11institutetext:  Observatoire astronomique de l’Université de Genève, Chemin
Pegasi 51, 1290 Versoix, Switzerland 22institutetext:  Weltraumforschung und
Planetologie, Physikalisches Institut, University of Bern,
Gesellschaftsstrasse 6, 3012 Bern, Switzerland 33institutetext:  Astrobiology
Research Unit, Université de Liège, Allée du 6 Août 19C, B-4000 Liège, Belgium
44institutetext:  Space sciences, Technologies and Astrophysics Research
(STAR) Institute, Université de Liège, Allée du 6 Août 19C, 4000 Liège,
Belgium 55institutetext:  Institute of Astronomy, KU Leuven, Celestijnenlaan
200D, 3001 Leuven, Belgium 66institutetext:  Mullard Space Science Laboratory,
University College London, Holmbury St Mary, Dorking, Surrey, RH5 6NT, UK
77institutetext:  Department of Astronomy, Stockholm University, AlbaNova
University Center, 10691 Stockholm, Sweden 88institutetext:  Center for Space
and Habitability, University of Bern, Gesellschaftsstrasse 6, 3012 Bern,
Switzerland 99institutetext:  Department of Physics and Kavli Institute for
Astrophysics and Space Research, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA 1010institutetext:  Department of Physics, University
of Warwick, Gibbet Hill Road, Coventry CV4 7AL, United Kingdom
1111institutetext:  Centre Vie dans l’Univers, Faculté des sciences,
Université de Genève, Quai Ernest-Ansermet 30, 1211 Genève 4, Switzerland
1212institutetext:  Observatoire Astronomique de l’Université de Genève,
Chemin Pegasi 51, CH-1290 Versoix, Switzerland 1313institutetext:  European
Space Agency (ESA), European Space Research and Technology Centre (ESTEC),
Keplerlaan 1, 2201 AZ Noordwijk, The Netherlands 1414institutetext:  Cavendish
Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK 1515institutetext:
Instituto de Astrofísica de Canarias, Vía Láctea s/n, 38200 La Laguna,
Tenerife, Spain 1616institutetext:  Departamento de Astrofísica, Universidad
de La Laguna, Astrofísico Francisco Sanchez s/n, 38206 La Laguna, Tenerife,
Spain 1717institutetext:  Departamento de Astronomía, Universidad de Chile,
Casilla 36-D, Santiago, Chile 1818institutetext:  Instituto de Astronomía,
Universidad Católica del Norte, Angamos 0610, Antofagasta 1270709, Chile
1919institutetext:  Department of Physics, University of Warwick, Gibbet Hill
Road, Coventry CV4 7AL, UK 2020institutetext:  Admatis, 5. Kandó Kálmán
Street, 3534 Miskolc, Hungary 2121institutetext:  Depto. de Astrofísica,
Centro de Astrobiología (CSIC-INTA), ESAC campus, 28692 Villanueva de la
Cañada (Madrid), Spain 2222institutetext:  Instituto de Astrofisica e Ciencias
do Espaco, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto,
Portugal 2323institutetext:  Departamento de Fisica e Astronomia, Faculdade de
Ciencias, Universidade do Porto, Rua do Campo Alegre, 4169-007 Porto, Portugal
2424institutetext:  Space Research Institute, Austrian Academy of Sciences,
Schmiedlstrasse 6, A-8042 Graz, Austria 2525institutetext:  INAF, Osservatorio
Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122 Padova, Italy
2626institutetext:  Centre for Exoplanet Research, School of Physics and
Astronomy, University of Leicester, Leicester LE1 7RH, UK 2727institutetext:
Centre for Exoplanet Science, SUPA School of Physics and Astronomy, University
of St Andrews, North Haugh, St Andrews KY16 9SS, UK 2828institutetext:
CFisUC, Department of Physics, University of Coimbra, 3004-516 Coimbra,
Portugal 2929institutetext:  Institute of Planetary Research, German Aerospace
Center (DLR), Rutherfordstrasse 2, 12489 Berlin, Germany 3030institutetext:
INAF, Osservatorio Astrofisico di Torino, Via Osservatorio, 20, I-10025 Pino
Torinese To, Italy 3131institutetext:  Centre for Mathematical Sciences, Lund
University, Box 118, 221 00 Lund, Sweden 3232institutetext:  Aix Marseille
Univ, CNRS, CNES, LAM, 38 rue Frédéric Joliot-Curie, 13388 Marseille, France
3333institutetext:  ELTE Gothard Astrophysical Observatory, 9700 Szombathely,
Szent Imre h. u. 112, Hungary 3434institutetext:  SRON Netherlands Institute
for Space Research, Niels Bohrweg 4, 2333 CA Leiden, Netherlands
3535institutetext:  Leiden Observatory, University of Leiden, PO Box 9513,
2300 RA Leiden, The Netherlands 3636institutetext:  Department of Space, Earth
and Environment, Chalmers University of Technology, Onsala Space Observatory,
439 92 Onsala, Sweden 3737institutetext:  Dipartimento di Fisica, Università
degli Studi di Torino, via Pietro Giuria 1, I-10125, Torino, Italy
3838institutetext:  National and Kapodistrian University of Athens, Department
of Physics, University Campus, Zografos GR-157 84, Athens, Greece
3939institutetext:  Astronomy Unit, Queen Mary University of London, Mile End
Road, London E1 4NS, UK 4040institutetext:  Department of Astrophysics,
University of Vienna, Türkenschanzstrasse 17, 1180 Vienna, Austria
4141institutetext:  Institute for Theoretical Physics and Computational
Physics, Graz University of Technology, Petersgasse 16, 8010 Graz, Austria
4242institutetext:  Instituto de Estudios Astrofísicos, Facultad de Ingeniería
y Ciencias, Universidad Diego Portales, Av. Ejército Libertador 441, Santiago,
Chile 4343institutetext:  NASA Ames Research Center, Moffett Field, CA 94035,
USA 4444institutetext:  Konkoly Observatory, Research Centre for Astronomy and
Earth Sciences, 1121 Budapest, Konkoly Thege Miklós út 15-17, Hungary
4545institutetext:  ELTE Eötvös Loránd University, Institute of Physics,
Pázmány Péter sétány 1/A, 1117 Budapest, Hungary 4646institutetext:  Lund
Observatory, Division of Astrophysics, Department of Physics, Lund University,
Box 43, 22100 Lund, Sweden 4747institutetext:  IMCCE, UMR8028 CNRS,
Observatoire de Paris, PSL Univ., Sorbonne Univ., 77 av. Denfert-Rochereau,
75014 Paris, France 4848institutetext:  Center for Astrophysics, Harvard and
Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 4949institutetext:
Institut d’astrophysique de Paris, UMR7095 CNRS, Université Pierre & Marie
Curie, 98bis blvd. Arago, 75014 Paris, France 5050institutetext:  Astrophysics
Group, Lennard Jones Building, Keele University, Staffordshire, ST5 5BG,
United Kingdom 5151institutetext:  Department of Physics and Astronomy,
McMaster University, 1280 Main Street West, Hamilton, Ontario, L8S 4L8
5252institutetext:  INAF, Osservatorio Astrofisico di Catania, Via S. Sofia
78, 95123 Catania, Italy 5353institutetext:  Institute of Optical Sensor
Systems, German Aerospace Center (DLR), Rutherfordstrasse 2, 12489 Berlin,
Germany 5454institutetext:  Dipartimento di Fisica e Astronomia ”Galileo
Galilei”, Università degli Studi di Padova, Vicolo dell’Osservatorio 3, 35122
Padova, Italy 5555institutetext:  ETH Zurich, Department of Physics, Wolfgang-
Pauli-Strasse 2, CH-8093 Zurich, Switzerland 5656institutetext:  Institut fuer
Geologische Wissenschaften, Freie Universitaet Berlin, Maltheserstrasse
74-100,12249 Berlin, Germany 5757institutetext:  Institut de Ciencies de
l’Espai (ICE, CSIC), Campus UAB, Can Magrans s/n, 08193 Bellaterra, Spain
5858institutetext:  Institut d’Estudis Espacials de Catalunya (IEEC), 08860
Castelldefels (Barcelona), Spain 5959institutetext:  Department of Earth,
Atmospheric and Planetary Sciences, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA 6060institutetext:  Department of Aeronautics and
Astronautics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA
6161institutetext:  HUN-REN-ELTE Exoplanet Research Group, Szent Imre h. u.
112., Szombathely, H-9700, Hungary 6262institutetext:  Institute of Astronomy,
University of Cambridge, Madingley Road, Cambridge, CB3 0HA, United Kingdom
6363institutetext:  Department of Astrophysical Sciences, Princeton
University, Princeton, NJ 08544, USA

