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: INAF -
Osservatorio Astrofisico di Catania, Oss. Astr. Catania, via S. Sofia 78,
95123 Catania Italy 77institutetext: INAF – Osservatorio Astronomico di Brera,
Via E. Bianchi 46, 23807 Merate (LC), Italy 88institutetext: Department of
Physics, University of Rome “Tor Vergata”, Via della Ricerca Scientifica 1,
I-00133, Rome, Italy 99institutetext: Max Planck Institute for Astronomy,
Königstuhl 17, D-69117, Heidelberg, Germany

# Orbital obliquity of the young planet TOI-5398 b
and the evolutionary history of the system ††thanks: Table A.1 is only
available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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. Mantovan ${}^{\href https://orcid.org/0000-0002-6871-6131}$ Orbital
obliquity of the young planet TOI-5398 b  and the evolutionary history of the
system ††thanks: Table A.1 is only available in electronic form at the CDS via
anonymous ftp to https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and
the evolutionary history of the system ††thanks: Table A.1 is only available
in electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and
the evolutionary history of the system ††thanks: Table A.1 is only available
in electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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${}^{\href
https://orcid.org/0000-0002-6492-2085}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and
the evolutionary history of the system ††thanks: Table A.1 is only available
in electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and
the evolutionary history of the system ††thanks: Table A.1 is only available
in electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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${}^{\href
https://orcid.org/0000-0002-9824-2336}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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${}^{\href
https://orcid.org/0000-0002-7657-7418}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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${}^{\href
https://orcid.org/0000-0002-1923-7740}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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${}^{\href
https://orcid.org/0000-0001-5154-6108}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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${}^{\href
https://orcid.org/0000-0001-8613-2589}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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. Spinelli Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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${}^{\href
https://orcid.org/0000-0002-4638-3495}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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${}^{\href
https://orcid.org/0000-0002-9937-6387}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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${}^{\href
https://orcid.org/0000-0001-5928-7251}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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. Marzari Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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${}^{\href
https://orcid.org/0000-0002-7504-365X}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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${}^{\href
https://orcid.org/0000-0001-9984-4278}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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${}^{\href
https://orcid.org/0000-0003-1149-3659}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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${}^{\href
https://orcid.org/0000-0002-5130-4827}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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${}^{\href
https://orcid.org/0009-0004-5914-7274}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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. Guilluy${}^{\href
https://orcid.org/0000-0002-1259-2678}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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. Mancini Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and
the evolutionary history of the system ††thanks: Table A.1 is only available
in electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and
the evolutionary history of the system ††thanks: Table A.1 is only available
in electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and
the evolutionary history of the system ††thanks: Table A.1 is only available
in electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and
the evolutionary history of the system ††thanks: Table A.1 is only available
in electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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. Micela${}^{\href
https://orcid.org/0000-0002-9900-4751}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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${}^{\href
https://orcid.org/0000-0001-9770-1214}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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${}^{\href
https://orcid.org/0000-0001-6880-5356}$ Orbital obliquity of the young planet
TOI-5398 b  and the evolutionary history of the system ††thanks: Table A.1 is
only available in electronic form at the CDS via anonymous ftp to
https://cdsarc.u-strasbg.fr (130.79.128.5) or via
http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and the
evolutionary history of the system ††thanks: Table A.1 is only available in
electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and
the evolutionary history of the system ††thanks: Table A.1 is only available
in electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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).Orbital obliquity of the young planet TOI-5398 b  and
the evolutionary history of the system ††thanks: Table A.1 is only available
in electronic form at the CDS via anonymous ftp to https://cdsarc.u-strasbg.fr
(130.79.128.5) or via http://cdsweb.u-strasbg.fr/cgi-bin/qcat?J/A+A/.††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: )

Multi-planet systems exhibit remarkable architectural diversity. However,
short-period giant planets are typically isolated. Compact systems like
TOI-5398, with an outer close-orbit giant and an inner small-size planet, are
rare among systems containing short-period giants. TOI-5398’s unusual
architecture coupled with its young age (650 $\pm$ 150 Myr) make it a
promising system for measuring the original obliquity between the orbital axis
of the giant and the stellar spin axis in order to gain insight into its
formation and orbital migration.

We collected in-transit (plus suitable off-transit) observations of TOI-5398 b
with HARPS-N at TNG on March 25, 2023, obtaining high-precision radial
velocity time series that allowed us to measure the Rossiter-McLaughlin (RM)
effect. By modelling the RM effect, we obtained a sky-projected obliquity of
$\lambda=3.0^{+6.8}_{-4.2}$ deg for TOI-5398 b, consistent with the planet
being aligned. With knowledge of the stellar rotation period, we estimated the
true 3D obliquity, finding $\psi=(13.2\pm 8.2)$ deg. Based on theoretical
considerations, the orientation we measure is unaffected by tidal effects,
offering a direct diagnostic for understanding the formation path of this
planetary system. The orbital characteristics of TOI-5398, with its compact
architecture, eccentricity consistent with circular orbits, and hints of
orbital alignment, appear more compatible with the disc-driven migration
scenario.

TOI-5398, with its relative youth (compared with similar compact systems) and
exceptional suitability for transmission spectroscopy studies, presents an
outstanding opportunity to establish a benchmark for exploring the disc-driven
migration model.

###### Key Words.:

techniques: radial velocities – stars: individual: BD+37 2118 – planet-star
interactions

##  1 Introduction

The obliquity between the orbital axis of a planet and the spin axis of its
host star is a key diagnostic for the mechanisms of formation and orbital
migration of exoplanets (e.g. Naoz et al. 2011). The sky-projected obliquity
of a transiting planet can be measured with in-transit radial velocities (RVs)
through the Rossiter-McLaughlin (RM) effect (Ohta et al. 2005).

Giant planets in close-in orbits are believed to form in situ close to the
final orbit, or in the outer regions and migrate inward (Dawson & Johnson,
2018). Different mechanisms are capable of shrinking the orbits, such as
dynamical interactions (high-eccentricity migration scenario) through planet-
planet scattering (Marzari et al., 2006) or the Kozai mechanism (Wu & Murray,
2003), and disc–planet interactions (Lin et al., 1996). These alternative
mechanisms are expected to imprint different signatures on the obliquity of
the planets. Scattering encounters should randomise the alignments of the
orbital planes, whereas a migration through disc–planet interactions should
keep the planetary orbits roughly coplanar throughout the entire process.

