probed by its CHEOPS phase
curveThe atmosphere and architecture of WASP-189 b probed by its CHEOPS phase
curve I. Walter The atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curveThe atmosphere and architecture of WASP-189 b probed by its
CHEOPS phase curve N. Walton The atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curveThe atmosphere and architecture of WASP-189 b
probed by its CHEOPS phase curve

(Received January 1, 2021; accepted January 1, 2021)

###### Abstract

Context. Gas giants orbiting close to hot and massive early-type stars can
reach dayside temperatures that are comparable to those of the coldest stars.
These ‘ultra-hot Jupiters’ have atmospheres made of ions and atomic species
from molecular dissociation and feature strong day-to-night temperature
gradients. Photometric observations at different orbital phases provide
insights on the planet’s atmospheric properties.

Aims. We aim to analyse the photometric observations of WASP-189 acquired with
the Characterising Exoplanet Satellite (CHEOPS) to derive constraints on the
system architecture and the planetary atmosphere.

Methods. We implemented a light-curve model suited for an asymmetric transit
shape caused by the gravity-darkened photosphere of the fast-rotating host
star. We also modelled the reflective and thermal components of the planetary
flux, the effect of stellar oblateness and light-travel time on transit-
eclipse timings, the stellar activity, and CHEOPS systematics.

Results. From the asymmetric transit, we measure the size of the ultra-hot
Jupiter WASP-189 b, $R_{p}=1.600^{+0.017}_{-0.016}\,R_{J}$, with a precision
of 1%, and the true orbital obliquity of the planetary system,
$\Psi_{p}=89.6\pm 1.2\deg$ (polar orbit). We detect no significant hotspot
offset from the phase curve and obtain an eclipse depth of
$\delta_{\text{ecl}}=96.5^{+4.5}_{-5.0}\,\text{ppm}$, from which we derive an
upper limit on the geometric albedo: $A_{g}<0.48$. We also find that the
eclipse depth can only be explained by thermal emission alone in the case of
extremely inefficient energy redistribution. Finally, we attribute the
photometric variability to the stellar rotation, either through superficial
inhomogeneities or resonance couplings between the convective core and the
radiative envelope.

Conclusions. Based on the derived system architecture, we predict the eclipse
depth in the upcoming Transiting Exoplanet Survey Satellite (TESS)
observations to be up to $\sim 165\,\text{ppm}$. High-precision detection of
the eclipse in both CHEOPS and TESS passbands might help disentangle
reflective and thermal contributions. We also expect the right ascension of
the ascending node of the orbit to precess due to the perturbations induced by
the stellar quadrupole moment $J_{2}$ (oblateness).

###### Key Words.:

techniques: photometric – planets and satellites: atmospheres – planets and
satellites: individual: WASP-189 b

##  1 Introduction

Extra-solar planets exhibit a wide range of sizes, compositions, temperatures,
and system architectures. Hot Jupiters are among the most extreme of these
worlds, orbiting so close to their host star that they can reach equilibrium
temperatures at their surfaces beyond 2000 K. The proximity of the star also
creates strong tidal forces causing the planet rotation and revolution periods
to synchronise. Once tidally locked, the planet always has the same hemisphere
facing the star, and this strongly impacts the atmospheric circulation.
Effects of stellar irradiation are further enhanced when the host is an early-
type F or A star, hotter and more massive than the Sun (e.g. Collier Cameron
et al., 2010; Gaudi et al., 2017). Close-in gas giants orbiting such stars,
dubbed ‘ultra-hot Jupiters’, have cloud-free daysides with temperatures
commensurate with the surface of cool stars, where most molecules are
thermally dissociated and atoms are ionised (Evans et al., 2017; Bell & Cowan,
2018; Kitzmann et al., 2018; Parmentier et al., 2018; Lothringer et al., 2018;
Fossati et al., 2021). Partially ionised atmospheres inhibit atmospheric
circulation from the dayside to the nightside of the planet (via Lorentz
forces), resulting in strong temperature contrasts of about $1000$ K (Komacek
& Showman, 2016). Colder nightside temperatures allow for various condensation
processes to occur, as exemplified by the measurement of a different iron
composition at the morning and evening twilights of the ultra-hot gas giant
WASP-76b (Ehrenreich et al., 2020; Kesseli & Snellen, 2021; Wardenier et al.,
2021). Due to their elevated temperatures and large day-to-night contrasts,
ultra-hot gas giants are especially amenable to mapping their atmospheres with
observations at various phase angles; that is, in transit (nightside),
occultation (dayside), and in-between (phase curve). Key insights on ultra-hot
gas giant atmospheres can thus be revealed by observing them at both infrared
and optical wavelengths. In fact, these exoplanet daysides emit thermal
radiation in the optical domain, giving rise to deep eclipses and large phase-
curve amplitudes (e.g. Bourrier et al., 2020).

Lendl et al. (2020) recently reported on occultations of the ultra-hot gas
giant WASP-189 b (Anderson et al., 2018) observed with the Characterising
Exoplanet Satellite (CHEOPS – Benz et al., 2021) in the visible wavelength
range (330-1100 nm). The occultation depth of $87.9\pm 4.3\,\text{ppm}$
appears compatible with an unreflective atmosphere heated to $3425\pm
27\,\text{K}$ when assuming inefficient heat redistribution (Lendl et al.,
2020). Using the ultra-high photometric precision of CHEOPS, Lendl et al.
(2020) also found the fast-rotating, gravity-darkened host star to cause an
asymmetric transit light curve, allowing a direct inference on the true
obliquity of the planet’s orbital spin axis.

Here, we report on the first full phase-curve observations of WASP-189 b,
which were obtained with CHEOPS. We jointly analyse them with the occultations
previously published by Lendl et al. (2020). Together, these observations
cover two full planetary orbits, six eclipses and 3.5 transits of WASP-189 b.
We complement these photometric observations with high-resolution spectroscopy
to refine the stellar properties (Sect. 2). We describe the CHEOPS
observations and their reduction in Sect. 3 and the light curve analysis in
Sect. 4. Finally, we discuss our results in Sect. 5.

##  2 Host star WASP-189

###  2.1 An oblate, gravity-darkened fast rotator

The host star WASP-189 (HD 133112; HR 5599) is a hot A4 star with an effective
temperature of 8000 K (see Table 1). Transits of a gas giant were reported by
Anderson et al. (2018) using WASP-South (Pollacco et al., 2006; Collier
Cameron et al., 2009) and TRAPPIST-South (Gillon et al., 2011; Jehin et al.,
2011). Follow-up spectroscopy with CORALIE (Queloz et al., 2000) and HARPS
(Mayor et al., 2003) allowed Anderson et al. (2018) to reveal the star as a
fast rotator ($v_{\star}\sin i_{\star}\sim 100\,\text{km}.\text{s}^{-1}$).
Rapid rotation is common to early-type stars (F, A, B, O – e.g. Royer et al.,
2007; Dufton et al., 2013) that tend to be radially distorted by the
centrifugal force resulting in oblate shapes. The surface gravity of an oblate
star hence varies as a function of latitude causing a change of local
temperature and brightness (von Zeipel, 1924): the equator appears darker than
the poles, a phenomenon known as gravity darkening (GD – Claret, 2000;
Espinosa Lara & Rieutord, 2011). When a planet on a misaligned orbit transits
such a star, the non-radially symmetric brightness distribution of the stellar
disk will create an asymmetry in the photometric transit light curve. The
asymmetry can be used to retrieve the absolute orientation of the system
(stellar inclination and orbital obliquity) as modelled by Barnes (2009) and
observed for several targets (Barnes et al., 2011; Szabó et al., 2012; Ahlers
et al., 2020; Lendl et al., 2020).

The CHEOPS phase-curve observations reported in this work furthermore reveal a
photometric variability attributed to WASP-189 (see Sect. 4.5 for details).

###  2.2 Refined stellar properties from spectroscopic measurements

To support our analysis of the CHEOPS observations of the WASP-189 system, we
computed the stellar parameters listed in Table 1 using the same methods as
described in Lendl et al. (2020) with the inclusion of a new validation
procedure detailed in Bonfanti et al. (2021).

For consistency, we briefly summarise the methods used for the derivation of
the stellar properties. The first method consists of using synthetic spectra
to fit spectral lines observed with the HARPS spectrograph (Mayor et al.,
2003, programmes 0100.C-0847 and 0103.C-0472, observed in 2018 and 2019,
respectively) with synthetic spectra. The modelling of the stellar atmosphere
and evolution then allows us to compute fundamental parameters. The first
output of the analysis is the projected rotation speed of the star
$v_{\star}\sin i_{\star}$. By assuming several conditions on iron atmospheric
content (excitation equilibrium of FeI and FeII, ionisation equilibrium of Fe
and minimum standard deviation of Fe abundance), we could compute,
respectively, the effective temperature $T_{\text{eff}}$, the surface gravity
$\log g$, and the microturbulent velocity $v_{\text{mic}}$. We also directly
derived the iron abundance $\left[\text{Fe}/\text{H}\right]$ from this method.
The second method we used is based on the infrared flux method (IRFM –
Blackwell & Shallis, 1977), which allows us to determine the angular diameter
of WASP-189 from its relationship with the bolometric flux and the flux
received on Earth. We used a Markov chain Monte Carlo (MCMC) formulation of
the IRFM method as described in Schanche et al. (2020). It consists of
building synthetic spectral energy distributions (SEDs) using stellar
parameters from the previous step ($T_{\text{eff}}$, $\log g$ and
$\left[\text{Fe}/\text{H}\right]$) and comparing them to photometry in various
passbands: Gaia passbands $G$, $G_{\text{BP}}$, and $G_{\text{RP}}$ (Gaia
Collaboration et al., 2021a); 2MASS passbands J, H, and K (Skrutskie et al.,
2006); and WISE passbands W1 and W2 (Wright et al., 2010). Once the angular
diameter is estimated, we derive the stellar radius $R_{\star}$ with the Gaia
EDR3 parallax after correction of the offset (Lindegren et al., 2021).
Finally, our third method is based on modelling the stellar evolution with two
different codes: PARSEC (Marigo et al., 2017) via the isochrone placement
algorithm (Bonfanti et al., 2015, 2016), and CLÉS (Scuflaire et al., 2008).
The input parameters are the previously determined effective temperature
$T_{\text{eff}}$, iron abundance $\left[\text{Fe}/\text{H}\right]$ and stellar
radius $R_{\star}$, from which the two models provide the stellar mass
$M_{\star}$ and the system age $t_{\star}$. We combine the posterior
probability distributions of the two results using the procedure detailed in
Bonfanti et al. (2021). All computed stellar parameters are listed in Table 1.

Table 1: Properties of the star WASP-189.

Parameter | Value | Source
---|---|---
Names & aliases | WASP-189 | Simbad 11$1$SIMBAD astronomical database from the Centre de Données astronomiques de Strasbourg111http://simbad.u-strasbg.fr/simbad/.
HD 133112
HR 5599
TIC 157910432
Gaia DR2 6339097679918871168
V-band magnitude | $6.618\pm 0.011$ | Simbad 11$1$SIMBAD astronomical database from the Centre de Données astronomiques de Strasbourg111http://simbad.u-strasbg.fr/simbad/.
Gaia G-band magnitude | $6.5537\pm 0.0004$ | Gaia archive 22$2$Archive of the Gaia mission of the European Space Agency222https://gea.esac.esa.int/archive/
$T_{\text{eff}}\ \left[K\right]$ | $8000\pm 80$ | spectroscopy
$M_{\star}\ \left[M_{\odot}\right]$ | $2.031^{+0.098}_{-0.096}$ | evolution model
$R_{\star}\ \left[R_{\odot}\right]$ | $2.365\pm 0.025$ | IRFM
$\log g\ \left[\log_{10}\left(\text{cm}.\text{s}^{-2}\right)\right]$ | $3.9\pm 0.2$ | spectroscopy
$v_{\text{mic}}\ \left[\text{km}.\text{s}^{-1}\right]$ | $2.70\pm 0.30$ | spectroscopy
$\left[\text{Fe}/\text{H}\right]$ | $0.29\pm 0.13$ | spectroscopy
$t_{\star}\ \left[\text{Gyr}\right]$ | $0.73^{+0.19}_{-0.20}$ | evolution model
$L_{\star}\ \left[L_{\odot}\right]$ | $20.64\pm 0.93$ | $L_{\star}=4\pi R_{\star}^{2}\sigma_{\text{SB}}T_{\text{eff}}^{4}$
$v_{\star}\sin i_{\star}\ \left[\text{km}.\text{s}^{-1}\right]$ | $93.1\pm 1.7$ | spectroscopy

333 The methods used to obtain the stellar parameters are described in the
text. The stellar luminosity $L_{\star}$ is computed following the
Stefan–Boltzmann law and using the Stefan–Boltzmann constant
$\sigma_{\text{SB}}$.