# Photo-dynamical characterisation of the TOI-178 resonant chain

Exploring the robustness of TTV and RV mass characterisation

A. Leleu
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1122 J.-B. Delisle
${}^{\href{https://orcid.org/0000-0001-5844-9888}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
11 L. Delrez
${}^{\href{https://orcid.org/0000-0001-6108-4808}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
334455 E. M. Bryant
${}^{\href{https://orcid.org/0000-0001-7904-4441}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
66 A. Brandeker
${}^{\href{https://orcid.org/0000-0002-7201-7536}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
77 H. P. Osborn
${}^{\href{https://orcid.org/0000-0002-4047-4724}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
8899 N. Hara
${}^{\href{https://orcid.org/0000-0001-9232-3314}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
11 T. G. Wilson
${}^{\href{https://orcid.org/0000-0001-8749-1962}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1010 N. Billot
${}^{\href{https://orcid.org/0000-0003-3429-3836}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
11 M. Lendl
${}^{\href{https://orcid.org/0000-0001-9699-1459}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
11 D. Ehrenreich
${}^{\href{https://orcid.org/0000-0001-9704-5405}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
111111 H. Chakraborty 1212 M. N. Günther
${}^{\href{https://orcid.org/0000-0002-3164-9086}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1313 M. J. Hooton
${}^{\href{https://orcid.org/0000-0003-0030-332X}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1414 Y. Alibert
${}^{\href{https://orcid.org/0000-0002-4644-8818}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
8822 R. Alonso
${}^{\href{https://orcid.org/0000-0001-8462-8126}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
15151616 D. R. Alves
${}^{\href{https://orcid.org/0000-0002-5619-2502}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1717 D. R. Anderson
${}^{\href{https://orcid.org/0000-0001-7416-7522}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1818 I. Apergis
${}^{\href{https://orcid.org/0009-0004-7473-4573}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1919 D. Armstrong
${}^{\href{https://orcid.org/0000-0002-5080-4117}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1919 T. Bárczy
${}^{\href{https://orcid.org/0000-0002-7822-4413}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2020 D. Barrado Navascues
${}^{\href{https://orcid.org/0000-0002-5971-9242}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2121 S. C. C. Barros
${}^{\href{https://orcid.org/0000-0003-2434-3625}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
22222323 M. P. Battley
${}^{\href{https://orcid.org/0000-0002-1357-9774}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1212 W. Baumjohann
${}^{\href{https://orcid.org/0000-0001-6271-0110}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2424 D. Bayliss
${}^{\href{https://orcid.org/0000-0001-6023-1335}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1919 T. Beck 22 W. Benz
${}^{\href{https://orcid.org/0000-0001-7896-6479}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2288 L. Borsato
${}^{\href{https://orcid.org/0000-0003-0066-9268}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2525 C. Broeg
${}^{\href{https://orcid.org/0000-0001-5132-2614}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2288 M. R. Burleigh
${}^{\href{https://orcid.org/0000-0003-0684-7803}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2626 S. L. Casewell
${}^{\href{https://orcid.org/0000-0003-2478-0120}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2626 A. Collier Cameron
${}^{\href{https://orcid.org/0000-0002-8863-7828}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2727 A. C. M. Correia 2828 Sz. Csizmadia
${}^{\href{https://orcid.org/0000-0001-6803-9698}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2929 P. E. Cubillos 30302424 M. B. Davies
${}^{\href{https://orcid.org/0000-0001-6080-1190}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
3131 M. Deleuil
${}^{\href{https://orcid.org/0000-0001-6036-0225}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
3232 A. Deline 11 O. D. S. Demangeon
${}^{\href{https://orcid.org/0000-0001-7918-0355}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
22222323 B.-O. Demory
${}^{\href{https://orcid.org/0000-0002-9355-5165}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
8822 A. Derekas 3333 B. Edwards 3434 A. Erikson 2929 A. Fortier
${}^{\href{https://orcid.org/0000-0001-8450-3374}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2288 L. Fossati
${}^{\href{https://orcid.org/0000-0003-4426-9530}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2424 M. Fridlund
${}^{\href{https://orcid.org/0000-0002-0855-8426}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
35353636 D. Gandolfi
${}^{\href{https://orcid.org/0000-0001-8627-9628}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
3737 K. Gazeas 3838 E. Gillen
${}^{\href{https://orcid.org/0000-0003-2851-3070}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
3939 M. Gillon
${}^{\href{https://orcid.org/0000-0003-1462-7739}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
33 M. R. Goad 2626 M. Güdel 4040 F. Hawthorn
${}^{\href{https://orcid.org/0000-0002-8675-182X}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1919 A. Heitzmann
${}^{\href{https://orcid.org/0000-0002-8091-7526}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
11 Ch. Helling 24244141 K. G. Isaak
${}^{\href{https://orcid.org/0000-0001-8585-1717}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1313 J. S. Jenkins
${}^{\href{https://orcid.org/0000-0003-2733-8725}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
4242 J. M. Jenkins 4343 A. Kendall
${}^{\href{https://orcid.org/0009-0006-0719-9229}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2626 L. L. Kiss 44444545 J. Korth 4646 K. W. F. Lam
${}^{\href{https://orcid.org/0000-0002-9910-6088}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2929 J. Laskar
${}^{\href{https://orcid.org/0000-0003-2634-789X}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
4747 D. W. Latham 4848 A. Lecavelier des Etangs
${}^{\href{https://orcid.org/0000-0002-5637-5253}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
4949 D. Magrin
${}^{\href{https://orcid.org/0000-0003-0312-313X}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2525 P. F. L. Maxted
${}^{\href{https://orcid.org/0000-0003-3794-1317}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
5050 J. McCormac
${}^{\href{https://orcid.org/0000-0003-1631-4170}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1919 C. Mordasini 2288 M. Moyano
${}^{\href{https://orcid.org/0000-0002-7927-9555}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1818 V. Nascimbeni
${}^{\href{https://orcid.org/0000-0001-9770-1214}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2525 G. Olofsson
${}^{\href{https://orcid.org/0000-0003-3747-7120}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
77 A. Osborn
${}^{\href{https://orcid.org/0000-0002-5899-7750}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
5151 R. Ottensamer 4040 I. Pagano
${}^{\href{https://orcid.org/0000-0001-9573-4928}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
5252 E. Pallé
${}^{\href{https://orcid.org/0000-0003-0987-1593}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
15151616 G. Peter
${}^{\href{https://orcid.org/0000-0001-6101-2513}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
5353 G. Piotto
${}^{\href{https://orcid.org/0000-0002-9937-6387}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
25255454 D. Pollacco 1010 D. Queloz
${}^{\href{https://orcid.org/0000-0002-3012-0316}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
55551414 R. Ragazzoni
${}^{\href{https://orcid.org/0000-0002-7697-5555}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
25255454 N. Rando 1313 H. Rauer
${}^{\href{https://orcid.org/0000-0002-6510-1828}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
29295656 I. Ribas
${}^{\href{https://orcid.org/0000-0002-6689-0312}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
57575858 G. Ricker 99 S. Saha
${}^{\href{https://orcid.org/0000-0001-8018-0264}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
4242 N. C. Santos
${}^{\href{https://orcid.org/0000-0003-4422-2919}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
22222323 G. Scandariato
${}^{\href{https://orcid.org/0000-0003-2029-0626}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
5252 S. Seager
${}^{\href{https://orcid.org/0000-0002-6892-6948}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
9959596060 D. Ségransan
${}^{\href{https://orcid.org/0000-0003-2355-8034}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
11 A. E. Simon
${}^{\href{https://orcid.org/0000-0001-9773-2600}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2288 A. M. S. Smith
${}^{\href{https://orcid.org/0000-0002-2386-4341}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2929 S. G. Sousa
${}^{\href{https://orcid.org/0000-0001-9047-2965}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
2222 M. Stalport 4433 S. Sulis
${}^{\href{https://orcid.org/0000-0001-8783-526X}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
3232 Gy. M. Szabó
${}^{\href{https://orcid.org/0000-0002-0606-7930}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
33336161 S. Udry
${}^{\href{https://orcid.org/0000-0001-7576-6236}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
11 V. Van Grootel
${}^{\href{https://orcid.org/0000-0003-2144-4316}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
44 R. Vanderspek 99 J. Venturini
${}^{\href{https://orcid.org/0000-0001-9527-2903}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
11 E. Villaver 15151616 J. I. Vinés
${}^{\href{https://orcid.org/0000-0002-1896-2377}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1818 N. A. Walton
${}^{\href{https://orcid.org/0000-0003-3983-8778}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
6262 R. G. West
${}^{\href{https://orcid.org/0000-0001-6604-5533}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
1919 J. Winn
${}^{\href{https://orcid.org/0000-0002-4265-047X}{\includegraphics[scale={0.5}]{figures/orcid.jpg}}}$
6363 T. Zivave 1919

###### Abstract

Context. The TOI-178 system consists of a nearby late K-dwarf transited by six
planets in the super-Earth to mini-Neptune regime, with radii ranging from
$\sim$1.2 to 2.9 $R_{\oplus}$ and orbital periods between 1.9 and 20.7 days.
All planets but the innermost one form a chain of Laplace resonances. The
fine-tuning and fragility of such orbital configurations ensure that no
significant scattering or collision event has taken place since the formation
and migration of the planets in the protoplanetary disc, hence providing
important anchors for planet formation models.

Aims. We aim to improve the characterisation of the architecture of this key
system, and in particular the masses and radii of its planets. In addition,
since this system is one of the few resonant chains that can be characterised
by both photometry and radial velocities, we aim to use it as a test bench for
the robustness of the planetary mass determination with each technique.

Methods. We perform a global analysis of all available photometry and radial
velocity using photo-dynamical modelling of the light curve. We also try
different sets of priors on the masses and eccentricity, as well as different
stellar activity models, to study their effects on the masses estimated by
transit timing variations and radial velocities.

Results.  We show how stellar activity is preventing us from obtaining a
robust mass estimation for the three outer planets using radial velocity data
alone. We also show that our joint photo-dynamical and radial velocity
analysis resulted in a robust mass determination for planets $c$ to $g$, with
precision of $\sim 12\%$ for the mass of planet $c$, and better than $10\%$
for planets $d$ to $g$. The new precisions on the radii range from $2$ to
$3\%$. The understanding of this synergy between photometric and radial
velocity measurements will be valuable during the PLATO mission. We also show
that TOI-178 is indeed currently locked in the resonant configuration,
librating around an equilibrium of the chain.

###### Key Words.:

Planetary systems – Stars: individual: TOI-178 – Techniques: photometric –
Techniques: radial velocity

##  1 Introduction

The observed architecture of planetary systems, defined as the orbit and
composition of their planets, is the outcome of their formation in their
proto-planetary disc and long-term evolution (typically Gyrs) after its
dispersal. In this context, planetary systems observed in chains of Laplace
resonances, where each consecutive pair of planets are in (or close to) a
2-body mean-motion resonance (MMR), are of particular interest. Indeed, the
fine-tuning and fragility of such orbital configurations ensure that no
significant scattering or collision event has taken place since the end of the
migration of the planets in the protoplanetary disc (e.g. Mills et al., 2016;
Izidoro et al., 2017). Hence, these systems are especially valuable for
constraining the outcome of protoplanetary discs and provide important anchors
for planet formation models.

To date, chains of Laplace resonances have only been observed for a few
systems: GJ 876 (Rivera et al., 2010), Kepler-60 (Goździewski et al., 2016),
Kepler-80 (MacDonald et al., 2016), Kepler-223 (Mills et al., 2016),
TRAPPIST-1 (Gillon et al., 2017; Luger et al., 2017), K2-138 (Lopez et al.,
2019), TOI-178 (Leleu et al. 2021a, hereafter L21), TOI-1136 (Dai et al.,
2023) and HD 110067 (Luque et al., 2023). All these systems, except GJ 876,
are transiting, which provides an opportunity to observe the effect of planet-
planet gravitational interactions, and thus constrain the masses and
eccentricities of the planets via their transit timing variations (TTVs). For
stars that are bright enough, it is also possible to obtain radial velocity
(RV) measurements, that can provide complementary constraints on the planetary
masses and orbital parameters. Out of the transiting systems cited above, only
K2-138, TOI-178, TOI-1136 and HD110067 have published RV measurements so far,
the other ones being too faint in the visible ($V$-mag$\gtrsim$14). Having the
possibility to measure planetary masses independently from RVs and TTVs is
especially valuable to understand how choices of noise models and degeneracies
between parameters affect the robustness of mass measurements for each
technique.

Using data from Transiting Exoplanet Survey Satellite (TESS, Ricker et al.
2015), CHaracterising ExOPlanets Satellite (CHEOPS, Benz et al. 2021) and Next
Generation Transit Survey (NGTS, Wheatley et al., 2018), L21 showed that the
nearby ($\sim$63 pc) late K-type star TOI-178 is a compact system of at least
six transiting planets in the super-Earth to mini-Neptune regime, with radii
ranging from $\sim$1.1 to 2.9 $R_{\oplus}$ and orbital periods of 1.91, 3.24,
6.56, 9.96, 15.23, and 20.71 days. The planetary radii were later refined by
Delrez et al. (2023), thereafter D23. The five outer planets form a 2:4:6:9:12
chain of Laplace resonances, while the innermost planets $b$ and $c$ are just
wide of the 3:5 MMR, which could indicate that it was previously part of the
chain but was then pulled away, possibly by tidal forces (L21).

