# TOI-1408: Discovery and Photodynamical Modeling of a Small Inner Companion
to a Hot Jupiter Revealed by TTVs

Judith Korth Lund Observatory, Division of Astrophysics, Department of Physics, Lund University, Box 118, 22100 Lund, Sweden  Priyanka Chaturvedi Thüringer Landessternwarte Tautenburg, Sternwarte 5, 07778 Tautenburg, Germany  Tata Institute of Fundamental Research (TIFR), Homi Bhabha Road, Colaba, Mumbai - 400005, India  Hannu Parviainen Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain  Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain  Ilaria Carleo  INAF – Osservatorio Astrofisico di Torino, Via Osservatorio 20, I-10025, Pino Torinese, Italy  Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain  Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain  Michael Endl  McDonald Observatory, The University of Texas, Austin Texas USA  Center for Planetary Systems Habitability, The University of Texas, Austin Texas USA  Eike W. Guenther Thüringer Landessternwarte Tautenburg, Sternwarte 5, 07778 Tautenburg, Germany  Grzegorz Nowak Institute of Astronomy, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudzia̧dzka 5, 87-100 Toruń, Poland  Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain  Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain  Carina Persson Department of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, SE-439 92 Onsala, Sweden  Phillip J. MacQueen  McDonald Observatory, The University of Texas, Austin Texas USA  Alexander J. Mustill Lund Observatory, Division of Astrophysics, Department of Physics, Lund University, Box 118, 22100 Lund, Sweden  Juan Cabrera Institut für Planetenforschung, Deutsches Zentrum für Luft- und Raumfahrt, Rutherfordstr. 2, 12489 Berlin, Germany  William D. Cochran McDonald Observatory, The University of Texas, Austin Texas USA  Center for Planetary Systems Habitability, The University of Texas, Austin Texas USA  Jorge Lillo-Box Centro de Astrobiologiía (CAB), CSIC-INTA, Camino Bajo del Castillo s/n, ESAC campus, 28692, Villanueva de la Cañada, Madrid, Spain  David Hobbs Lund Observatory, Division of Astrophysics, Department of Physics, Lund University, Box 118, 22100 Lund, Sweden  Felipe Murgas Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain  Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain  Michael Greklek-McKeon Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA, 91125, USA  Hanna Kellermann University Observatory Munich, Faculty of Physics, Ludwig-Maximilians-Universität München, Scheinerstr. 1, 81679 Munich, Germany  Guillaume Hébrard  Institut d’astrophysique de Paris, UMR7095 CNRS, Université Pierre & Marie Curie, 98bis boulevard Arago, 75014 Paris, France  Observatoire de Haute-Provence, CNRS, Université d’Aix-Marseille, 04870 Saint-Michel-l’Observatoire, France  Akihiko Fukui Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan  Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain  Enric Pallé  Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain  Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain  Jon M. Jenkins NASA Ames Research Center, Moffett Field, CA 94035, USA  Joseph D. Twicken SETI Institute, Mountain View, CA 94043 USA  NASA Ames Research Center, Moffett Field, CA 94035, USA  Karen A. Collins Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Samuel N. Quinn Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Ján Šubjak  Astronomical Institute, Czech Academy of Sciences, Fričova 298, 251 65, Ondřejov, Czech Republic  Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Paul G. Beck Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain  Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain  Davide Gandolfi Dipartimento di Fisica, Università degli Studi di Torino, via Pietro Giuria 1, I-10125, Torino, Italy  Savita Mathur Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain  Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain  Hans J. Deeg Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain  Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain  David W. Latham Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Simon Albrecht Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark  David Barrado Centro de Astrobiologiía (CAB), CSIC-INTA, Camino Bajo del Castillo s/n, ESAC campus, 28692, Villanueva de la Cañada, Madrid, Spain  Isabelle Boisse  Observatoire de Haute-Provence, CNRS, Université d’Aix-Marseille, 04870 Saint-Michel-l’Observatoire, France  Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France  Hervé Bouy Laboratoire d’astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, allée Geoffroy Saint-Hilaire, 33615 Pessac, France  Institut universitaire de France (IUF), 1 rue Descartes, 75231 Paris CEDEX 05  Xavier Delfosse  Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France  Olivier Demangeon  Instituto de Astrofísica e Ciências do Espaço, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal  Rafael A. García Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France  Artie P. Hatzes Thüringer Landessternwarte Tautenburg, Sternwarte 5, 07778 Tautenburg, Germany  Neda Heidari  Institut d’astrophysique de Paris, UMR7095 CNRS, Université Pierre & Marie Curie, 98bis boulevard Arago, 75014 Paris, France  Kai Ikuta Department of Multi-Disciplinary Sciences, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan  Petr Kabáth  Astronomical Institute, Czech Academy of Sciences, Fričova 298, 251 65, Ondřejov, Czech Republic  Heather A. Knutson Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA, 91125, USA  John Livingston  Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Astronomical Science Program, Graduate University for Advanced Studies, SOKENDAI, 2-21-1, Osawa, Mitaka, Tokyo, 181-8588, Japan  Eder Martioli Laboratório Nacional de Astrofísica, Rua Estados Unidos 154, 1006 37504-364, Itajubá - MG, Brazil  Institut d’astrophysique de Paris, UMR7095 CNRS, Université Pierre & Marie Curie, 98bis boulevard Arago, 75014 Paris, France  María Morales-Calderón Centro de Astrobiologiía (CAB), CSIC-INTA, Camino Bajo del Castillo s/n, ESAC campus, 28692, Villanueva de la Cañada, Madrid, Spain  Giuseppe Morello  Instituto de Astrofísica de Andalucía (IAA-CSIC), Gta. de la Astronomía s/n, 18008 Granada, Granada, Spain  Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain  Norio Narita Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan  Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain  Jaume Orell-Miquel  Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain  Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain  Hanna L. M. Osborne  Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK  European Southern Observatory, Karl-Schwarzschild-Straße 2, Garching bei München D-85748, Germany  Dinil B. Palakkatharappil Université Paris-Saclay, Université Paris Cité, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France  Viktoria Pinter  Nordic Optical Telescope, Rambla José Ana Fernandez Pérez 7, 38711, Breña Baja, Spain  University of Craiova, Alexandru Ioan Cuza 13, 200585, Craiova, Romania  Seth Redfield Astronomy Department and Van Vleck Observatory, Wesleyan University, Middletown, CT 06459, USA  Howard M. Relles Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Richard P. Schwarz Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Sara Seager Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Department of Aeronautics and Astronautics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA  Avi Shporer Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Marek Skarka  Astronomical Institute, Czech Academy of Sciences, Fričova 298, 251 65, Ondřejov, Czech Republic  Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Kotlářská 267/2, 611 37 Brno, Czech Republic  Gregor Srdoc  Kotizarovci Observatory, Sarsoni 90, 51216 Viskovo, Croatia  Monika Stangret INAF – Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122, Padova, Italy  Luis Thomas University Observatory Munich, Faculty of Physics, Ludwig-Maximilians-Universität München, Scheinerstr. 1, 81679 Munich, Germany  Max-Planck Institute for Extraterrestrial Physics, Giessenbachstrasse 1, D-85748 Garching, Germany  Vincent Van Eylen Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK  Noriharu Watanabe Department of Multi-Disciplinary Sciences, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan  Joshua N. Winn Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA

###### Abstract

We report the discovery and characterization of a small planet, TOI-1408 c, on
a 2.2-day orbit located interior to a previously known hot Jupiter, TOI-1408 b
(P=$4.42$ d, M=$1.86\pm 0.02$$M_{\mathrm{Jup}}$, R=$2.4\pm
0.5$$R_{\mathrm{Jup}}$) that exhibits grazing transits. The two planets are
near 2:1 period commensurability, resulting in significant transit timing
variations (TTVs) for both planets and transit duration variations (TDVs) for
the inner planet. The TTV amplitude for TOI-1408 c is 15% of the planet’s
orbital period, marking the largest TTV amplitude relative to the orbital
period measured to date. Photodynamical modeling of ground-based radial
velocity (RV) observations and transit light curves obtained with the
Transiting Exoplanet Survey Satellite (TESS) and ground-based facilities leads
to an inner planet radius of $2.22\pm 0.06$$R_{\oplus}$ and mass of $7.6\pm
0.2$$M_{\oplus}$ that locates the planet into the Sub-Neptune regime. The
proximity to the 2:1 period commensurability leads to the libration of the
resonant argument of the inner planet. The RV measurements support the
existence of a third body with an orbital period of several thousand days.
This discovery places the system among the rare systems featuring a hot
Jupiter accompanied by an inner low-mass planet.

Exoplanet dynamics (490)–Hot Jupiters (753)–Hot Neptunes (754)–Transit timing
variation method (1710)–Transit photometry (1709)–Radial velocity (1332)

††facilities: NOT, CTIO:1.3m, CTIO:1.5m, CXO, LCOGT, Hale, TNG, Smith, TLS,
CA0:2.2m, OHP:1.93m, WO:2m, FLWO:1.5m, OO:2, Mercator1.2m.††software: PyTTV
(Korth et al., 2023), PyTransit (Parviainen, 2015), Rebound (Rein & Liu,
2012), LDTk (Parviainen & Aigrain, 2015), emcee (Foreman-Mackey et al., 2013),
AstroImageJ (Collins et al., 2017), TAPIR (Jensen, 2013), Astropy (Astropy
Collaboration et al., 2013, 2018, 2022), NumPy (Van Der Walt et al., 2011),
SciPy (Virtanen et al., 2020), matplotlib (Hunter, 2007)

##  1 Introduction

The discovery of the first exoplanet around a main sequence star, 51 Peg b,
significantly advanced our understanding of planetary systems due to its
dissimilarity (e.g. short orbital period) from any known planets in our Solar
system (Mayor & Queloz, 1995). Nearly 30 years later, the origins of such hot
Jupiters (HJs)—gas giants with orbital periods less than 10 days and with
planet masses higher than 0.1$M_{\mathrm{Jup}}$ following the definition from
Wang et al. (2015)—remain elusive. Theories suggest that HJs could form in
situ (e.g., Batygin et al., 2016), migrate inward from beyond the ice line
through the interaction with the gas disk during formation (e.g, Lin et al.,
1996), or undergo high-eccentricity migration (HEM: Rasio & Ford, 1996) at a
later stage. For an overview, see Dawson & Johnson (2018) and references
therein.

The lack of detections of low-mass planets interior to HJs (Steffen et al.,
2012; Mustill et al., 2015; Huang et al., 2016; Hord et al., 2021) is a key
argument supporting HEM as the dominant formation channel. However,
exceptional systems where an inner low-mass planet accompanies an HJ have been
detected, such as WASP-47 (Hellier et al., 2012; Becker et al., 2015;
Nascimbeni et al., 2023), WASP-84 (Anderson et al., 2014; Maciejewski et al.,
2023), Kepler-730 (Zhu et al., 2018; Cañas et al., 2019), TOI-2000 (Sha et
al., 2023), WASP-132 (Hellier et al., 2017; Hord et al., 2022; Grieves et al.,
2024), and TOI-1130 (Huang et al., 2020a; Korth et al., 2023). These systems
cannot be explained by HEM and require dynamically quiet formation process,
such as disk migration (Fogg & Nelson, 2005, 2007; Wu et al., 2023; Wu & He,
2023; He et al., 2024), or a less quiet process, such as in situ formation
(Poon et al., 2021).

In this paper, we present the discovery of another of these rare systems
containing an HJ and a low-mass planet close to the 2:1 period
commensurability, resulting in measurable TTVs, similar to TOI-1130. We report
the discovery and characterization of a small planet, TOI-1408 c, interior to
a known grazing hot Jupiter, TOI-1408 b, discovered using the Transiting
Exoplanet Survey Satellite (TESS; Ricker et al., 2015) photometry and
confirmed by Galazutdinov et al. (2023). Our analysis also refines the orbital
and geometric properties of TOI-1408 b.

##  2 Observations

###  2.1 Photometric Observations

####  2.1.1 TESS Photometry

TOI-1408 (TIC 364186197) was observed at 2-min cadence by TESS in Sectors 16,
17, 18, 19, 24, 25, 52, 57, 58, 59, 73, and 76 from 2019-09-12 to 2024-03-25,
spanning 61 transits for TOI-1408 b and 114 transits for TOI-1408 c. We used
the publicly available Presearch Data Conditioning (PDC) light curves (Smith
et al., 2012; Stumpe et al., 2012, 2014) produced by the Science Processing
Operations Center (SPOC: Jenkins et al., 2016) at NASA Ames Research Center,
downloaded from the Mikulski Archive for Space
Telescopes.111https://mast.stsci.edu.

####  2.1.2 Ground-based Photometry

Table 1: Ground-based photometry. time [BJD] | Flux | e_Flux | Instrument
---|---|---|---
2459501.794176 | 1.001 | 0.002 | M3_g
2459501.794335 | 1.002 | 0.002 | M3_g
2459501.794457 | 0.999 | 0.002 | M3_g
2459501.794579 | 1.000 | 0.002 | M3_g
2459501.794702 | 0.999 | 0.002 | M3_g
2459501.794824 | 1.000 | 0.002 | M3_g

##### NOT/ALFOSC

We observed five TOI-1408 b transits in the $i$ band using the Alhambra Faint
Object Spectrograph and Camera (ALFOSC) instrument installed at the 2.56-m
Nordic Optical Telescope (NOT) at the Roque de los Muchachos Observatory on La
Palma, Spain, between 2021-07-08 and 2022-08-28. The photometry was reduced
with our pipeline following standard photometry practices (Parviainen et al.,
2019). The reduced photometry from NOT and all the other instruments is
available in Table 1.