##  3 CHEOPS observations and data reduction

###  3.1 Eclipse and phase-curve observations

CHEOPS made several photometric observations of the WASP-189 system in the
visible wavelength range (330-1100 nm – see Fig. 3), covering four eclipses
and two full orbits of WASP-189 b. The four eclipses and two transits out of
the first phase-curve observation were analysed in Lendl et al. (2020). In
this work, we present the joint analysis of the data previously published
together with both full phase curves.

As CHEOPS orbits the Earth on a low-altitude Sun-synchronous trajectory, the
target star is periodically occulted by our planet, causing interruptions in
the photometric sequence, referred to as Earth occultations and visible in the
light curve as gaps. In addition, the spacecraft regularly crosses the South
Atlantic anomaly (SAA) where the Earth’s magnetic field concentrates high-
energy particles that strongly deteriorate the quality of the images and make
them scientifically useless. In order to save the downlink bandwidth,
observations acquired during SAA crossings are not transferred to the ground.
Therefore, CHEOPS’ photometric light curves feature gaps on a nearly periodic
basis due to these two expected phenomena with typical durations per CHEOPS
orbit ranging from 25 to 40 min for the Earth occultation and from 0 to 18 min
for the SAA.

Each of the CHEOPS observations can be referred to using a file key that is a
unique identifier in the mission database. The observations log is shown in
Table 2 and lists the file keys of the data sets analysed in this work
together with additional information.

The earliest data acquired with CHEOPS are the eclipse light curves. The
images were taken at the very beginning of the mission, even before the start
of the nominal science mission, as part of the Early Science Programme (ESP),
which is dedicated to demonstrating the photometric capabilities of CHEOPS.
For this reason, the eclipse data sets were obtained in different conditions.
The first difference is the location of the target point spread function (PSF)
on the instrument detector. The CHEOPS detector is a charge-coupled device
(CCD) with an image section of 1024$\times$1024 pixels from which a
200$\times$200-pixel sub-array is extracted around the target PSF that is used
to compute the photometry. The plate scale of the instrument is 1 arcsecond
per pixel, and the PSF, defocused by design, spreads across a radius of about
16 pixels. The first eclipse observation was taken with the PSF in its initial
default position at the center of the detector,
$\left(x,y\right)=\left(512,512\right)$. To minimise the noise induced by
newly appearing hot pixels, the PSF location was changed to the lower right
part of the detector, $\left(x,y\right)=\left(664,296\right)$, for the other
three eclipse observations, and again to the upper left part of the CCD,
$\left(x,y\right)=\left(263,842\right)$, for the phase-curve observations.

The other change that occurred at the beginning of the CHEOPS mission and
affected the eclipse data sets is related to the temperature of the detector.
With the aim of reducing the impact of dark current and hot pixels on the
photometry, the CCD temperature was lowered during some ESP observations
including the four WASP-189 b eclipses (see Table 2). The nominal science CCD
temperature was then set to $-45^{\circ}\text{C,}$ and the WASP-189 phase
curves were obtained in these conditions. Finally, the capability of the
CHEOPS telescope to reject stray light varies with several parameters such as
the angular distance between the line of sight and the Earth limb, the
illumination of the Earth limb, the airglow of the Earth’s atmosphere, and the
brightness and location of the Moon. The main effect of stray light is an
increase of the background level of images that is expected to be corrected
during the data processing. We note that the second and third eclipse
observations of WASP-189 b have some images with very high background levels
compared to the other eclipse observations. For reasons that are not yet well
understood, correlations between photometric flux and background level start
to appear for extreme values of the background, and the two mentioned eclipse
observations are affected by this effect. Discarding frames with extremely
high background levels is a straightforward solution to this problem as
discussed in Section 4.1.

Apart from the differences described previously, the data sets analysed in
this work were obtained in the same conditions. The cadence of the
observations was one 200$\times$200-pixel sub-array every 4.8 s (upper limit
imposed by the saturation of the CCD) but, in order to be able to transfer
data to the Earth with the available downlink bandwidth, images were stacked
together on board by groups of seven. This resulted in an effective
integration time of 33.6 s ($=7\times 4.8\,\text{s}$) per downloaded frame.
The image read-out is performed simultaneously to the next exposure, leading
to an effective data cadence equal to the integration time (duty cycle of
100%). In addition, CHEOPS also provides smaller circular images, called
imagettes, with a radius of 25 pixels and at a higher cadence (one imagette
every 4.8 s). These imagettes do not allow us to perform precise aperture
photometry and were originally meant for visual inspection and cosmic ray
correction, but they can prove to be useful for PSF photometry, as described
in Section 3.2.2. The read-out frequency at which the CCD pixels were read is
100 kHz for all WASP-189 observations. The duration of the observations was
about 12.5 hours for each of the eclipses (about 3 eclipse durations for the
out-of-eclipse baseline) and about 75 hours for each of the phase curves
(encapsulating a full orbit plus an additional transit). The raw light curves
extracted by aperture photometry are shown on Figure 1.

Table 2: List of the CHEOPS observations.

File key 11$1$Each file key refers to a unique observation in the CHEOPS database. | UTC start 22$2$UTC start and end are the starting and ending dates of the observation in UTC. | UTC end 22$2$UTC start and end are the starting and ending dates of the observation in UTC. | Type | $N_{\text{frames}}$ | Efficiency 33$3$The efficiency represents the ratio between the observation time without interruptions (due to Earth occultation or SAA crossings) and the total observation duration. | PSF location | $T_{\text{CCD}}$
---|---|---|---|---|---|---|---
$\left[\%\right]$ | $\left(x,y\right)$ | $\left[{}^{\circ}\text{C}\right]$
CH_PR100041_TG000201_V0200 | 2020-03-19 02:50 | 2020-03-19 15:20 | Eclipse | 736 | 57.8 | $\left(512,512\right)$ | $-40$
CH_PR100041_TG000202_V0200 | 2020-03-27 07:27 | 2020-03-27 20:07 | Eclipse | 804 | 61.5 | $\left(664,296\right)$ | $-45$
CH_PR100041_TG000203_V0200 | 2020-03-30 01:05 | 2020-03-30 13:29 | Eclipse | 838 | 66.0 | $\left(664,296\right)$ | $-45$
CH_PR100041_TG000204_V0200 | 2020-04-07 04:08 | 2020-04-07 17:05 | Eclipse | 943 | 70.6 | $\left(664,296\right)$ | $-50$
CH_PR100036_TG000701_V0200 | 2020-06-15 18:07 | 2020-06-18 21:07 | Phase curve | 4597 | 59.5 | $\left(263,842\right)$ | $-45$
CH_PR100036_TG000702_V0200 | 2020-06-21 08:25 | 2020-06-24 10:23 | Phase curve | 4393 | 57.7 | $\left(263,842\right)$ | $-45$

444 The time spent accumulating photons for each frame (a single frame is here
the result of 7 stacked images) is referred to as the integration time and is
the same for all observations: $t_{\text{int}}=33.6\,\text{s}$. We note that
the image read-out is performed in parallel of the next exposure leading to an
effective data cadence equal to the integration time (duty cycle of 100%).

Figure 1: Raw light curves of the four eclipses and the two phase curves
measured by CHEOPS. The represented data were extracted by aperture photometry
by the mission data reduction pipeline for a circular aperture radius of 25
pixels. Each data set is shifted upward with respect to the one observed
before for visualisation purposes. The raw data points are shown in blue and
binned once per CHEOPS orbit (black). The orange solid line is the best-fit
model obtained with the modelling described in Section 4. The six observations
are represented as a function of time since inferior conjunction according to
the best-fit parameter values.

###  3.2 Data reduction

####  3.2.1 Aperture photometry

All CHEOPS observations were processed with the latest version of the Data
Reduction Pipeline (DRP), tagged version 13.1.0. We based the analysis
presented in this work on these output photometric light curves that were
produced following a procedure described in detail by Hoyer et al. (2020) and
summarised below.

The first step of the data processing is referred to as calibration and aims
to correct the images for the effects related to the detector and the optics.
It consists of subtracting the bias offset of the CCD, converting the digital
counts back to electrons (gain), correcting for the non-linear response of the
read-out electronics, subtracting the contribution of the dark current, and,
finally, correcting for the photo-response non-uniformity (flat field). The
reference files on which the calibration correction is based have been
produced during the on-ground calibration campaign performed on the CHEOPS
payload before launch (Chazelas et al., 2019; Deline et al., 2020).

The second step is called the correction step and removes the spurious effects
such as hits on the detector by high-energy particles (cosmic rays),
background level caused by stray light, and smearing trails of bright close-by
stars induced by the CCD read-out.

The CHEOPS DRP applies an aperture photometry method on the corrected images
that consists of determining the location of the PSF centre (using an
iterative centroiding technique) and counting the flux in electrons that falls
within a given radius from this centre. The DRP provides photometric light
curves for four different aperture radii: a default radius of 25 pixels, a
small radius of 22.5 pixels, a large radius of 30 pixels, and an optimal
radius that depends on the target observed555The plate scale of CHEOPS is 1
arcsecond per pixel.. The optimal aperture is designed to maximise the signal-
to-noise ratio of the photometry and to minimise the contamination from nearby
field stars based on simulations generated by the DRP using the Gaia catalogue
(Gaia Collaboration et al., 2021b) and a CHEOPS PSF template.

The photometric contamination from nearby stars as estimated by the DRP in the
simulations has two components: the direct contamination and the smearing
trails. The direct contamination is caused by nearby stars with PSF falling
entirely or partly inside the photometric aperture. The associated signal is a
positive offset with usually small variations that are in phase with the
rolling of CHEOPS around the line of sight. If the offset is significant, it
must be accounted for as it dilutes the flux from the target and might lead to
underestimate the transit depth for instance. The smearing trails are vertical
features above and below the nearby star PSF on the CCD. As the field of view
rotates, the nearby stars move on the CCD and the trails enter the aperture
periodically, contaminating the photometry. In most cases, both contributions
can be corrected as a function of the roll angle of the spacecraft (provided
in the light curve products of the DRP). In the case of WASP-189 observations,
the brightest nearby stars have magnitude differences larger than 7.8 in the
Gaia G band. They are expected to induce a direct contamination in the
aperture smaller than 0.076% with variations of the order of 0.001%, and a
signal due to smearing trails not greater than 0.021%. We thus consider that
the photometric dilution has a negligible effect on relative photometric
features (e.g. transits or eclipses). The associated variations, however, are
comparable to the estimated photometric precision of the data (signal-to-noise
ratio of the order of unity) and are corrected in the analysis as a function
of the roll angle of the spacecraft.

During the full reduction process, the DRP runs diagnosis to determine if the
data are valid according to several criteria: the location of CHEOPS with
respect to the SAA, the values of various thermal sensors, the angular
separations between the line of sight and three bodies (the Earth, the Moon,
and the Sun), and the number of cosmic ray hits on the detector. If one of
these criteria is out of range, the image is flagged accordingly. The DRP team
recommends discarding all flagged images from photometric analyses.

Figure 1 shows the raw light curves obtained for the default aperture (radius
of 25 pixels). For each observation, we estimate the photometric precision by
computing the median absolute deviation (MAD) of the flux jumps
$df=f_{i+1}-f_{i}$ to be robust to outliers and to remove correlated signals
(e.g. transit, eclipse, stellar activity). The root-mean-square (rms) noise
level $\tilde{\sigma}$ of the observation is then derived with the following
formula: $\tilde{\sigma}=\text{MAD}/2/\text{erf}^{-1}\\!\left(0.5\right)$,
with $\text{erf}^{-1}$ being the inverse error function. This technique allows
us to find results consistent with the ones computed from best-fit residuals
(see Sect. 5.4). We find that the photometric precision of all observations
are very similar, with values at the raw cadence (33.6 s) for the four
eclipses and the two phase curves of 81.3 ppm, 89.8 ppm, 90.5 ppm, 93.9 ppm,
92.6 ppm, and 93.3 ppm, respectively. When applying the same method to several
one-hour-long windows, we find median noise levels per one-hour bins of 11.3
ppm, 11.5 ppm, 10.8 ppm, 11.2 ppm, 12.1 ppm, and 12.1 ppm, respectively.

####  3.2.2 PSF photometry

In addition to the aperture photometry provided by the CHEOPS DRP, we cross-
checked our analysis and the results against another photometric extraction
technique, PSF photometry. This method consists of fitting the images with a
two-dimensional template PSF in order to determine the amplitude of the signal
and derive the flux.

We used a software package called PIPE, developed specifically for CHEOPS. The
code is available on GitHub666https://github.com/alphapsa/PIPE and will also
be extensively described in Brandeker et al. (in prep.).

