TOI-1670 b and c: An Inner Sub-Neptune with an Outer Warm Jupiter Unlikely to Have Originated from High-eccentricity Migration

TOI-1670 b and c: An Inner Sub-Neptune with an Outer Warm Jupiter Unlikely to Have Originated from High-eccentricity Migration Quang H. Tran 1 , Brendan P. Bowler 1 , Michael Endl 1,2 , William D. Cochran 1,2 , Phillip J. MacQueen 2 , Davide Gandolfi 3 , Carina M. Persson 4 , Malcolm Fridlund 4,5 , Enric Palle 6,7 , Grzegorz Nowak 6,7 , Hans J. Deeg 6,7 , Rafael Luque 8 , John H. Livingston 9,10,11 , Petr Kabáth 12 , Marek Skarka 12,13 , Ján Šubjak 12,14 , Steve B. Howell 15 , Simon H. Albrecht 16 , Karen A. Collins 17 , Massimiliano Esposito 18 , Vincent Van Eylen 19 , Sascha Grziwa 20 , Elisa Goffo 3,18 , Chelsea X. Huang 21,29 , Jon M. Jenkins 15 , Marie Karjalainen 12 , Raine Karjalainen 12 , Emil Knudstrup 16 , Judith Korth 22 , Kristine W. F. Lam 23 , David W. Latham 17 , Alan M. Levine 21 , H. L. M. Osborne 19 , Samuel N. Quinn 17 , Seth Redfield 24 , George R. Ricker 21 , S. Seager 21,25,26 , Luisa Maria Serrano 3 , Alexis M. S. Smith 23 , Joseph D. Twicken 15,27 , and Joshua N. Winn 28 1 Department of Astronomy, The University of Texas at Austin, 2515 Speedway, Stop C1400, Austin, TX 78712, USA; [email protected] 2 McDonald Observatory, The University of Texas at Austin, 2515 Speedway, Stop C1400, Austin, TX 78712, USA 3 Dipartimento di Fisica, Università degli Studi di Torino, via Pietro Giuria 1, I-10125, Torino, Italy 4 Department of Space, Earth and Environment, Chalmers University of Technology, Onsala Space Observatory, SE-439 92 Onsala, Sweden 5 Leiden Observatory, Leiden University, NL-2333 CA Leiden, The Netherlands 6 Instituto de Astrofísica de Canarias (IAC), E-38205 La Laguna, Tenerife, Spain 7 Departamento de Astrofísica, Universidad de La Laguna (ULL), E-38206, La Laguna, Tenerife, Spain 8 Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta de la Astronomía s/n, E-18008 Granada, Spain 9 Department of Astronomy, University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan 10 Astrobiology Center, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 11 National Astronomical Observatory of Japan, NINS, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan 12 Astronomical Institute of the Czech Academy of Sciences, Fričova 298, 25165, Ondr ejov, Czech Republic 13 Department of Theoretical Physics and Astrophysics, Masaryk University, Kotlárská 2, CZ-61137, Brno, Czech Republic 14 Astronomical Institute of Charles University, V Holešovičkách 2, 180 00, Praha, Czech Republic 15 NASA Ames Research Center, Moffett Field, CA 94035, USA 16 Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark 17 Center for Astrophysics |Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 18 Thüringer Landessternwarte Tautenburg, Sternwarte 5, D-07778 Tautenburg, Germany 19 Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, Surrey RH5 6NT, UK 20 Rheinisches Institut für Umweltforschung an der Universiät zu Köln, Aachener Straße 209, D-50931 Köln, Germany 21 Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 22 Department of Space, Earth and Environment, Astronomy and Plasma Physics, Chalmers University of Technology, SE-412 96 Gothenburg, Sweden 23 Institute of Planetary Research, German Aerospace Center (DLR), Rutherfordstraße 2, D-12489 Berlin, Germany 24 Astronomy Department and Van Vleck Observatory, Wesleyan University, Middletown, CT 06459, USA 25 Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 26 Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA 27 SETI Institute, Mountain View, CA 94043, USA 28 Department of Astrophysical Sciences, Peyton Hall, 4 Ivy Lane, Princeton, NJ 08544, USA Received 2021 September 25; revised 2022 February 7; accepted 2022 March 1; published 2022 April 21 Abstract We report the discovery of two transiting planets around the bright (V=9.9 mag)main-sequence F7 star TOI-1670 by the Transiting Exoplanet Survey Satellite. TOI-1670 b is a sub-Neptune (=- + R 2.06 b0.15 0.19 R ⊕ )on a 10.9 day orbit, and TOI-1670 c is a warm Jupiter (=- + R 0.987 c0.025 0.02 5 R Jup )on a 40.7 day orbit. Using radial velocity observations gathered with the Tull Coudé Spectrograph on the Harlan J. Smith telescope and HARPS-N on the Telescopio Nazionale Galileo, we find a planet mass of =- + M0.63 c0.08 0.09 M Jup for the outer warm Jupiter, implying a mean density of r =- + 0.81 c0.11 0.13 gcm −3 . The inner sub-Neptune is undetected in our radial velocity data (M b <0.13 M Jup at the 99% confidence level). Multiplanet systems like TOI-1670 hosting an outer warm Jupiter on a nearly circular orbit (=- + e0.09 c0.04 0.05)and one or more inner coplanar planets are more consistent with “gentle”formation mechanisms such as disk migration or in situ formation rather than high-eccentricity migration. Of the 11 known systems with a warm Jupiter and a smaller inner companion, eight (73%)are near a low-order mean-motion resonance, which can be a signature of migration. TOI-1670 joins two other systems (27% of this subsample)with period commensurabilities greater than 3, a common feature of in situ formation or halted inward migration. TOI1670 and the handful of similar systems support a diversity of formation pathways for warm Jupiters. Unified Astronomy Thesaurus concepts: Exoplanet astronomy (486);Radial velocity (1332);Exoplanet formation (492);Transit photometry (1709) 1. Introduction The origin of giant planets interior to the water-ice line remains an open question. A number of theories have been proposed to explain the closest-in (P<10 days)giant planets, The Astronomical Journal, 163:225 (16pp), 2022 May https://doi.org/10.3847/1538-3881/ac5c4f © 2022. The Author(s). Published by the American Astronomical Society. 