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

# CHEOPS and TESS view of the ultra-short period super-Earth TOI-561
b††thanks: Based in part on Guaranteed Time Observations on the European Space
Agency telescope CHEOPS under programme 10.0008 by the CHEOPS consortium. The
raw and detrended photometric time series data are available in electronic
form at the CDS via anonymous ftp to cdsarc.cds.unistra.fr (130.79.128.5) or
via https://cdsarc.cds.unistra.fr/cgi-bin/qcat?J/A+A/

J. A. Patel ${}^{\href https://orcid.org/0000-0001-5644-6624}$ 11
<EMAIL_ADDRESS>J. A. Egger ${}^{\href
https://orcid.org/0000-0003-1628-4231}$ 22 T. G. Wilson ${}^{\href
https://orcid.org/0000-0001-8749-1962}$ 3344 V. Bourrier ${}^{\href
https://orcid.org/0000-0002-9148-034X}$ 55 L. Carone 66 M. Beck ${}^{\href
https://orcid.org/0000-0003-3926-0275}$ 55 D. Ehrenreich ${}^{\href
https://orcid.org/https://www.notion.so/D-Ehrenreich-7e5007d8e19f46368c036b8dfbbd5a59}$
5577 S. G. Sousa ${}^{\href https://orcid.org/0000-0001-9047-2965}$ 88 W. Benz
${}^{\href https://orcid.org/0000-0001-7896-6479}$ 2299 A. Brandeker
${}^{\href https://orcid.org/0000-0002-7201-7536}$ 11 A. Deline ${}^{\href
https://orcid.org/https://www.notion.so/A-Deline-
TS1-9b818a0c88e04c45910c1e49e0a4397d}$ 55 Y. Alibert ${}^{\href
https://orcid.org/0000-0002-4644-8818}$ 22 K. W. F. Lam ${}^{\href
https://orcid.org/0000-0002-9910-6088}$ 1010 M. Lendl ${}^{\href
https://orcid.org/0000-0001-9699-1459}$ 55 R. Alonso 11111212 G. Anglada
${}^{\href https://orcid.org/0000-0002-3645-5977}$ 13131414 T. Bárczy
${}^{\href https://orcid.org/0000-0002-7822-4413}$ 1515 D. Barrado ${}^{\href
https://orcid.org/0000-0002-5971-9242}$ 1616 S. C. C. Barros ${}^{\href
https://orcid.org/0000-0003-2434-3625}$ 881717 W. Baumjohann ${}^{\href
https://orcid.org/0000-0001-6271-0110}$ 66 T. Beck 22 N. Billot ${}^{\href
https://orcid.org/0000-0003-3429-3836}$ 55 X. Bonfils ${}^{\href
https://orcid.org/0000-0001-9003-8894}$ 1818 C. Broeg ${}^{\href
https://orcid.org/0000-0001-5132-2614}$ 2299 M.-D. Busch 22 J. Cabrera 1010 S.
Charnoz ${}^{\href https://orcid.org/0000-0002-7442-491X}$ 1919 A. Collier
Cameron ${}^{\href https://orcid.org/0000-0002-8863-7828}$ 33 Sz. Csizmadia
${}^{\href https://orcid.org/0000-0001-6803-9698}$ 1010 M. B. Davies
${}^{\href https://orcid.org/0000-0001-6080-1190}$ 2020 M. Deleuil ${}^{\href
https://orcid.org/0000-0001-6036-0225}$ 2121 L. Delrez ${}^{\href
https://orcid.org/0000-0001-6108-4808}$ 22222323 O. D. S. Demangeon ${}^{\href
https://orcid.org/0000-0001-7918-0355}$ 881717 B.-O. Demory ${}^{\href
https://orcid.org/0000-0002-9355-5165}$ 99 A. Erikson 1010 A. Fortier
${}^{\href https://orcid.org/0000-0001-8450-3374}$ 2299 L. Fossati ${}^{\href
https://orcid.org/0000-0003-4426-9530}$ 66 M. Fridlund ${}^{\href
https://orcid.org/0000-0002-0855-8426}$ 24242525 D. Gandolfi ${}^{\href
https://orcid.org/0000-0001-8627-9628}$ 2626 M. Gillon ${}^{\href
https://orcid.org/0000-0003-1462-7739}$ 2222 M. Güdel 2727 K. Heng ${}^{\href
https://orcid.org/0000-0003-1907-5910}$ 9944 S. Hoyer ${}^{\href
https://orcid.org/0000-0003-3477-2466}$ 2121 K. G. Isaak ${}^{\href
https://orcid.org/0000-0001-8585-1717}$ 2828 L. L. Kiss 29293030 E. Kopp 3131
J. Laskar ${}^{\href https://orcid.org/0000-0003-2634-789X}$ 3232 A.
Lecavelier des Etangs ${}^{\href https://orcid.org/0000-0002-5637-5253}$ 3333
C. Lovis ${}^{\href https://orcid.org/0000-0001-7120-5837}$ 55 D. Magrin
${}^{\href https://orcid.org/0000-0003-0312-313X}$ 3434 P. F. L. Maxted
${}^{\href https://orcid.org/0000-0003-3794-1317}$ 3535 V. Nascimbeni
${}^{\href https://orcid.org/0000-0001-9770-1214}$ 3434 G. Olofsson ${}^{\href
https://orcid.org/0000-0003-3747-7120}$ 11 R. Ottensamer 2727 I. Pagano
${}^{\href https://orcid.org/0000-0001-9573-4928}$ 3636 E. Pallé ${}^{\href
https://orcid.org/0000-0003-0987-1593}$ 1111 G. Peter ${}^{\href
https://orcid.org/0000-0001-6101-2513}$ 3131 G. Piotto ${}^{\href
https://orcid.org/0000-0002-9937-6387}$ 34343737 D. Pollacco 44 D. Queloz
${}^{\href https://orcid.org/0000-0002-3012-0316}$ 38383939 R. Ragazzoni
${}^{\href https://orcid.org/0000-0002-7697-5555}$ 34343737 N. Rando 4040 F.
Ratti 4040 H. Rauer ${}^{\href https://orcid.org/0000-0002-6510-1828}$
101041414242 I. Ribas ${}^{\href https://orcid.org/0000-0002-6689-0312}$
13131414 N. C. Santos ${}^{\href https://orcid.org/0000-0003-4422-2919}$
881717 G. Scandariato ${}^{\href https://orcid.org/0000-0003-2029-0626}$ 3636
D. Ségransan ${}^{\href https://orcid.org/0000-0003-2355-8034}$ 55 A. E. Simon
${}^{\href https://orcid.org/0000-0001-9773-2600}$ 22 A. M. S. Smith
${}^{\href https://orcid.org/0000-0002-2386-4341}$ 1010 M. Steller ${}^{\href
https://orcid.org/0000-0003-2459-6155}$ 66 Gy. M. Szabó 43434444 N. Thomas 22
S. Udry ${}^{\href https://orcid.org/0000-0001-7576-6236}$ 55 B. Ulmer 3131 V.
Van Grootel ${}^{\href https://orcid.org/0000-0003-2144-4316}$ 2323 V. Viotto
3434 N. A. Walton ${}^{\href https://orcid.org/0000-0003-3983-8778}$ 4545

(Received XX; accepted XX)

###### Abstract

Context. Ultra-short period planets (USPs) are a unique class of super-Earths
with an orbital period of less than a day and hence subject to intense
radiation from their host star. These planets cannot retain a primordial H/He
atmosphere, and most of them are indeed consistent with being bare rocky
cores. A few USPs, however, show evidence for a heavyweight envelope, which
could be a water layer resilient to evaporation, or a secondary metal-rich
atmosphere sustained by outgassing of the molten, volcanic surface. Much thus
remains to be learned about the nature and formation of USPs.

Aims. The prime goal of the present work is to refine the bulk planetary
properties of the recently discovered TOI-561 b, through the study of its
transits and occultations. This is crucial to understand the internal
structure of this USP and assess the presence of an atmosphere.

Methods. We obtained ultra-precise transit photometry of TOI-561 b with
CHEOPS, and performed a joint analysis of this data along with three archival
visits from CHEOPS and four TESS sectors.