PIPE first derives a PSF template library from the observed sub-arrays and
imagettes, using a principal component analysis (PCA). The first five
principal components (PCs) together with a constant background are then used
to best fit the PSF of each image using a least-squares minimisation. The
number of PCs to use is a trade-off between following systematic PSF changes
and overfitting the noise, but the derived photometry changes only marginally.
The principal advantage is instead that the PC coefficients can be used to
track PSF changes and correct for the so-called ramp effect (Sect. 4.3).

Apart from serving as an independent extraction method, PIPE has the advantage
of being more robust against cosmic ray hits and telegraphic pixels, since
these can be identified and flagged as statistically inconsistent with the PSF
and thereby masked. Another advantage with PSF fitting is that accurate
photometry can be extracted at faster cadence from the smaller imagettes.
Since a large annulus around the target to measure the background is not
required, it is simultaneously fitted with the PSF.

PIPE provides the photometric light curves together with the computed PSF
coordinates, the background level and the relative weights $U_{X}$ of the
first five principal components of the PSF PCA. Similarly to the DRP, images
can be flagged for several reasons, such as a centroid far away from the
median centroid, high level of bad pixels, or poor PSF fit.

Following the method described in the previous section, we obtain an estimate
of the photometric precision that is very similar to the aperture photometry,
albeit slightly better. At the raw cadence (33.6 s), one obtains 76.9 ppm,
84.4 ppm, 86.8 ppm, 88.6 ppm, 86.6 ppm, and 86.6 ppm for the four eclipses and
the two phase curves, respectively. When considering one-hour-long windows,
one reaches 10.8 ppm, 11.0 ppm, 11.1 ppm, 10.6 ppm, 11.5 ppm, and 11.2 ppm,
respectively.

##  4 Light-curve analysis

###  4.1 Outlier removal

Prior to the analysis of the light curves, we perform a filtering of the data
by first discarding all points flagged during the photometric extraction
process, either aperture photometry from the DRP or PSF photometry from PIPE.
The flagged data points represent 458 / 12770 images (3.59%) for the aperture
photometry and 493 / 12770 (3.86%) and 2351 / 89390 (2.63%) for PSF photometry
on the sub-arrays and imagettes, respectively.

In addition, we apply two other criteria to identify data points as outliers
and discard them. The first criterion is related to the level of background in
the images. The change of background level is mostly due to variations in the
amount of stray light entering the telescope along the orbit of CHEOPS, with
the main sources of stray light being the Earth and the Moon. All observations
are affected by a periodic increase of background level before and after Earth
occultations. As mentioned in Section 3.1, the second and third eclipse
observations of WASP-189 show much higher background levels (up to 10 times
the nominal values) when compared to the other eclipse and phase-curve time
series. While inspecting the housekeeping parameters for possible
correlations, we noticed that these extreme background values resulted in an
unexpected effect on photometry. As shown in Figure 2, the measured flux
becomes suddenly strongly anti-correlated with the background level above a
given background value. This effect is present for all photometric apertures
(DRP) as well as in the PSF photometry of sub-arrays (PIPE). However, it is
not visible in the PSF photometry of imagettes, possibly due to the higher
photometric noise of the data linked to the shorter cadence. We checked the
moon separation for these observations and it appeared to be greater than
$140\deg$, and similar to the one of the first phase curve. The fourth eclipse
had the smallest moon separation of the whole data set ($\sim 35\deg$) but
shows no extreme background levels. As this correlation is not yet understood
and being investigated, we decided not to correct for it, but to clip out data
points with background levels above a threshold that varies with the
extraction technique (see Table 4). This step results in discarding 45 / 3321
points (1.36%) for the two smallest apertures (22.5 and the default 25
pixels), 65 / 3321 points (1.96%) for the two largest apertures (30 and the
optimal 40 pixels), and 52 / 3310 points (1.57%) for the PSF photometry on
sub-arrays.

Figure 2: Flux extracted with aperture photometry (DRP, aperture of 25 pixels)
as a function of the image background levels for the four eclipse observations
of WASP-189 b. This figure highlights the unexpected anti-correlation
occurring above a given background value. The vertical dotted black line shows
the threshold used to discard data points.

The second criterion is a sigma-clipping approach that differs slightly
between eclipses and phase curves. Both methods rely on an estimate
$\tilde{\sigma}$ of the noise level using a method robust against outliers
that fits a Gaussian distribution to the histogram of the data and computes
$\tilde{\sigma}$ from the Gaussian width. For the eclipses, we normalise each
individual observation by its median. We then perform a least-squares fit of
the data with a linear trend for each observation and a common eclipse model
from batman (Kreidberg, 2015). Points beyond $4\,\tilde{\sigma}$ from the
median value of the residuals are flagged as outliers. For the phase curves,
both time series are normalised by their global median value. The model used
to de-trend the data is a combination of a symmetric transit model (no GD)
from batman and a single slope. For the DRP and PIPE sub-array photometry,
data points are discarded when off by more than $6\,\tilde{\sigma}$ from the
residual median. The sigma-clipping limit is reduced to $5\,\tilde{\sigma}$
for the PIPE light curves obtained from imagettes. The number of outliers
flagged in the aperture photometry is 33 / 12267 points (0.27%), 39 / 12267
points (0.32%), 43 / 12247 points (0.35%), and 55 / 12247 points (0.45%) for
the 22.5-, 25- (default), 30-, and 40-pixel (optimal) aperture radii,
respectively. For the PSF photometry, we identify 10 / 12225 points (0.08%) as
outliers for sub-arrays and 15 / 87039 points (0.02%) for the imagettes.

###  4.2 Flux normalisation

The analysis of the photometric time series starts with the normalisation of
the measured flux before modelling the systematics and astrophysical signals.
As expected, the changes of observation conditions that occurred in-between
eclipse observations (see Section 3.1) modified the absolute stellar flux
measured with CHEOPS. The location of the PSF and the temperature of the
detector affect the flux repartition on the CCD (PSF shape) and the conversion
factor from photo-electrons to digital units (gain of the read-out
electronics), which in turn have an impact on the measured photometry. In
order to account for this effect, we decided to include in our model an
individual normalisation factor for each of the six eclipse and phase-curve
observations that is applied before any of the data analysis steps described
hereafter.

###  4.3 Systematic noise

The first systematic noise to be mentioned is specific to CHEOPS and is
related to the rotation of the spacecraft around the Earth. In order to
guarantee the thermal stability of the detector, the passively cooling
radiator of the instrument must never face the Sun nor the Earth, which
implies a continuous rolling of the spacecraft around its pointing direction
with one full rotation per orbit. This nadir-locked orientation implies a
rotation of the field of view around the target PSF, at a rate not necessarily
constant, which causes nearby stars and stray light sources to move around and
induce a periodic photometric noise. We implement a modelling of this
systematic effect using the first terms of the following Fourier series:

$\mathcal{S}_{\text{roll}}\\!\left(\theta_{\text{roll}}\right)=\sum_{i=1}^{N}{a_{i}\cos\\!\left(i\,\theta_{\text{roll}}\right)+b_{i}\sin\\!\left(i\,\theta_{\text{roll}}\right)},$
(1)

where $\theta_{\text{roll}}$ is the roll angle of CHEOPS provided in the data,
and $a_{i}$ and $b_{i}$ are free parameters. We chose to limit the model to
the first five terms of the series ($N=5$ in Eq. 1) as a trade-off between the
quality of the fit, the number of parameters and to avoid overfitting the
data. In addition, we implemented one model for the four eclipse observations
and another for the phase-curve observations, or in other words, one set of
coefficients $\left(a_{i},b_{i}\right)_{\text{ecl}}$ for the eclipses and one
set $\left(a_{i},b_{i}\right)_{\text{PC}}$ for the phase curves. This decision
is motivated by the fact that there is a gap of nearly 70 days between the
eclipse observations and the phase-curve observations. Over this two-month
time span, the Earth moves significantly around the Sun, modifying its
position with respect to WASP-189 in the field of view of CHEOPS. As a
consequence, the photometric systematics associated with the roll angle are
different and could not be modelled by the same set of coefficients.

Another well-known systematic effect in the CHEOPS data sets is the so-called
ramp effect. The source of this ramp is a thermo-mechanical distortion of the
telescope tube that induces variations in distance and orientation of various
optical components, including in particular the secondary mirror. This
distortion is hard to predict as it depends on the orientation of the
spacecraft with respect to the Sun during the observation preceding the one
being analysed. The ramp has been extensively studied, and it has been found
that (a) its duration can last up to more than ten hours with a behaviour
similar to that of a thermal relaxation (exponential decay), (b) its
photometric effect is correlated with the shape of the PSF, and (c) its
photometric effect is correlated to the outputs from several thermal sensors,
and in particular one referred to as _thermFront_2_. The study also concluded
on a possible modelling of the effect as a linear relationship between the
flux and the _thermFront_2_ temperature (Maxted et al., 2021). Based on this
conclusion, we included a modelling of the ramp effect as a linear
relationship between the flux and the telescope temperature as follows:

$\mathcal{S}_{\text{therm}}\\!\left(T_{\text{thermFront\\_2}}\right)=c_{\text{therm}}\,\Delta
T_{\text{thermFront\\_2}},$ (2)

where $T_{\text{thermFront\\_2}}$ and $\Delta T_{\text{thermFront\\_2}}$ are
the _thermFront_2_ temperature and its deviation from its median value
respectively, and $c_{\text{therm}}$ is a free parameter defining the strength
of the correlation.

To account for other long-term systematic effects, we included a trend model
for each observation. This model includes a linear slope with time for each
data set, plus an additional quadratic trend for each of the eclipse
observations, as shown in the following equation:

$\mathcal{S}_{\text{trend}}\\!\left(t\right)=c_{2}\,\left(t-t_{0}\right)^{2}+c_{1}\,\left(t-t_{0}\right),$
(3)

where $t$ is the time, $t_{0}$ is the mid-time of each observation, $c_{2}$
and $c_{1}$ are the quadratic and linear trend coefficients for each
observation ($c_{2}=0$ for the phase curves). The nature of the corrected
trends is not clearly determined and could be due to imperfect instrumental
long-term stability, stellar activity or both. However, in light of the
modelling of stellar activity in the phase curves discussed in Section 4.5, we
strongly suspect that most of the trends in the eclipse observations are of
stellar origin, but this could not be assessed given the short duration of
each observation.

Based on the available housekeeping data, we performed an extensive study on
possible correlations between the photometry and all the other parameters. We
found no correlation other than the ones mentioned before that required
modelling and correction.

###  4.4 Planetary model

The modelling of the photometric signal from WASP-189 b along its orbit is
decomposed in three contributions that are described below. The first and
second contributions are the transit model, for when the planet passes in
front of the gravity-darkened star, and the eclipse model, for when the planet
is hidden by the star. The third contribution is the phase-curve model that
describes the flux received by the observer from the planetary surface as a
function of its position around the star.

####  4.4.1 Transit model with stellar gravity darkening

The fast-rotating nature of WASP-189 causes the centrifugal force to have a
non-negligible effect with respect to the surface gravity. As a consequence,
the effective surface gravity at the stellar equator is smaller than the one
at the poles and the star becomes oblate. Based on the von Zeipel theorem (von
Zeipel, 1924), one can show that the radiative flux at a given latitude on the
rotating star is proportional to the local effective surface gravity (e.g.
Maeder, 2009), and this implies the following relation:

$T\\!\left(\vartheta\right)=T_{\rm pole}\left(\frac{g_{\rm
eff}\\!\left(\vartheta\right)}{g_{\rm eff,\,pole}}\right)^{\beta},$ (4)

where $T\\!\left(\vartheta\right)$ and $T_{\rm pole}$ are, respectively, the
temperatures at a given colatitude $\vartheta$ and at the poles
($\vartheta=0$), $g_{\rm eff}\\!\left(\vartheta\right)$ and $g_{\rm
eff,\,pole}$ are, respectively, the effective surface gravities at the
colatitude $\vartheta$ and at the poles, and $\beta$ is the GD exponent. The
value of $\beta$ is 0.25 for a purely radiative envelope but can deviate from
the theory as measured for the star Altair ($\beta=0.190\pm 0.012$, Monnier et
al., 2007). The previous equation shows that, as the stellar rotation reduces
the local effective gravity, the equator gets cooler and thus appears darker
than the poles. The GD leaves a peculiar photometric signature when a planet
transits in front of its star and hides regions with varying brightness,
leading to asymmetric transit light curves when the orbit is misaligned.