29 Juan Carlos Torres Fellow Original content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence. Any further distribution of this work must maintain attribution to the author(s)and the title of the work, journal citation and DOI. 1 or hot Jupiters (HJs; e.g., Dawson & Johnson 2018; Fortney et al. 2021). These scenarios are primarily divided between dynamically “violent”or “gentle”mechanisms. The former consists of three-body dynamical interactions such as planet– planet scattering or high-eccentricity tidal migration (e.g., Wu & Murray 2003; Fabrycky & Tremaine 2007; Triaud et al. 2010; Naoz et al. 2011; Batygin 2012). The latter refers to disk migration (e.g., Ward 1997; Albrecht et al. 2012; Kley & Nelson 2012)or in situ formation (e.g., Batygin et al. 2016; Boley et al. 2016; Huang et al. 2016; Anderson et al. 2020). These processes have also been used to explain part of the farther-out population of warm Jupiters (WJs; defined here to have 10 days <P<200 days). However, observed WJ demographics suggest that multiple processes are present in sculpting these more distant giant systems. WJs can be broadly divided into two classes. The first is a transient population that will likely evolve into HJs. In more disruptive formation mechanisms, such as high-eccentricity tidal migration, giant planets at comparatively wide separations are disturbed onto highly eccentric orbits by a third body via planet–planet scattering or von Zeipel–Lidov–Kozai oscillations and eventually circularize into orbits with shorter periods (Kozai 1962; Lidov 1962; Naoz 2016; Ito & Ohtsuka 2019). Eccentric giant planets undergoing this tidally damped inward migration are caught in a rapid, temporary state (Naef et al. 2001; Dawson & Johnson 2018; Dong et al. 2021; Jackson et al. 2021). They are expected to start their journeys with much higher eccentricities (e0.9; Vick et al. 2019), which can decay as rapidly as ∼1 Myr as they settle in near their host star (Patra et al. 2020; Mancini et al. 2022). The majority of WJs are not expected to belong to this transient classification. Instead, most WJs are a part of a “static”population that will remain stable over long time periods. This group consists of the apparently single systems with low-to-moderate eccentricities, as well as coplanar multiplanet systems containing WJs with low eccentricities. These WJs have periapses larger than what is required for efficient tidal damping of their orbits, which occurs at 0.05 au (Anderson et al. 2016; Dong et al. 2021),so these planets cannot be undergoing high-eccentricity migration. If most giant planets form beyond the water-ice line, other migration mechanisms must play a major role in sculpting these WJ orbital properties and demographics (e.g., Veras & Armitage 2005; Fogg & Nelson 2009; Dong et al. 2014; Ortiz et al. 2015; Huang et al. 2016; Anderson & Lai 2017; Anderson et al. 2020; Schlecker et al. 2020). However, the relative importance of these pathways is still unknown. Investigating WJ orbital eccentricities can place additional constraints on the dominant giant planet migration mechanism, since each scenario will produce different observed eccentricity distributions. WJs have an eccentricity distribution that peaks at e=0.0 with a tail that extends out to e∼0.8 (Kipping 2013; Dong et al. 2021). In order to produce the population of WJs with moderately eccentric orbits (e∼0.2–0.7), a mechanism is needed that can excite eccentricities. These potential excitation scenarios include interactions involving a disk (e.g., Goldreich & Sari 2003; Petrovich et al. 2019), secular eccentricity oscillations driven by interactions with a distant inclined giant planet (e.g., Anderson & Lai 2017), and planet–planet scattering events (e.g., Mustill et al. 2017; Frelikh et al. 2019; Marzari & Nagasawa 2019; Anderson et al. 2020).An important clue is the observed dependence on metallicity of the giant planet eccentricity distribution, where metal-rich systems (that may more favorably form multiple giant planets)are more likely to host eccentric gas giants (Dawson & MurrayClay 2013). Systems hosting WJs with low-mass inner companions on coplanar orbits are especially useful laboratories to test these planet formation and migration theories. Their small orbital eccentricities and low mutual inclinations suggest that disk migration or in situ formation likely helped create this population. WJs have a relatively high close companion rate of nearly 50% (Huang et al. 2016). However, their intrinsically low occurrence rate (∼1%–2%; Cumming et al. 2008) combined with the difficulty of detecting lower-mass inner planets means that only a handful of known multiplanet systems host a WJ (Johnson et al. 2010; Santerne et al. 2016; Fernandes et al. 2019). Increasing the number of systems with this multiplanet architecture may further distinguish this subsample into two WJ populations, each of which likely reflects different formation and migration routes. Here we present the discovery of the transiting multiplanet system TOI-1670 bc, a WJ (TOI-1670 c)with an inner subNeptune (TOI-1670 b)found with the Transiting Exoplanet Survey Satellite (TESS; Ricker et al. 2015). TOI-1670 b and c were originally identified by the TESS Science Processing Operations Center (SPOC; Jenkins et al. 2016)pipeline as two promising transiting signals that were subsequently promoted to TESS Object of Interest (TOI; Guerrero et al. 2021)status. TOI-1670 (TIC ID 441739020; 2MASS J17160415+7209402; Gaia DR2 1651911084230149248)is a relatively inactive (log ¢=- R 4.9 3 HK )old F7 dwarf with a TESS apparent magnitude of 9.5 mag and a moderate projected rotational velocity of ≈9kms −1 (see Section 3). In this work, we validate both planets and measure the mass of the outer planet, TOI1670 c. In Section 2, we describe the TESS photometric data and follow-up radial velocity (RV)observations used in the planet validation and mass measurement. Our characterization of the system, including the host star and a global fit to the RVs and light curve, is presented in Section 3. We conclude in Section 4by contextualizing TOI-1670 in the paradigm of WJs and their formation. 