# The Aligned Orbit of the Eccentric Proto Hot Jupiter TOI-3362b111Based on
observations made with ESO Telescopes at the La Silla Paranal Observatory
under programme ID 110.23Y8.004

Juan I. Espinoza-Retamal Instituto de Astrofísica, Pontificia Universidad
Católica de Chile, Av. Vicuña Mackenna 4860, 782-0436 Macul, Santiago, Chile
Rafael Brahm Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av.
Diagonal las Torres 2640, Peñalolén, Santiago, Chile  Millennium Institute for
Astrophysics, Chile  Data Observatory Foundation, Chile  Cristobal Petrovich
Instituto de Astrofísica, Pontificia Universidad Católica de Chile, Av. Vicuña
Mackenna 4860, 782-0436 Macul, Santiago, Chile  Millennium Institute for
Astrophysics, Chile  Andrés Jordán Facultad de Ingeniería y Ciencias,
Universidad Adolfo Ibáñez, Av. Diagonal las Torres 2640, Peñalolén, Santiago,
Chile  Millennium Institute for Astrophysics, Chile  Data Observatory
Foundation, Chile  El Sauce Observatory – Obstech, Chile  Guðmundur Stefánsson
NASA Sagan Fellow  Department of Astrophysical Sciences, Princeton University,
4 Ivy Lane, Princeton, NJ 08540, USA  Elyar Sedaghati European Southern
Observatory (ESO), Av. Alonso de Córdova 3107, 763 0355 Vitacura, Santiago,
Chile  Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av.
Diagonal las Torres 2640, Peñalolén, Santiago, Chile  Millennium Institute for
Astrophysics, Chile  Melissa J. Hobson Millennium Institute for Astrophysics,
Chile  Max-Planck-Institut für Astronomie, Königstuhl 17, D-69117 Heidelberg,
Germany  Diego J. Muñoz Facultad de Ingeniería y Ciencias, Universidad Adolfo
Ibáñez, Av. Diagonal las Torres 2640, Peñalolén, Santiago, Chile  Millennium
Institute for Astrophysics, Chile  Data Observatory Foundation, Chile  Center
for Interdisciplinary Exploration and Research in Astrophysics (CIERA) and
Department of Physics and Astronomy, Northwestern University, 2145 Sheridan
Road, Evanston, IL 60208, USA  Department of Astronomy and Planetary Science,
Northern Arizona University, Flagstaff, AZ 86011, USA  Gavin Boyle El Sauce
Observatory – Obstech, Chile  Cavendish Laboratory, J J Thomson Avenue,
Cambridge, CB3 0HE, UK  Rodrigo Leiva Millennium Institute for Astrophysics,
Chile  Instituto de astrofísica de Andalucía, CSIC, Glorieta de la Astronomía
s/n, 18008 Granada, Spain  Vincent Suc Millennium Institute for Astrophysics,
Chile  Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av.
Diagonal las Torres 2640, Peñalolén, Santiago, Chile  El Sauce Observatory –
Obstech, Chile

(Received June 1, 2019; Revised January 10, 2019)

###### Abstract

High-eccentricity tidal migration predicts the existence of highly eccentric
proto-hot Jupiters on the “tidal circularization track,” meaning that they
might eventually become hot Jupiters, but that their migratory journey remains
incomplete. Having experienced moderate amounts of the tidal reprocessing of
their orbital elements, proto-hot Jupiters systems can be powerful test-beds
for the underlying mechanisms of eccentricity growth. Notably, they may be
used for discriminating between variants of high-eccentricity migration, each
predicting a distinct evolution of misalignment between the star and the
planet’s orbit. We constrain the spin-orbit misalignment of the proto-hot
Jupiter TOI-3362b with high precision radial velocity observations using
ESPRESSO at VLT. The observations reveal a sky-projected obliquity
$\lambda=1.2_{-2.7}^{+2.8}$ deg and constrain the orbital eccentricity to
$e=0.720\pm 0.016$, making it one of the most eccentric gas giants for which
the obliquity has been measured. The large eccentricity and the striking orbit
alignment of the planet suggest that ongoing coplanar high-eccentricity
migration driven by a distant companion is a likely explanation for the
system’s architecture. This distant companion would need to reside beyond 5 au
at 95% confidence to be compatible with the available radial velocity
observations.