Young transiting planets represent a unique resource in their ability to help
us interpret yet inconclusive observational results. First, the obliquity may
be altered by tidal effects, and close-in planets orbiting stars of a few
gigayears old with deep convective envelopes are expected to show aligned
configurations regardless of the formation scenario, as is observed in most
cases (e.g. Albrecht et al., 2022, and references therein). Instead, when we
observe systems young enough to have avoided tidal alterations of obliquity,
we can access the original configuration. Second, a reliable rotation period
can be obtained for young, active stars. This enables us to measure the 3D
orbital obliquity with respect to the stellar rotation axis while avoiding the
bias reported by Masuda & Winn (2020) when the rotation period is included in
a Bayesian framework. Third, the amplitude of the RM signature increases with
the projected rotational velocity $v\sin{i_{\star}}$ of the star (Winn, 2010),
more than counterbalancing the adverse effect of rotation and activity on RV
errors. Indeed, some planets orbiting active stars have been confirmed by
detecting the RM effect rather than via dynamical mass measurements from
Keplerian orbit solutions (e.g. Mann et al. 2020).

Recent space missions led to the identification of many planets transiting
young stars (age $<$ 1 Gyr). The number of confirmed or validated systems is
about 70 (cf. NASA Exoplanet Archive). However, the RM effect has only been
measured for a limited number of objects, including AU Mic b (Palle et al.,
2020), DS Tuc A b (Benatti et al., 2021), TOI-942 b (Wirth et al., 2021),
V1298 Tau b and c (Johnson et al., 2022; Feinstein et al., 2021), HD 63433 b
and c (Mann et al., 2020; Dai et al., 2020), HIP 67522 b (Heitzmann et al.,
2021), TOI-1136 d (Dai et al., 2023), and TOI-2076 b (Frazier et al., 2023).
These measurements generally indicate aligned configurations, although with
moderately large uncertainties, usually ranging between 20 and 30 deg.
Consequently, the obliquity of young transiting planets remains significantly
unexplored.

Within the Global Architecture of Planetary Systems (GAPS, Covino et al. 2013;
Carleo et al. 2020) long-term program at TNG, many systems with young
transiting planets, first identified by space missions, are being
characterised through continued RV monitoring (e.g. Nardiello et al. 2022;
Damasso et al. 2023). Among the systems precisely characterised by GAPS, the
moderately young solar-analogue star TOI-5398 ($T_{\rm eff}=6000$K,
$R_{\star}=1.051\,R_{\odot}$, $M_{\star}=1.146\,M_{\odot}$) is of special
interest (Mantovan et al. 2022, 2024, hereafter MG24). TOI-5398 hosts a hot
sub-Neptune planet (TOI-5398 c, $R_{\rm p}=3.52\,R_{\oplus}$, $M_{\rm
p}=11.8\,M_{\oplus}$, $P\sim 4.77$ d) orbiting interior to a short-period warm
Saturn (TOI-5398 b, $R_{\rm p}$ = 10.30 $R_{\oplus}$, $M_{\rm
p}=58.7\,M_{\oplus}$, $P\sim 10.59$ d).

TOI-5398 is the youngest known compact system with a short-period giant, and
the giant TOI-5398 b has the highest transmission spectroscopy metric (TSM,
Kempton et al. 2018) value (TSM = 288) among all known warm giant planets (10
¡ $P$ ¡ 100 d, $M_{\rm p}$ ¿ 0.1 $M_{\rm J}$), making it ideal for atmospheric
characterisation with JWST (MG24). Given the unusual architecture of the
TOI-5398 system and its moderately young age ($650\pm 150$ Myr, $P_{\rm
rot}=7.34$ d), the system serves as a promising target for measuring the
original obliquity between the orbital axis of the giant and the spin axis of
the star (MG24), because it is young enough to have avoided tidal alterations
of obliquity. Moreover, unlike ordinary short-period giants, we can rule out
the high-eccentricity migration scenario for compact systems such as TOI-5398
(Mustill et al., 2015) and test the other formation and evolution models (e.g.
Lin et al., 1996) through detailed atmospheric characterisation.

Among the compact systems similar to TOI-5398 known to date, namely WASP-47
(Hellier et al., 2012), Kepler-730 (Zhu et al., 2018), TOI-1130 (Huang et al.,
2020), TOI-2000 (Sha et al., 2022), and WASP-132 (Hord et al., 2022), only
WASP-47 has an obliquity measurement (cf. TEPCat catalogue; Southworth 2011;
Sanchis-Ojeda et al. 2015), and the value obtained is consistent with the
system being aligned. Investigating obliquities in multi-planet systems,
focusing on uncommon systems such as the ones with an outer close-in giant and
an inner small-size planet (listed above), could play a pivotal role in
advancing our comprehension of their formation and their relationship to
ordinary isolated systems with short-period giant planets (Sanchis-Ojeda et
al., 2015).

##  2 Observations and data reduction

We observed TOI-5398 with the TNG telescope in the GIARPS configuration
(Claudi et al., 2018) by simultaneously measuring high-resolution spectra in
the optical (0.39-0.69 $\mu\textrm{m}$, HARPS-N, $R$ $\sim$ 115 000) and the
near-infrared (0.95-2.45 $\mu\textrm{m}$, GIANO-B, $R$ $\sim$ 50 000). In the
present work, our analysis is exclusively focused on the HARPS-N spectra. The
GIANO-B spectra will be presented in a forthcoming work dedicated to
conducting an in-depth atmospheric characterisation of the planet (D’Arpa et
al., in prep.).

We collected both in-transit observations and suitable off-transit
observations on March 25, 2023, and obtained a total of 47 spectra with an
exposure time of 600 s. This window covers the full 4.3 hour transit (26
spectra) and about 4 hours of off-transit baseline (21 spectra) subdivided
into $\sim$ 2 hours before and after the transit event. We monitored the
latter given the activity level of TOI-5398, for which the typical RV
dispersion due to stellar activity on rotation timescale is 29 m s-1 (MG24). A
long out-of-transit observation is required to correct potential RV trends
when fitting the RM effect on young stars (Benatti et al., 2021).
Coincidentally, TOI-5398 c was also transiting during our selected window for
TOI-5398 b (see following section). RV measurements of TOI-5398 collected on
March 25, 2023 are reported in Table A.1 available at the CDS.