2. Observations KESPRINT 30 is an international collaboration focused on the discovery, confirmation, and characterization of exoplanet candidates from space-based missions (e.g., Gandolfiet al. 2018; Persson et al. 2018; Livingston et al. 2019; Lam et al. 2020;Šubjak et al. 2020). As part of this consortium, a series of ground-based follow-up observations of TOI-1670 were taken. These data are primarily used to reject the possibility of a falsepositive scenario in which the observed transiting signal is caused by something other than a planet. For example, this includes a low-mass eclipsing binary (EB), a grazing transit of an EB, a background EB, or a transiting planet around a background star. Reconnaissance spectra are used to exclude an EB scenario by constraining the maximum amplitude of the RV signal. High-resolution speckle images are taken to exclude binary companions to TOI-1670 and nearby background stars. High-resolution spectra are used to characterize the host star and, when possible, measure the masses of the planets. 30 http://kesprint.science/ 2 The Astronomical Journal, 163:225 (16pp), 2022 May Tran et al. 2.1. TESS Photometry TOI-1670 was observed by TESS at 2 minute cadence over 11 sectors (15, 16, 18, 19, 20, 21, 22, 23, 24, 25, and 26)for a total of 323 days. Images were reduced and light curves were analyzed for transit signals with the TESS SPOC pipeline (Jenkins et al. 2016), which identified two potential transit signals (Jenkins 2002; Jenkins et al. 2010,2020)with periods of 40.7 (TOI-1670.01)and 10.9 (TOI-1670.02)days. The SPOC vetting tests (Twicken et al. 2018; Li et al. 2019) validated both signals as consistent with planets, and they were designated as TOIs (Guerrero et al. 2021)by the TESS Science Office. We downloaded the SPOC Pre-search Data Conditioning Simple Aperture Photometry (PDCSAP)light curve (Smith et al. 2012; Stumpe et al. 2012,2014)from the MAST data archive 31 using the lightkurve (Lightkurve Collaboration et al. 2018)software package. We removed all of the photometric measurements that are flagged as poor quality by the SPOC pipeline (DQUALITY >0)or where either the flux or flux error is listed as NaN. Outlier rejection was performed at 3σfor positive outliers and 10σfor negative outliers to allow for transit events. The light curve was flattened by removing low-frequency trends using a Savitzky–Golay filter (Savitzky & Golay 1964)after all transit events were masked out. The final light curve for TOI-1670 is shown in Figure 1. The photometric points used in the global model fit are shown in purple and pink. These cover the transit events for TOI-1670 b and c, respectively, and their times of transit are further denoted by the corresponding colored triangles along the time axis. 2.2. TRES Reconnaissance Spectroscopy We obtained six spectra of TOI-1670 with the Tillinghast Reflector Echelle Spectrograph (TRES; Fűrész 2008)on the 1.5 m Tillinghast telescope at the Fred L. Whipple Observatory on UT 2020 February 2 and 20; UT 2020 March 6, 9, and 16; and UT 2020 July 7. Exposure times ranged from 300 to 650 s and have an average signal-to-noise ratio (S/N)of 32 ±5. The RVs were extracted following Buchhave et al. (2010). The spectra have an average measurement error of 53 m s −1 and an rms of 54 m s −1 , which excludes the possibility of an EB scenario; however, these spectra are not used as part of the orbit fit. Table A1 in Appendix Areports the RV measurements. 2.3. OES Reconnaissance Spectroscopy We collected 32 spectra using the Ondr ejov Echelle Spectrograph (OES)on the 2 m Perek telescope at the Ondr ejov Observatory in the Czech Republic (Kabáth et al. 2020). These observations were obtained between UT 2020 February and UT 2020 September at a cadence of 3–5 RVs per month. We extracted the spectra and performed the bias, flatfield, and cosmic-ray corrections using standard IRAF 2.16 routines (Tody 1993). The RVs were extracted using the IRAF cross-correlation fxcor, taking the highest-S/N spectrum as a template. The average measurement error is 110 m s −1 , and the RV rms is 116 m s −1 . The Doppler signals for TOI-1670 b and c are not detected in this data set, so they are not used in the orbit fit. However, they are used to reject an EB scenario and justify further follow-up of TOI-1670 with precise RV measurements. The reconnaissance RV measurements are reported in Table A1 in Appendix A. 2.4. Tull CoudéSpectroscopy We used the Tull Coudé Spectrograph on the 2.7 m Harlan J. Smith telescope at McDonald Observatory to obtain 49 spectra of TOI-1670 between UT 2020 April and UT 2021 September. The Tull Coudé Spectrograph is a cross-dispersed echelle spectrograph with a wavelength coverage ranging from 3750 to 10200 Å(Tull et al. 1995). Our configuration uses a 1 2 slit, which yields a resolving power of R=60,000. Precise wavelength calibration and instrumental profile reconstruction are achieved with a temperature-controlled iodine vapor (I 2 ) cell that is mounted in front of the entrance slit. The RVs are extracted using the RV reduction pipeline Austral (Endl et al. 2000). The I 2 cell imprints a wellunderstood reference absorption spectrum onto the stellar spectra. Precise differential RVs are then calculated by Figure 1. Detrended TESS light curve of TOI-1670. The full light curve is shown in black. Purple and pink points are relative photometry within four transit durations before and after transits of TOI-1670 b and c, respectively, and are used in the global RV and light-curve model fit. Times of transit are also marked by triangles plotted along the time axis. Gaps in the light curve correspond to periods where TESS uploaded data, result from data quality cuts, or are during Sector 17, when TOI1670 was not observed. TESS sectors are shown in blue above the light curve. 31 https://archive.stsci.edu/missions-and-data/tess/ 3 The Astronomical Journal, 163:225 (16pp), 2022 May Tran et al. comparing each stellar-plus-iodine spectrum with a high-S/N stellar template devoid of iodine lines. The S-index activity metric for each spectrum is also calculated and calibrated onto the Mt. Wilson S-index system following the description in Paulson et al. (2002). Table A2 in Appendix Areports the resulting RVs, activity indices, and related measurement errors. 