11institutetext:  Instituto de Astrofísica de Canarias (IAC), E-38200 La
Laguna, Tenerife, Spain 22institutetext:  Departamento de Astrofísica,
Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain
33institutetext:  Centre for Exoplanet Science, SUPA School of Physics and
Astronomy, University of St Andrews, North Haugh, St Andrews KY16 9SS, UK
44institutetext:  Observatoire Astronomique de l’Université de Genève, Chemin
Pegasi 51, Versoix, Switzerland 55institutetext:  Center for Space and
Habitability, University of Bern, Gesellschaftsstrasse 6, 3012 Bern,
Switzerland 66institutetext:  Dipartimento di Fisica, Università degli Studi
di Torino, via Pietro Giuria 1, I-10125, Torino, Italy 77institutetext:
Physikalisches Institut, University of Bern, Sidlerstrasse 5, 3012 Bern,
Switzerland 88institutetext:  Aix Marseille Univ, CNRS, CNES, LAM, 38 rue
Frédéric Joliot-Curie, 13388 Marseille, France 99institutetext:  Division
Technique INSU, CS20330, 83507 La Seyne sur Mer cedex, France
1010institutetext:  Department of Physics, University of Warwick, Gibbet Hill
Road, Coventry CV4 7AL, United Kingdom 1111institutetext:  Instituto de
Astrofisica e Ciencias do Espaco, Universidade do Porto, CAUP, Rua das
Estrelas, 4150-762 Porto, Portugal 1212institutetext:  Departamento de Fisica
e Astronomia, Faculdade de Ciencias, Universidade do Porto, Rua do Campo
Alegre, 4169-007 Porto, Portugal 1313institutetext:  Space Research Institute,
Austrian Academy of Sciences, Schmiedlstrasse 6, A-8042 Graz, Austria
1414institutetext:  INAF, Osservatorio Astrofisico di Catania, Via S. Sofia
78, 95123 Catania, Italy 1515institutetext:  Institut de Ciencies de l’Espai
(ICE, CSIC), Campus UAB, Can Magrans s/n, 08193 Bellaterra, Spain
1616institutetext:  Institut d’Estudis Espacials de Catalunya (IEEC), 08034
Barcelona, Spain 1717institutetext:  Admatis, 5. Kandó Kálmán Street, 3534
Miskolc, Hungary 1818institutetext:  Depto. de Astrofisica, Centro de
Astrobiologia (CSIC-INTA), ESAC campus, 28692 Villanueva de la Cañada
(Madrid), Spain 1919institutetext:  Université Grenoble Alpes, CNRS, IPAG,
38000 Grenoble, France 2020institutetext:  Department of Astronomy, Stockholm
University, AlbaNova University Center, 10691 Stockholm, Sweden
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, F-75005
Paris, France 2323institutetext:  Science and Operations Department - Science
Division (SCI-SC), Directorate of Science, European Space Agency (ESA),
European Space Research and Technology Centre (ESTEC), Keplerlaan 1, 2201-AZ
Noordwijk, The Netherlands 2424institutetext:  Centre for Mathematical
Sciences, Lund University, Box 118, 221 00 Lund, Sweden 2525institutetext:
Astrobiology Research Unit, Université de Liège, Allée du 6 Août 19C, B-4000
Liège, Belgium 2626institutetext:  Space sciences, Technologies and
Astrophysics Research (STAR) Institute, Université de Liège, Allée du 6 Août
19C, 4000 Liège, Belgium 2727institutetext:  INAF, Osservatorio Astronomico di
Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy 2828institutetext:
Leiden Observatory, University of Leiden, PO Box 9513, 2300 RA Leiden, The
Netherlands 2929institutetext:  Department of Space, Earth and Environment,
Chalmers University of Technology, Onsala Space Observatory, 439 92 Onsala,
Sweden 3030institutetext:  University of Vienna, Department of Astrophysics,
Türkenschanzstrasse 17, 1180 Vienna, Austria 3131institutetext:  Konkoly
Observatory, Research Centre for Astronomy and Earth Sciences, 1121 Budapest,
Konkoly Thege Miklós út 15-17, Hungary 3232institutetext:  ELTE Eötvös Loránd
University, Institute of Physics, Pázmány Péter sétány 1/A, 1117
3333institutetext:  Institute of Optical Sensor Systems, German Aerospace
Center (DLR), Rutherfordstrasse 2, 12489 Berlin, Germany 3434institutetext:
IMCCE, UMR8028 CNRS, Observatoire de Paris, PSL Univ., Sorbonne Univ., 77 av.
Denfert-Rochereau, 75014 Paris, France 3535institutetext:  Institut
d’astrophysique de Paris, UMR7095, CNRS, Sorbonne Université, 98bis blvd.
Arago, 75014 Paris, France 3636institutetext:  Astrophysics Group, Keele
University, Staffordshire, ST5 5BG, United Kingdom 3737institutetext:
Department of Astrophysics, University of Vienna, Tuerkenschanzstrasse 17,
1180 Vienna, Austria 3838institutetext:  Dipartimento di Fisica e Astronomia
”Galileo Galilei”, Universita degli Studi di Padova, Vicolo dell’Osservatorio
3, 35122 Padova, Italy 3939institutetext:  ETH Zurich, Department of Physics,
Wolfgang-Pauli-Strasse 2, CH-8093 Zurich, Switzerland 4040institutetext:
Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK
4141institutetext:  ESTEC, European Space Agency, 2201AZ, Noordwijk, NL
4242institutetext:  Zentrum für Astronomie und Astrophysik, Technische
Universität Berlin, Hardenbergstr. 36, D-10623 Berlin, Germany
4343institutetext:  Institut für Geologische Wissenschaften, Freie Universität
Berlin, 12249 Berlin, Germany 4444institutetext:  ELTE Eötvös Loránd
University, Gothard Astrophysical Observatory, 9700 Szombathely, Szent Imre h.
u. 112, Hungary 4545institutetext:  MTA-ELTE Exoplanet Research Group, 9700
Szombathely, Szent Imre h. u. 112, Hungary 4646institutetext:  Institute of
Astronomy, University of Cambridge, Madingley Road, Cambridge, CB3 0HA, United
Kingdom

#  CHEOPS finds KELT-1b darker than expected in visible light

Discrepancy between the CHEOPS and TESS eclipse depths

H. Parviainen  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible light T. G. Wilson  CHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
light M. Lendl  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible light D. Kitzmann
CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b
darker than expected in visible light E. Pallé  CHEOPS finds KELT-1b darker
than expected in visible lightCHEOPS finds KELT-1b darker than expected in
visible lightCHEOPS finds KELT-1b darker than expected in visible lightCHEOPS
finds KELT-1b darker than expected in visible light L. M. Serrano  CHEOPS
finds KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker
than expected in visible light E. Meier Valdes  CHEOPS finds KELT-1b darker
than expected in visible lightCHEOPS finds KELT-1b darker than expected in
visible light W. Benz  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible light A. Deline CHEOPS finds KELT-1b darker than expected
in visible lightCHEOPS finds KELT-1b darker than expected in visible light D.
Ehrenreich  CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS
finds KELT-1b darker than expected in visible light P. Guterman CHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible light K. Heng
CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b
darker than expected in visible lightCHEOPS finds KELT-1b darker than expected
in visible lightCHEOPS finds KELT-1b darker than expected in visible light O.
D. S. Demangeon  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible light A. Bonfanti  CHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
light S. Salmon  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible light V. Singh
CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b
darker than expected in visible light N. C. Santos  CHEOPS finds KELT-1b
darker than expected in visible lightCHEOPS finds KELT-1b darker than expected
in visible lightCHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible light S. G. Sousa
CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b
darker than expected in visible light Y. Alibert  CHEOPS finds KELT-1b darker
than expected in visible lightCHEOPS finds KELT-1b darker than expected in
visible light R. Alonso  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible light G. Anglada  CHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds
KELT-1b darker than expected in visible light T. Bárczy  CHEOPS finds KELT-1b
darker than expected in visible lightCHEOPS finds KELT-1b darker than expected
in visible light D. Barrado y Navascues  CHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
light S. C. C. Barros  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible light W. Baumjohann  CHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
light M. Beck  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible light T. Beck CHEOPS
finds KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker
than expected in visible light N. Billot  CHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
light X. Bonfils  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible light A. Brandeker
CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b
darker than expected in visible light C. Broeg  CHEOPS finds KELT-1b darker
than expected in visible lightCHEOPS finds KELT-1b darker than expected in
visible lightCHEOPS finds KELT-1b darker than expected in visible lightCHEOPS
finds KELT-1b darker than expected in visible light J. Cabrera CHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible light S. Charnoz  CHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
light A. Collier Cameron  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible light C. Corral Van
Damme CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds
KELT-1b darker than expected in visible light Sz. Csizmadia  CHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible light M. B. Davies  CHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
light M. Deleuil  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible light L. Delrez
CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b
darker than expected in visible lightCHEOPS finds KELT-1b darker than expected
in visible lightCHEOPS finds KELT-1b darker than expected in visible light
B.-O. Demory  CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS
finds KELT-1b darker than expected in visible light A. Erikson CHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible light J. Farinato CHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
light A. Fortier  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible light L. Fossati  CHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
light M. Fridlund  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible light D. Gandolfi  CHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
light M. Gillon  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible light M. Güdel
CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b
darker than expected in visible light S. Hoyer  CHEOPS finds KELT-1b darker
than expected in visible lightCHEOPS finds KELT-1b darker than expected in
visible light K. G. Isaak  CHEOPS finds KELT-1b darker than expected in
visible lightCHEOPS finds KELT-1b darker than expected in visible light L. L.
Kiss CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
light E. Kopp CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS
finds KELT-1b darker than expected in visible light J. Laskar  CHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible light A. Lecavelier des Etangs  CHEOPS finds KELT-1b
darker than expected in visible lightCHEOPS finds KELT-1b darker than expected
in visible light C. Lovis  CHEOPS finds KELT-1b darker than expected in
visible lightCHEOPS finds KELT-1b darker than expected in visible light D.
Magrin  CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds
KELT-1b darker than expected in visible light P. F. L. Maxted  CHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible light M. Mecina CHEOPS finds KELT-1b darker than expected
in visible lightCHEOPS finds KELT-1b darker than expected in visible light V.
Nascimbeni  CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS
finds KELT-1b darker than expected in visible light G. Olofsson  CHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible light R. Ottensamer CHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
light I. Pagano  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible light G. Peter
CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b
darker than expected in visible light D. Piazza CHEOPS finds KELT-1b darker
than expected in visible lightCHEOPS finds KELT-1b darker than expected in
visible light G. Piotto  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible light D. Pollacco CHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
light D. Queloz  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible light R. Ragazzoni  CHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds
KELT-1b darker than expected in visible light N. Rando CHEOPS finds KELT-1b
darker than expected in visible lightCHEOPS finds KELT-1b darker than expected
in visible light H. Rauer  CHEOPS finds KELT-1b darker than expected in
visible lightCHEOPS finds KELT-1b darker than expected in visible lightCHEOPS
finds KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker
than expected in visible lightCHEOPS finds KELT-1b darker than expected in
visible lightCHEOPS finds KELT-1b darker than expected in visible light I.
Ribas  CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
light G. Scandariato  CHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible light D. Ségransan
CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b
darker than expected in visible light A. E. Simon  CHEOPS finds KELT-1b darker
than expected in visible lightCHEOPS finds KELT-1b darker than expected in
visible light A. M. S. Smith  CHEOPS finds KELT-1b darker than expected in
visible lightCHEOPS finds KELT-1b darker than expected in visible light M.
Steller  CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS
finds KELT-1b darker than expected in visible light Gy. M. Szabó CHEOPS finds
KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b darker than
expected in visible lightCHEOPS finds KELT-1b darker than expected in visible
lightCHEOPS finds KELT-1b darker than expected in visible light N. Thomas
CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds KELT-1b
darker than expected in visible light S. Udry  CHEOPS finds KELT-1b darker
than expected in visible lightCHEOPS finds KELT-1b darker than expected in
visible light V. Van Grootel  CHEOPS finds KELT-1b darker than expected in
visible lightCHEOPS finds KELT-1b darker than expected in visible light N. A.
Walton  CHEOPS finds KELT-1b darker than expected in visible lightCHEOPS finds
KELT-1b darker than expected in visible light

(Received September 15, 1996; accepted March 16, 1997)

Recent TESS-based studies have suggested that the dayside of KELT-1b, a
strongly-irradiated brown dwarf, is significantly brighter in visible light
than what would be expected based on Spitzer observations in infrared. We
observe eight eclipses of KELT-1b with CHEOPS (CHaracterising ExOPlanet
Satellite) to measure its dayside brightness temperature in the bluest
passband observed so far, and model the CHEOPS photometry jointly with the
existing optical and NIR photometry from TESS, LBT, CFHT, and Spitzer. Our
modelling leads to a self-consistent dayside spectrum for KELT-1b covering the
CHEOPS, TESS, H, Ks, and Spitzer IRAC 3.6 and 4.5 $\mu$m bands, where our
TESS, H, Ks, and Spitzer band estimates largely agree with the previous
studies, but we discover a strong discrepancy between the CHEOPS and TESS
bands. The CHEOPS observations yield a higher photometric precision than the
TESS observations, but do not show a significant eclipse signal, while a deep
eclipse is detected in the TESS band. The derived TESS geometric albedo of
$0.36^{+0.12}_{-0.13}$ is difficult to reconcile with a CHEOPS geometric
albedo that is consistent with zero because the two passbands have
considerable overlap. Variability in cloud cover caused by the transport of
transient nightside clouds to the dayside could provide an explanation for
reconciling the TESS and CHEOPS geometric albedos, but this hypothesis needs
to be tested by future observations.