Results. Our analysis of TOI-561 b transit photometry put strong constraints
on its properties. In particular, we restrict the uncertainties on the
planetary radius at $\sim 2\%$ retrieving $R_{p}=1.42\pm 0.02\,R_{\oplus}$.
This result informs our internal structure modelling of the planet, which
shows that the observations are consistent with negligible H/He atmosphere,
however requiring other lighter materials, in addition to pure iron core and
silicate mantle to explain the observed density. We find that this can be
explained by the inclusion of a water layer in our model. Additionally, we run
a grid of forward models with a water-enriched atmosphere to explain the
transit radius. We searched for variability in the measured $R_{p}/R_{\star}$
over time, which could trace changes in the structure of the planetary
envelope. No temporal variations are however recovered within the present data
precision. In addition to the transit event, we tentatively detect occultation
signal in the TESS data with an eclipse depth $L=27.40^{+10.87}_{-11.35}$ ppm.
We use models of outgassed atmospheres from the literature to explain this
eclipse signal. We find that the thermal emission from the planet can mostly
explain the observation. Based on this, we predict that near/mid-infrared
observations with the James Webb Space Telescope should be able to detect
silicate species in the atmosphere of the planet. This could also reveal
important clues about the planetary interior and help disentangle planet
formation and evolution models.

###### Key Words.:

techniques: photometric – planets and satellites: terrestrial planets –
planets and satellites: composition – planets and satellites: atmospheres –
planets and satellites: individual: TOI-561 b

##  1 Introduction

The advent of the new generation of exoplanet discovery missions, such as
Kepler/K2 (Borucki et al., 2010; Howell et al., 2014) and the Transiting
Exoplanet Survey Satellite (TESS; Ricker et al., 2014), has led to the
detection of the vast majority of known exoplanets. These planets and
planetary systems show a large diversity in terms of composition and orbital
architecture. In particular there exists one curious class of lower mass
planets with radius smaller than $\sim 2R_{\oplus}$ and that orbit their host
stars within a day, and hence dubbed as ultra-short period planets (USPs)
(see, Winn et al., 2018, for a review). USPs receive extreme radiation from
their host star, leading to the loss of any primary H/He atmosphere these
planet may have formed with (Lopez & Fortney, 2013; Owen & Wu, 2013, 2016).
These planets could, however, develop and sustain secondary atmospheres made
up of heavier species. This may consist of a variety of species such as Na, K,
SiO, SiO2, O2 etc., depending upon their dayside temperature and pressure
(Schaefer & Fegley, 2009; Miguel et al., 2011; Schaefer et al., 2012;
Wordsworth & Kreidberg, 2022). Since the surface of the planet can potentially
be molten (Léger et al., 2011), the composition of the atmosphere becomes
strongly dependent on that of the interior (see, e.g., Wordsworth & Kreidberg,
2022). The temperature gradient across the day- and night-side of the planet
can transport some of the species to the night-side, causing their
condensation (Kite et al., 2016). Additionally, some of the species can be
lost into space (see, Wordsworth & Kreidberg, 2022, for a review). These
various processes could potentially result in temporal variability for the
atmosphere of some USPs (Kite et al., 2016). In the most extreme cases, the
surface of the planet itself is thought to be disintegrating and escaping into
space, leading to the formation of dust tails that can be observed through
their highly asymmetric transit light curves (see, e.g., Rappaport et al.,
2012; Brogi et al., 2012; Rappaport et al., 2014). The photometric and
spectroscopic observations of such disintegrating planets, and more generally
of USPs, can not only give insights into the nature of their atmospheres but
also provide essential clues regarding their interior.

The formation mechanism of USPs is a topic of active research. Scarcity of
close-in planets with sub-Jovian mass in the so-called Neptunian desert (e.g.,
Lecavelier Des Etangs, 2007; Davis & Wheatley, 2009; Szabó & Kiss, 2011;
Beaugé & Nesvorný, 2013; Lundkvist et al., 2016) which is thought to be a
result of atmospheric escape (e.g., Lecavelier des Etangs et al., 2004; Owen &
Jackson, 2012; Owen, 2019), led to a conjecture that the USPs are remnants of
hot Neptunes that have gone through strong mass loss (e.g., Ehrenreich &
Désert, 2011; Lopez & Fortney, 2013; Venturini et al., 2020). Furthermore,
Winn et al. (2017) show that the metallicity distribution of USP hosts is very
similar to that of mini-Neptune hosts, which supports the suggested formation
pathway. Other hypotheses proposed to explain the formation of USPs include
the migration of rocky planets from outer orbits and in-situ formation
(Raymond et al., 2008; Winn et al., 2018).

In order to better understand the formation mechanism of USPs and to constrain
their internal structure and atmospheric composition, it is important to
perform systematic studies refining their mass and radius. TESS, with its
whole sky survey mode for exoplanet discovery, is a unique instrument to
discover new USPs around nearby stars. Weiss et al. (2021) and Lacedelli et
al. (2021) have uncovered a multiplanetary system around a small, metal-poor
G-dwarf TOI-561, which contains a USP, TOI-561 b, by analysing two of TESS
sectors along with other follow-up observations. Being a kinematic and
chemically thick disk star, TOI-561, is alpha-enhanced (Lacedelli et al.,
2022). This means that composition of the proto-planetary disk is likely
different to those around Solar-metallicity stars. This chemical-diversity is
likely propagated through to planetary internal structures (Asplund et al.,
2009; Thiabaud et al., 2014), as is starting to be seen observationally
(Adibekyan et al., 2021; Chen et al., 2022; Wilson et al., 2022). This could
makes the core of the planets it hosts smaller and silicate mantle larger.
This, in addition to potentially volatile envelope, is likely one of the main
reasons making TOI-561 b the lowest density USP observed up to today. The low
density composition ($0.69\,\rho_{\oplus}$, Lacedelli et al., 2022) is
drastically different from other well-known USPs, such as 55 Cnc e or K2-141
b, which have much higher densities ($1.21\,\rho_{\oplus}$ and
$1.48\,\rho_{\oplus}$, respectively, Bourrier et al., 2018; Malavolta et al.,
2018). This makes TOI-561 b an interesting target for follow-up observations.

Among the multiple planets hosted by TOI-561, the inner-most planet was found
to orbit the star in just $\sim 11$ hours. To probe the nature of this
extremely irradiated planet, which is also the lowest density USP observed
until today, we observed 13 new transits of TOI-561 b with the CHaracterising
ExOPlanet Satellite (CHEOPS; Benz et al., 2021, see Section 2) to refine its
radius, and thus improve our knowledge of its internal structure and putative
atmosphere. We combined the new CHEOPS observations along with three CHEOPS
archival transits and four TESS sectors to update the planetary parameters
(Section 2). We also searched for an occultation signal in the available
dataset to learn about the planetary atmosphere. Using the updated parameters,
we performed internal structure modelling of the planet (Section 3). Section 4
explores various theoretical models to understand the emission from the planet
(Section 4.1). Subsequently, we use these models to make predictions for
observations with the James Webb Space Telescope (JWST). We then carried out
an analysis to search for variability in the transit depth (Section 4.2). We
discuss our conclusions and their implications for future work in Section 5.

##  2 Observations and analysis

Table 1: Observation log for TESS and CHEOPS

Visit/Sector | Start Date | End Date | File Key
---|---|---|---
TESS Observations |  |  |
8 | 2019-02-02 20:49:29 | 2019-02-27 12:07:39 | Sector 8
35 | 2021-02-09 11:51:16 | 2021-03-06 11:36:59 | Sector 35
45 | 2021-11-07 11:45:08 | 2021-12-02 03:00:39 | Sector 45
46 | 2021-12-03 01:34:47 | 2021-12-30 04:54:14 | Sector 46
CHEOPS Observations |  |  |
37001 | 2021-01-23 15:29:08 | 2021-01-24 07:09:34 | CH_PR100031_TG037001
801 | 2021-02-28 20:16:29 | 2021-03-01 00:41:36 | CH_PR100008_TG000801
101 | 2021-03-01 01:12:29 | 2021-03-01 13:56:50 | CH_PR110050_TG000101
802 | 2021-03-06 04:53:31 | 2021-03-06 10:18:40 | CH_PR100008_TG000802
803 | 2021-03-08 10:30:28 | 2021-03-08 15:19:36 | CH_PR100008_TG000803
804 | 2021-03-12 11:02:08 | 2021-03-12 15:26:15 | CH_PR100008_TG000804
805 | 2021-03-16 00:55:08 | 2021-03-16 05:44:16 | CH_PR100008_TG000805
806 | 2021-03-20 02:04:08 | 2021-03-20 06:28:15 | CH_PR100008_TG000806
807 | 2021-03-20 22:11:48 | 2021-03-21 03:21:57 | CH_PR100008_TG000807
808 | 2021-03-23 03:48:32 | 2021-03-23 08:36:40 | CH_PR100008_TG000808
809 | 2021-03-24 02:09:09 | 2021-03-24 06:58:17 | CH_PR100008_TG000809
810 | 2021-03-25 00:00:07 | 2021-03-25 04:42:15 | CH_PR100008_TG000810
811 | 2021-03-29 10:19:08 | 2021-03-29 14:44:15 | CH_PR100008_TG000811
812 | 2021-03-30 08:23:29 | 2021-03-30 13:12:37 | CH_PR100008_TG000812
813 | 2021-03-31 06:01:29 | 2021-03-31 10:50:37 | CH_PR100008_TG000813
39301 | 2021-04-12 23:52:28 | 2021-04-15 05:37:57 | CH_PR100031_TG039301
111The file key is a unique identifier of each of the CHEOPS observations, and
helps in retrieving data from the archive.