2.5. FIES Spectroscopy We acquired seven spectra of TOI-1670 using the Fiber-fed Echelle Spectrograph (FIES; Frandsen & Lindberg 1999; Telting et al. 2014)at the 2.56 m Nordic Optical Telescope (Djupvik & Andersen 2010)of Roque de los Muchachos Observatory (La Palma, Spain). The observations were carried out between UT 2020 May 25 and UT 2020 September 6 as part of the Spanish CAT observing program 59–210. We used the FIES high-resolution mode, which provides a resolving power of R=67,000 in the spectral range 3760–8220 Å.We traced the RV drift of the instrument by acquiring long-exposed ThAr spectra (exposure time of 90 s)immediately before and after each science observation. The science exposure time was set to 1200–1800 s, depending on the sky conditions and scheduling constraints. The data reduction follows the steps described in Buchhave et al. (2010)and Gandolfiet al. (2015) and includes bias subtraction, flat-fielding, order tracing and extraction, and wavelength calibration. The RVs were derived via multiorder cross-correlations using the first stellar spectrum as a template. The S/N per pixel at 5500 Åranges between 40 and 65. The average RV uncertainty is 13.4 ±2.4 m s −1 . 2.6. HARPS-N Spectroscopy We observed TOI-1670 with the HARPS-N spectrograph (R≈115,000)on the 3.59 m Telescopio Nazionale Galileoat Roque de los Muchachos Observatory located in La Palma, Spain, between UT 2020 August and UT 2020 September (Cosentino et al. 2012,2014)during observing program A40TAC_22 (PI: Gandolfi). A total of eight spectra were taken; seven spectra had an exposure time of 1800 s, and one had an exposure time of 215 s. ThisresultedinanaverageS/N at 550 nm of 84 ±16 for the first sevenspectraandanS/N of 15 for the shorter exposure. We used the standard HARPS-N Data Reduction Software (DRS)with a G2 numerical mask to extract the RVs (Pepe et al. 2002). The RVs, their measurement errors, and the associated activity indicators, such as the bisector inverse slope (BIS), FWHM of the cross-correlation function, and S-index produced by the HARPS-N DRS, are listed in Table A2 of Appendix A. 2.7. High-resolution Imaging On the nights of UT 2021 April 5 and June 24, TOI-1670 was observed with the NESSI and ‘Alopeke speckle imagers (Scott et al. 2018; Scott 2019), mounted on the 3.5 m WIYN telescope at Kitt Peak and the 8.1 m Gemini North telescope on Maunakea, respectively. Both instruments simultaneously acquire data in two bands centered at 562 and 832 nm using high-speed electron-multiplying CCDs. Observations of TOI1670 were performed in the 562 and 832 nm bands following the procedures described in Howell et al. (2011). The resulting reconstructed images have a 5σdelta magnitude contrast of 4 to 8 magnitudes at angular separations from 20 mas to 1 2 in the 832 nm band (Figure 2). No other companion sources are detected in the reconstructed images within these angular limits down to the contrasts obtained. These angular limits correspond to spatial separations of 3.3–200 au at the distance of TOI-1670. 3. Analysis 3.1. Stellar Parameters Planetary parameters measured from the global joint fitof the transit and RV data depend on precise stellar mass and radius measurements. In particular, R p is determined from the transit depth and M p from the RV semiamplitude, which are dependent on R * and M * , respectively. The stellar mass and radius can be inferred with atmospheric and evolutionary models using the spectroscopic parameters (T eff , log g,[Fe/H], vsin i). We determine both of these spectroscopic and fundamental parameters for TOI-1670 using several approaches described below. 3.1.1. Spectral Analysis We analyzed the coadded HARPS-N (S/N=180)spectrum with the spectral analysis package Spectroscopy Made Easy (SME; Valenti & Piskunov 1996; Valenti & Fischer 2005; Piskunov & Valenti 2017). The spectral fitting technique of SME minimizes the χ 2 value by fitting synthetic spectra of stars based on grids of atmosphere models and observations. We fit Figure 2. Reconstructed speckle images of TOI-1670 from NESSI (top)and ‘Alopeke (bottom)in the 562 and 832 nm bands and their corresponding 5σ contrast curves. North is up, and east is to the left. 4 The Astronomical Journal, 163:225 (16pp), 2022 May Tran et al. the coadded HARPS-N spectrum with the ATLAS12 model spectra (Kurucz 2013)using the non-local thermodynamic equilibrium SME version 5.2.2 following the procedure described in Fridlund et al. (2017)to compute T eff , log g,v sin i, and chemical abundances. The stellar surface gravity, log g, was estimated using the spectral wings of the Ca I6102, 6122, 6162 Åtriplet and the Ca I6439 Åline. The microscopic and macroscopic turbulences, V mic and V mac , were held fixed to the values determined in the calibration for stars with similar T eff and log gfrom Bruntt et al. (2010)and Doyle et al. (2014), respectively. We also derive the stellar parameters using the publicly available SpecMatch-Emp software package (Yee et al. 2017).SpecMatch-Emp compares the HARPS-N template spectrum to a high-resolution (R∼55,000), high-S/N(>100) Keck/HIRES optical spectral library of 404 well-characterized earlyto late-type dwarfs (F1 to M5). The empirical spectra are calibrated using interferometry, so SpecMatch-Emp produces estimates for T eff ,[Fe/H], and R * (instead of log g). Prior to running the code, we convert the HARPS-N spectrum template onto the Keck/HIRES format following the procedure described in Hirano et al. (2018). The stellar parameters derived from SME and SpecMatchEmp are in good agreement with each other (Table 1). From SME, we find an effective temperature of T eff =6170 ±61 K and metallicity of [Fe/H]=0.09 ±0.07 dex, while SpecMatch-Emp gives T eff =6048 ±110 K and [Fe/ H]=0.05 ±0.09 dex; these are consistent with each other within 1σ. These results are also in good agreement with the photometrically derived effective temperature from Gaia DR2 (=- + T6162 eff 175 162 K)and agree at the 2σlevel with the TESS Input Catalog (TIC)v8 (Guerrero et al. 2021)value of 6345 ±121 K. For this work, we adopt the spectroscopic parameters from SME as it produces all atmospheric parameters. The final adopted stellar parameters are reported in Table 3. 