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Issue
A&A
Volume 710, June 2026
Article Number L38
Number of page(s) 7
Section Letters to the Editor
DOI https://doi.org/10.1051/0004-6361/202660771
Published online 30 June 2026

© The Authors 2026

Licence Creative CommonsOpen Access article, published by EDP Sciences, under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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1. Introduction

The spin-orbit obliquity is a key diagnostic for the mechanisms of formation and orbital migration of exoplanets (e.g. Naoz et al. 2011). The sky-projected obliquity (λ) can be measured with in-transit radial velocities (RVs) through the Rossiter-McLaughlin (RM) effect (e.g. Ohta et al. 2005). This can then be translated into the 3D orbital obliquity (ψ) with respect to the stellar rotation axis with knowledge of the stellar rotation period, projected rotational velocity, and stellar radius, yielding the stellar inclination, i★ (e.g. Winn et al. 2007).

Giant planets in close-in orbits are thought to form in situ close to the final orbit, or to form in the outer regions and migrate inwards (Dawson & Johnson 2018). Different mechanisms could shrink the orbits, such as dynamical interactions (high-eccentricity migration, HEM) through planet-planet scattering (Marzari et al. 2006) or the Kozai mechanism (Wu & Murray 2003) after disc dispersal, and disc–planet interactions (Lin et al. 1996). These mechanisms should imprint distinctive signatures in the observed obliquities. While scattering encounters or the HEM mechanism typically result in misaligned and eccentric orbits, migration through disc-planet interactions (disc migration) typically preserves low eccentricities and orbital coplanarity. However, star-planet tidal interactions may alter the obliquity, and close-in planets orbiting stars that are a few gigayears old and that have deep convective envelopes are expected to show aligned configurations regardless of the formation scenario, as is observed in most cases (e.g. Albrecht et al. 2012; Lai 2012). By observing systems that are still young enough to not have undergone significant tidal alterations of obliquity, we can access the original configuration in which the system formed (e.g. Albrecht et al. 2022). Hence, young transiting planets represent a unique resource for solving as-of-yet inconclusive observational results (e.g. Wright et al. 2023).

Space-based transit missions led to the identification of several planets transiting young stars (age < 1 Gyr), totalling about 100 confirmed or validated systems (cf. NASA Exoplanet Archive, Christiansen et al. 2025). However, the orbital obliquity has only been measured for 17 young systems (Table B.2). These measurements have generally indicated aligned configurations, although with moderately large uncertainties. Consequently, the obliquity of young transiting planets remains significantly unexplored. Moreover, among the very small sample of stars with ages ≤100 Myr (infant planets), each obliquity measurement is consistent with good alignment (e.g. Barber et al. 2025). It is crucial to extend the sample of such very young stars with obliquity measurements, as this trend would be an important clue about obliquity excitation for giant planets in close orbits. For instance, very young close-orbit giant planets could have formed in situ or undergone disc migration (both would preserve the initial alignment), while late-arriving ones could have formed through dynamical interactions (Albrecht et al. 2022).

TOI-837 b (Bouma et al. 2020) is a giant planet in the 35-Myr-old southern open cluster IC 2602 (Barragán et al. 2024; Damasso et al. 2024, B24, D24). Open clusters offer a unique opportunity to study exoplanets hosted by cluster members in extreme detail. It is possible to derive properties such as R★, M★, Teff, logg, distance, age, and metallicity with a high level of accuracy. Moreover, the analysis of exoplanets orbiting stars in different open clusters with different ages allows us to trace the temporal evolution of exoplanets, from their early formation stages to their old ages.

2. Target selection and properties

TOI-837 b was chosen from the pull of targets selected for the ongoing project ‘Obliquity of tidally – detached planets via the Rossiter-McLaughlin effect’ (ORME, Mantovan et al. in prep.). The ORME project – the first result of which was presented in Mantovan et al. (2024b) – aims to determine the orbital obliquity of young (age < 1 Gyr) or tidally detached (e.g. Rice et al. 2021) exoplanets by collecting in-transit RVs to measure the RM effect.