Using RV measurements obtained with the Echelle SPectrograph for Rocky
Exoplanets and Stable Spectroscopic Observations (ESPRESSO, Pepe et al. 2021)
installed at ESO’s Very Large Telescope, L21 were also able to derive
preliminary estimates for the masses of the planets, and thus their bulk
densities (when combined with the radii inferred from the transit photometry).
The planetary densities that they found show important variations from planet
to planet, jumping for example from $\sim$1 to 0.2 $\rho_{\oplus}$ between
planets $c$ and $d$. By doing a Bayesian internal structure analysis, they
showed that the two innermost planets are likely to be mostly rocky, which
could indicate that they have lost their primordial gas envelope through
atmospheric escape, while all the other planets appear to contain significant
amounts of water and/or gas (see also the independent internal structure
analysis by Acuña et al. 2022). However, the planetary densities on which
these results were based were constrained by a relatively low number of RV
points (46 ESPRESSO points).

In this paper, we re-analyse the data presented in L21 and D23, to which we
add CHEOPS observations taken in 2021, 2022 and 2023, TESS sector 69 (2023),
and NGTS observations taken in 2021, see section 2. The whole analysis is
performed by a photo-dynamical fit of all available photometry joint with the
fit of available RV measurments, see section 3. We then present the result of
our analysis in section 4. In particular, we discuss the robustness of the
extracted masses, exploring the effect of the mass-eccentricity degeneracy in
TTVs and the effect of activity modelling in RV. Finally, we conclude in
section 5.

##  2 Data

Figure 1:  Detrended light curves from CHEOPS as described in sec 2.1 and 3.3
Un-binned data are shown as blue points, and data in 30 min bins are shown as
green circles. The best fitting transit model for the system is shown in
black; the associated parameter values are from the final posterior shown in
Tables 2 and 3. Vertical lines indicate the planet that is transiting,
unflagged transits are caused by planet $b$. Each line contains about six days
of observation. Data published in L21 are indicated by a red upper line, the
one published in D23 by an orange upper line, and new data by a green upper
line. Raw data available online.

Figure 2:  Detrended light curves from TESS as described in sec 2.2 and 3.3
Un-binned data are shown as blue points, and data in 30 min bins are shown as
red circles. The best fitting transit model for the system is shown in black;
the associated parameter values are from the final posterior shown in Tables 2
and 3. Each line contains about 13 days of observation. Raw data available
online.

Figure 3:  Detrended light curves from NGTS as described in sec 2.3 and 3.3.
Data in 30 min bins are shown as purple circles. The best fitting transit
model for the system is shown in black; the associated parameter values are
from the final posterior shown in Tables 2 and 3. The line contains about 2
days of observation, each visit containing the data taken by respectively 7,
4, 3, 4, 4, 4, 3, 4 and 5 of the NGTS telescopes. Un-binned data are not shown
because the scatter is much larger than the than the flux range shown here.
Raw data available online.

In this study, we used the TESS, CHEOPS, ESPRESSO and NGTS data presented in
L21 and D23. In addition, we use 27 new CHEOPS visits, 8 new NGTS observations
and 4 EulerCam observations that were taken in order to monitor the TTVs of
the 5 outer planets of TOI-178. The new data is available online (link). We
also added the new data from the TESS sector 69.

###  2.1 CHEOPS

Following D23, the raw data of each visit were reduced with
PIPE111https://github.com/alphapsa/PIPE (Brandeker et al., 2022), a PSF
photometry package developed specifically for CHEOPS. PIPE first uses a
principal component analysis (PCA) approach to derive a PSF template library
from the data series. The first five principal components (PCs) together with
a constant background are then used to fit the individual PSFs of each image
using a least-squares minimisation and measure the target’s flux. The number
of PCs to use is a trade-off between following systematic PSF changes and
overfitting the noise. For faint stars such as TOI-178, the mean PSF (first
PC) is sufficient for a good extraction, and attempts to model the PSF better
with more PCs usually introduce noise in the extracted light curve. Some
advantages of using PSF photometry rather than aperture photometry for faint
targets are that: (1) the contributions to the signal of each pixel over the
PSF are weighted according to noise so that higher S/N photometry can be
extracted; (2) cosmic rays and bad pixels (both hot and telegraphic) are
easier to filter out or give lower weight in the fitting process; (3) PSF
photometry is less sensitive to contamination from nearby background stars;
(4) the background is fit simultaneously with the PSF for the same pixels,
which can be an advantage if there is some spatial structure. For each visit,
we then filter out all the data for which the background contamination rises
above 300 [electrons/pixel/exposure]. The detrended data is shown in Fig. 1.

###  2.2 TESS

TESS (Ricker et al., 2015) observed TOI-178 for the first time during Cycle
1/Sector 2 of its primary mission (22 August – 20 September 2018). These data,
obtained with a two-minute cadence, were previously presented in L21 and we
include them in our global analysis. TESS observed again TOI-178 during Cycle
3/Sector 29 and Cycle 5/Sector 69 of its extended mission, from 26 August to
22 September 2020 (presented in D23) and from 25 August to 20 September 2023.
The data were processed with the TESS Science Processing Operations Center
(SPOC) pipeline (Jenkins et al., 2016) at NASA Ames Research Center. We
retrieved the 2-minute cadence Pre-search Data Conditioning Simple Aperture
Photometry (PDCSAP, Stumpe et al. 2012; Smith et al. 2012; Stumpe et al. 2014)
from the Mikulski Archive for Space Telescopes222https://archive.stsci.edu
(MAST), using the default quality bitmask. The detrended data is shown in Fig.
2.

###  2.3 NGTS

We also included in our global analysis the light curves obtained with the
Next Generation Transit Survey (NGTS, Wheatley et al. 2018) that were
previously published in L21, as well as eight additional transit observations
that were obtained between the dates of 23 June 2021 and 3 November 2021. The
NGTS photometric facility consists of twelve independently operated robotic
telescopes each with a 20 cm diameter aperture and a field-of-view of 8
square-degrees. Using multiple NGTS telescopes to simultaneously observe the
same star has been shown to yield vast improvements in the photometric
precision compared to observing with a single telescope (Smith et al., 2020;
Bryant et al., 2020) and TOI-178 was observed using this high-precision multi-
telescope observing mode. We observed three transits of TOI-178 d, two
transits of TOI-178 e, two transits of TOI-178 f, and one transit of TOI-178
g. The NGTS observations were all performed using the custom NGTS filter (520
– 890 nm) at a cadence of 13 seconds and were reduced using a custom aperture
photometry pipeline which utilises the SEP package (Bertin & Arnouts, 1996;
Barbary, 2016) to perform the source extraction and also automatically selects
comparison stars which are similar to TOI-178 in magnitude, colour, and CCD
position using Gaia DR2 (Gaia Collaboration et al., 2018). For more details on
the reduction we refer the reader to Bryant et al. (2020). The NGTS light
curves used are displayed in Figure 3. We refer the reader to L21 and
references therein for more information about the previously published NGTS
data and their reduction.

###  2.4 EulerCam

Multiple transits of planets d, e and f were observed with the EulerCam
instrument between 2021-08-11 and 2021-10-20. EulerCam is a CCD imager
installed at the Cassegrain focus of the 1.2m Leonhard Euler telescope at La
Silla observatory (Lendl et al., 2012). We used the broadband NGTS filter for
our observations to maximise SNR. The data were reduced using the standard
EulerCam pipeline, which performs basic image calibration and relative
aperture photometry. The optimal aperture and reference stars are selected by
minimising the residual RMS of the final light curve. The EulerCam
observations took place simultaneously to CHEOPS and - as expected for ground-
based data - show much lower precision (RMS in the 900-1800 ppm range). We
eventually excluded them from our final analysis, but show them in Fig. 10 for
completeness.

###  2.5 ESPRESSO

The RV data, presented in L21, consist of 46 ESPRESSO points. Each measurement
was taken in high resolution (HR) mode with an integration time of 20 min
using a single telescope (UT) and slow read-out (HR 21). The source on fibre B
is the Fabry-Perot interferometer. Observations were made with a maximum
airmass of 1.8 and a minimum 30∘ separation from the Moon.

##  3 Data analysis

###  3.1 Stellar properties

Table 1: Properties of the host star TOI-178.

Property (unit) | Value | Source
---|---|---
Astrometric properties |
RA [J2000] | 00:29:12.49 | [1]
Dec [J2000] |  $-$30:27:14.86 | [1]
$\mathrm{\mu_{RA}}$ [mas $\mathrm{yr}^{-1}$] | $150.032\pm 0.028$ | [1]
$\mathrm{\mu_{Dec}}$ [mas $\mathrm{yr}^{-1}$] | $-87.132\pm 0.030$ | [1]
Parallax [mas] | $15.900\pm 0.031$ | [1]
Distance [pc] | $62.89\pm 0.12$ | from parallax
Photometric magnitudes |
$G$ [mag] | $11.1575\pm 0.0028$ | [1]
$G_{\rm BP}$ [mag] | $11.8398\pm 0.0029$ | [1]
$G_{\rm RP}$ [mag] | $10.3602\pm 0.0038$ | [1]
$J$ [mag] | $9.372\pm 0.021$ | [2]
$H$ [mag] | $8.761\pm 0.023$ | [2]
$K$ [mag] | $8.656\pm 0.021$ | [2]
$W$1 [mag] | $8.573\pm 0.022$ | [3]
$W$2 [mag] | $8.64\pm 0.02$ | [3]
Spectroscopic and derived properties |
$T_{\mathrm{eff}}$ (K) | $4316\pm 70$ | Spectroscopy [4]
log $g_{\star}$ (cgs) | $4.45\pm 0.15$ | Spectroscopy [4]
$\mathrm{[Fe/H]}$ (dex) | $-0.23\pm 0.05$ | Spectroscopy [4]
$v$ sin $i_{\star}$ (km $\mathrm{s}^{-1}$) | $1.5\pm 0.3$ | Spectroscopy [4]
$R_{\star}$ ($R_{\odot}$) | $0.662\pm 0.010$ | IRFM [5]
$M_{\star}$ ($M_{\odot}$) | $0.647_{-0.029}^{+0.030}$ | Isochrones [5]
$t_{\star}$ (Gyr) | $6.0_{-5.0}^{+6.8}$ | Isochrones [5]
$L_{\star}$ ($L_{\odot}$) | $0.136\pm 0.010$ | from $R_{\star}$ and $T_{\mathrm{eff}}$ [5]
$\rho_{\star}$ ($\rho_{\odot}$) | $2.23\pm 0.14$ | from $R_{\star}$ and $M_{\star}$ [5]

References. [1] Gaia EDR3 (Gaia Collaboration et al., 2021); [2] 2MASS
(Skrutskie et al., 2006); [3] WISe (Wright et al., 2010); [4] Leleu et al.
(2021a); [5] Delrez et al. (2023).

In this study, we use the stellar properties that were updated by D23. These
properties are summarized in Table 1.

###  3.2 Approach

The transit timing variations expected for TOI-178 were explained in section
6.4 and illustrated in Fig. 14 of L21. A dominant feature of these TTVs is the
effect of the proximity of 2-body MMRs (Lithwick et al., 2012) which induce
sinusoidal TTVs with a period of $\approx 260$ days (called super-period) on
the 5 planets that are part of the resonant chain (planets $c$ to $g$).
Transit timing variations due to the proximity of 2-body MMRs are known to
present a mass-eccentricity degeneracy at first order in the eccentricities
(Boué et al., 2012; Lithwick et al., 2012). This degeneracy can be broken if
either higher harmonics of the super-period (Hadden & Lithwick, 2016), the
short-term chopping effect (Deck & Agol, 2015), or the long-term evolution in
the resonance, can be constrained.

