11institutetext: Dipartimento di Fisica e Astronomia “Galileo Galilei”,
Università di Padova, Vicolo dell’Osservatorio 3, IT-35122, Padova, Italy
11email<EMAIL_ADDRESS>22institutetext: INAF - Osservatorio
Astronomico di Padova, Vicolo dell’Osservatorio 5, IT-35122, Padova, Italy
33institutetext: INAF - Osservatorio Astronomico di Palermo, Piazza del
Parlamento 1, I-90134, Palermo, Italy 44institutetext: INAF - Osservatorio
Astronomico di Trieste, Via Giambattista Tiepolo, 11, I-34131, Trieste (TS),
Italy 55institutetext: INAF - Osservatorio Astrofisico di Torino, via
Osservatorio 20, I-10025, Pino Torinese, Italy 66institutetext: Leibniz-
Institut für Astrophysik Potsdam (AIP), An der Sternwarte 16, 14482 Potsdam,
Germany 77institutetext: INAF – Osservatorio Astronomico di Roma, Via Frascati
33, 00078, Monte Porzio Catone (Roma), Italy 88institutetext: Department of
Physics and Astronomy, Vanderbilt University, Nashville, TN 37235, USA
99institutetext: Department of Physics, Fisk University, Nashville, TN 37208,
USA 1010institutetext: SUPA, School of Physics & Astronomy, University of St
Andrews, North Haugh, St Andrews KY169SS, UK; Centre for Exoplanet Science,
University of St Andrews, North Haugh, St Andrews KY169SS, UK
1111institutetext: Center for Astrophysics — Harvard & Smithsonian, 60 Garden
Street, Cambridge, MA, 02138, USA 1212institutetext: Space Science and
Astrobiology Division, NASA Ames Research Center, MS 245-3, Moffett Field, CA
94035, USA 1313institutetext: SETI Institute, Mountain View, CA 94043 USA/NASA
Ames Research Center, Moffett Field, CA 94035 USA 1414institutetext: NASA Ames
Research Center, Moffett Field, CA 94035, USA 1515institutetext: INAF -
Osservatorio Astrofisico di Catania, Oss. Astr. Catania, via S. Sofia 78,
95123 Catania Italy 1616institutetext: Department of Astrophysical Sciences,
Princeton University, Princeton, NJ 08544, USA 1717institutetext: ESO, Karl-
Schwarzschild-str. 2, 85748 Garching, Germany 1818institutetext: Sternberg
Astronomical Institute Lomonosov Moscow State University 1919institutetext:
Instituto de Astrofísica de Canarias (IAC), 38205 La Laguna, Tenerife, Spain
2020institutetext: Departamento de Astrofísica, Universidad de La Laguna
(ULL), Avd. Astrofísico Francisco Sánchez s/n, E-38206 La Laguna, Tenerife,
Spain 2121institutetext: Astrobiology Research Unit, Université de Liège, 19C
Allée du 6 Août, 4000 Liège, Belgium 2222institutetext: Department of Earth,
Atmospheric and Planetary Science, Massachusetts Institute of Technology, 77
Massachusetts Avenue, Cambridge, MA 02139, USA 2323institutetext: Acton Sky
Portal (private observatory), Acton, MA, USA 2424institutetext: INAF –
Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807 Merate (LC), Italy
2525institutetext: McDonald Observatory and Center for Planetary Systems
Habitability; The University of Texas, Austin Texas USA 2626institutetext:
Flatiron Research Fellow 2727institutetext: Center for Computational
Astrophysics, Flatiron Institute, 162 Fifth Avenue, New York, NY 10010, USA
2828institutetext: Department of Physics and Kavli Institute for Astrophysics
and Space Research, Massachusetts Institute of Technology, Cambridge, MA
02139, USA 2929institutetext: Fundación Galileo Galilei – INAF, Rambla José
Ana Fernandez Pérez 7, 38712 Breña Baja, TF, Spain 3030institutetext: NASA
Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA
3131institutetext: Astronomical Institute, Czech Academy of Sciences, Fričova
298, 251 65, Ondřejov, Czech Republic 3232institutetext: Department of
Astronomy & Astrophysics, University of Chicago, Chicago, IL 60637, USA
3333institutetext: Department of Astronomy, Wellesley College, Wellesley, MA
02481, USA 3434institutetext: Komaba Institute for Science, The University of
Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan 3535institutetext:
Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan
3636institutetext: Institute of Astronomy, Faculty of Physics, Astronomy and
Informatics, Nicolaus Copernicus University, Grudzia̧dzka 5, 87-100 Toruń,
Poland 3737institutetext: Royal Astronomical Society, Burlington House,
Piccadilly, London W1J 0BQ, UK 3838institutetext: Department of Physics and
Kavli Institute for Astrophysics and Space Research, Massachusetts Institute
of Technology, Cambridge, MA 02139, USA 3939institutetext: Department of
Aeronautics and Astronautics, MIT, 77 Massachusetts Avenue, Cambridge, MA
02139, USA 4040institutetext: Department of Physics, University of Warwick,
Gibbet Hill Road, Coventry CV4 7AL, UK 4141institutetext: Dipartimento di
Matematica e Fisica, Università Roma Tre, Via della Vasca Navale 84, 00146
Roma, Italy 4242institutetext: Department of Astronomy, The University of
Texas at Austin, Austin, TX 78712, USA 4343institutetext: NASA Sagan Fellow
4444institutetext: Dipartimento di Fisica, Universitá degli Studi di Torino,
via Pietro Giuria 1, I-10125, Torino, Italy

# The GAPS programme at TNG ††thanks: Based on observations made with the
Italian Telescopio Nazionale Galileo (TNG) operated by the Fundación Galileo
Galilei (FGG) of the Istituto Nazionale di Astrofisica (INAF) at the
Observatorio del Roque de los Muchachos (La Palma, Canary Islands, Spain).

XLIX. TOI-5398, the youngest compact multi-planet system composed of an inner
sub-Neptune and an outer warm Saturn

G. Mantovan ${}^{\href https://orcid.org/0000-0002-6871-6131}$ The GAPS
programme at TNG ††thanks: Based on observations made with the Italian
Telescopio Nazionale Galileo (TNG) operated by the Fundación Galileo Galilei
(FGG) of the Istituto Nazionale di Astrofisica (INAF) at the Observatorio del
Roque de los Muchachos (La Palma, Canary Islands, Spain).The GAPS programme at
TNG ††thanks: Based on observations made with the Italian Telescopio Nazionale
Galileo (TNG) operated by the Fundación Galileo Galilei (FGG) of the Istituto
Nazionale di Astrofisica (INAF) at the Observatorio del Roque de los Muchachos
(La Palma, Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). L. Malavolta${}^{\href
https://orcid.org/0000-0002-6492-2085}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). S. Desidera${}^{\href
https://orcid.org/0000-0001-8613-2589}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). T. Zingales${}^{\href
https://orcid.org/0000-0001-6880-5356}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). L. Borsato${}^{\href
https://orcid.org/0000-0003-0066-9268}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). G. Piotto${}^{\href
https://orcid.org/0000-0002-9937-6387}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). A. Maggio${}^{\href
https://orcid.org/0000-0001-5154-6108}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). D. Locci${}^{\href
https://orcid.org/0000-0002-9824-2336}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). D. Polychroni${}^{\href
https://orcid.org/0000-0002-7657-7418}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). D. Turrini${}^{\href
https://orcid.org/0000-0002-1923-7740}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). M. Baratella${}^{\href
https://orcid.org/0000-0002-1027-5003}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). K. Biazzo${}^{\href
https://orcid.org/0000-0002-1892-2180}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). D. Nardiello${}^{\href
https://orcid.org/0000-0003-1149-3659}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). K. Stassun${}^{\href
https://orcid.org/0000-0002-3481-9052}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). V. Nascimbeni${}^{\href
https://orcid.org/0000-0001-9770-1214}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). S. Benatti${}^{\href
https://orcid.org/0000-0002-4638-3495}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). A. Anna John${}^{\href
https://orcid.org/0000-0002-1715-6939}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). C. Watkins${}^{\href
https://orcid.org/0000-0001-8621-6731}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). A. Bieryla${}^{\href
https://orcid.org/0000-0001-6637-5401}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). J. J. Lissauer${}^{\href
https://orcid.org/0000-0001-6513-1659}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). J. D. Twicken${}^{\href
https://orcid.org/0000-0002-6778-7552}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). A. F. Lanza${}^{\href
https://orcid.org/0000-0001-5928-7251}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). J. N. Winn${}^{\href
https://orcid.org/0000-0002-4265-047X}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). S. Messina${}^{\href
https://orcid.org/0000-0002-2851-2468}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). M. Montalto${}^{\href
https://orcid.org/0000-0002-7618-8308}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). A. Sozzetti${}^{\href
https://orcid.org/0000-0002-7504-365X}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). H. Boffin${}^{\href
https://orcid.org/0000-0002-9486-4840}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). D. Cheryasov${}^{\href
https://orcid.org/0009-0003-4203-9667}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). I. Strakhov${}^{\href
https://orcid.org/0000-0003-0647-6133}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). F. Murgas${}^{\href
https://orcid.org/0000-0001-9087-1245}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). M. D’Arpa${}^{\href
https://orcid.org/0009-0004-5914-7274}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). K. Barkaoui${}^{\href
https://orcid.org/0000-0003-1464-9276}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). P. Benni${}^{\href
https://orcid.org/0000-0001-6981-8722}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). A. Bignamini${}^{\href
https://orcid.org/0000-0002-5606-6354}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). A. Bonomo${}^{\href
https://orcid.org/0000-0002-6177-198X}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). F. Borsa${}^{\href
https://orcid.org/0000-0003-4830-0590}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). L. Cabona${}^{\href
https://orcid.org/0000-0002-5130-4827}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). A. C. Cameron${}^{\href
https://orcid.org/0000-0002-8863-7828}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). R. Claudi${}^{\href
https://orcid.org/0000-0001-7707-5105}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). W. Cochran${}^{\href
https://orcid.org/0000-0001-9662-3496}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). K. A. Collins${}^{\href
https://orcid.org/0000-0001-6588-9574}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). M. Damasso${}^{\href
https://orcid.org/0000-0001-9984-4278}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). J. Dong${}^{\href
https://orcid.org/0000-0002-3610-6953}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). M. Endl${}^{\href
https://orcid.org/0000-0002-7714-6310}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). A. Fukui${}^{\href
https://orcid.org/0000-0002-4909-5763}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). G. Fűrész${}^{\href
https://orcid.org/0000-0001-8467-9767}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). D. Gandolfi${}^{\href
https://orcid.org/0000-0001-8627-9628}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). A. Ghedina${}^{\href
https://orcid.org/0000-0003-4702-5152}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). J. Jenkins${}^{\href
https://orcid.org/0000-0002-4715-9460}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). P. Kabáth${}^{\href
https://orcid.org/0000-0002-1623-5352}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). D. W. Latham${}^{\href
https://orcid.org/0000-0001-9911-7388}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). V. Lorenzi${}^{\href
https://orcid.org/0000-0002-1958-9930}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). R. Luque${}^{\href
https://orcid.org/0000-0002-4671-2957}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). J. Maldonado${}^{\href
https://orcid.org/0000-0002-2218-5689}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). K. McLeod${}^{\href
https://orcid.org/0000-0001-9504-1486}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). M. Molinaro${}^{\href
https://orcid.org/0000-0001-5028-6041}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). N. Narita${}^{\href
https://orcid.org/0000-0001-8511-2981}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). G. Nowak${}^{\href
https://orcid.org/0000-0002-7031-7754}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). J. Orell-Miquel${}^{\href
https://orcid.org/0000-0003-2066-8959}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). E. Pallé${}^{\href
https://orcid.org/0000-0003-0987-1593}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). H. Parviainen${}^{\href
https://orcid.org/0000-0001-5519-1391}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). M. Pedani${}^{\href
https://orcid.org/0000-0002-5752-6260}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). S. N. Quinn${}^{\href
https://orcid.org/0000-0002-8964-8377}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). H. Relles${}^{\href
https://orcid.org/0009-0009-5132-9520}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). P. Rowden${}^{\href
https://orcid.org/0000-0002-4829-7101}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). G. Scandariato${}^{\href
https://orcid.org/0000-0003-2029-0626}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). R. Schwarz${}^{\href
https://orcid.org/0000-0001-8227-1020}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). S. Seager${}^{\href
https://orcid.org/0000-0002-6892-6948}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). A. Shporer${}^{\href
https://orcid.org/0000-0002-1836-3120}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). A. Vanderburg${}^{\href
https://orcid.org/0000-0001-7246-5438}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain). T. G. Wilson${}^{\href
https://orcid.org/0000-0001-8749-1962}$ The GAPS programme at TNG ††thanks:
Based on observations made with the Italian Telescopio Nazionale Galileo (TNG)
operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).The GAPS programme at TNG ††thanks: Based on
observations made with the Italian Telescopio Nazionale Galileo (TNG) operated
by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di
Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma,
Canary Islands, Spain).

