# Evidence for Low-Level Dynamical Excitation in Near-Resonant Exoplanet
Systems111This paper includes data gathered with the 6.5 meter Magellan
Telescopes located at Las Campanas Observatory, Chile.

Malena Rice Department of Astronomy, Yale University, New Haven, CT 06511, USA
Department of Physics and Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Xian-Yu Wang
Department of Astronomy, Indiana University, Bloomington, IN 47405, USA
Songhu Wang Department of Astronomy, Indiana University, Bloomington, IN
47405, USA  Avi Shporer Department of Physics and Kavli Institute for
Astrophysics and Space Research, Massachusetts Institute of Technology,
Cambridge, MA 02139, USA  Khalid Barkaoui Astrobiology Research Unit,
University of Liège, Allée du 6 août, 19, 4000 Liège (Sart-Tilman), Belgium
Department of Earth, Atmospheric and Planetary Sciences, MIT, 77 Massachusetts
Avenue, Cambridge, MA 02139, USA  Instituto de Astrofísica de Canarias (IAC),
Calle Vía Láctea s/n, 38200, La Laguna, Tenerife, Spain  Rafael Brahm
Millennium Institute for Astrophysics, Chile  Facultad de Ingeniera y
Ciencias, Universidad Adolfo Ibáñez, Av. Diagonal las Torres 2640, Peñalolén,
Santiago, Chile  Data Observatory Foundation, Chile  Karen A. Collins Center
for Astrophysics, Harvard $\&$ Smithsonian, 60 Garden Street, Cambridge, MA
02138, USA  Andrés Jordán Millennium Institute for Astrophysics, Chile
Facultad de Ingeniera y Ciencias, Universidad Adolfo Ibáñez, Av. Diagonal las
Torres 2640, Peñalolén, Santiago, Chile  Data Observatory Foundation, Chile
Nataliea Lowson Centre for Astrophysics, University of Southern Queensland,
499-565 West Street, Toowoomba, QLD 4350, Australia  R. Paul Butler Carnegie
Institution for Science, Earth & Planets Laboratory, 5241 Broad Branch Road
NW, Washington, DC 20015, USA  Jeffrey D. Crane The Observatories of the
Carnegie Institution for Science, 813 Santa Barbara Street, Pasadena, CA
91101, USA  Stephen Shectman The Observatories of the Carnegie Institution for
Science, 813 Santa Barbara Street, Pasadena, CA 91101, USA  Johanna K. Teske
Carnegie Institution for Science, Earth & Planets Laboratory, 5241 Broad
Branch Road NW, Washington, DC 20015, USA  David Osip Las Campanas
Observatory, Carnegie Institution for Science, COlina el Pino, Casilla 601 La
Serena, Chile  Kevin I. Collins George Mason University, 4400 University
Drive, Fairfax, VA, 22030 USA  Felipe Murgas Instituto de Astrofísica de
Canarias (IAC), E-38205 La Laguna, Tenerife, Spain  Departamento de
Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife,
Spain  Gavin Boyle El Sauce Observatory – Obstech, Chile  Cavendish
Laboratory, JJ Thomson Avenue, Cambridge, CB3 0HE, UK  Francisco J. Pozuelos
Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta de la Astronomía
s/n, 18008 Granada, Spain  Astrobiology Research Unit, University of Liège,
Allée du 6 août, 19, 4000 Liège (Sart-Tilman), Belgium  Mathilde Timmermans
Astrobiology Research Unit, University of Liège, Allée du 6 août, 19, 4000
Liège (Sart-Tilman), Belgium  Emmanuel Jehin Space Sciences, Technologies and
Astrophysics Research (STAR) Institute, Université de Liège, Allée du 6 Août
19C, B-4000 Liège, Belgium  Michaël Gillon Astrobiology Research Unit,
University of Liège, Allée du 6 août, 19, 4000 Liège (Sart-Tilman), Belgium
Malena Rice<EMAIL_ADDRESS>
###### Abstract

The geometries of near-resonant planetary systems offer a relatively pristine
window into the initial conditions of exoplanet systems. Given that near-
resonant systems have likely experienced minimal dynamical disruptions, the
spin-orbit orientations of these systems inform the typical outcomes of
quiescent planet formation, as well as the primordial stellar obliquity
distribution. However, few measurements have been made to constrain the spin-
orbit orientations of near-resonant systems. We present a Rossiter-McLaughlin
measurement of the near-resonant warm Jupiter TOI-2202 b, obtained using the
Carnegie Planet Finder Spectrograph (PFS) on the 6.5m Magellan Clay Telescope.
This is the eighth result from the Stellar Obliquities in Long-period
Exoplanet Systems (SOLES) survey. We derive a sky-projected 2D spin-orbit
angle $\lambda=26^{+12}_{-15}\,{}^{\circ}$ and a 3D spin-orbit angle
$\psi=31^{+13}_{-11}\,{}^{\circ}$, finding that TOI-2202 b – the most massive
near-resonant exoplanet with a 3D spin-orbit constraint to date – likely
deviates from exact alignment with the host star’s equator. Incorporating the
full census of spin-orbit measurements for near-resonant systems, we
demonstrate that the current set of near-resonant systems with period ratios
$P_{2}/P_{1}\lesssim 4$ is generally consistent with a quiescent formation
pathway, with some room for low-level ($\lesssim 20\,^{\circ}$) protoplanetary
disk misalignments or post-disk-dispersal spin-orbit excitation. Our result
constitutes the first population-wide analysis of spin-orbit geometries for
near-resonant planetary systems.

planetary alignment (1243), orbital resonances (1181), exoplanet dynamics
(490), dynamical evolution (421), exoplanets (498), exoplanet systems (484)

††software: allesfitter (Günther & Daylan, 2021), AstroImageJ (Collins et al.,
2017), emcee (Foreman-Mackey et al., 2013), lightkurve (Lightkurve
Collaboration et al., 2018), matplotlib (Hunter, 2007), numpy (Oliphant, 2006;
Walt et al., 2011; Harris et al., 2020), pandas (McKinney, 2010), ptemcee
Vousden et al. (2016), scipy (Virtanen et al., 2020)

##  1 Introduction

The dynamical histories of planetary systems can, to some extent, be
reconstructed through their current orbital demographics. Near-resonant
systems, in which two or more planets exhibit near-exact integer ratio
commensurabilities of their orbital periods, offer an especially well-
constrained lens into the evolution of planetary systems (e.g. Goldreich &
Sciama, 1965; Lee & Peale, 2002; Millholland et al., 2018; Goyal et al.,
2023). In these cases, formation models must jointly account for both the
systems’ near-resonant configurations and their currently observed orbital
geometries.

One observable constraint on a system’s orbital geometry is the tilt of
companion planets’ orbits relative to the host star’s spin axis. The sky-
projection of these “spin-orbit” angles, $\lambda$, can be measured for
transiting planets through the Rossiter-McLaughlin effect (Rossiter, 1924;
McLaughlin, 1924), in which small radial velocity (RV) shifts are observed
across a transiting exoplanet’s passage in front of its host star. As the
planet transits, it sequentially blocks different red- and blue-shifted
components of the stellar disk, leading to a warped signal in the net observed
Doppler shift across the transit. The profile of this warped signal encodes
the degree of alignment between the planet’s transit path and the equator of
the host star.

To date, only a handful of near-resonant systems have had spin-orbit angles
measured to characterize the tilts of their constituent planetary orbits. In
this work, we add a new measurement to this sample: a Rossiter-McLaughlin
observation across a transit of the warm Jupiter TOI-2202 b. This is the
eighth result from the Stellar Obliquities in Long-period Exoplanet Systems
(SOLES) survey (Rice et al., 2021; Wang et al., 2022; Rice et al., 2022, 2023;
Hixenbaugh et al., 2023; Dong et al., 2023; Wright et al., 2023), which
examines the spin-orbit angles of relatively wide-orbiting transiting
exoplanets.

TOI-2202 b lies in a near-resonant configuration deduced through observations
of strong transit timing variations (TTVs) from an expected outer companion
near the 2:1 mean-motion resonance (MMR; Trifonov et al., 2021). Using newly
obtained RV measurements from the Carnegie Planet Finder Spectrograph (PFS;
Crane et al., 2006, 2008, 2010), together with archival data and new
photometry from an assortment of telescopes, we derive a moderate 2D spin-
orbit angle $\lambda=26^{+12}_{-15}\,{}^{\circ}$ and 3D spin-orbit angle
$\psi=31^{+13}_{-11}\,{}^{\circ}$ for TOI-2202 b.

Combining our measurement with archival results, we conduct the first
population study of the spin-orbit configurations of exoplanets with near-
resonant companions. Our findings support the hypothesis that near-resonant
planetary systems typically form quiescently – that is, within the disk plane
and without violent post-disk-dispersal interactions, such as planet-planet
scattering (Rasio & Ford, 1996), that would significantly displace orbits from
this initial plane – while simultaneously suggesting the prevalence of low-
level dynamical excitation even in near-resonant systems.

##  2 Observations

###  2.1 Photometric monitoring

Because TOI-2202 b exhibits strong TTVs, we obtained several photometric
transit observations leading up to the Rossiter-McLaughlin event to determine
the optimal observing window. We also obtained simultaneous photometry during
the scheduled Rossiter-McLaughlin observation to precisely constrain the
transit mid-time. Data from seven ground-based telescopes, described in the
following subsections and shown in Figure 1, were used in this transit
monitoring effort. Each transit of TOI-2202 b lasts 3.8 hours; however, most
of our photometric monitoring included observations only at ingress to
demonstrate the moving location of the transit start time. The derived transit
mid-times associated with each set of observations are provided in Table 1 for
reference.

