The Aligned Orbit of the Eccentric Proto Hot Jupiter TOI-3362b^*

Ever since the discovery of 51 Pegasi b (Mayor & Queloz 1995), the existence of hot Jupiters (HJs) has continued to elude a definitive explanation (see Dawson & Johnson 2018 for a review on the possible origins of HJs). Generally, the theoretical channels proposed to explain these close-in planets fall into two categories. One is high-eccentricity (high-e) tidal migration, in which proto HJs are first launched into highly eccentric orbits—either through planet–planet scattering (e.g., Rasio & Ford 1996; Beaugé & Nesvorný 2012), von Zeipel–Lidov–Kozai oscillations (e.g., Wu & Murray 2003; Fabrycky & Tremaine 2007; Naoz et al. 2011), or secular interactions (e.g., Wu & Lithwick 2011; Petrovich 2015)—after which tidal friction acts to circularize and shrink the orbits. The other proposed channel posits that HJs acquire their current orbits while still embedded in their parental gas disks, either by in situ formation (e.g., Batygin et al. 2016; Boley et al. 2016) or by formation beyond the ice line, followed by inward migration driven by nebular tides (e.g., Goldreich & Tremaine 1980; Lin & Papaloizou 1986).

If close-in giant planets are indeed the result of high-e tidal migration, we would expect to catch at least some planets on the tidal circularization track (Socrates et al. 2012). Arguably the clearest example of such a system is HD80606b (Naef et al. 2001), which has a distant stellar companion and exhibits extreme orbital eccentricity e = 0.93, and high stellar obliquity (Moutou et al. 2009; Pont et al. 2009; Winn et al. 2009; Hébrard et al. 2010), and is thus naturally explained by the von Zeipel–Lidov–Kozai mechanism (Wu & Murray 2003).

Similarly, TOI-3362b, with a reported orbital period P = 18.1 days and eccentricity e = 0.815 (Dong et al. 2021), may also be considered to be part of the small group of gas giants known to be undergoing high-e tidal migration. Nonetheless, due to the absence of detected (stellar or planetary) companions, coupled with the unresolved matter of stellar obliquity, conclusive identification of the mechanisms that have driven TOI-3362b toward high eccentricity has not been possible to date.

In this Letter, we present further orbital characterization of the TOI-3362b system, obtained from measurements of the Rossiter–McLaughlin (RM) effect and precise long-term radial velocity (RV) follow-up observations. Precise in-transit spectroscopic observations with ESPRESSO revealed that the planet's orbital axis is aligned with the stellar rotation axis with better than three-degree precision and the long-term RV follow-ups obtained with FEROS shows no clear features of long-period companions in the system.

2.1. Transit Spectroscopy

We observed a single transit of TOI-3362b with the ESPRESSO spectrograph (Pepe et al. 2021) on the night of the 2022 December 27, between 04:31 and 08:16 UT. ESPRESSO is the highly stabilized, fiber-fed cross-dispersed echelle spectrograph installed at the Incoherent Combined Coudé Focus of ESO's Paranal Observatory in Chile. It covers the wavelength range of 380–788 nm at a resolving power of R ≈ 140,000 in single-UT High-Resolution mode. We obtained 38 spectra of the host star during the primary transit with UT2 at an exposure time of 310 s. The observations were performed under clear sky conditions, with atmospheric seeing in the range of 058–195. The spectra have a median signal-to-noise ratio (S/N) of 54 at 550 nm and a median RV uncertainty of 12.4 m s⁻¹. The data were reduced with the dedicated data reduction pipeline (v. 2.4.0), including all standard reduction steps, which also provides RVs by fitting a Gaussian model to the calculated cross-correlation function (CCF). The CCF is calculated at steps of 0.5 km s⁻¹ (representing the sampling of the spectrograph) for ±100 km s⁻¹ centered on the estimated systemic velocity. The RM component of the RVs is presented in Figure 1.