###### Key Words.:

stars: individual: KELT-1 – brown dwarfs – stars: planetary systems – stars:
atmospheres – methods: observational

##  1 Introduction

KELT-1b is a strongly-irradiated transiting brown dwarf orbiting a 6500 K
F5-star on a short-period orbit of 1.2 days discovered by Siverd et al.
(2012). It has the third-shortest orbital period of the known transiting brown
dwarfs111According to the Transiting Brown Dwarfs list accessed 16.2.2022.
(Carmichael et al., 2021) after TOI-263b ($P=0.56$ d, Parviainen et al., 2020;
Palle et al., 2021) and NGTS-7Ab ($P=0.7$ d, Jackman et al., 2019), but
receives a significantly higher stellar insolation since TOI-263b and NGTS-7Ab
both orbit low-mass M dwarfs with $T_{\mathrm{eff}}\sim 3400$ K. The orbit and
insolation level of KELT-1b is similar to ultra-hot Jupiters (Parmentier et
al., 2018; Kitzmann et al., 2018; Lothringer et al., 2018) like MASCARA-2b
(Lund et al., 2017; Talens et al., 2018), WASP-121b (Delrez et al., 2016),
WASP-189b (Anderson et al., 2018), WASP-76b (West et al., 2016), and WASP-33b
(Cameron et al., 2010), but its surface gravity is $\sim$22 times larger than
that of Jupiter. This makes KELT-1b an interesting case to study how surface
gravity affects the atmospheres of strongly-irradiated sub-stellar objects.

KELT-1b’s emission spectrum222e use a very loose definition of ”emission
spectrum” that also includes the contribution from reflected light. has been
studied using ground- and space-based eclipse and phase curve observations
covering wavelengths from red-visible (TESS) to 4.5 $\mu$m (Spitzer). Soon
after KELT-1b’s discovery, Beatty et al. (2014) observed one secondary eclipse
using Spitzer (in IRAC 3.6 and 4.5 $\mu$m passbands) and four eclipses in
$z^{\prime}$-band using the 0.6 m RCOS telescope at Moore Observatory. Later,
Croll et al. (2015) observed an eclipse in the Ks-band using the WIRCam on the
Canada-France-Hawaii Telescope, Beatty et al. (2017) observed an eclipse in
the H-band using the LUCI1 spectrograph on the Large Binocular Telescope, and
Beatty et al. (2019) observed a full KELT-1b phase curve with Spitzer.
Finally, the TESS telescope observed KELT-1b continuously for 25 days in 2019,
and the photometry spanning 15 full orbits was studied by Beatty et al. (2020)
and von Essen et al. (2021).

The previous eclipse and phase curve studies have shown that KELT-1b’s eclipse
spectrum in near- and mid-IR agrees with a constant dayside brightness
temperature of $\sim$2900 K (Beatty et al., 2019). However, a study of the
TESS observations by Beatty et al. (2020) presented that the brown dwarf’s
dayside brightness temperature in the TESS band is significantly higher than
in the Spitzer bands. They proposed that this higher-than-expected brightness
in visible light could be due to a high albedo caused by silicate clouds
forming in the nightside of the brown dwarf. KELT-1b’s dayside is too hot for
clouds to form, but the nightside temperature is expected to be cool enough
for the formation of silicate clouds. These nightside clouds could then be
blown over to the dayside by winds, where they could survive all the way to
the local noon, boosting the otherwise low dayside albedo significantly.

However, the cloud hypothesis is not the only viable explanation for the
unexpected dayside brightness in the TESS band. von Essen et al. 2021 made an
independent analysis of the TESS phase curve and proposed that KELT-1b’s
dayside emission spectrum can be explained with thermal emission alone. von
Essen et al. (2021) suggest that the differences in the TESS and Spitzer
brightness temperatures can be explained by collision-induced absorption due
to H2-H2 and H2-He decreasing the brightness temperature in the Spitzer bands.
So, rather than the brightness of the TESS passband being boosted by
reflection from clouds, the Spitzer band brightness would be suppressed by
molecular absorption.

CHEOPS (CHaracterising ExOPlanet Satellite; Benz et al., 2021) is an ESA
S-class mission aiming to characterise transiting exoplanets around bright
($V<12$) stars based on ultra-high quality photometry (Broeg et al., 2013).
The satellite is equipped with a 30 cm telescope that allows it to reach a
photometric precision of 20 ppm in 6 h of integration time for a $V=9$ star
(as a design requirement), which is sufficient to detect Earth-sized planets
around G5 dwarf stars (Benz et al., 2021). CHEOPS was launched on 18 December
2019, the science observations started in April 2020, and the observations
have already been used to improve the characterisation of new and known
exoplanet systems (Bonfanti et al., 2021; Delrez et al., 2021; Szabó et al.,
2021; Barros et al., 2022; Lacedelli et al., 2022; Wilson et al., 2022),
improve the characterisation of eclipsing binaries with M-dwarf companions
(Swayne et al., 2021), search for transit timing variations (TTVs, Borsato et
al., 2021), search for planets transiting white dwarfs (Morris et al., 2021b),
and study secondary eclipses and full phase curves of super-Earths and hot
Jupiters (Lendl et al., 2020; Morris et al., 2021a; Hooton et al., 2022;
Deline et al., 2022). By now, the satellite has shown it can surpass its
design requirements by reaching 10-20 ppm photometric precision for one hour
of integration (Lendl et al., 2020).

In this paper, we present the phase curve analysis of KELT-1b based on new
CHEOPS secondary eclipse photometry and existing photometry from TESS,
Spitzer, LUCI1, and WIRCam. While the CHEOPS and TESS passbands have a
significant overlap, the CHEOPS band emphasises bluer wavelengths than the
TESS band. This difference allows us to test the two hypotheses about the
TESS-Spitzer dayside brightness temperature discrepancy. If the high-albedo
hypothesis is correct, both the CHEOPS\- and TESS-bands are expected to be
dominated by reflected light, and the CHEOPS-band can be expected to also show
a high albedo (albeit this cannot be taken for granted since there are
scenarios where KELT-1b could have a low albedo in the CHEOPS band and high in
the TESS-band, as will be discussed later). If the IR absorption hypothesis is
correct, CHEOPS passband brightness temperature could be expected to follow
closely the model presented in Fig. 9 in von Essen et al. (2021).

All the data and code for the analysis and figures are publicly available from
GitHub github, including additional Jupyter notebooks that work as appendices
to the paper.

##  2 Observations

###  2.1 CHEOPS

Figure 1: The detrended photometry from eight KELT-1b eclipses observed with
CHEOPS centred around the eclipse centre time ($T_{e}$). Each eclipse is
observed as a single continuous CHEOPS visit of $\approx$7 h (labelled as
V1-V8 in the figure), but the satellite’s orbital configuration leads to
periodic gaps in the photometry, as described in Sect. 2.1.

We observed eight eclipses of KELT-1b with the CHEOPS spacecraft between
2020-10-02 and 2020-12-01 as a part of the Guaranteed Time Observers (GTO)
programme (Fig. 1). Each eclipse observation consisted a CHEOPS visit of
$\approx$7 h of near-continuous observations with an exposure time of 60 s.
The length of the visits combined with the high precision of the KELT-1b
ephemeris (thanks to recent TESS observations) ensured that we obtained at
least 2 h of pre- and post-eclipse baseline for each visit. As CHEOPS is in a
low-Earth orbit, sections of the observations are unobtainable as they may
occur during Earth-target occultations, passages through the South Atlantic
Anomaly (SAA), or at times when the stray light level passes an acceptable
threshold. These effects manifest as gaps in the light curves and result in
lower efficiency observations. For the data presented here, we obtain an
average efficiency of 56%.

The data were automatically processed using the latest version of the CHEOPS
Data Reduction Pipeline (DRP v13; Hoyer et al. 2020) that performs image
calibration including bias, gain, non-linearity, dark current, and flat
fielding corrections, and amends of environmental and instrumental effects,
for example cosmic-ray hits, smearing trails, and background variations.
Subsequently, aperture photometry is conducted on the corrected images using a
set of standard apertures; $R$ = 22.5″(RINF), 25.0″(DEFAULT), and 30.0″(RSUP),
with an additional aperture that selects the radius based on contamination
level and instrumental noise (OPTIMAL), which for these observations yields
20.5″. Furthermore, the DRP calculates contamination estimates of the visits
that comes from background sources, as detailed in Section 6.1 of Hoyer et al.
(2020), which we remove from the data.

As the field of view of CHEOPS rotates due to the nadir-locked orbit of the
spacecraft, there may be short-term, non-astrophysical flux trends on the
order of a CHEOPS orbital timescale due to nearby contaminants, background
variations, or changes in instrumental environment. In previous studies these
trends have been corrected via linear decorrelation with instrumental basis
vectors (Bonfanti et al., 2021; Delrez et al., 2021) or Gaussian process
regression (Lendl et al., 2020), however a recent study has identified that
these roll angle trends can be corrected by a novel PSF detrending method
(Wilson et al., 2022). To summarise, this tool determines PSF shape changes of
a visit by computing vectors associated with these variations by conducting
principal component analyses (PCA) of the auto-correlation functions of the
CHEOPS subarray images. The components that contribute most to the PSF shape
changes are subsequently selected by a leave-one-out-cross-validation and are
used as covariates in the light curve model. For this study, we perform this
method on all visits using the DEFAULT aperture photometry fluxes as these
yield the lowest RMS (root mean square) noises.

###  2.2 TESS

The Transiting Exoplanet Survey Satellite (TESS, Ricker et al., 2014) observed
KELT-1b during Sector 17 for 25 days (2019-10-08–2019-11-02) with a two-minute
cadence. The observations cover 15 orbital periods with a high-enough
precision to measure all the significant phase curve components in the TESS
passband.

We use the Presearch Data Conditioning (PDC-SAP) light curve (Stumpe et al.,
2014, 2012; Smith et al., 2012) produced by the SPOC pipeline (Jenkins et al.,
2016). As in Beatty et al. (2020), we remove all the exposures with a non-zero
quality flag, but we do not apply any additional detrending or outlier
rejection. Instead, we use a Celerite-based (Foreman-Mackey et al., 2017)
time-dependent Gaussian process (GP) to calculate the likelihood as explained
in more detail later. Furthermore, we ignore the residual light contamination
of $\approx 0.82\%$ reported by von Essen et al. (2021), since it will not
have any practical effect on the results.

###  2.3 Spitzer

KELT-1 was observed with the Spitzer space telescope using the Infrared Array
Camera (IRAC; Fazio et al. 2004) in both S1 (3.6 $\mu$m) and S2 (4.5 $\mu$m)
passbands across multiple programs with the observations covering a secondary
eclipse (Beatty et al., 2017) and a full phase curve (Beatty et al., 2019).

Using these data, we extract light curves for all visits using our own custom
pipeline. Using the flux-calibrated and artefact-corrected CBCD (Corrected
Basic Calibrated Data) frames we constructed light curves by firstly using the
IRAC archive contributed IDL program
box_centroider.pro333https://irsa.ipac.caltech.edu/data/SPITZER/docs/irac/calibrationfiles/pixelphase/box_centroider.pro
to locate the target in the images. In addition to utilising these values as
decorrelation basis vectors during the joint fitting, these x- and y-centroid
positions were used as inputs into the aper.pro code to perform aperture
photometry using an aperture radius of 3 pixels and a sky annulus 12–20
pixels. Lastly, we perform the aperture corrections, correct intrapixel gain
using the
iracpc_pmap_corr.pro444https://irsa.ipac.caltech.edu/data/SPITZER/docs/dataanalysistools/tools/contributed/irac/iracpc_pmap_corr/
tool, and reject data with fluxes or x- and y-centroid positions 5$\sigma$
away from the respective, normalised mean.

###  2.4 Large Binocular Telescope H-band

Beatty et al. (2017) observed a spectrally resolved eclipse of KELT-1b in the
H-band using the LUCI1 multiobject spectrograph on the 2$\times$8.4 m Large
Binocular Telescope (LBT). The observations were carried out on the night of
2013-10-26 and lasted $\approx$5 h. The exposure time was 60 s, leading to 245
exposures in total.555The LBT H-band light curves used in Beatty et al.
(2017) were kindly provided by Dr. Beatty, personal communication.

###  2.5 Canada-France-Hawaii Telescope Ks-band

Croll et al. (2015) observed an eclipse of KELT-1b in the Ks-band ($\sim$2.15
$\mu$m) using the Wide-field Infrared Camera (WIRCam) on the 3.6 m Canada-
France-Hawaii Telescope (CFHT). The observations were carried out in
2012-10-10 and covered 6.8 h centred around the expected eclipse centre with
an exposure time of 8 s, leading to 1440 exposures in total.666The CFHT Ks-
band light curve used in Croll et al. (2015) was kindly provided by Dr. Croll,
personal communication. The reduction of the photometry is detailed
extensively in Croll et al. (2015, see especially the discussion about the
sensitivity to different data reduction approaches in Sect. 5.4).