Figure 1: Photometry obtained from the seven telescopes used within this work. Observations are ordered by the date on which they were taken, with shaded regions signifying the modeled transit durations and solid gray lines denoting the transit mid-times measured for each dataset. The transit models associated with our parameter solution are shown together with the data, and a dotted line marks the linear ephemeris mid-transit. The data behind this figure is provided in the digital version of this manuscript. Table 1: Transit numbers $N$, mid-transit times $t_{0}$, uncertainties $\sigma_{t_{0}}$, and offsets from the linear ephemeris prediction $\Delta_{\mathrm{lin}}$ for TOI-2202 b based on our collected photometry and new TESS transits collected since the publication of Trifonov et al. (2021). All listed $\Delta_{\mathrm{lin}}$ values in this table are provided relative to our derived transit mid-time epoch $T_{0}=2459577.9736362\pm 0.0039$ days (Table 3).  $N$ |  $t_{0}$ (BJDTDB) | $\sigma_{t_{0}}$ |  $\Delta_{\mathrm{lin}}$ (min) | Instrument
---|---|---|---|---
-45 | 2459041.9141860 | 0.0013 | 44.91 | TESS
-44 | 2459053.8287755 | 0.0012 | 46.98 | TESS
-43 | 2459065.7370344 | 0.0010 | 39.93 | TESS
-42 | 2459077.6478578 | 0.0010 | 36.58 | TESS
-41 | 2459089.5520912 | 0.0014 | 23.74 | TESS
-39 | 2459113.3622859 | 0.0016 | 0.54 | TESS
-24 | 2459292.0300834 | 0.0013 | -41.90 | TESS
-23 | 2459303.9535926 | 0.0015 | -26.98 | TESS
-18 | 2459363.5678775 | 0.0017 | 42.90 | TESS
-16 | 2459387.4048621 | 0.0016 | 58.28 | TESS
20 | 2459816.2811297 | 0.0035 | 62.34 |  Minerva-Australis
21 | 2459828.1830501 | 0.0029 | 46.16 | LCOGT SSO 0.4m
|  |  |  | LCOGT SSO 1m
24 | 2459863.8942826 | 0.0020 | 5.52 | LCOGT CTIO 0.4m
|  |  |  | LCOGT CTIO 1m
|  |  |  | TRAPPIST-South
25 | 2459875.7978660 | 0.0014 | -8.25 | LCOGT CTIO 0.4m
|  |  |  | Observatoire Moana ES
26 | 2459887.7015528 | 0.0006 | -21.88 | LCOGT CTIO 1m
|  |  |  | Observatoire Moana ES
36 | 2460006.8115743 | 0.0012 | -52.83 | TESS
37 | 2460018.7353190 | 0.0008 | -37.58 | TESS
38 | 2460030.6594249 | 0.0011 | -21.80 | TESS

####  2.1.1 MINERVA-Australis 0.7m photometry

Our team measured the ingress of TOI-2202 b’s transit on UT 8/24/2022 using
one of the 0.7-m Minerva-Australis telescopes (Addison et al., 2019) located
at the University of Southern Queensland’s Mount Kent Observatory. The
telescope is equipped with a $2000\times 2000$ pixel Andor CCD with pixel
scale $0.608\arcsec$, and we used a 15-pixel radius ($9.12\arcsec$) aperture
to extract the photometry. We obtained $2.65$ hours of pre-transit baseline
observations, as well as photometry during the first $2.25$ hours of transit.
Observations consisted of continuous 60-second broadband exposures. From this
set of observations, we derived a 62.34-minute late ingress of the transit
relative to this work’s fitted linear ephemeris of TOI-2202 b.

####  2.1.2 LCOGT SSO 0.4-m and 1-m photometry

We measured one partial transit of TOI-2202 b on UT 9/5/2022 using the Las
Cumbres Observatory Global Network (LCOGT; Brown et al., 2013) Siding Spring
Observatory (SSO) 0.4-m and 1-m telescopes, in New South Wales in Australia.
Observations were taken in the Sloan $i^{\prime}$ band with 170-second
exposures on the 0.4-m telescope and 43-second exposures on the 1-m telescope.
Photometry was extracted using the AstroImageJ software (Collins et al.,
2017).

We obtained 0.38 and 0.51 hours of pre-transit baseline observations, as well
as photometry during the first 2.88 and 2.95 hours of transit, with the 0.4-m
and 1-m LCOGT SSO telescopes, respectively. Conditions were poor during this
set of observations, impacting data obtained from both telescopes. Because
transparency losses became significant only 30 minutes after the observed
ingress, we were able to resolve a clear ingress that occurred $46.16$ minutes
late relative to this work’s fitted linear ephemeris of TOI-2202 b.

####  2.1.3 LCOGT CTIO 0.4-m and 1-m photometry

We also measured two transit ingress events for TOI-2202 b on UT 10/11/2022
and 10/23/2022, as well as one full transit on UT 11/4/2022, using the LCOGT
Cerro Tololo Inter-American Observatory (CTIO) 0.4-m telescope located 80 km
east of La Serena, Chile. The UT 10/11/2022 ingress was simultaneously
observed using the LCOGT CTIO 1-m telescope. Observations were taken in the
Sloan $i^{\prime}$ band, with 170-second exposures for the 0.4-m telescope
observations and 43-second exposures for the 1-m telescope observations.
Photometry was extracted using the AstroImageJ software (Collins et al.,
2017).

From the 0.4-m LCOGT CTIO telescope, we obtained 2.60, 2.41, and 2.05 hours of
pre-transit photometry observations, as well as 1.50, 2.08, and 5.74 hours of
post-ingress photometry observations during the evenings of UT 10/11/2022,
10/23/2022, and 11/4/2022, respectively. Our observations with the 1-m LCOGT
CTIO telescope on UT 10/11/2022 included 2.66 hours of pre-transit photometry
observations, as well as 1.69 hours of in-transit data.

Based on the obtained LCOGT CTIO photometry, we found that the 10/11/2022
transit event of TOI-2022 b occurred 5.52 minutes late relative to this work’s
fitted linear ephemeris of TOI-2202 b. The 10/23/2022 transit event occurred
8.25 minutes early, while the 11/4/2022 event occurred $21.88$ minutes early.

####  2.1.4 TRAPPIST-South 0.6-m photometry

We measured one transit egress event for TOI-2202 b on UT 10/11/2022 using the
TRAPPIST-South 0.6-m robotic telescope (Jehin et al., 2011; Gillon et al.,
2011) at La Silla Observatory in the Atacama Desert of Chile. Continuous
30-second bservations were taken with the Astrodon “I+z” filter. The TRAPPIST-
South observing sequence spanned $2.16$ hours prior to the transit ingress, as
well as $1.76$ in-transit hours of observations. The observed transit ingress
time was consistent with the times derived from the simultaneous LCOGT CTIO
0.4-m and 1-m observations taken on the same night (Section 2.1.3), with an
ingress $5.52$ minutes after the linear ephemeris prediction.

####  2.1.5 Observatoire Moana – El Sauce 0.6-m photometry

Our team measured two full transits of TOI-2202 b on UT 10/23/2022 and UT
11/4/2022 using the station of the Observatoire Moana located in El Sauce (ES)
Observatory (Ropert et al., 2021) in Chile. This station consists of a 0.6m
CDK robotic telescope coupled to an Andor iKon-L deep depletion $2000\times
2000$ CCD with a scale of $0.67\arcsec$ per pixel. The second of these
transits, on UT 11/4/2022, was obtained simultaneously with the presented
Rossiter-McLaughlin measurement across the transit of TOI-2202 b. Observations
were taken in the Sloan $r$ band with continuous 100-second exposures.

The 10/23/2022 observations included nearly the full planet transit ($3.76$
in-transit hours), as well as $4.20$ hours of pre-transit baseline photometry.
From this dataset, together with the LCOGT CTIO 1-m telescope observation
described in Section 2.1.3, we derived an 8.25-minute early ingress. The
11/4/2022 observation included $2.10$ hours of pre-transit and $1.91$ hours of
post-transit baseline observations, together with continuous observations
throughout the transit itself.

###  2.2 Radial velocity observations

We observed the Rossiter-McLaughlin effect across one full transit of TOI-2202
b, from UT 00:48-8:35 on November 4th, 2022, using the Carnegie Planet Finder
Spectrograph (Crane et al., 2006, 2008, 2010) on the 6.5m Magellan Clay
telescope at Las Campanas Observatory in the southern Atacama Desert of Chile.
Our team obtained 25 RV measurements, each with an exposure time of 1100
seconds, $3\times 3$ binning, and typical RV precision $\sim 3.1$ m/s.
Conditions were good throughout the observation, with typical seeing
$0.7-0.8\arcsec$ and a small spike in seeing about an hour before transit. The
airmass ranged from $z=1.40-1.77$ through the observing sequence.

In addition to the transit itself, the observing sequence included 1.99 hours
of pre-transit and 1.72 hours of post-transit baseline observations. The PFS
RV measurements obtained for this work are provided in Table 2 and shown in
Figure 2.