To check the robustness of the masses we derive, we follow Hadden & Lithwick
(2017) and try out priors that pull the solution toward opposite directions of
the degeneracy. We use their default prior which is log-uniform in planet
masses and uniform in eccentricities, and their high-mass prior which is
uniform in planet masses and log-uniform in eccentricities. In addition,
following Leleu et al. (2023), we performed a third fit, using log-uniform
mass prior and the Kipping (2013) prior for the eccentricity: a
$\beta$-distribution of parameters $\alpha=0.697$ and $\beta=3.27$. The
posterior associated with this set of priors is referred to as the final
posterior. Then, following Leleu et al. (2023), we quantify the robustness of
the mass determination by studying the difference between the mass posteriors
using the quantity $\Delta_{M}$:

$\Delta_{M}=\max(\Delta_{M+},\Delta_{M-}),$ (1)

which estimates ‘at how many sigmas’ of the final posterior the median of the
default and highmass posteriors are. More precisely

$\Delta_{M+}=\frac{m_{high,0}-m_{final,0}}{m_{final,+\sigma}-m_{final,0}},$
(2)

where $m_{high,0}$ is the maximum between the medians of the default and high-
mass posterior, $m_{final,0}$ is the median of the final posterior, and
$m_{final,+\sigma}$ is the $0.84$ quantile of the final posterior.
$\Delta_{M+}=0$ if $m_{high,0}<m_{final,0}$.

$\Delta_{M-}=\frac{m_{final,0}-m_{low,0}}{m_{final,0}-m_{final,-\sigma}},$ (3)

where $m_{low,0}$ is the minimum of the default and high-mass posterior
medians, and $m_{final,-\sigma}$ is the $0.16$ quantile of the final
posterior. $\Delta_{M-}=0$ if $m_{low,0}>m_{final,0}$. The values of
$\Delta_{M}$ obtained for each planet is given in Tables 2 and 3, and
discussed in sec. 4.3.

Transit timing variations can be studied by pre-extracting all the transit
timings of the planets, then fitting these transit timings. Alternatively, one
can analyse TTVs using a photo-dynamical model (Ragozzine & Holman, 2010); in
which the ideal light-curve, accounting for TTVs, is modelled and then fit to
the data. Leleu et al. (2023) showed that photo-dynamical analysis often
allows for a more robust mass estimate, especially for systems harbouring
planets whose individual transit signal-to-noise ratio (S/N) is low (typically
$\lesssim 3.5$). We therefore performed a photo-dynamical analysis of the
data.

###  3.3 Data modelling

We performed a joint analysis of all available transit events as well as the
RV data. The gravitational interaction of the 5 resonant planets ($c$ to $g$)
was also taken into account. For these planets, we used wide, flat priors for
the mean longitude, period, impact parameter, and ratio of the radius of the
planet over the radius of the star, $R_{p}/R_{\star}$. The mass, $e\cos\varpi$
and $e\sin\varpi$ priors were given in section 3.2. The stellar density and
stellar radius have Gaussian priors with the values given in Table 1.

####  3.3.1 Photometry

The photo-dynamical model of the planetary signals, presented in detail in
Leleu et al. (2021b, 2023), is computed by predicting transit timings using
the TTVfast package (Deck et al., 2014) for the outer five planets, and a
circular orbit for TOI-178b, and modeling the transits of each planet using
the batman package Kreidberg (2015). We model the instrumental systematics as
well as stellar activity using a linear combination of indicators and
B-splines. We use B-splines as functions of time for all photometric data, and
an additional B-spline is added with respect to the roll-angle for each CHEOPS
visit. The temporal B-splines have nodes separated by 0.4 days for all
photometry, and the periodic B-splines on the roll angle for CHEOPS have nodes
separated by 0.1 radian. For CHEOPS, we used as indicators the telescope tube
temperature, the position of the centroid along the Y direction, and the
background contamination. For NGTS we used the airmass. For TESS, the light
curves were pre-detrended so no indicator were used. This model (indicators
and B-splines) only introduces linear parameters. To accelerate the
exploration of the parameter space, we compute the likelihood marginalized
over these linear parameters, using the
linmarg333https://gitlab.unige.ch/delisle/linmarg python package (Leleu et
al., 2023). The limb-darkening parameters have Gaussian priors computed by
LDCU (Deline et al., 2022) for each photometric instrument. In addition, a
jitter term is added per instrument with a flat prior.

####  3.3.2 Radial velocities

The RV model of the planetary signals is also computed using the TTVfast
package (Deck & Agol, 2016) for the 5 outer planets, and a circular model for
TOI-178b. We fitted these data by modeling the planets’ orbits as well as
stellar activity using a Gaussian process (GP) model trained simultaneously on
the RVs and ancillary activity indicators using the
spleaf444https://gitlab.unige.ch/delisle/spleaf package with the FENRIR 2
modes - 4 harmonics - Matérn 1/2 kernel, see appendix C and Hara & Delisle
(2023). The GP is trained simultaneously on the RVs, the FWHM, and the
$H\alpha$ activity indicators.

##  4 Results of the data analysis

Figure 4:  Transit timing variations observed for TOI178 c, $d$ and e. The
filled area correspond to the 1$\sigma$ posterior of the photo-dynamical fits
presented in section 3.2. The error bars show an estimation of the 1$\sigma$
interval for the mid-transit timing of individual transits for each
instrument. See the text for more details.

Figure 5:  Transit timing variations observed for TOI178 f and g. See Fig. 4.

The transit timings estimated for each planet, as well as 300 samples of the
final posterior, can be found online. In particular, the transit timings are
propagated until 2030, to be usable for follow-up observations of the system.

###  4.1 Recovered TTV signal

Figures 4 and 5 show the TTVs of the full fit model for both the default and
highmass set of priors, along with estimations of individual transit timings
for each instrument. In these TTV plots, in addition to the super-period at
$\sim$260 days visible for all planets, the long-term evolution of the
resonant angles start to be visible for planets $c$, $d$ and $e$. The saw-
tooth shape of the chopping effect (Deck & Agol, 2015) can be seen on the
posterior of planets $f$ and $g$.

We also fit the individual transit timings to illustrate how each portion of
the TTVs curve is constrained. These individual transit timings were estimated
by an additional fit, using the full model described in section 3, but adding
each transit timing as a free parameter instead of using the timings of the
n-body integration. These individual transit timings are shown as colored
circles in Fig. 4 and 5. While individual transit timings follow the TTV
pattern recovered by the photo-dynamical fit for planets $d$ to $g$. For
planet $c$ the median error of the transit timings is larger than the
amplitude of the TTV signal. The median standard deviation of the timing
derived from the photo-dynamical model ($0.86$ minutes) highlights the
necessity of using photo-dynamical modelling for planets with low-S/N transits
(Leleu et al., 2023).

A downside of the photo-dynamical analysis is its model dependency: we have to
choose a priori how many planets we consider to model the light curves, and
the effect of potential missed planets can be hard to identify on the light
curve residuals. Having the individual transit timings also allows us to check
if there are hints of additional planets in the resonant chain, for example at
orbital periods larger than $P_{g}=20.7$ days. Indeed, the effect of this
additional planet, which is not modelled by the 5-resonant planet model, could
show when subtracting the timing from the photo-dynamical analysis to the
individual transit timings. However, beside a couple of strong outliers in the
TESS timing of TOI-178 $f$, we did not find strong evidence of a planet
significantly impacting the TTVs of the observed planets. We analysed the TESS
SAP and PDCSAP 2-minute light curves, as well as background and centroid time
series to attempt to understand whether systematic noise on the TESS detector
could be causing the two anomalously timed transits of TOI-178 $f$ at 1380.16
[BJD-2457000] and 2111.26 [BJD-2457000]. We found no sign of systematic noise
in the $\sim$12hrs around transit. These outliers might be due to the
relatively low SNR of the transits in the TESS data combined with an under
estimation of the error bars.