We applied the same RV data-reduction procedure outlined in MG24 to ensure
consistency in our analysis. In particular, we computed the RVs through the
cross-correlation function method (Pepe et al., 2002) and using a G2 mask. We
obtained RVs with average internal errors $\langle\rm RV_{\rm err}\rangle=$
5.5 m s-1 and average signal-to-noise ratio $\langle\rm S/N\rangle_{5460\AA}=$
42.

##  3 Analysis of the RM effect

Using PyORBIT111https://github.com/LucaMalavolta/PyORBIT (Malavolta et al.,
2016, 2018), a Python package designed for modelling planetary transits and
RVs while simultaneously accounting for stellar activity effects, we fit the
in-transit RV anomaly in a Bayesian framework following the method introduced
by Covino et al. (2013).

In the model, we relied on stellar and planetary parameters derived in MG24 to
determine the obliquity of TOI-5398 b. We used priors on orbital and stellar
parameters to properly account for their associated errors (see Appendix B for
more details). We show the data and best-fit RM model in Fig. 1, and the best-
fitting values of the parameters in Table 1. In particular, from the Bayesian
modelling of the RM effect due to planet b, we obtain a sky-projected
obliquity of $\lambda\,=3.0^{+6.8}_{-4.2}$ deg, which is consistent with the
planet being aligned.

Figure 1: Rossiter-McLaughlin model fit to the RV data collected during the
transit of TOI-5398 b. The top panel shows the best-fit RM model (black line)
superimposed on the RV observations (green dots). RV data have been corrected
for the Keplerian RV curve and systemic RV. The bottom panel displays the
residuals resulting from the fit. Table 1: Priors and outcomes of RM
modelling.

Planet
---
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}$) | BTJDaa$a$BTJD = BJDTDB \- 2457000.0. |  $\mathcal{N}$(2616.49232, 0.00022) | 2616.49232$\pm$0.00022
Orbital eccentricity ($e_{\rm b}$) |  |  $\mathcal{N}$(0.00, 0.098) |  $\leq$ 0.11bb$b$84th percentile.
RV semi-amplitude ($K_{\rm b}$) | m s-1 |  $\mathcal{N}$(15.7, 1.5) | 15.7$\pm$1.5
Planet-to-star radius ratio ($R_{\rm p}/R_{\star}$) |  |  $\mathcal{N}$(0.0899, 0.0007) | 0.0898$\pm$0.0007
Impact parameter ($b$) |  |  $\mathcal{N}$(0.27, 0.07) | 0.270${}^{+0.075}_{-0.089}$
Omega ($\omega$) | deg | … | -86${}^{+65}_{-67}$
Sky-projected obliquity ($\lambda$) | deg | … | 3.0${}^{+6.8}_{-4.2}$
True 3D obliquity ($\psi$) | deg | … | 13.2$\pm$8.2
Stellar parameters
Parameter | Unit | Prior | Value
Density ($\rho_{\star}$) | $\rho_{\sun}$ |  $\mathcal{N}$(0.971, 0.05) | 0.958${}^{+0.050}_{-0.049}$
Rotation period ($P_{\rm rot}$) | days |  $\mathcal{N}$(7.34, 0.05) | 7.34$\pm$0.05
Rotational velocity of the star ($v\sin{i_{\star}}$)cc$c$The uncertainty on the modelled $v\sin{i_{\star}}$ might be slightly underestimated (see e.g. Triaud 2018). | km s-1 |  $\mathcal{N}$(7.5, 0.6) | 7.02${}^{+0.17}_{-0.21}$
Quadratic limb-darkening coefficient ($u_{1}$) |  |  $\mathcal{N}$(0.5760, 0.0149) | 0.574$\pm$0.015
Quadratic limb-darkening coefficient ($u_{2}$) |  |  $\mathcal{N}$(0.1377, 0.0318) | 0.135${}^{+0.032}_{-0.031}$
Stellar inclination ($i_{\star}$) | deg | … | 76.4${}^{+8.3}_{-6.6}$
Convective blueshift (c1) |  |  $\mathcal{U}$(-2, 0) | -0.43${}^{+0.32}_{-0.59}$
Stellar equatorial velocity ($v_{\rm eq}$) | km s-1 |  $\mathcal{U}$(0, 20) | 7.238$\pm$0.100

222

As briefly mentioned in the previous section, the predicted central transit
times of planets b and c (Fig. 2) were very close to each other, implying that
the two transits overlapped, even when considering their ephemeris
uncertainties (propagated from MG24). We checked how the transit of TOI-5398 c
impacts our results. While this degeneracy does not alter our result regarding
the alignment of planet b, it slightly increases the associated uncertainty on
the measured obliquity (already taken into account).The current estimate of
the uncertainty in the obliquity of planet b can be thought of as an upper
limit on the possible contribution to the error budget due to the missing
modelling of planet c. Indeed, our simulations show that the RM signal
associated with planet c has an amplitude similar to the stellar noise
observed in our data (expected RV anomaly caused by the transit of planet
c333An order of magnitude lower than the one caused by the transit of planet
b, cf. MG24. $\sim$ 7 m s-1 vs. RMS $\sim$ 6 m s-1).

Figure 2: Combined RM effects of planets b and c simulated at different times
of inferior conjunction of planet c. For planet c, we assumed null obliquity,
while we fixed all the other parameters to the value reported in Table 1 and
in MG24. For comparison, the RM effect of planet b alone (see Fig. 1) is shown
as a black line in the upper plot, while black crosses denote the residuals.
The vertical lines denote the propagated ephemeris of planets b and c from
MG24.