(Compiled: )

###### Abstract

Context. Short-period giant planets ($P$ $\lesssim$ 10 days, $M_{\rm p}$ ¿ 0.1
$M_{J}$) are frequently found to be solitary compared to other classes of
exoplanets. Small inner companions to giant planets with $P$ $\lesssim$ 15
days are known only in five compact systems: WASP-47, Kepler-730, WASP-132,
TOI-1130, and TOI-2000. Here, we report the confirmation of TOI-5398, the
youngest compact multi-planet system composed of a hot sub-Neptune (TOI-5398
c, $P_{\rm c}$ = 4.77271 days) orbiting interior to a short-period Saturn
(TOI-5398 b, $P_{\rm b}$ = 10.590547 days) planet, both transiting around a
650 $\pm$ 150 Myr G-type star.

Aims. As part of the Global Architecture of Planetary System (GAPS) Young
Object project, we confirmed and characterised this compact system, measuring
the radius and mass of both planets, thus constraining their bulk composition.

Methods. Using multidimensional Gaussian processes, we simultaneously modelled
stellar activity and planetary signals from TESS Sector 48 light curve and our
HARPS-N radial velocity time series. We have confirmed the planetary nature of
both planets, TOI-5398 b and TOI-5398 c, alongside a precise estimation of
stellar parameters.

Results. Through the use of astrometric, photometric, and spectroscopic
observations, our findings indicate that TOI-5398 is a young, active G dwarf
star (650 $\pm$ 150 Myr), with a rotational period of $P_{\rm rot}$ = 7.34
days. The transit photometry and radial velocity measurements enabled us to
measure both the radius and mass of planets b, $R_{b}$ = 10.30$\pm$0.40
$R_{\oplus}$, $M_{b}$ = 58.7$\pm$5.7 $M_{\oplus}$, and c, $R_{c}$ = 3.52 $\pm$
0.19 $R_{\oplus}$, $M_{c}$ = 11.8$\pm$4.8 $M_{\oplus}$. TESS observed TOI-5398
during sector 48 and no further observations are planned in the current
Extended Mission, making our ground-based light curves crucial for ephemeris
improvement. With a Transmission Spectroscopy Metric value around 300,
TOI-5398 b is the most amenable warm giant (10 ¡ $P$ ¡ 100 days) for JWST
atmospheric characterisation.

###### Key Words.:

planetary systems – planets and satellites: fundamental parameters – stars:
fundamental parameters – stars: individual: BD+37 2118 – techniques:
photometric – techniques: radial velocities – planet-star interactions

##  1 Introduction

Multi-planet systems give the unique opportunity to investigate comparative
planetary science and understand the interactions and processes between their
planets (e.g., Dragomir et al. 2019; Kostov et al. 2019; Lacedelli et al.
2021; Trifonov et al. 2023). By studying the relative planet sizes and orbital
separations, the obliquity between the planetary orbital planes and the
stellar rotation axis, and other parameters, we can constrain the formation
and evolution processes of these systems (see, e.g., Mancini et al. 2022;
Grouffal et al. 2022). The precise measurement of the orbital architecture and
the bulk compositions of the planets is essential to fully characterise
similar systems. In this context, high-precision radial velocities (RVs) are
critical to deriving masses and eccentricities, which, combined with the radii
from transit, allow measuring precise inner bulk densities and, finally,
exploring the differences in planetary structure and evolution.

The architecture of multi-planet systems exhibits a significant amount of
diversity, with planets spanning the whole range of masses and grouped in a
wide variety of dynamical configurations. However, short-period giant planets
($P$ $\lesssim$ 10 days) are typically isolated planets compared to other
classes of exoplanets (Huang et al., 2016), and usually, their companions, if
present, are massive long-period planets with $P$ ¿ 200 days (Knutson et al.,
2014; Schlaufman & Winn, 2016). Compactness is a rare feature among known
multi-planet systems with inner giant planets. Moreover, there is a scarcity
of known compact systems composed of a close-orbit outer jovian-size planet
and an inner orbit small-size planet (e.g., Hord et al. 2022). Within this
family of unique planetary systems, we have WASP-47 (Hellier et al., 2012),
Kepler-730 (Zhu et al., 2018; Cañas et al., 2019), TOI-1130 (Huang et al.,
2020; Korth et al., 2023), TOI-2000 (Sha et al., 2022), and WASP-132 (Hord et
al., 2022).

Studying planets younger than 1 Gyr is crucial for understanding the
mechanisms at play during the early stages of planetary formation and
evolution, such as orbital migration, atmospheric evaporation, planetary
impacts, and so on (see, e.g., Terquem & Papaloizou, 2007; Bonomo et al.,
2019). However, it is challenging to identify and model such systems, because
of the magnetic activity of the host star. In fact, both the photometric and
spectroscopic time series are affected by the variations in star flux caused
by the numerous star spots, faculae, faculae network, and strong magnetic
activity on the stellar surface. Despite the stellar activity that hides
planetary signals, researchers have discovered planets orbiting members of
young stellar clusters (Malavolta et al., 2016; Newton et al., 2019; Mann et
al., 2020; Bouma et al., 2020), young stellar associations and moving groups
(Benatti et al., 2019; Damasso et al., 2023), and even young field stars
(Desidera et al., 2022). There is an increasing number of young exoplanets
with well-constrained ages, radii, and masses. Within the Global Architecture
of Planetary Systems (GAPS, Covino et al. 2013; Carleo et al. 2020) Young
Objects (YO) long-term program at Telescopio Nazionale Galileo (TNG), many
systems with young transiting planets, first identified by space missions such
as Kepler (Borucki et al., 2010), K2 (Howell et al., 2014), and TESS
(Transiting Exoplanet Survey Satellite, Ricker et al. 2015), are being
characterised through intense RV monitoring (e.g., Carleo et al. 2021;
Nardiello et al. 2022).

Among the systems under intensive scrutiny by GAPS, the moderately young
($\sim$ 650 Myr) solar-analogue star TOI-5398 (BD+37 2118) is of special
interest. In this paper, we characterise the compact multi-planet system
orbiting this star using a combination of TESS and ground-based photometry,
and RVs collected with the High Accuracy Radial velocity Planet Searcher
(HARPS-N, Cosentino et al. 2012) spectrograph (Section 2). The TESS pipeline
identified two candidate exoplanets: a transiting sub-Neptune (TOI-5398.02,
$P\sim$ 4.77 d) and a giant planet ($P\sim$ 10.59 d) that has been thoroughly
validated by Mantovan et al. (2022) and labelled TOI-5398 b. In Section 3, we
report the stellar properties determined using two independent methods.
Section 4 reports the procedures used to identify and confirm the two planets
in the system by outlining the detailed modelling of photometry and RV. In
Section 5, we discuss our results, provide suggestions for follow-up
observations, highlight the rarity of our system, and call attention to the
exquisite suitability of TOI-5398 b for future atmospheric characterisation.
Concluding remarks are in Section 6.

##  2 Observations

###  2.1 TESS Photometry

The TESS mission observed TOI-5398 (TIC 8260536) at 2 min cadence in Sector 48
from 28 January 2022 to 26 February 2022. In particular, we considered the
TESS Science Processing Operations Center (SPOC; Jenkins et al. 2016) Simple
Aperture Photometry (SAP; Twicken et al. 2010; Morris et al. 2020) flux light
curve and corrected it for time-correlated instrumental signatures. We
corrected the light curve using Cotrending Basis Vectors (CBVs) extracted
through the algorithm by Nardiello et al. (2020) as well as all SAP light
curves in the same Camera and CCD as TOI-5398. We did this rather than using
the Presearch Data Conditioning Simple Aperture Photometry (PDCSAP) light
curve from the SPOC because, with the latter, the target experienced numerous
systematic effects (Smith et al., 2012; Stumpe et al., 2012, 2014) as a result
of over-corrections and/or injection of spurious signals.