3.1.2. Stellar Mass and Radius We infer the stellar radius by fitting the spectral energy distribution (SED)of TOI-1670 using the software package ARIADNE. 32 ARIADNE utilizes a Bayesian model averaging framework that convolves four stellar atmosphere models— Phoenix v2 (Husser et al. 2013), BT-Settl (Allard et al. 2011), Kurucz (1993), and Castelli & Kurucz (2003)—with the response functions of commonly available broadband filters. For our SED fitting, we use the Two Micron All Sky Survey (2MASS)JHK s , Gaia DR2 (G,G BP ,B RP ), Johnson Vand B, and Wide-field Infrared Survey Explorer (WISE; W1 and W2) bandpasses. Synthetic SEDs are created by interpolating in T eff –glog –[Fe/H]space. Distance, radius, A V , and excess photometric uncertainty terms are free parameters in the fitting process. We set the priors for T eff , log g, and [Fe/H]to the values we found in Section 3.1.1, the distance prior to the Bailer-Jones et al. (2021)Bayesian-based value (- + 165.72 0.38 0.32 pc), and the stellar radius prior to the Gaia DR2 value (=- + R 1.38 0.07 0.08 *R e ). The extinction, A V , has a flat prior limited by the maximum line-of-sight reddening according to the recalibrated SFD galaxy dust map (Schlegel et al. 1998; Schlafly & Finkbeiner 2011). The excess photometric noise parameters all have Gaussian priors centered at zero with a standard deviation equal to 10 times the reported photometric error. The SED of TOI-1670 and best-fitting model are shown in Figure 3. We estimate the mass of TOI-1670 using the stellar isochrone software package isochrones (Morton 2015a) and the MESA Isochrones and Stellar Tracks (Dotter 2016; Choi et al. 2016)evolutionary model grids. The isochrones package infers fundamental stellar parameters by comparing a variety of observational inputs to interpolated model values. We input the Gaia DR2 parallax, broadband photometry (2MASS JHK s ; Gaia DR2 G,G BP , and B RP ; Johnson Vand B; and WISE W1 and W2), and the SME spectroscopic values (T eff , log g, and [Fe/H])as priors. The posteriors are sampled using the MultiNest (Feroz et al. 2009,2019)sampling algorithm. All values for the stellar radius and mass are reported in Table 2. We include values from the TIC and Gaia DR2, as well as the typical mass and radius for an F7V dwarf for reference (Cox 2000). The SpecMatch-Emp fit also derives a stellar radius, which we couple to the calibration equations from Torres et al. (2010)to infer a surface gravity of log g=4.14 ±0.07 dex and a stellar mass of 1.25 ±0.09 M e . All values are in good agreement with each other. We adopt the ARIADNE radius (R * =1.316 ±0.019 R e )and the isochrones mass (M * =1.21 ±0.02 M e )as the stellar parameters to be used in the global fit and report all adopted physical, photometric, and kinematic properties of TOI-1670 in Table 3. Table 1 Spectroscopic Parameters of TOI-1670 Derived Using SME and SpecMatch-Emp Method T eff (K)log g(gcm −3 )[Fe/H](dex)vsin i(km s −1 ) SME 6170 ±61 4.29 ±0.11 0.09 ±0.07 9.2 ±0.6 SpecMatch-Emp 6048 ±110 L0.05 ±0.09 L Figure 3. The SED of TOI-1670. Broadband photometry (Table 3)is shown with orange circles, with horizontal errors representing the bandpass width. The best-fitting model SED is shown in black, and the blue diamonds are the model flux integrated over each bandpass. The residuals normalized by the photometric errors are shown in the bottom panel. 32 https://github.com/jvines/astroARIADNE 5 The Astronomical Journal, 163:225 (16pp), 2022 May Tran et al. 3.2. Stellar Activity Stellar activity in the form of rotationally modulated starspots and granulation can both mimic and mask the signals of planets in light curves (Llama & Shkolnik 2015,2016)and RVs (e.g., Figueira et al. 2013). Thus, prior to running the global model fit, we first examine whether stellar activity significantly influences the light curve and RV time series of TOI-1670. We measured a low average value of log ¢=-  R 4.93 0.01 HK from the HARPS-N spectra, which suggests that TOI-1670 is a quiet star not dominated by stellar activity (Mamajek & Hillenbrand 2008). The TESS light curve prior to detrending also does not exhibit any significant rotation or activity-induced variability. A common statistical tool used to detect periodic signals in unevenly sampled time series data is the Lomb–Scargle periodogram (Lomb 1976; Scargle 1982). We utilize this algorithm to search for periodicity in both the TESS photometry and RV activity indicators to distinguish stellar activity–based signals from those induced by planetary motion. We compute the generalized Lomb–Scargle (GLS)periodograms (Zechmeister & Kürster 2009)for the “undetrended” PDCSAP light curve with the transit events removed, the RVs, the aforementioned activity indicators, and the spectral window function over the frequency range 0.0005–0.5 day −1 (2–2000 days)in Figure 4. The GLS power thresholds corresponding to false-alarm probability (FAP)levels of 1% and 0.1% computed via a bootstrap approach are shown as blue dotted lines (Kuerster et al. 1997). The GLS periodogram for the RVs was computed for the combined HARPS-N and Tull Coudé data after subtracting the systematic velocity offsets as reported in Table 4. The periodogram of the TESS photometry has very low power with no peaks that have significance higher than the 1% FAP level. This is consistent with the flat nature of the undetrended PDCSAP light curve and indicates that TOI-1670 does not have a large starspot coverage fraction. The strongest signal in the periodogram of the RVs is at the ∼40.7 day orbital period of TOI-1670 c, which has an FAP <0.1%. This peak has no counterparts in the periodograms of the activity indices, which would be the case if that signal originated from stellar activity. Activity signals can also appear at the frequency of the stellar rotation period. Using the stellar radius and vsin i,we can place a lower limit of P rot 7.2 days (f0.138 day −1 ).No significant peaks are visible in the GLS periodogram of the Sindices at this frequency. 