Table 2: PFS RV measurements for the TOI-2202 system, obtained across the transit of TOI-2202 b. Time (BJDTDB) | RV (m/s) |  $\sigma_{\rm RV}$ (m/s)
---|---|---
2459887.54038 | 2.96 | 3.65
2459887.55339 | -5.13 | 3.14
2459887.56648 | 0.41 | 3.30
2459887.57971 | -9.69 | 3.28
2459887.59266 | 4.42 | 3.13
2459887.60567 | -1.81 | 2.98
2459887.61865 | 7.04 | 2.89
2459887.63136 | 0.10 | 2.94
2459887.64460 | 16.42 | 2.75
2459887.65761 | 23.90 | 3.05
2459887.67055 | 8.11 | 3.12
2459887.68319 | 10.25 | 3.00
2459887.69672 | 0.13 | 3.52
2459887.70952 | -6.93 | 3.18
2459887.72273 | -15.10 | 3.30
2459887.73584 | -21.73 | 3.02
2459887.74891 | -31.60 | 3.47
2459887.76152 | -22.51 | 2.95
2459887.77497 | -8.69 | 3.75
2459887.78803 | 3.99 | 2.96
2459887.80100 | -3.67 | 3.26
2459887.81431 | 1.06 | 2.99
2459887.82690 | -16.80 | 3.00
2459887.84022 | -13.39 | 3.23
2459887.85314 | -12.86 | 3.33

##  3 Global fitting

###  3.1 Stellar parameters

The EXOFASTv2 Python package (Eastman, 2017; Eastman et al., 2019) was applied
to derive precise stellar parameter values by fitting the host star’s spectral
energy distribution (SED) using MESA Isochrones & Stellar Tracks (MIST; Choi
et al. 2016; Dotter 2016) models. We constructed the SED using photometric
measurements from 2MASS (Cutri et al., 2003), WISE (Cutri & et al., 2013), and
Gaia DR3 (Gaia Collaboration et al., 2022). Priors on stellar mass ($M_{*}$),
age, and effective temperature ($T_{\rm eff}$) were adopted from Trifonov et
al. (2021), while priors on parallax ($\varpi$) and upper limits on V-band
extinction ($A(V)$) were drawn from Gaia DR3 and Schlafly & Finkbeiner (2011),
respectively.

As noted by Eastman et al. (2022), incorporating transit-based densities can
provide further constraints on stellar radius and mass, surpassing the limits
of systematic error floors on stellar parameters (Tayar et al., 2020).
Therefore, we included an additional likelihood term for stellar density in
the stellar parameter fit to account for constraints imposed by the transit
fit. The output stellar parameters are provided in the top section of Table 3.
All derived stellar parameters are consistent with values reported in Trifonov
et al. (2021) within $1\sigma$.

###  3.2 Planetary system parameters

We applied the allesfitter Python package (Günther & Daylan, 2021) to refine
the TOI-2202 planetary system parameters and derive the sky-projected spin-
orbit angle of TOI-2202 b. Using allesfitter, we jointly modeled all
photometry and RV observations obtained in this work, together with ten
sectors of 30-minute cadence TESS (Ricker et al., 2015) photometry (Sectors 1,
2, 6, 9, 13, 27, 28, 29, 36, 39, 62, and 63)222TESS data used in this paper
can be found in MAST: http://dx.doi.org/10.17909/0cp4-2j79 (catalog
10.17909/0cp4-2j79) and all archival PFS, FEROS, and HARPS RVs and ground-
based transits reported in Trifonov et al. (2021).

The fitted parameters include the companion’s orbital period ($P_{b}$), the
reference mid-time epoch ($T_{0}$), all individual transit mid-times
($t_{0}$), the cosine of the planetary orbital inclination ($\cos{i_{b}}$),
the planet-to-star radius ratio ($R_{b}/R_{\star}$), the sum of radii divided
by the orbital semi-major axis ($(R_{\star}+R_{b})/a_{b}$), the RV semi-
amplitude ($K_{b}$), the parameterized orbital eccentricity and argument of
periastron ($\sqrt{e_{b}}\,\cos{\,\omega_{b}}$,
$\sqrt{e_{b}}\,\sin{\,\omega_{b}}$), the sky-projected spin-orbit angle
($\lambda$), the sky-projected stellar rotational velocity ($v\sin
i_{\star}$), and twelve limb-darkening coefficients, with two per photometric
band ($q_{1}$ and $q_{2}$ for each of LCO, El Sauce, Minerva, TRAPPIST-South,
and TESS) and two for the in-transit RV dataset ($q_{1;\mathrm{RM}}$ and
$q_{2;\mathrm{RM}}$). The systematic offsets between transit and RV datasets
obtained by different instruments were accounted for by fitting and
subtracting off a quadratic trend between each dataset. During the fit, the
jitter term for each RV dataset was added in quadrature. We also fitted for
the error scaling factor for each transit, normalized to the original
photometric errors to ensure that only the relative weights are important.
Parameters for planet c were held fixed at the values derived in Trifonov et
al. (2021).

Posterior distributions were derived for each free parameter using an affine-
invariant Markov Chain Monte Carlo (MCMC) analysis, with 100 walkers that were
each run to at least 30$\times$ the autocorrelation length ($\geq 500,000$
accepted steps per walker) to ensure convergence. The resulting planetary
parameters are provided in Table 3, while the transit mid-times $t_{0}$
derived for each light curve are listed in Table 1. The linear ephemeris was
derived by applying a weighted least-squares fit to the set of output transit
mid-times $t_{0}$. The reference epoch was optimized to minimize the
covariance between $T_{\rm 0}$ and $P_{b}$, and the resulting values are
listed in Table 3. From this analysis, we derived a moderate sky-projected
spin-orbit angle $\lambda=26^{+12}_{-15}\,{}^{\circ}$ for TOI-2202 b, with the
best-fitting model and residuals shown in Figure 2.

Trifonov et al. (2021) implemented a Gaussian Process (GP) analysis to derive
a stellar rotation period $P_{\mathrm{rot}}=24.1^{+2.3}_{-1.8}$ days for
TOI-2202 based on TESS light curve data from Sectors 1, 2, 6, 9, and 13. To
update this result, we reexamined the rotational period of TOI-2202 using the
full set of currently available TESS data from Sectors 1, 2, 6, 9, 13, 27, 28,
29, 36, 39, 62, and 63, collected over a span of 1716 days from July 25th,
2018 to April 6th, 2023. We applied the GP kernels SHOTerm and RotationTerm
that are encapsulated within the celerite2 Python package (Foreman-Mackey,
2018).

Four parallel chains were run using the PyMC3 Python package (Salvatier et
al., 2016) with an acceptance rate of 0.99, where each chain consisted of
10,000 tuning steps and 10,000 draws. Convergence was deemed to have been
achieved when the Gelman-Rubin diagnostic ($\hat{R}$; Gelman & Rubin, 1992)
fell below 1.01. The resulting stellar rotation period is $P_{\rm rot}$$=22\pm
1$ days, corresponding to a stellar equatorial velocity $v=(2\pi R_{*})/P_{\rm
rot}=1.86\pm 0.19$ km/s.

Lastly, we combined $v$ and $v\sin i_{*}$ to derive the stellar inclination
$i_{*}$ and the true stellar obliquity $\psi$ for TOI-2202. The Bayesian
inference method described in Masuda & Winn (2020) and Hjorth et al. (2021)
was applied to account for the interdependent parameters $v$ and $v\sin
i_{*}$, and uniform priors were adopted for the three input parameters
$R_{*}$, $P_{\mathrm{rot}}$, and $\cos i$. This analysis yielded a stellar
inclination estimate $i_{*}=89.77\pm 16.76\,^{\circ}$. Then, the true stellar
obliquity ($\psi$) was derived through Equation 9 of Fabrycky & Winn (2009),

$\cos\psi=\cos i_{*}\cos i+\sin i_{*}\sin i\cos\lambda,$ (1)

where $i_{*}$ is the stellar inclination and $i$ is the planet’s orbital
inclination. The resulting true stellar obliquity is
$\psi=31^{+13}_{-11}\,{}^{\circ}$.

Figure 2: PFS observations of the TOI-2202 b Rossiter-McLaughlin effect from
UT 11/4/22, together with the associated uncertainties and best-fitting model
(red dashed line). Two thousand Rossiter-McLaughlin model draws from the
posterior distribution are shown in gray. Residuals from the best-fitting
model are provided in the lower panel. Table 3: Priors and posteriors for the
TOI-2202 planetary system.