##  3 Stellar Characterisation

We revise the stellar radius, mass, and age and show the updated stellar
properties in Table 1. The stellar age and mass are compatible with the
previous estimates by Siverd et al. (2012) within $1\sigma$, but our new
radius is slightly larger than the original estimate of
$1.46^{+0.039}_{-0.030}\;R_{\odot}$.

Using the stellar parameters derived via our spectral analysis as priors, we
use a Markov-Chain Monte Carlo (MCMC) modified infrared flux method (IRFM;
Blackwell & Shallis 1977; Schanche et al. 2020) to compute the stellar radius
of KELT-1. We construct spectral energy distributions (SEDs) using atlas
Catalogue stellar atmospheric models (Castelli & Kurucz, 2003) and
subsequently compute synthetic fluxes by integrating the SEDs over passbands
of interest with attenuation of the SED included as a free parameter to
account for extinction. To calculate the stellar apparent bolometric flux,
hence the stellar angular diameter and effective temperature, we compare the
synthetic fluxes to the observed broadband photometry retrieved from the most
recent data releases for the following passbands; Gaia G, GBP, and GRP, 2MASS
J, H, and K, and WISE W1 and W2 (Gaia Collaboration et al., 2021; Skrutskie et
al., 2006; Wright et al., 2010). We convert the angular diameter into stellar
radius using the offset-corrected Gaia EDR3 parallax (Lindegren et al., 2021)
and determine a $R_{\star}=1.530\pm 0.009\,R_{\odot}$.

We then use $T_{\mathrm{eff}}$, [Fe/H], and $R_{\star}$ to compute the stellar
mass $M_{\star}$ and age $t_{\star}$ through two different sets of theoretical
evolutionary models. In detail, we retrieved a first pair of mass and age
estimates by applying the isochrone placement technique (Bonfanti et al.,
2015, 2016) to grids of stellar tracks and isochrones pre-computed by the
PARSEC777PAdova and TRieste Stellar Evolutionary Code:
http://stev.oapd.inaf.it/cgi-bin/cmd v1.2S code (Marigo et al., 2017). A
second pair of mass and age estimates, instead, was derived by directly
inputting $T_{\mathrm{eff}}$, [Fe/H], and $R_{\star}$ into the CLES888Code
Liègeois d’Évolution Stellaire code (Scuflaire et al., 2008), which computes
the best-fit evolutionary track following the Levenberg-Marquadt minimisation
scheme, as presented in Salmon et al. (2021). As described in Bonfanti et al.
(2021), once the two pairs of outcomes are available, we checked their mutual
consistency through a $\chi^{2}$-based criterion and finally merged the
posterior distributions obtaining
$M_{\star}=1.370_{-0.070}^{+0.047}\,M_{\odot}$ and $t_{\star}=1.8\pm 0.5$ Gyr.

Table 1: KELT-1 identifiers, coordinates, properties, and magnitudes.

_Main identifiers_
---
TIC | 432549364
GAIA DR2 | 2881784280929695360
2MASS | J00012691+3923017
_Equatorial coordinates_
RA (J2000) | $00^{h}\,01^{m}\,26\aas@@fstack{s}9$
Dec (J2000) | $39\degr\,23\arcmin\,01\aas@@fstack{\prime\prime}66$
_Stellar parameters_
Spectral type |  |  | F5 | (1)
Mass | $M_{\star}$ | [$M_{\odot}$ ] | $1.370_{-0.070}^{+0.047}$ | (2)
Radius | $R_{\star}$ | [$R_{\odot}$ ] | $1.530\pm 0.009$ | (2)
Density | $\rho$ | [$\mathrm{g\,cm^{-3}}$] | $0.539\pm 0.025$ | (2)
Eff. temperature | $T_{\mathrm{eff}}$ | [K] | $6516\pm 49$ | (1)
Surf. gravity | $\log g$ | [cgs] | $4.23\pm 0.02$ | (1)
Metallicity | [Fe/H] | [dex] | $0.05\pm 0.08$ | (1)
Age |  | [Gyr] | $1.8\pm 0.5$ | (2)
Proj. rot. velocity | $v\sin i$ | [km s-1] | $56\pm 2$ | (1)
Parallax |  | [mas] | 3.684 $\pm$ 0.014 | (3)
_Magnitudes_
Filter | Magnitude | Uncertainty |
TESS |  | 10.2242 | 0.0061 |
$B$ |  | 11.233 | 0.11 |
$V$ |  | 10.632 | 0.007 |
Gaia |  | 10.5905 | 0.00034 |
$J$ |  | 9.682 | 0.022 |
$H$ |  | 9.386 | 0.030 |
$K$ |  | 9.437 | 0.019 |

(1) Siverd et al. (2012); (2) This work; Gaia EDR3 (Gaia Collaboration et al.,
2021).

##  4 Theory: phase curve modelling

Figure 2: Components of a phase curve model (transit, eclipse, emitted light,
reflected light, Doppler beaming, and ellipsoidal variation) and the full
model. The figure is centred around the secondary eclipse, the horizontal
dotted line shows the baseline level (flux = 1), the vertical dashed line
shows the eclipse centre, and the solid grey line shows the transit centre.

###  4.1 Phase curve model

We model the light curves using a phase curve model implemented in PyTransit
v2.5.21 (Parviainen, 2015; Parviainen & Korth, 2020; Parviainen, 2020) that
includes a primary transit, secondary eclipse, thermal emission, reflection,
Doppler beaming, and ellipsoidal variation, as illustrated in Fig. 2. The
observed flux is presented as a sum of stellar and planetary components
divided by a baseline flux model as

$F=\frac{T\times\left(f_{\mathrm{EV}}+f_{\mathrm{DB}}\right)+E\times
k^{2}(f_{\mathrm{R}}+f_{\mathrm{E}})}{f_{b}},$ (1)

where $f_{b}$ represents the baseline flux, $T$ is the transit model,
$f_{\mathrm{EV}}$ is the ellipsoidal variation component, $f_{\mathrm{DB}}$ is
the Doppler beaming component, $E$ is the eclipse model, $k$ is the planet-
star radius ratio, $f_{\mathrm{R}}$ is the reflected flux, and
$f_{\mathrm{E}}$ is the thermal emission.999Note that the $f_{\mathrm{R}}$
and $f_{\mathrm{E}}$ represent the phase-dependent planet-star _flux ratios_
from reflection and thermal emission, where the _flux ratio_ stands for the
ratio of the radiation emitted towards the observer by the brown dwarf and the
star per projected unit area. That is, the _flux ratio_ is not the ratio of
the disk-integrated fluxes, but this can be obtained by multiplying the flux
ratio by the companion-star area ratio, $k^{2}$.

Table 2: Phase curve model parameters and priors used in the final joint
analysis. The global parameters are independent of passband or light curve,
the passband-dependent parameters are repeated for each passband, and the
light-curve-dependent parameters are repeated for each separate light curve.
N$(\mu,\sigma)$ stands for a normal prior with a mean $\mu$ and standard
deviation $\sigma$, and U$(a,b)$ stands for a uniform distribution from $a$ to
$b$.

Description | Parameter | Units | Prior
---|---|---|---
_Global parameters_
Mid-transit time | $t_{0}$ | d | N($T_{\mathrm{S12}}$)a
Orbital period | $p$ | d | N($P_{\mathrm{S12}}$)a
Area ratio | $k^{2}$ | - | N(0.078, 0.005)
Stellar density | $\rho$ | $\mathrm{g\,cm^{-3}}$ | N(0.54, 0.08)
Impact parameter | $b$ | - | U(0, 1)
| $\sqrt{e}\cos\omega$ | - | N(0, $10^{-6}$)
| $\sqrt{e}\sin\omega$ | - | N(0, $10^{-6}$)
EV offset | $o_{\mathrm{EV}}$ | deg | N(0, 5)
_Passband-dependent parameters_
Quad. LD coeff. 1 | $q_{1}$ | - |  LDTk b
Quad. LD coeff. 2 | $q_{2}$ | - |  LDTk b
EV amplitude | $A_{\mathrm{EV}}$ | - | Uc
DB amplitude | $A_{\mathrm{DB}}$ | - | Nd
Emission offset | $o_{\mathrm{e}}$ | deg | N(0, 10)
Log10 ds. emission | $\log_{10}f_{\mathrm{d}}$ | - | U(-3, 0)
Log10 ns. emission | $\log_{10}f_{\mathrm{n}}$ | - | U(-3, 0)
Geometric albedo | $a_{\mathrm{g}}$ | - | N($10^{-5}$, $10^{-7}$)
_Light-curve-dependent parameters_
Baseline level | $f_{0}$ | - | Ne
Log10 white noise | $\log_{10}\sigma$ | - | Nf
Ln GP input scale | $s_{i}$ | - | N(1, 1)g
Ln GP output scale | $s_{o}$ | - | Ng

101010 $a$$a$footnotetext: The zero epoch and orbital period priors are based
on the values reported in Siverd et al. (2012) with uncertainties multiplied
by three. $b$$b$footnotetext: The limb darkening coefficients have normal
priors calculated using LDTk. $c$$c$footnotetext: Ellipsoidal variation is
constrained based by EV amplitude ratios as explained in the text and listed
in Table 3 $d$$d$footnotetext: The Doppler beaming priors are given in Table
3. $e$$e$footnotetext: The baseline levels have wide normal priors that are
based on the light curve variability. $f$$f$footnotetext: The average log
white noise parameters have normal priors centred around a numerical white
noise estimate. $g$$g$footnotetext: The Gaussian process log input scales have
wide normal priors while the log output scales have normal priors based on
light curve variability.

The model parametrisation is shown in Table 2 and the parameter priors are
discussed in more detail in Sect. 5.2. We assume a circular orbit since
KELT-1b’s orbital eccentricity agrees with zero (Siverd et al., 2012), which
allows us to model the phase curve simply as a function of the orbital phase,
$\phi=2\pi(t-t_{0})/p$, where $t$ is the mid-exposure time, $t_{0}$ is the
mid-transit time, and $p$ is the orbital period.

The light curves are modelled jointly, and each light curve has its own
baseline and noise model. We use a simple constant baseline model (that is, a
normalisation factor), and model the systematics as a Gaussian process, as
detailed later. All the baseline and noise model parameters are free (or
loosely constrained) in the posterior optimisation and sampling.

###  4.2 Transits and eclipses

The transits are modelled using the quadratic limb darkening model by Mandel &
Agol (2002) as implemented in PyTransit (Parviainen, 2015), and the limb
darkening is parameterised using the triangular parameterisation by Kipping
(2013). The secondary eclipses are also modelled using PyTransit, using an
eclipse model that returns the fraction of visible planetary disk area to the
stellar disk area. We do not include light travel time into the eclipse
modelling since it is at maximum $\approx 25$ s.

###  4.3 Reflected light

The reflected light111111We fix the reflected light component to zero for the
main analyses due to its degeneracy with the thermal emission, but describe
the component here for consistency. is approximated using a Lambertian phase
function (Russell, 1916; Madhusudhan & Burrows, 2012). This choice is based on
simplicity since our data is not precise enough to justify a more realistic
(and flexible) phase function. The planet to star flux ratio from reflected
light is

$f_{\mathrm{R}}=\frac{a_{\mathrm{g}}}{a_{s}^{2}}\frac{\sin\alpha+(\pi-\alpha)\cos\alpha}{\pi}.$
(2)

where $a_{g}$ is the geometric albedo, $a_{\mathrm{s}}$ the scaled semi-major
axis, $a_{\mathrm{s}}=a/R_{\star}$, and $\alpha$ is the phase angle,
$\alpha=|\phi-\pi|$. The reflected light reaches its maximum near the eclipse
and goes to zero near the transit.

###  4.4 Thermal emission

The thermal emission from the companion is modelled as a sine wave between the
minimum flux ratio, $f_{\mathrm{n}}$, and maximum flux ratio,
$f_{\mathrm{d}}$, as

$f_{\mathrm{E}}=f_{\mathrm{n}}+\frac{f_{\mathrm{max}}-f_{\mathrm{min}}}{2}\left(1-\cos(\phi+o)\right),$
(3)

where $o$ is a phase offset that sets the location of the hot spot, moving the
minimum and maximum emission phases marked in Fig. 2.

###  4.5 Ellipsoidal variation

When a star is orbited by a massive companion on a short-period orbit, the
companion’s gravitational pull distorts the star’s shape from a sphere into an
ellipse. This causes an ellipsoidal variation (EV) signal (Morris, 1985;
Faigler & Mazeh, 2011; Barclay et al., 2012; Lillo-Box et al., 2014) that can
be approximated to a first order as

$f_{\mathrm{EV}}=-A_{\mathrm{EV}}\cos(2\phi+o),$ (4)

where $o$ is an angular offset, the amplitude, $A_{\mathrm{EV}}$, depends on
the companion-host mass ratio, orbital inclination, $i$, and scaled semi-major
axis, $a_{\mathrm{s}}$, as

$A_{\mathrm{EV}}=\beta\frac{M_{\mathrm{c}}}{a_{\mathrm{s}}^{3}M_{\star}}\sin^{2}(i),$
(5)

where $\beta$ is a factor that depends on the linear limb darkening
coefficient, $u$, and gravity darkening coefficient, $g$, as

$\beta=0.15\frac{(15+u)(1+g)}{3-u}.$ (6)

The EV signal is usually inconsequential when studying planetary phase curves,
but it can dominate over the other phase curve components when a star is
orbited by a massive companion on a short-period orbit, such as in the case of
KELT-1b. Including the EV component into an eclipse and phase curve model is
especially important because the EV signal has its minimum close to the mid-
eclipse time (Bell et al., 2019; Beatty et al., 2020, and Sect. 6.3 in von
Essen et al. 2021). Unaccounted for EV signal can artificially enhance the
eclipse depth estimated from secondary eclipse observations, as happened with
some of the previous KELT-1b observations.