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2.2. Photometry

2.2.1. Observatoire Moana

Simultaneously with ESPRESSO observations, we observed the transit using the station of the Observatoire Moana located in El Sauce (ES) Observatory in Chile. This station consists of a 0.6 m CDK robotic telescope coupled to an Andor iKon-L 936 deep depletion 2k × 2k CCD with a scale of 067 per pixel. We used a Sloan $r^{\prime}$ filter, and the exposure time was 14 s. The drift of the stars in the CCD during the sequence was smaller than 5 pixels, and the airmass ranged from 1.8 to 1.2. We processed the data with a dedicated pipeline that automatically performs the CCD reduction steps, followed by the measurement of the aperture photometry for the brightest stars in the field. The pipeline also generates the differential light curve of the target star by identifying the optimal comparison stars based on color, brightness, and proximity to the target. The light curve data are displayed in Figure 2 along with the optimal model.

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2.2.2. TESS

TOI-3362 was observed by the Transiting Exoplanet Survey Satellite (TESS; Ricker et al. 2015) in Sectors 9 and 10 (Year 1) at a cadence of 30 minutes, in Sectors 36 and 37 (Year 3) at a cadence of 10 minutes, and in Sectors 63 and 64 (Year 5) at a cadence of 2 minutes. We searched and downloaded all the light curves using lightkurve (Lightkurve Collaboration et al. 2018). TESS light curves were processed by the Science Processing Operations Center (SPOC) pipeline (Jenkins et al. 2016). The Presearch Data Conditioning component of the SPOC pipeline corrects the light curves for pointing- or focus-related instrumental signatures, discontinuities resulting from radiation events in the CCD detectors, outliers, and flux contamination. TESS data are displayed in Figure 2 along with the best-fit transit model.

2.2.3. Archival Observations

For this work, we also made use of the ground-based light curves published in Dong et al. (2021), who presented four photometric transit observations. Two of them were taken with the Antarctica Search for Transiting ExoPlanets (ASTEP) program (Guillot et al. 2015; Mékarnia et al. 2016), which observed the transits of TOI-3362b on the nights of 2020 August 10 and August 28. This 0.4 m telescope is equipped with an FLI Proline science camera with a KAF-16801E, 4096 × 4096 front-illuminated CCD with a scale of 093 per pixel, and it is located on the east Antarctic plateau. Observations were performed in a band that is similar to the R_c in transmission. Also, they observed one transit of TOI-3362b with the 1.0 m telescope at the Las Cumbres Observatory Global Telescope (LCOGT; Brown et al. 2013) Siding Spring Observatory (SSO) node in New South Wales, Australia. This telescope is equipped with a 4096 × 4096 Sinistro camera with a scale of 0389 per pixel. The transit was observed with the Pan-STARSS z-short and Bessell B filters on the night of 2021 February 7.

2.3. Spectroscopy

2.3.1. FEROS

We obtained seven spectra of TOI-3362 with FEROS (Kaufer et al. 1999) between 2021 November 23 and 2023 June 6. FEROS is an echelle spectrograph mounted at the MPG/ESO 2.2 m telescope located at La Silla Observatory, Chile. These observations were performed to identify long-period companions to TOI-3362b. The exposure times were set to 300 s, yielding an average S/N per spectral resolution element of 65. FEROS spectra were reduced, extracted, and analyzed with the ceres pipeline (Brahm et al. 2017a). The obtained FEROS RVs are displayed in Figure 3. No signatures of additional planets are identified from these observations.

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2.3.2. Archival Observations

For this work, we also made use of the RVs published in Dong et al. (2021). The first data set was obtained from the analysis of 21 spectra taken with the CHIRON echelle spectrograph (Tokovinin et al. 2013) mounted on the SMARTS 1.5 m telescope at Cerro Tololo in Chile. These observations were made between 2021 April 15 and May 12. The second data set was obtained from the analysis of nine spectra taken with the Minerva-Australis array (Addison et al. 2019). Minerva-Australis is an array of four identical 0.7 m telescopes linked via fiber feeds to a single KiwiSpec echelle spectrograph on Mt. Kent Observatory, Australia. These observations were made between 2021 May 16 and May 30.