3.3. Statistical Validation of TOI-1670 b Although TOI-1670 b is not significantly detected in the RV data set, our model is able to place an upper limit on its mass. The 3σupper limit of the fitted RV semiamplitude is 14.5 m s −1 , which corresponds to an upper limit of 0.18 M Jup for TOI1670 b, assuming an eccentricity of zero. An estimate of the RV precision required to robustly detect TOI-1670 b can be made from a predicted mass inferred from its radius. Inputting the stellar and planetary parameters measured in Section 3into a probabilistic mass–radius relation using the open software package forecaster (Chen & Kipping 2017)yields a mass estimate of - + 5 .2 2.0 4.0 M ⊕ for TOI1670 b. Assuming a circular orbit, this corresponds to an RV semiamplitude of ∼1.3 m s −1 . Robustly detecting an RV signal at this level requires instrument precision at the 1 m s −1 level and a well-behaved star. We can exclude false-positive scenarios to support TOI-1670 b as a likely planet using follow-up observations. From Gaia Table 2 Stellar Mass and Radius of TOI-1670 Derived from Different Methods Method M * (M e )R * (R e ) ARIADNE a 1.16 ±0.16 1.316 ±0.019 isochrones 1.21 ±0.02 1.316 ±0.007 SpecMatch-Emp+Torres b 1.25 ±0.09 1.57 ±0.18 TIC c 1.25 ±0.18 1.312 ±0.057 Gaia DR2 d L- + 1 .38 0.07 0.08 Typical F7V dwarf e 1.21 1.32 Adopted 1.21 ±0.02 1.316 ±0.019 Notes. a Mass calculated using derived radius and log g. b Mass calculated using SpecMatch-Emp parameters and calibration equations from Torres et al. (2010). c Stassun et al. (2019). d Gaia Collaboration et al. (2018). e Cox (2000). Table 3 Adopted Physical, Photometric, and Kinematic Properties of TOI-1670 Parameter Value Source TIC ID 441739020 1 TOI ID 1670 1 Gaia ID 1651911084230149248 2 2MASS ID J17160415 +7209402 3 Gaia α(J2000.0)17:16:04.16 2 Gaia δ(J2000.0)+72:09:40.17 2 Gaia epoch 2015.5 2 Gaia parallax (mas)5.92 ±0.02 2 Distance (pc)- + 1 65.72 0.38 0.32 4 Gaia μ α cos δ(mas yr −1 )−6.09 ±0.05 2 Gaia μ δ (mas yr −1 )5.154 ±0.05 2 B(mag)10.43 ±0.03 5 V(mag)9.89 ±0.03 5 T(mag)9.423 ±0.0061 1 G(mag)9.8232 ±0.0004 2 G RP (mag)9.4145 ±0.0014 2 G BP (mag)10.0747 ±0.0010 2 J(mag)8.97 ±0.02 3 H(mag)8.75 ±0.03 3 K s (mag)8.724 ±0.02 3 W1(mag)8.689 ±0.023 6 W2(mag)8.702 ±0.020 6 T eff (K)6170 ±61 This work log g(gcm −3 )4.29 ±0.11 This work [Fe/H](dex)0.09 ±0.07 This work vsin i(km s −1 )9.2 ±0.6 This work M * (M e )1.21 ±0.02 This work R * (R e )1.316 ±0.019 This work ρ * (gcm −3 )0.752 ±0.036 This work Age (Gyr)2.53 ±0.43 This work A V (mag)0.010 ±0.006 This work References. (1)Stassun et al. (2019),(2)Gaia Collaboration et al. (2018),(3) Cutri et al. (2003),(4)Bailer-Jones et al. (2021),(5)Høg et al. (2000),(6)Cutri et al. (2021). 6 The Astronomical Journal, 163:225 (16pp), 2022 May Tran et al. EDR3, we note that TOI-1670 has zero excess astrometric noise and a renormalized unit weight errorof 1.07, indicating that the single-star model is a good fit to the astrometric solution (Gaia Collaboration et al. 2018; Lindegren et al. 2018). From our RVs, we find that the overall RV variability is <110 m s −1 from OES, <54 m s −1 from TRES, <34 m s −1 from the Tull Coudé, and <16 m s −1 from HARPS-N, all of which robustly exclude an EB scenario for the host star. Finally, we use TRICERATOPS (Giacalone et al. 2021)to statistically evaluate the probability of possible false-positive scenarios involving nearby contaminant stars, including background EBs. TRICERATOPS is a Bayesian tool for validating transiting planet candidates by modeling and calculating the probability of different scenarios that produce transit-like light curves. Based on the lack of a close stellar companion from Gaia astrometry, our high-resolution imaging, and our RVs, we omit the optional false-positive calculations for the EB and unresolved stellar companion scenarios in the TRICERATOPS code. 33 TRICERATOPS returns a false-positive probability (the total probability of a false-positive scenario involving the primary star)of <0.015 and a nearby false-positive probability (the sum of all false-positive probabilities for scenarios involving nearby stars)of <10 −2 . The RV confirmation of the outer coplanar transiting WJ further supports the planetary nature of TOI-1670 b, as multiplanet systems are unlikely to be false positives (Lissauer et al. 2012; Rowe et al. 2014). 3.4. Joint Modeling of RVs and Photometry We perform a multiplanet global fit to the available RV and transit observations of TOI-1670 using the pyaneti modeling suite (Barragán et al. 2019). As the planetary Doppler signals are not recovered at a significant level in the TRES and Ondr ejov spectra, we only use the 49 Tull Coudé and eight HARPS-N RVs in the modeling. We limit the light-curve data to photometry spanning four full transit durations before and after all transit events of TOI-1670 b and c to improve computation efficiency; this results in a total of 24,772 photometric points. These regions are shown in purple and pink in Figure 1for TOI-1670 b and c, respectively. We simultaneously fit the Keplerian orbit and TESS light curve for eight parameters: orbital period (P), central time of transit (T 0 ), RV semiamplitude (K), transit impact parameter (b), planetary-to-stellar radius (R p /R * ), scaled semimajor axis (a/R * ), and parameterized forms of eccentricity and argument of periastron ( w esin and w ecos ). This last parameterization by Anderson et al. (2011)is used because the eccentricity posterior distribution for orbits with low eand broad ωis poorly sampled by Markov chains (e.g., Lucy & Sweeney 1971; Ford 2006; Wang & Ford 2011).Bydefining e and ωin a polar form, we avoid truncating the posterior distribution at zero and impose a uniform prior on e. We also adopt the parameterization of bas defined by Winn (2010), w =- + ⎜⎟ ⎛ ⎝⎞ ⎠ bai R e e cos 1 1sin,1 2 () * ** where i * is the stellar inclination, in order to impose priors that exclude nontransiting orbits >+b1R R p ( ) * . We set narrow uniform priors on both orbital period and time of transit based on visual inspection of the light curve and the SPOC preliminary parameters. For the inner sub-Neptune, the ranges are =T1721.92, 1721.99 b0, ()in units of (BJD TDB – 2,457,000)days, =P10.980, 10.988 b()days, and =K0.0, 10.0 b( ) ms −1 . For the outer Jupiter, the ranges are =T1750.82, 1750.92 c0, ()in units of (BJD TDB – 2,457,000)days, =P40.7485, 40.7505 c( ) days, and =K10.0, 100.0 c()ms −1 . The stellar mass and radius are also free parameters with Gaussian priors of = MM1.215, 0.023() *and = R R1.316, 0.019() *. 