Exoplanets (498) — Hot Jupiters (753) — Exoplanet dynamics (490) — Planetary
alignment (1243) — Exoplanet migration (2205)

††journal: ApJ

##  1 Introduction

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,
conclusively 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 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 Observations

###  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 27th of December 2022, 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 to 788 nm at a
resolving power of $R\approx$ 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 0.58 $-$ 1.95′′. The
spectra have a median S/N of 54 at 550 nm and a median RV uncertainty of 12.4
m/s. 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 (representing the sampling of the
spectrograph) for $\pm 100$ km/s centered on the estimated systemic velocity.
The RM component of the RVs is presented in Figure 1.

###  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 $\times$ 2k CCD with a scale of 0.67′′ 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 is displayed in
Figure 2 along with the optimal model.

####  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 (PDC)
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 is 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 4 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 10th August 2020 and 28th August 2020.
This 0.4 m telescope is equipped with an FLI Proline science camera with a
KAF-16801E, $4096\times 4096$ front-illuminated CCD with a scale of 0.93" 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 $4096\times 4096$ Sinistro camera with a scale of 0.389" per
pixel. The transit was observed with the Pan-STARSS $z$-short and Bessell $B$
filters on the night of 7th February 2021.

###  2.3 Spectroscopy

####  2.3.1 FEROS

We obtained 7 spectra of TOI-3362 with FEROS (Kaufer et al., 1999) between
23rd November 2021 and 6th June 2023. 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.

####  2.3.2 Archival Observations

For this work, we also made use of the radial velocities published in Dong et
al. (2021). The first dataset 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 15th April 2021 and 12th May 2021. The second dataset was
obtained from the analysis of 9 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 16th May
2021 and 30th May 2021.

##  3 Stellar Parameters

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 ($T_{eff}$, $\log{g}$,
[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 co-added 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 broad-band 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 Gaia DR2
(Gaia Collaboration et al., 2018) parallax. This procedure delivers a new
value of $\log{g}$ that is held fixed in a new run of zaspe. We iterate
between the two procedures until reaching convergence in the $\log{g}$. The
obtained parameters are presented in Table 1.

Table 1: Stellar propertiesa of TOI-3362.

Parameter | Value | Reference
---|---|---
RA (J2015.5) | 10h23m56.19s | Gaia DR2
Dec (J2015.5) | -56d50m35.22s | Gaia DR2
pmRA (mas/yr) | -24.12 $\pm$ 0.05 | Gaia DR2
pmDEC (mas/yr) | 7.72 $\pm$ 0.05 | Gaia DR2
$\pi$ (mas) | 2.79 $\pm$ 0.01 | Gaia DR2
T (mag) | 10.382 $\pm$ 0.006 | TICv8
B (mag) | 11.643 $\pm$ 0.12 | APASSb
V (mag) | 10.86 $\pm$ 0.010 | APASS
G (mag) | 10.7051 $\pm$ 0.002 | Gaia DR2c
GBP (mag) | 10.948 $\pm$ 0.005 | Gaia DR2
GRP (mag) | 10.335 $\pm$ 0.003 | Gaia DR2
J (mag) | 9.94 $\pm$ 0.02 | 2MASSd
H (mag) | 9.72 $\pm$ 0.02 | 2MASS
Ks (mag) | 9.69 $\pm$ 0.02 | 2MASS
$T_{eff}$ (K) | 6800 $\pm$ 100 | This work
$\log{g}$ (dex) | 4.14 $\pm$ 0.01 | This work
$[Fe/H]$ (dex) | 0.24 $\pm$ 0.05 | This work
$v\sin{i}$ (km/s) | 21.9 $\pm$ 0.8 | This work
$M_{\star}$ ($M_{\odot}$) | 1.53 $\pm$ 0.02 | This work
$R_{\star}$ ($R_{\odot}$) | 1.75 $\pm$ 0.02 | This work
L⋆ (L⊙) | 5.9 $\pm$ 0.3 | This work
AV (mag) | 0.26 $\pm$ 0.06 | This work
Age (Gyr) | 1.3 $\pm$ 0.2 | This work
$\rho_{\star}$ (g/cm3) | 0.40 $\pm$ 0.02 | This work

.