##### LCOGT/Sinistro

We observed six TOI-1408 b transits in the $i$ band using the Sinistro cameras
installed at 1-m telescopes from the Las Cumbres Observatory Global Telescope
(LCOGT; Brown et al., 2013) between 2021-07-21 and 2023-07-30. The photometry
was reduced with the same pipeline as the NOT photometry. We also observed two
transits of TOI-1408 b on 2021-05-21 and on 2022-09-11 in Pan-STARRS $z$-short
band. The $z$-short band images were calibrated by the standard LCOGT BANZAI
pipeline (McCully et al., 2018) and differential photometric data were
extracted using AstroImageJ (Collins et al., 2017).

##### LCOGT/MuSCAT3

We observed two transit of TOI-1408 b using the multi-color MuSCAT3 instrument
(Narita et al., 2020) installed at the 2-m Faulkes Telescope North of LCOGT in
Maui, Hawaii on 2021-10-14 and 2022-11-12. The transits were observed
simultaneously in the $g$, $r$, $i$, and $z_{s}$ bands. The images were
calibrated by the BANZAI pipeline (McCully et al., 2018), and aperture
photometry was performed with a custom pipeline described in Fukui et al.
(2011). We included only the first transit due to the low S/N of the second
observation.

##### Palomar/WIRC

We observed one transit of TOI-1408 b on 2022-09-02 in the K-continuum band
using the Wide-field Infrared Camera installed on the 5.1-m Hale Telescope at
Palomar Observatory in California. We calibrated the images with the pipeline
described in Vissapragada et al. (2020), and then performed aperture
photometry and detrended the light curve with the procedure described in
Greklek-McKeon et al. (2023).

###  2.2 Spectroscopic Observations

Table 2: RV measurements and their uncertainties. time [BJD] | RV [m/s] | e_RV [m/s] | Instrument
---|---|---|---
2459114.650183 | -33581.2 | 12.5 | McD HJST
2459115.814081 | -33383.9 | 19.8 | McD HJST
2459116.786824 | -33541.8 | 16.2 | McD HJST
2459133.725274 | -33413.8 | 19.9 | McD HJST
2459135.701365 | -33784.5 | 22.7 | McD HJST
2459143.690476 | -33627.5 | 8.7 | McD HJST

##### HARPS-N

We obtained 60 high-resolution spectra using the High Accuracy Radial velocity
Planet Searcher-North (HARPS-N: $\lambda$ $\in$ (378–691) nm, $R\approx
115\,000$; Cosentino et al., 2012) mounted at the 3.58-m Telescopio Nazionale
Galileo (TNG) of Roque de los Muchachos Observatory in La Palma, Spain, under
the observing program CAT21A_119 between 2021-05-21 and 2024-06-19. The
exposure time was set to 480–1800 seconds based on weather conditions and
scheduling constraints, leading to a signal-to-noise ratio (SNR) per pixel of
36–112 at 550 nm. The spectra were extracted using the off-line version of the
HARPS-N DRS pipeline (Cosentino et al., 2014), version HARPN_3.7. Absolute
radial velocities (RVs) were measured using an on-line version of the DRS, the
YABI tool,222Available at http://ia2-harps.oats.inaf.it:8000. by cross-
correlating the extracted spectra with a G2 mask (Baranne et al., 1996). The
DRS RVs from HARPS-N and all other instruments can be found in Table 2.

##### McDonald

We observed 89 high-precision spectra of TOI-1408 using the McDonald
Observatory 2.7-m Harlan J. Smith telescope (HJST) with its Tull Coudé
spectrometer ($\lambda$ $\in$ (340–1090) nm, $R\approx 60\,000$; Tull et al.,
1995). We pass the starlight through an I2 gas absorption cell in front of the
spectrograph entrance slit in order to impose our high-precision RV metric on
the spectrum. We used an exposure meter to terminate each observation when an
SNR of about 25-40 per pixel was achieved. All observations were reduced and
1-D spectra were extracted using standard IRAF routines. We computed the
radial velocities given in Table 2 using the AUSTRAL code (Endl et al., 2000).

##### TLS

We obtained 232 spectra of TOI-1408 using the Coudé-Échelle spectrograph of
the 2-m-Alfred Jensch telescope of the Thüringer Landessternwarte (TLS)
Tautenburg. The spectrograph’s old camera (CCD3, $\lambda$ $\in$ (452–765) nm,
$R=51\,000$) was updated to a new (Andor, $\lambda$ $\in$ (460–734) nm,
$R=63\,000$) during the observing period. When modeling the RVs, we allowed
for a velocity offset between the pre-upgrade and post-upgrade datasets
because of the difference between the two cameras. We obtained 74 spectra with
CCD3 and 158 spectra with the Andor-CCD with most exposure times of 1200 s.
The observations were carried out using an iodine cell for wavelength
calibrations. The RVs were obtained from the reduced spectra using viper
(Zechmeister et al., 2021) and the co-added HARPS-N spectrum as a template.

##### CAFE

We observed 20 spectra of TOI-1408 with the Calar Alto Fiber-fed Echelle
(CAFE) spectrograph ($\lambda$ $\in$ (407–925) nm, $R=60\,000$; Aceituno et
al., 2013) mounted at the 2.2-m telescope of the Calar Alto observatory
between 2021-11-10 and 2022-08-19 with a typical signal-to-noise ratio of 30.
We used the observatory pipeline described in Lillo-Box et al. (2020) to
perform the basic reduction and extraction of the spectra. This pipeline also
determines the radial velocity by performing cross-correlation against a solar
binary mask and correcting them from intra-night drifts using the Thorium-
Argon frames obtained in between each science frame. Usually, several epochs
were obtained per night with a mean individual radial velocity uncertainty of
17 m/s. The RVs were corrected for nightly zero points determined by observing
the same sample of radial velocity standards every observing night.

##### SOPHIE

We observed 15 high-precision spectra of TOI-1408 with the SOPHIE échelle
spectrograph ($\lambda$ $\in$ (387–694) nm, $R=75\,000$) between August 2020
and August 2023 installed at the 1.93-m telescope of Observatoire de Haute-
Provence, France (Perruchot et al., 2008; Bouchy et al., 2009). For each
spectrum, the exposure time was between 360 s and 1020 s, providing an SNR per
pixel at 550 nm between 38 and 53 depending on the weather conditions. The
radial velocities were extracted with the standard SOPHIE pipeline using
cross-correlation as presented by Bouchy et al. (2009) and refined by Heidari
et al. (2024). The derived radial velocities have typical uncertainties of the
order of $\pm 7$ m/s.