In this study, we used light curve points flagged with cadence quality flag
values equal to 0 and associated with local background values ¡ 4$\sigma$
above its mean111Mean of the sigma-clipped data, computed using Astropy. value
along the light curve. The only exceptions to this rule are the points before
BTJD222BTJD = $\mathrm{BJD_{TDB}}-2457000.0$ = 2609, which are flagged with a
quality flag value equal to 32768333“Insufficient Targets for Error Correction
Exclude”
https://outerspace.stsci.edu/display/TESS/2.0+-+Data+Product+Overview. and had
local background values up to 7$\sigma$ above the mean. We did this to
preserve one of the TOI-5398.02’s four genuine transits. The corrected light
curve shown in Fig. 1 displays a clear modulation that we attribute to the
stellar rotation. Therefore, to identify a rotation period of TOI-5398, we
computed the Generalised Lomb-Scargle (GLS) periodogram (Zechmeister &
Kürster, 2009) of the aforementioned light curve. The periodogram reveals the
most powerful peak at 7.18 $\pm$ 0.21 days, see Fig. 2. The link between this
peak and the rotational period of the star will be discussed in Sec. 3.5. The
light curve also exhibits two clear dips at BTJD $\sim$ 2616 and 2627, which
have been shown to be caused by a sub-stellar companion by Mantovan et al.
(2022).

Figure 1: SAP-corrected light curve of TOI-5398, observed in Sector 48. The
local background value is colour-coded. Yellow points – with local background
values $>$ 4$\sigma$ above its mean value along the light curve – have been
excluded from all analyses. Figure 2: GLS periodogram of the TESS SAP
photometry corrected for time-correlated instrumental signatures. The vertical
lines indicate the period at 7.18 days and its first harmonic.

###  2.2 ASAS-SN

We downloaded almost three years of archival data from ASAS-SN (Shappee et
al., 2014; Kochanek et al., 2017), spanning from October 2020 to May 2023. The
ASAS-SN images have a resolution of 8 arcsec/pixel ($\sim$15$\arcsec$ FWHM
PSF), and the observations of TOI-5398 were conducted in the Sloan $g-$band.
We checked the light curve to further constrain the stellar rotation period
identified from the TESS photometry. We extracted the GLS periodogram, which
reveals the most powerful peak with a period of 7.34$\pm$0.01 days, see Fig.
3.

Figure 3: GLS periodogram of the ASAS-SN photometry. The vertical line
indicates the stellar rotation period at 7.34 days.

###  2.3 Photometric Follow-up (TASTE)

A partial transit of TOI-5398 b was observed on January 31st, 2023 by The
Asiago Search for Transit timing variations of Exoplanets (TASTE) programme, a
long-term campaign to monitor transiting planets (Nascimbeni et al., 2011). To
prevent saturation and increase the photometric accuracy, the AFOSC camera,
mounted on the Copernico 1.82-m telescope at the Asiago Astrophysical
Observatory in northern Italy, was purposely defocused up to $8\arcsec$ FWHM.
A Sloan $r’$ filter was used to acquire 2916 frames with a constant exposure
time of 5 s. A pre-selected set of suitable reference stars was always imaged
in the same field of view in order to apply precise differential photometry.
At the end of the series, the sky transparency was highly variable, and thick
clouds passed by during the off-transit, leaving us with a subset of 2721
images with a good S/N ratio.

We reduced the Asiago data with the STARSKY code (Nascimbeni et al., 2013), a
software pipeline specifically designed to perform differential photometry on
defocused images. After a standard correction through bias and flat-field
frames, the size of the circular apertures and the weights assigned to each
reference star were automatically chosen by the code to minimise the
photometric scatter of our target. The time stamps were consistently converted
to the $\mathrm{BJD_{TDB}}$ standard following Eastman et al. (2010), as done
for the following photometric data sets as well.

###  2.4 Photometric Follow-up (TFOP)

The TESS pixel scale is $\sim 21\arcsec$/ pixel and photometric apertures
typically extend out to roughly 1 arcminute, generally causing multiple stars
to blend in the TESS aperture. To determine the true source of transit signals
in the TESS data, improve the transit ephemerides, monitor for transit timing
variations, and check the SPOC pipeline transit depth after accounting for the
crowding metric, we conducted ground-based lightcurve follow-up observations
of the field around TOI-5398 as part of the TESS Follow-up Observing
Program444https://tess.mit.edu/followup Sub Group 1 (TFOP; Collins, 2019). We
used the TESS Transit Finder, which is a customised version of the Tapir
software package (Jensen, 2013), to schedule our transit observations. All the
image data were calibrated, and all photometric data were extracted using
AstroImageJ unless otherwise noted. We used circular photometric apertures
centred on TOI-5398, and checked also the flux from the nearest known
neighbour in the Gaia DR3 (Gaia Collaboration et al., 2023) and TICv8
catalogues (TIC 8260534), which is $\sim 37\arcsec$ north-east of TOI-5398.

####  2.4.1 LCOGT

We observed four and two partial transit windows of TOI-5398 b and
TOI-5398.02, respectively, in Pan-STARRS $z$-short band using the Las Cumbres
Observatory Global Telescope (LCOGT; Brown et al., 2013) 1.0 m network nodes
at Teide Observatory (TEID) on the island of Tenerife, McDonald Observatory
(MCD) near Fort Davis, Texas, United States, and Cerro Tololo Inter-American
Observatory in Chile (CTIO). We observed an ingress and egress of TOI-5398 b
on UTC 2022 April 21 from TEID and MCD, respectively, an egress on UTC 2023
March 04 from TEID, and an ingress on UTC 2023 March 15 from MCD. We observed
an egress of TOI-5398.02 on UTC 2022 April 02 from CTIO, and an ingress on UTC
2022 May 10 from MCD. The 1 m telescopes are equipped with $4096\times 4096$
Sinistro cameras having an image scale of $0.389\arcsec$/ pixel, resulting in
a $26\arcmin\times 26\arcmin$ field of view. The images were calibrated by the
standard LCOGT BANZAI pipeline (McCully et al., 2018). We used circular
photometric apertures having radii in the range $5.1\arcsec$ to $7.4\arcsec$,
which excluded all of the flux from TIC 8260534.

####  2.4.2 Acton Sky Portal

A full transit window of TOI-5398 b was observed in Sloan $i^{\prime}$ band on
UTC 2022 April 21 from the Acton Sky Portal private observatory in Acton, MA,
USA. The 0.36 m telescope is equipped with a SBIG Alumna CCD4710 camera having
an image scale of $1.0$ arcsec/ pixel, resulting in a $17.1\times 17.1$ arcmin
field of view. We used a circular photometric aperture with radius $9\arcsec$,
which excluded all of the flux from TIC 8260534.

####  2.4.3 Whitin

A full transit window of TOI-5398 b was observed in Sloan $r^{\prime}$ band on
UTC 2022 April 21 using the Whitin observatory 0.7 m telescope in Wellesley,
MA, USA. The $2048\times 2048$ FLI ProLine PL23042 detector has an image scale
of $0.68$$\arcsec$/pixel, resulting in a $23.2\times 23.2$ arcmin field of
view. We used a circular photometric aperture with a radius of $8.2$ arcsec,
which excluded all of the flux from TIC 8260534.

####  2.4.4 KeplerCam

We obtained photometric observations using the KeplerCam CCD on the 1.2m
telescope at the Fred Lawrence Whipple Observatory (FLWO) at Mount Hopkins,
Arizona to observe an egress of TOI-5398 b. Observations were taken in the
Sloan z′ band on UT April 21, 2022. KeplerCam is a $4096\times 4096$ Fairchild
detector which has a field of view of 23$\arcmin$x23$\arcmin$ and an image
scale of 0.672$\arcsec$/pixel when binned by 2. We used circular photometric
apertures with radius $6.7\arcsec$, which excluded all of the flux from TIC
8260534.

####  2.4.5 MuSCAT2

TOI-5398 was observed by MuSCAT2 on the night of 30 March 2022. MuSCAT2 is a
multi-band imager (Narita et al., 2019) mounted on the Telescopio Carlos
Sánchez (TCS, 1.52 m) at Teide Observatory, Spain. The instrument is capable
of obtaining simultaneous images in Sloan $g^{\prime}$, $r^{\prime}$,
$i^{\prime}$, and $z_{s}$ bands with little readout time. Each camera has a
field of view of $7.4\arcmin\times 7.4\arcmin$ and a pixel scale of
0.44$\arcsec$/ pixel.

The observation was made with the telescope defocused to avoid the saturation
of the target. On the night of 30 March 2022, the $i^{\prime}$-band camera
presented technical issues and could not be used; the exposure times were set
to 10, 5, and 10 seconds in $g^{\prime}$, $r^{\prime}$, and $z_{s}$,
respectively. Standard data reduction, aperture photometry, and transit model
fit including systematic noise was done using the MuSCAT2 pipeline
(Parviainen, 2015; Parviainen et al., 2019).

####  2.4.6 Results

Due to their low S/N and to reduce the computational cost, we decided not to
include light curves observed with LCOGT-CTIO and Whitin in our analysis. All
other observations resulted in clear transit detections and the light curve
data are included in the joint modelling in Sect. 4.2 of this work.

###  2.5 HARPS-N Spectroscopic Follow-up

We collected observations of TOI-5398 with HARPS-N at TNG spanning the period
between May 2022 and June 2023 and obtained a total of 86 spectra, with
exposure times ranging from 900 to 1200 s. These spectra cover the wavelength
range 383-693 nm with a resolving power of R $\sim$ 115 000. We performed the
observations within the framework of the GAPS project and incorporated 10
hours coming from a time-sharing agreement (proposal A46TAC_32, PI: G.
Mantovan). Additionally, we included eight off-transit spectra from the DDT
proposal A46DDT4 (PI: G. Mantovan) in our analysis, which we binned from 600
to 1200 s of exposure time. Six spectra taken in Spanish time (CAT22A_48, PI:
E. Pallé) are also included in a comprehensive analysis of the object. We
excluded from our final analysis one of these six spectra as it was collected
during the in-transit phase of planet b. We decided to proceed in this way
after considering the expected amplitude of the Rossiter-McLaughlin (Ohta et
al., 2005) signature, which is further discussed in Sect. 5.4. We report the
details about the observations and typical S/N in Table 1.