Note. — 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, bMunari et al.
(2014),cGaia Collaboration et al. (2018),dSkrutskie et al. (2006)

##  4 Photometric Analysis

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 $R_{p}/R_{\star}$ 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 $q_{1}$ and $q_{2}$ from Kipping
(2013a) with uniform priors. We placed Gaussian priors for each transit mid-
point based on the expected values calculated from the orbital period and
$t_{0}$ from Dong et al. (2021), with uncertainties of 0.1 days. 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. Year 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 $\sim 5$ minutes
and we obtained improved orbital ephemeris of TOI-3362b.

Table 2: Summary of priors and resulting posteriors of the joint fit.

Parameter | Description | Prior | Posterior
---|---|---|---
$\lambda$ | Sky-projected obliquity (deg) | $U(-180,180)$ | $1.2_{-2.7}^{+2.8}$
$v\sin{i}$ | Projected rotational velocity (km/s) | $U(0,30)$ | $20.2_{-1.6}^{+1.7}$
$P$ | Orbital period (days) | $N(18.095368,0.000012)$ | $18.095367_{-0.000008}^{+0.000008}$
$t_{0}$ | Transit midpoint (BJD) | $N(2458529.3277,0.0007)$ | $2458529.3279_{-0.0006}^{+0.0006}$
$\rho_{\star}$ | Stellar density (g/cm3) | $N(0.40,0.02)$ | $0.40_{-0.02}^{+0.02}$
$b$ | Impact parameter | $U(0,1)$ | $0.57_{-0.05}^{+0.04}$
$R_{p}/R_{\star}$ | Radius ratio | $U(0,1)$ | $0.070_{-0.001}^{+0.001}$
$e$ | Eccentricity | $U(0,0.95)$ | $0.720_{-0.016}^{+0.016}$
$\omega$ | Argument of periastron (deg) | $U(0,360)$ | $60.6_{-7.1}^{+8.0}$
$K$ | RV semiamplitude (m/s) | $U(0,1000)$ | $338_{-27}^{+27}$
$a/R_{\star}$ | Scaled semimajor axis | - | $19.1_{-0.3}^{+0.3}$
$i$ | Orbital inclination (deg) | - | $84.25_{-0.31}^{+0.34}$
$a$ | Semimajor axis (au) | - | $0.155_{-0.003}^{+0.003}$
$R_{p}$ | Planet radius ($R_{J}$) | - | $1.2_{-0.02}^{+0.02}$
$M_{p}$ | Planet mass ($M_{J}$) | - | $4.0_{-0.4}^{+0.4}$
$\rho_{p}$ | Planet mean density (g/cm3) | - | $3.0_{-0.3}^{+0.3}$
$q_{1}^{\rm ESPRESSO}$ | ESPRESSO linear limb darkening parameter | $U(0,1)$ | $0.42_{-0.17}^{+0.22}$
$q_{2}^{\rm ESPRESSO}$ | ESPRESSO quadratic limb darkening parameter | $U(0,1)$ | $0.58_{-0.34}^{+0.29}$
$\beta$ | Intrinsic stellar line width (km/s) | $N(7.8,2.0)$ | $7.6_{-1.9}^{+1.9}$
$\gamma_{\rm ESPRESSO}$ | ESPRESSO RV offset (m/s) | $U(7500,8000)$ | $7637_{-33}^{+35}$
$\gamma_{\rm CHIRON}$ | CHIRON RV offset (m/s) | $U(6200,6600)$ | $6443_{-35}^{+35}$
$\gamma_{\rm MINERVA}$ | Minerva-Australis RV offset (m/s) | $U(7400,7800)$ | $7604_{-56}^{+55}$
$\gamma_{\rm FEROS}$ | FEROS RV offset (m/s) | $U(7800,8200)$ | $8029_{-35}^{+35}$
$\sigma_{\rm ESPRESSO}$ | ESPRESSO RV jitter (m/s) | $LU(10^{-3},100)$ | $0.1_{-0.1}^{+1.0}$
$\sigma_{\rm CHIRON}$ | CHIRON RV jitter (m/s) | $LU(10^{-3},100)$ | $0.3_{-0.3}^{+15.6}$
$\sigma_{\rm MINERVA}$ | Minerva-Australis RV jitter (m/s) | $LU(10^{-3},100)$ | $0.4_{-0.3}^{+13.2}$
$\sigma_{\rm FEROS}$ | FEROS RV jitter (m/s) | $LU(10^{-3},100)$ | $0.2_{-0.2}^{+7.2}$
$q_{1}^{\rm TESS}$ | TESS linear limb darkening parameter | $U(0,1)$ | $0.08_{-0.05}^{+0.10}$
$q_{2}^{\rm TESS}$ | 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 | $U(0,1)$ | $0.37_{-0.27}^{+0.37}$
$q_{1}^{z_{s}}$ |  $z_{s}$ linear limb darkening parameter | $U(0,1)$ | $0.12_{-0.08}^{+0.15}$
$q_{2}^{z_{s}}$ |  $z_{s}$ quadratic limb darkening parameter | $U(0,1)$ | $0.30_{-0.22}^{+0.35}$
$q_{1}^{R_{c}}$ |  $R_{c}$ linear limb darkening parameter | $U(0,1)$ | $0.08_{-0.06}^{+0.12}$
$q_{2}^{R_{c}}$ |  $R_{c}$ quadratic limb darkening parameter | $U(0,1)$ | $0.32_{-0.23}^{+0.37}$
$\sigma_{\rm TESS}^{\rm Y1}$ | TESS Year 1 photometric jitter | $LU(10^{-6},50)$ | $0.00142_{-0.00012}^{+0.00014}$
$\sigma_{\rm TESS}^{\rm Y3}$ | TESS Year 3 photometric jitter | $LU(10^{-6},50)$ | $0.00089_{-0.00006}^{+0.00006}$
$\sigma_{\rm TESS}^{\rm Y5}$ | TESS Year 5 photometric jitter | $LU(10^{-6},50)$ | $0.00002_{-0.00001}^{+0.00010}$
$\sigma_{r^{\prime}}$ |  $r^{\prime}$ photometric jitter | $LU(10^{-6},50)$ | $0.00351_{-0.00021}^{+0.00022}$
$\sigma_{B}$ |  $B$ photometric jitter | $LU(10^{-6},50)$ | $0.00140_{-0.00010}^{+0.00011}$
$\sigma_{z_{s}}$ |  $z_{s}$ photometric jitter | $LU(10^{-6},50)$ | $0.00063_{-0.00013}^{+0.00012}$
$\sigma_{R_{c}}$ |  $R_{c}$ 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
$b$. $N(m,\sigma)$ denotes a normal prior with mean $m$, and standard
deviation $\sigma$. $LU(a,b)$ denotes a log-uniform prior with a start value
$a$ and end value $b$.