While both confirming studies (B24; D24) found that TOI-837 b has a mass of ≈120M⊕, they estimated slightly different radii ( R p = 9 . 2 − 0.3 + 0.4 R ⊕ Mathematical equation: $ R_{\mathrm{p}}=9.2^{+0.4}_{-0.3}\,R_\oplus $; R p = 9 . 7 − 0.6 + 0.9 R ⊕ Mathematical equation: $ R_{\mathrm{p}}=9.7^{+0.9}_{-0.6}\ \,R_\oplus $) and densities (ρp = 0.9 ± 0.2 g cm−3; ρp = 0.7 ± 0.2 g cm−3). They found solutions that are consistent with TOI-837 b having a nearly circular orbit, with a 1σ upper limit on the eccentricity of e < 0.27 (B24). The discrepancy in Rp measurements is likely due to different dilution treatments in crowded environments, with the work by D24 potentially affected by overcorrection of stellar contamination. Our reanalysis of TESS light curves presented in Sect. 4 reconciles this discrepancy.

3. Observations and data reduction

3.1. ESPRESSO observations

We observed TOI-837 with ESPRESSO (Pepe et al. 2021) 1-UT mode at ESO’s Very Large Telescope (VLT) on March 29, 2025 (proposal P114.27GY, PI: G. Mantovan), obtaining 53 spectra with 300 s of exposure time, with an average seeing of 0.4 arcseconds, an ⟨S/N⟩ 5500 Å = 46 and ⟨RVerr⟩ = 7.4 m s−1. We collected in-transit (2 hours) and suitable out-of-transit (3 hours) observations. The wavelength range of these spectra is 378–789 nm with a resolving power of R ∼ 140 000.

We used the ESO’s official ESPRESSO pipeline through the EsoReflex workflow (Freudling et al. 2013), and computed the RVs using the cross-correlation function (CCF) method (Pepe et al. 2002). The strong stellar activity of TOI-837 distorts the core of the average line profile, while its fast stellar rotation broadens the line. Hence, we selected a half-window large enough to include the continuum when fitting the CCF profile. Specifically, we used an F9 mask and a grid size of 120 km s−1.

3.2. TESS photometry

TESS observed TOI-837 at 2 min cadence in sectors 10, 11, 37, 38, 63, 64, and 90. We extracted light curves using the PATHOS approach (Nardiello et al. 2020), which allowed us to minimise neighbour flux contamination (down to 0.5 TESS pixels) and preserve stellar rotation information more effectively (e.g. Nardiello et al. 2022; Mantovan et al. 2024a). Sector 90 coincided with the ESPRESSO observing window, enabling a simultaneous observation crucial for better modelling stellar activity by combining both datasets.

4. Analysis

To refine the planetary and stellar parameters reported in Bouma et al. (2020), B24 and D24, we examined our light curves and spectroscopic time series within a Bayesian framework using PyORBIT1 (Malavolta et al. 2016, 2018), a public software that allows planetary transits and RVs to be modelled while considering stellar activity effects. We also did a transmission spectroscopy analysis to probe the atmospheric composition of TOI-837 b; the results and methodology are summarised in Appendix A.