##### MaHPS

We observed 61 spectra between May and October 2022 using the 2.1 m Fraunhofer
Telescope at the Wendelstein Observatory (WO) in the German Alps. Our Manfred-
Hirt-Planet Spectrograph (MaHPS) comprises the high-resolution spectrograph
FOCES combined with the Menlo Systems Astrofrequency comb as a calibration
light source ($\lambda$ $\in$ (380–880) nm, $R=60\,000$). For the data
reduction from 2D to 1D, we used GAMSE. For the comb calibration, creation of
the b-spline optimized templates, and RV extraction via a fit, we used our
pipeline MARMOT. Descriptions of both can be found in Kellermann (2021). We
provide two data sets one with Thar and one with comb calibration.

##### TRES

We obtained 18 spectra with the Tillinghast Reflector Echelle Spectrograph
(TRES; $\lambda$ $\in$ (384–910) nm, $R=44\,000$; Fűrész, 2008) mounted on the
1.5-m Tillinghast Reflector at the Fred Lawrence Whipple Observatory on Mount
Hopkins in southern Arizona between 2019-12-05 and 2024-06-17. The reduction
and RV analysis followed the procedures described in Buchhave et al. (2010)
and Quinn et al. (2012). The main difference is that the template spectrum for
the RV extraction was created by median-combining all of the out-of-transit
observed spectra of TOI-1408 (after shifting to align them). Thus, these are
relative velocities where the internal precision of each observation has been
based on the scatter in the velocities between the individual echelle orders.

##### OES

We observed 14 spectra with the Ondrejov Echelle Spectrograph (OES; $\lambda$
$\in$ (380–900) nm, $R=50\,000$; Kabáth et al., 2020) installed on a 2-m Perek
telescope in Ondřejov, Czech Republic. Exposure times varied from 2700 s to
3600 s depending on the weather conditions. The data are reduced via standard
spectroscopic IRAF routines. OES is used for the KESPRINT follow-up of TESS
targets (Šubjak et al., 2020; Tran et al., 2022; Kabáth et al., 2022).

##### HERMES

We monitored the system for ten days in August 2020 with the High Efficiency
and Resolution Mercator Echelle Spectrograph (HERMES; $\lambda$ $\in$
(375–900) nm, $R=90\,000$; Raskin et al., 2011), mounted on the 1.2 m Mercator
telescope at the Spanish Observatorio del Roque de los Muchachos of the
Instituto de Astrofísica de Canarias. We obtained these observations with
simultaneous thorium-argon emission spectra, which allows a precision of 2 to
3 m/s (Beck et al., 2015).

##  3 Analysis

###  3.1 Stellar characterization

We carried out a stellar analysis using the co-added HARPS-N spectra following
an approach described in Fridlund et al. (2017) and Persson et al. (2018)
using the empirical SpecMatch-Emp code (Yee et al., 2017) and the Spectroscopy
Made Easy (SME) analysis package (Valenti & Piskunov, 1996; Piskunov &
Valenti, 2017) version 5.22 to obtain the effective temperature,
$T_{\mathrm{eff}}$, the logarithm of the surface gravity, $\log g$,
$V\sin{i_{\star}}$, and the abundance of iron relative to hydrogen, [Fe/H].
The derived stellar parameters from SME were used to model the stellar radius
and mass with the spectral energy distribution Bayesian model averaging fitter
(ARIADNE; Vines & Jenkins, 2022) using the Phoenix v2 (Husser et al., 2013),
BtSettl (Allard et al., 2012), Castelli & Kurucz (2003), and Kurucz (1979)
atmospheric model grids. The stellar rotation period, estimated from the
spectroscopically derived $V\sin{i_{\star}}$ and the stellar radius assuming
that the star is seen equator-on, is $7.9\pm 0.6$ d. This period is also
identified from the TESS photometry with a Lomb-Scargle periodogram as well as
by running the rotation pipeline (Mathur et al., 2010; García et al., 2014;
Santos et al., 2021) on the Quick Look Pipeline (Huang et al., 2020b) data, we
derived a rotation period of $7.5\pm 0.62$ d, indicating that the planetary
orbital axis is indeed well-aligned with the spin of the star. This supports a
more quiet formation scenario, such as disk migration. The results from the
models are listed in Table 3 alongside further stellar properties.

Table 3: Stellar parameters of TOI-1408. Parameter | Value | Reference
---|---|---
RA [J2000, epoch 2016] | 20:54:02.653 | 1
Dec [J2000, epoch 2016] | +72:34:50.34 | 1
parallax [mas] | $7.16\pm 0.01$ | 1
$\rm\mu_{\alpha\star}$ [mas $\mathrm{yr}^{-1}$] | $11.85\pm 0.01$ | 1
$\rm\mu_{\delta\star}$ [mas $\mathrm{yr}^{-1}$] | $28.73\pm 0.01$ | 1
RUWE | 0.88 | 1
Distance [pc] | $139.13\pm 0.2$ | 1
V [mag] | $9.27\pm 0.02$ | 2
J [mag] | $8.37\pm 0.02$ | 3
Spectral type | F8 V | this work
Age [Gyr] | $2.7\pm 0.3$ | this work
$T_{\mathrm{eff}}$ [K] | $6117\pm 31$ | this work
[Fe/H] [dex] | $0.25\pm 0.06$ | this work
$\log g$ [cm $\mathrm{s}^{-2}$] | $4.10\pm 0.06$ | this work
$V\sin{i_{\star}}$ [km $\mathrm{s}^{-1}$] | $9.8\pm 0.7$ | this work
$M_{\star}$ [$M_{\odot}$ ] | $1.31\pm 0.01$ | this work
$R_{\star}$ [$R_{\odot}$ ] | $1.53\pm 0.02$ | this work
$\rho_{\star}$ [g $\mathrm{cm}^{-3}$] | $0.51\pm 0.02$ | this work
$L_{\star}$ [$L_{\odot}$] | $2.96\pm 0.10$ | this work

###  3.2 Discovery of TOI-1408 c

A transit timing variation (TTV) search using Python Tool for Transit
Variation (PyTTV; Korth et al., 2023) revealed a clear TTV signal for TOI-1408
b, suggesting the presence of an additional body in the TOI-1408 system. We
used the Open Transit Search pipeline (OpenTS; Pope et al., 2016) to search
for additional transit signals from the TESS Sectors 16, 17, 18, 19, 24, and
25. This analysis identified a transit-like signal with a period of 2.22 d,
depth of 100 ppm, and transit duration of 0.62 h. This signal has not been
reported as a TOI by the TESS mission. We also identified the transiting
signal using the Détection Spécialisée de Transits code (DST; Cabrera et al.,
2012).