##  5 Joint fit

Figure 1: ESPRESSO observations of the RM effect of TOI-3362b with the
Keplerian signal removed. The red line shows the best fit, and the red areas
show 1, 2, and 3$\sigma$ models. The bottom panel shows the residuals. The
ESPRESSO RVs of this figure without the Keplerian signal removed are available
as the Data behind the Figure. Figure 2: TESS and ground-based transit
photometry of TOI-3362b overplotted with best-modeled lightcurves. Residuals
are shown for all transits. TESS light curves of different years are shown all
together. Observatoire Moana - ES light curve is also shown with 10min-binned
data in black, and it is available as the Data behind the Figure. Figure 3:
Left: Out-of-transit radial velocities of TOI-3362 taken with different
instruments. The red curve shows the best model. Residuals are shown at the
bottom. Right: Out-of-transit radial velocity curve in phase. The red curve
shows the best model, and the red regions show 1, 2, and 3$\sigma$ models.
FEROS data is available as the Data behind the Figure.

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 lightcurve in phases
between 0.48 and 0.52 ($\sim$ 5 times the transit duration, see Figure 2). For
the limb darkening, we considered a quadratic law using the parametrization
from Kipping (2013a). To model the out-of-transit radial velocities 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.

We placed uninformative priors for almost all parameters, except for the
orbital period and time of mid-transit, which had been constrained in Section
4, and for the stellar density that was constrained in Section 3. Likewise,
for $\beta$, which is the total line width accounting for both instrumental
resolution and macroturbulence velocity. We considered an instrumental
broadening of 2.15 km/s 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 $\sim 7.5$ km/s for $T_{eff}\sim$ 6,800
K. We added those broadening values in quadrature to set our prior for
$\beta$, with an uncertainty of 2 km/s. We also tested with an uninformative
prior, which resulted in the same posterior values, demonstrating that the
posteriors are insensitive to $\beta$. 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 3,500 live points to ensure convergence.
Figure 6 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, 2, and 3 show the different datasets together with the best-fit
model. We found a sky-projected obliquity $\lambda=1.2_{-2.7}^{+2.8}$ deg,
which means that the TOI-3362b is perfectly aligned with its host star. This
result is surprising given the high eccentricity $e=0.720\pm 0.016$ 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
$89.140_{-0.668}^{+0.584}$ deg, respectively. That is mainly a result of the
additional datasets 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 $\sim
2$ years, 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 $a\lesssim 4.5$ au (see Section
6.1).