We obtained the stellar parameters by following the procedure presented in Brahm et al. (2019). We use a two-step iterative process. The first process consists of obtaining the atmospheric parameters (T_eff, $\mathrm{log}g$, [Fe/H], and $v\sin i$) of the host star from a high-resolution spectrum using the zaspe code (Brahm et al. 2017b). This code compares the observed spectrum to a grid of synthetic spectra and determines reliable uncertainties that take into account systematic mismatches between the observations and the imperfect theoretical models. For this analysis, we used the coadded out-of-transit ESPRESSO spectra to obtain the atmospheric parameters. The second step consists of obtaining the physical parameters of the star by using publicly available broadband magnitudes of the star and comparing them with those produced by different PARSEC stellar evolutionary models (Bressan et al. 2012) by taking into account the distance to the star computed from the Gaia DR2 (Gaia Collaboration et al. 2018) parallax. This procedure delivers a new value of $\mathrm{log}g$ that is held fixed in a new run of zaspe. We iterate between the two procedures until reaching convergence in the $\mathrm{log}g$. The obtained parameters are presented in Table 1.

Table 1. Stellar Properties ^a of TOI-3362

Parameter	Value	Reference
R.A. (J2015.5)	10h23m5619	Gaia DR2
Decl. (J2015.5)	−56d50m3522	Gaia DR2
pmR.A.(mas yr−1)	−24.12 ± 0.05	Gaia DR2
pmdecl. (mas yr−1)	7.72 ± 0.05	Gaia DR2
π (mas)	2.79 ± 0.01	Gaia DR2
T (mag)	10.382 ± 0.006	TICv8
B (mag)	11.643 ± 0.12	APASS b
V (mag)	10.86 ± 0.010	APASS
G (mag)	10.7051 ± 0.002	Gaia DR2 c
GBP (mag)	10.948 ± 0.005	Gaia DR2
GRP (mag)	10.335 ± 0.003	Gaia DR2
J (mag)	9.94 ± 0.02	2MASS d
H (mag)	9.72 ± 0.02	2MASS
Ks (mag)	9.69 ± 0.02	2MASS
Teff (K)	6800 ± 100	This work
$\mathrm{log}g$ (dex)	4.14 ± 0.01	This work
[Fe/H] (dex)	0.24 ± 0.05	This work
$v\sin i$ (km s−1)	21.9 ± 0.8	This work
M⋆ (M⊙)	1.53 ± 0.02	This work
R⋆ (R⊙)	1.75 ± 0.02	This work
L⋆ (L⊙)	5.9 ± 0.3	This work
AV (mag)	0.26 ± 0.06	This work
Age (Gyr)	1.3 ± 0.2	This work
ρ⋆ (g cm−3)	0.40 ± 0.02	This work

Notes.

^a The stellar parameters computed in this work do not consider possible systematic differences among different stellar evolutionary models (Tayar et al. 2022) and have underestimated uncertainties. ^b Munari et al. (2014). ^c Gaia Collaboration et al. (2018). ^d Skrutskie et al. (2006).

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To update the orbital ephemeris of TOI-3362b and look for transit timing variations (TTVs), we modeled all the photometric data presented in Section 2.2 using the juliet code (Espinoza et al. 2019). juliet uses batman (Kreidberg 2015) for the transit model and the dynesty dynamic nested sampler (Speagle 2020) to perform Bayesian analysis and explore the likelihood space to obtain posterior probability distributions. We placed uninformative priors on the transit parameters R_p /R_⋆ and b, with an informative prior on the stellar density, that was constrained in Section 3. We sampled the limb-darkening parameters using the quadratic q₁ and q₂ from Kipping (2013a) with uniform priors. We placed Gaussian priors for each transit midpoint based on the expected values calculated from the orbital period and t₀ from Dong et al. (2021), with uncertainties of 0.1 day. To account for variability in TESS light curves, we included a Matern-3/2 Gaussian process (GP) as implemented in celerite (Foreman-Mackey et al. 2017) and available in juliet. Years 1, 3, and 5 had their own GP kernels to account for differences in variability captured in different epochs and cadences. From this analysis, we ruled out the presence of TTVs greater than ∼5 minutes and we obtained improved orbital ephemeris of TOI-3362b.