34 These parameters are further constrained by the stellar mean density, which is affected by Pand a/R * (Seager & MallénOrnelas 2003; Winn 2010). We assumed a quadratic limbdarkening law following the equations from Mandel & Agol (2002), who defined the linear and quadratic coefficients as u 1 and u 2 , respectively. The parameterization of =+quu 112 2 () and =+ - quuu0.5 211 21 () from Kipping (2013)is adopted. We set broad uniform priors for all other parameters and report them in Table 4.A“jitter”term is added to the RVs to account Figure 4. Lomb–Scargle periodograms of the undetrended TESS photometry after removing transit signals (first panel), combined Tull Coudé and HARPSN RVs after subtracting the systemic velocities (second panel), combined Mt. Wilson S-index from Tull Coudé and HARPS-N spectra (third panel), and spectral window function (fourth panel). The 1% and 0.1% FAP (blue dashed and dotted, respectively)lines are calculated using bootstrap resampling. The purple and pink vertical lines are the ∼10.9 and ∼40.7 day planetary signals determined by the TESS SPOC, respectively. The highest peak of each periodogram is shown as a dashed black vertical line. There are no counterpart peaks in the periodogram of the S-index that correspond to the ∼40.7 day signal seen in the periodogram of the RV, and no peaks rise above the 1% FAP threshold. No peaks are visible in the spectral window function periodogram. The only significant period (<0.1% FAP)in the RVs is at 41.1 days, consistent with the WJ (TOI-1670 c). 33 Giacalone et al. (2021)noted that using follow-up observations to rule out unresolved stellar companion scenarios produces similar results for both TRICERATOPS and the target validation code vespa (Morton 2015b; Morton et al. 2016). 34 Here and  refer to the uniform and normal distributions, respectively, where the latter is defined as ms,( ) . 7 The Astronomical Journal, 163:225 (16pp), 2022 May Tran et al. for any systematic and astrophysical variance not reported in the observational uncertainties. 35 Posterior distributions of fitted and derived parameters were sampled using a Markov Chain Monte Carlo Metropolis– Hasting algorithm following the description by Sharma (2017) as implemented by pyaneti. The distributions were sampled using 50 chains for 10,000 iterations with a thinning factor of 10. The convergence of each chain was determined with the Gelman–Rubin diagnostic test (Gelman & Rubin 1992). Using the TESS photometry and RVs, we jointly model the transits of TOI-1670 b and c and the RV curve of TOI-1670 c using the priors as previously described. 36 The posterior values of the fitted and derived system parameters from pyaneti for TOI-1670 are given in Table 4. The best-fitting phased TESS light curves for TOI-1670 b and c and RV model for TOI-1670 c are plotted in Figures 5–7. Figure C1 in Appendix Cdisplays the posterior distributions for the fitted parameters. We find a mass, radius, and density of TOI-1670 c of =- + M0.63 c0.08 0.09 M Jup ,=- + R 0.987 c0.025 0.02 5 R Jup , and r =- + 0.81 c0.11 0.13 gcm −3 , respectively. For TOI-1670 b, we find a radius of - + 2 .06 0.15 0.19 R ⊕ and a 3σmass upper limit of M b <0.13 M Jup . 4. Discussion The existence of WJ systems hosting one or more smaller coplanar inner companions such as TOI-1670 is inconsistent Table 4 Priors and Posteriors on the Global System Parameters of TOI-1670 b and c Parameter Adopted Prior Posterior Values Fitted Parameters bcbc T 0 (BJD TDB −2,457,000)1721.92, 1721.99( ) 1750.82, 1750.92( ) - + 1 721.9423 0.0062 0.0071 - + 1 750.88286 0.00083 0.00085 P(days)10.980, 10.988()40.7485, 40.7505( ) - + 1 0.98462 0.00051 0.0004 6 - + 4 0.74976 0.00021 0.000022 K(ms −1 )0.0, 10.0( ) 10.0, 100( ) - + 4 .6 3.0 3.3 - + 3 2.7 4.3 4.7 b0.0, 1.0( ) 0.0, 1.0( ) - + 0.61 0.37 0.22 - + 0.76 0.04 0.02 a/R * 1.1, 20.0( ) 1.1, 50.0( ) - + 1 6.88 0.27 0.27 - + 4 0.68 0.66 0.6 6 R p /R * 0.0, 0.05( ) 0.0, 0.15( ) - + 0.014 0.001 0.001 - + 0.077 0.002 0.002 esin ω-1.0, 1.0( ) -1.0, 1.0( ) - + 0.18 0.44 0.3 6 - + 0.27 0.1 0 0.0 8 ecos ω-1.0, 1.0( ) -1.0, 1.0( ) -- + 0.63 0.21 0.62 -- + 0.07 0.13 0.1 4 Derived Parameters M p LL - + 1 3.8 8.7 9.5 M ⊕ - + 0.63 0.08 0.09 M Jup R p LL - + 2 .06 0.15 0.19 R ⊕ - + 0.987 0.025 0.025 R Jup ρ p (gcm −3 )LL - + 8.6 5.6 6.9 - + 0.81 0.11 0.13 eLL - + 0.59 0.26 0.17 - + 0.09 0.04 0.05 ω(deg)LL - + 1 63.6 53.7 41.7 - + 1 05.5 29.4 28. 6 i(deg)LL - + 86.87 1.07 1.1 6 - + 88.84 0.04 0.0 4 a(au)LL - + 0.103 0.002 0.002 - + 0.249 0.005 0.005 T 14 (hr)LL - + 2 .80 0.19 0.16 - + 5 .40 0.05 0.0 6 T eq (K)LL - + 1 062 13 1 4 - + 684 9 9 Additional Parameters M * (from scaled parameters)(M e )1.215, 0.023( ) - + 1 .218 0.07 6 0.081 - + 1 .239 0.079 0.08 4 ρ * (from transit)(gcm −3 )L- + 0.754 0.035 0.037 - + 0.767 0.037 0.038 q 1 0.0, 1.0( ) - + 0.35 0.11 0.19 q 2 0.0, 1.0( )  - + 0.32 0.23 0.39 γ Tull (km s −1 )--3.6849, 3.3299( ) -- + 3.5132 0.0043 0.0042 γ FIES (km s −1 )-0.1217, 0.1139( ) -- + 0.0108 0.0061 0.0061 γ HARPS−N (km s −1 )--11.6134, 11.3602( ) -- + 11.4805 0.0027 0.0025 RV jitter (Tull)(ms −1 )L- + 2 2.5 3.6 3.9 RV jitter (FIES)(ms −1 )L- + 5 .3 3.8 7.2 RV jitter (HARPS-N)(ms −1 )L- + 4 .3 2.8 4.5 Figure 5. Transit light curve folded to the orbital period of TOI-1670 b. The TESS photometry is shown in purple, and the solid black line is the best-fitting transit model. The black circles are the photometric data binned over 20 minute intervals. The middle panel zooms in on the best-fit transit model and binned data points. The fit residuals are shown in the lower panel. 