##  4 Discussion

###  4.1 Three-dimensional orbital obliquity

Thanks to information on the stellar rotation period —crucial for probing the
stellar inclination $i_{\star}$ while avoiding the bias on its inference (see
Masuda & Winn 2020)— and orbital inclination $i_{\rm p}$ provided by MG24, we
used Eq. 14 from Cegla et al. (2016):

$\psi=\cos{(\sin{i_{\star}}\cos{\lambda}\sin{i_{\rm
p}}+\cos{i_{\star}}\cos{i_{\rm p}})^{-1}},$ (1)

and estimated the true 3D obliquity $\psi=(13.2\pm 8.2)$ deg. Following the
findings from Mancini et al. (2022), which highlight that most exoplanets
orbiting cool stars ($T_{\rm eff}$ $<$ 6100 K, known as the Kraft break, Kraft
1967) tend to have true 3D obliquity $\psi$ $<$ 30 deg, we produced a
preliminary polar plot (see Fig. 3). This plot shows the 3D obliquity of each
known system with a $\psi$ measurement (cf. TEPCat catalogue). The radial
coordinate represents the stellar effective temperature $T_{\rm eff}$, while
the stellar age is colour coded. The 3D obliquities and $T_{\rm eff}$ come
from the TEPCat catalogue, whereas the ages are taken from the GAPS internal
database for young stars and from the NASA Exoplanet Archive. It is worth
noting that since ages have been estimated using different techniques, they
might lack homogeneity. If we focus on the stellar temperature, there appears
to be a potential trend around low obliquity values among cool stars (first
noted by Winn et al. 2010); though a significant fraction still possess values
of between 80 and 125 degrees, as previously noted by Albrecht et al. 2021 and
Attia et al. 2023. When considering also stellar ages, cool systems ($T_{\rm
eff}$ $<$ 6100 K) with obliquities of between 80 and 125 degrees tend to be
old (age $>$ 1 Gyr), with the only exception being Kepler-63 b (Sanchis-Ojeda
et al., 2013). All these cool systems with obliquities of between 80 and 125
degrees have scaled distances $a/R_{\star}>$ 10\. Conversely, young cool
systems tend to have low obliquity values. Our analysis confirms the trends
noted by Albrecht et al. (2022), yet the sample is too small to draw
conclusions about post-formation misalignment mechanisms.

Figure 3: True 3D obliquity of known exoplanets as a function of the effective
temperature of the host star, $T_{\rm eff}$. Each exoplanet except for
TOI-5398 (star marker) is represented by a circle. The stellar age is colour
coded. The data were taken from TEPCat, NASA Exoplanet Archive, and GAPS
(priv. comm.) in January 2024.

The orbital characteristics of TOI-5398, with its compact architecture,
eccentricity values consistent with circular orbits (and stable dynamics; see
Appendix F), and hints of orbital alignment, appear more compatible with the
disc-driven migration scenario. This crucial result reveals the potential for
exploring the disc-driven migration model through detailed atmospheric
characterisation of this system (see Sect. 1).

###  4.2 Timescale of the tidal decay of obliquity

Both the equilibrium tide models of Zahn (1977) or Albrecht et al. (2012) and
the dynamical tide model of Lai (2012) —this latter having a modified tidal
quality factor ($Q^{\prime}_{\star}$) of $10^{5}$ for the
star444$Q^{\prime}_{\star}=(3/2)(Q_{\star}/k_{2})$, where $Q_{\star}$ is the
tidal quality factor of the star, that is, the ratio of the total maximum
energy of its tidal distortion to the energy dissipated by the tides in one
cycle, while $k_{2}$ is the Love number that depends on the internal
stratification of the star (Zahn, 2008).— give an obliquity decay timescale
exceeding the Hubble time. The alignment timescale result is longer than the
age of TOI-5398 ($650\pm 150$ Myr), which implies that the obliquity measured
should still be unaltered by tidal effects, thus providing a direct diagnostic
for the formation path of the planetary system.

We verified the expected tidal realignment timescales for each of the giant
planets in the other five compact systems: WASP-47 b, Kepler-730 b, TOI-1130
c, TOI-2000 c, and WASP-132 b. These planets orbit cool stars with convective
envelopes, placing them below the Kraft break. Therefore, we can calculate the
convective tidal realignment timescale following the same methodology as
outlined in Wright et al. (2023). Notably, we find that the decay time greatly
exceeds the Hubble time by several orders of magnitude for all checked
systems, which confirms that the use of the above approach is well justified.
This result shows that RM observations could be used to measure the unaltered
obliquities in each of these compact systems and advance our understanding of
their formation path.

###  4.3 Formation history of the system

We investigated the possible formation pathways of the two planets, expanding
on the preliminary exploration of MG24. Based on the spin–orbit alignment of
the system, we assumed the architecture as primordial and shaped by disc-
driven migration during the planet growth process. We looked for formation
tracks resulting in the current planetary masses in the framework of the
pebble-accretion scenario. We performed Monte Carlo explorations using a
modified version of the GroMiT code (Polychroni et al., 2023), complementing
the growth and migration treatments of solid planets and planetary cores from
Johansen et al. (2019) and those of gaseous planets from Tanaka et al. (2020)
with the condensation sequence treatment in protoplanetary discs from Turrini
et al. (2023) (see Appendix C for details).

Figure 4: Illustrative example of planetary growth tracks satisfying the
selection conditions associated with the black box in Fig. 5 (see main text
for details). The growth tracks are projected in the semimajor axis–planetary
mass space. The red curves are for planets formed in discs dominated by
millimetre-sized pebbles, while the blue curves are for planets formed in
discs dominated by centimetre-sized pebbles. All successful tracks are
associated with formation regions comprised between 1 and 2 AU and have
planetary cores of about 2 M⊕. The core mass is identified by the mass value
where the growth tracks intercept the pebble isolation mass curve (purple
dotted line).

All successful synthetic planets emerge from planetary seeds that formed
between 1 and 2 AU within the first 1 Myr of the lifetime of the disc; that
is, they grew from first-generation planetesimals born close to the host star
(see Fig. 4 for illustrative examples). Their formation takes about 2–2.5 Myr,
of which 1.5–2 Myr are needed for the cores to reach the pebble isolation
mass. All these planets are dominated in mass by their gaseous envelopes and
possess small cores (about 2 $M_{\oplus}$), which means they underwent some
level of runaway gas accretion. In the case of planet c, this gas-dominated
interior structure appears to be at odds with that expected for planets in its
mass range (Fortney et al., 2007; Lopez & Fortney, 2014) and with its modal
density value estimated by MG24. However, GroMiT formation tracks provide only
lower limits on the interior abundance of heavy elements as they do not yet
account for the accretion of planetesimals and high-metallicity disc gas (see
e.g. Turrini et al. 2023 for a discussion).