Table 1: Observations from TESS and HARPS-N summarised. TOI-5398
---
Dataset | Parameter | Value
TESS | Sector | 48
Camera | 1
CCD | 1
HARPN-N | N∘ spectra | 86
Time-span (days) | 439
$\sigma_{\rm RV}$ (m s-1) | 29
$\langle\rm RV_{\rm err}\rangle$ (m s-1) | 3.5
$\langle\rm S/N\rangle_{5460\AA}$ | 64

We reduced the data collected using the HARPS-N Data Reduction Software (DRS
3.7.0), and we computed the RV through the Cross-Correlation Function (CCF)
method (Pepe et al. 2002 and references therein). With this method, the
scientific spectra are cross-correlated with a binary mask describing the
typical features of a star with a chosen spectral type. We used a G2 mask for
TOI-5398. The resulting CCFs provide us with a representation of the mean line
profile of each spectrum. However, the high levels of stellar activity might
distort the core of the average line profile, and the stellar rotation
broadens the line. Therefore, we needed to proceed with care in selecting the
half-window for the evaluation of the CCF and use a width large enough to
include the continuum when fitting the CCF profile (see Damasso et al. 2020).
We decided to use the G2 mask with a half-window of 40 km s-1 (instead of the
default value of 20 km s-1) and reprocessed our data using the DRS version
implemented through the YABI workflow interface (Hunter et al., 2012) at the
Italian Center for Astronomical Archives555https://www.ia2.inaf.it/. We
obtained RVs with a dispersion of a few tens of m s-1 ($\sim$29 m s-1), while
their internal errors are approximately a few m s-1 ($\sim$3.5 m s-1).

To assess the jitter in the RV series caused by the stellar activity, we
additionally extracted a set of activity indices. The value of the CCF
bisector span (BIS), as well as the CCF’s full width at half-maximum (FWHM)
depth and the CCF’s equivalent width ($W_{\rm CCF}$, see Collier Cameron et
al. 2019 for further details), are provided by the HARPS-N DRS, while the log
$R^{\prime}_{HK}$ index from the Ca II H&K lines was obtained using a method
available on YABI (based on the prescriptions of Lovis et al. 2011 and
references therein) and using the $(B-V)_{0}$ colour index quoted in Sect. 3.
Finally, we extracted the H$\alpha$ index using the ACTIN 2
Code666https://github.com/gomesdasilva/ACTIN2 (Gomes da Silva et al. 2018,
2021). Figure 4 displays the spectroscopic time series.

Figure 4: Left: HARPS-N spectroscopic time series used in this work. The time
series for RV, BIS, $\log R^{{}^{\prime}}_{\rm HK}$, H$\alpha$, FWHM, and
$W_{\rm CCF}$ are shown in the panels in order from top to bottom. Right: GLS
periodogram of the RVs and the spectroscopic activity indicators under
analysis. The primary peak of each periodogram is indicated by a vertical
orange line. The dotted green lines represent the stellar rotation period
described in Sect. 2.2. The signals along the red dotted vertical lines
correspond to the transit-like signals with periods 4.77 and 10.59 d.

We also obtained medium precise RVs with the Tull spectrograph at the Harlan
J. Smith 2.7m telescope at McDonald Observatory. They can be found in the
Appendix.

###  2.6 High angular resolution data

TOI-5398 was also observed on 2 December 2022 with the speckle polarimeter on
the 2.5 m telescope at the Caucasian Observatory of Sternberg Astronomical
Institute (SAI) of Lomonosov Moscow State University. Speckle polarimeter uses
high–speed low–noise CMOS detector Hamamatsu ORCA–quest (Strakhov et al.,
2023). The atmospheric dispersion compensator was active, which allowed using
the $I_{c}$ band. The respective angular resolution is $0.083\arcsec$, while
the long–exposure atmospheric seeing was $0.6\arcsec$. We did not detect any
stellar companions brighter than $\Delta I_{c}=4.0$ and 6.8 mag at
$\rho=0.25\arcsec$ and $1.0\arcsec$, respectively, where $\rho$ is the
separation between the source and the potential companion (Fig. 5).

Figure 5: High angular resolution speckle imaging of TOI- 5398 in $I_{c}$
filter using SAI-2.5 m telescope.

##  3 The host star

###  3.1 Atmospheric parameters and metallicity

Given the relatively young age of TOI-5398, we analysed the HARPS-N combined
spectrum following the same methodology as, for example, in Nardiello et al.
(2022) and Damasso et al. (2023). Specifically, we adopted an innovative
approach to derive the stellar parameters with the equivalent width (EW)
method using a combination of iron (Fe) and titanium (Ti) lines. In this way,
we avoid issues related to the effect of the stellar activity that can shape
the stellar spectrum at young ages (for a detailed explanation we refer the
reader to Baratella et al., 2020a, b). Our initial guesses of the atmospheric
parameters were estimated with Gaia DR3 (Gaia Collaboration et al., 2016) and
2MASS photometry (Cutri et al., 2003) using the tool colte (Casagrande et al.,
2021) and adopting $E(B-V)=0.008\pm 0.016$ (Lallement et al., 2014; Capitanio
et al., 2017). The photometric estimates vary from $5978\pm 65$ K in $(G_{\rm
BP}-J)$ to $6039\pm 34$ K in $(G_{\rm RP}-H)$. From these initial values and
taking the Gaia parallax into account, we derived $4.44\pm 0.09$ dex as input
surface gravity, while an initial microturbulence $\xi=1.07\pm 0.05$ km s-1
was estimated from the relation by Dutra-Ferreira et al. (2016).

The final stellar parameters ($T_{\mathrm{eff}}$, $\log g$, $\xi$, [Fe/H])
were then derived through the MOOG code (Sneden 1973) and adopting the ATLAS9
grid of model atmospheres with new opacities (Castelli & Kurucz 2003). Our
spectroscopic analysis provides as final atmospheric parameters
$T_{\mathrm{eff}}=6000\pm 75$ K, $\log g=4.44\pm 0.10$ dex and $\xi=1.12\pm
0.12$ km s-1, in excellent agreement with the initial guesses. The derived
iron abundance (computed with respect to the solar Fe abundance as in
Baratella et al. 2020a) is [Fe/H]$=+0.09\pm 0.06$, while the titanium
abundance is [Ti/H]$=+0.08\pm 0.05$, where the errors include the scatter due
to the EWs measurements and the uncertainties in the stellar parameters.
Despite the error bars, there is an indication of a slightly super-solar
metallicity.

###  3.2 Projected rotational velocity

With the same code and model atmospheres described in Sect. 3.1, we adopted
the atmospheric parameters ($T_{\mathrm{eff}}$, $\log g$, $\xi$, [Fe/H])
previously derived to measure the stellar projected rotational velocity
($v\sin{i_{\star}}$). In particular, fixing the macroturbulence velocity to
the value of 3.8 km s-1 from the relationship by Doyle et al. (2014), we
applied the spectral synthesis method within MOOG for three spectral regions
around 5400, 6200, and 6700 Å. Our final value of $v\sin{i_{\star}}$ is
$7.5\pm 0.6$ km s-1. We refer the readers to Biazzo et al. (2022) for the
procedure based on spectral synthesis and the description of the uncertainty
measurement.

###  3.3 Lithium Abundance

Since the lithium line at $\lambda$ 6707.8 Å in the co-added spectrum of
TOI-5398 resulted to be blended with the iron line at $\lambda$ 6707.4 Å, we
applied the spectral synthesis technique as done in Sect. 3.2. Using MOOG and
the Castelli & Kurucz (2003) model atmospheres, we fixed the stellar
parameters and $v\sin{i_{\star}}$ to the values derived in the previous steps.
Then, adopting two different line lists by Carlberg et al. (2012) and Chris
Sneden (priv. comm.) and applying the non-LTE calculations of Lind et al.
(2009), we obtained a lithium abundance of $\log n$(Li)${}_{\rm NLTE}=2.82\pm
0.11$, where the error bar considers uncertainties in the line list, in
stellar parameters and in the definition of the continuum position around the
Li line. With its effective temperature and lithium abundance, the position of
our target in the $\log n$(Li)$-T_{\rm eff}$ appears to be in between that of
the M35 cluster ($\sim$ 200 Myr) and that of the Hyades cluster ($\sim$ 650
Myr, see, e.g., Sestito & Randich 2005, Cummings et al. 2017).

###  3.4 Chromospheric activity

The lower chromosphere Ca II H&K emission was measured on HARPS-N spectra
using YABI (see Sect. 2.5). The average value of the S-index, calibrated in
the Mt. Wilson scale (Baliunas et al., 1995), is 0.325, corresponding to $\log
R^{\prime}_{\rm HK}$ = -4.43 $\pm$ 0.02 (arithmetic average and standard
deviation).

For the $\log R^{\prime}_{\rm HK}$ determination, we adopted $(B-V)_{0}$ =
0.58 mag, which was derived from $T_{\rm eff}$ through Casagrande et al.
(2006) calibration, as the observed values from Tycho2 and APASS suffer from
large uncertainties. Our determination of $\log R^{\prime}_{\rm HK}$ implies a
stellar age of 370 Myr using the activity-age calibration and an expected
rotation period of 4.25 d (corresponding to a gyrochronology age of 260 Myr),
through Mamajek & Hillenbrand (2008) relationships. The difference with the
observed photometric period (see Sect. 3.5) is not significant considering
observational errors and intrinsic scatter of magnetic activity (see e.g. Fig.
7 of Mamajek & Hillenbrand 2008). It is possible that the star is caught by
chance at a level of chromospheric activity higher than the average one or
that the activity is somewhat enhanced by the presence of planetary
companions.