###  4.6 Doppler beaming

Doppler beaming is calculated following Loeb & Gaudi (2003), Barclay et al.
(2012), and (Claret et al., 2020) as

$f_{\mathrm{DB}}=A_{\mathrm{DB}}\sin\phi,$ (7)

where the Doppler beaming amplitude is

$A_{\mathrm{DB}}=\frac{\hat{B}}{c}\left(\frac{2\pi
G}{p}\right)^{1/3}\frac{M_{p}\sin i}{M_{\star}^{2/3}},$ (8)

and $c$ is the speed of light in vacuum, $G$ is the gravitation constant, $p$
is the orbital period, and $\hat{B}$ is the photon-weighted passband-
integrated beaming factor (Bloemen et al., 2010). The passband-integrated
beaming factor can be calculated from a stellar spectrum for any given
passband as

$\hat{B}=\frac{\int
B\,\lambda\,F(\lambda)T(\lambda)\,\mathrm{d}\lambda}{\int\lambda\,F(\lambda)T(\lambda)\,\mathrm{d}\lambda},$
(9)

where $T(\lambda)$ is the passband transmission, $F_{\lambda}$ is the stellar
flux at wavelength $\lambda$, and $B=5+\mathrm{d}\log
F_{\lambda}/\mathrm{d}\log\lambda$ is the beaming factor.

Unlike ellipsoidal variations, Doppler beaming has only a minimal effect on
the observed eclipse depth. This is because the beaming signal behaves as a
linear slope at the vicinity of the eclipse. In addition, beaming is expected
to have a much lower amplitude for KELT-1b than EV.

##  5 Phase curve analysis

###  5.1 Overview

The goal of our study is to estimate the day- and nightside flux ratios
between KELT-1b and its host star in the CHEOPS, TESS, H, Ks, and Spitzer 3.6
and 4.5 $\mu$m passbands. We do this by modelling the light curves described
in Sect. 2 jointly using the phase curve model defined in Sect. 4, which will
also yield improved orbital and geometric parameter estimates as a by-product.
The analysis follows the standard procedures of Bayesian inference (Gelman et
al., 2013; Parviainen, 2018), where we aim to estimate the posterior
probability distributions (posteriors) for the model parameters given a model,
observations, and prior probability distributions (priors) for the model
parameters.

We carry out three analyses that are each further divided into separate
scenarios (summarised in Appendix B) with different priors on the phase curve
model parameters. The analyses are:

  1. 1.

External data analysis models the TESS, LBT, CFHT, and Spitzer observations
jointly, and is implemented in ExternalDataLPF github Python class. The
external data analysis was carried out to create priors for the CHEOPS
analysis, and also allows us to test how including the CHEOPS observations
affects the final parameter estimates.

  2. 2.

CHEOPS analysis models only the CHEOPS-observed eclipses, and is implemented
in the CHEOPSLPF github class. The main motivation for the CHEOPS analysis is
to create detrended CHEOPS light curves to be used in the final analysis. As
mentioned in Sect. 2.1, the CHEOPS light curves contain strong instrument-
related systematics that are captured as basis vectors (covariates) using the
approach described in Wilson et al. (2022). Modelling the CHEOPS observations
jointly using a linear baseline model leads to a model with 180 free
parameters, most of which are baseline model coefficients. This is a lot
considering that the external data analysis has 79 free parameters, most of
which are physically interesting. After some tests, we decided to carry out
the CHEOPS analysis with a full baseline model and simplify the final analysis
by using the CHEOPS analysis posterior baseline model to remove the
systematics from the CHEOPS light curves included in the final analysis.

  3. 3.

Final analysis models all the data jointly, and is implemented in the FinalLPF
github class.

The posterior estimates from the final analysis are adopted as the final
results, but the differences between the analyses and scenarios are discussed
in Appendix B.

All the analyses are implemented as Python classes that inherit
pytransit.lpf.PhaseCurveLPF, where the base class implements the functionality
to model phase curves jointly for a set of heterogeneous light
curves121212Heterogeneous in the sense that they may have been observed in
different passbands, exposure times (some light curves possibly needing model
supersampling and others not), and each possibly with their own noise and
systematics models. (including optimisation and Markov Chain Monte Carlo
sampling), and the derived classes mainly read in the different light curves,
set up the noise models, and set up the model parameter priors based on the
analysis scenario.

###  5.2 Priors

####  5.2.1 Orbit, geometry, and limb darkening

For the final and external dataset analyses, we set uninformative or only
slightly informative priors on the geometric and orbital parameters, such as
the orbital period, impact parameter, and companion-star radius ratio, as
listed in Table 2. We also assume that the radius ratio is constant from
passband to passband because of a small atmospheric scale height caused by
KELT-1b’s high surface gravity (Beatty et al., 2020). Further, we assume a
circular orbit and force eccentricity to zero, since both the RV and eclipse
observations agree with a circular orbit (Siverd et al., 2012).

The transit model adopts a quadratic limb darkening law with the Kipping
(2013) parametrisation. We set uninformative priors on the TESS band limb
darkening coefficients since the TESS light curve does not feature any strong
systematics and the transit SNR is high. However, we set LDTk-calculated
normal priors on the limb darkening coefficients for the two Spitzer passbands
because the strong systematics in the Spitzer light curves will not allow us
to constrain limb darkening well.

####  5.2.2 Doppler beaming

Table 3: Theoretical Doppler beaming and ellipsoidal variation amplitudes, and ellipsoidal variation amplitude ratios relative to the TESS passband. Passband | Doppler beaming amplitude [ppm] | EV amplitude [ppm] | EV amplitude ratio
---|---|---|---
CHEOPS | $63\pm 3.2$ | $479\pm 30$ | $1.083\pm 0.007$
TESS | $41\pm 2.1$ | $442\pm 28$ | 1.00
H | $25\pm 1.3$ | $358\pm 23$ | $0.809\pm 0.004$
Ks | $19\pm 9.2$ | $352\pm 22$ | $0.796\pm 0.004$
Spitzer 3.6 $\mu$m | $17\pm 8.3$ | $343\pm 22$ | $0.774\pm 0.003$
Spitzer 4.5 $\mu$m | $16\pm 7.9$ | $341\pm 22$ | $0.772\pm 0.003$

Since we have good estimates for the orbital parameters and the brown dwarf to
star mass ratio (from Siverd et al. 2012 RV observations), we can set strict
priors on the Doppler beaming amplitudes for each passband. We calculate the
beaming factors and beaming amplitude priors for each passband with PyTransit
(namely using the doppler_beaming_factor and doppler_beaming_amplitude
functions), using the BT-Settl (Allard, 2013) model spectra to derive the
beaming factors rather than a black body approximation. The beaming amplitude
estimates are listed in Table 3, and they are computed in the Doppler beaming
github notebook.

####  5.2.3 Ellipsoidal variation

As with Doppler beaming, we can calculate theoretical EV amplitudes for all
the passbands by combining our prior knowledge about KELT-1b and its host star
with stellar spectrum models. We use the companion-star mass ratio estimate
from Siverd et al. (2012) and the semi-major axis estimate from Beatty et al.
(2020). We estimate the linear limb darkening coefficients for each passband
using LDTk (Parviainen & Aigrain, 2015), and estimate the gravity darkening
coefficients for the CHEOPS passband from the tables in Claret et al. (2020),
for the TESS passband from the tables in Claret (2017), and for the H, Ks, and
Spitzer passbands from the tables in Claret & Bloemen (2011).131313The EV
amplitudes are calculated in the EV notebook github available from GitHub. As
shown in Table 3, the EV signal amplitude is 10-20 times larger than the DB
signal amplitude, being roughly equal to the amplitude of the brown dwarf’s
phase curve signal in the TESS passband (Beatty et al., 2020; von Essen et
al., 2021).

The EV amplitude scales with the semi-major axis as $a_{s}^{-3}$, which makes
the theoretical EV amplitudes sensitive on our prior $a_{s}$ choice. This
causes a potential problem because EV directly affects our eclipse depth
measurements. A biased prior $a_{s}$ estimate could also significantly bias
the eclipse depths measured for passbands with only near-eclipse photometry
(CHEOPS, H, Ks) if we were to base the EV amplitude priors on the theoretical
amplitudes. Thus, it would be better to try to find a way to constrain the EV
amplitudes in a way that is independent of prior mass ratio and semi-major
axis estimates.

Fortunately, the TESS light curve covers the whole orbital phase with a
precision that allows us to estimate the EV amplitude in the TESS passband.
The EV amplitude ratio between a passband $i$ and the TESS passband $t$ is
simply

$\frac{A_{\mathrm{EV,i}}}{A_{\mathrm{EV,t}}}=\frac{\beta_{\mathrm{i}}}{\beta_{\mathrm{t}}},$
(10)

where the $\beta$ are the factors in Eq. (6) that depend on stellar gravity
and limb darkening (which both are passband-dependent), but have no dependence
on KELT-1b’s mass or orbit.

Thus, we can constrain the EV signal by constraining the EV amplitudes
relative to TESS passband rather than constraining the absolute EV amplitudes
themselves. This approach is less likely to introduce biases to EV amplitudes,
and so also less likely to bias the secondary eclipse signal. In the final
analysis, we set uninformative priors on the absolute EV amplitudes on all
passbands, and set informative priors on the EV amplitude ratios between the
TESS passband any every other passband

$\displaystyle\frac{A_{\mathrm{EV,\,i}}}{A_{\mathrm{EV,\,t}}}$
$\displaystyle\sim N(\frac{\beta_{\mathrm{i}}}{\beta_{\mathrm{t}}},\sigma),$

where the $\beta$ factors are calculated using the LDTk and gravity darkening
tabulations described earlier and $\sigma$ is the uncertainty in the $\beta$
ratio estimate. Further, instead of using the uncertainties listed in Table 3,
we set $\sigma$ to 0.01 to reduce our sensitivity to any biases in the $\beta$
factors (the chosen value is rather arbitrary but in general 3-10 times larger
than the theoretical uncertainties).

###  5.3 Noise and systematics model

In addition to the planetary signal and the EV and Doppler boosting signals
from the star, the photometry contains _systematics_ from astrophysical and
instrumental sources, as well as from changes in the observing conditions.
These systematics need to be taken into account during the phase curve
modelling.

We model the variability in the photometry not explained by the phase curve
model as a Gaussian process (Rasmussen & Williams, 2006; Gibson et al., 2012;
Roberts et al., 2013). We use time as the only GP input for the TESS data,
which allows us to use Celerite (Foreman-Mackey et al., 2017) to evaluate the
GP log likelihood for the whole unbinned TESS data set. The other data sets
use more complex covariance functions with multiple input variables (e.g., x
and y centroids, airmass and seeing), and for those we evaluate the GP log
likelihood using George (Ambikasaran et al., 2014).

Each GP adds a set of hyperparameters into the joint model. We use simple
covariance kernels (Matern-3/2 and squared exponential) which add only a log
input scale parameter per input variable and a log output scale parameter per
GP (detailed later separately for each data set). These GP hyperparameters are
constrained only loosely during the optimisation and MCMC sampling. We
standardise the GP input variables (that is, we divide each input variable
time series with its standard deviation and remove its mean) and constrain the
log input scale parameters using normal priors centred at one with a standard
deviation of one. The priors for the log output scale parameters are also
normal and centred at log variance of the light curve with a standard
deviation of 1.5.

We also fit the average white noise for each light curve. The white noise is
parametrised using its log variance, and constrained with loose normal prior
centred around a white noise estimate computed from the standard deviation of
the flux point-to-point differences as

$\sigma=\mathrm{median}(\mathrm{sd}(\mathrm{diff}(\@vec{f})))\;/\sqrt{2},$
(11)

where $\@vec{f}$ is a vector containing the fluxes, sd stands for standard
deviation, and $\mathrm{diff}$ stands for discrete difference.

The noise models are defined in the _init_lnlikelihood method of the FinalLPF
github and ExternalDataLPF github analysis classes, and the noise model
parameter priors are set in the classes’ _post_initialisation method. The
posterior MCMC sampling marginalises over the whole GP hyperparameter space
allowed by the priors and the data to ensure we are not overfitting the data.

#####  TESS

We model the variability in the TESS data as a Gaussian process with time as
the only input variable using the Celerite package. The GP uses basic
Matern-3/2 covariance kernel that yields three additional parameters to the
joint model, the log input and output scales and the log white noise variance,
and these parameters are constrained with relatively loose normal priors as
described earlier.

##### LBT H

The systematics in the H band light curve are modelled as a GP with eight
input variables (including, for example, the airmass and the target and
comparison star locations). The covariance kernel is a product of eight
squared exponential kernels where each kernel provides an independent input
scale parameter into the joint model. This brings the total number of
parameters from the H band data set noise model to ten.

##### CFHT Ks

The systematics in the Ks band light curve are modelled as a GP with two input
variables, the x and y centroids. The covariance kernel is a product of two
squared exponential kernels where each kernel provides an independent input
scale parameter into the joint model, leading in total to four additional
parameters.