Gas-dominated exoplanets in the mass range of planets b and c show indications
of being significantly enriched in heavy elements, enriched by 20–40 times on
average with respect to their host stars (Thorngren et al., 2016). Taking into
account the slightly supersolar metallicity of TOI-5398 ([Fe/H] = 0.09 $\pm$
0.06, MG24) and the observed mass–metallicity trend in exoplanets (Thorngren
et al., 2016), the gaseous envelopes of planets b and c could incorporate
about 20 and 6 $M_{\oplus}$ of heavy elements, respectively (even more if the
envelope of planet c was originally larger; MG24). Due to the limited orbital
migration characterising the successful formation tracks, if both planets were
formed by pebble accretion, the most likely source for this enrichment is the
accretion of high-metallicity gas (Booth & Ilee, 2019) rather than
planetesimals (Turrini et al., 2021). As suggested by the observations of the
Juno mission for Jupiter, such high-metallicity envelopes can mimic the
effects of larger cores (Wahl et al., 2017; Stevenson, 2020), resulting in an
interior structure for planet c more compatible with its density value
($\rho_{\rm p}=1.5$ g cm-3, MG24, ).

Alternatively, if the circumstellar disc of TOI-5398 hosted a significant
population of planetesimals, the migrating core of planet b could have
accreted the planetesimals along its path and grown by a mixture of pebble and
planetesimal accretion without being limited by the pebble isolation mass (see
Fig. 4). This would have delayed the onset of its gas accretion (Alibert et
al., 2018; Turrini et al., 2023), resulting in a smaller gaseous envelope and
the core/envelope ratio characteristic of Neptune-like planets (Fortney et
al., 2007; Lopez & Fortney, 2014).

###  4.4 Photoevaporation history

The young age of TOI-5398 (650 Myr) stands out among similar compact systems
(aged between 3 and 10 Gyr), making it a prime target for photoevaporation
studies. We evaluated the mass-loss rate of the planetary atmospheres with the
new ATES photoionisation hydrodynamics code (Caldiroli et al., 2021, 2022),
coupled with the planetary core–envelope models by Fortney et al. (2007) and
Lopez & Fortney (2014), the MESA Stellar Tracks (MIST; Choi et al. 2016), and
the extreme-ultraviolet (XUV) luminosity time evolution (Sanz-Forcada et al.,
2022). To perform simulations of the past evolution of the planetary
atmospheres, we determined the planetary core mass and radius of the planets
in TOI-5398 at their present age (details in Appendices D and E). The best
agreement with the measured values was obtained by assuming high-metallicity
atmospheric envelopes.

Specifically, assuming cores composed of 50% rocks and 50% ices, we find core
sizes larger than predicted by the formation models if simple pebble accretion
is assumed. Higher initial core/envelope ratios could be obtained if we were
to postulate dense high-metallicity envelopes or pebble+planetesimal accretion
scenarios (as proposed in Sect. 4.3). Alternatively, we were able to find
configurations with smaller rock–iron cores (i.e. Earth-like core composition
with 67% rocks and 33% iron) and very large envelopes at the present age of
the system, in line with the formation scenario, but only for planet b.

We investigated the hydrodynamic atmospheric evolution of the two planets,
from $t_{0}=3$ Myr (i.e. the end time of planetary formation models) to 650
Myr. We performed several simulations for both planets, assuming cores of
different sizes. Our results, which are shown in Figs. 8, 9, and in Table 3,
confirm that planet b experienced a relatively minor atmospheric escape during
its lifetime, and it remains dominated by a massive and extended envelope, in
agreement with the GroMiT results (Sect. 4.3). Assuming a small or a large
core does not affect the evolution appreciably. The reconstructed planetary
radius at $t_{0}$ was a factor 2.2–2.4 times the value at the current age.

In the case of planet c, the evolution was more important. In particular,
assuming a large core, the atmospheric mass fraction decreased from $\sim 38$%
at $t_{0}$ to 1.5% at the present age; hence, the initial planetary mass was
$\sim 60$% larger than the current value. The mass-loss rate was a factor
$\sim 6$–7 higher than for planet b at $t_{0}$, but now it is slightly lower.
This difference can be explained by considering that the XUV irradiation
remained a factor $\sim 3$ higher during the evolution of planet c, while its
mean density increased from 0.01 g cm-3 at $t_{0}$ to 1.5 g cm-3 today due to
a decrease in the planet radius by a factor 5.8. We cannot find a reasonable
evolutionary path for planet c if a small core is assumed because the high
efficiency of the photoevaporation in the past implies a relatively thin
atmospheric envelope at the present age, and therefore the core size cannot be
much smaller than the planet.

##  5 Conclusions

We conclude by outlining the key findings from this study:

  1. 1.

The system’s orbital plane is well aligned with the equatorial plane of the
star, indicating its architecture is shaped by disc-driven migration.

  2. 2.

The orientation we measure is unaffected by tidal effects, meaning that a
direct diagnostic of the formation path is possible for this planetary system.

  3. 3.

Planet b experienced little mass loss. Hence, the current size and structure
of this gas giant are likely primordial. Its current radius suggests an
envelope with enhanced opacity.

  4. 4.

Planet c was more significantly shaped by photoevaporation. Its modal density
(MG24) is suggestive of an atmosphere and appears incompatible with the small
core set by pebble accretion alone, as the efficient photoevaporation would
have shrunk the planetary radius beyond what is observed.

  5. 5.

The presence of a large core inside planet c argues in favour of formation in
a planetesimal-rich disc, while a small core surrounded by high-metallicity
gas favours a pebble-rich disc. As both planets were born in the same
circumstellar disc, these two formation scenarios also have different
implications for the composition of planet b.

  6. 6.

Future observations with JWST can help discriminate between the two scenarios;
for example by constraining the carbon–sulphur ratio (Turrini et al., 2021;
Crossfield, 2023): a large C/S ratio would point to a pebble-rich disc, while
a lower one to a planetesimal-rich disc.