35 We also fit for a noise term in the photometry and find a value 2 orders of magnitude less than the typical uncertainty with no appreciable change in other model parameters. We choose not to include this term in the final joint model fit. 36 We also consider less complex models in Appendix B. We ultimately choose to apply a full global fit to robustly assess the parameter uncertainties. 8 The Astronomical Journal, 163:225 (16pp), 2022 May Tran et al. with dynamical migration routes. During high-eccentricity tidal migration, multiple close-in planets would likely interact with each other and potentially lead to ejections or collisions (e.g., Rasio & Ford 1996; Chatterjee et al. 2008; Mustill et al. 2015). Similarly, planet–planet scattering and von Zeipel–Lidov– Kozai interactions require an outer companion (Veras & Armitage 2005; Anderson & Lai 2017). Multiplanet systems hosting a WJ with low eccentricity represent another type of system that experienced comparatively gentle dynamical histories, such as inward disk migration or in situ formation. TOI-1670 joins a handful of confirmed systems with an outer warm giant exoplanet (M p >0.25 M Jup , 10 days <P<200 days)and at least one inner smaller companion (see Table 5). Figure 8shows the eccentricity versus semimajor axis of the confirmed WJs. The WJs in all 11 systems (including TOI1670)with similar configurations have low eccentricities, whereas the eccentricities of other WJs without known inner companions are widely distributed. This divergence further suggests that this group of TOI-1670-like systems may have formed and migrated along a similar evolutionary pathway. One way to disentangle whether disk migration or in situ formation plays the dominant role in sculpting these multiplanet systems is by examining their period ratios in search of near mean-motion resonances (MMRs). Disk migration is expected to efficiently capture giant planets into MMRs close to small integer period ratios such as 2:1, 3:1, 3:2, and 4:3 (e.g., Goldreich & Tremaine 1980;Lee&Peale2001;Armitage2010;Winn& Fabrycky 2015). In situ formation can also create planets in orbital resonances, either coincidentally or by eccentricity damping via interactions with the protoplanetary or planetesimal disk (Dawson et al. 2016;Morrisonetal.2020). In this formation scenario, there should be a population of systems that congregate at or near these different integer ratios. Within the sample of 11 known systems that have a giant planet with a small inner companion, eight WJs (73%)are in or near a 2:1 or 3:1 resonance with the inner planet (Table 5). With a period ratio of 3.7, TOI-1670 joins two other systems, K2-290 and HIP 57274, that have non-MMR orbital period ratios greater than 3. The planets in these systems may have formed in situ or migrated inward together. Alternatively, that the planets in these systems are not locked in an MMR could also indicate that they formed independently and did not migrate together or became unstable over time (Petitetal.2020; Pichierri & Morbidelli 2020;Izidoro et al. 2021). This may hint at a division within this small class of WJs in multiplanet systems in which some migrate into place via disk migration (those with integer period ratios), while others formed where we see them today or experienced further dynamical interaction later in their lifetime. This hypothesis can be further investigated by increasing the number of warm giant planets with smaller inner companions and examining population trends within this sample. We thank Benjamin Tofflemire, Daniel Krolikowski, Michael Gully-Santiago, and Erik Petigura for insightful discussions on the generalized Lomb–Scargle periodogram, gas giant occurrence rate, and light-curve analysis. Q.H.T. and B.P.B. acknowledge support from a NASA FINESST grant (80NSSC20K1554). This work benefited from involvement in ExoExplorers, which is sponsored by the Exoplanets Program Analysis Group (ExoPAG)and NASAʼs Exoplanet Exploration Program Office (ExEP). B.P.B. acknowledges support from National Science Foundation grant AST-1909209 and NASA Exoplanet Research Program grant 20-XRP20_2-0119. C.M.P. and M.F. gratefully acknowledge the support of the Swedish National Space Agency (DNR 65/19 and 177/19). J.K. gratefully acknowledges the support of the Swedish National Space Agency (SNSA; DNR 2020-00104). P.K., M.S., J.S., and R.K. acknowledge the financial support of Intertransfer grant No. LTT-20015. M.K. acknowledges support from ESAs PEA4000127913. M.E. acknowledges the support of the DFG priority program SPP 1992 “Exploring the Diversity of Extrasolar Planets”(HA 3279/12-1). Funding for the Stellar Astrophysics Centre is provided by the Danish National Research Foundation (grant agreement No. DNRF106). D.G. and L.M.S. gratefully acknowledge financial support from the Cassa di Risparmio di Torino (CRT) foundation under grant No. 2018.2323 “Gaseous or rocky? Unveiling the nature of small worlds.”This work is partly supported by JSPS KAKENHI grant No. JP20K14518 and SATELLITE Research from Astrobiology Center (AB022006). Funding for the TESS mission is provided by NASA’s Science Mission Directorate. This research has made use of the Exoplanet Follow-up Observation Program website, which is operated by the California Institute of Technology, under contract with the Figure 6. Transit light curve folded to the orbital period of TOI-1670 c. The TESS photometry is shown in pink, and the solid black line is the best-fitting transit model. Black points are the photometric data binned over 20 minute intervals. Figure 7. The RV curve of TOI-1670, phase-folded to the orbital period of the WJ, TOI-1670 c, with the contributions of the inner companion removed. The different colored points denote the different spectrographs, and the best-fitting RV model is shown by the solid black line. The fit residuals are shown in the lower panel. 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