| Description (units) | Priors | Fitted Value |
---|---|---|---|---
Stellar Parameters:
$M_{*}$ | Mass ($\,M_{\sun}$) | - | $0.841^{+0.034}_{-0.032}$ |
$R_{*}$ | Radius ($\,R_{\sun}$) | - | $0.808^{+0.024}_{-0.022}$ |
$T_{\rm eff}$ | Effective temperature (K) | $\mathcal{N}(5144,104)$ | $5169^{+80}_{-78}$ |
$\log{g}$ | Surface gravity (cgs) | $\mathcal{N}(4.55,0.20)$ | $4.548^{+0.030}_{-0.031}$ |
$[{\rm Fe/H}]$ | Metallicity (dex) | - | $0.059\pm 0.043$ |
$v\sin{i_{*}}$ | Projected rotational velocity (km/s) | $\mathcal{U}(1.7;0;10)$ | $2.14_{-0.25}^{+0.28}$ |
Age | Age (Gyr) | - | $6.4^{+4.4}_{-3.9}$ |
$A_{V}$ | V-band extinction (mag) | $\mathcal{U}(0.242,0.054)$ | $0.282\pm 0.047$ |
$\varpi$ | Parallax (mas) | $\mathcal{U}(4.226,0.019)$ | $4.225^{+0.016}_{-0.017}$ |
$d$ | Distance (pc) | - | $236.69^{+0.95}_{-0.90}$ |
Planetary Parameters:
$R_{b}/R_{*}$ | Planet-to-star radius ratio | $\mathcal{U}(0.125;0;1)$ | $0.1265_{-0.0018}^{+0.0017}$ |
$(R_{*}+R_{b})/a_{b}$ | Sum of radii divided by orbital semimajor axis | $\mathcal{U}(0.043;0;1)$ | $0.04336_{-0.00056}^{+0.00059}$ |
$\cos{i_{b}}$ | Cosine of the orbital inclination | $\mathcal{U}(0.011;0;1)$ | $0.0125_{-0.0029}^{+0.0020}$ |
$P_{b}$ | Orbital period (days) | $\mathcal{U}(11.913;10.913;12.913)$ | $11.9126075\pm 0.00011$ |
$T_{0;b}$ | Mid-time epoch (BJDTDB) - 2458000 (days) | $\mathcal{U}(1578;1577;1579)$ | $1577.9736362\pm 0.0038$ |
$K_{b}$ | Radial velocity semi-amplitude (km/s) | $\mathcal{U}(0;0;10)$ | $0.0907_{-0.0100}^{+0.0083}$ |
$\sqrt{e_{b}}\cos{\omega_{b}}$ | Eccentricity parameter 1 | $\mathcal{U}(0;-1;1)$ | $-0.059_{-0.075}^{+0.089}$ |
$\sqrt{e_{b}}\sin{\omega_{b}}$ | Eccentricity parameter 2 | $\mathcal{U}(0;-1;1)$ | $-0.04_{-0.13}^{+0.14}$ |
$P_{c}$ | Orbital period (days) | fixed | $24.7557$ |
$T_{0;c}$ | Mid-time epoch (BJDTDB) - 2458000 (days) | fixed | 327.103 |
$K_{c}$ | Radial velocity semi-amplitude (km/s) | fixed | $0.0196$ |
$\sqrt{e_{c}}\cos{\omega_{c}}$ | Eccentricity parameter 1 | fixed | $-0.023$ |
$\sqrt{e_{c}}\sin{\omega_{c}}$ | Eccentricity parameter 2 | fixed | $-0.099$ |
$\lambda$ | Sky-projected spin-orbit angle ($\,{}^{\circ}$) | $\mathcal{U}(0;-180;180)$ | $26_{-15}^{+12}$ |
$q_{1;\mathrm{LCO}}$ | Linear limb-darkening coefficient for LCO | $\mathcal{U}(0.5;0;1)$ | $0.64\pm 0.24$ |
$q_{2;\mathrm{LCO}}$ | Quadratic limb-darkening coefficient for LCO | $\mathcal{U}(0.5;0;1)$ | $0.32_{-0.13}^{+0.21}$ |
$q_{1;\mathrm{El\,Sauce}}$ | Linear limb-darkening coefficient for El Sauce | $\mathcal{U}(0.5;0;1)$ | $0.60_{-0.36}^{+0.29}$ |
$q_{2;\mathrm{El\,Sauce}}$ | Quadratic limb-darkening coefficient for El Sauce | $\mathcal{U}(0.5;0;1)$ | $0.28_{-0.20}^{+0.31}$ |
$q_{1;\mathrm{\textsc{Minerva}}}$ | Linear limb-darkening coefficient for Minerva | $\mathcal{U}(0.5;0;1)$ | $0.43_{-0.18}^{+0.24}$ |
$q_{2;\mathrm{MINERVA}}$ | Quadratic limb-darkening coefficient for Minerva | $\mathcal{U}(0.5;0;1)$ | $0.37_{-0.24}^{+0.32}$ |
$q_{1;\mathrm{TRAPPIST}}$ | Linear limb-darkening coefficient for TRAPPIST | $\mathcal{U}(0.5;0;1)$ | $0.43_{-0.13}^{+0.19}$ |
$q_{2;\mathrm{TRAPPIST}}$ | Quadratic limb-darkening coefficient for TRAPPIST | $\mathcal{U}(0.5;0;1)$ | $0.56_{-0.20}^{+0.23}$ |
$q_{1;\rm TESS}$ | Linear limb-darkening coefficient for TESS | $\mathcal{U}(0.5;0;1)$ | $0.32_{-0.13}^{+0.20}$ |
$q_{2;\rm TESS}$ | Quadratic limb-darkening coefficient for TESS | $\mathcal{U}(0.5;0;1)$ | $0.45_{-0.18}^{+0.27}$ |
$q_{1;\rm PFS,RM}$ | Linear limb-darkening coefficient for the PFS RM | $\mathcal{U}(0.5;0;1)$ | $0.72_{-0.28}^{+0.20}$ |
$q_{2;\rm PFS,RM}$ | Quadratic limb-darkening coefficient for the PFS RM | $\mathcal{U}(0.5;0;1)$ | $0.56_{-0.32}^{+0.28}$ |
Derived Parameters:
$M_{b}$ | Planetary mass (MJup) | - | $0.904_{-0.10}^{+0.087}$ |
$R_{b}$ | Planetary radius (RJup) | - | $0.977\pm 0.016$ |
$a_{b}/R_{*}$ | Semi-major axis over host radius | - | $25.98\pm 0.34$ |
$i_{b}$ | Inclination ($\,{}^{\circ}$) | - | $89.29_{-0.12}^{+0.17}$ |
$e_{b}$ | Eccentricity | - | $0.022_{-0.015}^{+0.022}$ |
$\omega_{b}$ | Argument of periastron ($\,{}^{\circ}$) | - | $212_{-94}^{+59}$ |
$T_{\mathrm{tot;b}}$ | Total transit duration (days) | - | $3.768_{-0.030}^{+0.032}$ |
$b$ | Impact parameter | - | $0.323_{-0.080}^{+0.055}$ |
$u_{\mathrm{1;\mathrm{LCO}}}$ | Linear limb-darkening coefficient 1 for LCO | - | $0.52\pm 0.19$ |
$u_{\mathrm{2;\mathrm{LCO}}}$ | Linear limb-darkening coefficient 2 for LCO | - | $0.28_{-0.33}^{+0.27}$ |
$u_{\mathrm{1;El\,Sauce}}$ | Linear limb-darkening coefficient 1 for El Sauce | - | $0.73_{-0.18}^{+0.15}$ |
$u_{\mathrm{2;El\,Sauce}}$ | Linear limb-darkening coefficient 2 for El Sauce | - | $-0.08_{-0.24}^{+0.30}$ |
$u_{\mathrm{1;\textsc{Minerva}}}$ | Linear limb-darkening coefficient 1 for Minerva | - | $0.38_{-0.27}^{+0.42}$ |
$u_{\mathrm{2;\textsc{Minerva}}}$ | Linear limb-darkening coefficient 2 for Minerva | - | $0.38_{-0.27}^{+0.42}$ |
$u_{\mathrm{1;TRAPPIST}}$ | Linear limb-darkening coefficient 1 for TRAPPIST | - | $0.48\pm 0.30$ |
$u_{\mathrm{2;TRAPPIST}}$ | Linear limb-darkening coefficient 2 for TRAPPIST | - | $0.17\pm 0.37$ |
$u_{\mathrm{1;TESS}}$ | Linear limb-darkening coefficient 1 for TESS | - | $0.51_{-0.14}^{+0.13}$ |
$u_{\mathrm{2;TESS}}$ | Linear limb-darkening coefficient 2 for TESS | - | $0.06_{-0.25}^{+0.28}$ |
$u_{\mathrm{1;PFS,RM}}$ | Linear limb-darkening coefficient 1 for the PFS RM | - | $0.88_{-0.52}^{+0.46}$ |
$u_{\mathrm{2;PFS,RM}}$ | Linear limb-darkening coefficient 2 for the PFS RM | - | $-0.09_{-0.43}^{+0.52}$ |

††footnotetext: Note: $P_{b}$ and $T_{0}$ were derived from the weighted
least-square fit to the transit mid-times $t_{0}$.

##  4 Population analysis of near-resonant systems

###  4.1 The distribution of spin-orbit angles for near-resonant exoplanets

To place this measurement into context, we examined the full set of transiting
exoplanet systems with (1) a sky-projected spin-orbit measurement and (2)
evidence that the transiting planet lies near a low-order mean-motion
resonance with a neighboring companion. We initialized our sample by cross-
matching the set of all exoplanets with $\lambda$ measurements in the TEPCat
catalogue (Southworth, 2011) with the set of exoplanets with one or more
confirmed planetary companions around the same host star in the NASA Exoplanet
Archive’s Planetary Systems table (NASA Exoplanet Archive, 2023).333Both
catalogues were accessed on 7/20/2023. Thirty-two planets were identified that
fit these two criteria.444Because WASP-18 c is a contested planet, the WASP-18
system was removed from our sample.

We also searched for any planets showing clear sinusoidal TTVs attributable to
a resonant or near-resonant planetary companion that has not yet been directly
confirmed. We identified all planets with (1) a spin-orbit measurement in the
TEPCat catalogue and (2) ttv_flag = True in the NASA Exoplanet Archive. In
addition to TOI-2202 b, nine further candidate near-resonant planets were
recovered in this manner. However, after closer examination, we concluded
that, other than TOI-2202 b, none of the identified TTV planets without
confirmed, nearby companions showed compelling sinusoidal signals (see
Appendix A for more details). Therefore, only TOI-2202 was added to the
initial set of 32 identified systems with confirmed planetary companions.