The upper chromosphere H$\alpha$ line, extracted following Sect. 2.5, has a
value of 0.151 $\pm$ 0.002 (arithmetic average and standard deviation).

The tidal evolution of the stellar rotation has a timescale much longer than
the age of the system, even considering an extremely strong tidal interaction
with a stellar modified tidal quality factor $Q^{\prime}_{\star}=10^{5}$
(e.g., Mardling & Lin, 2002). Therefore, we do not expect that tides can
affect the estimate of the stellar age based on gyrochronology.

###  3.5 Rotation period

The extensive datasets we collected allowed us to obtain several estimates of
the rotation period of the star, a key parameter for age determination.

As mentioned in Sect. 2.1 and 2.2, from the analysis of TESS and ASAS-SN light
curves we obtained photometric periodicities of 7.18$\pm$0.21 days and
7.34$\pm$0.01 days, respectively. The larger error of the determination based
on TESS data is due to the short baseline of the monitoring (single sector).

We also exploited the HARPS-N time series of RVs and several spectroscopic
indicators (Sect. 2.5). Figure 4 displays the GLS periodograms, computed for
the frequency range 0.0001 - 0.5 days-1, or 2 - 10000 days. We removed a
linear trend from $\log R^{\prime}_{\rm HK}$ and H$\alpha$ time series before
running the GLS. The RVs, BIS, and FWHM periodograms show a clear peak (orange
line, normalised GLS power $>$ 0.2) close to the first harmonic of the stellar
rotation period detected using the ASAS-SN photometry (7.34 days, dotted green
lines). As the period is compatible with the modulation observed in the
photometry and consistently recovered in both the radial velocity and the
activity indexes, we associated this signal to the stellar activity. This
result could be a sign that RV variations are dominated by dark spots rather
than the quenching of convective motions in the magnetised regions of the
photosphere, as explained in the study by Lanza et al. (2010). Moreover, the
periodograms of H$\alpha$ and $W_{\rm CCF}$ show peaks consistent with the
photometric period. On the other hand, the second-highest peak in the RVs
periodogram reveals the signal of the validated gas giant planet. We also add
that the detailed modelling of the RV time series with Gaussian processing
regression (Sect. 4.3) yields a value of 7.37$\pm$0.03 days for the rotation
period.

In short, the available photometric and spectroscopic time series consistently
indicate a periodicity of 7.34$\pm$0.05 days, which we interpret as the
rotation period of the star. The shorter periodicity seen in some
spectroscopic indicators is fully compatible with being the first harmonic of
the true periodicity, as observed in several other cases (see, e.g., TOI-1807
Nardiello et al. 2022). Conversely, we think it is unlikely that the true
period is the shorter one (3.66 d), with this periodicity arising from the
presence of active regions of comparable areas on opposite stellar
hemispheres, especially considering the long-time baseline of the ASAS-SN time
series.

###  3.6 Kinematics

Kinematic space velocities were derived adopting Gaia kinematic parameters and
using the formalism by Johnson & Soderblom (1987). The resulting $U$, $V$, and
$W$ (Table 2) are at the boundary of the kinematic space of young stars (age
younger than about 500 Myr, Montes et al., 2001; Maldonado et al., 2010) in
agreement with the other age diagnostics.

TOI-5398 is not a member of known young moving groups and a dedicated search
of comoving objects within a few degrees in the Gaia DR3 catalogue does not
yield convincing candidates.

###  3.7 Age, mass and radius

The indirect methods discussed above point towards an age of a few hundred
Myr, similar to the Hyades and Praesepe open clusters, which have also super-
solar metallicity (Hyades with +0.15 dex, Cummings et al. 2017, and Praesepe
with +0.21 dex, D’Orazi et al. 2020). In the case of a G dwarf star with a few
hundred million years, the most robust age indicator is the rotation period.
An age of 680 Myr is obtained with the Mamajek & Hillenbrand (2008)
calibration for our adopted rotation period. Using different photometric
colours and calibrations (e.g., G-K Messina et al., 2022) yields similar
results. The Lithium abundance is intermediate between members of Hyades
($\sim$ 650 Myr) and M35 ($\sim$ 200 Myr) of similar colours, pointing to a
somewhat younger age. The level of chromospheric activity also suggests a
younger age than the gyrochronology, but the discrepancy of both lithium and
$\log R^{{}^{\prime}}_{\rm HK}$ with the expectations for an age similar to
that of the Hyades cluster is marginal. Finally, kinematic parameters are
fully consistent with an age of a few hundred Myr and the lack of comoving
objects prevents a more precise age estimate. We further add that isochrone
fitting, performed using the param777 http://stev.oapd.inaf.it/cgi-
bin/param_1.3 tool (da Silva et al., 2006) does not add further relevant
information (nominal age 1.6$\pm$1.6 Gyr). We then adopt a system age of
650$\pm$150 Myr from the indirect indicators.

Through the param tool and by imposing the age range allowed by indirect
methods to avoid the inclusion of solutions not compatible with the above
results (Desidera et al., 2015), we derived the stellar mass and radius. We
obtained in this way a stellar mass of 1.146$\pm$0.013 $M_{\odot}$ and a
stellar radius $R=$ 1.051$\pm$0.013 $R_{\odot}$, where the uncertainties are
those provided by the param interface and do not include possible systematics
of the adopted stellar models.

From the combination of $R_{\star}$, $P_{\rm rot}$ and $v\sin{i_{\star}}$, we
infer a system orientation fully compatible with edge-on. Indeed, for the
nominal parameters, $\sin i_{\star}$ is just below unity, and taking error
bars into account we estimate $i_{\star}\geq 69$ deg.

The stellar parameters outlined in this and the preceding subsections serve as
the reference for this study. We present them in Table 2 for reference.

Table 2: Stellar properties of TOI-5398

Parameter | TOI-5398 | Ref
---|---|---
$\alpha$ (J2000) | 10 47 31.09 |  Gaia DR3
$\delta$ (J2000) | +36 19 45.86 |  Gaia DR3
$\mu_{\alpha}$ (mas yr-1) |  1.360$\pm$0.014 |  Gaia DR3
$\mu_{\delta}$ (mas yr-1) |  7.003$\pm$0.012 |  Gaia DR3
RV (km s-1) |  -9.95$\pm$0.30 |  Gaia DR3
$\pi$ (mas) |  7.6190$\pm$0.0143 |  Gaia DR3
$U$ (km s-1) |  4.07$\pm$0.14 |  This paper (Sect. 3.6)
$V$ (km s-1) |  4.90$\pm$0.02 |  This paper (Sect. 3.6)
$W$ (km s-1) |  -8.83$\pm$0.27 |  This paper (Sect. 3.6)
V (mag) |  10.06$\pm$0.03 |  Tycho-2 (Høg et al., 2000)
$B-V$ (mag) | 0.58 |  This paper (Sect. 3.4)
$G$ (mag) |  9.9875$\pm$0.0004 |  Gaia DR3
$G_{BP}-G_{RP}$ (mag) | 0.7582 |  Gaia DR3
$J_{\rm 2MASS}$ (mag) |  9.026$\pm$0.021 | 2MASS
$H_{\rm 2MASS}$ (mag) |  8.772$\pm$0.023 | 2MASS
$K_{\rm 2MASS}$ (mag) |  8.713$\pm$0.018 | 2MASS
$T_{\rm eff}$ (K) |  6000$\pm$75 |  This paper (spec) (Sect. 3.1)
$\log g$ |  4.44$\pm$0.10 |  This paper (Sect. 3.1)
${\rm[Fe/H]}$ (dex) |  +0.09$\pm$0.06 |  This paper (Sect. 3.1)
$E(B-V)$ (mag) |  $\leq$ 0.024aa$a$84th percentile. |  PIC (Montalto et al., 2021)
$S_{\rm MW}$ |  0.325 $\pm$0.008 |  This paper (Sect. 3.4)
$\log R^{{}^{\prime}}_{\rm HK}$ |  -4.43$\pm$0.02 |  This paper (Sect. 3.4)
$v\sin{i_{\star}}$ (km s-1) |  7.5$\pm$0.6 |  This paper (Sect. 3.2)
$P_{\rm rot}$ (d) |  7.34$\pm$0.05 |  This paper (Sect. 3.5)
A(Li) |  2.82$\pm$0.11 |  This paper (Sect. 3.3)
Mass ($M_{\odot}$) |  1.146$\pm$0.013 |  This paper (Sect. 3.7)
Radius ($R_{\odot}$) |  1.051$\pm$0.013 |  This paper (Sect. 3.7)
Age (Myr) |  650$\pm$150 |  This paper (Sect. 3.7)
$i_{\star}$ (deg) |  $\geq$69 |  This paper (Sect. 3.7)
888

###  3.8 Spectral Energy Distribution

As an independent determination of the basic stellar parameters, we performed
an analysis of the broadband spectral energy distribution (SED) of the star
together with the Gaia DR3 parallax (see, e.g., Stassun & Torres, 2021), in
order to determine an empirical measurement of the stellar radius, following
the procedures described in Stassun & Torres (2016); Stassun et al. (2017,
2018). We pulled the $JHK_{S}$ magnitudes from 2MASS, the W1–W4 magnitudes
from WISE (Wright et al., 2010), and the $G_{\rm BP}\,G_{\rm RP}$ magnitudes
from Gaia (Gaia Collaboration et al., 2016), and the FUV and NUV magnitudes
from GALEX (Martin et al., 2005). Together, the available photometry spans the
full stellar SED over the wavelength range 0.2–22 $\mu$m (see Fig. 6).

Figure 6: Spectral energy distribution of TOI-5398. Red symbols represent the
observed photometric measurements, where the horizontal bars represent the
effective width of the passband. Blue symbols are the model fluxes from the
best-fit PHOENIX atmosphere model (black). The Gaia spectrum is overlaid as a
grey swathe and shown in closer detail in the inset plot.