  7. 7.

TOI-5398’s youth (650 $\pm$ 150 Myr) stands out among similar compact systems
(aged between 3 and 10 Gyr), making it an exceptional target for
photoevaporation studies. The possible survival of a primordial atmosphere of
the inner companion is crucial for yielding comparative planetology and
exploring the disc-driven migration model through detailed atmospheric
characterisation.

###### Acknowledgements.

We greatly acknowledge the fundamental support provided by the TNG director
Adriano Ghedina in approving the allocated time for DDT proposal A46DDT4 (PI:
G. Mantovan). For the same reason, we also acknowledge the support provided by
Vania Lorenzi, Aldo Fiorenzano, and Walter Boschin. GMa acknowledges support
from CHEOPS ASI-INAF agreement n. 2019-29-HH.0. This work benefited from the
2023 Exoplanet Summer Program in the Other Worlds Laboratory (OWL) at the
University of California, Santa Cruz, a program funded by the Heising-Simons
Foundation and NASA. DP acknowledges the support from the Istituto Nazionale
di Oceanografia e Geofisica Sperimentale (OGS) and CINECA through the program
“HPC-TRES (High Performance Computing Training and Research for Earth
Sciences)” award number 2022-05 as well as the support of the ASI-INAF
agreement n 2021-5-HH.1-2022. AMa acknowledges partial support from the PRIN-
INAF 2019 (project HOT-ATMOS). RSp acknowledges the support of the ARIEL
ASI/INAF agreement n. 2021-5-HH.0. RSp, GMi acknowledge the support of the
ASI-INAF agreement n 2021.5-HH.1-2022. This work has been supported by the
PRIN-INAF 2019 “Planetary systems at young ages (PLATEA)”. The authors
acknowledge financial contribution from PRIN MUR 2022 (project No.
2022CERJ49), PRIN-INAF 2019, and the ASI-INAF agreement n.2018-16-HH.0 (THE
StellaR PAth project). GPi, LMa acknowledge partial support by DOR2023 DFA,
UNIPD. L.M. acknowledges financial contribution from PRIN MUR 2022 project
2022J4H55R. TZi acknowledges NVIDIA Academic Hardware Grant Program for the
use of the Titan V GPU card and the support by the CHEOPS ASI-INAF agreement
n. 2019-29-HH.0 and the Italian MUR Departments of Excellence grant 2023-2027
“Quantum Frontiers”.

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##  Appendix A Radial velocities

The RV spectroscopic time series will be available in electronic format as
supplementary material for the Letter at the CDS.

##  Appendix B Bayesian fit

To properly take into account the errors associated with the orbital
parameters, we used Gaussian priors on the orbital period ($P_{\rm b}$), the
central time of the first transit ($T_{0,{\rm b}}$), the planet-to-star radius
ratio ($R_{\rm b}/R_{\star}$), and the impact parameter ($b$). We imposed a
Gaussian prior on the eccentricity following Van Eylen et al. (2019) and
included the RV semi-amplitude $K_{\rm b}$ in the modelling. We also imposed
Gaussian priors on the host star density ($\rho_{\star}$), rotational period
($P_{\rm rot}$), and projected rotational velocity ($v\sin{i_{\star}}$). In
particular, we treated the stellar equatorial velocity ($v_{\rm eq}$, obtained
from $P_{\rm rot}$) and inclination ($i_{\star}$) as independent variables.
Therefore, we are not subject to the bias described in Masuda & Winn (2020).
We treated the stellar limb-darkening (LD) contribution by estimating $\rm
u_{1}$ and $\rm u_{2}$ using PyLDTk555https://github.com/hpparvi/ldtk (Husser
et al. 2013; Parviainen & Aigrain 2015) and assuming a boxcar filter as the
passband in the HARPS-N spectral range. We also modelled and fit the effect of
the stellar convective blueshift (CB) on the in-transit RV curve (e.g. Cegla
et al. 2016) using a linear law as a function of the limb angle. Short-term
stellar activity was included in the model through a jitter term to be added
in quadrature to the errors of the RV.

We performed a global optimisation of the parameters by running PyDE (Storn &
Price 1997) and performing a Bayesian analysis of the RM signal in the RV time
series using emcee Foreman-Mackey et al. (2013). This choice was motivated by
the fact that MCMC samplers allow the specification of a prior on derived
parameters, as in the case of $v\sin{i_{\star}}$, which is not possible in the
case of nested sampling algorithms.

We performed a preliminary analysis including a Gaussian Process with a
Matern-3/2 kernel (e.g. Palle et al. 2020) to model the stellar activity, and
the simultaneous modelling of the RM signals associated with both planets. The
results are perfectly consistent with those presented in Table 1 with
comparable error bars, while the obliquity of planet c remains unconstrained.
For this reason, we preferred the simpler model presented in Sect. 3. A more
complex modelling will be presented after securing more spectroscopic and
photometric in-transit data.

##  Appendix C Simulating planet-formation tracks

We considered a template protoplanetary disc similar to the nominal one from
Turrini et al. (2023) (see Table 2 for a summary of its main physical
parameters) and sampled 4$\times$104 planet formation tracks. Half of our
samples assume centimetre(cm)-sized pebbles and the other half
millimetre(mm)-sized pebbles, to explore the impact of faster and slower
growth rates of the planetary cores (Johansen et al. 2019). The resulting
populations of synthetic planets are shown in Fig. 5. Each formation track is
characterised by a different combination of the initial time and orbital
distance of the planetary seed, both values being obtained using uniform
extractions (see Table 2 for the relevant ranges and see Fig. 4 for
illustrative examples of the growth tracks).

Because of the uncertainties of the real characteristics and lifetime of
TOI-5398’s now-dispersed disc, we considered as successful all planets between
8 and 80 $M_{\oplus}$, that is roughly between 20% lower than the mass of
planet c and 20% higher than the mass of planet b at their current age, and
whose final orbits fall within 0.2 AU. This region in the space of the
solutions is identified by the black box in Fig. 5. Consistently with Johansen
et al. (2019), we find that forming planets in this mass range is more
challenging than forming less massive super-Earths ($<5$ $M_{\oplus}$) and
more massive gas giants ($>100$ $M_{\oplus}$), and that it is comparatively
easier to form synthetic counterparts of planet b than of planet c (see Fig.
5).