To precisely constrain the parameters of TOI-3362b and its orbit, we jointly modeled all observations presented in Section 2. In total, our model considered 38 parameters, 10 physical and 28 instrumental (see Table 2). We used batman (Kreidberg 2015) to model all the light curves. To reduce the computational cost, we only considered the detrended TESS light curve in phases between 0.48 and 0.52 (∼5 times the transit duration; see Figure 2). For the limb darkening, we considered a quadratic law using the parameterization from Kipping (2013a). To model the out-of-transit RVs and RM effect, we used the rmfit package (Stefansson et al. 2020, 2022), which uses the framework from Hirano et al. (2010) to model the anomalous RVs seen during the RM effect and radvel (Fulton et al. 2018) to model the Keplerian out-of-transit RV curve. We removed one FEROS and two CHIRON measurements that were taken during the transits of the planet, and we did not use them during the analysis. To account for the instrumentation offsets and systematics, we included independent RV offsets and log-uniform jitter terms for each instrument.

Table 2. Summary of Priors and Resulting Posteriors of the Joint Fit

Parameter	Description	Prior	Posterior
λ	Sky-projected obliquity (deg)	U( − 180, 180)	${1.2}_{-2.7}^{+2.8}$
$v\sin i$	Projected rotational velocity (km s−1)	U(0, 30)	${20.2}_{-1.6}^{+1.7}$
P	Orbital period (days)	N(18.095368, 0.000012)	${18.095367}_{-0.000008}^{+0.000008}$
t0	Transit midpoint (BJD)	N(2458529.3277, 0.0007)	${2458529.3279}_{-0.0006}^{+0.0006}$
ρ⋆	Stellar density (g cm−3)	N(0.40, 0.02)	${0.40}_{-0.02}^{+0.02}$
b	Impact parameter	U(0, 1)	${0.57}_{-0.05}^{+0.04}$
Rp /R⋆	Radius ratio	U(0, 1)	${0.070}_{-0.001}^{+0.001}$
e	Eccentricity	U(0, 0.95)	${0.720}_{-0.016}^{+0.016}$
ω	Argument of periastron (deg)	U(0, 360)	${60.6}_{-7.1}^{+8.0}$
K	RV semiamplitude (m s−1)	U(0, 1000)	${338}_{-27}^{+27}$
a/R⋆	Scaled semimajor axis	⋯	${19.1}_{-0.3}^{+0.3}$
i	Orbital inclination (deg)	⋯	${84.25}_{-0.31}^{+0.34}$
a	Semimajor axis (au)	⋯	${0.155}_{-0.003}^{+0.003}$
Rp	Planet radius (RJ)	⋯	${1.2}_{-0.02}^{+0.02}$
Mp	Planet mass (MJ)	⋯	${4.0}_{-0.4}^{+0.4}$
ρp	Planet mean density (g cm−3)	⋯	${3.0}_{-0.3}^{+0.3}$
${q}_{1}^{\mathrm{ESPRESSO}}$	ESPRESSO linear limb-darkening parameter	U(0, 1)	${0.42}_{-0.17}^{+0.22}$
${q}_{2}^{\mathrm{ESPRESSO}}$	ESPRESSO quadratic limb-darkening parameter	U(0, 1)	${0.58}_{-0.34}^{+0.29}$
β	Intrinsic stellar line width (km s−1)	N(7.8, 2.0)	${7.6}_{-1.9}^{+1.9}$