As shown by Lendl et al. (2020), the transit light curve of WASP-189 b shows
such GD features and one must account for this effect in the modelling of the
data. In our analysis, we make use of
pytransit777https://github.com/hpparvi/PyTransit (Parviainen, 2015), version
2.5.13, which provides gravity-darkened transit models implemented based on
Barnes (2009)888We note that there is a typo in Eq. 14 of Barnes (2009), and
the terms $(1-f^{2})$ should be replaced by $(1-f)^{2}$.. The base assumption
of various equations of the code is that the gravitational potential follows a
Roche model, which is equivalent to assuming that only the outer layers of the
star are distorted by rotation, meaning that the inner layers are spherical,
hence producing the same gravitational potential as if the whole mass were
concentrated at the centre of the star. pytransit represents the star as a
discretised oblate sphere and computes the transit luminosity dip with a
discretised planetary disc crossing and partially hiding the stellar object
projected onto the plane of the sky. The effective surface gravity $g_{\rm
eff}$ at each point on the stellar surface is evaluated from the Newtonian
gravity force and the centrifugal force:

$\overrightarrow{g_{\rm
eff}}\\!\left(\vartheta\right)=-\frac{G\,M_{\star}}{r_{\vartheta}^{2}}\,\overrightarrow{u_{r}}+\left(\frac{2\pi}{P_{\star}}\right)^{2}R_{\star}\sin\\!\left(\vartheta\right)\,\overrightarrow{u_{\rm
x}},$ (5)

where $G$ is the universal gravitational constant, $M_{\star}$ is the stellar
mass, $r_{\vartheta}$ is the distance from the stellar centre of the point
considered, $P_{\star}$ is the rotation period of the star, $R_{\star}$ is the
stellar radius, and $\vartheta$ is the colatitude of the point. The unit
vectors $\overrightarrow{u_{r}}$ and $\overrightarrow{u_{\rm x}}$ point
outwards, in the opposite direction to the stellar centre and perpendicularly
to the spin axis of the star, respectively. The local stellar radius
$r_{\vartheta}$ equals $R_{\star}$ at the equator and decreases down to
$R_{\rm pole}=R_{\star}\left(1-f_{\star}\right)$ at the poles ($\vartheta=0$),
where $f_{\star}$ is the stellar oblateness that can be expressed as a
function of the stellar parameters:

$f_{\star}=1-\frac{R_{\rm
pole}}{R_{\star}}=\frac{2\pi^{2}R_{\star}^{3}}{2\pi^{2}R_{\star}^{3}+G\,M_{\star}P_{\star}^{2}}=\frac{3\pi}{2\,G\,\rho_{\star}P_{\star}^{2}}$
(6)

The temperature map of the star is computed from Eqs. 4 and 5 for every
discretised surface point. The conversion from temperature to measured flux
requires two additional elements, which are the flux emission spectrum
$\mathcal{S}\\!\left(\lambda,T\right)$ for a given temperature $T$ and the
instrument sensitivity or passband $\mathcal{T}_{\rm
inst}\\!\left(\lambda\right)$. pytransit provides the option to use synthetic
spectra from the PHOENIX library (Husser et al., 2013), which are more suited
than black-body laws to approximate the emission spectra of hot stars such as
WASP-189. We compute the CHEOPS passband by combining the optical throughput
of the telescope and the quantum efficiency of the detector that are both
available as reference files in the CHEOPS mission archive999https://cheops-
archive.astro.unige.ch/archive_browser/. The measured flux from a given point
can then be computed after including the limb-darkening effect at the
considered location:

$\mathcal{F}\\!\left(\vartheta,\mu\right)=\int_{\lambda=0}^{+\infty}\\!\mathcal{S}\\!\left(\lambda,T\\!\left(\vartheta\right)\right)\mathcal{T}_{\rm
inst}\\!\left(\lambda\right)d\lambda\,\times\,\mathcal{I}\\!\left(\mu\right),$
(7)

where $\mathcal{F}\\!\left(\vartheta,\mu\right)$ is the local flux, $\lambda$
is the wavelength, $\vartheta$ is the colatitude of the point, and
$\mu=\sqrt{1-x^{2}}$ with $x$ the normalised radial coordinate of the point.
The term $\mathcal{I}\\!\left(\mu\right)$ represents the local attenuation due
to the limb darkening and is implemented in pytransit with the quadratic law
$\mathcal{I}\\!\left(\mu\right)=1-u_{1}\left(1-\mu\right)-u_{2}\left(1-\mu\right)^{2}$,
where $u_{1}$ and $u_{2}$ are the two limb-darkening parameters.

The implementation of pytransit limits the wavelength range of the PHOENIX
spectra $\mathcal{S}\\!\left(\lambda,T\\!\left(\vartheta\right)\right)$ from
300 to 1000 nm, while the CHEOPS passband $\mathcal{T}_{\rm
inst}\\!\left(\lambda\right)$ covers the wavelength range from 330 to 1100 nm.
As shown in Fig. 3, the combination of low stellar flux emission and poor
CHEOPS sensitivity in the 1000-1100 nm range makes the contribution of this
part of the spectrum in the integrated stellar flux small ($<0.2\%$). Using
pytransit and a black-body approximation for WASP-189, we estimated the effect
of not accounting for wavelengths longer than 1000 nm to have an impact on the
transit depth smaller than 0.2 ppm, which we considered to be negligible.

Figure 3: Synthetic SED from the PHOENIX library (Husser et al., 2013) of a
star similar to WASP-189. The stellar properties are an effective temperature
$T_{\text{eff}}=8000\,K$, a surface gravity $\log g=4.0,$ and a solar
metallicity. The stellar SED in blue is represented over the CHEOPS passband
(330-1100 nm) shown as a black dotted line. The effective SED as seen by the
CHEOPS instrument is the solid orange line, for which one can see that most of
the energy lies within the 400-800 nm range and the contribution beyond 1000
nm is marginal.

The pytransit model we implemented has the following parametrisation: the time
of inferior conjunction $T_{0}$; the period of the planetary orbit $P$; the
planet-to-star radii ratio $k=R_{p}/R_{\star}$; the normalised semi-major axis
of the planetary orbit $a/R_{\star}$; the inclination of the planetary orbit
$i_{p}$; the eccentricity $e$ and the argument of periastron $\omega$ combined
into two parameters $e\cos\\!\left(\omega\right)$ and
$e\sin\\!\left(\omega\right)$; the two coefficients $u_{1}$ and $u_{2}$ of the
quadratic limb-darkening law $\mathcal{I}\\!\left(\mu\right)$; the projected
stellar rotation speed $v_{\star}\sin\\!\left(i_{\star}\right)$; the
temperature of the stellar poles $T_{\rm pole}$; the stellar inclination
$i_{\star}$; the projected orbital obliquity $\lambda_{p}$; the GD exponent
$\beta$; the stellar equatorial radius $R_{\star}$; and the stellar mass
$M_{\star}$. One must note that the time of inferior conjunction $T_{0}$ might
differ from the mid-transit time (mid-time between first and fourth contacts)
in the case of oblate stars. For configurations where the orbit is misaligned
but not perpendicular to the stellar equatorial plane ($\lambda_{p}\neq
90\deg$) and the impact parameter is not zero ($i_{p}\neq 90\deg$), the
oblateness of the star will cause a shift in mid-transit time with respect to
$T_{0}$ due to a late ingress or an early egress depending on the system
orientation. The parameter $R_{\star}$ is the stellar radius at the equator
and is the one used to normalise the planet radius $R_{p}$ and the semi-major
axis $a$ and to compute two key parameters of the GD effect: the rotation
period and the density of the star. The orientation of the planetary system is
fully described by the three angles $i_{p}$, $i_{\star},$ and $\lambda_{p}$,
which need to be unambiguously defined in the case of GD. The convention on
angular geometry used in this work is detailed in Fig. 14. The stellar
inclination is allowed to vary from $0\deg$ (north pole on) to $180\deg$
(south pole on). The orbital inclination $i_{p}$ is also defined on the
interval $\left[0,180\right]\deg$ with $i_{p}=90\deg$ corresponding to a
transit through the center of the stellar oblate disc. The projected orbital
obliquity $\lambda_{p}$ varies in the $\left[-180,180\right]\deg$ range and
affects the orientation of the planet transit path around the centre of the
stellar oblate disc.

####  4.4.2 Eclipse model with an oblate star

Similarly to transits (see Section 4.4.1), the shape of the light curve during
the occultation of the planet by its host will be affected by the oblateness
of the star, causing a shift of the mid-eclipse time with respect to the time
of superior conjunction and a change of shape of the ingress and egress. We
note that this effect will be even more pronounced for grazing eclipses (and
transits), but this does not concern the case of WASP-189 b.

For our analysis, we implemented an eclipse model based on the gravity-
darkened transit model from pytransit to be consistent with the oblateness
inferred by a given set of parameters. The model is the same as the transit
model described in the previous section, with a few changes in the parameter
values, as detailed below. The value of the time of inferior conjunction
$T_{0}$ is replaced by the time of superior conjunction, which is half an
orbital period after $T_{0}$ for circular orbits and can be computed from
Kepler’s equation, relating the mean and eccentric anomalies, for eccentric
orbits. The values of the limb-darkening coefficients $u_{1}$ and $u_{2}$ and
the GD exponent $\beta$ are set to zero to generate a uniform oblate stellar
disc. The value of the argument of periastron $\omega$, the inclination of the
planetary orbit $i$ and the projected orbital obliquity $\lambda_{p}$ are
modified as if the system were being observed from the other side. This
transformation leads to the following sets of parameters:
$\omega_{\text{ecl}}=\omega+\pi$, $i_{p,\,\text{ecl}}=\pi-i_{p}$, and
$\lambda_{p,\,\text{ecl}}=-\lambda_{p}$.

The generated light curve corresponds to a transit in front of a uniform
oblate stellar disc and has to be normalised to obtain the eclipse model for a
fast-rotating star. The normalisation is performed so that the flux when the
planet is fully occulted is 0, and the out-of-eclipse flux is equal to 1 (see
Fig. 4). The normalised eclipse light curve is then multiplied by the phase-
curve signal computed in the next section, which defines the eclipse depth.

Figure 4: Eclipse models with and without accounting for the oblateness of the
host star. The parameters used in this example are the ones reported in Lendl
et al. (2020) except for the stellar rotation speed (increased by a factor of
2 to enhance the effect of rotation), the projected orbital obliquity (set to
$40\deg$), and the orbital inclination (set to $93\deg$). The left panel shows
the light curves normalised by the out-of-eclipse flux for the same system
orientation with an oblate stellar disc (solid black line) and a spherical
star (orange dashed line). On the right we show a projected view of the system
during the eclipse event. The black ellipse and the orange dashed circle show
the limb of the star in both cases with the same colour code. The path of the
planet is represented by the straight black line and arrows, with the planet
being to scale (small black disc).

####  4.4.3 Phase-curve signal

The phase-curve model used in this work is analytical and assumes the planet
is a perfect Lambertian reflector (isotropic scattering) with a given
geometric albedo $A_{g}$, and the thermal emission is approximated by a
sinusoidal function of the planetary phase. To describe the orbital phase of
the planet, we defined the phase angle of the planet
$\alpha=\arccos\\!\left[-\sin\\!\left(\omega+\nu\right)\sin\\!\left(i_{p}\right)\right]$,
where $\omega$ is the argument of periastron, $\nu$ is the true anomaly, and
$i_{p}$ is the orbital inclination. The values of the phase angle range
between 0 at superior conjunction and $\pi$ at inferior conjunction. The
reflected flux component due to the isotropic reflection of the stellar light
off the planetary atmosphere can be written analytically (Sobolev, 1975;
Charbonneau et al., 1999):

$\frac{F_{\text{refl}}}{F_{\star}}=A_{g}\,\left(\frac{R_{p}}{a}\frac{1+e\cos\\!\left(\nu\right)}{1-e^{2}}\right)^{2}\,\frac{\sin\\!\left(\alpha\right)+\left(\pi-\alpha\right)\cos\\!\left(\alpha\right)}{\pi},$
(8)

where $F_{\star}$ is the stellar flux, $A_{g}$ is the geometric albedo,
$R_{p}$ is the planetary radius, $a$ is the semi-major axis, $e$ is the
eccentricity, $\nu$ is the true anomaly, and $\alpha$ is the phase angle. The
thermal emission flux is approximated by the following function of the phase
angle:

$F_{\text{therm}}=\left(F_{\text{day}}-F_{\text{night}}\right)\frac{1+\cos\\!\left(\alpha_{\text{therm}}\right)}{2}+F_{\text{night}},$
(9)

where $F_{\text{day}}$ and $F_{\text{night}}$ are the planetary fluxes of the
dayside and the nightside, respectively, and

$\alpha_{\text{therm}}=\arccos\\!\left[-\sin\\!\left(\omega+\nu-\phi_{\text{therm}}\right)\sin\\!\left(i_{p}\right)\right],$
(10)

with $\phi_{\text{therm}}$ being the phase shift of the thermal emission
accounting for hotspot offset. In total, our phase-curve model makes use of
four parameters in addition to the ones already provided to the transit model
described in Section 4.4.1: the geometric albedo $A_{g}$, the planet dayside
and nightside fluxes, and the hotspot offset $\phi_{\text{therm}}$.