###  2.1 Observations and data reduction

We observed precise transit photometry of TOI-561 with CHEOPS, which is an
S-class mission launched by the European Space Agency to obtain precision
photometry of exoplanetary systems. We observed 13 transits of TOI-561 b with
CHEOPS within the Guaranteed Time Observation (GTO) program during February-
March 2021 (see, Table 1 for the observation log). In addition to this, we
included in our analysis three archival visits of CHEOPS for the same system
from Lacedelli et al. (2022). Since the target is moderately bright
($m_{\textnormal{Gaia}}=10.01$), we observed it with the longest available
exposure time of 60 s to reduce the instrumental noise. To save bandwidth,
only the so-called sub-arrays, a cropped circular subsections of 100 pixel
radius, are downloaded instead of the full frame images. The photometry from
these sub-arrays was extracted with the CHEOPS Data Reduction Pipeline (DRP;
Hoyer et al., 2020), which uses aperture photometry to derive fluxes from the
sub-arrays. We also extracted photometry independently using the PSF
photometry package PIPE222https://github.com/alphapsa/PIPE (see Szabó et al.
2021 for more information). When comparing the extracted lightcurves, we found
PIPE to typically produce a $\sim$10% lower scatter at around 300 part per
million (ppm) as measured by the median absolute deviation (MAD) at the
observed time cadence of 1 min. In the following we therefore report on the
analysis of the PIPE photometry, although the results are consistent with
independent analysis of the DRP photometry that was also made but that is not
reported here.

Prior to performing the transit analysis, we discarded all points from our
CHEOPS dataset which are flagged by PIPE to have poor photometry. This is
mostly because those points are contaminated by, e.g., strong cosmic rays or
trails from satellites crossing the field. Additionally, we discarded frames
with high background (higher than 400 e- pix-1, mostly when pointing near
Earth’s limb) as the background often shows a temporally variable spatial
structure that is difficult to remove and significantly increases the noise.
The photometry also shows a trend with background at high values, possibly due
to a non-linearity in the charge transfer efficiency. This filtering of data
points is independent of the photometric values, so does not bias the light
curve. About 12% of data points are discarded in this way, almost all from
observations with a line of sight within 20∘ from the Earth limb.

In addition to the CHEOPS datasets we used TESS data from the four sectors in
which it observed TOI-561 (Sectors 8, 35, 45 and 46). We used the PDC-SAP
photometry (Smith et al., 2012; Stumpe et al., 2014) reduced by the TESS
Science Processing Operations Center (SPOC; Jenkins et al., 2016).

###  2.2 Transit lightcurve analysis

Although we take advantage of both the CHEOPS and TESS datasets to refine the
planetary properties of TOI-561 b, we first analysed these datasets
individually. This was mainly to determine and constrain the astrophysical and
systematic noise models for CHEOPS and TESS data. We use juliet (Espinoza et
al., 2019) to perform this analyses. We detail this process in the sub-
sections below.

TOI-561 b is the innermost body in the four-planet system around TOI-561.
Since we focus on TOI-561 b in the present work we decided to set wide
uninformative priors for its planetary parameters, except for the orbital
period and the transit time on which we set Gaussian priors based on Lacedelli
et al. (2022). Since TOI-561 b orbits the star in around 11 hours, we expect
its orbit to be circularised. Accordingly we fixed its orbital eccentricity
($e_{b}$) to zero and argument of periastron ($\omega_{b}$) to $90^{\circ}$.
For the other three planets we set Gaussian priors on most of the planetary
parameters, with mean and standard deviation based on their values from
Lacedelli et al. (2022). We fixed their eccentricities and arguments of
periastron passage to the values from Lacedelli et al. (2022), as it is
difficult to retrieve these properties from the photometric data alone. The
values from Lacedelli et al. (2022) are expected to be robust as they use
plenty of radial velocity (RV) data points along with photometric data to
estimate those parameters. The latter is also the reason, in addition to the
lack of further public RV data, we decided not to use RV data in our analysis.

Instead of using different priors for the scaled semi-major axis
($a/R_{\star}$) of each planet, we fitted the stellar density to take
advantage of our knowledge of stellar properties. The value of the stellar
density, which can be connected to the scaled semi-major axis through Kepler’s
third law, was computed from the stellar mass and radius reported in Lacedelli
et al. (2022). Doing this we account for stellar properties while modelling
planetary transits, reduce the number of free parameters in the fit by three,
and enforce that all planets are orbiting a star with the same stellar
density. For the sake of completeness, we re-did the whole analysis without
using the stellar density prior (see, Appendix A). The results are in
agreement with our main approach albeit with increased uncertainties, as
expected.

We use quadratic limb darkening law in our analysis. We parametrize the limb
darkening coefficients (LDCs) as suggested by Kipping (2013) and use this
parametrization in our model. This was to ensure that they yield physically
plausible stellar intensity profile. As the theoretical LDCs, computed from
model stellar atmospheres, could have discrepancies with the empirical LDCs
(see, e.g., Espinoza & Jordán, 2015; Patel & Espinoza, 2022), we set uniform
priors between 0 and 1 to the transformed quadratic LDCs, for both CHEOPS and
TESS bandpasses. The full list of priors on the planetary parameters can be
found in Table 3.

####  2.2.1 Analysis of the CHEOPS photometry

The orbit of CHEOPS is designed to be nadir locked, with the satellite
revolving continuously over the day-night terminator of the Earth. As a
result, the field-of-view of CHEOPS rotates around the target star in the
image frame. The rotation can introduce correlations with background stars.
These environmental effects, in addition to other instrumental effects, makes
CHEOPS photometry correlated with instrumental parameters (Lendl et al., 2020;
Wilson et al., 2022). The transit lightcurves that we obtained for TOI-561 b
were no exception — we therefore detrend the data against spacecraft roll
angle in order to provide robust estimates of the planetary properties.

In most of the visits we analysed, the trend with roll angle was too
complicated to model with sinusoidal functions. Thus, a Gaussian Process (GP)
model, implemented from celerite (Foreman-Mackey et al., 2017) in juliet,
built from an Exponential-Matérn kernel, was used to model the trend with roll
angle. Each visit was decorrelated individually against the roll angle in this
way. However, as the number of data points in an individual visit is not
enough to constrain the GP model properly, we fitted all visits simultaneously
to obtain a well-constrained GP model that accounts for the trends with roll
angle. In this whole dataset fit, the priors on GP hyperparameters were chosen
from our previous analysis of individual visits to help convergence and a
better optimization of parameter space. In the end, we subtracted the modelled
trend with roll angle from our dataset to produce roll angle detrended
photometry. To prevent an underestimation of uncertainties in the subsequent
analysis, we propagated the uncertainties in the GP model by adding them in
quadrature to the errorbars on fluxes of each visit (similar to the treatment
of CHEOPS data with PSF-basis vectors in Wilson et al., 2022).

While the resulting photometry is corrected for the roll angle trend, it still
shows the correlations with other parameters, such as the PSF centroid
position, background, time etc. It is important to choose a suitable set of
detrending vectors for each visit, otherwise we may overfit or underfit the
dataset. Our systematic search for a set of decorrelation vectors was done
using pycheops (Maxted et al., 2022). In this search we added one by one
decorrelation vector to our model and kept the vector only if it yielded a
higher Bayes’ factor. The optimal sets of detrending vectors obtained in this
way contain background and up to second order polynomials in PSF centroid
position. In our analysis, we use a linear model to detrend against these
vectors.

Some of the visits also show the well understood “ramp” effect in the dataset
(Morris et al., 2021). The ramp effect happens because of the change in the
shape of PSF related to the thermal effects in the telescope tube that arises
mainly when re-pointing the satellite. This has been seen in other datasets
and PSF-based methods have been seen to remove such trends (Wilson et al.,
2022). Since this effect is directly linked to the shape of the PSF, we can
correct for it using the principal components of the PSF model extracted from
PIPE. We added these components in our linear model when required.

In addition to these linear models, we also added a GP model (produced using
Exponential-Matérn kernel from celerite implemented in juliet) to account for
temporal astrophysical/systematic trends, meaning that our final model
includes linear and GP models for a decorrelation and a transit model (batman;
Kreidberg, 2015). As previously, we first analysed each visit independently
and then used the derived hyperparameters as priors in our joint photometry
analysis.