4.1. Joint Bayesian analysis

We simultaneously modelled all TESS light curves and the in-transit RV anomaly. In this way, we modelled the orbital obliquity, the planetary transits, and the stellar activity using Gaussian processes (GPs, Rasmussen & Williams 2006; Rajpaul et al. 2015). We fitted the central time of transit (T0,  b), orbital period (Pb), impact parameter (b), planet-to-star radius ratio (Rp/R★), and sky-projected obliquity (λ). Following Mantovan et al. (2022), we determined the dilution factor (df) caused by stars within 0.5 pixels of TOI-837 (i.e. those not accounted for by PATHOS) and applied a fixed correction term of df = 0.069 in the modelling. We imposed Gaussian priors on the projected rotational velocity (v sin i★), the host star density (ρ★), and stellar radius (R★) from D24, while leaving the stellar rotation period (Prot) free to vary. We treated the stellar equatorial velocity (veq, obtained from Prot) and inclination (i★) as independent variables. Therefore, we are not subject to the bias described in Masuda & Winn (2020). The v sin i★ is a derived parameter. It was calculated from veq and i★ at each step, and then compared with the prior. To explore the parameter space efficiently, we restricted cos i★ to the range [0, 1], effectively selecting a single stellar pole orientation. We treated the stellar limb darkening (LD) contribution by estimating u1 and u2 using PyLDTk2 (Husser et al. 2013; Parviainen & Aigrain 2015), taking into account the spectroscopic Teff and log g from D24, assuming a boxcar filter as the passband in the ESPRESSO spectral range and adding 0.1 in quadrature to their Gaussian errors to take into account the known model underestimation. Additionally, we used a linear law as a function of limb angle to model and fit the effect of stellar convective blueshift (CB) on the in-transit RV curve (e.g. Cegla et al. 2016). Short-term stellar activity was incorporated into the model using a jitter term, which was added in quadrature to the RV and photometry errors. An extended description of how priors and boundaries have been defined can be found in Mantovan et al. (2026).

Our modelling of the RV anomaly foresees the calculation of a synthetic CCF, assuming a Gaussian profile for the local spectral line emerging from the surface of the star, and accounting for the missing contribution from the stellar surface covered by the planet. The model reproduces a CCF with characteristics similar to the observed one (such as the RV range and step), and fits a Gaussian to derive the predicted RV anomaly. This method was introduced in Covino et al. (2013) and implemented in PyORBIT in Mantovan et al. (2024b). To minimise the differences between the RV measurement techniques and our modelling approach, we decided to analyse the RV measured from the CCF. We note that the rotational profile differs from a Gaussian; we accounted for this bias by using a Gaussian profile on both the observed and modelled CCFs, thereby introducing the same systematic error on both sides.

We modelled stellar activity in the full 27.4 days TESS photometric time series using a unidimensional GP. We adopted a rotation kernel as defined in Foreman-Mackey (2018). In contrast, due to the shorter (5-hour) time frame, we modelled the upward trend due to stellar activity observed in the spectroscopic time series using a second-degree polynomial function. We preferred a second-order over a linear one because the posterior of the quadratic coefficient is not centred on zero, though only marginally. Further technical details are given in Appendix C and the best-fitting values in Table B.1.

4.2. Three-dimensional orbital obliquity

By jointly modelling the space-based photometry and RM effect, we obtained a sky-projected obliquity, λ = 341 . 1 − 2.5 + 2.3 Mathematical equation: $ \lambda\, = 341.1^{+2.3}_{-2.5} $ deg, and a reliable stellar rotation period, Prot  = 3.00 ± 0.02 d, the latter consistent with the values reported in Bouma et al. (2020) and D24. We then translated λ into the true three-dimensional orbital obliquity (ψ). We did so by sampling from the posterior distributions of the projected obliquity (λ), stellar inclination (i★) and planetary orbital inclination (ip), using Eq. (7) from Winn et al. (2007), to estimate the three-dimensional orbital obliquity, ψ = 25 . 9 − 6.3 + 7.5 Mathematical equation: $ \psi = 25.9 ^{+7.5}_{-6.3} $ deg. This result confirms the prograde motion of the planet and suggests a mildly misaligned orbit (Fig. 1).

Thumbnail: Fig. 1. Refer to the following caption and surrounding text. Fig. 1.

Simultaneous photometric and spectroscopic transit fit of TOI-837 b, observed on 29 March 2025. The top panels show the best-fit transit model (black line) overlaid on TESS observations (green dots), with residuals in parts per thousand. The bottom panels show the best-fit RM model (black line) superimposed on ESPRESSO RVs (orange dots), with residuals in meter seconds−1. The RV data have been corrected for the systemic RV, while the upward trend caused by stellar activity has been modelled using a second-order polynomial.

5. Discussion

5.1. Tidal decay of obliquity timescale

The equilibrium tide model of Albrecht et al. (2012) gives an obliquity decay timescale exceeding the Hubble time for TOI-837. Such a result is confirmed also by considering the dynamical obliquity tides excited by the planet inside the star (see Appendix D). Therefore, the measured obliquity should still be unaltered by tidal effects, resulting in a direct diagnostic for the formation path of this planetary system.