γESPRESSO	ESPRESSO RV offset (m s−1)	U(7500, 8000)	${7637}_{-33}^{+35}$
γCHIRON	CHIRON RV offset (m s−1)	U(6200, 6600)	${6443}_{-35}^{+35}$
γMINERVA	Minerva-Australis RV offset (m s−1)	U(7400, 7800)	${7604}_{-56}^{+55}$
γFEROS	FEROS RV offset (m s−1)	U(7800, 8200)	${8029}_{-35}^{+35}$
σESPRESSO	ESPRESSORV jitter (m s−1)	LU(10−3, 100)	${0.1}_{-0.1}^{+1.0}$
σCHIRON	CHIRON RV jitter (m s−1)	LU(10−3, 100)	${0.3}_{-0.3}^{+15.6}$
σMINERVA	Minerva-Australis RV jitter (m s−1)	LU(10−3, 100)	${0.4}_{-0.3}^{+13.2}$
σFEROS	FEROS RV jitter (m s−1)	LU(10−3, 100)	${0.2}_{-0.2}^{+7.2}$
${q}_{1}^{\mathrm{TESS}}$	TESS linear limb-darkening parameter	U(0, 1)	${0.08}_{-0.05}^{+0.10}$
${q}_{2}^{\mathrm{TESS}}$	TESS quadratic limb-darkening parameter	U(0, 1)	${0.38}_{-0.27}^{+0.37}$
${q}_{1}^{r^{\prime} }$	$r^{\prime}$ linear limb-darkening parameter	U(0, 1)	${0.73}_{-0.26}^{+0.19}$
${q}_{2}^{r^{\prime} }$	$r^{\prime}$ quadratic limb-darkening parameter	U(0, 1)	${0.72}_{-0.33}^{+0.20}$
${q}_{1}^{B}$	B linear limb-darkening parameter	U(0, 1)	${0.05}_{-0.04}^{+0.10}$
${q}_{2}^{B}$	B quadratic limb-darkening parameter	U(0, 1)	${0.37}_{-0.27}^{+0.37}$
${q}_{1}^{{z}_{s}}$	zs linear limb-darkening parameter	U(0, 1)	${0.12}_{-0.08}^{+0.15}$
${q}_{2}^{{z}_{s}}$	zs quadratic limb-darkening parameter	U(0, 1)	${0.30}_{-0.22}^{+0.35}$
${q}_{1}^{{R}_{c}}$	Rc linear limb-darkening parameter	U(0, 1)	${0.08}_{-0.06}^{+0.12}$
${q}_{2}^{{R}_{c}}$	Rc quadratic limb-darkening parameter	U(0, 1)	${0.32}_{-0.23}^{+0.37}$
${\sigma }_{\mathrm{TESS}}^{{\rm{Y}}1}$	TESS Year 1 photometric jitter	LU(10−6, 50)	${0.00142}_{-0.00012}^{+0.00014}$
${\sigma }_{\mathrm{TESS}}^{{\rm{Y}}3}$	TESS Year 3 photometric jitter	LU(10−6, 50)	${0.00089}_{-0.00006}^{+0.00006}$
${\sigma }_{\mathrm{TESS}}^{{\rm{Y}}5}$	TESS Year 5 photometric jitter	LU(10−6, 50)	${0.00002}_{-0.00001}^{+0.00010}$
${\sigma }_{r^{\prime} }$	$r^{\prime}$ photometric jitter	LU(10−6, 50)	${0.00351}_{-0.00021}^{+0.00022}$
σB	B photometric jitter	LU(10−6, 50)	${0.00140}_{-0.00010}^{+0.00011}$
${\sigma }_{{z}_{s}}$	zs photometric jitter	LU(10−6, 50)	${0.00063}_{-0.00013}^{+0.00012}$
${\sigma }_{{R}_{c}}$	Rc photometric jitter	LU(10−6, 50)	${0.00174}_{-0.00007}^{+0.00007}$

Note. U(a, b) denotes a uniform prior with a start value a and end value b. N(m, σ) denotes a normal prior with mean m, and standard deviation σ. LU(a, b) denotes a log-uniform prior with a start value a and end value b.