Figure 4: Sky-projected obliquity versus the eccentricity of all giant planets
($M_{p}>0.2\,M_{J}$ and $R_{p}>8\,R_{\oplus}$) with $e>0.1$ colored by their
final semimajor axes $a_{\rm final}=a(1-e^{2})$ from TEPCat (Southworth,
2011). TOI-677b was also measured using ESPRESSO (Sedaghati et al., 2023). The
left panel shows single stars, while the right panel shows known binary
systems. Figure 5: Mass versus semimajor axis diagram for the possible
coplanar outer companions to TOI-3362b. Red regions indicate zones where we
can rule out the companion at 1 and $2\sigma$ confidence levels. The black
dashed line shows the mass of TOI-3362b. The line $\epsilon_{\rm gr}=1$ (Eq.
2) is shown in red dashed, demarking the regions where secular excitation is
quenched by relativistic precession ($\epsilon_{\rm gr}>1$). Since the
companion might lie in the region where secular eccentricity excitation is
forbidden, we discard it as responsible for the current high eccentricity of
TOI-3362b.

##  6 Discussion

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 $a_{\rm final}=a(1-e^{2})\approx
9.2\,R_{\star}$ or $0.07$ au, corresponding to an orbital period of $P_{\rm
final}\approx 6$ days. We estimate a tidal circularization timescale of the
planet following Goldreich & Soter (1966):

$\tau_{\rm
circ}\equiv-\frac{e}{\dot{e}}=\frac{2P}{63\pi}Q^{\prime}_{p}\frac{M_{p}}{M_{\star}}\left(\frac{a}{R_{p}}\right)^{5}F(e),$
(1)

where $Q^{\prime}_{p}$ 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 pseudo-
synchronous rotation of the planet, we have $F(e)\approx 0.004$. Further,
assuming that $Q^{\prime}_{p}=10^{5}-10^{6}$, we have $\tau_{\rm circ}\sim
0.8-8$ Gyr for TOI-3362b. This timescale is longer than that reported by Dong
et al. (2021), who inferred $\tau_{\rm circ}\sim 2.7$ Gyr assuming an
eccentricity of 0.815 and $Q^{\prime}_{p}=10^{6}$. Given that the age of
TOI-3362 is $1.3\pm 0.2$ 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
integration222We integrate the eccentricity evolution at constant $a_{\rm
final}$, from $e=0.72$ to $e=0.53$ at which point the orbit shrunk to 0.11 au
or an orbital period of 10 days. shows that the orbit can shrink down to 0.11
au in a timespan 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 $0.1\lesssim
e\lesssim 0.4$ are spreading from well-aligned to moderately misaligned
($\lambda\sim 45$ deg). All six systems with $e\gtrsim 0.4$ are well aligned,
with TOI-3362b being the most eccentric one and having $\lambda<10$ deg at a
$3\sigma$ confidence level thanks to ESPRESSO precision. This trend disappears
when we consider binary stars, as shown in the right panel of Figure 4.

###  6.1 Limits on potential outer companions

The baseline of the RV measurements used in this work is $\sim 2$ years, 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). Given that
Coplanar Coplanar High- Eccentricity Migration (CHEM, Petrovich, 2015) is the
most likely explanation for the system’s current architecture (see Section
6.2), the orbital inclinations of the companions were between $70^{\circ}$ and
$90^{\circ}$. 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 $\sim 4.5$ au ($2\sigma$) and $\sim 7$ au ($1\sigma$).

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 time scale
for relativistic precession. The ratio $\epsilon_{\rm gr}$ between these
timescales can be calculated as:

$\epsilon_{\rm
gr}=\frac{GM_{\star}^{2}a_{c}^{3}(1-e_{c}^{2})^{3/2}}{c^{2}a^{4}M_{c}},$ (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 $\epsilon_{\rm gr}=1$ for $e_{c}=0.5$.