5.2. The mildly misaligned orbit of TOI-837 b

Our analysis clearly shows that the orbit of the Saturn-size planet TOI-837 b is mildly misaligned. To contextualise this result, we selected all planets with a measured three-dimensional obliquity, ψ, that orbit stars younger than 1 Gyr (see Fig. 2). Moreover, we divided the sample into young (< 1 Gyr) and infant (< 100 Myr) planets, as we are interested in investigating whether obliquity excitation for giant planets could occur during the early stages of a planetary system. TOI-837 b is the first planet younger than 100 Myr with accessible ψ with a statistically significant, mildly misaligned orbit (≈30 deg). The mild but statistically significant misalignment of TOI-837 b, together with its very young age of around 35 Myr, supports the possibility of obliquity excitation taking place in the early stages of a planetary system. The obliquity of TOI-837 b has an extremely long tidal damping timescale, and it produces a precession of its orbital plane that should be measurable (see Appendix D for details).

Thumbnail: Fig. 2. Refer to the following caption and surrounding text. Fig. 2.

True obliquity for the youngest stars (< 1 Gyr). The planetary orbital period is colour-coded, while the size of each dot indicates the radius of the given planet.

5.3. Bound binary companion and dynamical implications

TOI-837 has a co-moving stellar companion that is 4.92 mag fainter (in the G band) and located at a separation of 2.3″. D24 confirmed that the co-moving star is bound (TOI-837 B, binary separation about 328 au). The gravitational pull of TOI-837 B might excite a high orbital inclination, consequently altering the obliquity through tidal and dynamical effects, and significantly influence the system’s evolution. The accuracy and precision of our observations are unaffected, as the companion is too faint and distant to contaminate the spectra of TOI-837. Moreover, the RV variations are negligible in the time span of our observations. The presence of a bound stellar companion helps us to place a few constraints on the formation and dynamical history of TOI-837 b. First, we inspected post-formation mechanisms involving a close stellar companion, capable of explaining mild misalignments. We estimated the timescale that stellar Kozai-Lidov oscillations (Wu & Murray 2003) would take if triggered by TOI-837 B. Taking into account the stellar mass of the companion (MB = 0.39 M⊙) as reported in B24, we estimated a timescale of around 320 Myr (Ford et al. 2000), which is an order of magnitude longer than the age of the system. This result and the much faster relativistic precession of the apsidal line of the orbit make the stellar Kozai scenario improbable (Appendix D).

Conversely, a primordial misalignment scenario, such as that proposed by Batygin (2012), could explain the moderate misalignment. In this framework, a massive distant companion exerts a secular torque on the protoplanetary disc, causing it to precess and become tilted with respect to the stellar spin axis, which is assumed to remain approximately fixed. Planets forming within the disc inherit this misalignment. This picture was later refined by Picogna & Marzari (2015), who showed that, once the planet becomes sufficiently massive, it can decouple dynamically from the disc due to the perturbations of the inclined companion. The planet and the disc then evolve almost independently, developing a significant mutual inclination. Nevertheless, inward migration is not halted: the planet continues to drift towards the star because its inclined orbit repeatedly crosses the disc, where interaction with the gas leads to angular momentum loss through dynamical friction. An alternative scenario has been proposed by Albrecht et al. (2022), in which the misalignment arises from a resonance crossing driven by a wide-orbiting, highly inclined stellar companion. Recent observations show that about one third of isolated systems have misaligned outer discs (>10 au) from birth (Biddle et al. 2025), which challenges certain primordial obliquity excitation mechanisms. Nevertheless, the inner regions may remain aligned (e.g. Espinoza-Retamal et al. 2026, and references therein), and formation simulations by D24 demonstrate that TOI-837 b formed in the inner disc between 2 and 4 au.