###  4.2 Recovered planetary parameters

Table 2: Fitted and derived properties of the planets of TOI-178

Parameter | Prior | photo-dynamical | photo-dynamical + RV
---|---|---|---
TOI-178b
$t_{0}$ [day] |  $\mathcal{U}$(-1.0e+30,1.0e+30) | $1931.17893_{-6.6e-04}^{+7.7e-04}$ | $1931.1793_{-0.0010}^{+0.0012}$
$P$ [day] |  $\mathcal{U}$(0.0e+00,1.0e+04) | $1.9145603_{-2.3e-06}^{+3.9e-06}$ | $1.9145601_{-2.9e-06}^{+4.4e-06}$
$M/M_{\sun}$ | ∗ | $2.2e-06_{-2.0e-06}^{+1.3e-05}$ | $2.9e-06_{-2.0e-06}^{+2.1e-06}$
$R/R_{\star}$ |  $\mathcal{U}$(0.0e+00,2.0e-01) | $0.01691_{-6.3e-04}^{+4.4e-04}$ | $0.01663_{-4.9e-04}^{+5.3e-04}$
$b$ |  $\mathcal{U}$(0.0e+00,1.5e+00) | $0.417_{-0.039}^{+0.006}$ | $0.336_{-0.095}^{+0.081}$
dF [ppm] | derived | $285.9_{-20.9}^{+15.1}$ | $276.6_{-15.9}^{+17.8}$
$M[M_{\text{Earth}}]$ | derived | - | $0.96_{-0.65}^{+0.70}$
$R[R_{\text{Earth}}]$ | derived | $1.227_{-0.046}^{+0.034}$ | $1.200_{-0.037}^{+0.041}$
$\rho[\rho_{\text{Earth}}]$ | derived | - | $0.57_{-0.39}^{+0.40}$
TOI-178c
$\lambda$ [deg] |  $\mathcal{U}$(-360.00,360.00) | $54.895_{-0.012}^{+0.015}$ | $54.85_{-0.13}^{+0.15}$
$P$ [day] |  $\mathcal{U}$(3.14,3.34) | $3.2384871_{-7.3e-06}^{+8.9e-06}$ | $3.238486_{-1.1e-05}^{+1.2e-05}$
$e\cos\varpi$ | ∗ | $2.4e-05_{-2.3e-04}^{+4.4e-04}$ | $2.5e-05_{-3.7e-04}^{+5.4e-04}$
$e\sin\varpi$ | ∗ | $8.5e-06_{-3.0e-04}^{+5.8e-04}$ | $3.5e-05_{-3.8e-04}^{+7.9e-04}$
$M/M_{\sun}$ | ∗ | $1.5e-05_{-2.6e-06}^{+2.5e-06}$ | $1.4e-05_{-1.6e-06}^{+1.5e-06}$
$R/R_{\star}$ |  $\mathcal{U}$(0.0e+00,2.0e-01) | $0.02447_{-4.6e-04}^{+4.2e-04}$ | $0.02431_{-4.4e-04}^{+3.8e-04}$
$b$ |  $\mathcal{U}$(0.0e+00,1.5e+00) | $0.422_{-0.008}^{+0.014}$ | $0.394_{-0.042}^{+0.039}$
dF [ppm] | derived | $599.0_{-22.3}^{+20.9}$ | $590.8_{-21.0}^{+18.7}$
$e$ | derived | $3.6e-04_{-3.0e-04}^{+6.6e-04}$ | $3.2e-04_{-2.0e-04}^{+6.5e-04}$
$\varpi$ [deg] | derived | $15_{-110}^{+96}$ | $26_{-128}^{+90}$
$M[M_{\text{Earth}}]$ | derived | $5.00_{-0.87}^{+0.83}$ | $4.64_{-0.53}^{+0.52}$
$R[R_{\text{Earth}}]$ | derived | $1.777_{-0.035}^{+0.034}$ | $1.754_{-0.040}^{+0.032}$
$\rho[\rho_{\text{Earth}}]$ | derived | $0.91_{-0.18}^{+0.12}$ | $0.87_{-0.10}^{+0.11}$
$\Delta_{M}$ |  | 0.72 | 0.55
TOI-178d
$\lambda$ [deg] |  $\mathcal{U}$(-360.00,360.00) | $28.525_{-0.013}^{+0.011}$ | $28.36_{-0.16}^{+0.15}$
$P$ [day] |  $\mathcal{U}$(6.36,6.75) | $6.557593_{-8.4e-05}^{+9.0e-05}$ | $6.557569_{-5.6e-05}^{+6.5e-05}$
$e\cos\varpi$ | ∗ | $-0.00637_{-4.2e-04}^{+5.0e-04}$ | $-0.0052_{-0.0014}^{+0.0012}$
$e\sin\varpi$ | ∗ | $-0.0047_{-0.0020}^{+0.0025}$ | $-0.0028_{-0.0026}^{+0.0021}$
$M/M_{\sun}$ | ∗ | $1.7e-05_{-1.2e-06}^{+1.1e-06}$ | $1.6e-05_{-1.3e-06}^{+1.2e-06}$
$R/R_{\star}$ |  $\mathcal{U}$(0.0e+00,2.0e-01) | $0.03755_{-3.5e-04}^{+3.7e-04}$ | $0.03737_{-4.1e-04}^{+3.5e-04}$
$b$ |  $\mathcal{U}$(0.0e+00,1.5e+00) | $0.511_{-0.004}^{+0.012}$ | $0.501_{-0.027}^{+0.023}$
dF [ppm] | derived | $1409.8_{-26.3}^{+28.0}$ | $1396.4_{-30.6}^{+26.6}$
$e$ | derived | $0.0080_{-0.0014}^{+0.0015}$ | $0.0068_{-0.0016}^{+0.0016}$
$\varpi$ [deg] | derived | $-143.8_{-16.5}^{+11.0}$ | $-146.3_{-15.6}^{+20.8}$
$M[M_{\text{Earth}}]$ | derived | $5.70_{-0.40}^{+0.36}$ | $5.20_{-0.43}^{+0.39}$
$R[R_{\text{Earth}}]$ | derived | $2.728_{-0.037}^{+0.033}$ | $2.695_{-0.046}^{+0.041}$
$\rho[\rho_{\text{Earth}}]$ | derived | $0.280_{-0.020}^{+0.023}$ | $0.265_{-0.024}^{+0.024}$
$\Delta_{M}$ |  | 0.84 | 0.52
555 results of the photo-dynamical and photo-dynamical+RV fits for the final
set of priors. The orbital elements are given at the date 2458352.55018382
BJD. $\lambda$ is the mean longitude of the planet, $\varpi$ its longitude of
periastron. $\Delta_{M}$ is the robustness criterion defined in equation (1).
$b$, $e$, $M$ and $R$ are the planet’s impact parameter, eccentricity, mass
and radius respectively. $M_{\odot}$ is the mass of the sun and $R_{\star}$ is
the radius of the star. ∗ The mass and eccentricity priors depend on the case,
see section 3.2.

Table 3: Fitted and derived properties of the planets of TOI-178

Parameter | Prior | photo-dynamical | photo-dynamical + RV
---|---|---|---
TOI-178e
$\lambda$ [deg] |  $\mathcal{U}$(-360.00,360.00) | $74.265_{-0.017}^{+0.017}$ | $74.23_{-0.11}^{+0.11}$
$P$ [day] |  $\mathcal{U}$(9.66,10.26) | $9.96336_{-1.5e-04}^{+1.6e-04}$ | $9.96318_{-1.2e-04}^{+1.5e-04}$
$e\cos\varpi$ | ∗ | $-9.8e-05_{-5.1e-04}^{+1.7e-04}$ | $-2.0e-05_{-6.7e-04}^{+4.3e-04}$
$e\sin\varpi$ | ∗ | $-2.7e-05_{-5.2e-04}^{+3.5e-04}$ | $1.0e-07_{-5.7e-04}^{+5.7e-04}$
$M/M_{\sun}$ | ∗ | $1.0e-05_{-1.0e-06}^{+1.1e-06}$ | $1.0e-05_{-8.6e-07}^{+8.6e-07}$
$R/R_{\star}$ |  $\mathcal{U}$(0.0e+00,2.0e-01) | $0.03205_{-4.7e-04}^{+4.5e-04}$ | $0.03191_{-3.7e-04}^{+3.8e-04}$
$b$ |  $\mathcal{U}$(0.0e+00,1.5e+00) | $0.5842_{-0.0073}^{+0.0099}$ | $0.574_{-0.027}^{+0.015}$
dF [ppm] | derived | $1026.9_{-30.0}^{+28.9}$ | $1018.1_{-23.5}^{+24.7}$
$e$ | derived | $4.3e-04_{-3.6e-04}^{+7.5e-04}$ | $3.8e-04_{-2.5e-04}^{+6.7e-04}$
$\varpi$ [deg] | derived | $-49_{-95}^{+175}$ | $2_{-134}^{+120}$
$M[M_{\text{Earth}}]$ | derived | $3.34_{-0.33}^{+0.37}$ | $3.48_{-0.29}^{+0.29}$
$R[R_{\text{Earth}}]$ | derived | $2.325_{-0.038}^{+0.034}$ | $2.301_{-0.039}^{+0.038}$
$\rho[\rho_{\text{Earth}}]$ | derived | $0.264_{-0.023}^{+0.029}$ | $0.287_{-0.026}^{+0.028}$
$\Delta_{M}$ |  | 0.54 | 0.49
TOI-178f
$\lambda$ [deg] |  $\mathcal{U}$(-360.00,360.00) | $157.735_{-0.005}^{+0.021}$ | $157.67_{-0.09}^{+0.11}$
$P$ [day] |  $\mathcal{U}$(14.78,15.69) | $15.23351_{-1.6e-04}^{+1.4e-04}$ | $15.23335_{-2.5e-04}^{+2.4e-04}$
$e\cos\varpi$ | ∗ | $-2.1e-04_{-4.6e-04}^{+2.9e-04}$ | $-8.4e-06_{-6.7e-04}^{+8.1e-04}$
$e\sin\varpi$ | ∗ | $8.8e-04_{-0.0009}^{+0.0015}$ | $0.0017_{-0.0016}^{+0.0016}$
$M/M_{\sun}$ | ∗ | $1.9e-05_{-1.5e-06}^{+1.2e-06}$ | $1.7e-05_{-1.2e-06}^{+1.4e-06}$
$R/R_{\star}$ |  $\mathcal{U}$(0.0e+00,2.0e-01) | $0.03370_{-5.9e-04}^{+5.7e-04}$ | $0.03351_{-6.0e-04}^{+5.6e-04}$
$b$ |  $\mathcal{U}$(0.0e+00,1.5e+00) | $0.7399_{-0.0042}^{+0.0098}$ | $0.734_{-0.014}^{+0.015}$
dF [ppm] | derived | $1135.6_{-39.3}^{+38.6}$ | $1123.1_{-40.0}^{+38.0}$
$e$ | derived | $0.0011_{-0.0009}^{+0.0014}$ | $4.5e-04_{-2.6e-04}^{+7.0e-04}$
$\varpi$ [deg] | derived | $98.2_{-73.8}^{+28.8}$ | $87.3_{-40.1}^{+28.5}$
$M[M_{\text{Earth}}]$ | derived | $6.24_{-0.49}^{+0.40}$ | $5.63_{-0.41}^{+0.45}$
$R[R_{\text{Earth}}]$ | derived | $2.443_{-0.053}^{+0.047}$ | $2.417_{-0.048}^{+0.041}$
$\rho[\rho_{\text{Earth}}]$ | derived | $0.426_{-0.037}^{+0.040}$ | $0.402_{-0.036}^{+0.033}$
$\Delta_{M}$ |  | 0.93 | 1.15
TOI-178g
$\lambda$ [deg] |  $\mathcal{U}$(-360.00,360.00) | $55.329_{-0.019}^{+0.007}$ | $55.27_{-0.10}^{+0.06}$
$P$ [day] |  $\mathcal{U}$(20.09,21.34) | $20.71700_{-3.4e-04}^{+4.1e-04}$ | $20.71663_{-3.0e-04}^{+3.1e-04}$
$e\cos\varpi$ | ∗ | $-2.0e-04_{-2.9e-04}^{+2.1e-04}$ | $1.2e-04_{-4.2e-04}^{+7.9e-04}$
$e\sin\varpi$ | ∗ | $-9.3e-05_{-6.9e-04}^{+2.6e-04}$ | $-1.7e-04_{-0.0012}^{+0.0005}$
$M/M_{\sun}$ | ∗ | $1.4e-05_{-7.9e-07}^{+8.8e-07}$ | $1.3e-05_{-1.1e-06}^{+1.2e-06}$
$R/R_{\star}$ |  $\mathcal{U}$(0.0e+00,2.0e-01) | $0.04086_{-5.4e-04}^{+4.4e-04}$ | $0.04075_{-4.4e-04}^{+5.1e-04}$
$b$ |  $\mathcal{U}$(0.0e+00,1.5e+00) | $0.8591_{-0.0029}^{+0.0027}$ | $0.8549_{-0.0054}^{+0.0055}$
dF [ppm] | derived | $1669.2_{-43.7}^{+36.0}$ | $1660.5_{-35.5}^{+41.8}$
$e$ | derived | $4.5e-04_{-3.5e-04}^{+5.4e-04}$ | $4.3e-04_{-2.8e-04}^{+6.5e-04}$
$\varpi$ [deg] | derived | $-98_{-44}^{+227}$ | $-44_{-63}^{+120}$
$M[M_{\text{Earth}}]$ | derived | $4.58_{-0.26}^{+0.29}$ | $4.40_{-0.37}^{+0.39}$
$R[R_{\text{Earth}}]$ | derived | $2.964_{-0.048}^{+0.042}$ | $2.939_{-0.055}^{+0.057}$
$\rho[\rho_{\text{Earth}}]$ | derived | $0.177_{-0.012}^{+0.013}$ | $0.174_{-0.015}^{+0.015}$
$\Delta_{M}$ |  | 1.34 | 0.79
666 results of the photo-dynamical and photo-dynamical+RV fits for the final
set of priors. The orbital elements are given at the date 2458352.55018382
BJD. $\lambda$ is the mean longitude of the planet, $\varpi$ its longitude of
periastron. $\Delta_{M}$ is the robustness criterion defined in equation (1).
$b$, $e$, $M$ and $R$ are the planet’s impact parameter, eccentricity, mass
and radius respectively. $M_{\odot}$ is the mass of the sun and $R_{\star}$ is
the radius of the star. ∗ The mass and eccentricity priors depend on the case,
see section 3.2.