Following the detection of the tentative transit signal, we carried out a
dynamical stability analysis using Rebound (Rein & Liu, 2012), which confirmed
that the orbits are compatible with long-term stability and predicted large
TTVs. Further analysis of individual TESS sectors supported the existence of
large TTVs for TOI-1408 c. The transit durations and depths estimated from
single sectors differed from those estimated by combining multiple sectors,
leading to smearing of the transit shape. The orbital period was estimated to
vary between 2.18 to 2.22 d from sector to sector. The individual transits
have too low SNR to measure their transit durations, but we estimated the
average transit durations combining two TESS sectors. Sectors 16 and 17 yield
an average transit duration of 2.75 h, while Sectors 73 and 76 yield an
average transit duration of 2.42 h.

After our detection, the SPOC conducted an independent multi-sector transit
search of the light curves up to Sector 59, using an adaptive, noise-
compensating matched filter (Jenkins, 2002; Jenkins et al., 2010, 2020). The
4.42-day signal of TOI-1408 b was identified with a high significance, and
after removing TOI-1408 b’s transits, the next strongest signal ($8.16\sigma$)
had an orbital period of $\sim 2.22$ days. Despite exceeding the $7.1\sigma$
threshold, this detection was vetoed by the transit search pipeline’s
$\chi^{2}$ discriminator (Seader et al., 2013), likely due to smearing from
large TTVs.

###  3.3 Radial Velocities

We carried out a two-planet RV analysis using all the RV data described in
Sect. 3.3 with PyTransit’s RVLPF RV modeling class (Parviainen, 2015). This
code models the RV signals from two planets, incorporating an additional free
RV offset and jitter term for each RV data set, a sinusoidal stellar
rotational signal, and a quadratic trend in time to account for possible long-
period companions. An analysis with wide normal priors on the planet b and c
periods and transit centers identified with OpenTS leads to a posterior period
estimate of $2.1613\pm 0.0002$ d and an RV semi-amplitude of $5.0\pm 1.0$ m/s
that correspond to a minimum mass of $13.0\pm 3.0$ $M_{\oplus}$. The RV
information about TOI-1408 c comes from the HARPS-N data combining high
precision with a long observing time span. For TOI-1408 b, we obtain a close-
to-circular orbit with an RV semi-amplitude of $191.3\pm 0.9$ m/s, translating
to a minimum planet mass of $1.846\pm 0.009$ $M_{\mathrm{Jup}}$.

The RV analysis also detected a clear non-linear trend showing evidence of an
outer companion with an orbital period of thousands of days. Including the
first two TRES points observed $\sim$200 days before the main observing
campaign started gives a Keplerian orbit with a period of $\sim$2530 d with
$K=195\pm 4$m $\mathrm{s}^{-1}$, $M_{\mathrm{p}}=14.6\pm
0.3$$M_{\mathrm{Jup}}$ and an eccentricity of $0.35\pm 0.02$. However, since
the two points responsible for the solution are separated from the rest, we
leave the characterization of the outer planet as future work requiring
significantly longer RV follow-up.

Gaia DR3 astrometry indicates that the source is a primary star with 25
visibility periods and a ruwe of 0.88, indicating the solution is a good fit
to linear space motion. At first sight, this would indicate it is a single
star (Gaia astrometry could not detect planets with periods of a few days).
However, the astrometry also shows an excess noise of 0.062 mas with a high
significance of 4.7, which indicates there is a significant but small
disturbance remaining. Gaia DR3 uses 34 months of data and may be sampling a
substantial fraction of the suggested orbital period; however, very long
orbital periods would require much more than 34 months of data to detect a
companion.

We can estimate the astrometric signature of the long-period outer companion
using the parameters given in this paper and assuming an almost circular
orbit. The period can be used to estimate a semi-major axis for a roughly
circular orbit, and the astrometric signature is obtained using

$\alpha=\left(\frac{M_{p}}{M_{\star}}\right)\left(\frac{a_{p}}{1~{}{\rm
AU}}\right)\left(\frac{d}{1~{}{\rm pc}}\right)~{}\left[{\rm arsec}\right]\,.$
(1)

The suspected third planet gives an astrometric signature of 0.364 mas which
should be easily detectable by Gaia, but with a period of almost 7 years
compared to the Gaia mission duration of only 2.8 years, it could remain
undetected until Gaia DR4 or DR5, especially if the planet is near apogee or
if its period is underestimated.

###  3.4 Photodynamical analysis

Figure 1: Photometry from TESS (top) and the RV measurements from different
instruments (bottom) with the posterior models from the photodynamical
analysis. The photometry is centered around the transit centers, with light
gray points showing the raw photometry and black points with error bars show
the photometry binned to 7 min. The RV figure shows the RV observations as
black symbols with error bars, the photodynamical RV model as a green line,
and the quadratic trend as a black dashed line. The gray symbols mark
observations with uncertainties $>25$ m/s, and the black symbols observations
with uncertainties $<25$ m/s. See Fig. 5 in Appendix A for the ground-based
photometry of TOI-1408 b and Fig. 6 in Appendix B for a detailed illustration
of the RV observations and the RV model. Figure 2: Posterior TTV model from
the photodynamical modeling of TOI-1408 c (upper panel) and TOI-1408 b (lower
panel). The colored lines span the 0.5 to 99.5 percentiles of the TTV model
posterior distribution from the photodynamical analysis, but the uncertainties
are mostly smaller than the line width. The dashed lines mark the subtracted
mean orbital periods and the TTVs with their individual uncertainties measured
by fitting each transit center separately are shown as black (TESS) and gray
(ground-based) points with error bars for comparison.

Since the planets strongly interact gravitationally, producing significant
TTVs, we performed a photodynamical analysis by modeling the TESS photometry,
the ground-based photometry, and the ground-based RVs from various facilities.
This was done using PyTTV following the approach described in Korth et al.
(2023). We assumed a two-planet model, a sinusoidal RV signal to account for
the stellar rotation based on the results from the stellar characterization in
Sec. 3.1, and a quadratic RV trend based on the results from the RV analysis
in Sec. 3.3. The model is parametrized as described in Korth et al. (2023),
with the exception of the impact parameter, $b$. Instead, we used the grazing
parameter, $g$ $=b/(1+k)$, where $k$ is the radius ratio, due to the grazing
geometry of TOI-1408 b. The longitudes of the ascending notes, $\Omega$, are
fixed to $\pi$ for both planets. The model is parameterized using the sampling
parameters $\sqrt{e}\cos{\omega}$ and $\sqrt{e}\sin\omega$, but we also set a
prior on the orbital eccentricities. The model parameters and their priors are
listed in Table 4.

The code models the photometry and RVs simultaneously using Rebound (Rein &
Liu, 2012; Rein & Spiegel, 2015; Tamayo et al., 2020) for dynamical
integration, including General Relativity effects as implemented in Reboundx
(Tamayo et al., 2020), and PyTransit (Parviainen, 2015, 2020; Parviainen &
Korth, 2020) for transit modeling. The analysis begins with a global
optimization using the differential evolution method (Storn & Price, 1997;
Price et al., 2005), followed by Markov Chain Monte Carlo (MCMC) sampling
starting from the global optimization results with the emcee sampler (Foreman-
Mackey et al., 2013).