If such a misalignment is indeed primordial, then it is likely that TOI-837 b underwent disc-driven migration rather than HEM. This is further suggested by the circular orbit of TOI-837 b and a circularisation timescale greater than 1 Gyr (e.g. Adams & Laughlin 2006), which once again supports a primordial circular orbit (in agreement with D24, conclusions). This result is in line with the discussion first presented in Albrecht et al. (2022), but it is an important addition given the statistically significant mild misalignment. Nevertheless, given the long eccentricity damping timescale, the orbit of TOI-837 b could have a small eccentricity that could be primordial or, in principle, be excited by the fast rotation of the star according to Leconte et al. (2010)’s model because (Pb/Prot) cos ψ > 18/11. However, we show in Appendix D that the excitation timescale is too long to be relevant, leaving any primordial eccentricity virtually unaffected.

6. Key findings

  1. TOI-837 b is an infant giant planet (35 Myr) in the southern open cluster IC 2602 on a mildly misaligned orbit, and it is the only one known with such a configuration among infant (age < 100 Myr) planets with accessible ψ;

  2. Both the obliquity and circular orbit of TOI-837 b are unaltered by tidal effects, supporting the hypothesis that obliquity excitation occurred early on in the formation and evolution of the system;

  3. A primordial misalignment scenario involving the bound stellar companion TOI-837 B, such as the exertion of a secular torque on the protoplanetary disc or resonance crossing, could explain the mild misalignment observed. This supports that TOI-837 b underwent disc-driven migration;

  4. Our transmission spectroscopy analysis yields neither statistically significant absorption features nor evidence of detectable species in the planetary atmosphere.

Acknowledgments

Based on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme ID P114.27GY. This paper includes data collected with the TESS mission, obtained from the MAST archive at the Space Telescope Science Institute (STScI). Funding for the TESS mission is provided by the NASA Explorer Program. STScI is operated by the Association of Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555. G. M. acknowledges support by the Space It Up project funded by the Italian Space Agency, ASI, and the Ministry of University and Research, MUR, under contract n. 2024-5-E.0 – CUP n. I53D24000060005. L.M. acknowledges financial support from BIRD funds-PRD 2025, DFA-University of Padova.

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Appendix A: Probing the atmospheric composition

Given its large transmission spectroscopy metric (TSM) value (see Kempton et al. (2018); D24), we tried to look at any possible atmospheric signal of TOI-837 b detectable in our data. We extracted the planetary transmission spectrum following the classic procedure used in high-resolution transmission spectroscopy (e.g. Wyttenbach et al. 2015; Borsa et al. 2021). Telluric correction was already performed at DRS level, since for our analysis we used the "S1D_TELL_CORR" products of the pipeline. We shifted the spectra to the stellar restframe by using the Keplerian model of the system with the values in Table B.1, then we normalised each spectrum. Then we divided all the spectra by a master stellar spectrum created by averaging the out-of-transit spectra. At the end we moved to the planetary restframe by shifting all the residual spectra for the theoretical planetary RV, and the transmission spectrum was then created averaging all the full-in-transit residual spectra. We searched for possible absorption given by Na D doublet, Hα, K, Mg, Li, without finding any trace of significant absorption beyond noise. We also tried to look for elements/molecules by using the cross-correlation technique. We generated model templates for V, Cr, Mg, K, Ca, H2O by using petitRADTRANS (Mollière et al. 2019), assuming an isothermal atmospheric profile with T = 1000 K, continuum pressure level of 10 mbar and solar abundances. The atmospheric models were then translated into flux (Rp/Rs)2, convolved at the ESPRESSO resolving power and continuum normalised. Cross-correlation between model templates and the single transmission spectra was then performed in the same way as in Borsa et al. (2021). We find no evidence in the planetary atmosphere for the presence of any of the species we looked for. It is interesting to note the map relative to the H2O non-detection (Fig. A.1). No spurious signal is present, a sign that the automatic telluric correction done by the ESPRESSO DRS is performing very well.

Thumbnail: Fig. A.1. Refer to the following caption and surrounding text. Fig. A.1.

CCF signal for H2O in TOI-837 b. Left panel: 2D tomography of the in-transit CCF residuals. Central panel: Kp-Vel map. The green dotted lines mark the theoretical planetary position. Right panel: planetary absorption signal averaged at the theoretical Kp.