Next, we identified systems within our sample with compact configurations near
low-order resonances. The sample was restricted to include only systems for
which the planet with a spin-orbit measurement has a small period ratio
$P_{2}/P_{1}\lesssim 4$ relative to at least one of its nearest neighbors.
This limit was selected for direct comparison with Figure 4 of Fabrycky et al.
(2014). Both inner and outer planetary companions were considered, and the
default parameter solution orbital periods from the NASA Exoplanet Archive
were adopted for all planets. This cut left 19 planets in 16 systems, with
properties described in Table 4.

The associated period ratio distribution is shown in Figure 3. The inner and
outer period ratios for a single planet were included separately in cases
where both met the criterion $P_{2}/P_{1}\lesssim 4$. For systems in which
more than one planet has a spin-orbit measurement (HD 3167, TRAPPIST-1, and
V1298 Tau), each planet was separately considered and each relevant period
ratio was included only once, for a total of 24 neighboring period ratios with
$P_{2}/P_{1}\lesssim 4$.

Figure 3: Period ratios of planet pairs in compact systems
($P_{\mathrm{out}}/P_{\mathrm{in}}\lesssim 4$) in which at least one planet
has a spin-orbit measurement.

Lastly, we identified planets within the sample with at least one neighboring
companion near a low-order orbital commensurability. Specifically, we searched
for planet pairs that fall within 5% of the 2:1, 3:1, 4:1, 3:2, or 5:3 mean
motion commensurabilities. Most near-resonant pairs within the sample were
found to lie just wide of the 2:1 and 3:2 resonances (see Figure 3), with a
distribution comparable to that of the Kepler multi-transiting systems
examined in Fabrycky et al. (2014). In total, thirteen near-resonances were
identified across twelve planet pairs.

The stellar obliquity distribution for these pairs as a function of host star
$T_{\mathrm{eff}}$ is shown in Figure 4. As displayed in the top panel of
Figure 4 and in Table 4, TOI-2202 b is the first exoplanet in a near-resonant
configuration for which the measured sky-projected spin-orbit angle has not
been consistent with exact alignment ($|\lambda|=0\,^{\circ}$) within
$1\sigma$.

Figure 4: Sky-projected stellar obliquity $|\lambda|$ and 3D stellar obliquity
$\psi$ for near-resonant systems. The maximum expected level of stellar
obliquity excitation in the convergent migration framework is shown in gray.
All $\psi$ values have been measured for separate planetary systems, while
V1298 Tau and TRAPPIST-1 include two and three separate displayed $|\lambda|$
values, respectively.

For six near-resonant systems, the 3D spin-orbit angle $\psi$ has also been
derived. These systems are shown in the bottom panel of Figure 4. The
distribution of 3D angles reveals that a few more systems, in addition to
TOI-2202 b, are likely offset from exact alignment. While no systems with
near-resonant configurations have been found with strong misalignments
indicative of polar or retrograde orbits, the scatter in the 3D distribution
suggests some range in true stellar obliquities even in near-resonant systems.

To quantify this deviation from alignment, we drew 10,000 iterations of random
values from each of the six systems with measured $\psi$ values, using the
reported Gaussian uncertainties from each measurement. Each iteration was then
compared with an “aligned” Rayleigh distribution of 100,000 values, with scale
parameter $\sigma=1.8$ such that $\sim 98$% of draws fall within the range
$\psi<5\,^{\circ}$. A Kolmogorov-Smirnov (K-S) test quantifying the difference
between these two distributions returns $p<0.05$ for 99.4% of random draws,
whereas 0.6% of random draws return $p>0.05$. As a result, we reject the null
hypothesis that the observed $\psi$ distribution exhibits consistent, near-
exact ($\psi<5\,^{\circ}$) alignment.

###  4.2 Additional relevant systems

A few systems were excluded from this population that may also serve to inform
the distribution of spin-orbit angles for near-resonant systems. While these
systems do not fit the criteria used to develop the uniform sample in Section
4.1, they each offer further relevant insights into the dynamical evolution of
near-resonant exoplanet pairs.

One notable omission is a 1D spin-orbit measurement – obtained in the
inclination direction – that indicates a misalignment in the compact multi-
planet system Kepler-56 (Huber et al., 2013). This system was excluded because
it does not have a reported $\lambda$ constraint. Instead, an asteroseismic
analysis conducted by Huber et al. (2013) revealed a stellar spin axis
inclined at $i_{*}\sim 45\,^{\circ}$. The Kepler-56 system includes two
transiting planets that, by definition, have $i_{b}\sim i_{c}\sim
90\,^{\circ}$, such that the stellar spin axis at $i_{*}\sim 45\,^{\circ}$
indicates a substantial misalignment in the line-of-sight direction. Kepler-56
b and c lie near a 2:1 (c:b) mean motion commensurability.

Another relevant system is 55 Cancri, which is an aligned multiplanet system
that includes a near-resonance (McArthur et al., 2004; Fischer et al., 2008;
Nelson et al., 2014). This system was not included within our analysis because
the only planet with a spin-orbit measurement in the system, 55 Cancri e, has
a large period ratio $P_{b}/P_{e}\sim 20$ with its nearest neighbor. However,
a 3:1 near-resonance exists elsewhere in the system between 55 Cancri b and c,
suggesting that this system may have formed in a similar manner to the the
systems in our sample. 55 Cancri e has a measured spin-orbit angle
$\lambda=10^{+17}_{-20}\,{}^{\circ}$ and $\psi=23^{+14}_{-12}\,{}^{\circ}$
(Zhao et al., 2023).

A third relevant system is HIP 41378, with a previously reported misalignment
$|\lambda|=57^{+26}_{-18}\,{}^{\circ}$ for the planet HIP 41378 d (Grouffal et
al., 2022). This system was excluded from the sample because the orbital
period of HIP 41378 d has not been precisely confirmed. Only a partial
Rossiter-McLaughlin observation has been obtained for this system due to the
planet’s long transit duration, with measurement uncertainties in the acquired
dataset that are comparable to the signal amplitude (Grouffal et al., 2022).
HIP 41378 d provides an especially interesting case study as one of the few
long-period transiting exoplanets that is amenable to spin-orbit measurements.
Additional measurements would be helpful to more clearly establish whether
this planet lies within a near-resonance and to more precisely constrain its
spin-orbit configuration.

Table 4: Systems with a spin-orbit measurement and $P_{2}/P_{1}\lesssim 4$.
$N_{\rm pl}$ is the number of confirmed planets in the system to date. $N_{\rm
pl}$, orbital period, and $T_{\rm eff,*}$ have values drawn from the default
parameter solution listed in the NASA Exoplanet Archive on 7/20/23. Identified
MMRs are provided for commensurabilities in which the listed planet is the
outer (longer-period) planet of the pair, as well as those in which the listed
planet is the inner (shorter-period) planet of the pair. References are
provided for $\lambda$ measurements. $\psi$ values were drawn from Albrecht et
al. (2021) when possible for systems with no measurement reported in TEPCat.

System | Planet | $N_{\rm pl}$ | Orbital period (days) | MMRout | MMRin |  $T_{\mathrm{eff},*}$ (K) | $|\lambda|\,(\arcdeg)$ | Reference, $|\lambda|$ | $\psi\,(\arcdeg)$
---|---|---|---|---|---|---|---|---|---
AU Mic | b | 2 | $8.463000\pm 0.000002$ | - | - | $3588\pm 87$ | $3.0_{-10.4}^{+10.3}$ | Pallé et al. (2020) | $12.1^{+11.3}_{-7.5}$
HD 3167 | c | 4 | $29.8454\pm 0.0012$ | - | - | $5261\pm 60$ | $108.9_{-5.4}^{+5.5}$ | Bourrier et al. (2021) | $107.7^{+5.1}_{-4.9}$
HD 63433 | b | 2 | $7.10793^{+0.00040}_{-0.00034}$ | - | - | $5640\pm 74$ | $8_{-45}^{+33}$ | Mann et al. (2020) | $25.6^{+22.5}_{-15.3}$
HD 106315 | c | 2 | $21.05704\pm 0.00046$ | - | - | $6327\pm 48$ | $2.7_{-2.7}^{+2.6}$ | Bourrier et al. (2023) | -
Kepler-9∗ | b | 3 | $19.23891\pm 0.00006$ | - | 2:1 (c:b) | $5774\pm 60$ | $13\pm 16$ | Wang et al. (2018) | $28.1^{+13.0}_{-13.6}$
Kepler-25∗ | c | 3 | $12.72070^{+0.00011}_{-0.00010}$ | 2:1 (c:b) | - | $6354\pm 27$ | $0.5\pm 5.7$ | Albrecht et al. (2013) | $5.7^{+4.2}_{-3.2}$
Kepler-30∗ | b | 3 | $29.33434\pm 0.00815$ | - | 2:1 (c:b) | $5498\pm 54$ | $1\pm 10$ | Sanchis-Ojeda et al. (2012) | -
Kepler-89 | d | 4 | $22.3429890\pm 0.0000067$ | 2:1 (d:c) | - | $6182\pm 58$ | $6_{-13}^{+11}$ | Hirano et al. (2012) | -
TOI-942 | b | 2 | $4.324210\pm 0.000019$ | - | - | $4969\pm 100$ | $1_{-33}^{+41}$ | Wirth et al. (2021) | $2^{+27}_{-33}$
TOI-1136∗ | d | 6 | $12.51937^{+0.00037}_{-0.00041}$ | 2:1 (d:c) | 3:2 (e:d) | $5770\pm 50$ | $5\pm 5$ | Dai et al. (2023) | $<28$
TOI-2076∗ | b | 3 | $10.35509^{+0.00020}_{-0.00014}$ | - | 2:1 (c:b) | $5187_{-53}^{+54}$ | $3_{-15}^{+16}$ | Frazier et al. (2023) | $18^{+10}_{-9}$
TOI-2202∗ | b | 2 | $11.9126075\pm 0.00011$ | - | 2:1 (c:b) | $5169^{+80}_{-78}$ | $26^{+15}_{-12}$ | this work | $31^{+13}_{-11}$
TRAPPIST-1∗ | b | 7 | $1.510826\pm 0.000006$ | - | 4:1 (e:b) | $2566\pm 26$ | $15_{-30}^{+26}$ | Hirano et al. (2020) | ${}^{{\dagger}}23.3^{+17.0}_{-13.6}$
TRAPPIST-1∗ | e | 7 | $6.101013\pm 0.000035$ | 3:2 (e:d) | 3:2 (f:e) | $2566\pm 26$ | $9_{-51}^{+45}$ | Hirano et al. (2020) | ${}^{{\dagger}}23.3^{+17.0}_{-13.6}$
TRAPPIST-1∗ | f | 7 | $9.207540\pm 0.000032$ | 3:2 (f:e) | 2:1 (h:f) | $2566\pm 26$ | $21\pm 32$ | Hirano et al. (2020) | ${}^{{\dagger}}23.3^{+17.0}_{-13.6}$
V1298 Tau∗ | b | 4 | $24.1396\pm 0.0018$ | 2:1 (b:d) | 2:1 (e:b) | $5050\pm 100$ | $4_{-10}^{+7}$ | Johnson et al. (2022) | $8^{+4}_{-7}$
V1298 Tau∗ | c | 4 | $8.24958\pm 0.00072$ | - | 3:2 (d:c) | $5050\pm 100$ | $4.9_{-15.1}^{+15.0}$ | Feinstein et al. (2021) | -
WASP-47 | b | 4 | $4.1591492\pm 0.0000006$ | - | - | $5552\pm 75$ | $0\pm 24$ | Sanchis-Ojeda et al. (2015) | $29^{+11}_{-13}$
WASP-148∗ | b | 2 | $8.803809\pm 0.000043$ | - | 4:1 (c:b) | $5555\pm 90$ | $8.2_{-8.7}^{+9.7}$ | Wang et al. (2022) | -