We performed a fit using PHOENIX stellar atmosphere models (Husser et al.,
2013), with the free parameters being the effective temperature ($T_{\rm
eff}$) and metallicity ([Fe/H]), as well as the extinction $A_{V}$, which we
limited to maximum line-of-sight value from the Galactic dust maps of Schlegel
et al. (1998). The resulting fit (Fig. 6) has a best-fit $A_{V}=0.05\pm 0.02$
mag, $T_{\rm eff}=6025\pm 100$ K, [Fe/H] = $0.0\pm 0.5$, with a reduced
$\chi^{2}$ of 1.3 (excluding the FUV and W4 fluxes, which suggest modest
excesses in both the UV and mid-IR). Integrating the (unreddened) model SED
gives the bolometric flux at Earth, $F_{\rm bol}=2.470\pm 0.029\times 10^{-9}$
erg s-1 cm-2. Taking the $F_{\rm bol}$ and $T_{\rm eff}$ together with the
Gaia parallax, gives the stellar radius, $R_{\star}=1.059\pm 0.036$ R⊙. In
addition, we can estimate the stellar mass from the empirical relations of
Torres et al. (2010), obtaining $M_{\star}=1.12\pm 0.07$ M⊙.

Finally, we may estimate the stellar rotation period from the above radius
together with the spectroscopically measured $v\sin{i_{\star}}$, giving
$P_{\rm rot}/\sin i_{\star}=7.1\pm 0.6$ d.

These results are fully compatible with those presented in Sects. 3.1 and 3.7.

##  4 Analysis

###  4.1 Planets detection and vetting tests

Two candidate exoplanets orbiting TOI-5398 were identified in Sector 48 light
curves in both SPOC (Jenkins et al., 2016) and QLP (Huang et al., 2020)
pipeline: one giant and one sub-Neptune ($P$ $\sim$ 10.59 d and $P$ $\sim$
4.77 d, respectively). The SPOC performed a transit search with an adaptive,
noise-compensating matched filter (Jenkins 2002; Jenkins et al. 2010, 2020),
producing Threshold Crossing Events (TCEs) for which an initial limb-darkened
transit model was fitted (Li et al., 2019) and diagnostic tests were conducted
to help assess the planetary nature of the signal (Twicken et al., 2018). The
QLP performed its transit search with the Box Least Squares Algorithm (Kovács
et al., 2002). The two transit signatures passed all TESS data validation
diagnostic tests, and the TESS Science Office issued alerts for TOI-5398.01
(10.59 days) and TOI-5398.02 (4.77 days) on 24 March 2022. The SPOC difference
image centroid offsets (Twicken et al., 2018) localised the transit source for
TOI 5398.01 within 0.41 $\pm$ 2.55 arcsec and for TOI 5398.02 within 2.69
$\pm$ 2.67 arcsec; all TIC v8 (Stassun et al., 2019) objects other than
TOI-5398 were excluded as the source of each transit signature.

Mantovan et al. (2022) thoroughly validated the planetary nature of the giant,
which they label TOI-5398 b, by ruling out any false positive (FP) scenarios
capable of mimicking the observed transit signal. In fact, mainly due to the
low spatial resolution of TESS cameras ($\approx$ 21 arcsec/pixel), some
objects initially identified as sub-stellar candidates might be FPs. As a
result, vetting and validation tests are critical. Therefore, in order to
better understand also the nature of the sub-Neptune candidate, TOI-5398.02,
and to exclude FP scenarios, we followed the procedure adopted in Mantovan et
al. (2022). The latter approach, further described in Appendix A, considers
the major concerns reported in Morton et al. (2023) and ensures reliable
results when using VESPA (Morton, 2012, 2015). This procedure allowed us to
statistically validate the sub-Neptune exoplanet and label it as TOI-5398 c.

###  4.2 Photometry time-series analysis

To characterise the properties of TOI-5398 b and c, we investigated all the
ground-based photometry simultaneously with the two transits of TOI-5398 b and
the four transits of TOI-5398 c observed by TESS in a Bayesian framework using
PyORBIT999https://github.com/LucaMalavolta/PyORBIT. (Malavolta et al. 2016,
Malavolta et al. 2018), a Python package for modelling planetary transits and
radial velocities while simultaneously accounting for stellar activity
effects. The ground-based observations rule out false positive scenarios
caused by BEBs and further confirm the transits of two planetary companions
orbiting TOI-5398.

In the present case, we took our TESS corrected light curve (see Sect. 2.1)
and carefully considered the influence of stellar contamination from neighbour
stars. We verified the stellar dilution by measuring a dilution factor, which
defines the total flux from contaminants that fall into the photometric
aperture divided by the flux contribution of the target star. Following Sect.
2.2.2 of Mantovan et al. (2022), we determined the dilution factor (and its
associated error) by calculating the contribution of the flux that falls into
the TESS aperture for each star. We determined its value to be 0.00735 $\pm$
0.00005, and we imposed it as a Gaussian prior in the modelling. Then, we
selected each space-based transit event from the corrected light curve – and
an out-of-transit part as long as the corresponding transit duration (both
before the ingress and after the egress) – and created a mask that flags each
transit and cuts the corresponding portions of the light curve. The transits
of the two planets do not overlap.

We simultaneously modelled each transit (ground- and space-based) using the
code BATMAN (Kreidberg, 2015), fitting the following parameters: the central
time of transit ($T_{0}$), the planetary-to-star radius ratio
($R_{p}/R_{\star}$), the impact parameter, $b$, the stellar density
($\rho_{\star}$, in solar units), the quadratic limb-darkening (LD)
coefficients $u_{1}$ and $u_{2}$ adopting the LD parametrization ($q_{1}$ and
$q_{2}$) introduced by Kipping (2013), a second-order polynomial trend to take
into account the local stellar variability (with c0 as the intercept, $c_{1}$
as the linear coefficient, and $c_{2}$ as the quadratic coefficient), and a
jitter term to be added in quadrature to the errors of the photometry to
account for any effects that were not included in our model (e.g., short-term
stellar activity) or any underestimation of the error bars. We applied an
airmass detrending technique to each ground-based light curve, and we
estimated $u_{1}$ and $u_{2}$ using
PyLDTk101010https://github.com/hpparvi/ldtk (Husser et al., 2013; Parviainen &
Aigrain, 2015) and applying the specific filters used for the observations. We
imposed a Gaussian prior on the stellar density, whereas we imposed uniform
priors on the period and $T_{0}$. We then imposed a Gaussian prior on the
eccentricities following Van Eylen et al. (2019).

We performed a global optimisation of the parameters by executing a
differential evolution algorithm (Storn & Price 1997,
PyDE111111https://github.com/hpparvi/PyDE) and performing a Bayesian analysis
of each selected light curve around each transit. The latter was achieved
using the affine-invariant ensemble sampler (Goodman & Weare 2010) for Markov
Chain Monte Carlo (MCMC) implemented within the package emcee (Foreman-Mackey
et al. 2013). We used $4n_{dim}$ walkers (with $n_{dim}$ being the
dimensionality of the model) for 50 000 generations with PyDE and then with
100 000 steps with emcee – where we applied a thinning factor of 200 to reduce
the effect of the chain auto-correlation. We discarded the first 25 000 steps
(burn-in) after checking the convergence of the chains with the Gelman–Rubin
(GR) statistics (Gelman & Rubin 1992, threshold value R̂ = 1.01). Unless
specifically mentioned otherwise, the same sampling configuration and process
have been used throughout all occurrences of PyDE and emcee. Figures 7, 8, 9,
and Table 3 present the results of the modelling.

Figure 7: Photometric modelling of TOI-5398 b planetary signal. In the top
panel, we display the TESS phase-folded transits after normalisation along
with the transit model (black line). In the panel below, we show the
residuals.  Figure 8: As in Fig. 7, but for TOI-5398 c. Table 3: Priors and
outcomes of the model of planet b and c from the analysis of the photometric
time series. In Appendix E, we show the second part of this Table, moved for
improved readability of the main text.

Photometric time-series fit
---
Parameter | Unit | Prior | Value
Stellar density ($\rho_{\star}$) | $\rho_{\sun}$ |  $\mathcal{U}$(0.75, 10.25) | 0.971${}^{+0.042}_{-0.049}$
Planet b
Parameter | Unit | Prior | Value
Orbital period ($P_{\rm b}$) | days |  $\mathcal{U}$(10.57, 10.61) | 10.590547${}^{+0.000012}_{-0.000011}$
Central time of the first transit ($T_{\rm 0,b}$) | BTJD |  $\mathcal{U}$(2616.4, 2616.6) | 2616.49232${}^{+0.00022}_{-0.00021}$
Semi-major axis to stellar radius ratio ($a_{\rm b}/R_{\star}$) |  | … | 20.0$\pm$0.6
Orbital semi-major axis ($a_{\rm b}$) | au | … | 0.098$\pm$0.005
Orbital inclination ($i$) | deg | … | 89.21${}^{+0.31}_{-0.21}$
Orbital eccentricity ($e_{\rm b}$) |  |  $\mathcal{N}$(0, 0.098) |  $\leqslant$ 0.094aa$a$84th percentile.
Impact parameter ($b$) |  |  $\mathcal{U}$(0, 1) | 0.272${}^{+0.069}_{-0.110}$
Transit duration ($T_{14}$) | days | … | 0.1774${}^{+0.0062}_{-0.0043}$
Planetary radius ($R_{\rm b}$) | $R_{\oplus}$ | … | 10.30$\pm$0.40
Planet c
Parameter | Unit | Prior | Value
Orbital period ($P_{\rm c}$) | days |  $\mathcal{U}$(4.770, 4.776) | 4.77271${}^{+0.00016}_{-0.00014}$
Central time of the first transit ($T_{\rm 0,c}$) | BTJD |  $\mathcal{U}$(2628.5, 2628.7) | 2628.61781${}^{+0.00090}_{-0.00086}$
Semi-major axis to stellar radius ratio ($a_{\rm c}/R_{\star}$) |  | … | 11.8$\pm$0.4
Orbital semi-major axis ($a_{\rm c}$) | au | … | 0.057$\pm$0.003
Orbital inclination ($i$) | deg | … |  $\geqslant$ 88.40bb$b$16th percentile.
Orbital eccentricity ($e_{\rm c}$) |  |  $\mathcal{N}$(0, 0.098) |  $\leqslant$ 0.117aa$a$84th percentile.
Impact parameter ($b$) |  |  $\mathcal{U}$(0, 1) |  $\leqslant$ 0.34aa$a$84th percentile.
Transit duration ($T_{14}$) | days | … | 0.1303${}^{+0.0042}_{-0.0053}$
Planetary radius ($R_{\rm c}$) | $R_{\oplus}$ | … | 3.52$\pm$0.19

121212

Figure 9: Ground-based photometric data, simultaneously modelled with TESS
transits. Different datasets are represented by distinct colours. Top:
Individual transits of TOI-5398 b on the left and TOI-5398 c on the right.
Bottom: Simultaneous transit of TOI-5398 b and c, observed during the night of
21 April 2022.