Appendix B: Extra figures and tables

This section includes Table B.1 presenting all the priors and outcomes of the RM modelling and Figure B.1 showing the retrieved posterior distributions.

Thumbnail: Fig. B.1. Refer to the following caption and surrounding text. Fig. B.1.

Posterior distributions of stellar parameters and obliquities that were retrieved in this work.

Table B.1.

Priors and outcomes of RM modelling.

Table B.2.

List of young systems with obliquity measurements.

Appendix C: Global optimisation and Bayesian analysis

We performed a global optimisation of the parameters by running PyDE (Storn & Price 1997; Parviainen 2016) for 100 000 generations and a Bayesian analysis of the RM signal in the RV time series using EMCEE Foreman-Mackey et al. (2013) for 150 000 steps. The motivation behind the choice of using MCMC samplers is described in depth in Mantovan et al. (2024b). We used 4 × ndim walkers, with ndim being the model dimensionality, and discarded the first 50 000 steps (burn-in). The Gelman-Rubin statistics (Gelman & Rubin 1992, threshold value R ̂ = 1.01 Mathematical equation: $ \hat{R} = 1.01 $) were used to check the convergence of the chains.

Appendix D: Tides and precession

The obliquity of the star in our system can be effectively damped by dynamical obliquity tides according to the mechanism proposed by Lai (2012). Considering a stellar modified tidal quality factor Q ′ s = 10 6 Mathematical equation: $ Q\prime_{\mathrm{s}} = 10^{6} $, that corresponds to an efficient tidal dissipation inside the star itself, we obtain an e-folding decay time of the stellar obliquity of ∼9 × 1012 years, much longer than the Hubble time. To compute such a timescale, we used the constant tidal time lag model by Leconte et al. (2010) translating the modified tidal quality factor into a time lag with the relationship Δ t = ( 3 / 2 ) ( k L 2 n Q ′ s ) − 1 Mathematical equation: $ \Delta t = (3/2)(k_{\mathrm{L2}} n Q\prime_{\mathrm{s}})^{-1} $, where kL2 is the stellar potential Love number of degree two and n is the orbital mean motion.

TOI-837 b is one of the few giant close-in planets for which the orbital period is longer than the rotation period of the host star, making tides transfer angular momentum from the stellar spin to the orbit of the planet. However, such a transfer is very slow because of the relatively large value of a/Rs = 17.62 ± 0.14. Therefore, the timescales for the increase of the semimajor axis and for the excitation of the eccentricity of the orbit are remarkably longer than the Hubble time, even considering dynamical tides inside the star. Dynamical tides in the form of inertial waves, having the Coriolis force as their restoring force, are expected to be excited inside the star because the tidal frequency 2|n − Ωs|< 2Ωs, where Ωs is the stellar spin frequency (e.g. Ogilvie 2014).

The tidal damping of the orbital eccentricity is dominated by the dissipation of tides inside the planet. Considering a modified tidal quality factor of the planet Q ′ p = 10 5 Mathematical equation: $ Q\prime_{\mathrm{p}} =10^{5} $, that is similar to that of Jupiter, we obtain an eccentricity damping timescale τe ≡ e/|de/dt| = 3.4 Gyr indicating that any primordial small eccentricity could have not been appreciably damped along the lifetime of the system (∼35 Myr). On the other hand, the timescale for the synchronization of the planetary rotation with the orbital period and the damping of any primordial obliquity of the planetary spin is only ∼0.14 Myr, much shorter than the age of the system.

The fast rotation of TOI-837 and the obliquity of the orbit of its planet produce a precession of the line of the nodes of its orbit that can be evaluated using, for example, the model by Damiani & Lanza (2011). Using their Eq. (8), we find a period of the precession of the line of the nodes of 0.85 or 1.7 Myr for a stellar apsidal motion constant k2 = 0.02 or k2 = 0.01, respectively. According to the models computed by Claret (2019), the former value of the apsidal motion constant is close to that of the present Sun, while the latter is for a star on the zero-age main sequence, ∼20% more massive than the Sun and with a solar chemical composition.