####  2.2.2 Analysis of the TESS photometry

We analysed data from the four TESS sectors. The main source of noise in TESS
data is found to be systematic and astrophysical trends. To account for these
trends we introduce in our fitting procedure a GP model based on an
Exponential-Matérn kernel. Furthermore, our global model includes a four-
planet transit model (batman; Kreidberg, 2015), a jitter term and an out-of-
transit flux offset. Since the PDC-SAP flux is expected to correct for the
dilution from the nearby light sources, we set the dilution factor (see
Espinoza et al., 2019, for details) to 1 in our model, meaning that no
dilution was assumed from nearby sources.

Similar to our CHEOPS analysis, we analysed individual TESS sectors before
performing the joint analysis. The priors on the noise model hyperparameters
in our joint analysis were selected based on the posteriors from the analysis
of individual sectors.

Table 2: Planetary and stellar parameters used in the transit analysis

Parameters | Symbols | Values | Priors | Units
---|---|---|---|---
Planetary parameters (Planet b) |  |  |  |
Orbital period | $P$ | $0.4465689683^{+0.0000002381}_{-0.0000003152}$ | $\mathcal{N}(0.4465688,7.5e-7)$ | days
Transit time | $T_{0}$ | $2459578.546253^{+0.000219}_{-0.000205}$ | $\mathcal{N}(2459578.545979,0.000499)$ | BJD
Planet-to-star radius ratio | $R_{p}/R_{\star}$ | $0.015439^{+0.000218}_{-0.000226}$ | $\mathcal{U}(0,1)$ | -
Impact parameter | $b$ | $0.088469^{+0.050057}_{-0.047097}$ | $\mathcal{U}(0,1)$ | -
Scaled semi-major axis | $a/R_{\star}$ | $2.692202^{+0.017130}_{-0.021185}$ | - | -
Eccentricity | $e_{b}$ | - | Fixed to 0 | -
Argument of periastron passage | $\omega_{b}$ | - | Fixed to 90 | deg
Stellar Parameters |  |  |  |
Stellar density | $\rho_{\star}$ | $1850.964990^{+35.557225}_{-43.352182}$ | $\mathcal{N}(1896.89,91.17)$ | kg/m3
Limb darkening coefficients |  |  |  |
CHEOPS passband | $u_{1_{CHEOPS}}$ | $0.291191^{+0.150360}_{-0.149110}$ | - | -
$u_{2_{CHEOPS}}$ | $0.397271^{+0.242151}_{-0.238942}$ | - | -
TESS passband | $u_{1_{TESS}}$ | $0.357040^{+0.146761}_{-0.170364}$ | - | -
$u_{2_{TESS}}$ | $0.226039^{+0.259467}_{-0.232569}$ | - | -
Derived planetary Parameters††footnotemark: † |  |  |  |
Radius | $R_{p}$ | $1.4195^{+0.0217}_{-0.0224}$ | - | $R_{\oplus}$
Density | $\rho_{p}$ | $4.3049^{+0.4411}_{-0.4216}$ | - | g/cm3
$\rho_{p}$ | $0.7834^{+0.0803}_{-0.0767}$ | - | $\rho_{\oplus}$
Orbital distance | $a_{b}$ | $0.0106^{+0.0001}_{-0.0001}$ | - | AU
Inclination | $i_{b}$ | $88.1178^{+1.0045}_{-1.0820}$ | - | deg

333The Gaussian priors with mean $\mu$ and variance $\sigma^{2}$ are displayed
as $\mathcal{N}(\mu,\sigma^{2})$. $\mathcal{U}(a,b)$ shows the uniform prior
between $a$ and $b$.
$$\dagger$$$$\dagger$$footnotetext: In the calculation of the derived
planetary parameters we used following stellar and planetary parameters from
Lacedelli et al. (2022) and Brinkman et al. (2023): $R_{\star}=0.843\pm
0.005\,R_{\odot}$ and $M_{p}=2.24\pm 0.20\,M_{\oplus}$.

####  2.2.3 Joint photometry analysis

Our final analysis consists in a joint fit to the TESS and CHEOPS datasets, in
order to refine the planetary parameters for TOI-561 b as much as possible. As
previously, our global model contains, in addition to a four-planets transit
model, a jitter term, a mean out-of-transit offset and linear and GP models to
account for various systematic and astrophysical correlations in the data. We
used our earlier analysis of individual CHEOPS visits and TESS sectors to set
informative priors on the nuisance parameters in the joint CHEOPS-TESS
analysis. Adapting this two-step method not only helps in determining and
constraining the noise model, but also allows for a better and faster sampling
of the parameter space. The second point is crucial when the total number of
free parameters becomes large, which is the case in the present analysis.

Indeed, the four-planets transit model and different noise models for each
visit/sector yields a large number of free parameters in the analysis, for a
grand total of 168. Among these, 143 are nuisance parameters accounting for
various decorrelations in the datasets. The other parameters are either
planetary or stellar. We decided to use nested sampling methods (Skilling,
2004, 2006) to sample the distribution from the posterior. Since we are
dealing with a high dimensional parameter space, we follow the recommendations
from Espinoza et al. (2019) and use dynamic nested sampling (Higson et al.,
2019) as included in juliet via dynesty (Speagle, 2020).

The fitted median transit model for planet b, along with detrended data and
residuals from the fit, are shown in Figure 1 for CHEOPS and TESS. It can be
seen from the residuals that all of the instrumental and astrophysical trends
were effectively removed, demonstrating the quality of our fitting procedure.
The corresponding median posteriors and their 1-$\sigma$ credible intervals
for various planetary parameters are tabulated in Table 3. Raw and detrended
data, along with the best-fitted models, for individual CHEOPS visits are
displayed in Figure 12 in the Appendix B. The correlation plot, made using
corner.py (Foreman-Mackey, 2016), for fitted planetary parameters is presented
in Figure 13 in the Appendix C.

Figure 1: Phase-folded lightcurve for planet b, over all of the (a) CHEOPS
visits and (b) TESS sectors. Top sub-panels: the dark blue curve represents
the median fitted model, and the orange curves are models computed from
randomly drawn samples from the posterior. The light blue and dark blue points
are the original points and the binned data points. Bottoms sub-panels
residuals from the median model.

In Figure 2 we show an inter-comparison of $P$, $a/R_{\star}$, $b$ and
$R_{p}/R_{\star}$ between our various analysis, and with their literature
values. It can readily be observed that our CHEOPS and TESS analysis agree
with each other and with the joint analysis, and that they are all in
excellent agreement with the values reported from Lacedelli et al. (2022).
Thanks to the additional CHEOPS and TESS data we are able to put stronger
constraints on the planetary parameters, in particular the planet-to-star
radius ratio with a precision improved to $\sim 2\%$.

Figure 2: Comparison of retrieved planetary parameters ($P$, $a/R_{\star}$,
$b$ and $R_{p}/R_{\star}$) from our analyses of CHEOPS (blue points), TESS
(red points) and joint dataset (green points) with the literature values from
Lacedelli et al. (2022). The latter is represented by the dashed purple line
and $1-\sigma$ uncertainty band. The horizontal axis gives the arbitrary
transit time. Note that the scale on the vertical axis is relative and
centered on the literature value of the given parameter.

###  2.3 Eclipse analysis

Figure 3: Top panel: The upper plot shows the median fitted model (in dark
blue), which includes the planet transit and eclipse, and the original (in
light blue) and binned datapoints (dark blue) from four TESS sectors. The
orange curves are the models computed from the randomly chosen samples from
the posteriors. The lower plot shows the residuals for the fit. Bottom panel:
Posterior distribution of the eclipse depth. The dark orange line and the
light orange band gives the median eclipse depth and the $1-\sigma$ credible
interval. The dashed line represents the null hypothesis.

TOI-561 b orbits its host star at a short orbital distance of $0.0105$ AU
(see, Table 3). We expect the dayside temperature444Calculated using,
$T_{\textnormal{day}}=T_{\star}\sqrt{\frac{R_{\star}}{a}}(1-A_{B})^{1/4}f^{1/4}$,
while using the bond albedo $A_{B}=0$ and the heat re-distribution factor
$f=2/3$, for a bare rock with no heat re-distribution and $f=1/4$ for uniform
heat distribution (Burrows, 2014; Koll et al., 2019). of this planet to be
around $T_{\textnormal{day}}\sim 2963.51$ K for instant re-radiation of the
heat to the space (i.e., no heat re-distribution) and
$T_{\textnormal{day}}\sim 2319.07$ K for a uniform heat distribution
(effective temperature of the host star $T_{\star}=5372$ K; Lacedelli et al.,
2022). Based on these estimates we foresee the planet to produce strong
thermal emission, even in the optical CHEOPS passband. In the light of
previous CHEOPS measurements of planetary occultations (e.g. Lendl et al.,
2020; Hooton et al., 2022), we thus search for a secondary eclipse signal in
our dataset.