We show the TESS photometry and RV measurements with their corresponding model
in Fig. 1, the ground-based photometry in Fig. 5 in Appendix A, the TTV model
in Fig. 2, and the full RV model in Fig. 6 in Appendix B. The posteriors for
model parameters and the derived planetary parameters are listed in Table 4.
The posterior mean ephemeris for TOI-1408 c is $T_{0,c}=2458739.845\pm 0.002$
d and $P_{c}=2.19592\pm 0.00002$ d, while for TOI-1408 b, the ephemeris is
$T_{0,b}=2458740.8584\pm 0.0002$ d and $P_{b}=4.424703\pm 0.000001$ d.
TOI-1408 c exhibits a TTV period of 138 d with an amplitude of 8 h, and
TOI-1408 b shows TTVs with an amplitude of 8 min. TOI-1408 c’s TTV amplitude
is 15% of its orbital period, the largest TTV amplitude relative to a planet’s
orbital period known at the time of writing. The TDVs for TOI-1408 c
calculated from the photodynamical model agree with the measured values.

In a previous study by Galazutdinov et al. (2023), TOI-1408 b was found to
have a mass of $1.69\pm 0.20$ $M_{\mathrm{Jup}}$ and an eccentric orbit
($e=0.259\pm 0.026$). While their mass value agrees within 1$\sigma$ with our
(largely more precise) value of $1.86\pm 0.02$ $M_{\mathrm{Jup}}$, we find the
TOI-1408 b’s orbit to be nearly circular. Additionally, our photodynamical
modeling could constrain the TOI-1408 b’s radius more reliably.

Table 4: Photodynamical model parameters with their priors and posteriors, and the posteriors for derived planetary parameters. $\mathcal{N}(\mu,\sigma)$ stands for a normal prior with a mean $\mu$ and standard deviation $\sigma$, and $\mathcal{U}(a,b)$ stands for a uniform distribution from $a$ to $b$. The osculating orbital elements are valid for the reference time $T_{0}=2458739.84$. Stellar parameter |  | Prior | Posterior |
---|---|---|---|---
$R_{\star}$ [$R_{\odot}$ ] |  | $\mathcal{N}(1.534,0.021)$ | $1.53\pm 0.02$ |
$M_{\star}$ [$M_{\odot}$ ] |  | $\mathcal{N}(1.309,0.012)$ | $1.312\pm 0.009$ |
Limb darkening ${q}_{1}$ |  | $\mathcal{U}(0,1)$ | $0.5\pm 0.1$ |
Limb darkening ${q}_{2}$ |  | $\mathcal{U}(0,1)$ |  $<0.85$aaThe $q_{2}$ limb darkening coefficient is not well constrained so we give only its 99th percentile upper limit. |
$\rho_{\star}$ [g $\mathrm{cm}^{-3}$] |  |  | $0.52\pm 0.02$ |
RV parameter |  |  |  |
$\gamma$ [m $\mathrm{s}^{-1}$] |  |  $\mathcal{N}$bbThe instrument-specific RV shifts, $\gamma$, have loosely informative priors based on the RV measurement medians and scatters. |  |
$\dot{\gamma}$ [m $\mathrm{s}^{-1}$ $\mathrm{days}^{-1}$] |  | $\mathcal{N}(0,1)$ | $-0.344\pm 0.002$ |
$\ddot{\gamma}$ [m $\mathrm{s}^{-1}$ $\mathrm{days}^{-2}$] |  | $\mathcal{N}(0.000,0.001)$ | $0.000143\pm 0.000004$ |
$\log_{10}$ RV-jitter [$\log_{10}$ m $\mathrm{s}^{-1}$] |  $\mathcal{N}$ccAll RV jitter terms have the same prior $\mathcal{N}(-1,1)$. |  |
$P_{\mathrm{rot}}$ [days] |  | $\mathcal{N}(7.90,0.05)$ | $7.91\pm 0.02$ |
$A_{\mathrm{rot}}$ [m $\mathrm{s}^{-1}$] |  | $\mathcal{U}(0,40)$ | $2.3\pm 0.7$ |
| TOI-1408 c | TOI-1408 b
Fitted planet parameter | Prior | Posterior | Prior | Posterior
$P$ [days] | $\mathcal{N}(2.170,0.005)$ | $2.1664\pm 0.0001$ | $\mathcal{N}(4.424,0.005)$ | $4.42587\pm 0.00003$
$T_{0}$ [BJD] | $\mathcal{N}(2458739.84,0.01)$ | $2458739.847\pm 0.004$ | $\mathcal{N}(2458740.85,0.01)$ | $2458740.8581\pm 0.0002$
$\log_{10}$ $M_{\mathrm{p}}$ [$\log_{10}M_{\odot}$] | $\mathcal{U}(-5.3,-3.9)$ | $-4.64\pm 0.01$ | $\mathcal{N}(-2.7,0.03)$ | $-2.749\pm 0.003$
$R_{\mathrm{p}}/R_{\star}$ | $\mathcal{N}(0.014,0.004)$ | $0.0134\pm 0.0003$ | $\mathcal{U}(0.05,0.35)$ | $0.15\pm 0.02$
$\sqrt{e}\cos{\omega}$ | $\mathcal{U}(-0.5,0.5)$ | $-0.353\pm 0.0005$ | $\mathcal{U}(-0.5,0.5)$ | $0.012\pm 0.007$
$\sqrt{e}\sin{\omega}$ | $\mathcal{U}(-0.5,0.5)$ | $0.103\pm 0.002$ | $\mathcal{U}(-0.5,0.5)$ | $-0.046\pm 0.006$
$g$ | $\mathcal{U}(0,1)$ | $0.73\pm 0.02$ | $\mathcal{U}(0,1)$ | $0.939\pm 0.004$
Derived planet parameter |  |  |  |
$M_{\mathrm{p}}$ [$M_{\oplus}$ ] |  | $7.6\pm 0.2$ |  | $593\pm 4$
$R_{\mathrm{p}}$ [$R_{\oplus}$ ] |  | $2.22\pm 0.06$ |  | $25\pm 4$
$\rho_{\mathrm{p}}$ [g $\mathrm{cm}^{-3}$] |  | $3.8\pm 0.3$ |  | $0.18^{+0.17}_{-0.08}$
$\mathrm{T_{14}}$ [h] |  | $2.67\pm 0.06$ |  | $1.65\pm 0.03$
$e$ | $\mathcal{N}(0,0.083)$ | $0.1353\pm 0.0001$ | $\mathcal{N}(0,0.083)$ | $0.0023\pm 0.0005$
$\omega$ [∘] |  | $286.3\pm 0.3$ |  | $170\pm 10$
$i$ [∘] |  | $82.6\pm 0.3$ |  | $82.4\pm 0.2$
$a/R_{\star}$ |  | $5.04\pm 0.06$ |  | $8.13\pm 0.09$
$a$ [AU] |  | $0.03587\pm 0.00008$ |  | $0.05778\pm 0.0001$