The precession of the orbital plane of TOI-837 b causes a change in the inclination of its orbit to the line of sight of ∼0.″69/yr for k2 = 0.01. This produces a change in the duration of the transit of ∼14 s in ten years. Detecting such a change might in principle be possible with the CHEOPS space-craft, where e.g. Borsato et al. (2021) obtained a precision of 13-16 s, although we note that this was for a bright, less active system.

The period of precession of the spin of the planet is only ∼67 years, considering synchronization with the orbital motion, and a planetary Love number and normalized moment of inertia equal to those of Jupiter. This implies that the planetary spin precession cannot be captured into a resonance with the nodal precession during the evolution of the system (Millholland & Laughlin 2019). Therefore, the planetary spin is predicted to stay normal to the orbital plane during all the future evolution of the system and a large planetary obliquity is not expected to be excited by such a mechanism.

If the orbit of TOI-837 b has a small eccentricity, say of 0.05, its apsidal line will precede under the action of the general relativistic effects and the stellar quadrupole moment. We estimate an apsidal precession period of ∼45400 years with more than 80% of the precession rate produced by general relativity effects (e.g. Mardling & Lin 2002). Such a relativistic precession does not affect the line of the nodes that precedes under the effect of the stellar gravitational quadrupole moment only (J2 ∼ 10−5 for k2 = 0.01). As a consequence of the apsidal precession, the mid transit times of the planetary transits and occultations oscillate along the precession cycle (e.g. Damiani & Lanza 2011). However, the effect is very small and can probably not be detected because the transit time variation in ten years is only 0.05 s for an eccentricity of 0.05. By considering an eccentricity of 0.1, the transit time variation doubles to ∼0.10 s, while all the precession periods and the transit duration variation remain the same within a few percent.

Thumbnail: Fig. D.1. Refer to the following caption and surrounding text. Fig. D.1.

Figure similar to Fig. 1 but with the RV data corrected for both systemic RV and stellar activity.

We conclude that the apsidal precession is always much faster than the Kozai cycles potentially induced by the distant stellar companion TOI-837 B, which implies that any excitation of the eccentricity of TOI-837 b by the gravitational perturbation of that companion with a Kozai mechanism is negligible. The eccentricity excited by TOI-837 B along the main-sequence lifetime of the system is also negligible in the case of nearly coplanar orbits, according to the model by Mardling (2007), because of the large separation between the planet and the distant companion.

All Tables

Table B.1.

Priors and outcomes of RM modelling.

In the text
Table B.2.

List of young systems with obliquity measurements.

In the text

All Figures

Thumbnail: Fig. 1. Refer to the following caption and surrounding text. Fig. 1.

Simultaneous photometric and spectroscopic transit fit of TOI-837 b, observed on 29 March 2025. The top panels show the best-fit transit model (black line) overlaid on TESS observations (green dots), with residuals in parts per thousand. The bottom panels show the best-fit RM model (black line) superimposed on ESPRESSO RVs (orange dots), with residuals in meter seconds−1. The RV data have been corrected for the systemic RV, while the upward trend caused by stellar activity has been modelled using a second-order polynomial.
In the text
Thumbnail: Fig. 2. Refer to the following caption and surrounding text. Fig. 2.

True obliquity for the youngest stars (< 1 Gyr). The planetary orbital period is colour-coded, while the size of each dot indicates the radius of the given planet.
In the text
Thumbnail: Fig. A.1. Refer to the following caption and surrounding text. Fig. A.1.

CCF signal for H2O in TOI-837 b. Left panel: 2D tomography of the in-transit CCF residuals. Central panel: Kp-Vel map. The green dotted lines mark the theoretical planetary position. Right panel: planetary absorption signal averaged at the theoretical Kp.
In the text
Thumbnail: Fig. B.1. Refer to the following caption and surrounding text. Fig. B.1.

Posterior distributions of stellar parameters and obliquities that were retrieved in this work.
In the text
Thumbnail: Fig. D.1. Refer to the following caption and surrounding text. Fig. D.1.

Figure similar to Fig. 1 but with the RV data corrected for both systemic RV and stellar activity.
In the text

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