#####  Spitzer

The systematics in the Spitzer data are also modelled using a GP with the x
and y centroids as the input variables. The Beatty et al. (2019) Spitzer phase
curve observations were split into three 12 h long stares and the telescope
was repointed between the stares. This repointing led to strong
discontinuities in the raw photometry, so we decided to treat each 12 h stare
as a separate light curve. This gives us four light curves per Spitzer
passband (one covering the Beatty et al. 2014 eclipse and three covering the
phase curve). However, all the eight light curves use the same GP kernel and
hyperparameters (we assume the centroid-related systematics in both Spitzer
passbands have similar input and output scales), so the Spitzer GP model adds
only four parameters to the joint model.

The Spitzer light curves consist of 10 000 (eclipse) and 64 000 (phase curve)
flux measurements. This is too much for a standard brute-force GP that
requires inversion of the covariance matrix and scales as
$\mathcal{O}(n^{3})$. Splitting the phase curve observations helps, but we
still needed to bin the Spitzer observations to a two-minute cadence to make a
GP noise model computationally feasible.

#####  CHEOPS

The CHEOPS data is modelled differently in the final analysis and in the
initial CHEOPS-only analysis. In the CHEOPS analysis we use a linear model to
model the systematics with basis vectors determined by the pipeline of Wilson
et al. (2022) used as covariates. The number of basis vectors used per visit
varies from 13 to 29, and the total number of basis vectors (and thus free
parameters from the baseline model) is 156.

The noise model for the CHEOPS data assumes i.i.d. (independent and
identically distributed) normally distributed noise (that is, white noise),
and introduces only the logarithm of the average standard deviation of the
noise distribution for each visit as an additional free parameter into the
full model (that is, we assume the noise does not vary significantly from
observation to observation within a single light curve, and marginalise over
the average observation uncertainty).

In the final analysis, we detrend the CHEOPS data using the posterior median
baseline model from the CHEOPS analysis and give each visit only a free
normalisation term. We do this to simplify the final model, since using the
full CHEOPS baseline model would add 156 free parameters into the final model.
There is a danger that this approach could lead to underestimated
uncertainties in some of our quantities of interest, but a comparison of the
CHEOPS and final analysis results (see Sect. B) shows that any effects from
not marginalising over the baseline model coefficients in the final analysis
are insignificant.

##  6 Results and Discussion

###  6.1 Results from the phase curve analysis

Figure 3: The dayside (orange) and nightside (blue) planet-star flux ratio
posterior distributions for all passbands. The central 68% posterior interval
is marked with a dark shade and the previous dayside flux ratio estimates are
shown as diamonds with errorbars showing their reported uncertainties. For
TESS, the Beatty et al. (2020) estimate is shown on the left and the von Essen
et al. (2021) estimate on the right. Figure 4: Dayside flux ratio posterior
distribution for the CHEOPS passband. The dashed lines shows the posterior
percentile limits.

We show the dayside and nightside planet-star flux ratio posteriors from the
final joint modelling for all passbands in Fig. 3, give a more detailed view
of the CHEOPS band dayside flux ratio posterior in Fig. 4, and show the
observations and the model posteriors for all the passbands in Fig. 5. The
final flux ratio estimates are reported in Table 4, and the rest of the
parameter estimates in Table 5. The results from the external dataset and
CHEOPS-only analysis scenarios are presented in Appendix B.

Except for the CHEOPS band, the dayside flux ratio posteriors are
approximately normal with a well-defined non-zero mode, and for these
posteriors we report their median values with uncertainties based on the 16th
and 84th central percentiles in Table 4. The CHEOPS band flux ratio posterior
has its mode at zero, and we report its posterior median and 99th percentile
upper limits.

The nightside flux ratio posteriors all have their modes at zero, and we
report only the 99th posterior percentile upper limits. The nightside flux
ratio upper limit for the TESS band is 4.6%, while for the Spitzer bands it is
$\approx 12\%$.

Table 4: Final day- and nightside planet-star flux ratios and eclipse depths
with their uncertainties and upper limits.

Passband | Ns. flux ratio [%] | Ns. $T_{\mathrm{b}}$ [K] | Ds. flux ratio [%] | Eclipse depth [ppm] | Ds. $T_{\mathrm{b}}$ [K]
---|---|---|---|---|---
CHEOPS |  |  | $<1.50$ | $<90$ | 2100 – 2900
TESS | $<4$ | ¡ 3000 | $6.1\pm 0.9$ | $360\pm 50$ | 3100 – 3500
H |  |  | $17.3\pm 3.2$ | $1000\pm 190$ | 2100 – 3400
Ks |  |  | $24.9\pm 1.9$ | $1460\pm 110$ | 2900 – 3400
Spitzer 3.6 $\mu$m | $<11$ | ¡ 1900 | $34.9\pm 1.4$ | $2040\pm 80$ | 3000 – 3400
Spitzer 4.5 $\mu$m | $<12$ | ¡ 1800 | $35.3\pm 1.5$ | $2070\pm 80$ | 2800 – 3200

141414The nightside estimates give an upper limit corresponding to the 99th
posterior percentiles. The dayside estimates correspond to posterior medians
with uncertainties based on the 16th and 84th posterior percentiles (except
for CHEOPS, for which we only give the upper limit). The reported dayside
brightness temperature values are lower and upper limits corresponding to
2.5th and 97.5th posterior percentiles.

Table 5: Final estimates for the geometric and orbital parameters of KELT-1b
and its host star.

_Ephemeris_
---
Transit epoch | $T_{0}$ | [BJD] | $2455914.1624\pm 4\times 10^{-4}$
Orbital period | $P$ | [days] | $1.2174942\pm 2\times 10^{-7}$
Transit duration | $T_{14}$ | [h] | $2.74\pm 0.01$
_Relative properties_
Area ratio | $k^{2}$ | $[A_{\star}]$ | $0.00585\pm 7\times 10^{-5}$
Radius ratio | $k$ | $[R_{\star}]$ | $0.0765\pm 5\times 10^{-4}$
Semi-major axis | $a$ | $[R_{\star}]$ | $3.53\pm 0.09$
Impact parameter | $b$ | $[R_{\star}]$ | $0.35\;(-0.08)\;(+0.06)$
_Absolute properties_
KELT-1b radius | $R_{\mathrm{c}}$ | [$R_{\mathrm{Jup}}$ ] | $1.138\pm 0.010$
KELT-1b surface gravity | $\log g_{\mathrm{c}}$ | [cgs] | $4.73\pm 0.02$
KELT-1b eq. temperaturea | $T_{\mathrm{eq}}$ | [K] | $2550\pm 170$
Semi-major axis | $a$ | [AU] | $0.0251\pm 0.0006$
Orbital inclination | $i$ | [deg] | $84.3\;(-1.1)\;(+1.4)$
Stellar density | $\rho_{\star}$ | $[\mathrm{g\,cm^{-3}}]$ | $0.56\pm 0.04$
TESS emission offset | $O_{\mathrm{T}}$ | [deg] | $3\pm 9$
Spitzer 3.6 $\mu$m emission offset | $O_{\mathrm{S1}}$ | [deg] | $-29\pm 5$
Spitzer 4.5 $\mu$m emission offset | $O_{\mathrm{S2}}$ | [deg] | $4\pm 6$

151515The estimates correspond to the posterior median with the uncertainties
based on the 16th and 84th posterior percentiles. $a$$a$footnotetext: The
equilibrium temperature is calculated using the stellar $T_{\mathrm{eff}}$
estimate, scaled semi-major axis distribution, heat redistribution factor
distributed uniformly between 0.25 and 0.5, and albedo distributed uniformly
between 0 and 0.4.

Figure 5: The observed light curves from CHEOPS, TESS, LBT, CFHT, and Spitzer
(points) and the posterior median phase curve model (black line). The
systematics model has been removed from the observations, the TESS and CHEOPS
observations have been folded over the phase, and the data has been binned to
10 min for visualisation purposes. The figure is produced by the final light
curve plot github notebook.

The dayside flux ratios for the TESS and Spitzer passbands (Fig. 3) agree
fairly well with the previous studies by Beatty et al. (2020), von Essen et
al. (2021), and Beatty et al. (2019). The flux ratios for the H and Ks bands
are lower than what reported by Beatty et al. (2017) and Croll et al. (2015),
but this is expected since neither of the original studies included an
ellipsoidal variation signal into their models.

The dayside flux ratio for the 3.6 $\mu$m Spitzer band is higher than the
estimate by Beatty et al. (2019). The Spitzer light curves use relatively
flexible GP systematics models and all the Spitzer light curves are well-fit
visually (Fig. 5). Even then, we cannot rule out poorly modelled systematics
as the source of the anomalously high 3.6 $\mu$m flux ratio since the pre-
eclipse baseline is relatively short for the full-phase Spitzer observations.

Curiously, the CHEOPS and TESS bands show a strong discrepancy in their
dayside flux ratios. Panel a) of Fig. 6 shows the CHEOPS observations with the
median posterior model and panel b) shows the same for TESS. The combined
CHEOPS observations have a noise estimate of $\approx 50$ ppm over a 20-min
bin, half of the TESS noise of $\approx 97$ ppm, but we do not detect a
significant eclipse signal in the CHEOPS band. Instead, we are only able to
set an upper limit of 1.4% for the CHEOPS dayside flux ratio, while in the
TESS band we detect a clear eclipse corresponding to a flux ratio of $6\pm
1\%$.

Figure 6: The CHEOPS eclipse observations binned to a 20-minute cadence with
the phase curve model (panel a) and TESS observations with a similar binning
(panel b). The median posterior phase curve model is shown as a black line and
its 68% central posterior interval is shown as a shaded blue region.

###  6.2 Dayside brightness temperatures

Figure 7: Day- and nightside brightness temperatures estimated from the flux
ratios using the BT-Settl stellar model spectra by Allard (2013). The shading
marks the 68% and 99.7% central posterior intervals, from the darkest to the
lightest shade, and the dashed horizontal line shows the best-fitting constant
brightness temperature. The previous dayside $T_{\mathrm{b}}$ estimates for
the TESS and Spitzer passbands by Beatty et al. (2019), Beatty et al. (2020),
and von Essen et al. (2021) are shown as points with errorbars.

We estimate the day- and nightside brightness temperatures, $T_{\mathrm{b}}$,
separately for each passband using the flux ratio estimates from the final
joint modelling and BT-Settl stellar spectrum models by Allard (2013). We
calculate the $T_{\mathrm{b}}$ estimates as averages over three approaches: a)
physical-physical, where both the brown dwarf and its host star spectra are
modelled with BT-Settl models; b) blackbody-physical, where KELT-1b is
modelled as a black body and the star uses a BT-Settl model; and c) blackbody-
blackbody, where both KELT-1b and the star are modelled as black bodies.

Figure 7 shows the approach-averaged brightness temperatures together with the
best-fitting constant dayside temperature (the $T_{\mathrm{b}}$ estimates are
also included in Table 4, and the per-approach estimates are shown in Appendix
C). The dayside temperatures mostly agree with each other within the
uncertainties. In agreement with Beatty et al. (2020), the TESS passband has a
somewhat higher $T_{\mathrm{b}}$ than the Spitzer bands, but the disagreement
is less significant for us (for example, the TESS and Spitzer 3.6 $\mu$m band
68% central posterior intervals overlap). Our $T_{\mathrm{b}}$ estimate for
the TESS passband agrees well with the Beatty et al. (2020) result, but our
Spitzer $T_{\mathrm{b}}$ estimates are higher than what reported in Beatty et
al. (2019).

As expected from the differences in the dayside flux ratios, the dayside
$T_{\mathrm{b}}$ estimate for CHEOPS is significantly lower than for TESS.

###  6.3 Nightside brightness temperatures

We can measure the nightside brightness temperatures only for the passbands
where we have observations over the full orbital phase (TESS and Spitzer), and
the estimates are rather poorly constrained. Both Spitzer passbands support
cool night sides, with 99th $T_{\mathrm{b}}$ posterior percentiles $\approx
1850\leavevmode\nobreak\ K$, while in the TESS passband we can only set a 3000
K upper limit. These results support a significant temperature difference
between the night- and dayside, and agree with the previous Spitzer and TESS
analyses by Beatty et al. (2020) and von Essen et al. (2021)

###  6.4 Atmospheric modelling

For the atmospheric modelling, we follow the same basic approach as done in
Lendl et al. (2020). In particular, we employ the self-consistent atmosphere
model HELIOS (Malik et al., 2017, 2019) to generate a grid of possible
atmospheres for KELT-1b. As discussed in Lendl et al. (2020), in the context
of 1D models, the flux of the host star impinging upon the top of the
atmosphere (TOA) of KELT-1b is (Cowan & Agol, 2011)

$F_{\mathrm{TOA}}=F_{*}\left(\frac{R_{*}}{a}\right)^{2}\left(1-A_{\mathrm{B}}\right)\left(\frac{2}{3}-\frac{5\epsilon}{12}\right)\
,$ (12)

where $F_{*}$ is the stellar photometric flux, $R_{*}$ the radius of the host
star KELT-1, $a$ the orbital distance of KELT-1b, $A_{\mathrm{B}}$ its Bond
albedo, and $\epsilon$ the heat redistribution efficiency ($0\leq\epsilon\leq
1$). In terms of the atmosphere model HELIOS, the term

$E_{*}=\left(1-A_{\mathrm{B}}\right)\left(\frac{2}{3}-\frac{5\epsilon}{12}\right)$
(13)

is fully degenerate with respect to $A_{\mathrm{B}}$ and $\epsilon$, such that
we only vary $E_{*}$ rather than $A_{\mathrm{B}}$ and $\epsilon$
independently.