As our CHEOPS observations are focused on the transit event, most of the
exposures do not cover the secondary eclipse of the planet. Fortunately
though, two of the archival visits (Visit No. 39301 and 101, see, Table 1)
cover orbital phases during which occultation occurs five times. However we
found that these five occurences are not enough to recover any significant
signal from the CHEOPS data. Nonetheless, we derive a 99.7 percentile limit of
98.93 ppm on eclipse depth for this dataset.

On the other hand, TESS observed the event more than 150 times during its four
observation sectors. Furthermore, we anticipate a larger occultation signal in
the TESS data considering that the thermal contribution from TOI-561 b would
be larger in the TESS bandpass than in the CHEOPS bandpass. With our upper
estimation for the planet temperature $T_{\textnormal{day}}\sim 2963.51$ K we
expect the magnitude of the occultation to be at least $14$ ppm in the TESS
bandpass assuming blackbody emission. To test this hypothesis, we modelled the
TESS data with a joint transit-eclipse model and an only-transit model using
batman models implemented in juliet. We masked out transit signals from all
other planets to simplify the analysis. We use the same method as described in
Section 2.2 to model the dataset, but this time we provide informative priors
on all planetary parameters except the eclipse depth. We set a uniform prior
between $-100$ ppm to $100$ ppm for the eclipse depth. We compare the Bayesian
evidence for a model comparison between the model with and without eclipse.

We find that the model with eclipse is strongly favoured statistically
($\Delta\ln{Z}\sim 1.43$). The phase-folded light curve, including the median
model with randomly selected models, is shown in Figure 3. We are only able to
put an upper limit on the eclipse depth (a 99.7 percentile limit of $60.15$
ppm) because the value we derive from its posterior distribution (Figure 3) is
consistent with zero at 3-$\sigma$: $L=27.40^{+10.87}_{-11.35}$ ppm. The
derived posterior can still put some constraints on the planetary atmosphere.

Figure 4: Relation between the geometric albedo and the brightness temperature
of the planet, given the posteriors of the occultation depth (with mean and
$1-\sigma$ intervals $27.40^{+10.87}_{-11.35}$ ppm) in the TESS bandpass.

It should be noted here that both reflection and thermal emission could
contribute to the occultation signal we are detecting in the TESS data. With
TESS data alone we cannot determine the fractional contribution to the eclipse
depth from these components, so that we would need similar observations at
longer wavelengths. With the measured posterior of eclipse depth from the TESS
data we can, however, estimate the geometric albedo ($A_{g}$) over a range of
brightness temperatures (a measure of the thermal emission) for this planet.
We followed the method from Mallonn et al. (2019) to find the relation between
the geometric albedo and the temperature, which is plotted in Figure 4. As
expected, the median solution (depicted by an orange line) is degenerate. The
posterior of eclipse depth (as shown in Figure 3, with
$L=27.40^{+10.87}_{-11.35}$ ppm), assuming a zero contribution from
reflection, would result in a dayside temperature of $\sim 3325$ K. On the
other hand, to explain the extracted eclipse depth with reflection alone, the
geometric albedo of the planet should be around $0.83$. Note that both of
these values are the extreme cases and the true solution should lie somewhere
in between.

##  3 Internal structure modelling

Table 3: Posteriors of the internal structure parameters

Internal structure parameter | Model with water layer | Model without water layer
---|---|---
Mcore/Mtotal | $0.09^{+0.10}_{-0.08}$ | $0.12^{+0.11}_{-0.11}$
Mwater/Mtotal | $0.22^{+0.16}_{-0.15}$ | -
log Mgas [M$\oplus$] | $-9.48^{+2.20}_{-2.27}$ | $-7.06^{+0.24}_{-0.28}$
555Posteriors parameters for models with and without an included fully
distinct water layer. The errors are the 5 and 95 percentile of the
corresponding posterior distributions.

Table 4: Posteriors of the internal structure parameters for models with fixed
(to stellar values) and varying Si/MG/Fe ratios in the models

Internal structure parameter | Model with fixed Si/Mg/Fe ratios | Model with free Si/Mg/Fe ratios
---|---|---
Mcore/Mtotal | $0.09^{+0.10}_{-0.08}$ | $0.06^{+0.14}_{-0.05}$
Mwater/Mtotal | $0.22^{+0.16}_{-0.15}$ | $0.27^{+0.18}_{-0.20}$
log Mgas [M$\oplus$] | $-9.48^{+2.20}_{-2.27}$ | $-9.40^{+2.23}_{-2.34}$
Fecore | $0.90^{+0.09}_{-0.08}$ | $0.90^{+0.09}_{-0.08}$
Simantle | $0.41^{+0.07}_{-0.06}$ | $0.42^{+0.08}_{-0.07}$
Mgmantle | $0.48^{+0.10}_{-0.10}$ | $0.49^{+0.12}_{-0.23}$

666Posteriors for two models, the first model assumes the Si/Mg/Fe ratios of
the planet match the ones measured for the star exactly and the other model
where a wide variety of Si/Mg/Fe ratios are allowed. The errors are the 5 and
95 percentile of the corresponding posterior distributions.

In the following, we discuss the internal structure of TOI-561 b. The method
used is described in more depth in Leleu et al. (2021) and is based on Dorn et
al. (2017), but we will briefly summarise the most important aspects below.
The Bayesian inference scheme we applied takes as input parameters the stellar
(mass, radius, effective temperature, age and [Si/H], [Mg/H] and [Fe/H]) and
planetary observables (mass relative to the star, transit depth and period) of
the system. The likelihood of a given structure is calculated based on an
internal structure model. We assume that the planet is spherically symmetric
and consists of four fully distinct layers: an inner iron core (Hakim et al.,
2018), a silicate mantle (Sotin et al., 2007), a water layer (Haldemann et
al., 2020) and a pure H/He atmosphere (Lopez & Fortney, 2014). Furthermore, we
assume that the Si/Mg/Fe ratios of the planet match the ones of the star
exactly. Although this is supported e.g. by Thiabaud et al. (2015), recent
work by Adibekyan et al. (2021) suggests that the correlation might not be
1:1. Implementing this possibility in the model is the subject of future work.

For the priors of the internal structure parameters, we chose a prior that is
uniform in log for the gas mass. For the mass fractions of the inner core,
mantle and water layers with respect to the solid planet, our chosen prior is
uniform with the added condition that they add up to one and with an upper
limit of the water mass fraction of 50% (Thiabaud et al., 2014; Marboeuf et
al., 2014). We stress that the results of the internal structure modelling do
depend to a certain extent on the chosen priors.

We ran two different versions of our model for TOI-561 b. Given its high
equilibrium temperature, any water layer would have evaporated and formed a
thick water vapour atmosphere. However, with the current version of our model,
it is not possible to include such a critical steam layer. We therefore
choose, on the one hand, to run a dry model of the planet without an added
water layer. On the other hand, we also run the full version of the model with
an included water layer, as this is the most general way to model the planet
that our model allows and any additional assumptions would in the end just be
mirrored in the resulting posteriors. The resulting differences in the
posteriors of the internal structure parameters are summarised in Table 3.

Figure 5: Corner plot of the posteriors of the internal structure parameters
of TOI-561 b. Shown are the mass fractions with respect to the solid planet of
the inner core and water layer, the molar fractions of Si and Mg in the mantle
layer and Fe in the inner core and the gas mass in Earth masses in a
logarithmic scale. The titles on top of each column are the median and the 5
and 95 percentiles of each distribution, which are also shown by the dashed
lines.

For the dry model, the posterior of the gas mass is quite well constrained
with a median at around $10^{-7}$ M$\oplus$. However, such a planet is
unphysical, as any H/He envelope would get evaporated very fast. Figure 5
shows the resulting posterior distributions of the internal structure
parameters of TOI-561 b when running the full version of the model. For both
versions of the model, a planet that only consists of an iron core and a
silicate mantle seems to be unlikely given the observations. The best model
converges toward a water mass fraction that is constrained quite well, while
the gas mass fraction is negligibly small, as expected from the strong
irradiation of the planet. This finding and our core mass fraction is in
agreement with a recent independent study that found a water- and gas-devoid
planet can only reproduce the observed density with a composition much lower
than the stellar value and the inclusion of a high melt fraction (Brinkman et
al., 2023).

We point out that in our model, the inner layers of the planet are not
influenced by the pressure or temperature of the atmosphere, which does have
an influence on the modelled radius of the solid planet. To investigate the
influence a higher temperature could have on the radius of the iron core and
silicate mantle, we selected a sub-sample of points from our posterior
distribution and used the forward model to re-compute their radius when
setting the temperature at the boundary between the atmosphere and the solid
planet to the equilibrium temperature of TOI-561 b. This showed that raising
the temperature of the iron core and the silicate mantle only results in an
increase in radius of less than 3%, which is not enough to explain the
difference in radius between the observations and a planet consisting only of
an iron core and a silicate mantle.