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We placed uninformative priors for almost all parameters, except for the orbital period and time of midtransit, which had been constrained in Section 4, and for the stellar density that was constrained in Section 3. Likewise, for β, which is the total line width accounting for both instrumental resolution and macroturbulence velocity. We considered an instrumental broadening of 2.15 km s⁻¹ because of the ESPRESSO resolution, and following Albrecht et al. (2012) we used the macroturbulence law for hot stars from Gray (1984), which yields a broadening of ∼7.5 km s⁻¹ for T_eff ∼ 6800 K. We added those broadening values in quadrature to set our prior for β, with an uncertainty of 2 km s⁻¹. We also tested with an uninformative prior, which resulted in the same posterior values, demonstrating that the posteriors are insensitive to β. All priors and resulting posteriors are shown in Table 2. To estimate the Bayesian posteriors and evidences of our model, we used the dynesty dynamic nested sampler. Since the number of parameters was 38, we considered 3500 live points to ensure convergence. Figure A1 shows the joint posterior distributions and histograms for the physical parameters of our model. All parameters appear well behaved and approximately Gaussian in distribution.

We performed a series of tests to verify our results. First, we tried emcee (Foreman-Mackey et al. 2013) to sample the posteriors. Second, we performed a joint fit of the photometry and the out-of-transit RVs with juliet. Finally, we carried out a fully independent analysis using the exoplanet (Foreman-Mackey et al. 2021) package implementing the RM effect using the formalism of Hirano et al. (2011). All of them returned a completely consistent set of orbital and stellar obliquity parameters.

Figures 1–3 show the different data sets together with the best-fit model. We found a sky-projected obliquity $\lambda ={1.2}_{-2.7}^{+2.8}$°, which means that the TOI-3362b is perfectly aligned with its host star. This result is surprising given the high eccentricity e = 0.720 ± 0.016 of its orbit. We also found lower values of eccentricity and inclination than the ones reported by Dong et al. (2021) of ${0.815}_{-0.032}^{+0.023}$° and ${89.140}_{-0.668}^{+0.584}$°, respectively. That is mainly a result of the additional data sets included in this work. In particular, FEROS, ESPRESSO, and TESS Year 5 measurements allowed us to better constrain the orbit of the planet. But in general, the parameters are consistent with the previously reported ones. The baseline of the RV measurements used in this work is ∼2 yr, and residuals show no clear evidence of companions (see Figure 3). Therefore, based on the residuals, we rule out the presence of a companion with at least the same mass in orbits with a ≲ 4.5 au (see Section 6.1).

The values of the semimajor axis and eccentricity obtained here are consistent with TOI-3362b being a proto HJ undergoing high-e tidal migration. The approximate final orbital separation is a_final = a(1 − e²) ≈ 9.2 R_⋆ or 0.07 au, corresponding to an orbital period of P_final ≈ 6 days. We estimate a tidal circularization timescale of the planet following Goldreich & Soter (1966):

$$\begin{eqnarray}&&{\tau }_{\mathrm{circ}}\equiv -\displaystyle \frac{e}{\dot{e}}=\displaystyle \frac{2P}{63\pi }{Q}_{p}^{{\prime} }\displaystyle \frac{{M}_{p}}{{M}_{\star }}{\left(\displaystyle \frac{a}{{R}_{p}}\right)}^{5}F(e),\end{eqnarray} \tag{ 1 }$$

where ${Q}_{p}^{{\prime} }$ is the planet's modified tidal quality factor (e.g., Goldreich & Soter 1966; Ogilvie & Lin 2007) and F(e) is an eccentricity-dependent correction factor (Hut 1981). For e = 0.720, and assuming pseudosynchronous rotation of the planet, we have F(e) ≈ 0.004. Further, assuming that ${Q}_{p}^{{\prime} }={10}^{5}\mbox{--}{10}^{6}$, we have τ_circ ∼ 0.8–8 Gyr for TOI-3362b. This timescale is longer than that reported by Dong et al. (2021), who inferred τ_circ ∼ 2.7 Gyr assuming an eccentricity of 0.815 and ${Q}_{p}^{{\prime} }={10}^{6}$. Given that the age of TOI-3362 is 1.3 ± 0.2 Gyr, it is not surprising that we observe the planet in an eccentric orbit. Moreover, the planetary orbit may not have enough time to fully circularize before the host star exits the main sequence. Nevertheless, it could undergo sufficient orbital decay to meet the criteria for classification as an eccentric HJ (P < 10 days). Indeed, direct numerical integration ¹⁵ shows that the orbit can shrink down to 0.11 au in a time span of 0.3–3 Gyr.