Since KELT-1b is a brown dwarf, its emission might also have a considerable
contribution from an internal energy flux, described by an internal
temperature $T_{\mathrm{int}}$. In case of an isolated object without exterior
insolation, $T_{\mathrm{int}}$ would be the object’s effective temperature.
For the construction of the model grid, we therefore use $T_{\mathrm{int}}$ as
an additional free parameter. Given the lack of any spectrally-resolved data
for KELT-1b and the fact that the host star’s [Fe/H] value is consistent with
0 (see Table 1), we use solar element abundances for the model grid, for
simplicity. The general modelling parameters for KELT-1b (surface gravity,
stellar radius, orbital distance) are taken from Tables 1 & 5.

Using HELIOS, we generate a model grid of self-consistent forward atmosphere
models for KELT-1b in terms of the free parameters $T_{\mathrm{int}}$ and
$E_{*}$. In total we compute 165 different atmospheric model structures for
internal temperatures between 0 K and 3000 K and $E_{*}$ values from 0.667 to
0.06. The former value is the highest possible one for $E_{*}$, with a Bond
albedo and heat redistribution $\epsilon$ of 0. The latter value represents a
case with a high Bond albedo and/or very efficient heat redistribution.

Besides the atmospheric structure, we also obtain the brown dwarf’s emission
spectrum for each model in a postprocess procedure. For this postprocess, we
exclude any scattering contribution, such that the obtained emission spectra
describe only the atmosphere’s thermal emission. These spectra are then
integrated over the passbands shown in Table 4

$F_{\mathrm{pb}}=\int\mathcal{F}F_{\lambda}\,\mathrm{d}\lambda$ (14)

to calculate the flux $F_{\mathrm{pb}}$ inside each passband. Here,
$\mathcal{F}$ is the corresponding filter transmission function and
$F_{\lambda}$ the calculated emission spectrum of KELT-1b.

Since these $F_{\mathrm{pb}}$ are a smooth function of $T_{\mathrm{int}}$ and
$E_{*}$, we parameterise them as a two-dimensional, fifth-order polynomial in
each passband. This allows us to evaluate the passband-integrated fluxes
$F_{\mathrm{pb}}(T_{\mathrm{int}},E_{*})$ as a continuous function of
$T_{\mathrm{int}}$ and $E_{*}$ directly, rather than having to perform an
interpolation in two dimensions.

These passband-integrated fluxes are then added to a forward model in the
Bayesian retrieval framework Helios-r2 (Kitzmann et al., 2020) to constrain
the geometric albedos in the TESS and CHEOPS passbands ($A_{\mathrm{g,T}}$ and
$A_{\mathrm{g,C}}$). This forward model calculates the secondary eclipse
depths $\mathrm{d}F_{\mathrm{se}}$

$\mathrm{d}F_{\mathrm{se}}=A_{g}\left(\frac{R_{\mathrm{p}}}{a}\right)^{2}+\frac{F_{\mathrm{pb}}(T_{\mathrm{int}},E_{*})}{\int\mathcal{F}F_{*}\,\mathrm{d}\lambda}\left(\frac{R_{\mathrm{p}}}{R_{*}}\right)^{2}$
(15)

as a function of the free parameters $A_{g}$, $R_{\mathrm{p}}/a$,
$R_{\mathrm{p}}/R_{*}$, $T_{\mathrm{int}}$, and $E_{*}$ in each photometric
passband. The stellar photospheric spectrum $F_{*}$ is taken from the grid of
PHOENIX stellar atmosphere models (Husser et al., 2013), based on the stellar
parameters given in Table 1.

Within the retrieval forward model we assume that only the CHEOPS and TESS
passbands have contributions by a geometric albedo, whereas the other ones are
dominated by thermal emission rather than reflected light. This is based on
the fact that the stellar spectrum peaks at lower wavelengths, where the
CHEOPS and TESS passbands are located. The TESS band receives 64% of the
stellar flux received by the CHEOPS band, and the H, Ks, 3.6 $\mu$m, and 4.5
$\mu$m bands receive 7%, 3%, 1%, and 0.5% of the flux received by the CHEOPS
band, respectively. The four photometric measurements (H, Ks, Spitzer) in the
infrared are dominated by the brown dwarf’s own thermal emission rather than
by scattering of its host star’s incident flux.

Figure 8: Posterior distributions for the retrieval calculations using
Helios-r2. We show only the four most important parameters. The rest are
strongly constrained by priors and do not show significant correlations
between other parameters. Figure 9: Posterior model spectrum for the retrieval
calculations using Helios-r2. The light gray line shows the high resolution
version of the posterior thermal emission model median, and the darker gray
line shows the same spectrum in medium resolution. The black points show the
retrieval eclipse depths with their corresponding 1$\sigma$ intervals, and the
orange shaded areas show the eclipse depth posteriors estimated from the
photometry as given in Table 4. The black labelled lines show the passbands,
where the Spitzer passbands are simplified for visualisation.

The posterior distributions of our retrieval calculations are shown in Fig. 8
while the posterior photometric points are depicted in Fig. 9. The posterior
distributions show a strong correlation between the parameters $E_{*}$ and
$T_{\mathrm{int}}$. When the retrieval is performed only on the TESS and
Spitzer data points, bimodal solutions for these two quantities are obtained
(see Appendix D). This bimodality is broken primarily by the H and Ks data
points, while the CHEOPS data point further narrows the posterior distribution
on $E_{*}$. The retrieved value of $T_{\mathrm{int}}=2670_{-150}^{+190}$ K
implies that the brown dwarf has a very strong contribution from an internal
heat flux to the total energy budget of the atmosphere and corroborates the
findings of Beatty et al. (2020). As already noted by Beatty et al. (2019), an
isolated brown dwarf with the same age and mass would have an internal
temperature of about 850 K. The relatively low value of $E_{*}=0.22\pm 0.08$,
on the other hand, suggests that the heat recirculation $\epsilon$ must very
high, or that the atmosphere has a large Bond albedo $A_{\mathrm{B}}$.

For the TESS passband, we obtain a geometric albedo of about
$0.36^{+0.12}_{-0.13}$, while in the CHEOPS passband the results are
consistent with a geometric albedo of 0.

The resulting photometry posteriors in comparison to the measured data are
shown in the lower panel of Fig. 8. The results imply that atmosphere forward
models can explain most of the measured data points while especially the
Spitzer 3.6 $\mu$m band shows larger deviations. This mismatch in the Spitzer
3.6 $\mu$m band might point to non-equilibrium chemistry effects. The
predicted CHEOPS photometry secondary eclipse depth also deviates from the
measured one. In fact, even without including an additional geometric albedo
contribution, the measured eclipse depth cannot be described by the expected
thermal emission of the planet.

Figure 10: The upper panel depicts the values of $E_{*}$ as function of the
heat recirculation parameter $\epsilon$ and Bond albedo $A_{\mathrm{B}}$. The
solid white line represents the retrieved median value of $E_{*}$, the dashed
lines are its corresponding 1$\sigma$ interval, and the shaded region
corresponds to the parameter space excluded by the upper nightside
$T_{\mathrm{b}}$ limit of 1900 K. The lower panel shows the temperature-
pressure profile of a HELIOS model (solid line), computed for the median
values of $E_{*}$ and $T_{\mathrm{int}}$. Besides the temperature-pressure
profile, the lower panel also includes a set of condensate stability curves
(dashed lines) for a selected set of species.

The upper panel of Fig. 10 depicts the heat recirculation efficiencies
$\epsilon$ and Bond albedos $A_{\mathrm{B}}$ that are consistent with the
retrieved value of $E_{*}=0.22\pm 0.08$ and the upper nightside
$T_{\mathrm{b}}$ limit of 1900 K. If the heat redistribution is very efficient
($\epsilon\approx 1$), then the Bond albedo has to be close to 0. On the other
hand, if only very little heat gets transported to the nightside
($\epsilon\approx 0$), then the Bond albedo has to be in the range of 0.7,
implying that potentially clouds scatter almost 70% of the incident stellar
radiation back to space. In principle, given constraints on $T_{\rm int}$, the
dayside temperature and the nightside temperature, the degeneracy between
$A_{\rm B}$ and $\epsilon$ could be broken.

Additionally, we calculate a separate HELIOS model for the above stated
retrieved median values of $E_{*}$ and $T_{\mathrm{int}}$. The resulting
temperature-pressure profile is shown in the bottom panel of Fig. 10. Due to
the very high $T_{\mathrm{int}}$, the temperature increases very strongly in
the lower atmosphere, while being close to isothermal around 0.1 bar, and
showing a temperature inversion in the upper atmosphere, originating from the
absorption of stellar light. The plot also depicts a set of stability curves
for selected high-temperature condensates. These stability curves in
combination with the atmosphere’s temperature profile imply that most likely
condensates do not form on the dayside of KELT-1b. This is not surprising
given its high equilibrium temperature.

The derived geometric albedo of $0.36^{+0.12}_{-0.13}$ in the TESS passband is
challenging to explain, since even the most refractory mineral (corundum) does
not condense at the derived dayside temperatures. However, it agrees with the
scenario where silicate clouds form on the nightside of the brown dwarf and
are then transported to the dayside by winds, as well as with geometric
albedos for silicate cloud-covered planets of $\sim 0.4$ predicted by Sudarsky
et al. (2000). It is nevertheless difficult to reconcile this high value of
the TESS geometric albedo with a CHEOPS geometric albedo of essentially zero,
because the two passbands have considerable overlap.

###  6.5 Chromatic albedo variability as a possible source of discrepancy
between CHEOPS and TESS

The albedo spectrum of a planet (or, as in our case, a brown dwarf) can
feature clear distinct regions of low and high reflectivity in wavelength
space (Sudarsky et al., 2000; Burrows et al., 2008). In theory, a low blue-
optical albedo combined with a high NIR albedo could be explained with
silicate clouds and a strong optical absorber above the silicate cloud deck
(such as gaseous TiO/VO or S2/HS, Schwartz & Cowan, 2015), and there is a
possibility that the discrepancy between the CHEOPS and TESS dayside flux
ratios could be explained by a high-contrast feature in KELT-1b’s albedo
spectrum.

We modelled whether the CHEOPS-TESS discrepancy could be explained by
chromatic variations in KELT-1b’s albedo using a simple two-level albedo
spectrum model parametrised by the geometric albedo in blue,
$A_{\mathrm{g,0}}$, geometric albedo in red, $A_{\mathrm{g,1}}$, and step
location, $\lambda_{0}$, dictating where the blue albedo changes to red. The
modelling is detailed in Appendix A and carried out in the chromatic albedo
variation test github notebook.

As a conclusion, the discrepancy is challenging to explain with a simple two-
level albedo spectrum model due to the large overlap between the CHEOPS and
TESS bands. The largest TESS to CHEOPS reflected light contrast,
$C=(F_{\mathrm{T}}-F_{\mathrm{C}})/(F_{\mathrm{T}}+F_{\mathrm{C}})$, of $0.41$
is obtained for a step model with $A_{\mathrm{g}}=0.01$ for $\lambda<860$ nm
and $A_{\mathrm{g}}=0.4$ for $\lambda\geq 860$ nm, as illustrated in Fig. 12.
A geometric albedo of $0.01$ is not unheard of for a highly irradiated body,
but is similar to what has been estimated for TrES-2b (Kipping & Spiegel,
2011), while the geometric albedo of $0.4$ corresponds to the silicate cloud
models by Sudarsky et al. (2000), and is in line with the previous KELT-1b
study by Beatty et al. (2020). However, the maximum contrast of 0.4 is at the
lower tail of the measured $C$ posterior distribution that has its mode at 0.9
(Fig. 13). Even a blue $A_{\mathrm{g}}$ of 0.001 with $\lambda_{0}=940$ nm
would lead to $C$ of only 0.6, so it is unlikely that a physically plausible
albedo spectrum would be able to reproduce the observed CHEOPS-TESS
discrepancy.

###  6.6 Temporal variability as a possible source of discrepancy between
CHEOPS and TESS

Since the upper limits on the nightside temperature are low enough that clouds
could conceivably form, it is plausible that nightside clouds could be
transported to the dayside (Beatty et al., 2020), where they could survive
transiently. This raises the possibility of dayside cloud cover being
transient. Variability in cloud cover could be one possible explanation for
the dayside flux ratio discrepancy between the CHEOPS and TESS passbands. von
Essen et al. (2021) reported possible temporal variation in the eclipse depths
in the TESS data, but concluded that the variations were most likely
associated with stellar variability. We likewise carried out per-eclipse
analyses for the TESS and CHEOPS data, but also need to conclude that the per-
eclipse precision is not high enough to study whether the eclipse depths show
systematic variations that would securely not be caused by stellar variability
or systematics.

The spatial distribution of transient dayside clouds transported from the
brown dwarf’s nightside can be expected to be strongly nonuniform, which would
lead to asymmetric reflection phase curve as with Kepler-7b (Demory et al.,
2013). While our analyses simplify the KELT-1b’s flux contribution by using
only the emission component, the emission component has a loosely constrained
phase offset that allows it to catch any existing phase curve asymmetries.