$*$$*$footnotetext: At least one low-order period commensurability has been
identified in this system. The TRAPPIST-1 and TOI-1136 systems are each
resonant chains, with additional commensurabilities within each system that
are not listed in this table.

$\textdagger$$\textdagger$footnotetext: A single $\psi$ value was derived for
the TRAPPIST-1 system in Albrecht et al. (2021), such that these three values
all correspond to just one measurement.

##  5 Discussion

The measured spin-orbit angle of TOI-2202 b, together with the full census of
spin-orbit measurements for near-resonant exoplanets, indicates that even
quiescently formed systems may experience low-level dynamical excitation that
produces some dispersion in their spin-orbit orientations. The root of this
excitation is intertwined with the underlying formation and prevalence of
resonances in planetary systems.

At the population level, most near-resonant planets identified by Kepler have
been observed to lie just wide of true mean motion resonances (Lissauer et
al., 2011; Fabrycky et al., 2014). This finding has been generally interpreted
as evidence that such systems began in resonant configurations and were later
displaced from deep resonance. Within this framework, near-resonant systems
constitute a population that has successfully retained the imprints of past
dynamical capture into mean motion resonance – a delicate configuration that
easily diverges from exact commensurability through post-disk-dispersal
dynamical perturbations (e.g. Michtchenko et al., 2008a, b; Deck et al., 2012;
Izidoro et al., 2017; Leleu et al., 2021) – such that they offer key clues
into their host systems’ primordial architectures.

Convergent migration (e.g. Goldreich & Sciama, 1965; Snellgrove et al., 2001;
Lee & Peale, 2002; Rein, 2012; Bitsch et al., 2019) could feasibly produce
mean motion resonances across a broad range of planetary systems. Planet pairs
may be gently pushed away from nominal mean motion resonances in the post-
disk-dispersal stage through eccentricity damping and orbital circularization
(Terquem & Papaloizou, 2007; Lithwick & Wu, 2012; Batygin & Morbidelli, 2013;
Goldreich & Schlichting, 2014; Delisle et al., 2014; Chatterjee & Ford, 2015)
or through in situ mass growth by planets on circular orbits (Petrovich et
al., 2013).

Convergent migration and gentle resonance divergence mechanisms are expected
to operate within the plane of the host protoplanetary disk. Even in the more
dynamically violent instability framework modeled by Izidoro et al. (2017),
the spin-orbit distribution of planets formed within aligned protoplanetary
disks is expected to peak at $\psi\leq 5\,^{\circ}$ (Esteves et al., 2023)
from the process of resonance disruption alone. Therefore, nonzero spin-orbit
misalignments may trace small primordial tilts of the systems’ natal
protoplanetary disks. Previous studies examining the mutual inclinations
between stellar rotation axes and their surrounding protoplanetary (Davies,
2019) and debris (Hurt & MacGregor, 2023) disks have each demonstrated
evidence consistent with a prevalence of low-level ($\lesssim 20\,^{\circ}$)
disk misalignments that could be attributed solely to chaotic accretion in
turbulent molecular cloud cores (Takaishi et al., 2020).

Alternatively, low-level misalignments in near-resonant systems may be
produced by a mechanism that does not require a primordially misaligned inner
disk. For example, Gratia & Fabrycky (2017) showed that planet-planet
scattering in the outer region of a planetary system may gently produce small
misalignments of up to $\sim 20\,^{\circ}$ in the inner planetary system,
offering a possible avenue to tilt a system without disrupting near-
resonances. A misaligned outer planet, produced through either planet-planet
scattering in the outer system or through a misaligned outer disk (e.g. Nealon
et al., 2019), could also potentially tilt its inner companions (Zhang et al.,
2021) while preserving a near-resonant configuration.

The planets in compact, near-resonant systems considered within this work span
a range of masses, from sub-Earth-mass (TRAPPIST-1 e at $0.692\pm
0.022M_{\oplus}$; Agol et al., 2021) to Jovian-mass planets (TOI-2202 b at
$0.904^{+0.087}_{-0.10}M_{J}$; this work). This wide range of planet masses
may encompass multiple regimes of planet formation and migration that have not
been disentangled within our analysis. The current set of observations does
not demonstrate a clear distinction between the spin-orbit angles of lower-
and higher-mass planets in near-resonant configurations. An expanded sample
may unveil population differences, if present, across mass regimes.

Further monitoring of the TOI-2202 system is needed to more precisely pinpoint
the properties of the TOI-2202 c planet and to constrain the presence of
additional companions within the system. A direct confirmation of TOI-2202 c
would offer the opportunity to demonstrate whether the system lies within, or
only near, a “true” resonance, such that a critical angle in the system
librates about a fixed point. More broadly, additional high-precision
Rossiter-McLaughlin measurements for near-resonant systems offer a promising
path forward to constrain the origins of low-level misalignments in
quiescently formed systems.

Across the broader population of exoplanets with spin-orbit measurements –
including those that are not in near-resonant systems – previous work has
found evidence for a nonzero mean stellar obliquity with significant scatter
$\psi=19\pm 10\,^{\circ}$ (Muñoz & Perets, 2018). The apparent persistence of
this deviation from alignment, even in near-resonant systems, suggests the
universality of low-level dynamical excitation – a pattern well exemplified by
the TOI-2202 system.

##  6 Acknowledgements

We thank the anonymous referee for their valuable feedback on this manuscript.
We thank Chelsea Huang for her support in organizing observations for this
work. M.R. and S.W. thank the Heising-Simons Foundation for their generous
support. M.R. acknowledges support from Heising-Simons Foundation Grant
#2023-4478, as well as the 51 Pegasi b Fellowship Program. This paper includes
data gathered with the 6.5 meter Magellan Telescopes located at Las Campanas
Observatory, Chile. This research has made use of the NASA Exoplanet Archive,
which is operated by the California Institute of Technology, under contract
with the National Aeronautics and Space Administration under the Exoplanet
Exploration Program. A.J. acknowledges support from ANID – Millennium Science
Initiative – ICN12_009 and FONDECYT project 1210718. R.B. acknowledges support
from FONDECYT project 11200751 and additional support from ANID – Millennium
Science Initiative – ICN12_009.

This work makes use of observations from the LCOGT network. Part of the LCOGT
telescope time was granted by NOIRLab through the Mid-Scale Innovations
Program (MSIP). MSIP is funded by NSF. This research has made use of the
Exoplanet Follow-up Observation Program (ExoFOP; DOI: 10.26134/ExoFOP5)
website, which is operated by the California Institute of Technology, under
contract with the National Aeronautics and Space Administration under the
Exoplanet Exploration Program. K.A.C. acknowledges support from the TESS
mission via subaward s3449 from MIT.

This work makes use of data from the Minerva-Australis facility. Minerva-
Australis is supported by Australian Research Council LIEF Grant LE160100001,
Discovery Grants DP180100972 and DP220100365, Mount Cuba Astronomical
Foundation, and institutional partners University of Southern Queensland, UNSW
Sydney, MIT, Nanjing University, George Mason University, University of
Louisville, University of California Riverside, University of Florida, and The
University of Texas at Austin. We respectfully acknowledge the traditional
custodians of all lands throughout Australia, and recognise their continued
cultural and spiritual connection to the land, waterways, cosmos, and
community. We pay our deepest respects to all Elders, ancestors and
descendants of the Giabal, Jarowair, and Kambuwal nations, upon whose lands
the Minerva-Australis facility at Mt Kent is situated.