Choosing different constraints for the Si/Mg/Fe ratios of the planet will also
influence the radius of the iron core and silicate mantle. As an extreme case,
we used our forward model to calculated the thickness of these two layers for
an iron-free planet, again with a temperature equal to the equilibrium
temperature of TOI-561 b at the outer boundary. This gives a radius very close
to the one derived for TOI-561 b. Since a planet that is completely iron-free
seems unlikely, we conclude that some heavier elements are in fact necessary.
Additionally, we also performed another Bayesian analysis where we lifted the
compositional constraints on the planet by allowing the code to sample from a
wide range of stellar abundances, i.e. [Si/H]=[Mg/H]=[Fe/H]=$0^{+1}_{-1}$. The
posteriors of the internal structure parameters resulting from this analysis
are summarised in Table 4. For the Si/Fe and Mg/Fe ratios of the planet, the
model gives posteriors of Si/Fe = $2.45^{+3.06}_{-1.75}$ and Mg/Fe =
$2.98^{+4.18}_{-2.51}$, while the corresponding ratios derived from the
stellar abundances are Si/Fe = $1.74^{+0.42}_{-0.39}$ and Mg/Fe = $2.02\pm
0.51$. Also with this version of the model, a pure iron-silicate structure
seems to be unlikely given the observations.

Figure 6: Transit radius as a function of the total envelope mass fraction and
the molar water fraction in the envelope for a planetary structure with a
fixed core mass and composition in agreement with the values for a dry model
in Table 3. The red lines show the radius values that are in agreement with
the measurements for TOI-561 b.

However, this analysis still assumes the water layer to be in a condensed
state and fully distinct from the H/He atmosphere. We therefore ran an
additional exploratory internal structure analysis using a new version of our
model that is fully self-consistent and features an H2O enriched envelope.
More specifically, we assumed the planet to have an inner iron core and a
silicate mantle with a fixed combined mass of $2.24\pm 0.20$ M⊕ and a
composition corresponding to the median of the posterior distribution obtained
using the dry model without water layer described above. This assumption is
justified since the gas mass we obtain when running a dry model is very small
(see Table 3). Note that this new model version uses the atmosphere model of
Parmentier & Guillot (2014) instead of the one of Lopez & Fortney (2014) used
in the remaining analysis. Then, we ran a grid of planetary forward models
consisting of this fixed core and gas envelopes with different masses (going
from mass fraction of $10^{-9}$ up to $10^{-6}$) and different envelope
enrichments (with molar water fractions between 0 and 1). The results of this
study can be seen in Fig. 6. For each grid point, the transit radius of the
resulting structure is shown (in color), with the red line depicting the
measured radius of TOI-561 b. Any combination of envelope mass and enrichment
$Z$ that lies within the two other red lines is consistent with the
measurements.

##  4 Prospects of an atmosphere

###  4.1 Interpreting the eclipse signal

Figure 7: Theoretical models of outgassed atmosphere of TOI-561 b. (a)
Temperature-Pressure profile of the planet for a Bulk-Silicate-Earth (BSE)
composition (in orange) and an evolved BSE composition (in blue). (b)
Theoretical emission spectra for BSE (orange curve) and evolved BSE (blue
curve) composition. The TESS observation of the eclipse depth is shown as a
green star, where errorbars on the wavelength axis show the extent of the TESS
bandpass. Other data points are simulated observations for NIRSpec (circles),
MIRI (squares) and CHEOPS (triangles) for both models (represented by the
colors). The white-light eclipse depths are shown for NIRSpec and MIRI with
errors showing their wavelength coverage. Two inset plots display the zoom-in
version of the main plot for NIRSpec and MIRI wavelength ranges. The
spectroscopic simulated eclipse depths (at $R=7$) are also shown for both
instruments in these plots.

The absence of thick ($>0.1$ bar) atmospheres on the inner two TRAPPIST-1
planets as observed by JWST (Zieba et al., 2023; Greene et al., 2023) and even
on a temperate rocky world like LHS 3844 b as evidenced by Spitzer (Kreidberg
et al., 2019) appears at a first glance to make the presence of an atmosphere
on a USP like TOI-561 b even less likely.

USPs, however, have such large surface temperatures ($>2000$ K) that their
surfaces are expected to be at least partially molten giving rise to tenuous,
evaporating atmospheres of more exotic compositions than on rocky planets in
the Solar System. A tenuous rock vapour atmosphere was e.g. suggested on
K2-141 b based on Kepler and Spitzer observations (Zieba et al., 2022). Thus,
constraining such evaporating atmospheres on USPs raised the tantalizing
prospect to assess surface properties on rocky exoplanets more directly by
observing spectral features of outgassed molecules (see e.g. Chao et al.,
2021)

Recently, Zilinskas et al. (2022) comprised an extensive catalogue of
outgassed atmospheres for various USPs, including TOI-561 b, under different
assumptions for surface composition and outgassing efficiency. These authors
provided pressure temperature profiles for these different atmospheres that
were calculated self-consistently, assuming irradiation at the substellar
point without any heat distribution and assuming an albedo of 0. In the case
of TOI-561 b, the model yields about 0.4 bar for 60% outgassing efficiency.

We use two planetary compositions, bulk silicate (oxidized) Earth (BSE) and
evolved BSE (with 60% outgassing efficiency), to estimate the eclipse depth in
the TESS bandpass to explain the measured eclipse signal described in Section
2.3777Models are publicly available at https://github.com/zmantas/LavaPlanets.
These two models yield SiO as the main volatile in the outgassed atmosphere.
The predicted eclipse depths for these models are 27.81 ppm and 26.59 ppm,
which is very close to the observed eclipse depth of 27.40 ppm. This points to
the fact that the emission coming from the planet in the TESS bandpass is the
result of thermal emission only. However, we note here that models assume zero
albedo and no heat re-distribution resulting in a hotter temperature profile
(see, e.g., Figure 7 (a)) which was already noted by Zilinskas et al. (2022)
as a limitation of their methods. It has been pointed out, however, that lava
surfaces should have very low albedos (Essack et al., 2020). Further, K2-141
b, the only USP so far with hints of an outgassed rock vapour atmospheres,
exhibits a phase-curve consistent with a scenario without any heat re-
distribution.

Keeping in mind these caveats, we make predictions for potential CHEOPS and
upcoming JWST observations of this planet. The BSE and evolved BSE models
predict the eclipse depth of 18.74 ppm and 17.18 ppm in the CHEOPS bandpass
(orange and blue triangles in Figure 7 (b)). A larger observed depth in the
CHEOPS bandpass could be a sign of a significant reflective component.
However, it would take an impractically large number of observations (62 and
73 for two models) to constrain the above-mentioned eclipse depths at
3$\sigma$ as the target is faint with at $G=10.0$ mag888We used CHEOPS
Exposure Time Calculator (https://cheops.unige.ch/pht2/exposure-time-
calculator/) to estimate this.. Furthermore, it is questionable if lava worlds
exhibit large albedos, at least if an oxidized silicate surface like in the
BSE scenario is assumed (Essack et al., 2020). It would take ultramafic
surfaces composed of olivine and enstatite to assume larger albedos (¿0.2) (Hu
et al., 2012).

On the other hand, the James Webb Space Telescope (JWST) would be ideal to
observe the thermal emission from the planetary surface or its atmosphere.
Indeed, a Cycle 2 GO program (GO 3860, PI: J. Teske) aims to do the same by
observing phase curves of the planet with NIRSpec/G395H. Here we use Zilinskas
et al. (2022) models with PandExo (Batalha et al., 2017) to simulate
observations for NIRSPec/G395H and MIRI/LRS. NIRSpec/G395H instrument, which
has a wavelength range of $2.87-5.18\,\mu$m, covers a SiO feature between
$4\,\mu$m and $5\ \mu$m. With 4 occultations of the planet (as asked for in GO
3860) we can expect to detect this feature as a rise in eclipse depth after
$\sim 4\,\mu$m as shown in upper-left inset panel of Figure 7 (b) (orange and
blue circles). Unfortunately, both BSE and evolved BSE models are very similar
in this range. Therefore the NIRSpec/G395H observations cannot distinguish
between both models. There is an additional SiO feature near $\sim 9\,\mu$m
covered by MIRI/LRS wavelength range, $5.02-13.86\,\mu$m. This feature is
interesting because the amplitude (along with the baseline) of the feature
changes depending on the outgassing efficiency. As shown in Figure 7 (b), the
evolved BSE composition (in blue) has a lower amplitude and high baseline of
the feature compared to that of the BSE composition (in orange). Additionally,
there is a small SiO2 feature near $\sim 7\,\mu$m which is only present in
spectrum of BSE composition (see the lower-right inset panel in Figure 7 (b)).
Both of these features can be helpful in distinguishing the two models. We,
however, note that with the current precision of the MIRI/LRS instrument, it
would be quite challenging to reach the required level of noise floor to
differentiate two models at higher statistical confidence. We used 10 eclipse
observations for our simulation here.