Figure 6:  Mass-radius relationship of the TOI-178 planets (final posterior),
compared with the super-Earth to Mini-Neptune populations for which the mass
was derived through RV (Otegi et al., 2020) and TTVs (Hadden & Lithwick, 2017;
Leleu et al., 2023). The composition lines ‘terrestrial Earth-like’, in solid
brown, and pure MgSiO3 ‘rocky’, in dashed brown, are taken from Zeng et al.
(2016).  Figure 7:  Mean density of the TOI-178 planets as a function of their
equilibrium temperature.

The planetary masses, radii, transit parameters and Jacobi orbital elements at
epoch 1352.55018 [BJD-2457000] are given in tables 2 and 3 for the final
posterior of the photo-dynamical fit and the combined photo-dynamical and
radial-velocity fits, while Table 5 shows the fitted stellar and noise
parameters.

In this joint analysis, the final posterior reaches a precision of 12% for the
mass of planet $c$, while the precision on the mass of planets $d$ to $g$ is
better than $10\%$. The precision we reach for planet $b$ remains similar to
the one published in L21 since the planet does not induce any significant TTVs
on the other planets of the system. The precision on the planetary radius
ranges from $\approx 3\%$ for planet $b$ to better than $2\%$ for the largest
planet of the system. Figure 6 shows the newly-determined mass-radius relation
of TOI-178 planets with respect to the planets whose mass was characterised by
either TTVs (green) or RV (blue). These reference planets were chosen for
their robustness to the choice of prior for the TTV population and for their
precision for the RV population. The RV-characterised planets appear to be
denser than the TTV-characterised ones, although we note that selecting RV-
characterised planets based on their precision might bias that population
toward denser mass-radius combinations. The apparent discrepancy between the
TTV and RV-characterised planets has been heavily discussed (e.g. Wu &
Lithwick, 2013; Weiss & Marcy, 2014; Mills & Mazeh, 2017; Hadden & Lithwick,
2017; Cubillos et al., 2017; Millholland et al., 2020; Leleu et al., 2023;
Adibekyan et al., 2024). In particular, Hadden & Lithwick (2017) put forward
as a possible explanation a selection bias, since TTVs tend to allow the
characterisation of small planets on larger orbital periods, hence cooler
orbits, than the bulk of the RV characterisation. It has also been proposed
that the systems formed in different environments, such as a different amount
of available iron during the planets’ formation (Adibekyan et al., 2024). The
gaseous planets of TOI-178 appear to be on the lower end of density for their
respective radius, more akin to the sub-population that was characterised
using TTVs. The possibility to characterise TOI-178 both by RV and TTVs makes
this system especially relevant to study this discrepancy. With the
preliminary masses obtained by the RV data, L21 showed that the density of the
planets appeared to not evolve monotonically with respect to the equilibrium
temperature of the planets. With our updated planetary masses and radii, we
retrieve a similar trend, see Fig. 7.

Finally, L21 claimed that the system is in a chain of Laplace resonances.
Using the preliminary mass determination, averaged periods and assuming zero
initial eccentricities, they showed that the system was within the stability
domain of the 3-body resonance chain. Thanks to the joint photo-dynamical and
RV fit, we now have estimations of the instantaneous orbital elements of each
planet at a given date, see Tables 2 and 3. Using 300 randomly-selected
samples of the final posterior, we can simulate the future evolution of the
Laplace angles $\psi_{1}=1\lambda_{c}-4\lambda_{d}+3\lambda_{e}$,
$\psi_{2}=2\lambda_{d}-5\lambda_{e}+3\lambda_{f}$, and
$\psi_{3}=1\lambda_{e}-3\lambda_{f}+2\lambda_{g}$, where $\lambda$ is the mean
longitude of the planet. Figure 8 shows the 3-sigma envelope of the evolution
of these angles over the next 1000 years. Although we lose the information on
the phase of the libration after a few decades, we can see that our full
posterior yields librating angles, confirming that the system is indeed
currently librating in the chain of Laplace resonances, around the equilibrium
described in Table 6 of L21.

Figure 8:  3-sigma envelope of the evolution of the Laplace angles $\Psi_{1}$,
$\Psi_{2}$ and $\Psi_{3}$ over 1000 years for the final posterior.

###  4.3 Robustness of retrieved masses

Figure 9:  Mass determination for different datasets and choices of priors. In
black are the posterior from the RV analysis with the FENRIR 2 modes - 4
harmonics - Matérn 1/2 model, see appendix C. The blue posteriors are from the
photo-dynamical model alone, while the green posteriors are from the joint
photo-dynamical and RV fits. The default, highmass, and final set of priors
are defined in section 3.2. The cyan posterior synthesise the best fits of all
our models with additional planets in the resonant chain, see appendix D. The
photodyn final and photodyn+RV final are the cases reported in Tables 2 and 3.

To estimate the robustness of mass determination with each technique, we
performed RV fits, photo-dynamical fits, and joint fits with the sets of
priors defined in section 3.2. Note that for the RV fit the eccentricity of
the planets is set to zero and therefore the default and final sets of prior
are identical. For the RV fits, we additionally tested several activity models
(see Appendix C), as it might have an impact on the mass determination (e.g.,
Bonfils et al., 2018; Ahrer et al., 2021).

We show in Fig. 9, a summary of the mass posteriors from these analyses. We
see in Fig. 9 that the RV mass determination (black and gray) is robust for
the three inner planets, in the sense that the posteriors for the default and
the high-mass priors agree well. The three outer planets’ masses are more
sensitive to the priors. We also note that these planets’ masses are also more
sensitive to a change of stellar activity model (see Appendix C). We recall
that the stellar rotation period was estimated to be about 35 d by L21, and
that we only have 46 RV measurements over 113 days. It is thus very
challenging to train a GP at the same time as fitting for the 6 planets’
masses on these data. Since we noticed a strong model sensitivity for the 3
outer planets’ mass, we selected one of the activity models that exhibited
large uncertainties for these 3 parameters for the subsequent analyses (joint
fit), in order to avoid biasing the final mass estimates for these planets
(see Appendix C). This does not mean that the chosen activity model is
intrinsically better, and better constraining the activity model and the
planets’ masses from RV would require more measurements.

The posteriors of the masses from the photo-dynamical fit for the different
sets of priors are shown in blue in Fig. 9. As explained in section 3.2, the
$\Delta_{M}$ criterion quantifies the prior-dependency of these masses.
$\Delta_{M}=0$ indicates that the posteriors of the default, highmass and
final fit all perfectly agree, while $\Delta_{M}=1$ indicates that the median
of the default or highmass posteriors lies exactly at $1-\sigma$ from the
median of the final posterior. Leleu et al. (2023) set an arbitrary limit at
$\Delta_{M}=1.3$ to estimate if the retrieved mass was degenerate or not. From
the photo-dynamical fit alone, planets $c$, $d$, $e$ and $f$ have robust
masses, with values of $\Delta_{M}=0.72$, $0.84$, $0.54$ and $0.93$, while the
mass of planet $g$ shows a larger prior dependency, with $\Delta_{M}=1.34$
(see Tables 2 and 3).

Overall, the photo-dynamical model gives better mass precision than the RV fit
for planets $d$ to $e$, while having a much lower prior dependency for the
three outer planets. The mass of TOI-178c is fully consistent between the two
techniques, which is important to note, since this mass is also the least
sensitive to the choice of activity model or mass prior in the RV analysis,
see appendix C. For TOI-178d the median of the default photo-dynamic posterior
is 3-$\sigma$ away from the RV default posterior. However, while the
posteriors shown in Fig. 9 display a low dependency on the mass prior, the
dependency to the choice of activity model is bigger, see Appendix C.

When looking at the posteriors of the joint fit, in green in Fig 9, we can see
that most of the constraints on the mass come from the photo-dynamical model,
as the joint mass posteriors (green) remain close to the photo-dynamical
posteriors (blue). We nonetheless note a general improvement in the robustness
of the determined mass, as the prior dependency ($\Delta_{M}$ in Tables 2 and
3) is generally smaller in the joint fit than in the photo-dynamical fit
alone, beside a slight increase to 1.15 for TOI-178f.

Finally, we estimate the robustness of the derived masses to the existence of
a 7th planet in the resonant chain, see appendix D for more detail. The
outcome of this test is reported by the cyan error bars in Fig. 9. Overall,
adding an additional planet to the model does not change significantly the
derived mass for the six known planets. These results are in agreement with
the findings of Leleu et al. (2023), i.e., that having the proper TTV model
(number of planets in the system) goes hand in hand with having a low prior
dependency (low value of $\Delta_{M}$) for the retrieved masses of the system.

##  5 Summary and conclusion

In this study we present a joint analysis of all available data for the
TOI-178 system. We combine a photo-dynamical analysis of the data taken by
TESS, CHEOPS, and NGTS, with the RVs taken by ESPRESSO. In addition, we
explore the robustness of the derived masses by comparing the results of the
joint fit to the mass constraints obtained by RV and photometry independently.

We show that the available RVs on their own do not enable us to fully
distinguish the planetary signal from stellar activity for the three outer
planets. In particular, for planet $g$ fitting the RV data with different
activity models and mass priors results in a large range of unphysical mass
measurements, an issue that was also reported for example by Lopez et al.
(2019) in the case of K2-138f. Arguably, the case of TOI-178d is the most
intriguing, with all activity models and choice of priors giving relatively
consistent masses using the RV data alone, and these masses being at least
2-$\sigma$ away from the results of the photo-dynamical fit. On the other
hand, the techniques fully agree for the mass of TOI-178c, which is reassuring
since it is the mass that is the most robust to the choice of activity model
and mass prior across all our tests (see appendix C).

We show that the photo-dynamical analysis is able to retrieve robust masses on
its own (low dependency on the chosen set of priors) for planets $c$ to $f$,
and our joint photo-dynamical+RV analysis is able to retrieve robust masses
for planets $c$ to $g$. This highlights the need for thorough follow-up
campaigns for the characterisation of multi-planetary systems, as well as the
development and calibration of robustness criteria for the derived planetary
masses. For these criteria, bright resonant systems such as TOI-178 have an
important role to play since its planetary masses can be obtained
independently by both RV and TTVs. Obtaining these two independent mass
measurements for the same planets can ensure the accuracy of these
measurements and therefore help with the calibration of the robustness
criteria. In the case of TOI-178, additional transit measurements would
further increase our confidence in the robustness of the TTV masses, while
additional RV measurements are necessary both to obtain better constraints on
planet $b$ which is outside of the chain, and to distinguish the signal of the
three outer planets from the stellar activity. These observations also enable
a better understanding of the synergies between photometric and RV
measurements on a given system, which will be key to achieve the full
potential of the upcoming PLATO mission (Rauer et al., 2014).

Thanks to the intensive follow-up effort, TOI-178 now has one of the best
characterised architectures of resonant chains of sub-Neptunes. We were able
to show that the system is indeed librating inside the Laplace resonant chain,
around the equilibrium that was described in section 6.2 of L21. We also
reached a precision of $\sim 12\%$ for the mass of planet $c$, and better than
$10\%$ for planets $d$, $e$, $f$ and $g$. The new precisions on the radii
range from $\approx 3\%$ for planet $b$ to better than $2\%$ for planet $g$.
We note that we were not able to improve the mass of planet $b$ as it is not
interacting with the resonant chain. The newly derived mass and radius of $c$
appear to depart from the 100% rocky composition that was favoured in L21,
implying that a gaseous envelope is necessary to reproduce the observed
density. Planet $d$ to $g$ remain akin to the TTV-characterised Mini-Neptunes
(see Fig. 6), making of TOI-178 a key system in the study of the apparent
mass-radius discrepancy between TTV and RV-characterised population (Wu &
Lithwick, 2013; Weiss & Marcy, 2014; Mills & Mazeh, 2017; Cubillos et al.,
2017; Hadden & Lithwick, 2017; Millholland et al., 2020; Leleu et al., 2023;
Adibekyan et al., 2024). We also retrieve the non-monotonous density
variations as function as the distance to the star that was hinted at by L21.