TRAPPIST-South is funded by the Belgian National Fund for Scientific Research
(F.R.S.-FNRS) under grant PDR T.0120.21, with the participation of the Swiss
National Science Fundation (SNF). M.G. and E.J. are FNRS Senior Research
Associates. The postdoctoral fellowship of K.B. is funded by F.R.S.-FNRS grant
T.0109.20 and by the Francqui Foundation. F.J.P. acknowledges financial
support from the grant CEX2021-001131-S funded by MCIN/AEI/
10.13039/501100011033. This publication benefits from the support of the
French Community of Belgium in the context of the FRIA Doctoral Grant awarded
to MT. This research was supported in part by Lilly Endowment, Inc., through
its support for the Indiana University Pervasive Technology Institute.

Magellan II: PFS; Minerva-Australis 0.7m; LCOGT SSO 0.4m and 1m; LCOGT CTIO
0.4m and 1m; TRAPPIST-South 0.6m; Observatoire Moana – El Sauce 0.6m; NASA
Exoplanet Archive; TEPCat

##  Appendix A Vetting TTV Planets

Here, we discuss the vetting process applied in Section 4 to identify planets
with observed TTVs that are likely indicative of near-resonant companions. The
candidate planets with spin-orbit measurements and with TTV signals, but with
no confirmed planet attributable to the TTV signal, include CoRoT-2 b,
HAT-P-13 b, KELT-19 A b, KOI-12 b, KOI-13 b, Kepler-17 b, Qatar-1 b, WASP-12
b, and WASP-43 b.

None of the identified TTV planets without confirmed, nearby companions
included clear sinusoidal signals. WASP-12 b and Qatar-1 b were removed from
the sample, since both systems have been monitored and found to show no
sinusoidal TTVs (Collins et al., 2017). We note that the Qatar-1 b TTV
detection has also been contested (von Essen et al., 2013; Maciejewski et al.,
2015). Likewise, WASP-12 b demonstrates transit timing variations that
previous studies have found are most consistent with a decaying orbit (Yee et
al., 2020; Turner et al., 2021; Wong et al., 2022). The TTV observations of
WASP-43 b have also been attributed to orbital decay (Jiang et al., 2016),
which was later ruled out (Hoyer et al., 2016; Garai et al., 2021); however,
the WASP-43 b TTVs show no clear signs of periodicity. No clear periodicities
were identified in the CoRoT-2 b, HAT-P-13 b, or KELT-19 A b TTVs based on the
results reported in Ivshina & Winn (2022). Holczer et al. (2016) found that
the frequency of Kepler-17 b’s TTVs may be attributable to the star’s
rotational frequency, and that the TTVs of KOI-13 b show a strong stroboscopic
effect such that they may not be associated with a companion planet. Holczer
et al. (2016) also identified no clearly sinusoidal periodicity in the TTVs
observed for KOI-12 b.