In the framework of this analysis, we also used another phase-curve model with
a more complex implementation. The reflective component was more generic and
allowed a divergence from a Lambertian profile as described in Heng et al.
(2021). The thermal emission of the planet was computed from 2D temperature
maps and integrated in the CHEOPS passband as detailed in Morris et al.
(2021). This approach involved more free parameters and provided inconclusive
results: the reflective component was consistent with a Lambertian profile and
the thermal map was not constrained mostly due to the strong degeneracy
between reflected light and thermal flux. We thus opted for the model with a
Lambertian reflector and a sinusoidal thermal phase curve.

In addition to the reflective and thermal flux of the phase-curve model, we
implemented the possibility to fit for the ellipsoidal variations (Mazeh,
2008) and the Doppler beaming (Maxted et al., 2000), both approximated by
sinusoidal functions:

$\displaystyle
F_{\text{ell}}=2\,A_{\text{ell}}\,\cos^{2}\\!\left(\omega+\nu\right)\,\sin\\!\left(i_{p}\right),$
(11) $\displaystyle
F_{\text{beam}}=A_{\text{beam}}\,\cos\\!\left(\omega+\nu+\pi\right)\,\sin\\!\left(i_{p}\right),$
(12)

where $\nu$ is the true anomaly, $\omega$ is the argument of periastron,
$i_{p}$ is the orbital inclination, and $A_{\text{ell}}$ and $A_{\text{beam}}$
are the semi-amplitudes of the ellipsoidal variations and the Doppler beaming,
respectively.

The combined model of the light curve including the transit and eclipse models
can be expressed as follows:

$F_{p}=F_{\text{tra}}+\left(F_{\text{refl}}+F_{\text{therm}}\right)\times
F_{\text{ecl}}+F_{\text{ell}}+F_{\text{beam}},$ (13)

where $F_{\text{tra}}$ is the gravity-darkened transit light curve,
$F_{\text{refl}}$ and $F_{\text{therm}}$ are the reflected light and thermal
emission from the planet, $F_{\text{ecl}}$ is the normalised eclipse model,
and $F_{\text{ell}}$ and $F_{\text{beam}}$ represent the contributions from
ellipsoidal variations and Doppler beaming.

####  4.4.4 Light-travel time

The light-travel time (LTT) across the planetary system is accounted for in
the model used in this work. The observation times are converted into
reference times by correcting for the light-travel time along the projected
distance between the current planet position and its position at inferior
conjunction. This choice of reference frame allows us to synchronise the time
of inferior conjunction $T_{0}$ in both time frames. For eccentric orbits,
such a correction is slow as it has to be solved numerically, but fortunately
it simplifies into an analytical formula for circular orbits:

$t_{\text{ref}}=t_{\text{obs}}-\frac{a}{c}\,\left[1-\cos\\!\left(2\pi\frac{t_{\text{obs}}-T_{0}}{P}\right)\right]\,\sin\\!\left(i_{p}\right),$
(14)

where $t_{\text{ref}}$ is the time corrected for LTT, $t_{\text{obs}}$ is the
observation time, $a$ is the semi-major axis, $c$ is the speed of light,
$T_{0}$ is the time of inferior conjunction, and $P$ and $i_{p}$ are the
orbital period and inclination. In this work, we always use Eq. 14 for LTT
correction, even in the cases of non-zero eccentricity, as the orbit of
WASP-189 b is expected to be close to circular ($e\sim 0$). This approximation
avoids the use of slow numerical implementation. The expected amplitude of the
correction is of the order of 50 seconds, which is the LTT between superior
and inferior conjunctions. In practice, the transit, eclipse, and phase-curve
models (Sections 4.4.1, 4.4.2, and 4.4.3) of data points observed at times
$t_{\text{obs}}$ are computed using the corresponding LTT-corrected times
$t_{\text{ref}}$.

###  4.5 Stellar variability

In addition to the instrumental systematics and the planet-related signal, the
photometric time series feature another flux variability that we studied
carefully before including a model for it. Fig. 5 shows the raw flux
variations after the removal of outliers (see Section 4.1) and the de-trending
of a linear slope, where the variability is visible on top of the phase-curve
signal from the planet. The first striking aspect is the absence of
variability in the second phase curve while its detection is unambiguous in
the first one, as revealed by the Lomb-Scargle periodograms (Lomb, 1976;
Scargle, 1982) visible in the same figure. In addition, the maximum of the
peak is located at a period of 1.19 days with a full width at half maximum of
0.43 days, which is consistent with the stellar rotation period of 1.24 days
computed based on Lendl et al. (2020). We computed the spectral window of the
observations to see how the sampling and gaps in the data impact the
periodogram. The spectral window shows peaks corresponding to the orbital
period of CHEOPS and its harmonics. This means these peaks are induced jointly
by the photometric modulation with the roll angle of the spacecraft and the
observation gaps due to Earth occultation and SAA crossings. Other peaks at
longer periods are visible in the spectral window of the observations but none
matches the signal at 1.19 days (see Fig. 5).

Figure 5: Photometric variability observed in the two WASP-189 b phase-curve
time series. _Top._ Raw flux (blue points) variations in ppm around the median
value after removing outliers, de-trending for linear slope and hiding in-
transit (light red shaded area) and in-eclipse (light yellow shaded area)
data. Black points are raw flux variations binned once per CHEOPS orbit (98.90
min). _Bottom._ Lomb-Scargle periodograms of the raw flux variations shown
above. The top panel represents the power spectrum of the first phase curve
observed for BJD ¡ 2 459 020, while the mid panel corresponds to the second
phase-curve observation (BJD ¿ 2 459 021). The bottom panel is the periodogram
of the combined phase-curve time series. The period range covered by each
periodogram goes from the Nyquist period (twice the sampling time) to the
duration of the time series, and thus encompasses all detectable
periodicities. The coloured triangles in the top panel mark the periods of
interest. The four leftmost ones (yellow) represent the fundamental (filled
triangle) and harmonic (empty triangles) frequencies of the CHEOPS orbital
period. The rightmost triangle (black) and the associated vertical dashed line
correspond to the orbital period of WASP-189 b. The red triangle and dashed
line mark the stellar rotation period of 1.24 days computed from Lendl et al.
(2020). The grey shaded area corresponds to the spectral window of the
observation (spectral power induced by the sampling and the gaps of the data).

The spectral type and temperature of WASP-189 places it at the limit between
stars with and without chromospheres and coronae (Fossati et al., 2018).
Therefore, the nature of the stellar outer envelope is not well defined and
might be either convective or radiative, which in turn implies the presence or
absence of stellar spots, respectively. In the former case and the presence of
spots, the stellar rotation is expected to create a photometric signature that
could resemble the one observed in this time series. On the other hand, if the
outer envelope is radiative, the lack of strong superficial magnetic activity
leads to an absence of stellar spots and another mechanism is necessary to
explain photometric variability in phase with the stellar rotation. Recent
asteroseismologic studies (Balona, 2019; Trust et al., 2020) based on long-
term observations with the Kepler space telescope (Koch et al., 1998; Borucki
et al., 2010) and the Transiting Exoplanet Survey Satellite (TESS – Ricker et
al., 2015) argue in favour of the presence of inhomogeneities of unknown
origin at the surface of hot stars that could explain photometric variability
matching the stellar rotation. As shown in the bottom panel of Fig. 2 of Trust
et al. (2020), the photometric variability of the A-type star KIC 6192566
measured by Kepler is remarkably similar to the one observed for WASP-189,
with a strong modulation matching the stellar rotation period that nearly
disappears on a timescale of a few days, before reappearing. Lee & Saio (2020)
recently provided another possible explanation for such a photometric
variability. They show that, when the convective core of an early-type star
rotates slightly faster than its radiative envelope, it can excite non-radial
pulsations (gravity modes) through resonance couplings. The frequency of the
oscillation would then be at the rotating speed of the core, which is almost
identical to the one of the outer layers.

The properties of WASP-189, namely its surface gravity $\log g$ and its
effective temperature $T_{\text{eff}}$, places it in the instability strip
where about 50% of the stars are expected to be hybrid $\delta$ Scuti -
$\gamma$ Doradus pulsators (Uytterhoeven et al., 2011). $\delta$ Scuti stars
exhibit pulsations with periods typically ranging from dozens of minutes to
several hours, while the $\gamma$ Doradus variability is on longer timescales
ranging from half a day to a few days. Therefore, $\gamma$ Doradus pulsation
modes could also explain the photometric variability seen in the CHEOPS phase
curves. However, the surface gravity and effective temperature of the star are
compatible with those of hybrid pulsators presenting both types of pulsations,
albeit more in favour of $\delta$ Scuti modes in this case. Based on the Lomb-
Scargle periodograms shown in Fig. 5, peaks with periods shorter than six
hours are exact harmonics of the CHEOPS orbital period and cannot be
attributed to $\delta$ Scuti pulsations. This leaves only the mode at 1.19
days to be compatible with $\gamma$ Doradus pulsations, but it seems unlikely
that only a single frequency would be visible. For completeness, we ran a non-
adiabatic oscillation computation of a stellar model of WASP-189 to determine
which types of modes are expected to be excited, as well as their oscillation
frequencies. We used the non-adiabatic code MAD (Dupret, 2001; Dupret et al.,
2005), with the inclusion of a time-dependent treatment for the interaction of
convection with pulsation. The results show $\delta$ Scuti oscillation modes
are unstable for this stellar model at frequencies between 225 and 500 µHz, or
periods between 30 and 75 min, but none in the $\gamma$ Doradus regime that
could explain the peak observed in the data.

Based on this study on the possible origin of the photometric variability
observed in the CHEOPS phase curves of WASP-189 b, we attributed the signal to
the stellar rotation and decided to model it with a Gaussian process (GP) to
account for its imperfect periodic nature. We implemented the GP model based
on the code celerite (Foreman-Mackey et al., 2017) and, more specifically,
using the kernel corresponding to a stochastically driven damped harmonic
oscillator (SHO kernel). This kernel is efficient at modelling quasi-periodic
oscillations and is defined by three hyper-parameters $S_{0}$, $Q$ and $P_{0}$
that drive the amplitude, the damping and the period of the oscillations,
respectively. $Q$ is called the quality factor and has to be greater than 0.5
for oscillations to occur. In this case, the damped period of the oscillations
is given by $P_{\text{damped}}=2Q/\\!\sqrt{4Q^{2}-1}\,P_{0}$, which can be
used to estimate the period really fitted by the GP model.

The GP is fitted simultaneously with all other components of the model
described previously. However, a Gaussian process is not a parametric function
as its output also depends on the fitted data. Therefore, at each iteration
the GP model is applied to the data after normalisation and subtraction of all
parametric components of the model. The likelihood of the global model to
represent the data is computed during this last step and involves the
inversion of the covariance matrix generated from the kernel and the error
bars of the data points. During the likelihood computation, an additional
parameter $\sigma_{w}$ acting as a jitter term is added quadratically to the
residual error bars (diagonal of the covariance matrix) in order to account
for possible underestimation of uncertainties in the data.

###  4.6 Prior probabilities of the model parameters

To fit our model to the data, we compute the probability of the full model
(normalisation factor, systematic noise, planet-related signal and stellar
variability) considering the measured data set. The probability is computed by
combining the likelihood of all points to be represented by the model for a
given set of parameter values, and the prior probabilities placed on the model
parameters. We sample the posterior probability using a MCMC algorithm based
on the code emcee described in Foreman-Mackey et al. (2013).

We describe here the choice of prior probabilities made for the models
described in Sections 4.2, 4.3, 4.4, and 4.5. Most of the parameters have
broad uniform priors around the expected values with the only aim of reducing
the size of the explorable parameter space and facilitate convergence of the
fit. Choices of parameter prior probabilites motivated by other factors are
detailed below. All prior probabilities are listed in Table 3 alongside the
best-fit parameter values.

####  4.6.1 Eccentricity and argument of periastron

In the final analyses presented in this work, the eccentricity $e$ used for
the computation of the planet-related model is fixed to zero. This is
justified by the fact that the orbits of ultra-hot Jupiters are expected to
reach long-term stability on circular configuration. To support this choice of
setting the eccentricity to zero, we ran a fit with free eccentricity values.
We obtained from the results an upper limit at $3\sigma$ (99.93% confidence)
on the eccentricity of 0.027. The largest values of $e$ were reached for
specific orbital configurations with the line of apsides along the line of
sight ($\omega=90\pm 180\deg$). When discarding these cases, the upper limit
at $3\sigma$ on the eccentricity dropped to 0.005.

