We report the discovery and characterization of a small planet TOI1408 c on a 2.2 day orbit located interior to a previously known hot Jupiter TOI1408 b P  4.42 days M  1.86  0.02 M SUBJupSUB R  2.4  0.5 R SUBJupSUB that exhibits grazing transits. The two planets are near 21 period commensurability resulting in significant transit timing variations TTVs for both planets and transit duration variations for the inner planet. The TTV amplitude for TOI1408 c is 15 of the planets orbital period marking the largest TTV amplitude relative to the orbital period measured to date. Photodynamical modeling of groundbased radial velocity RV observations and transit light curves obtained with the Transiting Exoplanet Survey Satellite and groundbased facilities leads to an inner planet radius of 2.22  0.06 R SUBSUB and mass of 7.6  0.2 M SUBSUB that locates the planet into the subNeptune regime. The proximity to the 21 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 lowmass planet.
['Exoplanet dynamics', 'Hot Jupiters', 'Hot Neptunes', 'Transit timing variation method', 'Transit photometry', 'Radial velocity', '490', '753', '754', '1710', '1709', '1332', 'Astrophysics - Earth and Planetary Astrophysics']
TOI1408 Discovery and Photodynamical Modeling of a Small Inner Companion to a Hot Jupiter Revealed by Transit Timing Variations
{'TOI-1408: Discovery and Photodynamical Modeling of a Small Inner Companion to a Hot Jupiter Revealed by TTVs': "Judith Korth , 1 Priyanka Chaturvedi , 2, 3 Hannu Parviainen , 4, 5 Ilaria Carleo, 6, 5, 4 Michael Endl, 7, 8 Eike W. Guenther , 2 Grzegorz Nowak , 9, 5, 4 Carina Persson , 10 Phillip J. MacQueen, 7 Alexander J. Mustill , 1 Juan Cabrera , 11 William D. Cochran , 7, 8 Jorge Lillo-Box , 12 David Hobbs , 1 Felipe Murgas , 5, 4 Michael Greklek-McKeon , 13 Hanna Kellermann , 14 Guillaume H'ebrard, 15, 16 Akihiko Fukui , 17, 5 Enric Pall'e, 5, 4 Jon M. Jenkins , 18 Joseph D. Twicken , 19, 18 Karen A. Collins , 20 Samuel N. Quinn , 20 J'an ˇ Subjak, 21, 20 Paul G. Beck , 4, 5 Davide Gandolfi , 22 Savita Mathur , 5, 4 Hans J. Deeg , 5, 4 David W. Latham , 20 Simon Albrecht , 23 David Barrado , 12 Isabelle Boisse, 16, 24 Herv'e Bouy , 25, 26 Xavier Delfosse, 27 Olivier Demangeon, 28 Rafael A. Garc'ıa , 29 Artie P. Hatzes , 2 Neda Heidari, 15 Kai Ikuta , 30 Petr Kab'ath, 21 Heather A. Knutson , 13 John Livingston, 31, 32, 33 Eder Martioli , 34, 15 Mar'ıa Morales-Calder'on , 12 Giuseppe Morello, 35, 5 Norio Narita , 17, 31, 5 Jaume Orell-Miquel, 5, 4 Hanna L. M. Osborne, 36, 37 Dinil B. Palakkatharappil , 29 Viktoria Pinter, 38, 39 Seth Redfield , 40 Howard M. Relles , 20 Richard P. Schwarz , 20 Sara Seager , 41, 42, 43 Avi Shporer , 41 Marek Skarka, 21, 44 Gregor Srdoc, 45 Monika Stangret , 46 Luis Thomas , 14, 47 Vincent Van Eylen , 36 Noriharu Watanabe , 30 and Joshua N. Winn 48  \n1  \nLund Observatory, Division of Astrophysics, Department of Physics, Lund University, Box 118, 22100 Lund, Sweden 2 Thuringer Landessternwarte Tautenburg, Sternwarte 5, 07778 Tautenburg, Germany 3 Tata Institute of Fundamental Research (TIFR), Homi Bhabha Road, Colaba, Mumbai - 400005, India 4 Departamento de Astrof'ısica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain 5 Instituto de Astrof'ısica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain 6 INAF - Osservatorio Astrofisico di Torino, Via Osservatorio 20, I-10025, Pino Torinese, Italy 7 McDonald Observatory, The University of Texas, Austin Texas USA  \n8 Center for Planetary Systems Habitability, The University of Texas, Austin Texas USA  \n9 Institute of Astronomy, Faculty of Physics, Astronomy and Informatics, Nicolaus Copernicus University, Grudzi¸adzka 5, 87-100 Toru'n, Poland  \n10  \nDepartment of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, SE-439 92 Onsala,  \nSweden  \n- 11 Institut fur Planetenforschung, Deutsches Zentrum fur Luft- und Raumfahrt, Rutherfordstr. 