## References

  * Addison et al. (2019) Addison, B., Wright, D. J., Wittenmyer, R. A., et al. 2019, PASP, 131, 115003
  * Agol et al. (2021) Agol, E., Dorn, C., Grimm, S. L., et al. 2021, , 2, 1
  * Albrecht et al. (2013) Albrecht, S., Winn, J. N., Marcy, G. W., et al. 2013, The Astrophysical Journal, 771, 11
  * Albrecht et al. (2021) Albrecht, S. H., Marcussen, M. L., Winn, J. N., Dawson, R. I., & Knudstrup, E. 2021, ApJ, 916, L1
  * Batygin & Morbidelli (2013) Batygin, K., & Morbidelli, A. 2013, AJ, 145, 1
  * Bitsch et al. (2019) Bitsch, B., Izidoro, A., Johansen, A., et al. 2019, Astronomy & Astrophysics, 623, A88
  * Bourrier et al. (2021) Bourrier, V., Lovis, C., Cretignier, M., et al. 2021, Astronomy & Astrophysics, 654, A152
  * Bourrier et al. (2023) Bourrier, V., Attia, O., Mallonn, M., et al. 2023, Astronomy & Astrophysics, 669, A63
  * Brown et al. (2013) Brown, T., Baliber, N., Bianco, F., et al. 2013, Publications of the Astronomical Society of the Pacific, 125, 1031
  * Chatterjee & Ford (2015) Chatterjee, S., & Ford, E. B. 2015, ApJ, 803, 33
  * Choi et al. (2016) Choi, J., Dotter, A., Conroy, C., et al. 2016, ApJ, 823, 102
  * Collins et al. (2017) Collins, K. A., Kielkopf, J. F., & Stassun, K. G. 2017, AJ, 153, 78
  * Collins et al. (2017) Collins, K. A., Kielkopf, J. F., Stassun, K. G., & Hessman, F. V. 2017, AJ, 153, 77
  * Crane et al. (2006) Crane, J. D., Shectman, S. A., & Butler, R. P. 2006, in Ground-based and Airborne Instrumentation for Astronomy, Vol. 6269, SPIE, 972–982
  * Crane et al. (2010) Crane, J. D., Shectman, S. A., Butler, R. P., et al. 2010, in Ground-based and Airborne Instrumentation for Astronomy III, Vol. 7735, SPIE, 1909–1923
  * Crane et al. (2008) Crane, J. D., Shectman, S. A., Butler, R. P., Thompson, I. B., & Burley, G. S. 2008, in Ground-based and Airborne Instrumentation for Astronomy II, Vol. 7014, SPIE, 2484–2493
  * Cutri & et al. (2013) Cutri, R. M., & et al. 2013, VizieR Online Data Catalog, II/328
  * Cutri et al. (2003) Cutri, R. M., Skrutskie, M. F., van Dyk, S., et al. 2003, 2MASS All Sky Catalog of point sources.
  * Dai et al. (2023) Dai, F., Masuda, K., Beard, C., et al. 2023, The Astronomical Journal, 165, 33
  * Davies (2019) Davies, C. L. 2019, MNRAS, 484, 1926
  * Deck et al. (2012) Deck, K. M., Holman, M. J., Agol, E., et al. 2012, ApJ, 755, L21
  * Delisle et al. (2014) Delisle, J.-B., Laskar, J., & Correia, A. 2014, Astronomy & Astrophysics, 566, A137
  * Dong et al. (2023) Dong, J., Wang, S., Rice, M., et al. 2023, arXiv preprint arXiv:2305.16495
  * Dotter (2016) Dotter, A. 2016, ApJS, 222, 8
  * Eastman (2017) Eastman, J. 2017, EXOFASTv2: Generalized publication-quality exoplanet modeling code, , , ascl:1710.003
  * Eastman et al. (2022) Eastman, J. D., Diamond-Lowe, H., & Tayar, J. 2022, arXiv e-prints, arXiv:2209.14301
  * Eastman et al. (2019) Eastman, J. D., Rodriguez, J. E., Agol, E., et al. 2019, arXiv e-prints, arXiv:1907.09480
  * Esteves et al. (2023) Esteves, L., Izidoro, A., Winter, O. C., Bitsch, B., & Isella, A. 2023, MNRAS, arXiv:2303.05635
  * Fabrycky & Winn (2009) Fabrycky, D. C., & Winn, J. N. 2009, ApJ, 696, 1230
  * Fabrycky et al. (2014) Fabrycky, D. C., Lissauer, J. J., Ragozzine, D., et al. 2014, ApJ, 790, 146
  * Feinstein et al. (2021) Feinstein, A. D., Montet, B. T., Johnson, M. C., et al. 2021, The Astronomical Journal, 162, 213
  * Fischer et al. (2008) Fischer, D. A., Marcy, G. W., Butler, R. P., et al. 2008, ApJ, 675, 790
  * Foreman-Mackey (2018) Foreman-Mackey, D. 2018, Research Notes of the American Astronomical Society, 2, 31
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306
  * Frazier et al. (2023) Frazier, R. C., Stefánsson, G., Mahadevan, S., et al. 2023, The Astrophysical Journal Letters, 944, L41
  * Gaia Collaboration et al. (2022) Gaia Collaboration, Vallenari, A., Brown, A. G. A., et al. 2022, arXiv e-prints, arXiv:2208.00211
  * Garai et al. (2021) Garai, Z., Pribulla, T., Parviainen, H., et al. 2021, MNRAS, 508, 5514
  * Gelman & Rubin (1992) Gelman, A., & Rubin, D. B. 1992, Statistical Science, 7, 457
  * Gillon et al. (2011) Gillon, M., Jehin, E., Magain, P., et al. 2011, EPJ Web of Conferences, 11, 06002\.  https://doi.org/10.1051/epjconf/20101106002
  * Goldreich & Schlichting (2014) Goldreich, P., & Schlichting, H. E. 2014, AJ, 147, 32
  * Goldreich & Sciama (1965) Goldreich, P., & Sciama, D. 1965, Monthly Notices of the Royal Astronomical Society, 130, 159
  * Goyal et al. (2023) Goyal, A. V., Dai, F., & Wang, S. 2023, arXiv e-prints, arXiv:2307.15875
  * Gratia & Fabrycky (2017) Gratia, P., & Fabrycky, D. 2017, MNRAS, 464, 1709
  * Grouffal et al. (2022) Grouffal, S., Santerne, A., Bourrier, V., et al. 2022, A&A, 668, A172
  * Günther & Daylan (2021) Günther, M. N., & Daylan, T. 2021, ApJS, 254, 13
  * Harris et al. (2020) Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357
  * Hirano et al. (2012) Hirano, T., Narita, N., Sato, B., et al. 2012, The Astrophysical Journal Letters, 759, L36
  * Hirano et al. (2020) Hirano, T., Gaidos, E., Winn, J. N., et al. 2020, The Astrophysical Journal Letters, 890, L27
  * Hixenbaugh et al. (2023) Hixenbaugh, K., Wang, X.-Y., Rice, M., & Wang, S. 2023, The Astrophysical Journal Letters, 949, L35
  * Hjorth et al. (2021) Hjorth, M., Albrecht, S., Hirano, T., et al. 2021, Proceedings of the National Academy of Sciences, 118, e2017418118
  * Holczer et al. (2016) Holczer, T., Mazeh, T., Nachmani, G., et al. 2016, ApJS, 225, 9
  * Hoyer et al. (2016) Hoyer, S., Pallé, E., Dragomir, D., & Murgas, F. 2016, AJ, 151, 137
  * Huber et al. (2013) Huber, D., Carter, J. A., Barbieri, M., et al. 2013, Science, 342, 331
  * Hunter (2007) Hunter, J. D. 2007, Computing in science & engineering, 9, 90
  * Hurt & MacGregor (2023) Hurt, S. A., & MacGregor, M. A. 2023, ApJ, 954, 10
  * Ivshina & Winn (2022) Ivshina, E. S., & Winn, J. N. 2022, ApJS, 259, 62
  * Izidoro et al. (2017) Izidoro, A., Ogihara, M., Raymond, S. N., et al. 2017, Monthly Notices of the Royal Astronomical Society, 470, 1750
  * Jehin et al. (2011) Jehin, E., Gillon, M., Queloz, D., et al. 2011, The Messenger, 145
  * Jiang et al. (2016) Jiang, I.-G., Lai, C.-Y., Savushkin, A., et al. 2016, AJ, 151, 17
  * Johnson et al. (2022) Johnson, M. C., David, T. J., Petigura, E. A., et al. 2022, AJ, 163, 247
  * Lee & Peale (2002) Lee, M. H., & Peale, S. J. 2002, ApJ, 567, 596
  * Leleu et al. (2021) Leleu, A., Alibert, Y., Hara, N., et al. 2021, Astronomy & Astrophysics, 649, A26
  * Lightkurve Collaboration et al. (2018) Lightkurve Collaboration, Cardoso, J. V. d. M., Hedges, C., et al. 2018, Lightkurve: Kepler and TESS time series analysis in Python, Astrophysics Source Code Library, record ascl:1812.013, , , ascl:1812.013
  * Lissauer et al. (2011) Lissauer, J. J., Ragozzine, D., Fabrycky, D. C., et al. 2011, ApJS, 197, 8
  * Lithwick & Wu (2012) Lithwick, Y., & Wu, Y. 2012, ApJ, 756, L11
  * Maciejewski et al. (2015) Maciejewski, G., Fernández, M., Aceituno, F. J., et al. 2015, A&A, 577, A109
  * Mann et al. (2020) Mann, A. W., Johnson, M. C., Vanderburg, A., et al. 2020, The Astronomical Journal, 160, 179
  * Masuda & Winn (2020) Masuda, K., & Winn, J. N. 2020, AJ, 159, 81
  * McArthur et al. (2004) McArthur, B. E., Endl, M., Cochran, W. D., et al. 2004, ApJ, 614, L81
  * McKinney (2010) McKinney, W. 2010, in Proceedings of the 9th Python in Science Conference, Vol. 445, Austin, TX, 51–56
  * McLaughlin (1924) McLaughlin, D. B. 1924, ApJ, 60, 22
  * Michtchenko et al. (2008a) Michtchenko, T. A., Beaugé, C., & Ferraz-Mello, S. 2008a, MNRAS, 387, 747
  * Michtchenko et al. (2008b) —. 2008b, MNRAS, 391, 215
  * Millholland et al. (2018) Millholland, S., Laughlin, G., Teske, J., et al. 2018, AJ, 155, 106
  * Muñoz & Perets (2018) Muñoz, D. J., & Perets, H. B. 2018, AJ, 156, 253
  * NASA Exoplanet Archive (2023) NASA Exoplanet Archive. 2023, Planetary Systems Composite Parameters, vVersion: 2023-07-23, NExScI-Caltech/IPAC, doi:10.26133/NEA12.  https://catcopy.ipac.caltech.edu/dois/doi.php?id=10.26133/NEA12
  * Nealon et al. (2019) Nealon, R., Pinte, C., Alexander, R., Mentiplay, D., & Dipierro, G. 2019, Monthly Notices of the Royal Astronomical Society, 484, 4951
  * Nelson et al. (2014) Nelson, B. E., Ford, E. B., Wright, J. T., et al. 2014, MNRAS, 441, 442
  * Oliphant (2006) Oliphant, T. E. 2006, A guide to NumPy, Vol. 1 (Trelgol Publishing USA)
  * Pallé et al. (2020) Pallé, E., Oshagh, M., Casasayas-Barris, N., et al. 2020, Astronomy & Astrophysics, 643, A25
  * Petrovich et al. (2013) Petrovich, C., Malhotra, R., & Tremaine, S. 2013, ApJ, 770, 24
  * Rasio & Ford (1996) Rasio, F. A., & Ford, E. B. 1996, Science, 274, 954
  * Rein (2012) Rein, H. 2012, MNRAS, 427, L21
  * Rice et al. (2023) Rice, M., Wang, S., Gerbig, K., et al. 2023, AJ, 165, 65
  * Rice et al. (2021) Rice, M., Wang, S., Howard, A. W., et al. 2021, AJ, 162, 182
  * Rice et al. (2022) Rice, M., Wang, S., Wang, X.-Y., et al. 2022, AJ, 164, 104
  * Ricker et al. (2015) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003
  * Ropert et al. (2021) Ropert, S., Rojas, R., Suc, V., et al. 2021, in Revista Mexicana de Astronomia y Astrofisica Conference Series, Vol. 53, 47–49
  * Rossiter (1924) Rossiter, R. A. 1924, ApJ, 60, 15
  * Salvatier et al. (2016) Salvatier, J., Wiecki, T. V., & Fonnesbeck, C. 2016, PeerJ Computer Science, 2, e55
  * Sanchis-Ojeda et al. (2012) Sanchis-Ojeda, R., Fabrycky, D. C., Winn, J. N., et al. 2012, Nature, 487, 449
  * Sanchis-Ojeda et al. (2015) Sanchis-Ojeda, R., Winn, J. N., Dai, F., et al. 2015, ApJ, 812, L11
  * Schlafly & Finkbeiner (2011) Schlafly, E. F., & Finkbeiner, D. P. 2011, ApJ, 737, 103
  * Snellgrove et al. (2001) Snellgrove, M., Papaloizou, J., & Nelson, R. 2001, Astronomy & Astrophysics, 374, 1092
  * Southworth (2011) Southworth, J. 2011, MNRAS, 417, 2166
  * Takaishi et al. (2020) Takaishi, D., Tsukamoto, Y., & Suto, Y. 2020, Monthly Notices of the Royal Astronomical Society, 492, 5641
  * Tayar et al. (2020) Tayar, J., Claytor, Z. R., Huber, D., & van Saders, J. 2020, arXiv e-prints, arXiv:2012.07957
  * Terquem & Papaloizou (2007) Terquem, C., & Papaloizou, J. C. 2007, ApJ, 654, 1110
  * Trifonov et al. (2021) Trifonov, T., Brahm, R., Espinoza, N., et al. 2021, AJ, 162, 283
  * Turner et al. (2021) Turner, J. D., Ridden-Harper, A., & Jayawardhana, R. 2021, AJ, 161, 72
  * Virtanen et al. (2020) Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261
  * von Essen et al. (2013) von Essen, C., Schröter, S., Agol, E., & Schmitt, J. H. M. M. 2013, A&A, 555, A92
  * Vousden et al. (2016) Vousden, W. D., Farr, W. M., & Mandel, I. 2016, MNRAS, 455, 1919
  * Walt et al. (2011) Walt, S. v. d., Colbert, S. C., & Varoquaux, G. 2011, Computing in Science & Engineering, 13, 22
  * Wang et al. (2018) Wang, S., Addison, B., Fischer, D. A., et al. 2018, AJ, 155, 70
  * Wang et al. (2022) Wang, X.-Y., Rice, M., Wang, S., et al. 2022, ApJ, 926, L8
  * Wirth et al. (2021) Wirth, C. P., Zhou, G., Quinn, S. N., et al. 2021, The Astrophysical Journal Letters, 917, L34
  * Wong et al. (2022) Wong, I., Shporer, A., Vissapragada, S., et al. 2022, AJ, 163, 175
  * Wright et al. (2023) Wright, J., Rice, M., Wang, X.-Y., Hixenbaugh, K., & Wang, S. 2023, arXiv e-prints, arXiv:2308.07532
  * Yee et al. (2020) Yee, S. W., Winn, J. N., Knutson, H. A., et al. 2020, ApJ, 888, L5
  * Zhang et al. (2021) Zhang, J., Weiss, L. M., Huber, D., et al. 2021, AJ, 162, 89
  * Zhao et al. (2023) Zhao, L. L., Kunovac, V., Brewer, J. M., et al. 2023, Nature Astronomy, 7, 198