Evidence for Low-level Dynamical Excitation in Near-resonant Exoplanet Systems

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 on the 6.5 m Magellan Clay Telescope. This is the eighth result from the Stellar Obliquities in Long-period Exoplanet Systems survey. We derive a sky-projected 2D spin–orbit angle λ=26−15+12° and a 3D spin–orbit angle ψ=31−11+13° , 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 ≲ 4 is generally consistent with a quiescent formation pathway, with some room for low-level (≲20°) 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.


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, λ, 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;Rice et al. 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. 2006Crane et al. , 2008Crane et al. , 2010)), together with archival data and new photometry from an assortment of telescopes, we derive a moderate 2D spin-orbit angle λ = 26 +12 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 nearresonant planetary systems typically form quiescentlythat is, within the disk plane and without violent postdisk-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.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 × 2000 pixel Andor CCD with pixel scale 0.608 ′′ , and we used a 15-pixel radius (9.12 ′′ ) 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.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 ′ 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.

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 ′ 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  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.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.

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(Crane et al. , 2008(Crane et al. , 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 × 3 binning, and typical RV precision ∼ 3.1 m/s.Conditions were good throughout the observation, with typical seeing 0.7−0.8′′ 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 posttransit baseline observations.The PFS RV measurements obtained for this work are provided in Table 2 and shown in Figure 2.
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 1σ.
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 ⋆ ), the sum of radii divided by the orbital semi-major axis ((R ⋆ + R b )/a b ), the RV semi-amplitude (K b ), the parameterized orbital eccentricity and argument of periastron ( √ e b cos ω b , √ e b sin ω b ), the skyprojected spin-orbit angle (λ), the sky-projected stellar rotational velocity (v sin i ⋆ ), and twelve limb-darkening coefficients, with two per photometric band (q 1 and q 2 for each of LCO, El Sauce, Minerva, TRAPPIST-1 TESS data used in this paper can be found in MAST: 10.17909/0cp4-2j79 South, and TESS) and two for the in-transit RV dataset (q 1;RM and q 2;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× the autocorrelation length (≥ 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 leastsquares fit to the set of output transit mid-times t 0 .The reference epoch was optimized to minimize the covariance between T 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 λ = 26 +12 −15 • 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 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 ( R; Gelman & Rubin 1992) fell below 1.01.The resulting stellar rotation period is P rot = 22 ± 1 days, corresponding to a stellar equatorial velocity v = (2πR * )/P rot = 1.86 ± 0.19 km/s.
Lastly, we combined v and v sin i * to derive the stellar inclination i * and the true stellar obliquity ψ 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 rot , and cos i.This analysis yielded a stellar inclination estimate i * = 89.77±16.76• .Then, the true stellar obliquity (ψ) was derived through Equation 9 of Fabrycky & Winn (2009), where i * is the stellar inclination and i is the planet's orbital inclination.The resulting true stellar obliquity is ψ = 31 +13 −11 • .

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 meanmotion resonance with a neighboring companion.We initialized our sample by cross-matching the set of all exoplanets with λ 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). 2 2 Both catalogues were accessed on 7/20/2023.
Thirty-two planets were identified that fit these two criteria. 3e 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 spinorbit measurement in the TEPCat catalogue and (2) ttv_flag = True in the NASA Exoplanet Archive.In addition to TOI-2202 b, nine further candidate nearresonant 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 ≲ 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 ≲ 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 ≲ 4.
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 nearresonant 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 multitransiting systems examined in Fabrycky et al. (2014).Note: P b and T0 were derived from the weighted least-square fit to the transit mid-times t0.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 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 spinorbit angle has not been consistent with exact alignment (|λ| = 0 • ) within 1σ.
For six near-resonant systems, the 3D spin-orbit angle ψ 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 nearresonant systems.
To quantify this deviation from alignment, we drew 10,000 iterations of random values from each of the six systems with measured ψ 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 σ = 1.8 such that ∼ 98% of draws fall within the range ψ < 5 • .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 ψ distribution exhibits consistent, near-exact (ψ < 5 • ) alignment.

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 λ constraint.Instead, an asteroseismic analysis conducted by Huber et al. (2013) revealed a stellar spin axis inclined at i * ∼ 45 • .The Kepler-56 system includes two transiting planets that, by definition, have i b ∼ i c ∼ 90 • , such that the stellar spin axis at i * ∼ 45 • 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 nearresonance (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 ∼ 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 λ = 10 +17 −20 • and ψ = 23 +14 −12 • (Zhao et al. 2023).A third relevant system is HIP 41378, with a previously reported misalignment |λ| = 57 +26 −18 • 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 longperiod transiting exoplanets that is amenable to spinorbit 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.

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, nearresonant 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 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 ψ ≤ 5 • (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 (≲ 20 • ) disk misalignments that could be attributed solely to chaotic accre-Table 4. Systems with a spin-orbit measurement and P2/P1 ≲ 4. N pl is the number of confirmed planets in the system to date.N pl , orbital period, and T 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 λ measurements.ψ values were drawn from Albrecht et al. (2021)  Wang et al. ( 2022) -* 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.† A single ψ value was derived for the TRAPPIST-1 system in Albrecht et al. (2021), such that these three values all correspond to just one measurement.
Alternatively, low-level misalignments in nearresonant 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 ∼ 20 • 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 ± 0.022M ⊕ ; 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 spinorbit measurements -including those that are not in near-resonant systems -previous work has found evidence for a nonzero mean stellar obliquity with significant scatter ψ = 19 ± 10 • (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.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.
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

Figure 1 .
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.
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 × 2000 CCD with a scale of 0.67 ′′ 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.

Figure 2 .
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.

Figure 3 .
Figure 3. Period ratios of planet pairs in compact systems (Pout/Pin ≲ 4) in which at least one planet has a spin-orbit measurement.

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

Table 1 .
Trifonov et al. (2021)d-transit times t0, uncertainties σt 0 , and offsets from the linear ephemeris prediction ∆ lin for TOI-2202 b based on our collected photometry and new TESS transits collected since the publication ofTrifonov et al. (2021).All listed ∆ lin values in this table are provided relative to our derived transit mid-time epoch T0 = 2459577.9736362±0.0039days (Table3).

Table 2 .
PFS RV measurements for the TOI-2202 system, obtained across the transit of TOI-2202 b.

Table 3 .
Priors and posteriors for the TOI-2202 planetary system.