2, 12489 Berlin, Germany 12 Centro de Astrobiologi'ıa (CAB), CSIC-INTA, Camino Bajo del Castillo s/n, ESAC campus, 28692, Villanueva de la Ca˜nada, Madrid, Spain  \n13 Division of Geological and Planetary Sciences, California Institute of Technology, Pasadena, CA, 91125, USA 14 University Observatory Munich, Faculty of Physics, Ludwig-Maximilians-Universitat Munchen, Scheinerstr. 1, 81679 Munich, Germany  \n15 Institut d'astrophysique de Paris, UMR7095 CNRS, Universit'e Pierre & Marie Curie, 98bis boulevard Arago, 75014 Paris, France 16 Observatoire de Haute-Provence, CNRS, Universit'e d'Aix-Marseille, 04870 Saint-Michel-l'Observatoire, France 17 Komaba Institute for Science, The University of Tokyo, 3-8-1 Komaba, Meguro, Tokyo 153-8902, Japan 18 NASA Ames Research Center, Moffett Field, CA 94035, USA  \n19  \nSETI Institute, Mountain View, CA 94043 USA  \n- 20 Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA\n- 21 Astronomical Institute, Czech Academy of Sciences, Friˇcova 298, 251 65, Ondˇrejov, Czech Republic\n- 22 Dipartimento di Fisica, Universit'a degli Studi di Torino, via Pietro Giuria 1, I-10125, Torino, Italy\n- 23 Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark  \n24 Aix Marseille Univ, CNRS, CNES, LAM, Marseille, France  \n25 Laboratoire d'astrophysique de Bordeaux, Univ. Bordeaux, CNRS, B18N, all'ee Geoffroy Saint-Hilaire, 33615 Pessac, France 26 Institut universitaire de France (IUF), 1 rue Descartes, 75231 Paris CEDEX 05  \n27 Universit'e Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France  \n- 28 Instituto de Astrof'ısica e Ciˆencias do Espa¸co, Universidade do Porto, CAUP, Rua das Estrelas, 4150-762 Porto, Portugal 29 Universit'e Paris-Saclay, Universit'e Paris Cit'e, CEA, CNRS, AIM, 91191, Gif-sur-Yvette, France\n- Department of Multi-Disciplinary Sciences, Graduate School of Arts and Sciences, The University of Tokyo, 3-8-1 Komaba, Meguro,\n- 30 Tokyo 153-8902, Japan  \n31 Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  \n32 National Astronomical Observatory of Japan, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan  \n- 33 Astronomical Science Program, Graduate University for Advanced Studies, SOKENDAI, 2-21-1, Osawa, Mitaka, Tokyo, 181-8588, Japan  \n34 Laborat'orio Nacional de Astrof'ısica, Rua Estados Unidos 154, 1006 37504-364, Itajub'a - MG, Brazil  \n- 35 Instituto de Astrof'ısica de Andaluc'ıa (IAA-CSIC), Gta. de la Astronom'ıa s/n, 18008 Granada, Granada, Spain\n- 36 Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK  \n37  \nEuropean Southern Observatory, Karl-Schwarzschild-Straße 2, Garching bei Munchen D-85748, Germany  \n- 38 Nordic Optical Telescope, Rambla Jos'e Ana Fernandez P'erez 7, 38711, Bre˜na Baja, Spain  \n39 University of Craiova, Alexandru Ioan Cuza 13, 200585, Craiova, Romania  \n40 Astronomy Department and Van Vleck Observatory, Wesleyan University, Middletown, CT 06459, USA  \n- 41 Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA\n- 42 Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  \n43  \nDepartment of Aeronautics and Astronautics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA  \n- 44 Department of Theoretical Physics and Astrophysics, Faculty of Science, Masaryk University, Kotl'aˇrsk'a 267/2, 611 37 Brno, Czech Republic  \n45 Kotizarovci Observatory, Sarsoni 90, 51216 Viskovo, Croatia  \n46 INAF - Osservatorio Astronomico di Padova, Vicolo dell'Osservatorio 5, 35122, Padova, Italy  \n47 Max-Planck Institute for Extraterrestrial Physics, Giessenbachstrasse 1, D-85748 Garching, Germany 48 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 ± 0 . 02 M Jup , R=2 . 4 ± 0 . 5 R 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 ± 0 . 06 R ⊕ and mass of 7 . 6 ± 0 . 2 M ⊕ 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.  \nKeywords: Exoplanet dynamics (490)-Hot Jupiters (753)-Hot Neptunes (754)-Transit timing variation method (1710)-Transit photometry (1709)-Radial velocity (1332)", '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 Jup following the defini-  \ntion 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.  \nThe 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 WASP47 (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; Ca˜nas 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).  \nIn 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.', '2.1. Photometric Observations': '2.1.1. TESS Photometry  \nTOI-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 TOI1408 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. 1', '2.1.2. Ground-based Photometry': '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,  \nTable 1. Ground-based photometry.  \nNote -Table 1 is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content.  \nSpain, 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.  \nLCOGT/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).  \nLCOGT/MuSCAT3 -We observed two transit of TOI1408 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.  \nPalomar/WIRC -We observed one transit of TOI1408 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).  \nTable 2. RV measurements and their uncertainties.  \nNote -Table 2 is published in its entirety in the machine-readable format. A portion is shown here for guidance regarding its form and content.', '2.2. Spectroscopic Observations': 'HARPS-N -We obtained 60 high-resolution spectra using the High Accuracy Radial velocity Planet SearcherNorth (HARPS-N: λ ∈ (378-691) nm, R ≈ 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 202406-19. 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 36112 at 550 nm. The spectra were extracted using the offline 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, 2 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.  \nMcDonald -We observed 89 high-precision spectra of TOI-1408 using the McDonald Observatory 2.7-m Harlan J. Smith telescope (HJST) with its Tull Coud\'e spectrometer ( λ ∈ (340-1090) nm, R ≈ 60 000; Tull et al. 1995). We pass the starlight through an I 2 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).  \nTLS -We obtained 232 spectra of TOI-1408 using the Coud\'e-\' Echelle spectrograph of the 2-m-Alfred Jen-  \nh telescope of the Thuringer Landessternwarte (TLS) Tautenburg. The spectrograph\'s old camera (CCD3, λ ∈ (452-765) nm, R = 51000) was updated to a new (Andor, λ ∈ (460-734) nm, R = 63000) during the observing period. When modeling the RVs, we allowed for a velocity offset between the pre-upgrade and postupgrade 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.  \nCAFE -We observed 20 spectra of TOI-1408 with the Calar Alto Fiber-fed Echelle (CAFE) spectrograph ( λ ∈ (407-925) nm, R = 60000; 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.  \nSOPHIE -We observed 15 high-precision spectra of TOI-1408 with the SOPHIE \'echelle spectrograph ( λ ∈ (387-694) nm, R = 75000) 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 crosscorrelation 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 ± 7 m/s.  \nMaHPS -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 ( λ ∈ (380-880) nm, R = 60000).  \nFor 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.  \nTRES -We obtained 18 spectra with the Tillinghast Reflector Echelle Spectrograph (TRES; λ ∈ (384910) nm, R = 44000; F"ur\'esz 2008) mounted on the 1.5m 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 outof-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.  \nOES -We observed 14 spectra with the Ondrejov Echelle Spectrograph (OES; λ ∈ (380-900) nm, R = 50 000; Kab\'ath et al. 2020) installed on a 2-m Perek telescope in Ondˇrejov, 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 ( ˇ Subjak et al. 2020; Tran et al. 2022; Kab\'ath et al. 2022).  \nHERMES -We monitored the system for ten days in August 2020 with the High Efficiency and Resolution Mercator Echelle Spectrograph (HERMES; λ ∈ (375900) nm, R = 90000; 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 Astrof\'ısica 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).', '3.1. 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 eff , the  \nTable 3. Stellar parameters of TOI-1408.  \nReferences -1: Gaia DR3 (Gaia Collaboration et al. 2016, 2023; Babusiaux et al. 2023), 2: Tycho-2 (Høg et al. 2000), 3: 2MASS (Cutri et al. 2003).  \nlogarithm of the surface gravity, log g , V sin i ⋆ , 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 ⋆ and the stellar radius assuming that the star is seen equator-on, is 7 . 9 ± 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; Garc'ıa 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 ± 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.", '3.2. Discovery of TOI-1408 c': "A transit timing variation (TTV) search using Python Tool for Transit Variation ( PyTTV ; Korth et al. 2023) re-  \nvealed 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 D'etection Sp'ecialis'ee de Transits code ( DST; Cabrera et al. 2012).  \nFollowing 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 TOI1408 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.  \nAfter 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 σ ) had an orbital period of ∼ 2 . 22 days. Despite exceeding the 7 . 1 σ threshold, this detection was vetoed by the transit search pipeline's χ 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 longperiod 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 ± 0 . 0002 d and an RV semi-amplitude of 5 . 0 ± 1 . 0 m/s that correspond to a minimum mass of  \n13 . 0 ± 3 . 0 M ⊕ . 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 semiamplitude of 191 . 3 ± 0 . 9 m/s, translating to a minimum planet mass of 1 . 846 ± 0 . 009 M Jup .  \nThe 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 ∼ 200 days before the main observing campaign started gives a Keplerian orbit with a period of ∼ 2530 d with K = 195 ± 4ms -1 , M p = 14 . 6 ± 0 . 3 M Jup and an eccentricity of 0 . 35 ± 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.  \nGaia 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.  \nWe can estimate the astrometric signature of the longperiod 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  \nα = ( M p M ⋆ ) ( a p 1 AU ) ( d 1 pc ) [arsec] . (1)  \nThe 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': "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  \nFigure 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.  \n<!-- image -->  \nfrom 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  \nimpact 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, Ω, are fixed to π for both planets. The model is parameterized using the sampling parameters √ e cos ω and √ e sin ω , but we also set a prior on the orbital eccentricities. The model parameters and their priors are listed in Table 4.  \nFigure 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.  \n<!-- image -->  \nThe 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 (ForemanMackey et al. 2013).  \nWe 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 ± 0 . 002 d and P c = 2 . 19592 ± 0 . 00002 d, while for TOI-1408 b, the ephemeris is T 0 ,b = 2458740 . 8584 ± 0 . 0002 d and P b = 4 . 424703 ± 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  \nrelative 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.  \nIn a previous study by Galazutdinov et al. (2023), TOI-1408 b was found to have a mass of 1 . 69 ± 0 . 20 M Jup and an eccentric orbit ( e = 0 . 259 ± 0 . 026). While their mass value agrees within 1 σ with our (largely more precise) value of 1 . 86 ± 0 . 02 M 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.", '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 (Nesvorn'y et al. 2013). The largest contribution arises from the variability in e cos ω with a period of ∼ 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 ∼ 15 y and a ptp amplitude of 0.2 h. The third contribution comes from the variation in tangential velocity  \nTable 4. Photodynamical model parameters with their priors and posteriors, and the posteriors for derived planetary parameters. N ( µ, σ ) stands for a normal prior with a mean µ and standard deviation σ , and 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.  \nat mid-transit, which has a period similar to the TTV period and a ptp amplitude of 0.08 h.  \nThe two planets of TOI-1408 lie close to the 2:1 period commensurability ( P b /P c ≈ 2 . 