In Figure 4, we compare the orbital properties of TOI-3362b to a larger population of planets, plotting the sky-projected obliquity versus eccentricity for all giant planets with e > 0.1. In the left panel, for single-star systems, we can see an emerging trend: sky-projected obliquities tend to be lower with increasing eccentricity. Systems with 0.1 ≲ e ≲ 0.4 are spreading from well aligned to moderately misaligned (λ ∼ 45 deg). All six systems with e ≳ 0.4 are well aligned, with TOI-3362b being the most eccentric one and having λ < 10° at a 3σ confidence level thanks to ESPRESSO precision. This trend disappears when we consider binary stars, as shown in the right panel of Figure 4.

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6.1. Limits on Potential Outer Companions

The baseline of the RV measurements used in this work is ∼2 yr, and residuals show no clear evidence of companions (see Figure 3). To estimate what kind of planets can produce RV signals consistent with the residuals of the RV model for TOI-3362b, we performed a population synthesis study.

We considered cold Jupiters with masses between 1 and 30 M_J, and with semimajor axes between 1 and 15 au. The eccentricity of the planets was drawn from a uniform distribution between 0.2 and 0.5, considering that cold Jupiters tend to be moderately eccentric (e.g., Kipping 2013b). The orbital inclinations of the companions were between 0° and 90°. With those distributions, we estimated the RV signals of different companions. Figure 5 shows regions of the mass versus semimajor axis diagram where planets produced RV signals that are inconsistent with the residuals of Figure 3. We discard the existence of similar mass companions to TOI-3362b within ∼4.5 au (2σ) and ∼7 au (1σ).

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We also estimated the region of the diagram in Figure 5 where secular eccentricity excitation is forbidden by relativistic precession. This occurs when the secular timescale due to the companion is longer than the timescale for relativistic precession. The ratio _gr between these timescales can be calculated as

$$\begin{eqnarray}&&{\epsilon }_{\mathrm{gr}}=\displaystyle \frac{{{GM}}_{\star }^{2}{a}_{c}^{3}{\left(1-{e}_{c}^{2}\right)}^{3/2}}{{c}^{2}{a}^{4}{M}_{c}},\end{eqnarray} \tag{ 2 }$$

where a_c , e_c , and M_c are the semimajor axis, eccentricity, and mass of the companion, respectively. In Figure 5, the red dashed line corresponds to _gr = 1 for e_c = 0.5.

6.2. Origin

We would like to thank Jiayin Dong for useful discussions on the implementation of the RM effect in exoplanet. J.I.E.R. acknowledges support from the National Agency for Research and Development (ANID) Doctorado Nacional grant 2021-21212378. A.J., R.B., C.P., and M.H. acknowledge support from ANID—Millennium Science Initiative—ICN12_009. C.P. acknowledges support from CATA-Basal AFB-170002, ANID BASAL project FB210003, FONDECYT Regular grant 1210425, CASSACA grant CCJRF2105, and ANID+REC Convocatoria Nacional subvención a la instalación en la Academia convocatoria 2020 PAI77200076. R.B. acknowledges support from FONDECYT project 11200751. A.J. acknowledges support from FONDECYT project 1210718. G.S. acknowledges support provided by NASA through the NASA Hubble Fellowship grant HST-HF2-51519.001-A awarded by the Space Telescope Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc., for NASA, under contract NAS5-26555.

Figure A1 shows the joint posterior distributions and histograms for the physical parameters of our model.

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