The emission offset (see Table 5) agrees with zero for the TESS and Spitzer
4.5 $\mu$m bands, but is significantly non-zero for the Spitzer 3.6 $\mu$m
band. Of these three, only the TESS band phase curve is expected to have a
significant contribution from reflection, and the lack of measurable phase
curve asymmetry speaks against the cloud transport hypothesis. Then again,
were the dayside cloud coverage to be variable, the variability could blur the
asymmetry when averaged over the TESS observations of 25 days.

The TESS and Spitzer phase offsets disagree with the ones estimated by Beatty
et al. (2019) and Beatty et al. (2020). They observe slight eastward shifts
for the TESS and Spitzer 4.5 $\mu$m bands and a strong eastward shift for the
Spitzer 3.6 $\mu$m band, while our shifts are westwards.

The phase curve asymmetry in Spitzer 3.6 $\mu$m band is curious. The Spitzer
phase curve observations were carried out eight days apart of each other, so
the differences could be explained by temporal variability in emission and
reflection. This could also explain the higher-than expected dayside eclipse
depth for the 3.6 $\mu$m band (Fig. 9). Our eclipse depth estimate differs
from the result from Beatty et al. (2019) (whose estimate agrees with the
theoretical models) and we extracted the Spitzer light curves using our own
pipeline, so we cannot completely rule out untreated systematics. However, the
time-scale of the variability (smooth over KELT-1b phase) combined with our
use of a flexible GP to model the Spitzer systematics lead us to believe the
differences in the two Spitzer phase curves arise from true variability in the
KELT-1b’s phase curve.

###  6.7 Comparison between emission spectrum models

Figure 11: Comparison between different emission spectrum models. The shaded
areas show the 68%, 95%, and 99.7% central posterior intervals for the eclipse
depths as estimated from the joint modelling, and the black points show the
passband-integrated emission spectrum models. Helios-r2 corresponds to the
model presented here, VE21a and VE21b to the models in von Essen et al.
(2021), BT-CN, BT-DS, BT-NG, and BT-ST to the best-fitting BT-Cond, BT-Dusty,
BT-NextGen, and BT-Settl models by Allard (2013), respectively, and DRIFT-PH
to the best-fitting Drift-PHOENIX model (Witte et al., 2011).

We compare our retrieved emission spectrum with the spectra by von Essen et
al. (2021), the BT model spectra (Allard, 2013)161616Downloaded from F.
Allard’s Star, Brown Dwarf, and Planet simulator. and DRIFT-PHOENIX spectra
(Witte et al., 2011)171717Obtained from the SVO Theoretical spectra web
server. for isolated brown dwarfs in Fig. 11. The ”Helios-r2” results
correspond to the modelling described earlier in Sec. 6.4 and the ”VE21a” and
”VE21b” results correspond to the modelling detailed in von Essen et al.
(2021). The rest of the results correspond to best-fitting BT-SETTL, BT-
NextGen, BT-COND, BT-Dusty, and DRIFT-PHOENIX models where the host star was
modelled using a BT-SETTL spectrum with $T_{\mathrm{Eff}}=6500$ K, $\log
g=4.0$ and $z=0.0$.

The best fitting BT and DRIFT-PHOENIX models correspond to a dayside
temperatures of 3000 K (DRIFT-PHOENIX) and 3100 K (BT models) and fit even the
CHEOPS passband surprisingly well. However, none of the isolated brown dwarf
models can simultaneously explain both the CHEOPS and TESS eclipse depths.

###  6.8 Sensitivity on parametrisation and priors

The choice of parametrisation and priors generally affects the parameter
posteriors, and this effect can be significant when studying weak signals. We
parameterise the day- and nightside emission using log flux ratios and set
uniform priors on these, which equals to setting log-uniform (reciprocal)
priors on the day- and nightside emission flux ratios. This differs from
previous studies using uniform priors on the flux ratios.

We repeated our final joint analysis parametrising it with flux ratios rather
than log flux ratios to study how sensitive our posteriors are on the choice
of parametrisation. The most notable difference was in nightside flux ratios,
where uniform priors led to distributions with non-zero modes and the maximum
flux ratios (99th posterior percentile) were somewhat higher than when using
log-uniform priors. The second notable difference was in the CHEOPS band
dayside flux ratio posterior, which again had a non-zero mode with slightly
larger maximum values. However, the difference in the maximum flux ratio was
not large enough to change the main outcome of the analysis. The dayside flux
ratio posteriors for other passbands were not significantly affected.

We adopt the log flux ratio parametrisation since this should be more suitable
for a ”scale” parameter with an unknown magnitude. This ensures that we do not
give too much weight on apparent posterior modes if in reality we can only
estimate the upper limit securely.

###  6.9 The role of ellipsoidal variation in the CHEOPS band

Our CHEOPS band eclipse depth estimate is very sensitive on the amplitude of
the ellipsoidal variation signal, as illustrated in Fig. 16 in Appendix B.
However, the two approaches leading to physically expected EV amplitudes yield
similar CHEOPS band eclipse depths, and only nonphysically low EV amplitudes
can lead to eclipse depths that would agree with the TESS measurement.

##  7 Conclusions

We have estimated a self-consistent dayside emission spectrum of KELT-1b
covering the CHEOPS, TESS, H, Ks, and Spitzer IRAC 3.6 and 4.5 $\mu$m
passbands by modelling new CHEOPS secondary eclipse photometry of KELT-1b
jointly with the existing ground- and space-based light curves. The study adds
so-far the bluest point to KELT-1b’s emission spectrum, improves the two
ground-based NIR measurements by using a phase curve model that includes a
stellar ellipsoidal variation signal, and improves the Spitzer measurements by
modelling the Beatty et al. (2017) and Beatty et al. (2019) observations
jointly with all the other light curves.

The emission spectrum largely agrees with the previous studies. The H and Ks
dayside flux ratios are lower than the prior estimates but this was expected
since the previous modelling ignored the stellar ellipsoidal variation signal.
The TESS and Spitzer 4.5 $\mu$m bands also agree with the results by Beatty et
al. (2019), Beatty et al. (2020), and von Essen et al. (2021). The 3.6 $\mu$m
Spitzer band dayside flux ratio estimate deviates both from previous results
by Beatty et al. (2019) and theoretical expectations. This difference can
either be due to insufficiently modelled systematics or non-equilibrium
chemistry effects in the brown dwarf atmosphere, but we cannot securely
determine its cause.

Another, significantly more difficult-to-explain discrepancy occurs between
the CHEOPS and TESS passbands: the TESS observations show a deep secondary
eclipse with a depth of $360\pm 60$ ppm, but the CHEOPS observations strictly
rule out an eclipse with a depth larger than 80 ppm (Fig. 6). The CHEOPS
observations yield nearly twice higher photometric precision than the TESS
observations, so an eclipse similar to the TESS band one would be easily
detected.

Atmospheric models can reproduce KELT-1b’s emission spectrum fairly well in
most of the passbands considered, but generally manage to manage to fit only
either the CHEOPS or TESS band well. Atmospheric modelling with HELIOS leads
to a solution where KELT-1b’s geometric albedo is $\approx 0.3$ in the TESS
passband but consistent with 0 in the CHEOPS band. A contrast like this in the
geometric albedo is difficult to explain due to the overlap between the two
passbands. The models by von Essen et al. (2021) explain the TESS band without
reflection but cannot explain the CHEOPS band eclipse depth (Fig. 11).
Finally, the BT and DRIFT-PHOENIX models (Allard, 2013; Witte et al., 2011)
for isolated brown dwarfs also reproduce the observed KELT-1b emission
spectrum fairly well, but have trouble reconciling the CHEOPS and TESS eclipse
depths.

It is challenging to explain the discrepancy between the CHEOPS and TESS band
eclipse depths. Temporal variability in cloud cover caused by the transport of
transient nightside clouds to the dayside (as suggested by Beatty et al. 2020)
could provide a potential explanation. High contrast features in KELT-1b’s
albedo spectrum could also play a role in explaining a part of the
discrepancy, and could be caused by a layer of strong optical absorber such as
gaseous TiO/VO or S2/HS residing above a silicate cloud layer(Schwartz &
Cowan, 2015), but are unable to explain the discrepancy fully.

All in all, instead of shedding light on which of the previously proposed
theories might work best to explain KELT-1b’s atmosphere, our additional blue-
optical eclipse depth measurement has introduced a new open question that is
challenging to explain with our current knowledge. Further observations
combined with comprehensive modelling are required to study whether KELT-1b’s
dayside brightness spectrum is explained by transient silicate clouds (Beatty
et al., 2020), thermal emission (von Essen et al., 2021), or something else,
and how the discrepancy between the CHEOPS and TESS band eclipse depths can be
explained.

###### Acknowledgements.

We thank the anonymous referee for their insight and useful comments. We
kindly thank T.G. Beatty and B. Croll for providing their observations to our
use, and C. von Essen, M. Mallonn, N. Madhusudhan, and A. Piette for providing
their model spectra from von Essen et al. (2021). CHEOPS is an ESA mission in
partnership with Switzerland with important contributions to the payload and
the ground segment from Austria, Belgium, France, Germany, Hungary, Italy,
Portugal, Spain, Sweden, and the United Kingdom. The CHEOPS Consortium would
like to gratefully acknowledge the support received by all the agencies,
offices, universities, and industries involved. Their flexibility and
willingness to explore new approaches were essential to the success of this
mission. ACC and TGW acknowledge support from STFC consolidated grant numbers
ST/R000824/1 and ST/V000861/1, and UKSA grant number ST/R003203/1. ML
acknowledges support of the Swiss National Science Foundation under grant
number PCEFP2_194576. This project has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (project Four Aces. grant agreement No 724427). It has
also been carried out in the frame of the National Centre for Competence in
Research PlanetS supported by the Swiss National Science Foundation (SNSF). DE
acknowledges financial support from the Swiss National Science Foundation for
project 200021_200726. This work was supported by FCT - Fundação para a
Ciência e a Tecnologia through national funds and by FEDER through COMPETE2020
- Programa Operacional Competitividade e Internacionalizacão by these grants:
UID/FIS/04434/2019, UIDB/04434/2020, UIDP/04434/2020, PTDC/FIS-AST/32113/2017
& POCI-01-0145-FEDER- 032113, PTDC/FIS-AST/28953/2017 &
POCI-01-0145-FEDER-028953, PTDC/FIS-AST/28987/2017 &
POCI-01-0145-FEDER-028987, O.D.S.D. is supported in the form of work contract
(DL 57/2016/CP1364/CT0004) funded by national funds through FCT. LMS
gratefully acknowledges financial support from the CRT foundation under Grant
No. 2018.2323 ‘Gaseous or rocky? Unveiling the nature of small worlds’. S.G.S.
acknowledge support from FCT through FCT contract nr. CEECIND/00826/2018 and
POPH/FSE (EC). YA and MJH acknowledge the support of the Swiss National Fund
under grant 200020_172746. We acknowledge support from the Spanish Ministry of
Science and Innovation and the European Regional Development Fund through
grants ESP2016-80435-C2-1-R, ESP2016-80435-C2-2-R, PGC2018-098153-B-C33,
PGC2018-098153-B-C31, ESP2017-87676-C5-1-R, MDM-2017-0737 Unidad de Excelencia
Maria de Maeztu-Centro de Astrobiología (INTA-CSIC), as well as the support
of the Generalitat de Catalunya/CERCA programme. The MOC activities have been
supported by the ESA contract No. 4000124370. S.C.C.B. acknowledges support
from FCT through FCT contracts nr. IF/01312/2014/CP1215/CT0004. XB, SC, DG, MF
and JL acknowledge their role as ESA-appointed CHEOPS science team members.
ABr was supported by the SNSA. ACC acknowledges support from STFC consolidated
grant numbers ST/R000824/1 and ST/V000861/1, and UKSA grant number
ST/R003203/1. This project was supported by the CNES. The Belgian
participation to CHEOPS has been supported by the Belgian Federal Science
Policy Office (BELSPO) in the framework of the PRODEX Program, and by the
University of Liège through an ARC grant for Concerted Research Actions
financed by the Wallonia-Brussels Federation. L.D. is an F.R.S.-FNRS
Postdoctoral Researcher. B.-O.D. acknowledges support from the Swiss National
Science Foundation (PP00P2-190080). MF and CMP gratefully acknowledge the
support of the Swedish National Space Agency (DNR 65/19, 174/18). DG
gratefully acknowledges financial support from the CRT foundation under Grant
No. 2018.2323 “Gaseousor rocky? Unveiling the nature of small worlds”. M.G. is
an F.R.S.-FNRS Senior Research Associate. SH gratefully acknowledges CNES
funding through the grant 837319. KGI is the ESA CHEOPS Project Scientist and
is responsible for the ESA CHEOPS Guest Observers Programme. She does not
participate in, or contribute to, the definition of the Guaranteed Time
Programme of the CHEOPS mission through which observations described in this
paper have been taken, nor to any aspect of target selection for the
programme. This work was granted access to the HPC resources of MesoPSL
financed by the Region Ile de France and the project Equip@Meso (reference
ANR-10-EQPX-29-01) of the programme Investissements d’Avenir supervised by the
Agence Nationale pour la Recherche. PM acknowledges support from STFC research
grant number ST/M001040/1. GSc, GPi, IPa, LBo, VNa and RRa acknowledge the
funding support from Italian Space Agency (ASI) regulated by “Accordo ASI-INAF
n. 2013-016-R.0 del 9 luglio 2013 e integrazione del 9 luglio 2015 CHEOPS Fasi
A/B/C”. This work was also partially supported by a grant from the Simons
Foundation (PI Queloz, grant number 327127). IRI acknowledges support from the
Spanish Ministry of Science and Innovation and the European Regional
Development Fund through grant PGC2018-098153-B- C33, as well as the support
of the Generalitat de Catalunya/CERCA programme. GyMSz acknowledges the
support of the Hungarian National Research, Development and Innovation Office
(NKFIH) grant K-125015, a PRODEX Institute Agreement between the ELTE Eötvös
Loránd University and the European Space Agency (ESA-D/SCI-LE-2021-0025), the
Lendület LP2018-7/2021 grant of the Hungarian Academy of Science and the
support of the city of Szombathely. V.V.G. is an F.R.S-FNRS Research
Associate. NAW acknowledges UKSA grant ST/R004838/1. M.F. gratefully
acknowledge the support of the Swedish National Space Agency (SNSA; DNR
177/19, 174/18, and 65/19). S.S. has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (grant agreement No 833925, project STAREX).