These new constraints will allow better comparison with models of formation
and evolution of planetary systems (e.g. Emsenhuber et al., 2020; Izidoro et
al., 2022), as well as enabling the exploitation of the JWST transmission
spectrum of the system to their full potential.

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###### Acknowledgements.

CHEOPS is an ESA mission in partnership with Switzerland with important
contributions to the payload and the ground segment from Austria, Belgium,
France, Germany, Hungary, Italy, Portugal, Spain, Sweden, and the United
Kingdom. The CHEOPS Consortium would like to gratefully acknowledge the
support received by all the agencies, offices, universities, and industries
involved. Their flexibility and willingness to explore new approaches were
essential to the success of this mission. CHEOPS data analysed in this article
will be made available in the CHEOPS mission archive
(https://cheops.unige.ch/archive_browser/). This work has been carried out
within the framework of the NCCR PlanetS supported by the Swiss National
Science Foundation under grants 51NF40_182901 and 51NF40_205606. AL
acknowledges support of the Swiss National Science Foundation under grant
number TMSGI2_211697. 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.
thanks the Belgian Federal Science Policy Office (BELSPO) for the provision of
financial support in the framework of the PRODEX Programme of the European
Space Agency (ESA) under contract number 4000142531. The contributions at the
Mullard Space Science Laboratory by E.M.B. have been supported by STFC through
the consolidated grant ST/W001136/1. ABr was supported by the SNSA. This work
has been carried out within the framework of the NCCR PlanetS supported by the
Swiss National Science Foundation under grants 51NF40_182901 and
51NF40_205606. TWi acknowledges support from the UKSA and the University of
Warwick. ML acknowledges support of the Swiss National Science Foundation
under grant number PCEFP2_194576. This project has received funding from the
Swiss National Science Foundation for project 200021_200726. It has also been
carried out within the framework of the National Centre of Competence in
Research PlanetS supported by the Swiss National Science Foundation under
grant 51NF40_205606. The authors acknowledge the financial support of the
SNSF. ML and HC acknowledge support of the Swiss National Science Foundation
under grant number PCEFP2_194576. MNG is the ESA CHEOPS Project Scientist and
Mission Representative, and as such also responsible for the Guest Observers
(GO) Programme. MNG does not relay proprietary information between the GO and
Guaranteed Time Observation (GTO) Programmes, and does not decide on the
definition and target selection of the GTO Programme. YAl acknowledges support
from the Swiss National Science Foundation (SNSF) under grant 200020_192038.
We acknowledge financial support from the Agencia Estatal de Investigación of
the Ministerio de Ciencia e Innovación MCIN/AEI/10.13039/501100011033 and the
ERDF “A way of making Europe” through projects PID2019-107061GB-C61,
PID2019-107061GB-C66, PID2021-125627OB-C31, and PID2021-125627OB-C32, from the
Centre of Excellence “Severo Ochoa” award to the Instituto de Astrofísica de
Canarias (CEX2019-000920-S), from the Centre of Excellence “María de Maeztu”
award to the Institut de Ciències de l’Espai (CEX2020-001058-M), and from the
Generalitat de Catalunya/CERCA programme. This research was funded in whole or
in part by the UKRI, (Grants ST/X001121/1, EP/X027562/1). We acknowledge
financial support from the Agencia Estatal de Investigación of the Ministerio
de Ciencia e Innovación MCIN/AEI/10.13039/501100011033 and the ERDF “A way of
making Europe” through projects PID2019-107061GB-C61, PID2019-107061GB-C66,
PID2021-125627OB-C31, and PID2021-125627OB-C32, from the Centre of Excellence
“Severo Ochoa” award to the Instituto de Astrofísica de Canarias
(CEX2019-000920-S), from the Centre of Excellence “María de Maeztu” award to
the Institut de Ciències de l’Espai (CEX2020-001058-M), and from the
Generalitat de Catalunya/CERCA programme. S.C.C.B. acknowledges support from
FCT through FCT contracts nr. IF/01312/2014/CP1215/CT0004. The contributions
from MPB were carried out within the framework of the National Centre of
Competence in Research PlanetS supported by the Swiss National Science
Foundation under grants 51NF40_182901 and 51NF40_205606. The authors
acknowledge the financial support of the SNSF. LBo, GBr, VNa, IPa, GPi, RRa,
GSc, VSi, and TZi acknowledge support from CHEOPS ASI-INAF agreement n.
2019-29-HH.0. C.B. acknowledges support from the Swiss Space Office through
the ESA PRODEX program. ACC acknowledges support from STFC consolidated grant
number ST/V000861/1, and UKSA grant number ST/X002217/1. P.E.C. is funded by
the Austrian Science Fund (FWF) Erwin Schroedinger Fellowship, program
J4595-N. This project was supported by the CNES. This work was supported by
FCT - Fundação para a Ciência e a Tecnologia through national funds and by
FEDER through COMPETE2020 through the research grants UIDB/04434/2020,
UIDP/04434/2020, 2022.06962.PTDC. O.D.S.D. is supported in the form of work
contract (DL 57/2016/CP1364/CT0004) funded by national funds through FCT.
B.-O. D. acknowledges support from the Swiss State Secretariat for Education,
Research and Innovation (SERI) under contract number MB22.00046. 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”. E.G. gratefully acknowledges support from the UK Science and
Technology Facilities Council (STFC. project reference ST/W001047/1). M.G. is
an F.R.S.-FNRS Senior Research Associate. FH is supported by an STFC
studentship. CHe acknowledges support from the European Union H2020-MSCA-
ITN-2019 under Grant Agreement no. 860470 (CHAMELEON). 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. JSJ greatfully acknowledges support by FONDECYT grant
1201371 and from the ANID BASAL project FB210003. K.W.F.L. was supported by
Deutsche Forschungsgemeinschaft grants RA714/14-1 within the DFG Schwerpunkt
SPP 1992, Exploring the Diversity of Extrasolar Planets. This work was granted
access to the HPC resources of MesoPSL financed by the Region Ile de France
and the project Equip@Meso (reference ANR-10-EQPX-29-01) of the programme
Investissements d’Avenir supervised by the Agence Nationale pour la Recherche.
PM acknowledges support from STFC research grant number ST/R000638/1. This
work was also partially supported by a grant from the Simons Foundation (PI
Queloz, grant number 327127). NCSa acknowledges funding by the European Union
(ERC, FIERCE, 101052347). Views and opinions expressed are however those of
the author(s) only and do not necessarily reflect those of the European Union
or the European Research Council. Neither the European Union nor the granting
authority can be held responsible for them. A. S. acknowledges support from
the Swiss Space Office through the ESA PRODEX program. S.G.S. acknowledge
support from FCT through FCT contract nr. CEECIND/00826/2018 and POPH/FSE
(EC). The Portuguese team thanks the Portuguese Space Agency for the provision
of financial support in the framework of the PRODEX Programme of the European
Space Agency (ESA) under contract number 4000142255. 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
Lendulet 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. JV acknowledges support from the Swiss National Science Foundation
(SNSF) under grant PZ00P2_208945. NAW acknowledges UKSA grant ST/R004838/1.
ACMC acknowledges support from the FCT, Portugal, through the CFisUC projects
UIDB/04564/2020 and UIDP/04564/2020, with DOI identifiers
10.54499/UIDB/04564/2020 and 10.54499/UIDP/04564/2020, respectively.

##  Appendix A EulerCam

The detrended Eulercam data is shown in Fig. 10

Figure 10: Detrended light curves from EulerCam as described in Sec. 2.4. Un-
binned data are shown as blue points, and data in 30 min bins are shown as
purple circles. The best fitting transit model for the system is shown in
black.

##  Appendix B Posteriors of stellar and noise parameters

The posteriors of the stellar and noise parameters are given in tables 7 and
8.

Table 4: Fitted properties of the star and noise parameters

Parameter | Prior | photo-dynamical | photo-dynamical + RV
---|---|---|---
Star |
$\rho_{\star}$ [$\rho_{\odot}$] |  $\mathcal{N}$(2.23,0.14) | $2.244_{-0.019}^{+0.006}$ | $2.34_{-0.10}^{+0.07}$
$R_{\star}$ [$R_{\odot}$] |  $\mathcal{N}$(0.66,0.01) | $0.6659_{-0.0075}^{+0.0039}$ | $0.6592_{-0.0094}^{+0.0084}$
TESS
$\sigma$ |  $\mathcal{U}$(0.0e+00,1.0e+30) | $1.7e-05_{-1.1e-05}^{+1.7e-05}$ | $1.7e-05_{-1.1e-05}^{+1.5e-05}$
$u1$ |  $\mathcal{N}$(0.49,0.02) | $0.485_{-0.009}^{+0.011}$ | $0.485_{-0.018}^{+0.015}$
$u2$ |  $\mathcal{N}$(0.19,0.05) | $0.144_{-0.005}^{+0.021}$ | $0.167_{-0.035}^{+0.047}$
CHEOPS
$\sigma$ |  $\mathcal{U}$(0.0e+00,1.0e+30) | $3.0e-04_{-5.3e-06}^{+5.9e-06}$ | $3.0e-04_{-5.8e-06}^{+5.8e-06}$
$u1$ |  $\mathcal{N}$(0.54,0.02) | $0.550_{-0.018}^{+0.006}$ | $0.551_{-0.022}^{+0.019}$
$u2$ |  $\mathcal{N}$(0.16,0.03) | $0.153_{-0.007}^{+0.017}$ | $0.174_{-0.022}^{+0.024}$
NGTS
$\sigma$ |  $\mathcal{U}$(0.0e+00,1.0e+30) | $4.0e-05_{-2.5e-05}^{+3.1e-05}$ | $3.9e-05_{-2.7e-05}^{+3.3e-05}$
$u1$ |  $\mathcal{N}$(0.53,0.03) | $0.524_{-0.012}^{+0.017}$ | $0.514_{-0.021}^{+0.030}$
$u2$ |  $\mathcal{N}$(0.17,0.04) | $0.174_{-0.014}^{+0.008}$ | $0.139_{-0.030}^{+0.037}$
777 Stellar and noise parameters, with $\sigma$ the jitter terms and $uk$ the
limb-darkening coefficients.