###  6.2 Origin

The orbital properties of TOI-3362b and the limits for the presence of an
outer companion pose a major challenge to most dynamical paths as we describe
next.

  * •

In-situ formation or disk-driven migration could naturally explain the small
sky-projected obliquity, but does not naturally account for the high
eccentricity of the orbit. The mass ratio $M_{p}/M_{\star}\simeq 2.5\times
10^{-3}$, may not be large enough to carve a wide and deep enough gap to drive
significant eccentricity growth (Papaloizou et al., 2001; Bitsch et al.,
2013). If, in turn, the planet resides inside a cavity, simulations show that
the eccentricity may grow up to $\sim 0.4$ (Debras et al., 2021), still well
below the planet’s eccentricity of 0.72.

  * •

Planet-planet interactions at the planet’s current orbital location may have
excited the planet’s eccentricity either by scattering or secular excitation
(e.g., Anderson & Lai 2017) from a yet undetected companion. Scattering could
account for the planetary eccentricity by ejecting planets (e.g., Anderson et
al. 2020), though such outcome is not expected to be dominant because the
ratio between the planets’ escape velocity and its circular velocity a the
current semi-major axis is $\sim 1$ (Petrovich et al., 2014). An added
complication is to retain the small obliquity after scattering. In turn, the
secular excitation is largely disfavored by our radial velocity measurements.
As shown in Figure 5, our measurements exclude more massive companions inside
$\sim 4-5$ au (at 2$\sigma$), while the potential planet allowed at larger
orbital separations would be unable to drive secular excitation as
relativistic precession would quench it (red dashed line).

  * •

Tidal high-eccentricity migration can naturally account for the planetary
eccentricity and semi-major axis. Assuming an equilibrium tide model, the
planet would rapidly migrate from $\sim 1$ au (the circularization timescale
from $e$ = 0.9 to 0.72 is $\sim 0.15-1.5$ Gyr) and would stall as the
eccentricity decreases (the circularization timescale from $e=$ 0.72 to 0.5 is
$\sim 0.3-3$ Gyr). The small obliquity rules out most high-eccentricity
migration paths including vo Zeipel-Lidov-Kozai oscillations (Fabrycky &
Tremaine, 2007), planet-planet scattering (Beaugé & Nesvorný, 2012), and
secular chaos (Wu & Lithwick, 2011), while favoring CHEM (Petrovich, 2015)
that can naturally account for the small obliquity.

In summary, the most likely explanation for the system is CHEM driven by a yet
undetected massive and distant companion. If found using longer-term
observations, this model would predict that the companion should have a small
to moderate mutual inclination ($\lesssim 20$ deg), potentially measurable
using upcoming Gaia astrometric data. Using the method333The method uses the
Fisher matrix method to propagate the expected error of Gaia’s astrometric and
RV measurements into the parameters of different kinds of possible companions,
thus constraining what signals can be detected. The relatively low probability
of $\sim 30\%$ comes mostly from the fact that the star is massive and lies at
$d\approx 360$ pc, reducing the amplitude of the expected signal as observed
by Gaia. from Espinoza-Retamal et al. (submitted to AJ) we estimate a
probability of $\sim 30\%$ to measure the mutual inclinations if the planet
lies at 5 au and use the current RV data. To improve our chances of detecting
the potential planet and measuring the mutual inclination through a combined
joint RV and Gaia astrometry analysis, long-term radial velocity monitoring
will be necessary.

We would like to thank Jiayin Dong for useful discussions on the
implementation of the RM effect in exoplanet. JIER acknowledges support from
the National Agency for Research and Development (ANID) Doctorado Nacional
grant 2021-21212378. AJ, RB, CP, and MH acknowledge support from ANID –
Millennium Science Initiative – ICN12_009. CP acknowledges support from CATA-
Basal AFB-170002, ANID BASAL project FB210003, FONDECYT Regular grant 1210425,
CASSACA grant CCJRF2105, and ANID+REC Convocatoria Nacional subvencion a la
instalacion en la Academia convocatoria 2020 PAI77200076. RB acknowledges
support from FONDECYT project 11200751. AJ acknowledges support from FONDECYT
project 1210718. GS 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.

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##  Appendix A Corner plot

Figure 6: Joint posterior distributions and histograms of the posteriors for
the physical parameters of our model.