###  4.3 RV time-series analysis

Using PyORBIT, we investigated the RV time series data in a Bayesian
framework. We tried various approaches to model the stellar activity through
the use of Gaussian processes (GP, Rasmussen et al. 2006, Haywood et al.
2014). We experimented with a number of data set combinations to constrain the
GP hyper-parameters and consider various planetary system architectures (one,
two, or more planets). Here we outline the three most notable cases.

Due to the high computational cost of the Gaussian Processes, we only modelled
the spectroscopic time series in each test. However, we included the
inclination from the photometry model to determine the true masses.

####  4.3.1 Case 1: Two-planet system and activity modelling trained on
spectroscopy – uni-dimensional GP

In the first case, we tested a multi-planet system model formed by TOI-5398 b
and TOI-5398 c.

PyORBIT simultaneously modelled the stellar activity and the signals of the
two planets in the RV series. In the model, we included the inclination
measured from photometry and the stellar mass $M_{\star}$ as derived in Sect.
3 to determine the true masses of the planets. We then imposed a Gaussian
prior on the eccentricities following Van Eylen et al. (2019) and included the
RV semi-amplitudes $K_{\rm b}$ and $K_{\rm c}$ in the RV modelling. Moreover,
we used Gaussian priors on both orbital periods ($P_{\rm b}$, $P_{\rm c}$) and
central time of the first transits ($T_{0,{\rm b}}$, $T_{0,{\rm c}}$) by
considering the parameters outlined in Sect. 4.2.

We modelled the stellar activity in the RV, BIS, and
$\log{R^{{}^{\prime}}_{HK}}$ series simultaneously, through a Gaussian process
(GP) regression. We used a quasi-periodic kernel as defined by Grunblatt et
al. (2015). As part of this modelling, we set the stellar rotation period
$P_{\rm rot}$ (Gaussian prior, as defined in Sec. 2.2), the characteristics
decay time scale $P_{\rm dec}$, and the coherence scale $w$. In accordance
with Eastman et al. (2013), we fitted the periods and semi-amplitudes of the
RV signals in the linear space, and we determined the eccentricity $e$ and the
argument of periastron $\omega$ by fitting $\sqrt{e}\cos{\omega}$ and
$\sqrt{e}\sin{\omega}$. We performed a global optimisation of the parameters
by running PyDE and performing a Bayesian analysis of the planetary signals
and activity in the RV time series using emcee. The results of this analysis
will be discussed after presenting each respective Case and in Table 4.

####  4.3.2 Case 2: Two-planet system and activity modelling trained on
spectroscopy – multidimensional GP

In the present case, we modelled the stellar activity of TOI-5398 using the
multidimensional GP framework developed by Rajpaul et al. (2015) and re-
implemented in PyORBIT in accordance with the prescriptions in the paper (see
also Barragán et al. 2022). Again, we relied on the quasi-periodic kernel and
its derivatives. We modelled RV, BIS, and $\log{R^{{}^{\prime}}_{HK}}$
spectroscopic time series. The three-dimensional GP model is the following:

$\displaystyle\begin{split}&\Delta RV=V_{\rm c}\,G(t)+V_{\rm
r}\,\dot{G}(t),\\\ &\log R^{\prime}_{\rm HK}=L_{\rm c}\,G(t),\\\ &BIS=B_{\rm
c}\,G(t)+B_{\rm r}\,\dot{G}(t),\end{split}$ (1)

where $G(t)$ is the GP and $\dot{G}(t)$ its time derivative. The constants
denoted by subscripts $r$ and $c$ represent free parameters linking the
individual time series to $G(t)$ and $\dot{G}(t)$ (Barragán et al., 2023).

To perform the modelling with PyORBIT, we assigned the same priors described
in Case 1 to planets and stellar parameters. Then, we run a global
optimisation of the parameters with PyDE and a Bayesian analysis of the
planetary signals and activity in the RV time series with emcee. The results
of this analysis will be discussed after presenting each respective Case and
in Table 4.

####  4.3.3 Case 3: Extended list of activity indexes and search for a third
planet

To better disentangle planetary and stellar signals, we extended the list of
activity indexes that PyORBIT simultaneously models with the two keplerian
signals (described in Case 2). In particular, we included the chromospheric
activity indicator H$\alpha$ (Gomes da Silva et al., 2011) as well as the two
CCF asymmetry diagnostics FWHM and equivalent width $W_{\rm CCF}$.

On the one hand, the H$\alpha$ line complements the lower chromosphere
indicator $\log{R^{{}^{\prime}}_{\rm HK}}$ by providing information about the
conditions in the upper chromosphere of a star (Gomes da Silva et al., 2011).
In fact, H$\alpha$ and Ca H & K are emitted from different depths – and formed
at different temperatures – in the chromosphere (Robertson et al., 2013; Gomes
da Silva et al., 2014). Therefore, it is helpful to study the H$\alpha$ and
$\log{R^{{}^{\prime}}_{\rm HK}}$ indices simultaneously (Gomes da Silva et
al., 2011, 2014) to learn more about the presence of distinct activity-related
features and disentangle their signals from keplerian ones in the RV time
series. On the other hand, according to three years of RV monitoring of the
Sun (Collier Cameron et al., 2019), the line-shape parameters of the CCF
appear to respond to different components of the active regions. Moreover,
they help to track global temperature changes in the photosphere (see also
Malavolta et al. 2017).

For the reasons listed, we decided to include in the modelling both the
chromospheric indicators H$\alpha$ and $\log{R^{{}^{\prime}}_{\rm HK}}$, as
well as the three CCF asymmetry diagnostics BIS, FWHM, and $W_{\rm CCF}$. We
followed the multidimensional GP formalism introduced by Rajpaul et al. (2015)
to examine the RV and BIS time series, using the first derivative of the GP.
Conversely, we did not use the first derivative for the remaining four time
series, as suggested also in Barragán et al. (2023). The six-dimensional GP
model is an extension of Eq. LABEL:eq:model, with the addition of the
following supplementary terms:

$\displaystyle\begin{split}&{\rm H}\alpha=L2_{\rm c}\,G(t)\\\ &{\rm
FWHM}=L3_{\rm c}\,G(t),\\\ &W_{\rm CCF}=L4_{\rm c}\,G(t).\end{split}$ (2)

We performed the modelling with PyORBIT in the same way as described in the
previous cases, and we show the outcomes in Figs. 10, 11, and Table 5.

We emphasise that the inclusion of the additional activity indicators reduces
the uncertainties in the keplerian signals and the RV jitter term with respect
to Case 1 and Case 2. Notably, the orbital parameters remain consistent across
all cases. We used the Bayesian Information Criterion (BIC, Schwarz 1978) to
compare the first two cases. Our analysis revealed a strong preference for
Case 2 over Case 1, with a substantial $\Delta$BIC12 value of 148 (Kass &
Raftery, 1995). In general, the BIC may not be the optimal estimator for the
Bayesian evidence; in this specific case, however, we believe that the extreme
difference between the BIC values ($\Delta$BIC = 148) largely overcomes any
possible bias, thus favouring Case 2 over Case 1. As for Case 3, a direct BIC
comparison is not possible due to the different datasets employed in the
analysis. Instead, based on logical grounds, we selected Case 3 as our
reference model. Each additional activity indicator provides valuable
information on specific aspects of stellar activity, as evidenced in previous
paragraphs, and their inclusion is justified by the amplitude parameter of the
covariance matrix being significantly different from zero for every extra
activity indicator, i.e. those activity indicators further constrain the
activity model independently from the observed radial velocities. The adopted
masses for planet b and c are $58.7^{+5.7}_{-5.6}$ and
$11.8^{+4.8}_{-4.7}\,M_{\oplus}$, respectively. We include the final
parameters of planets b and c in Table 6, while Table 4 lists the differences
between this model and the other two. The significantly reduced jitter
observed in Case 2 and 3, compared to Case 1, may be attributed to the limited
capability of the uni-dimensional GP framework to model the different
periodicities present in the spectroscopic time series. In particular, the RV
and chromospheric indexes exhibit different periodicities (see also Sec. 3.5
and Fig. 4) when the RV variations are dominated by dark spots. In contrast,
Case 2 and 3 use the formalism outlined in Rajpaul et al. (2015), which
effectively handles variations in periodicity.

Table 4: Comparison of the three different models Parameter | Case 1 | Case 2 | Case 3
---|---|---|---
$K_{\rm b}$ (m s-1) | 14.0$\pm$2.7 | 14.8$\pm$1.7 | 15.7$\pm$1.5
$K_{\rm c}$ (m s-1) | 3.9${}^{+2.7}_{-2.4}$ | 3.1${}^{+1.7}_{-1.6}$ | 4.1${}^{+1.6}_{-1.7}$
$\sigma^{RV}_{jitter}$ (m s-1) | 8.3${}^{+7.3}_{-6.2}$ | 1.8${}^{+2.0}_{-1.2}$ | 1.5${}^{+1.6}_{-1.1}$

In addition to the modelling mentioned above, we looked for the presence of a
third planet in our RV dataset, by applying wide uniform priors on its period
$P$ and RV semi-amplitude $K$. While we did not impose the orbit of the third
planet to be circular, we applied a Gaussian prior on the eccentricity
following Van Eylen et al. (2019). Initially, the global optimization
algorithm and the first ten thousand steps of the MCMC exploration suggested a
significant detection of a third planet. However, the chains of the planet’s
orbital period diverged quite soon, preventing us from claiming a third planet
detection. We underline that the solutions for planets b and c in the system
show little variation, which further strengthens the validity of their
detections.