##  4 Discussion

###  4.1 Dynamics

Besides the large TTVs, TOI-1408 c also exhibits transit duration variations
(TDVs). Forward modeling of the posterior solution from Table 4 allowed us to
identify three main contributions to the TDVs, similar to those observed in
KOI-142 (Nesvorný et al., 2013). The largest contribution arises from the
variability in $e\cos{\omega}$ with a period of $\sim 1$ y and a peak-to-peak
(ptp) amplitude of 0.4 h. The second largest contribution is due to variations
in the orbital inclination, with a period of $\sim 15$ y and a ptp amplitude
of 0.2 h. The third contribution comes from the variation in tangential
velocity at mid-transit, which has a period similar to the TTV period and a
ptp amplitude of 0.08 h.

Figure 3: Resonant behavior of the TOI-1408 b--c pair. Left: behavior of the
single-planet resonant argument, $\theta_{\mathrm{c}}$, from the
photodynamical simulations, showing libration about zero. Right: the same
solution, transformed to the canonical resonance model of Nesvorný &
Vokrouhlický (2016), where $\Psi$ is a function of both planets’
eccentricities, and $\psi$ of both planets’ resonant arguments, although both
are dominated by the inner planet c. This numerical solution is marked in
salmon, and $\psi$ librates about $\pi$. However, a comparison to level curves
of the Hamiltonian (grey, with black marking the analytical curve
corresponding to the numerical solution) shows that no resonant separatrix
exists for these system parameters.

The two planets of TOI-1408 lie close to the 2:1 period commensurability
($P_{\mathrm{b}}/P_{\mathrm{c}}\approx 2.04$), raising the question of whether
the system is dynamically in resonance. We first checked for libration of the
resonant argument of the lower-mass interior planet
$\theta_{\mathrm{c}}=2\lambda_{\mathrm{b}}-\lambda_{\mathrm{c}}-\varpi_{\mathrm{c}}$.
This resonant argument shows libration about $0^{\circ}$, suggesting a
resonant state (Fig. 3, left). To understand the resonant dynamics more
deeply, we then compared the system’s behavior to the Hamiltonian resonance
model of Nesvorný & Vokrouhlický (2016). The resonant argument in this model
still librates (about $180^{\circ}$), but when we examine the full phase
portrait, the resonant separatrix does not exist (Fig. 3, right). The
separation to resonance, quantified by the parameter $\delta$ of Nesvorný &
Vokrouhlický (2016), is around $\delta=0.3$ for all posterior draws (mean
$\delta=0.297$ with a standard deviation of $0.002$). The resonant separatrix
exists only for $\delta\geq 1$, and hence this system is by this definition
just wide of resonance, similar to KOI-142 and TOI-2202 (Nesvorný et al.,
2022). This could imply that convergent migration in the protoplanetary disc
did not proceed far enough to drive the system deeply into resonance, or
alternatively, that tidal forces or other effects caused the orbits to diverge
and escape from the resonant state (Delisle et al., 2012).

###  4.2 Stability

We performed a numerical stability analysis using Rebound and its
implementation of the Mean Exponential Growth factor of Nearby Orbits (MEGNO;
Cincotta & Simó, 2000) indicator to determine if the planetary system lies in
a stable configuration, which we define as $|\text{MEGNO}-2|<0.4$. We mapped
the system’s stability in the $P_{\mathrm{b}}-P_{\mathrm{c}}$, and
$e_{\mathrm{c}}-P_{\mathrm{c}}$ parameter spaces by drawing samples from the
photodynamical model posterior and replacing the mapped parameters with
samples created using a quad-tree based importance sampler. We simulated the
system for 1.6 million TOI-1408 c orbits for each sample using the WHFast
integrator (Rein & Tamayo, 2015) and saved the MEGNO indicator. We visualize
the results as a stability map shown in Fig. 4, where the value of a cell
represents the fraction of stable orbits within that cell. It is worth noting
that the system is close to a narrow zone of instability.

Figure 4: Two-dimensional MEGNO maps probing the stability near the posterior
solution from the photodynamical analysis. The color indicates the orbital
stability, and the dashed lines show the parameter posterior median values
listed in Table 4.

###  4.3 Implications from Occurrence rates

The results highlight the unique position of TOI-1408 as a stable island in a
chaotic environment, particularly given the low occurrence rate of inner
companions to HJs, estimated at $1.4\%\pm 1.0\%$ by He et al. (2024). Their
findings, based on N-body simulations spanning a period of $10^{8}$ years,
agree with the lower limit from a previous study by Wu et al. (2023), which
reported a rate of non-aligned nearby planetary companion to hot Jupiters at
$12\%\pm 6\%$ based on a search for TTVs in the Kepler sample. He et al.
(2024) also found that the occurrence rate of inner companions to HJ
significantly decreases as the HJ’s orbital period shortens, resulting in only
a few stable systems with HJ orbital periods less than 6 d.

##  5 Conclusions

In this study, we report the discovery and photodynamical characterization of
TOI-1408 c, a transiting sub-Neptune on a 2.2-day orbit located interior to a
transiting hot Jupiter, TOI-1408 b (P=$4.42$ d, M=$1.86\pm
0.02$$M_{\mathrm{Jup}}$, R=$2.4\pm 0.5$$R_{\mathrm{Jup}}$). The planets are
near a 2:1 period commensurability with librating resonant arguments, yet both
remain outside the resonant configuration. This configuration leads to
remarkable TTVs and TDVs, with TOI-1408 c exhibiting the largest TTV amplitude
relative to its orbital period recorded thus far.

The existence of a small inner planet in such a tight orbit around a hot
Jupiter adds a valuable data point against the backdrop of current planet
formation theories, challenging the typical scenarios suggested for close-in
giant planets. The planets in the TOI-1408 system are both transiting and
exhibit measurable TTVs, similar to TOI-1130, but in a much tighter orbit
configuration.

Additionally, RV measurements suggest a third, long-period outer body in the
system, indicating a complex dynamical environment, such as WASP-47 and
WASP-132. The system’s dynamic nature and the likely presence of a third body
invite further observational campaigns to refine the orbital parameters and
investigate long-term stability.

This research not only deepens our understanding of multi-planet systems
involving hot Jupiters with inner low-mass companions but also underscores the
need for continued exploration to uncover the diverse architectures of
exoplanetary systems.