####  4.6.2 Limb-darkening coefficients

The values of the limb-darkening coefficients $u_{1}$ and $u_{2}$ are
constrained by boundaries ensuring physically meaningful values of the
intensity on the stellar disc, which correspond to a positive intensity
everywhere on the stellar disc and a monotonically decreasing intensity
towards the stellar limb. These conditions translate into three inequalities
in the case of the quadratic limb-darkening law as detailed in Kipping (2013):
$u_{1}+u_{2}\leq 1$, $u_{1}\geq 0$ and $u_{1}+2u_{2}\geq 0$. In addition to
these boundaries, we included a prior probability on the coefficients, which
corresponds to the likelihood of the quadratic limb-darkening law
$\mathcal{I}\\!\left(\mu\right)$ for a given pair $\left(u_{1},u_{2}\right)$
to represent a theoretical stellar intensity profile. This intensity profile
is generated using the code LDCU101010https://github.com/delinea/LDCU , which
computes limb-darkening profiles and coefficients from two libraries of
synthetic stellar spectra, ATLAS (Kurucz, 1979) and PHOENIX (Husser et al.,
2013), following the method detailed by Espinoza & Jordán (2015). In order to
account for uncertainties $\sigma_{i}$ on the stellar parameters, several
profiles are generated by drawing stellar parameter set from normal
distributions $\mathcal{N}\\!\left(\mu_{i},\sigma_{i}^{2}\right)$. Each
profile is interpolated with a cubic spline over 100 evenly-spaced $\mu$
points as done by Claret & Bloemen (2011) to uniformly spread the weighting of
the profile across the stellar disc. The likelihood $\mathcal{L}_{\text{LD}}$
is computed as a function of the $\chi^{2}$, which is the sum of the quadratic
distance of the theoretical profile points to the evaluated profile
$\mathcal{I}\\!\left(\mu\right)$, and the minimum $\chi^{2}$ value
corresponding to the best possible fit of the theoretical profiles:
$\ln\\!\left(\mathcal{L}_{\text{LD}}\right)=\left(1-\chi^{2}/\chi^{2}_{\text{min}}\right)/2$.
This value is added to the global log-likelihood value and acts as a penalty
for coefficients that would generate a limb-darkening profile completely
different from the best-fit to the synthetic atmospheric models. The inclusion
of the limb-darkening likelihood in the MCMC runs was tested and validated as
it resulted in a faster convergence of the exploration and a negligible effect
on the final parameter values.

####  4.6.3 Gravity-darkening parameters

Prior probabilities are set on the parameters used to constrain the GD effects
in the transit and eclipse models. The projected stellar rotation speed
$v_{\star}\sin\\!\left(i_{\star}\right)$, the temperature of the stellar poles
$T_{\rm pole}$, the stellar equatorial radius $R_{\star},$ and the stellar
mass $M_{\star}$ have normal priors corresponding to the values and
uncertainties listed in Table 1, with the temperature of the poles equal to
the effective temperature. We let the stellar inclination $i_{\star}$ vary
uniformly in the range $\left[0-180\right]\deg$. We place a normal prior on
the projected orbital obliquity $\lambda_{p}$ based on the value derived from
Doppler tomography by Anderson et al. (2018) ($\lambda_{p}=89.3\pm 1.4\deg$).
Finally, we fix the GD exponent $\beta$ to the value provided in Table 2 of
Claret (2016) based on the stellar effective temperature, which gives
$\beta=\beta_{1}/4=0.22$.

We note that, when allowed to vary freely, the parameters $\beta$ and
$\lambda_{p}$ converge to values that are unphysical and inconsistent with
Doppler tomography, respectively. This degeneracy justifies the use of prior
probabilities from Doppler tomography as detailed in Sect. 5 of Hooton et al.
(2021).

####  4.6.4 Phase-curve parameters

In the phase-curve model, we let the geometric albedo $A_{g}$ vary freely
within the theoretical Lambertian range $\left[0,2/3\right]$ and the hotspot
offset $\phi_{\text{therm}}$ within the range $\left[-60,60\right]\,\deg$. We
constrain the dayside flux $F_{\text{day}}$ to positive values. We first ran
an analysis allowing non-zero nightside flux that yields
$F_{\text{night}}=9.1_{-6.8}^{+14.3}\,\text{ppm}$. We therefore fix its value
to zero.

####  4.6.5 Ellipsoidal variations and Doppler beaming

The amplitudes of the ellipsoidal variations and the Doppler beaming are both
fixed to zero. When fitted for, the amplitude of the ellipsoidal variations
are degenerate with the GP modelling the stellar variability, especially in
the second phase curve where the variability is nearly absent, and we obtain
$A_{\text{ell}}=25_{-16}^{+21}$ ppm. Following the formula from Esteves et al.
(2013) and using values from Lendl et al. (2020), we can estimate the order of
magnitude of the semi-amplitude of the ellipsoidal variations:

$A_{\text{ell}}\approx\alpha_{\text{ell}}\frac{M_{p}\sin\\!\left(i_{p}\right)}{M_{\star}}\left(\frac{R_{\star}}{a}\right)^{3}\approx
10\,\text{ppm},$ (15)

where $\alpha_{\text{ell}}$ is a coefficient of the order of unity and $M_{p}$
is the mass of WASP-189 b. The value we obtain by fitting CHEOPS data is much
larger than this estimation, and it can be explained by the fact that large
amplitudes $A_{\text{ell}}$ create a signal in the second phase curve that
mimics the variability modelled by the GP in the first phase curve. The global
model has a higher likelihood given the ability of the GP to correct for this
induced signal, even though its amplitude is not realistic. We thus decide to
fix $A_{\text{ell}}$ to zero to prevent this effect from biasing the results
of the final analyses. The amplitude of the Doppler beaming is consistent with
zero when left free, so we remove a degree of freedom by fixing it to zero as
well. We validate this choice by computing the expected amplitude following
Esteves et al. (2013): $A_{\text{beam}}\approx\alpha_{\text{beam}}\,K/c\approx
1\,\text{ppm,}$ where $\alpha_{\text{beam}}$ is of the order of unity, $K$ is
the radial-velocity semi-amplitude, and $c$ is the speed of light.

In addition, we performed model comparison by including Ellipsoidal
variations, Doppler beaming, or both phase-curve components. We compared the
different best-fit models using the Bayesian information criterion (BIC) and
the Akaike information criterion (AIC). The model that minimises both the BIC
and AIC is the one without Ellipsoidal variations or Doppler beaming. We
report here the BIC and AIC differences with the best model:
$\Delta\text{BIC}=20.5$ and $\Delta\text{AIC}=13.1$ for Ellipsoidal variations
only, $\Delta\text{BIC}=9.4$ and $\Delta\text{AIC}=2.0$ for Doppler beaming
only, and $\Delta\text{BIC}=18.0$ and $\Delta\text{AIC}=3.2$ when including
both. These numbers further validate the choice to discard the contribution of
Ellipsoidal variations and Doppler beaming from our final model.

####  4.6.6 Gaussian process

The last set of prior probabilities are placed on the Gaussian process hyper-
parameters. The amplitude and damping hyper-parameters $S_{0}$ and $Q$ are
sampled logarithmically and allowed to vary in an interval aimed uniquely to
reduce the size of the parameter space and improve convergence speed of the
MCMC run. The undamped period of the oscillations $P_{0}$ is constrained by a
normal prior determined from the oscillation peak identified in the Lomb-
Scargle periodogram (see Fig. 5), which corresponds to a distribution of
$1.2\pm 0.2$ days.

##  5 Results and discussion

In this section, we present the results we obtained from the analyses of the
data sets. We performed several fits on all available light curves obtained
with either aperture or PSF photometry. This represented a total of six light
curves, with four aperture sizes provided by the DRP and two PIPE time series
extracted from sub-array images and at higher cadence from imagettes. The
results obtained from all light curves lead to consistent values for every
planetary parameter. The best precision was reached in the cases of aperture
photometry with the so-called default aperture size (aperture radius of 25
pixels) and PSF photometry on the sub-array images. Unfortunately, the gain in
cadence provided by the imagettes did not provide additional constraints on
the parameter values due to a too significant loss of precision. Even though
the outcomes of the default aperture photometry and PSF photometry on sub-
arrays were extremely similar, we noticed a slight improvement with the latter
with some of the system properties better constrained. Therefore, in this work
we present the best outcome of our analysis obtained with the MCMC analysis of
the light curves extracted with PIPE from the 200$\times$200-pixel subarray
images. The MCMC sampling of the posterior distribution was performed using
128 chains with burn-in phases longer than 20 000 steps to ensure convergence
of the algorithm, and sampling phases of 16 384 steps.

###  5.1 Planetary parameters

The values of the main parameters of our best fit are listed in Table 3. They
are related to the architecture of the planetary system, the stellar activity
or the ramp effect. The other parameters used to normalise and de-trend the
light curve against time and roll angle have their best-fit values listed in
Table 5.

The system orientation is illustrated in Fig. 6, where the star is represented
at its most probable inclination, and several planetary orbits sampled from
the posterior distribution are shown.

Table 3: Planetary parameters obtained with the model assuming a Lambertian
reflector and approximating the thermal emission as a sinusoidal function of
the phase angle.

Fitted parameter | Symbol | Value | Prior | Units
---|---|---|---|---
Time of inferior conjunction | $T_{0}$ | $16.434866\pm 0.000060$ | $\mathcal{U}\\!\left(16.4333,16.4363\right)$ | $\text{BJD}-2\,459\,000$
Orbital period | $P$ | ${2.724035}_{-0.000023}^{+0.000022}$ | $\mathcal{U}\\!\left(2.7213,2.7268\right)$ | days
Planet-to-star radii ratio | $k=R_{p}/R_{\star}$ | ${0.06958}\pm{0.00016}$ | $\mathcal{U}\\!\left(0.069,0.072\right)$ | -
Normalised semi-major axis | $a/R_{\star}$ | ${4.587}_{-0.034}^{+0.037}$ | $\mathcal{U}\\!\left(4.4,4.9\right)$ | -
Orbital inclination | $i_{p}$ | ${84.58}_{-0.22}^{+0.23}$ | $\mathcal{U}\\!\left(82.48,85.59\right)$ | deg
Eccentricity / argument of periastron | $e\cos\\!\left(\omega\right)$ | 0 | fixed | -
$e\sin\\!\left(\omega\right)$ | 0 | fixed | -
Geometric albedo††footnotemark: † |  $A_{g}$††footnotemark: † |  ${0.26}_{-0.17}^{+0.12}$††footnotemark: † | $\mathcal{U}\\!\left(0,2/3\right)$ | -
Dayside thermal flux††footnotemark: † |  $F_{\text{day}}/F_{\star}$††footnotemark: † |  ${42}_{-30}^{+38}$††footnotemark: † | $\mathcal{U}\\!\left(0,+\infty\right)$ | ppm
Nightside thermal flux | $F_{\text{night}}/F_{\star}$ | $0$ | fixed | ppm
Hotspot offset‡‡footnotemark: ‡ |  $\phi_{\text{therm}}$‡‡footnotemark: ‡ |  ${-7}\pm{17}$‡‡footnotemark: ‡ | $\mathcal{U}\\!\left(-60,60\right)$ | deg
Limb-darkening coefficients | $u_{1}$ | ${0.414}_{-0.022}^{+0.024}$ | - | -
$u_{2}$ | ${0.155}_{-0.034}^{+0.032}$ | - | -
Projected rotation speed | $v_{\star}\sin\\!\left(i_{\star}\right)$ | ${92.5}_{-1.7}^{+1.8}$ | $\mathcal{N}\\!\left(93.1,1.7\right)$ | $\text{km}/\text{s}$
Temperature of the stellar poles | $T_{\rm pole}$ | ${7967}_{-67}^{+69}$ | $\mathcal{N}\\!\left(8000,80\right)$ | K
Stellar inclination | $i_{\star}$ | ${68.2}\pm{1.6}$ | $\mathcal{U}\\!\left(0,180\right)$ | deg
Projected orbital obliquity | $\lambda_{p}$ | ${91.7}\pm{1.2}$ | $\mathcal{N}\\!\left(89.3,1.4\right)$ | deg
Gravity-darkening exponent | $\beta$ | 0.22 | fixed | -
Stellar radius | $R_{\star}$ | ${2.363}_{-0.024}^{+0.025}$ | $\mathcal{N}\\!\left(2.365,0.025\right)$ | $R_{\odot}$
Stellar mass | $M_{\star}$ | ${2.073}_{-0.098}^{+0.090}$ | $\mathcal{N}\\!\left(2.031,0.098\right)$ | $M_{\odot}$
Ellipsoidal variations amplitude | $F_{\text{ell}}/F_{\star}$ | 0 | fixed | ppm
Doppler beaming amplitude | $F_{\text{beam}}/F_{\star}$ | 0 | fixed | ppm
Ramp effect coefficient | $c_{\text{therm}}$ | ${-52}_{-33}^{+34}$ | $\mathcal{U}\\!\left(-1000,1000\right)$ | $\text{ppm}/\text{K}$
GP amplitude | $\ln\\!\left(S_{0}\right)$ | ${-22.21}_{-0.69}^{+0.84}$ | $\mathcal{U}\\!\left(-25,-14\right)$ | -
GP quality factor | $\ln\\!\left(Q\right)$ | ${0.85}_{-0.83}^{+0.89}$ | $\mathcal{U}\\!\left(-4,12\right)$ | -
GP undamped period | $P_{0}$ | ${1.27}_{-0.10}^{+0.12}$ | $\mathcal{N}\\!\left(1.2,0.2\right)$ | days
Jitter white noise | $\sigma_{w}$ | ${17.0}_{-3.4}^{+2.8}$ | $\mathcal{U}\\!\left(0,+\infty\right)$ | ppm
Derived parameter |  |  |  |
Optimal time of inferior conjunction | $T_{0,\,\text{opt}}$ | ${21.882937}\pm{0.000048}$ |  | $\text{BJD}-2\,459\,000$
Planetary radius | $R_{p}$ | ${1.600}^{+0.017}_{-0.016}$ |  | $R_{J}$
Semi-major axis | $a$ | ${0.05040}^{+0.00064}_{-0.00060}$ |  | AU
Impact parameter | $b$ | ${0.433}^{+0.014}_{-0.015}$ |  | -
Eccentricity | $e$ | 0 (fixed) |  | -
Eclipse depth | $\delta_{\text{ecl}}$ | ${96.5}^{+4.5}_{-5.0}$ |  | ppm
Stellar rotation period | $P_{\star}$ | ${1.198}^{+0.026}_{-0.025}$ |  | days
Stellar oblateness | $f_{\star}$ | ${2.88}^{+0.15}_{-0.12}$ |  | $\%$
Stellar density | $\rho_{\star}$ | ${0.1617}^{+0.0088}_{-0.0089}$ |  | $\rho_{\odot}$
True orbital obliquity | $\Psi_{p}$ | ${89.6}\pm{+1.2}$ |  | deg
GP damped period | $P_{\text{damped}}$ | ${1.32}^{+0.22}_{-0.11}$ |  | days