Figure 10: Phase-folded RV fit of TOI-5398 b planetary signal. The reported
error bars include the jitter term, added in quadrature. The shaded area
represents the RV model’s $\pm$1$\sigma$ uncertainties. The bottom panel
displays the residuals of the fit. Figure 11: As in Fig. 10, but for planet c.
Table 5: Priors and outcomes of the model of planet b and c from analysing
spectroscopic series with a multidimensional GP framework (Case 3). GP
framework parameters
---
Parameter | Unit | Prior | Value
Uncorrelated RV jitter ($\sigma^{\rm RV}_{\rm jitter,0}$) | m s-1 | … | 1.5${}^{+1.6}_{-1.1}$
RV offset ($\gamma^{\rm RV}_{0}$) | m s-1 | … | -9432.9${}^{+2.9}_{-2.8}$
Uncorrelated BIS jitter ($\sigma^{\rm BIS}_{\rm jitter,0}$) | m s-1 | … | 17.1${}^{+1.7}_{-1.6}$
BIS offset ($\gamma^{\rm BIS}_{0}$) | m s-1 | … | 10.6${}^{+2.6}_{-2.6}$
Uncorrelated log$R^{{}^{\prime}}_{HK}$ jitter ($\sigma^{\rm logR^{{}^{\prime}}_{HK}}_{\rm jitter,0}$) |  | … | 0.0127${}^{+0.0014}_{-0.0013}$
log$R^{{}^{\prime}}_{HK}$ offset ($\gamma^{\rm logR^{{}^{\prime}}_{HK}}_{0}$) |  | … | -4.4137$\pm$0.0030
Uncorrelated H${\alpha}$ jitter ($\sigma^{\rm H{\alpha}}_{\rm jitter,0}$) |  | … | 0.0017$\pm$0.0002
H${\alpha}$ offset ($\gamma^{\rm H\alpha}_{0}$) |  | … | 0.1513$\pm$0.0004
Uncorrelated FWHM jitter ($\sigma^{\rm FWHM}_{\rm jitter,0}$) | km s-1 | … | 0.046$\pm$0.004
FWHM offset ($\gamma^{\rm FWHM}_{0}$) | km s-1 | … | 11.37$\pm$0.01
Uncorrelated $W_{\rm CCF}$ jitter ($\sigma^{W_{\rm CCF}}_{\rm jitter,0}$) | km s-1 | … | 0.0013${}^{+0.0013}_{-0.0009}$
$W_{\rm CCF}$ offset ($\gamma^{W_{\rm CCF}}_{0}$) | km s-1 | … | 3.505$\pm$0.003
Multidimensional GP parameters (Rajpaul et al., 2015) |  |  |
$V_{c}$ | m s-1 |  $\mathcal{U}$(-100.0, 100.0) | -11.2${}^{+2.3}_{-2.8}$
$V_{r}$ | m s-1 |  $\mathcal{U}$(0.0, 100.0) | 27.1${}^{+4.3}_{-3.6}$
$B_{c}$ | m s-1 |  $\mathcal{U}$(-100.0, 100.0) | 4.3${}^{+3.3}_{-3.2}$
$B_{r}$ | m s-1 |  $\mathcal{U}$(-100.0, 100.0) | -35.6${}^{+4.8}_{-6.0}$
$L_{c}$ (log$R^{{}^{\prime}}_{HK}$) |  |  $\mathcal{U}$(-0.1, 0.1) | -0.011$\pm$0.002
$L2_{c}$ (H${\alpha}$) |  |  $\mathcal{U}$(-0.1, 0.1) | -0.0015$\pm$0.0003
$L3_{c}$ (FWHM) | km s-1 |  $\mathcal{U}$(-0.5, 0.5) | -0.043${}^{+0.007}_{-0.008}$
$L4_{c}$ ($W_{\rm CCF}$) | km s-1 |  $\mathcal{U}$(-0.02, 0.02) | -0.014$\pm$0.002
Stellar activity (RV + activity indexes)
Parameter | Unit | Prior | Value
Rotational period ($P_{\rm rot}$) | days |  $\mathcal{N}$(7.34, 0.15) | 7.37$\pm$0.03
Decay Timescale of activity ($P_{\rm dec}$) | days |  $\mathcal{U}$(10.0, 2000.0) | 26.1${}^{+3.3}_{-3.0}$
Coherence scale ($w$) |  |  $\mathcal{U}$(0.01, 0.60) | 0.36$\pm$0.03
Planet b
Parameter | Unit | Prior | Value
Orbital period ($P_{\rm b}$) | days |  $\mathcal{N}$(10.590547, 0.000012) | 10.590547$\pm$0.000012
Central time of the first transit ($T_{\rm 0,b}$) | BTJD |  $\mathcal{N}$(2616.49232, 0.00022) | 2616.49232$\pm$0.00022
Orbital eccentricity ($e_{\rm b}$) |  |  $\mathcal{N}$(0.00, 0.098) |  $\leqslant$ 0.13
Argument of periastron ($\omega_{\rm b}$) | deg | … | 92${}^{+82}_{-45}$
Semi-major axis to stellar radius ratio ($a_{\rm b}/R_{\star}$) |  | … | 20.2$\pm$0.3
Orbital semi-major axis ($a_{\rm b}$) | au | … | 0.0988$\pm$0.0004
RV semi-amplitude ($K_{\rm b}$) | m s-1 |  $\mathcal{U}$(0.01, 100.0) | 15.7${}^{+1.5}_{-1.5}$
Planetary mass ($M_{\rm p,b}$) | $M_{\oplus}$ | … | 58.7${}^{+5.7}_{-5.6}$
Planet c
Parameter | Unit | Prior | Value
Orbital period ($P_{\rm c}$) | days |  $\mathcal{N}$(4.77271, 0.00016) | 4.77270$\pm$0.00016
Central time of the first transit ($T_{\rm 0,c}$) | BTJD |  $\mathcal{N}$(2628.6178, 0.0009) | 2628.6178$\pm$0.0009
Orbital eccentricity ($e_{\rm c}$) |  |  $\mathcal{N}$(0.00, 0.098) |  $\leqslant$ 0.14
Argument of periastron ($\omega_{\rm c}$) | deg | … | 172${}^{+79}_{-107}$
Semi-major axis to stellar radius ratio ($a_{\rm c}/R_{\star}$) |  | … | 11.9$\pm$0.2
Orbital semi-major axis ($a_{\rm c}$) | au | … | 0.0581$\pm$0.0002
RV semi-amplitude ($K_{\rm c}$) | m s-1 |  $\mathcal{U}$(0.01, 100.0) | 4.1${}^{+1.7}_{-1.6}$
Planetary mass ($M_{\rm p,c}$) | $M_{\oplus}$ | … | 11.8${}^{+4.8}_{-4.7}$

###  4.4 Search for TTVs

To investigate the potential presence of dynamical interactions between
TOI-5398 b and c we performed a search for Transit Timing Variations

(TTVs; e.g., Agol et al., 2005; Holman & Murray, 2005; Borsato et al., 2014,
2019a, 2021) of planet b and c.

We computed the observed (O) - calculated (C) diagrams for both planets
removing the linear ephemeris (in Table 6) to each transit times. See the O-C
diagrams in Fig. 12 and 13 for planets b and c, respectively. The possible TTV
amplitude ($A_{\mathrm{TTV}}$), computed as the semi-amplitude of the O-C, is
of $2.9_{-1.0}^{+1.2}$ minutes for planet b and $4_{-2}^{+7}$ minutes for
planet c. The associated error is derived from the subtraction of
$A_{\mathrm{TTV}}$ from the high-density interval at 95% of a Monte-Carlo
sampling of 10 000 repetitions.

Figure 12: O-C plot representing the observed (O) and calculated (C) transit
times for the linear ephemeris of TOI-5398 b (see Table 6). Each dataset is
shown in a distinct colour. Figure 13: As in Fig. 12, but for TOI-5398 c.

Although our data do not cover a sufficient portion of the super-period to
offer direct evidence, they do suggest a possible TTV due to the gravitational
interaction of planets b and c (see Fig. 12). The sparse sampling of the TTV
signals did not allow us to run a dynamical fit, so we decided to run a
forward dynamical model with
TRADES131313https://github.com/lucaborsato/trades(Borsato et al., 2014, 2019b,
2021), similar to what has been done in Tuson et al. (2023). We took the
masses and the orbital parameters from Tables 5 and 6 and we integrated the
orbits for about 500 days and computed the transit times of each planet. We
computed the O-C diagrams (see Fig. 14) and we found that the simulated
$A_{\mathrm{TTV}}$ is of the order of $\sim 40$ seconds and of $\sim 2$
minutes for planet b and c, respectively.

We used the orbital parameters from Table 6 as the starting point of the
dynamical simulation, i.e., we assumed that the value we obtained from our
global fit spanning about 500 days represents a specific configuration in
time, which may not be true. The amplitude of the simulated TTVs may also be
dependent on the eccentricity of the two planets. Nevertheless, the outcome of
the simulation is compatible (at $2\sigma$ for b and at $1\sigma$ for c) with
the measured TTVs. The scope of this analysis is to show that TTVs can indeed
be present in this system, while a full dynamical analysis is outside the
scope of this paper and it is left to future works.

Figure 14:  Synthetic O-C diagrams (top-panel planet b, bottom-panel planet c)
computed from the dynamical simulation with TRADES and with parameters from
Table 5.

##  5 Discussion

Table 6: Final parameters of TOI-5398 multi-planet system.

Parameter | TOI-5398 b | TOI-5398 c
---|---|---
$P$ (days) | 10.590547${}^{+0.000012}_{-0.000011}$ | 4.77271${}^{+0.00016}_{-0.00014}$
$T_{\rm 0}$ (BTJD) | 2616.49232${}^{+0.00022}_{-0.00021}$ | 2628.61781${}^{+0.00090}_{-0.00086}$
$a/R_{\star}$ | 20.0$\pm$0.6 | 11.8$\pm$0.4
$a$ (au) | 0.098$\pm$0.005 | 0.057$\pm$0.003
$R_{\rm p}/R_{\star}$ | 0.0899${}^{+0.0007}_{-0.0006}$ | 0.0308${}^{+0.0011}_{-0.0012}$
$R_{\rm p}$ ($R_{\oplus}$) | 10.30$\pm$0.40 | 3.52$\pm$0.19
$b$ | 0.272${}^{+0.069}_{-0.110}$ |  $\leqslant$ 0.34aa$a$84th percentile.