Table 2: Input parameters used to run the modified GroMiT code to produce
simulated planets for the TOI 5398 system’s planets b and c.  Simulation
Parameters
---
N∘ of simulations | 4$\times 10^{4}$
Seed formation time | 0.1–1.0$\,\times\,10^{6}$ yr
Disc lifetime |  $3.0\times 10^{6}$ yr
Star, planet, and disc properties
Stellar mass | 1.15 M⊙
Seed mass | 0.01 M⊕
Initial envelope mass | 0.0 M⊕
Initial semimajor axis | 0.5–20.0 au
Disc characteristic radius | 50.0 au
Temperature at 1 au | 200 K
Surface density at 1 au | 420 kg m-2
Disc accretion coefficient, $\alpha$ | 0.01
Turbulent viscosity, $\alpha$ν | 0.0001
Pebble size666Half of the simulations where run using pebble sizes of 1 cm and the other half 1 mm. | 1 cm & 1 mm
Figure 5: Synthetic populations of planets resulting from the two Monte Carlo
runs of 20000 extractions each with GroMiT. The plot shows the final masses
and orbital positions of the simulated growth tracks (see Fig. 4 for
illustrative examples of the growth tracks). The red symbols are associated
with planets formed in discs dominated by mm-sized pebbles, and the blue
symbols with planets formed in discs dominated by cm-sized pebbles. TOI-5398 b
and c are represented by the two larger cyan-filled circles. The black box
highlights the region of the parameter space populated by the successful
solutions (see main text for details).

##  Appendix D Planetary structure

Figure 6: Solutions of core–envelope models for planet TOI-5398 b. (Left) Core
radius vs. theoretical planetary radius for total mass fixed at the measured
value and a rock-ice core (locus of purple points) or a rock-iron core (green
points). The vertical line indicates the measured planetary radius. The two
black points mark two different configurations selected for exploring the
photoevaporation history of the planet. (Right) Similar to the previous plot
but for the atmospheric mass fraction.

To determine the atmospheric mass fraction at present age, we adopted the
core-envelope structure by Fortney et al. (2007) and Lopez & Fortney (2014).
This core-envelope model has four unknowns: the mass and radius of the core,
$M_{\rm core}$ and $R_{\rm core}$, the radius of the envelope, $R_{\rm env}$,
and the atmospheric mass fraction, $f_{\rm atm}$. These quantities are linked
by four relations and two observational constraints: the measured planet
radius and total mass, $R_{\rm p}$ and $M_{\rm p}$; the Lopez & Fortney (2014)
relation which links $R_{\rm env}$, $f_{\rm atm}$, and $M_{\rm p}$; and a
relation between $R_{\rm core}$, $R_{\rm p}$, and $M_{\rm p}$ at the given age
and star–planet distance. For the latter, we adopted the internal structure
models by Fortney et al. (2007), where cores can be composed of different ice-
rock or rock-iron mixtures.

We note that the relation by Lopez & Fortney (2014) was developed for H–He-
dominated atmospheres, and it also takes into account the cooling and
contraction of the envelope as a consequence of its thermal evolution (Lopez
et al. 2012). For atmospheres with solar metallicity, the radius of the
envelope decreases with age as $t^{-0.11}$, while for enhanced opacities the
variation is $\propto t^{-0.18}$.

Assuming cores composed of 50% rocks and 50% ices, and a high-metallicity
envelope, we found for planet b a first closed solution with $M_{\rm
core,b}=16.9$ $M_{\oplus}$ and $R_{\rm core,b}=2.8$ $R_{\oplus}$,
corresponding to the measured planetary mass and optical radius (Fig.6). The
alternative assumption of solar metallicity yields a maximum value of the
planetary radius that is about 7% lower than the available measurement. The
core and envelope densities result around $4.2$ g cm-3 and $0.55$ g cm-3,
respectively. The mean planetary density is around $0.3$ g cm-3.

For planet c, we obtained $M_{\rm core,c}=11.7$ $M_{\oplus}$ and $R_{\rm
core,c}=2.6$ $R_{\oplus}$, a solution self-consistent with the assumed core-
envelope structure and the measured mass and optical radius. These values
imply a core density of $\sim 3.6$ g cm-3, an envelope density $\sim 1.3$ g
cm-3, and a mean planetary density $\sim 1.5$ g cm-3. These core sizes and
masses are larger than predicted by the planetary formation models described
in Sect. 4.3, while the densities are lower.

Alternatively, we searched for a second possible configuration assuming a
denser Earth-like core composition with 67% rocks and 33% iron. In this case,
we found a solution only for planet b with a smaller and less massive core
(Fig.6), having $M_{\rm core,b}=2.0$ $M_{\oplus}$ and $R_{\rm core,b}=1.2$
$R_{\oplus}$, more in line with the prediction of the planetary formation
models.

Figure 7:  (Left) Evolutionary track of TOI 5398 up to an age of 5 Gyr in the
effective temperature–bolometric luminosity plane. A black cross marks the
current location of the star on the track. (Right) Time evolution of X-ray
(5–100 Å), EUV (100–920 Å), and total XUV luminosity of TOI 5398, according to
Penz et al. (2008) and the X-ray/EUV scaling by Sanz-Forcada et al. (2022).
The grey area is the observed $1\sigma$ spread around the median (dark grey
line) of the X-ray luminosity distributions for dG stars in the Hyades and
Pleiades open clusters. Uncertainties on the age and X-ray luminosity of TOI
5398 are also indicated.

##  Appendix E Photoevaporation modelling

We adopted the ATES photoionisation hydrodynamics code by Caldiroli et al.
(2021, 2022). To speed up the computation, we resorted to their analytical
approximation, which provides the planetary mass-loss rate as a function of
planetary mass and radius, star–planet distance, and stellar high-energy flux.
This is achieved by evaluating the effective photoevaporation efficiency,
$\eta_{\rm eff}$, which enters into the classical energy-limited formulation
(Erkaev et al. 2007):

$\dot{M}=\eta_{\rm eff}\frac{3F_{\rm XUV}}{4GK\rho_{\rm
p}}\leavevmode\nobreak\ ,$ (2)

where $F_{\rm XUV}$ is the X-EUV flux at the (average) orbital distance,
$\rho_{\rm p}$ is the mean planetary mass density, and the factor K accounts
for the host star tidal forces (Erkaev et al. 2007). In all cases, we consider
the stellar evolutionary track and the long-term change of the XUV
irradiation.