111111 The upper part of the table lists the fitted parameters with their
corresponding prior probabilities. Uniform prior probabilities are represented
with $\mathcal{U}\\!\left(x_{\text{min}},x_{\text{max}}\right)$, where
$x_{\text{min}}$ and $x_{\text{max}}$ are the minimum and maximum allowed
values, respectively. Normal (Gaussian) prior probabilities are written as
$\mathcal{N}\\!\left(\mu,\sigma\right)$, where $\mu$ and $\sigma$ are the mean
and standard deviation of the normal distribution, respectively. The lower
part of the table shows the values of parameters derived from the sampled
parameter space. $$\dagger$$$$\dagger$$footnotetext: The values of $A_{g}$ and
$F_{\text{day}}$ both define the eclipse depth and are thus degenerate. This
induces a strong linear correlation between the two parameters and the values
reported in the table are not representative of any convergence. The eclipse
depth obtained from $A_{g}$ and $F_{\text{day}}$ is a well-defined quantity
and should be the one to refer to. $$\ddagger$$$$\ddagger$$footnotetext: The
hotspot offset is unconstrained for small values of $F_{\text{day}}$ where
$A_{g}$ solely contributes to the phase curve amplitude. The value of
$\phi_{\text{therm}}$ reported in the table is computed for values of
$A_{g}<0.05$ to ensure reliable estimates of uncertainties.

Figure 6: Representation of the two possible orientations of the WASP-189
planetary system. The star is in its most probable state with the grey dashed
curves marking constant stellar latitudes and with the north pole upward. In
this representation, the stellar rotation causes the left and right
hemispheres to appear blueshifted and redshifted, respectively. The stellar
oblateness $f_{\star}=2.88^{+0.15}_{-0.12}\,\%$ is highlighted by the grey
dotted circle. The colour scale represents the stellar flux as seen by CHEOPS,
including the limb darkening, the gravity darkening and the instrument
passband. The black disc shows the planet to scale, and the black shaded area
is made of several orbits drawn from the posterior probability distribution
and highlights the uncertainty we obtain on the orbital orientation of
WASP-189 b. The black arrows indicate the direction in which the planet
crosses the stellar disc. The values of the stellar inclination $i_{\star}$,
the orbital inclination $i_{p}$ and the projected orbital obliquity
$\lambda_{p}$ are displayed.

All the fitted planetary parameters are consistent with values obtained in
Lendl et al. (2020). We note that the quantities normalised by the stellar
radius ($k$ and $a/R_{\star}$) are different, however. This is actually due to
the fact that the stellar radius used for normalisation in Lendl et al. (2020)
is the polar radius, while it is the equatorial radius in our approach.
Comparing absolute values of the planetary radius $R_{p}$ and the semi-major
axis $a$ show consistencies within $0.1\,\sigma$.

We note that the time of inferior conjunction $T_{0}$ and the orbital period
$P$ are fully consistent with the values reported in Anderson et al. (2018).
We obtain an exquisite precision on both parameters with fewer than two
seconds for the period and an optimum $T_{0,\,\text{opt}}$ at about four
seconds.

From the fitted parameters, we derive a series of other useful parameter
values, including the eclipse depth that is the amplitude of the phase curve
(reflected and thermal) at superior conjunction ($\omega+\nu=270\deg$). It is
important to mention that the amplitudes of the reflected and thermal parts of
the phase curve are degenerate, especially where the hotspot offset is zero
and the full amplitude is the sum of both contributions. This is the case for
WASP-189 b, and the degeneracy is strong, with either a high geometric albedo
$A_{g}$ and a low dayside thermal flux $F_{\text{day}}$ with unconstrained
$\phi_{\text{therm}}$ values, or a more constrained offset for low $A_{g}$
(see Fig. 15). In order to provide reliable uncertainties on
$\phi_{\text{therm}}$, the value reported in Table 3 is computed for
$A_{g}<0.05$, corresponding to a fully thermal phase curve. We find no
indication for a hotspot offset from the sub-stellar point with the phase
curve peaking at occultation.

The true orbital obliquity $\Psi_{p}$ describes the relative orientation of
the planetary orbit with respect to the spin axis of the star, and we compute
it with the following angular relationship (see Fig. 14):

$\Psi_{p}=\arccos\\!\left[\cos\\!\left(i_{p}\right)\cos\\!\left(i_{\star}\right)+\cos\\!\left(\lambda_{p}\right)\sin\\!\left(i_{p}\right)\sin\\!\left(i_{\star}\right)\right],$
(16)

where $i_{p}$ is the orbital inclination, $i_{\star}$ is the stellar
inclination and $\lambda_{p}$ is the projected orbital obliquity. From the
values of our MCMC runs, we obtain $\Psi_{p}=89.6\pm 1.2\deg,$ which is fully
consistent with a polar orbit.

Despite the convergence of the fit towards a well-defined system orientation,
there remains a degeneracy inherent to gravity-darkened transit photometry.
The following four sets of angular parameters will produce the same
photometric signal:

  1. 1.

$\left(i_{\star},\ i_{p},\ \lambda_{p}\right),$

  2. 2.

$\left(i_{\star},\ 180^{\circ}-i_{p},\ 180^{\circ}-\lambda_{p}\right),$

  3. 3.

$\left(180^{\circ}-i_{\star},\ 180^{\circ}-i_{p},\ -\lambda_{p}\right),$

  4. 4.

$\left(180^{\circ}-i_{\star},\ i_{p},\ 180^{\circ}+\lambda_{p}\right).$

From the observer’s point of view, the difference between configurations 1 and
2 (and between 3 and 4) is that one is the symmetric of the other with respect
to the ‘vertical’ plane containing the line of sight and the spin axis of the
star. Similarly, configuration 1 (respectively 2) is symmetrical to 3
(respectively 4) with respect to the ‘horizontal’ plane containing the line of
sight and perpendicular to the vertical plane mentioned previously.
Spectroscopic observations such as Doppler tomography can rule out two out of
four configurations by measuring if the transiting planet is above the
blueshifted or the redshifted hemisphere of the rotating star. In the case of
WASP-189 b, Anderson et al. (2018) show by Doppler tomography that the polar
orbit only goes over the redshifted hemisphere, which leaves only two possible
scenarios, which are represented in Fig. 6. The second scenario (right-most
panel of Fig. 6), where $i_{\star}>90\deg$ and the south stellar pole is
visible, cannot be ruled out based on the data available, and it gives the
same true orbital obliquity.

We constrained the stellar oblateness $f_{\star}$ of WASP-189 and were able to
determine that its polar radius is about 2.9% smaller than its equatorial
radius due to its fast rotation. This corresponds to a temperature difference
of about 200 K between the poles ($8000\,\text{K}$ by construction) and the
equator ($\sim 7800\,\text{K}$).

###  5.2 Phase-curve constraints on the planetary atmosphere

The light curve obtained after removing systematic noise and stellar activity
assuming a Lambertian reflector and a sinusoidal thermal emission is shown in
Fig. 7. The phase curve highlights the exquisite photometric precision of
CHEOPS while the planetary eclipse and the asymmetric transit are visible.
Fig. 8 shows the corrected phase-folded data around the mid-transit time and
clearly reveals the effect of GD on the transit shape. The uncertainty on
transit and eclipse parameters is small compared to the uncertainty found on
the phase-curve signal. This is due to the photometric variability attributed
to the stellar activity that is not perfectly periodic and thus hard to
disentangle unambiguously from the planetary signal.

Figure 7: Phase-folded phase curve of WASP-189 b after correction of
instrumental systematics and stellar activity. This data set is obtained from
a model assuming the planet is a Lambertian reflector and approximating the
thermal emission with a sinusoid function. The data points de-trended from all
systematics and stellar variability are shown in blue in the two upper panels.
The mid-panel shows a zoomed-in view of the upper one to better visualise the
phase-curve signal. The lower panel shows the residual after subtraction of
the best-fit planet model. The black points represent the binned data with 100
bins per orbital period, which corresponds to a bin duration of about 40 min.
The faded orange lines are 100 samples drawn from the posterior probability
obtained from the MCMC run, and thus highlight the uncertainty we obtain on
the best-fit model. Figure 8: Phase-folded transit light curve of WASP-189 b
after correction of instrumental systematics and stellar activity. The top
panel shows the normalised flux and two best-fit transit models with and
without GD. The mid-panel (blue-point residuals) shows the best-fit residuals
for the model accounting for GD. The bottom panel displays the residuals after
removing a transit light curve without accounting for the gravity-darkened
stellar photosphere. The black dashed line is the deviation of the model
without GD from the model with GD.

As described in Section 5.1, we computed the hotspot offset
$\phi_{\text{therm}}$ reported in Table 3 for parameter sets probing a phase-
curve model dominated by thermal emission ($A_{g}<0.05$). We derived a hotspot
offset consistent with zero ($\phi_{\text{therm}}=-7\pm 17\,\deg$). The
precision on this parameter value is likely limited by the uncertainty on the
phase curve induced by stellar activity.

As reported in Section 4.6, we fitted the data with the nightside flux as a
free parameter and found a value consistent with zero. However, we also
obtained a large upper uncertainty on $F_{\text{night}}$ with a $3\,\sigma$
upper limit matching a uniform temperature distribution on the planet surface
($F_{\text{night}}\approx F_{\text{day}}$), which means a heat redistribution
efficiency of $\varepsilon=F_{\text{night}}/F_{\text{day}}\approx 1$. This
results from the use of Gaussian processes to model the stellar activity: its
flexibility was necessary to properly correct the associated signal, but with
the downside of adding considerable degeneracies in the shape of the phase-
curve model. Using the nightside flux we obtain when this parameter is free,
we can nevertheless compute the heat redistribution efficiency in the CHEOPS
passband $\varepsilon=0.23_{-0.17}^{+0.35}$. Assuming a fully thermal phase
curve ($A_{g}\to 0$), we derive a smaller limit of $\varepsilon\to
0.13_{-0.10}^{+0.21}$ (see Fig. 9). Both values are less than $1.5\,\sigma$
away from 0 and have a $3\,\sigma$ upper limit consistent with 1.

Figure 9: Heat redistribution efficiency $\varepsilon$ as a function of the
geometric albedo $A_{g}$. The median value of $\varepsilon$ is represented by
the blue curve and the shaded areas highlight the $1\,\sigma$, $2\,\sigma,$
and $3\,\sigma$ uncertainty range. The value of $\varepsilon$ is computed as
the nightside-to-dayside flux ratio measured by CHEOPS. Low values of $A_{g}$
indicates low reflectivity and a phase-curve amplitude mostly due to thermal
emission. The thermal contribution gets smaller for higher $A_{g}$ where
reflected light starts to be important.