##  6 Acknowledgments

We thank the anonymous referee for their timely and positive feedback. This
work was supported by the KESPRINT collaboration, an international consortium
devoted to the characterization and research of exoplanets discovered with
space-based missions (https://kesprint.science/). This paper includes data
collected with the TESS mission, obtained from the MAST data archive at the
Space Telescope Science Institute (STScI). Funding for the TESS mission is
provided by the NASA Explorer Program. STScI is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555.
We acknowledge the use of public TESS Alert data from pipelines at the TESS
Science Office and at the TESS Science Processing Operations Center. Resources
supporting this work were provided by the NASA High-End Computing (HEC)
Program through the NASA Advanced Supercomputing (NAS) Division at Ames
Research Center for the production of the SPOC data products. Based on
observations made with the Nordic Optical Telescope, owned in collaboration by
the University of Turku and Aarhus University, and operated jointly by Aarhus
University, the University of Turku and the University of Oslo, representing
Denmark, Finland and Norway, the University of Iceland and Stockholm
University at the Observatorio del Roque de los Muchachos, La Palma, Spain, of
the Instituto de Astrofisica de Canarias. The data presented here were
obtained [in part] with ALFOSC, which is provided by the Instituto de
Astrofisica de Andalucia (IAA) under a joint agreement with the University of
Copenhagen and NOT. This article is based on observations made with the LCOGT
telescopes, one of whose nodes is located at the Observatorios de Canarias del
IAC on the island of Tenerife in the Observatorio del Teide. This paper is
based on observations made with the MuSCAT3 instrument, developed by the
Astrobiology Center and under financial supports by JSPS KAKENHI (JP18H05439)
and JST PRESTO (JPMJPR1775), at Faulkes Telescope North on Maui, HI, operated
by the Las Cumbres Observatory. This research has made use of the Exoplanet
Follow-up Observation Program (ExoFOP; DOI: 10.26134/ExoFOP5) website, which
is operated by the California Institute of Technology, under contract with the
National Aeronautics and Space Administration under the Exoplanet Exploration
Program. This paper includes data taken at The McDonald Observatory of The
University of Texas at Austin. Based on observations made with the Italian
Telescopio Nazionale Galileo (TNG) operated on the island of La Palma by the
Fundación Galileo Galilei of the INAF (Istituto Nazionale di Astrofisica) at
the Spanish Observatorio del Roque de los Muchachos of the Instituto de
Astrofisica de Canarias under programmes CAT19A_162, ITP19_1 and A41TAC_49.
Based partly on observations collected at the Centro Astronómico Hispano en
Andalucía (CAHA) at Calar Alto, operated jointly by Junta de Andalucía and
Consejo Superior de Investigaciones Científicas (IAA-CSIC). This work has made
use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement J. K. gratefully acknowledges
the support of the Swedish National Space Agency (SNSA; DNR 2020-00104) and of
the Swedish Research Council (VR: Etableringsbidrag 2017-04945) H.P.
acknowledges support from the Spanish Ministry of Science and Innovation with
the Ramon y Cajal fellowship number RYC2021-031798-I, and funding from the
University of La Laguna and the Spanish Ministry of Universities. AJM
acknowledges financial support from the Swedish National Space Agency (Career
Grant 2023-00146). We acknowledge financial support from the Agencia Estatal
de Investigación of the Ministerio de Ciencia e Innovación (MCIN/AEI, DOI:
10.13039/501100011033) and the ERDF “A way of making Europe” through project
PID2021-125627OB-C32, and from the Centre of Excellence “Severo Ochoa” award
to the Instituto de Astrofisica de Canarias. Also from MCIN/AEI, HJD
acknowledges funding under grant PID2019-107061GB-C66 and G. M. under the
Severo Ochoa grant CEX2021-001131-S and the Ramón y Cajal grant
RYC2022-037854-I. J.L-B. acknowledges financial support received from "la
Caixa" Foundation (ID 100010434) and from the European Unions Horizon 2020
research and innovation programme under the Marie Slodowska-Curie grant
agreement No 847648, with fellowship code LCF/BQ/PI20/11760023. This research
has also been partly funded by the Spanish State Research Agency (AEI)
Projects No. PID2019-107061GB-C61 and MDM-2017-0737 Unidad de Excelencia
“María de Maeztu”- Centro de Astrobiología, CSIC/INTA. G.N. thanks for the
research funding from the Ministry of Education and Science programme the
"Excellence Initiative - Research University" conducted at the Centre of
Excellence in Astrophysics and Astrochemistry of the Nicolaus Copernicus
University in Toruń, Poland. This work is partly supported by JSPS KAKENHI
Grant Numbers JP24H00017, JP24H00248, JP24K00689, JP24K17082 and JSPS
Bilateral Program Number JPJSBP120249910. P.G.B. acknowledges support by the
Spanish Ministry of Science and Innovation with the Ramón y Cajal fellowship
number RYC-2021-033137-I and the number MRR4032204. E.M. acknowledges funding
from FAPEMIG under project number APQ-02493-22 and a research productivity
grant number 309829/2022-4 awarded by CNPq, Brazil. D.G. gratefully
acknowledges the financial support from the grant for internationalization
(GAND_GFI_23_01) provided by the University of Turin. M.S. acknowledges the
support of the Italian National Institute of Astrophysics (INAF) through the
project ’The HOT-ATMOS Project: characterizing the atmospheres of hot giant
planets as a key to understand the exoplanet diversity’ (1.05.01.85.04). J.Š.
acknowledges the support from GACR grant 23-06384O. PK and MS are thankful for
the support from grant LTT-20015. P.C. acknowledges the DFG through grant CH
2636/1-1 and the support of the Department of Atomic Energy, Government of
India. E.W.G acknowledges support from the Thüringer Ministerium für
Wirtschaft, Wissenschaft und Digitale Gesellschaft. R.A.G. and D.B.P.
acknowledge the support from PLATO CNES grant. S.M. acknowledges support by
the Spanish Ministry of Science and Innovation with the grant no.
PID2019-107061GB-C66 and through AEI under the Severo Ochoa Centres of
Excellence Programme 2020--2023 (CEX2019-000920-S).

##  Appendix A Ground-based transits for TOI-1408 b

Figure 5: Ground-based transit measurements observed with the different
facilities shown as gray points, the same points binned to 15-min as black
points with error bars, and the photodynamical model in blue.

##  Appendix B Radial velocity model from the photodynamical analysis

Figure 6: Radial velocity observations together with the model from the
photodynamical analysis. One line covers 20 TOI-1408 b periods, after which
the data and the model are offset down by 800 m/s. The vertical bar at line 1
shows the scale, spanning the RV range from -400 to 400 m/s. The light gray
symbols mark RV observations with uncertainties larger than 25 m/s, and the
black symbols observations with uncertainties smaller than 25 m/s.

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