04), raising the question of whether the system is dynamically in resonance. We  \nfirst checked for libration of the resonant argument of the lower-mass interior planet θ c = 2 λ b -λ c -ϖ c . This resonant argument shows libration about 0 · , suggesting a resonant state (Fig. 3, left). To understand the resonant dynamics more deeply, we then compared the  \n<!-- image -->  \nFigure 3. Resonant behavior of the TOI-1408 b-c pair. Left: behavior of the single-planet resonant argument, θ c , from the photodynamical simulations, showing libration about zero. Right: the same solution, transformed to the canonical resonance model of Nesvorn'y & Vokrouhlick'y (2016), where Ψ is a function of both planets' eccentricities, and ψ of both planets' resonant arguments, although both are dominated by the inner planet c. This numerical solution is marked in salmon, and ψ librates about π . 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.  \n<!-- image -->  \nsystem's behavior to the Hamiltonian resonance model of Nesvorn'y & Vokrouhlick'y (2016). The resonant argument in this model still librates (about 180 · ), 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 δ of Nesvorn'y & Vokrouhlick'y (2016), is around δ = 0 . 3 for all posterior draws (mean δ = 0 . 297 with a standard deviation of 0 . 002). The resonant separatrix exists only for δ ≥ 1, and hence this system is by this definition just wide of resonance, similar to KOI-142 and TOI-2202 (Nesvorn'y 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 & Sim'o 2000) indicator to determine if the planetary system lies in a stable configuration, which we define as | MEGNO -2 | < 0 . 4. We mapped the system's stability in the P b -P c , and e c -P c parameter spaces by drawing samples from the photodynamical model posterior  \nand 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.", '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% ± 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 nonaligned nearby planetary companion to hot Jupiters at 12% ± 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': "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.  \n<!-- image -->  \nIn 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 ± 0 . 02 M Jup , R=2 . 4 ± 0 . 5 R 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.  \nThe 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.  \nAdditionally, RV measurements suggest a third, longperiod outer body in the system, indicating a complex dynamical environment, such as WASP-47 and WASP132. 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.  \nThis 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  \ndiscovered 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.  \nThis research has made use of the Exoplanet Followup 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 Fundaci'on 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 Astron'omico Hispano en Andaluc'ıa (CAHA) at Calar Alto, operated jointly by Junta de Andaluc'ıa and Consejo Superior de Investigaciones Cient'ıficas (IAACSIC). 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 Investigaci'on of the Ministerio de Ciencia e Innovaci'on (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 Ram'on 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  \nState Research Agency (AEI) Projects No. PID2019107061GB-C61 and MDM-2017-0737 Unidad de Excelencia 'Mar'ıa de Maeztu'- Centro de Astrobiolog'ıa, 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 Toru'n, 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 Ram'on y Cajal fellowship number RYC-2021033137-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. ˇ S. 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 Thuringer Ministerium fur 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).  \nFacilities: 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.  \nSoftware: 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)  \nFigure 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.  \n<!-- image -->", 'REFERENCES': 'Aceituno, J., S\'anchez, S. 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