Table 5: Fitted properties of the star and noise parameters

Parameter | Prior | photo-dynamical | photo-dynamical + RV
---|---|---|---
offset RV |  $\mathcal{U}$(-1.0e+300,1.0e+300) | $-$ | $57159.10_{-0.16}^{+4.14}$
offset $H\alpha$ |  $\mathcal{U}$(-1.0e+300,1.0e+300) | $-$ | $0.3242_{-0.0081}^{+0.0084}$
offset FWHM |  $\mathcal{U}$(-1.0e+300,1.0e+300) | $-$ | $6729.20_{-0.22}^{+0.79}$
$\sigma$ RV |  $\mathcal{U}$(0.0e+00,3.0e+01) | $-$ | $0.94_{-0.11}^{+0.30}$
$\sigma$ H$\alpha$ |  $\mathcal{U}$(0.0e+00,3.0e+01) | $-$ | $0.00107_{-6.5e-04}^{+4.6e-04}$
$\sigma$ FWHM |  $\mathcal{U}$(0.0e+00,3.0e+01) | $-$ | $3.84_{-0.16}^{+0.81}$
$GP\alpha_{000}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $10.83_{-5.49}^{+0.06}$
$GP\alpha_{010}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $0.003_{-0.021}^{+0.021}$
$GP\alpha_{011}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-0.003_{-0.022}^{+0.020}$
$GP\alpha_{020}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-12.06_{-4.27}^{+0.15}$
$GP\alpha_{021}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $21.57_{-0.49}^{+0.25}$
$GP\alpha_{100}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $16.35_{-1.25}^{+0.28}$
$GP\alpha_{110}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $0.016_{-0.014}^{+0.013}$
$GP\beta_{110}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-0.014_{-0.013}^{+0.012}$
$GP\alpha_{111}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-9.1e-04_{-0.009}^{+0.010}$
$GP\alpha_{120}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $28.73_{-0.05}^{+2.06}$
$GP\beta_{120}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-56.46_{-0.19}^{+0.57}$
$GP\alpha_{121}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-12.46_{-0.13}^{+1.92}$
$GP\beta_{121}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-9.09_{-1.64}^{+0.42}$
$GP\alpha_{200}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-2.33_{-0.97}^{+0.35}$
$GP\alpha_{210}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-0.008_{-0.014}^{+0.009}$
$GP\beta_{210}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $0.006_{-0.011}^{+0.012}$
$GP\alpha_{211}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-0.005_{-0.016}^{+0.013}$
$GP\alpha_{220}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-14.33_{-0.08}^{+4.16}$
$GP\beta_{220}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $29.22_{-4.11}^{+0.12}$
$GP\alpha_{221}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-38.19_{-0.79}^{+1.16}$
$GP\beta_{221}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-11.72_{-0.23}^{+3.19}$
$GP\alpha_{300}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $12.43_{-2.97}^{+0.14}$
$GP\alpha_{310}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $0.009_{-0.010}^{+0.010}$
$GP\beta_{310}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-0.008_{-0.012}^{+0.011}$
$GP\alpha_{311}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $1.3e-04_{-0.0031}^{+0.0033}$
$GP\alpha_{320}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $53.01_{-3.67}^{+0.31}$
$GP\beta_{320}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-24.08_{-0.64}^{+0.89}$
$GP\alpha_{321}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $0.68_{-0.15}^{+4.75}$
$GP\beta_{321}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-5.31_{-3.48}^{+0.12}$
$GP\alpha_{400}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-6.51_{-0.85}^{+0.51}$
$GP\alpha_{410}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-0.0084_{-0.0084}^{+0.0081}$
$GP\beta_{410}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-0.0043_{-0.0060}^{+0.0071}$
$GP\alpha_{411}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $0.0025_{-0.0051}^{+0.0051}$
$GP\alpha_{420}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $-9.96_{-0.28}^{+0.27}$
$GP\beta_{420}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $7.21_{-0.99}^{+0.55}$
$GP\alpha_{421}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $9.25_{-1.16}^{+0.32}$
$GP\beta_{421}$ |  $\mathcal{N}$(0.0e+00,2.0e+02) | $-$ | $1.26_{-0.10}^{+2.55}$
GP P |  $\mathcal{LU}$(20.00,60.00) | $-$ | $59.75_{-0.55}^{+0.10}$
GP $\rho$ |  $\mathcal{LU}$(10.00,100000.00) | $-$ | $1051.07_{-4.97}^{+0.16}$
888 Noise parameters for the RV fit. $\sigma$ are the jitter term for the
velocity and indicators. $P$ is the period of the GP. $\rho$ is the decay
timescale. $\alpha_{klm}$ and $\beta_{klm}$ are amplitudes of the cosine and
sine terms in the spleaf.term.MultiFourierKernel
(https://obswww.unige.ch/\~delisle/spleaf/doc/\\_autosummary/spleaf.term.MultiFourierKernel.html\\#spleaf.term.MultiFourierKernel),
respectively, with $k$ the harmonic (f = k/P), l the timeseries, and m the
mode.

##  Appendix C Robustness of RV mass estimation and activity modeling

Figure 11: Comparison of RV mass estimates for various activity models and
priors.

Poorly modelled noise sources can introduce biases, in particular in
parameters which are always positive such as mass and eccentricity (Hara et
al. 2019). In order to test the robustness of the mass estimates of the six
planets obtained from the ESPRESSO RVs, we analyzed the RV time series using
different activity models and different mass priors. The results are shown in
Fig. 11. In all cases, we fixed the periods and phases of the planets to the
values of Delrez et al. (2023, median values from Tables 3 and 4). We assumed
circular orbits and only let the semi-amplitudes of the RV signals to vary. We
tested each activity model with a log-uniform (default) prior on the semi-
amplitudes and with a uniform (high-mass) prior. In the case of the high-mass
prior, we also allow for negative semi-amplitudes (hence negative planetary
masses). A negative semi-amplitude means that the signal is found in opposite
phase with what is expected from the transit. We allow for this to better
assess the reliability of the mass constraints we get from RVs and the
dependency of the results with respect to the activity model and the mass
prior. We used the spleaf GP package (Delisle et al. 2020), with either the
ESP kernel (Delisle et al. 2022) or the FENRIR kernel (Hara & Delisle 2023).
In both cases, the GP is trained simultaneously on the RVs, the FWHM, and the
H$\alpha$ activity indicators.

The ESP kernel is an approximation of the widely spread SEP (squared-
exponential periodic) kernel (e.g. Aigrain et al. 2012). The approximation is
truncated at a given number of harmonics of the rotation period. We used from
2 (rotation period + semi-period) to 5 harmonics.

The FENRIR kernel is, in its data-driven form that we used here, a more
flexible kernel which is also decomposed into harmonics of the rotation
period, but with free coefficients. As for the ESP kernel we used from 2 to 5
harmonics. We also tested different decay envelopes (Matérn 1/2 and 3/2), as
well as one or two modes. Having two modes is equivalent to having two
independent FENRIR processes with the same hyper-parameters but different
amplitudes. It is thus a much more flexible model than the one-mode FENRIR
kernel.

From Fig. 11, we conclude that the mass of the three inner planets seems
robust to changes in the activity model and in the mass prior. The mass of
planet e is more prior dependent. Finally planets f and g are both prior and
activity model dependent. The mass of planet g seems especially challenging to
constraint from the existing RV data. Some models (e.g., ESP kernel, FENRIR
with a Matérn 3/2 envelope) prefer a negative mass for planet g, which means
that they probably use the planetary signal in opposite phase to model a
component of the stellar activity. When only allowing for positive masses, the
mass posteriors for these models is still pushed toward zero. Obtaining a
negative mass is a clear diagnostic that the model is not behaving correctly,
but a bias in the opposite direction (i.e., artificially increasing the planet
mass) is much more difficult to detect. Planet f might be in this latter case
(i.e., overestimated mass).

In order to avoid biasing the final mass estimates for the combined RV +
photodynamical fit, but still benefiting from the RV constraints, we select
from this analysis the FENRIR 2 modes, 4 harm, Matérn 1/2 activity model.
Indeed, for the most model-sensitive planets (e, f, and g) this model exhibits
large uncertainties on the mass posteriors and strong prior-sensitivity which
reflects well the overall lack of constraint we have on these masses. However,
this does not mean that this an intrinsically better model for stellar
activity. In particular, we noticed that the rotation period is not well
constrained with this model and reaches the upper-bound of the prior interval.
We stress out here the need for additional RV measurements to better constrain
the masses and activity model.

##  Appendix D Adding a planet in the chain

Figure 12:  Best solutions of the joint photo-dynamical + RV fit when adding
an hypothetical planet $h$ near various MMR with respect to planet $g$.

Here we check if additional planets in the resonant chain, on an external
orbit to the known planet g, could either better explain the available data,
or invalidate the masses that we found for the 6-planet solution. We consider
a set of first- and second-order mean-motion resonances and compute what their
orbital period should be to fit in the resonant chain. For each of these
potential periods for planet $h$, we ran 10 fits of the individual transit
timings shown in Figs. 4 and 5. Each of these fits were initialised as follow:
the orbital parameters and masses of the planets $c$ to $g$ were set to the
best fit of the photo-dynamical analysis, the mean longitude of planet $h$ was
randomly picked in the [$0^{\circ}$,$360^{\circ}$] range, the period was
randomly picked in order to continue the chain with a super-period of $260\pm
20$ days, see eq.C.6 of L21. $k_{h}=e_{h}\cos\varpi_{h}$ and
$h_{h}=e_{h}\sin\varpi_{h}$ were initially set at $3e-4$ and $M_{h}/M_{\sun}$
was initially set to $6*10^{-7}$. The set of priors is identical to the
highmass case described in section 3.2. Once these fits converged, the best
solution of each fit is used to initialise a joint photo-dynamical+RV fit,
identical in all point to those presented in section 3.2, only adding the
additional planet.

In Fig. 12, we show the best solution of each of these fits, as a function of
the MMR near which the $g$-$h$ pair is, the mass of planet $h$, and the
logprob of the solution. From these, we find that several resonances have
similar probabilities, with the general trend that the larger $P_{h}$, the
more massive $m_{h}$ is. Such degeneracy is well understood when considering
the TTV induced at the super-period for nearly-resonant pair of planets (e.g.
Lithwick et al. 2012; Hadden & Lithwick 2016). However, there are additional
arguments that could have allowed us to favour some solutions with respect to
others. Indeed, small period ratios can also lead to short-term sawtooth-like
TTVs known as chopping (Deck & Agol 2015, e.g.), that could better explain the
TTVs of planet $g$. In addition, some of the tried periods for $h$ are also
close to significant MMR with planet $f$: for example, $g$ and $h$ close to a
$3:2$ or a $5:4$, MMR implies that $f$ and $h$ are close to a $2:1$ or a $5:3$
MMR, which could also impact the overall goodness of the fit.

Our best solutions are for an additional planet between a 5/4 and 3/2 MMR with
planet $g$, and masses of the order of 1$M_{Earth}$ or less. We would like to
point out that the parameter space for this potential additional planet is
much larger than the ones explored here, and therefore we cannot exclude that
an additional more massive planet is orbiting TOI-178, especially if it is not
in direct continuation of the resonant chain. Nonetheless, this preliminary
study allows us to draw two conclusions: there are no obvious signals in the
available data for an additional planet in the chain, and the best fits of the
available data with the additional planet do not significantly impact the
derived mass for the known planets in the system. Indeed, we aggregated the
mass of the planets $b$ to $g$ for the best fits of the configurations in
which the added planet has a period in either 5/4, 4/3, 7/5 or 3/2 MMR with
planet $g$. These aggregated results are represented in cyan in the Fig. 9. As
one can see, these aggregated masses remain close to the 6-planet solution
with the highmass prior.