Due to the degeneracy between reflective and thermal components, we could not
determine the contribution of each part in the absence of a significant
hotspot offset. Therefore, we followed an approach assuming two extreme cases
for the phase-curve amplitude.

We first considered the case where the amplitude of the eclipse is only due to
reflected light without any thermal contribution from the planet. The
conversion of the eclipse depth $\delta_{\text{ecl}}$ into reflected light is
done with Eq. 8 for a circular orbit ($e=0$) and at superior conjunction
($\alpha=0$). It provides a maximum geometric albedo $A_{g,\text{max}}=0.42\pm
0.02$, from which we compute a 3 $\sigma$ upper limit (99.93% confidence) and
determine that the geometric albedo of WASP-189 b is $A_{g}<0.48$. This value
is not inconsistent with other measured geometric albedos of ultra-hot
Jupiters, even though these objects usually feature dark atmospheres with low
reflectivity (Wong et al., 2021).

We then explored how the thermal emission contributes to the flux as seen
through the CHEOPS passband. The power irradiating the planet is given by the
following expression:

$P_{\text{irr}}=\pi
R_{p}^{2}\,\left(\frac{R_{\star}}{a}\right)^{2}\int_{\lambda=0}^{+\infty}\\!\mathcal{S}\\!\left(\lambda,T_{\star}\right)d\lambda,$
(17)

where $R_{p}$ is the planet radius, $R_{\star}$ is the stellar radius, $a$ is
the semi-major axis, and $\mathcal{S}\\!\left(\lambda,T_{\star}\right)$ is the
flux emission spectrum of the star as a function of its temperature
$T_{\star}$ and the wavelength $\lambda$. We can compute the power absorbed by
the planetary atmosphere $P_{\text{abs}}=\left(1-A_{B}\right)P_{\text{irr}}$,
with $A_{B}$ being the Bond albedo. The power radiated away by the planet must
equal the one absorbed if it is to be at equilibrium. Assuming the planet is a
black-body radiator, we can derive a relationship between the average
effective temperature of the planet $\tilde{T}_{p}$ and the irradiating power
$P_{\text{irr}}$ and obtain the following:

$\tilde{T}_{p}^{4}=\frac{1-A_{B}}{4\,\sigma_{\text{SB}}}\left(\frac{R_{\star}}{a}\right)^{2}\int_{\lambda=0}^{+\infty}\\!\mathcal{S}\\!\left(\lambda,T_{\star}\right)d\lambda,$
(18)

where $A_{B}$ is the Bond albedo and $\sigma_{\text{SB}}$ is the Stefan-
Boltzmann constant. The average effective temperature of the planet is given
by $\tilde{T}_{p}^{4}=1/{4\pi}\,\iint_{\text{planet}}T_{p}^{4}\,d\Omega$,
where $T_{p}$ is the local effective temperature in the planet atmosphere and
$d\Omega$ describes a surface element of the atmosphere. Following the
parametrisation of Cowan & Agol (2011), we define the dayside and nightside
effective temperatures as functions of the heat redistribution efficiency
$\varepsilon$ and $\tilde{T}_{p}$:

$\displaystyle
T_{\text{day}}=\left(\frac{8-5\varepsilon}{3}\right)^{\frac{1}{4}}\tilde{T}_{p}\,,$
(19) $\displaystyle
T_{\text{night}}=\varepsilon^{\frac{1}{4}}\,\tilde{T}_{p}\,.$ (20)

We computed the ranges of values of both temperatures when the Bond albedo and
the heat redistribution efficiency are allowed to vary within
$\left[0,1\right]$ (see Fig. 10). We used the spectral energy distribution
represented in Fig. 3 for the star and the median value reported in Table 3
for $a/R_{\star}$. In the limit where no energy is absorbed by the planet
($A_{B}\to 1$), we retrieve a temperature of $0\,\text{K}$ everywhere on the
planet. When all the irradiating power is absorbed ($A_{B}=0$) and the heat
redistribution is perfect ($\varepsilon=1$), we obtain a uniform atmospheric
temperature $T_{\text{uni}}=\tilde{T}_{p}=2625\pm 11\,\text{K}$. The maximum
dayside temperature is reached when $A_{B}=\varepsilon=0,$ and one obtains
$T_{\text{day, max}}=3355\pm 14\,\text{K}$ for WASP-189 b.

Figure 10: Dayside (top) and nightside (bottom) effective temperatures of the
atmosphere of WASP-189 b as a function of the Bond albedo $A_{B}$ and the heat
redistribution efficiency $\varepsilon$. The lower limit of the colour scale
is set to $1000\,\text{K}$ for a better visualisation. White dotted lines show
isothermal profile across the ($A_{B},\varepsilon$) plane. The dayside and
nightside temperatures are equal when $\varepsilon=1$ and both go down to
$0\,\text{K}$ in the limit $A_{B}\to 1$. If the dayside temperature is
$3000\,\text{K}$, then the Bond albedo values range from 0 to $\sim\\!0.38$,
with corresponding values of the heat redistribution efficiency from
$\sim\\!0.59$ to 0, respectively. In such a case, the maximum nightside
temperature would be $\sim\\!2250\,\text{K}$.

From the dayside temperatures, we were able to calculate the corresponding
eclipse depth in the CHEOPS passband with the following relationship:

$\frac{F_{\text{day}}}{F_{\star}}=\left(\frac{R_{p}}{R_{\star}}\right)^{2}\frac{\int_{\lambda=0}^{+\infty}\\!\mathcal{S}\\!\left(\lambda,T_{\text{day}}\right)\mathcal{T}_{\rm
inst}\\!\left(\lambda\right)d\lambda}{\int_{\lambda=0}^{+\infty}\\!\mathcal{S}\\!\left(\lambda,T_{\star}\right)\mathcal{T}_{\rm
inst}\\!\left(\lambda\right)d\lambda},$ (21)

where $R_{p}$ and $R_{\star}$ are the planetary and stellar radii,
respectively, $\mathcal{S}\\!\left(\lambda,T_{\text{day}}\right)$ and
$\mathcal{S}\\!\left(\lambda,T_{\star}\right)$ are the flux emission spectra
of the planet and the star, respectively, and $\mathcal{T}_{\rm
inst}\\!\left(\lambda\right)$ is the CHEOPS passband. Similarly to the
computation of the effective temperatures, we used a black-body approximation
for the planet, the spectral energy distribution of Fig. 3 for the star, and
the median value of $k=R_{p}/R_{\star}$ from Table 3. The values of
$F_{\text{day}}/F_{\star}$ are displayed in Fig. 11. We retrieve expected
trends such as a low flux for high Bond albedo values and maximal dayside
thermal flux when $\varepsilon\to 0$.

Figure 11: Dayside thermal flux from WASP-189 b as a function of the Bond
albedo $A_{B}$ and the heat redistribution efficiency $\varepsilon$. The flux
$F_{\text{day}}$ is normalised by the stellar flux $F_{\star}$. Both fluxes
are computed as seen through CHEOPS passband. White dotted lines show constant
flux ratios across the ($A_{B},\varepsilon$) plane. If the measured eclipse
depth contains a thermal emission component with an amplitude 40 ppm, the Bond
albedo may be as large as $\sim\\!0.29$ and as small as zero, and the
corresponding heat redistribution efficiency would be zero and $\sim\\!0.45,$
respectively.

It is noteworthy that the maximum dayside temperature $T_{\text{day, max}}$
computed in the case of black-body emission only produces an eclipse depth of
$68.5\pm 1.8\,\text{ppm}$ in the CHEOPS passband, which is not enough to
explain the measured $\delta_{\text{ecl}}=96.5^{+4.5}_{-5.0}\,\text{ppm}$. The
dayside temperature necessary to generate such a thermal flux is $3542\pm
14\,\text{K}$ and corresponds to a negative Bond albedo ($A_{B}\approx-0.24$
for $\varepsilon=0$). From these values, it seems that a fully thermal
emission with no reflected light cannot explain the eclipse depth measured
with CHEOPS and this in turn implies a minimum geometric albedo $A_{g}>0.12\pm
0.02$.

To further explore the limits on the dayside thermal flux, we computed a
synthetic planetary emission spectrum using the radiative transfer code HELIOS
(Malik et al., 2017, 2019). The model assumes a cloud-free atmosphere in
chemical equilibrium with several sources of opacity as detailed in Lendl et
al. (2020). The inputs used for the computation are a stellar emission
spectrum from the PHOENIX library (Husser et al., 2013) as shown in Fig. 3 and
the median values of the planetary parameters listed in Table 3. We also
assumed that all the irradiating stellar power is absorbed by the planet
($A_{B}=0$) and that there is no heat redistribution ($\varepsilon=0$). We
obtain the SED represented in Fig. 12 with many emission lines in the blue end
of the spectrum leading, in turn, to an increase of flux in the CHEOPS
passband with respect to the black-body approximation.

Figure 12: Spectral energy distribution of the planet WASP-189 b. The high-
resolution synthetic SED computed with HELIOS (Malik et al., 2017, 2019) is
represented in light blue and assumes the irradiating power is fully absorbed
and not redistributed ($A_{B}=\varepsilon=0$). The dark blue line is the
HELIOS data averaged over bins of 5 nm. The black-body profile of the maximum
dayside temperature $T_{\text{day, max}}$ is shown in orange. We note that
accounting for emission lines in the planetary spectrum leads to a different
SED with respect to the black-body approximation, with more energy in the blue
end. This affects the flux measured in the CHEOPS passband (black dotted
line), with the dayside of WASP-189 b appearing slightly brighter.

We computed the dayside flux in the CHEOPS passband using the HELIOS emission
spectrum for the planet in Eq. 21. The resulting eclipse depth is slightly
larger than the black-body case with $77.0\pm 2.5\,\text{ppm}$. This value
remains inconsistent with the measured eclipse depth by $3.5\,\sigma$ and
implies $A_{g}>0.08\pm 0.02$.

In an attempt to maximise the modelled eclipse depth, we performed a final
computation using the same HELIOS model for the planet but accounting for a
cooler and oblate star. We used a stellar emission spectrum from the PHOENIX
library (Husser et al., 2013) for a stellar temperature of 7800 K (minimum
temperature reached at the equator) and reduced the size of the stellar disc
by a factor $1-f_{\star}$ in Eq. 21 to account for its oblateness. These
settings represent a lower limit of the received stellar flux (smaller cooler
star) and maximise the detected eclipse depth up to $87.1\pm 3.1\,\text{ppm}$,
which is marginally consistent ($1.6\,\sigma$) with a fully thermal phase
curve and no reflected light. However, this latest result corresponds to an
extreme limit where $A_{B}=\varepsilon=0$ and the stellar flux is expected to
be underestimated as we assumed a uniform photospheric temperature equal to
the equatorial minimum. As a consequence, the numbers we obtain seem to point
toward a non-zero geometric albedo, with a lower limit of $A_{g}>0.041\pm
0.026$ in the most extreme case. These results are consistent with the
temperatures reported by Lendl et al. (2020), where the dayside temperature
computed with $A_{g}=0$ exceeds the limit of stellar irradiating power.

In light of this study, it is difficult to explain the observed eclipse depth
by pure thermal emission from the atmosphere of WASP-189 b. The conditions we
considered to reach marginal consistency are extreme with the smallest and
coolest possible star, and a full absorption of the stellar irradiation by the
planet. Moreover, the heat redistribution inefficiency maximising the dayside
temperature also implies a nightside temperature of nearly $0\,\text{K}$.
Therefore, it seems very likely that the planetary signal is not purely
thermal and that the atmosphere has a non-negligible reflective component in
the CHEOPS passband. As we have shown in this study, deriving a precise lower
limit on the geometric albedo strongly depends on the assumptions made on the
star, the system architecture and the atmospheric properties of the planet.
Nevertheless, it is noteworthy that the limit on the geometric albedo derived
using the HELIOS emission spectrum is consistent with $A_{g}\sim 0.1$. Similar
geometric albedos have been reported for comparable planets (Wong et al.,
2021), even if the current atmospheric models struggle to explain the presence
of reflective material at temperatures larger than 3000 K.

TESS (Ricker et al., 2015) will observe the WASP-189 system in April-May 2022
when covering sector 51. With its redder passband, TESS is more sensitive to
planetary thermal flux than CHEOPS, and the planet-to-star flux ratio is
expected to be even larger in the case of hot blue stars such as WASP-189.
Thus, if the flux from WASP-189 b is mostly of thermal origin, the eclipse
observed with TESS should be much deeper than the one reported in this work.
Following the method described in this section, we computed the dayside
thermal flux in the TESS passband as a function of $A_{B}$ and $\varepsilon$,
similarly to Fig. 11. We estimate that TESS could measure an eclipse depth due
to thermal emission up to $165\pm 4\,\text{ppm}$. Such a signal should be