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##  Appendix A Chromatic albedo variability

###  A.1 Albedo spectrum model

We use a simple two-level albedo spectrum model to study whether the
discrepancy between the CHEOPS and TESS band dayside flux ratios could be
caused by chromatic variability in albedo. The model is parametrised by two
geometric albedo levels, $A_{\mathrm{g,0}}$ (blue) and $A_{\mathrm{g,1}}$
(red), and a step location $\lambda_{0}$, so that

$A_{\mathrm{g}}(\lambda)=\cases{A_{\mathrm{g,0}}}&\quad\text{if
}\lambda<\lambda_{0}\\\ A_{\mathrm{g,1}}\quad\text{if
}\lambda\geq\lambda_{0}{},$ (16)

as illustrated in the middle panel of Fig. 12.

Our main quantity of interest is the contrast between the reflected light in
the CHEOPS and TESS bands,

$C=\frac{F_{\mathrm{T}}-F_{\mathrm{C}}}{F_{\mathrm{T}}+F_{\mathrm{C}}},\quad\text{where
}F=\int S(\lambda)\,A_{\mathrm{g}}(\lambda)\,T(\lambda)\,\mathrm{d}\lambda,$
(17)

$S$ is the stellar spectrum (a BT-Settl spectrum by Allard 2013),
$A_{\mathrm{g}}$ is the albedo spectrum model, and $T$ is the transmission
function for either TESS ($F_{\mathrm{T}}$) or CHEOPS ($F_{\mathrm{C}}$).

We assume $A_{\mathrm{g,1}}$ that matches the geometric albedo for a ”roaster”
planet with silicate clouds from Sudarsky et al. (2000), compute $C$ for a set
of $A_{\mathrm{g,0}}$ values between 0.01 (similar to TrES-2b, Kipping &
Spiegel 2011) and 0.05, and $\lambda_{0}$ values between 400 and 1000 nmn and
show the results in the lower panel of Fig. 12. The maximum $C$ is achieved
for $\lambda_{0}\sim 800$ nm and varies from 0.17 to 0.41. We obtain $C=0.41$
for the lowest included blue $A_{\mathrm{g}}$ (as expected),
$A_{\mathrm{g,0}}=0.01$, and $C$ decreases quickly with increasing
$A_{\mathrm{g,0}}$.

Figure 12: Top: a model spectrum of KELT-1 (light blue line), the CHEOPS and
TESS passband transmission functions (black lines). Middle: the step-function
albedo spectrum model. Bottom: TESS to CHEOPS reflected light contrast as a
function of $A_{\mathrm{g,0}}$ and $\lambda_{0}$. The step locations
corresponding to the maximum flux ratios are marked as dashed vertical lines.

###  A.2 Comparison between modelled and observed reflection ratio

Before we can compare our theoretical $C$ values to the $C$ distribution from
the light curve analysis, we need to estimate and remove the contribution from
thermal emission in the latter. This contribution should be small for the
CHEOPS and TESS bands, but is not necessarily negligible.

We calculate a set of ”reflection only” flux ratio samples from the original
dayside flux ratio samples using a Monte Carlo approach. We draw a set of
KELT-1b temperatures from a uniform distribution from 2800 to 3200 K
(corresponding to the KELT-1b Spitzer 4.5 $\mu$m $T_{\mathrm{b}}$ ranges given
in Table 4), we calculate the emission contribution in the CHEOPS and TESS
bands using BT-Settl model spectra by Allard (2013) for each temperature
sample, and finally we remove these contributions from a random sample of
CHEOPS and TESS dayside flux ratios . The full process is carried out in the
chromatic albedo variation test notebook github. Figure 13 shows the original
dayside flux ratios with emission and reflection, and the ”reflection-only”
flux ratios. The correction is rather small for the CHEOPS band, but
significant for the TESS band. The bottom panel of Fig. 13 shows the final
observed $R$ distribution.

Our maximum $C$ of 0.41 is at the lower tail of the observed $C$ distribution.
The maximum $C$ corresponds to a low but still physically realistic red
$A_{\mathrm{g}}$ of 0.01, which is similar to what has been measured for
TrES-2b (Kipping & Spiegel 2011). However, while both $A_{\mathrm{g,0}}$ and
$A_{\mathrm{g,1}}$ are physically plausible separately, it is unclear if this
is the case for an albedo spectrum combining the two.

Figure 13: Top and middle: original CHEOPS and TESS dayside flux ratio
distribution estimates together with versions of the distributions with the
thermal emission removed as described in Appendix A.2. Bottom: the final TESS
to CHEOPS contrast ($C$) distribution and the maximum $C$ of $\approx 0.4$
from the albedo spectrum modelling marked with a dashed vertical line.

##  Appendix B Comparison between EV scenarios

In addition to the final joint analysis, we carried out a set of analyses for
the external dataset (ED) consisting of the TESS, H, Ks, and Spitzer
photometry, and another set of analyses for the CHEOPS photometry alone. The
ED analysis was carried out to provide priors on the orbital and geometric
parameters for the CHEOPS analysis and to study how different approaches to
constrain the EV amplitudes affect our parameter estimates. The CHEOPS
analysis was carried out to also study how the EV amplitude constraints affect
the eclipse depth estimate, and to provide detrended CHEOPS light curves for
the final joint analysis.

Both the ED and CHEOPS-only analyses consider three scenarios that differ in
the priors set on the EV amplitude:

  * a)

_Theoretical EV:_ the EV amplitudes are given normal priors based on
theoretical estimates in Table 3. This is the most constraining scenario, but
can lead to biased eclipse depth estimates if the semi-major axis estimate
used to calculate the amplitudes is biased.

  * b)

_Constrained EV ratios:_ the EV amplitudes are constrained relative to the EV
amplitude in the TESS passband and the TESS passband EV amplitude is given a
uninformative prior. This scenario works as a stepping stone between the
strongly constrained and completely unconstrained EV cases, and is not
sensitive on our prior semi-major axis estimate.

  * c)

_Unconstrained EV :_ all the EV amplitudes are given uninformative priors to
see whether the EV amplitudes from the TESS and Spitzer passbands (where we
have coverage over the full orbital phase) agree with the theoretical
expectations.

The CHEOPS-only analysis considers also an additional scenario with EV
amplitude forced to zero:

  * d)

_No EV:_ the EV amplitude is forced to zero. The motivation for this scenario
is to see how ignoring EV would affect our CHEOPS eclipse depth estimate.

Except for the data included, the models are identical to the full joint
model. The baseline and noise models for the ED analyses are the same as in
the final analysis. However, the baseline in the CHEOPS-only analysis is
modelled differently than in the final analysis. In the CHEOPS analysis, we
use the basis vectors determined by the pipeline by Wilson et al. as
covariates and each visit is given its own set of baseline coefficients. The
number of basis vectors per visit varies from 13 to 29, and the total number
of basis vectors (and thus free parameters from the baseline model) is 156.
This approach allows us to detrend the CHEOPS light curves together with the
phase curve model to be used in the final joint analysis (this is safe and
does not affect the final parameter estimates). The CHEOPS analysis uses the
posteriors from the external dataset analysis as priors for the zero epoch,
orbital period, stellar density, impact parameter, and planet-star area ratio.

We show the EV amplitude posteriors for all the analyses in Fig. 14, the
dayside flux ratio posteriors in Fig. 15, and the CHEOPS dayside posteriors in
Fig. 16. All in all, the dayside flux ratio posteriors all agree with each
other for the final joint modelling and EV prior scenarios a and b. The flux
ratio posteriors also agree for scenario d for passbands with photometry
covering a full orbital phase, but the free- and no-EV scenarios (c and d)
lead to overestimated flux ratios for CHEOPS photometry.

Figure 14: Ellipsoidal variation amplitude posteriors for f) final joint
analysis with relative prior on the EV amplitude, a) theoretical prior on EV
amplitude, b) relative prior on EV amplitude, and c) uninformative prior on EV
amplitude. The case c with an uninformative prior on EV is shown only for the
TESS and Spitzer passbands with full phase coverage. Figure 15: As in Fig. 14,
but for the dayside flux ratio. Figure 16: Dayside flux ratios the CHEOPS-only
analysis scenarios. The scenario labels are the same as in Fig. 14 with an
additional no-EV scenario d.

##  Appendix C Comparison between brightness temperature calculation
approaches

We calculate the brightness temperatures discussed in Sect. 6.2 using three
approaches: physical-physical, where both the star and the brown dwarf are
modelled using the BT-Settl spectra by Allard (2013); blackbody-physical,
where KELT-1b is modelled as a black body and its host star using a BT-Settl
spectrum; and blackbody-blackbody, where both the brown dwarf and its host
star are modelled as black bodies. Our reported brightness temperature values
correspond to averages over the three approaches, but we also tested whether
the different approaches agree with each other or not.

We present the day- and nightside brightness temperatures separately for each
approach in Fig. 17. All the three approaches agree with each other within
uncertainties.

Figure 17: Day- and nightside brightness temperatures estimated from the flux
ratios using the p) physical-physical, x) blackbody-physical, and b)
blackbody-blackbody approaches. The shading marks the 68% and 99.7% central
posterior intervals.

##  Appendix D Additional retrievals to explore dependence of outcomes on data

Figure 18: Eclipse depths in ppm as a function of the internal temperature
$T_{\mathrm{int}}$ and the parameter $E_{*}$ based on the HELIOS model
calculations described in Sect. 6.4. The six panels show the eclipse depths in
the various passbands. White, solid lines depict the measured occultation
depths from Table 4, dashed lines refer to their $1\sigma$ error bars. The red
lines for the two Spitzer passbands refer to the eclipse depths reported by
Beatty et al. (2019).

Figure 18 shows the degeneracy between $T_{\rm int}$ and $E_{*}$ when the
model accounts for the TESS secondary eclipse. Within this family of
solutions, the solution of von Essen et al. (2021), which explains the dayside
emission spectrum without the need for reflected light, is included.

Figure 19: Posterior distributions for a retrieval with Helios-r2 using only
the TESS secondary eclipse measurement and the two Spitzer points from Beatty
et al. (2019). The left panel shows the results for the full range of prior
distributions discussed in Sect. 6.4, which yields a bimodal solution with
respect to $T_{\mathrm{int}}$ and the parameter $E_{*}$. The two other panels
isolate the two solution modes.

In Fig. 19, we perform a retrieval with just the TESS, Spitzer 3.6 $\mu$m and
4.5 $\mu$m data points reported by Beatty et al. (2019). The left panel of 19
shows a a retrieval with the full prior ranges. This yields a bimodal solution
for $T_{\rm int}$ and $E_{*}$: a high value of $E_{*}$ and a low value of
$T_{\rm int}$, or a low value of $E_{*}$ and a high value of $T_{\rm int}$.
The two different solutions are isolated in the other two panels of Fig. 19.

When the H and Ks data points are added to the retrieval, the bimodality is
broken (not shown). This can be understood by looking at the ranges of $T_{\rm
int}$ and $E_{*}$ values that are consistent with the measurements in this
passbands depicted in 18. The results presented in this figure suggest that
the H and Ks secondary eclipse measurements are only consistent with high
values of $T_{\rm int}$

Figure 20: Posterior distributions for a retrieval with Helios-r2 using all
secondary eclipse measurement from Table 4 but with the two Spitzer points
from Beatty et al. (2019).

Finally, in Fig. 20 we perform a retrieval using all available measurements
but with the Spitzer secondary eclipse depths from Beatty et al. (2019). The
addition of the CHEOPS data point now further narrows the posterior on $E_{*}$
down to a median value of 0.2. this behaviour can also be easily understood by
comparison with the theoretically calculated eclipse depths in Fig. 18. For
CHEOPS, the results indicate that the measured eclipse depth is only
compatible with a maximum $E_{*}$ of about 0.2.

We also note that the results using the Spitzer data from Beatty et al. (2019)
essentially provides the same posterior distribution than using the reduction
of the Spitzer of this work (see Fig. 8). . In particular, the very high
geometric albedo in the TESS passband is consistently obtained with either
dataset. Despite the difference in the reported Spitzer 3.6 $\mu$m eclipse
depth, the retrieval outcome is, thus, unaffected because the solution with
$T_{\rm int}\approx 2700$ K has a blackbody peak at shorter wavelengths,
within the H and Ks passbands.