The white-light eclipse depth could be used to estimate the temperature of the
planet and solve the degeneracy between thermal and reflective components
(also shown in Figure 4). As already mentioned, models from Zilinskas et al.
(2022) suggest that the emission in the TESS bandpass is only because of
thermal radiation. If this is the case, then the white-light eclipse depth
should be around 121.9 and 123.7 ppm for NIRSpec/G395H and 154.7 and 154.8 ppm
for MIRI/LRS, respectively for BSE and evolved BSE compositions (see, Figure 7
(b)). A lower observed white-light eclipse depths in these bands would be a
hint of lower dayside temperature and a non-zero bond albedo.

###  4.2 Search for variability

Figure 8: Difference in ppm between the $R_{p}/R_{\star}$ measured in
individual (for CHEOPS data) or group of five (for TESS data) transits and the
value of $R_{p}/R_{\star}$ from the joint CHEOPS-TESS analysis. The orange and
blue points are the results from TESS and CHEOPS analysis, respectively.

As mentioned in Section 2.3 the dayside temperature of the planet can reach as
high as $\sim 3000$ K, resulting in the surface of the planet being partially
or completely molten. At this point, the planet can support a secondary
outgassed atmosphere made up of heavier species such as Na, O2, SiO etc.
(Schaefer & Fegley, 2009; Miguel et al., 2011; Schaefer et al., 2012;
Wordsworth & Kreidberg, 2022). Driven by winds, the species could travel to
the nightside and condensate, or even escape into space (Kite et al., 2016;
Wordsworth & Kreidberg, 2022). Both phenomena, influenced by the surface
activity of the planet and the stellar irradiation, could lead to temporal
variability in the atmospheric properties, which could potentially be
imprinted on the $R_{p}/R_{\star}$ measured over several transit events.

Motivated by this rationale, we re-analysed all of the TESS and CHEOPS data,
but now fixed all parameters except $R_{p}/R_{\star}$ to the values derived
from our joint photometric analysis (Table 3). While for the CHEOPS data we
model each transit individually, for TESS we group $\sim 5$ transit events
together to achieve a better signal-to-noise ratio. The result of this
analysis is presented in Figure 8. It shows the difference in ppm between each
measured $R_{p}/R_{\star}$ and the $R_{p}/R_{\star}$ from our joint analysis.
As can be seen from Figure 8, we find no significant variations of
$R_{p}/R_{\star}$ over time. Although this does not rule out the possibility
of the atmosphere being variable, our current dataset does not have the
required precision.

##  5 Discussion and conclusions

Figure 9: Mass-radius diagram of exoplanets with $M_{p}\leq 10\,M_{\oplus}$
and $R_{p}\leq 4\,R_{\oplus}$ known with a precision better than 30%. Gray
points are all known exoplanets as of June 19, 2023 taken from the NASA
Exoplanet Archive (2023). The black dashed line is the iso-density curve for
the value we derived for TOI-561 b. Mass of TOI-561 b is taken from Brinkman
et al. (2023) and radius from the present work. Also shown are theoretical
models for various chemical compositions as computed by Zeng et al. (2019).
Mass and radius of 55 Cnc e, K2-141 b and CoRoT-7 b are taken from Bourrier et
al. (2018), Malavolta et al. (2018) and John et al. (2022).

USPs are a singular class of planets orbiting their host star within a day.
This property makes them appealing targets to study various compelling
phenomena caused by the extreme proximity and radiation from the star, ranging
from their internal structure to their atmospheres. Indeed, well-known USPs
like 55 Cnc e, K2-141 b, or LHS 3844 b display a diverse range of composition
from a thick, possibly variable atmosphere to no atmosphere at all (Demory et
al., 2016; Kreidberg et al., 2019; Zieba et al., 2022). Among the USP
population, the newly discovered TOI-561 b stands apart by its unusually low
density of $4.3049^{+0.4411}_{-0.4216}$ g/cm3 when other USPs show
comparatively higher densities (see the position of TOI-561 b in the Mass-
Radius diagram in Figure 9).

Figure 10: Population of USPs in (a) planetary surface gravity - equilibrium
temperature and (b) host star metallicity - surface gravity spaces. In the top
panel, the data points are color-coded with the metallicity of the host star.
The USP population is defined as all confirmed planets with an orbital period
of less than a day and $M_{p}\leq 10\ M_{\oplus}$. The data of the systems is
taken from NASA Exoplanet Archive (2023) on June 19, 2023. Additionally, the
bottom planets only include planets hotter than 1700 K (see, text).

To further place TOI-561 b in the context of known USPs and understand how the
metal-poor nature of the host star may have affected its formation and
evolution we retrieve planet properties for all well-characterised bodies with
an orbital period of less than a day and $M_{p}\leq 10\ M_{\oplus}$ from NASA
Exoplanet Archive (2023). To test how stellar composition may impact USPs, we
first make the assumption that these bodies have entirely lost their
primordial Hydrogen atmosphere and as such, we are only observing the
refractory components of the planets. To assess this assumption, we computed
the minimum mean molecular weight (MMMW) of a potential atmosphere of our USP
sample using a formulation of the Jeans escape criteria (Jeans, 1925) that
includes Roche lobe effects (Erkaev et al., 2007; Fossati et al., 2017) in a
similar manner to Wilson et al. (2022). Given their current physical and
orbital parameters, we find that all USPs with the exception of TOI-1075 b
(Essack et al., 2023) cannot have held onto a primordial envelope and thus our
assumption is valid. We recognise that other processes may have induced
further atmospheric mass loss, and whilst the inclusion of this additional
modelling is beyond the scope of the paper, any additional atmospheric escape
would strengthen our assumption that the USPs are primarily rocky.
Furthermore, we find that the minimum mean molecular weight of an atmosphere
that TOI-561 b can retain is 11 amu, providing evidence that the planet has
likely lost its primordial atmosphere, but may have a heavier, secondary
envelope.

Assured that the planets in our USP sample do not have primordial atmospheres,
we assess the physical properties of these well-characterised bodies to probe
underlying processes. In the upper panel of Figure 10, we plot the planetary
surface gravity of rocky USPs as determined by our MMMW analysis above against
their equilibrium temperature that reveals a clear trend. To substantiate the
identified correlation, we used a Bayesian correlation tool (Figueira et al.,
2016) and find the median and standard deviation of the correlation metric
distribution (similar to Spearman’s rank value) to be 0.45$\pm$0.16. This
strong trend is interpreted as lower surface gravity planets have lower
gravitational potential wells that allow for the escape of envelopes at lower
temperatures and hence appearing as the rocky USPs identified by our MMMW
analysis. Adibekyan et al. (2021) found evidence for stellar composition
influencing planetary internal structure for terrestrial bodies. If this
finding is universal, one would expect planets around metal-poor stars to have
smaller iron cores and thus lower surface gravities for bodies of a similar
radius. Interestingly, if we remove metal-poor planets, [Fe/H] $<$ -0.1,
(TOI-561 b, Kepler-10 b, Kepler-78 b, and TOI-1685 b) from our USP sample, the
correlation between surface gravity and equilibrium temperature increases to
0.65$\pm$0.13. This is because metal-poor USP planets have lower surface
gravities at similar equilibrium temperatures. This supports previous findings
that host star metallicity alters planet interior structure (Adibekyan et al.,
2021; Wilson et al., 2022). This is highlighted in the lower panel of Figure
10 by removing the equilibrium temperature dependence by selecting USPs hotter
than 1700 K and finding a strong trend between surface gravity and host-star
metallicity.

Figure 10 also clearly shows the uniqueness of TOI-561 b in temperature-
metallicity-surface gravity space. All these motivated our follow-up of this
planet with the state-of-the-art photometric telescope CHEOPS to measure
precisely its radius, and therefore constrain its composition and structure.
We acquired 13 new transit observations of the planet with CHEOPS, and
combined them with three archival visits from CHEOPS and four TESS sectors.
This joint dataset allowed us to set strong constraints on the planet-to-star
radius ratio, reducing its uncertainty to $\sim 2\%$. We used our updated
parameters for TOI-561 b to model its internal structure. We find that the
structure of planet is consistent with negligible primary volatile atmosphere
made up of H/He which is expected from the atmospheric escape from such a
highly irradiated planet. Additionally, a planet built only from an iron core
and silicate mantle is not enough to explain the observed density, supporting
the presence for lighter material in the planetary structure. This could be a
thin layer of high mean molecular weight (e.g. a water layer, as shown by our
internal structure modelling), or a larger silicate mantle than what is
predicted using a one-to-one scaling between the stellar and planet
composition, or a combination of both. We also ran a fully self-consistent
exploratory forward model with water enriched envelope and demonstrate that
such a model can indeed explain the transit radius. We show that a range of
water fraction in the atmosphere is possible for a varying atmospheric mass
fraction. Given the high equilibrium temperature of the planet, a plausible
scenario is that it sustains a secondary atmosphere containing heavy metal
species. We searched for variations in $R_{p}/R_{\star}$ over time, which
could trace the variability expected from such an envelope, but found no such
evidence.