The track followed by TOI-5398 in the theoretical temperature–luminosity
diagram is shown in Fig. 7, and was computed using the web-based
interpolator777https://waps.cfa.harvard.edu/MIST/interp_tracks.html of the
MESA Isochrones and Stellar Tracks (MIST, Choi et al. 2016).

Figure 8: Evolutionary history of TOI-5398 planets. Panels (a) and (b) show
the planetary mass and radius vs. time. Dotted segments indicate the evolution
backwards from the current age and solid lines evolution forward in time.
These simulations were constructed with the initial conditions given by the
planetary mass and radius measured at the present age of 650 Myr.

Figure 9: Similar to Fig. 8 but for the time evolution of the mass-loss rate (a) and the atmospheric mass fraction (b).  Table 3: Photoevaporation modelling outcome. Core Radius | Core Mass | Mass | Radius | $f_{\rm atm}$ | Mass loss rate | Mass | Radius | $f_{\rm atm}$ | Mass loss rate
---|---|---|---|---|---|---|---|---|---
($R_{\oplus}$) | ($M_{\oplus}$) | ($M_{\oplus}$) | ($R_{\oplus}$) | (%) | (g/s) | ($M_{\oplus}$) | ($R_{\oplus}$) | (%) | (g/s)
Planet b | current age | at 3 Myr
2.8 | 16.9 | 58.7 | 10.3 | 71.3 | $3.2\times 10^{11}$ | 61.1 | 22.3 | 72.4 |  $5.8\times 10^{12}$
1.2 | 2.0 | 58.7 | 10.2 | 96.6 | $3.1\times 10^{11}$ | 61.2 | 24.4 | 96.7 | $6.9\times 10^{12}$
Planet c | current age | at 3 Myr
2.6 | 11.7 | 11.8 | 3.5 | 1.5 | $2.2\times 10^{11}$ | 18.7 | 20.4 | 37.6 |  $4.0\times 10^{13}$

For the stellar X-ray emission at different ages, we adopted the analytic
description by Penz et al. (2008) anchored to the current value of the X-ray
luminosity, $L_{\star,\,X}=10^{29}$ erg s-1 in the band 5–100 Å, derived from
the rotation–activity relationships by Pizzolato et al. (2003). To evaluate
the stellar irradiation in the XUV band, we adopted the scaling law between
EUV (100–920 Å) and X-ray (5–100 Å) luminosities derived by Sanz-Forcada et
al. (2022):

$\log L_{\rm EUV}=(0.793\pm 0.058)\log L_{\rm x}+(6.53\pm
1.61)\leavevmode\nobreak\ ,$ (3)

which is an updated version of the widely used relationship by Sanz-Forcada et
al. (2011). The predicted evolution of the X-ray and EUV stellar luminosities
are shown in Fig. 7.

To determine the evolution of the planetary radius associated with the
variation of the planetary mass, we adopted the core–envelope structure model
described in Appendix D. With this model, we incorporate a time dependence of
the envelope size linked to the variation of the surface equilibrium
temperature during the evolution.

We then proceeded with the time evolution of the photoevaporation. For each
time step of the simulation, we compute the mass-loss rate and update $f_{\rm
atm}$ and the planetary mass, obtaining a new value of $R_{\rm env}$ with the
relation by Lopez & Fortney (2014). The latter quantity added to the core
radius (assumed constant) provides the updated planetary radius.

According to the aforementioned scenario and assuming a core that does not
change in size or mass, we followed the planetary evolution back in time. We
stopped our simulations in the past at 3 Myr, the end time of planetary
formation models, that is, when the circumstellar disc has already disappeared
and each planet is in its final, stable orbit. The results of the simulations,
described in Sect. 4.4, are shown in Figures 8 and 9, and in Table 3.

The mass-loss rate for planet b closely follows the broken power-law decrease
in the XUV irradiation over time (Fig. 7). The variation for planet c appears
smoother and faster. The reason for the different behaviour is that the mean
density of planet b changes little during the evolution, while it increases
substantially for planet c. This also explains why the mass-loss rate of
planet c becomes lower than that of planet b at an age of around $530$ Myr.

We checked the deviations of the actual numerical results of the ATES code
from the analytic approximation we employed. Our computed mass-loss rates are
systematically lower by a factor of 1.5–2 than the more precise values. This
difference is negligible for the evolution of planet b but may imply an
underestimation of the initial mass and radius for planet c (by less than a
factor of 2). The mass-loss rate could be overestimated by a similar factor if
the EUV flux were predicted with the scaling law proposed by Johnstone et al.
(2021) instead of the adopted Eq. 3.

We stress that for the backward-in-time modelling, we are considering as
initial conditions the current planetary core–envelope structure (Appendix D).
These initial conditions provide strong constraints on the possible
evolutionary histories of the two planets.

##  Appendix F Dynamics of the system

The system’s dynamics is currently dominated by the mutual secular
perturbations of planets and tides. In the long term, the system is stable
even if the outer planet (b) were closer than the nominal orbit (up to a limit
of $a_{b}>0.07$ AU) and a high eccentricity (0.2) is assumed. At present, the
two planets are very close to a 9:4 mean motion resonance, which may have been
broken after the dissipation of the gaseous disc. It is a weak resonance, but
it may be responsible for an eccentricity value slightly different from zero
for both planets, built during the resonance capture induced by the migration
of the planets while embedded in the disc.

If planet b were in an almost circular orbit, the secular perturbations on
planet c would cause only a low forced eccentricity (of the order of 0.001),
which might trigger some tidal deformation — as in the case of Io and Jupiter.
On the other hand, if planet b were on an eccentric orbit with a value equal
to the upper boundary (0.12) given by the data analysis (see Table 1) then the
forced eccentricity of planet c would be about 0.08 with a sinusoidal
oscillation amplitude of 0.16. In the latter case, planet c would be likely to
undergo strong tidal deformations, which would possibly represent an
additional heat source.