In addition to the transit observations, we also find a weak detection of the
secondary eclipse in the TESS data, with an eclipse depth
$L=27.40^{+10.87}_{-11.33}$ ppm. Since this emission signal would be
contaminated from the reflective and thermal radiation from the planet we
cannot uniquely determine its geometric albedo and dayside temperature. The
measured occultation depth would correspond to a dayside temperature of $\sim
3325$ K if the signal is assumed to originate entirely from thermal radiation;
on the contrary, a hundred per cent contribution from reflection would imply a
geometric albedo of $\sim 0.83$. Given that the planet could possibly have a
secondary metal-rich atmosphere, we use models of outgassed atmospheres from
Zilinskas et al. (2022) to explain this eclipse signal. The expected eclipse
depths from the two model compositions, bulk silicate (oxidized) Earth (BSE)
and evolved BSE composition with 60% outgassing efficiency, are 27.81 ppm and
26.59 ppm, which is very close to the observed value. This implies that the
emission in TESS bandpass is mainly of thermal origin.

The composition of the atmosphere is essential in constraining the surface
properties, given the possible atmosphere-interior interactions. Our
photometric observations cannot constrain the presence of a specific species
in the atmosphere of the planet. While ground-based spectroscopy could be used
to search for light species escaping the planet, such as sodium or oxygen,
space-based observations in the infrared could allow us to detect a mineral
atmosphere. The recently launched James Webb Space Telescope (JWST) would be
the ideal facility to search for dust and metals in the atmosphere of TOI-561
b, which is one of the main goals of the recently accepted Cycle 2 GO program
(GO 3860, PI: J. Teske). Using the models from Zilinskas et al. (2022) we
anticipate that these observations, which will use NIRSpec/G395H instrument,
should be able to detect SiO in the atmosphere. Observations in mid-infrared
using MIRI/LRS (mode not used by GO 3860) could, in principle, not only detect
the mineral species in the atmosphere but also infer the surface evolution.
Infrared observations have the other advantage of breaking the degeneracy
between reflective and thermal contributions in the observed emission. Hence,
with the help of observations at longer wavelengths, we can not only identify
the composition of the atmosphere, but also derive the temperature of the
planet.

###### Acknowledgements.

We would like to thank an anonymous referee for their detailed referee report
and suggestions which significantly improved the manuscript. 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. This research has made use of the NASA Exoplanet
Archive, which is operated by the California Institute of Technology, under
contract with the National Aeronautics and Space Administration under the
Exoplanet Exploration Program. JAP would like to thank Yamila Miguel for an
insightful discussion on the atmosphere of lava planets and for kindly
providing theoretical models for our planet. JAP and ABr were supported by the
SNSA. JAE and YA acknowledge the support of the Swiss National Fund under
grant 200020_172746. This project has received funding from the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme (project Spice Dune, grant agreement No 947634). ACC and
TGW acknowledge support from STFC consolidated grant numbers ST/R000824/1 and
ST/V000861/1, and UKSA grant number ST/R003203/1. LCa acknowledges financial
support from the Österreichische Akademie der Wissenschaften; LCa acknowledges
support from the European Union H2020-MSCA-ITN-2019 under Grant Agreement no.
860470 (CHAMELEON). S.G.S. acknowledge support from FCT through FCT contract
nr. CEECIND/00826/2018 and POPH/FSE (EC). 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). D.Eh. and A.De. acknowledge financial support from the SNSF
for project 200021_200726. KWFL acknowledge support by DFG grants RA714/14-1
within the DFG Schwerpunkt SPP 1992, “Exploring the Diversity of Extrasolar
Planets”. ML acknowledges support of the Swiss National Science Foundation
under grant number PCEFP2_194576. We acknowledge support from the Spanish
Ministry of Science and Innovation and the European Regional Development Fund
through grants ESP2016-80435-C2-1-R, ESP2016-80435-C2-2-R,
PGC2018-098153-B-C33, PGC2018-098153-B-C31, ESP2017-87676-C5-1-R,
MDM-2017-0737 Unidad de Excelencia Maria de Maeztu-Centro de Astrobiología
(INTA-CSIC), as well as the support of the Generalitat de Catalunya/CERCA
programme. The MOC activities have been supported by the ESA contract No.
4000124370. S.C.C.B. acknowledges support from FCT through FCT contracts nr.
IF/01312/2014/CP1215/CT0004. XB, SC, DG, MF and JL acknowledge their role as
ESA-appointed CHEOPS science team members. This project was supported by the
CNES. The Belgian participation to CHEOPS has been supported by the Belgian
Federal Science Policy Office (BELSPO) in the framework of the PRODEX Program,
and by the University of Liège through an ARC grant for Concerted Research
Actions financed by the Wallonia-Brussels Federation. L.D. is an F.R.S.-FNRS
Postdoctoral Researcher. This work was supported by FCT - Fundação para a
Ciência e a Tecnologia through national funds and by FEDER through COMPETE2020
- Programa Operacional Competitividade e Internacionalizacão by these grants:
UID/FIS/04434/2019, UIDB/04434/2020, UIDP/04434/2020, PTDC/FIS-AST/32113/2017
& POCI-01-0145-FEDER- 032113, PTDC/FIS-AST/28953/2017 &
POCI-01-0145-FEDER-028953, PTDC/FIS-AST/28987/2017 &
POCI-01-0145-FEDER-028987, O.D.S.D. is supported in the form of work contract
(DL 57/2016/CP1364/CT0004) funded by national funds through FCT. B.-O.D.
acknowledges support from the Swiss National Science Foundation
(PP00P2-190080). MF 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. VNa, IPa, GPi, RRa and
GSc acknowledge support from CHEOPS ASI-INAF agreement n. 2019-29-HH.0. 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
a PRODEX Experiment Agreement No. 4000137122, the Lendület LP2018-7/2021 grant
of the Hungarian Academy of Science and the support of the city of
Szombathely. V.V.G. is an F.R.S-FNRS Research Associate. NAW acknowledges UKSA
grant ST/R004838/1.

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##  Appendix A Photometric analysis without assuming prior knowledge on the
stellar density

As mentioned in Section 2.2, in our joint transit fit for the full dataset of
CHEOPS and TESS, we used informative priors on the stellar density. The
stellar density adopted in the procedure was computed from the stellar mass
and radius estimated from the stellar spectroscopic analysis performed in
Lacedelli et al. (2022). For completeness, and to test the validity of our
results, we perform another analysis. This time, we do not assume any prior
knowledge on the stellar density. Therefore, in this analysis, we put a wide
uninformative priors on $a/R_{\star}$ (scaled semi-major axis, uniformally
distributed priors in the log-space between 1 and 10) instead of using the
stellar density. We illustrate the result of this analysis in Figure 11 by
comparing them with the analysis performed in Section 2.2.3. Given the
precision and quantity of the data at hand, the planetary parameters agree
very well (with $3-\sigma$, even $1-\sigma$ for some of them) with our
analysis when we use priors on the stellar density and with their literature
counterparts from Lacedelli et al. (2022). However, as expected, the
uncertainty on the parameters are increased in this analysis owing to our lack
of knowledge of semi-major axis/stellar density.

Figure 11: Comparison of some of the retrieved planetary parameters (Period
$P$, scaled semi-major axis $a/R_{\star}$, impact parameter $b$ and the
planet-to-star radius ratio $R_{p}/R_{\star}$) between our two analysis: the
main analysis in which we used informative priors on the stellar density
(orange points) and the analysis in which we assumed no prior knowledge on the
stellar density (green points). The purple dashed line with the band show the
literature values of the parameters along with uncertainties from Lacedelli et
al. (2022). Note that the scale on the y-axis is the relative.

##  Appendix B Raw and detrended photometry from CHEOPS

Figure 12: Raw and detrended CHEOPS observations. (a): Raw (top panel) and
detrended photometry (bottom panel) for our new CHEOPS observations. The blue
points and orange points are the original and binned data points. The fitted
median full model (in top panel) and transit model (in bottom panel) is shown
with dark green lines. (b): Same as (a), but for archival CHEOPS visits. Note
that, for clarity, we masked the transit signals from other planets.

##  Appendix C Correlation plot of the fitted transit parameters

Figure 13: Correlation plot of the fitted transit parameters of planet b in
our joint photometric analysis.