NEID Rossiter–McLaughlin Measurement of TOI-1268b: A Young Warm Saturn Aligned with Its Cool Host Star

The TESS data for TOI-1268 are available as 10 × 10 subimages with 2 minute time sampling, and as part of full-frame images (FFIs) with 30 minute sampling. We obtained three sectors of TESS Primary Mission data from 2019 August 15 to 2019 September 11 (Sector 15) and from 2020 January 21 to 2020 March 18 (Sectors 21 and 22), and one sector of TESS extended mission data from 2021 July 23 to 2021 August 20 (Sector 41). The target will have at least two more sectors of TESS observations in Sector 48 (2022 January 28 to 2022 February 26) and Sector 49 (2022 February 26 to 2022 March 26).

The transit signal was detected with a period of ∼8.16 days at high significance independently by the NASA Science Processing Operations Center (SPOC) pipeline (Jenkins et al. 2016) and the MIT Quick-Look Pipeline (QLP; Huang et al. 2020a, 2020b), and was released to the public for follow-up observations as TOI-1268.01. In total, 14 transits of TOI-1268b were observed by TESS. The TESS light curves do not show any strong instrument systematics. We used the Pre-search Data Conditioning SAP flux (PDC_SAPFLUX; Stumpe et al. 2012, 2014) for the light-curve analysis.

Through the TESS Follow-up Observing Program (TFOP) collaboration (Collins et al. 2018), we observed eight full or partial transits of TOI-1268b with ground-based seeing-limited telescopes, including three transits observed simultaneously with the TESS observations. We used the TESS Transit Finder, which is a customized version of the Tapir software package (Jensen 2013), to schedule our transit observations. These observations confirm that the transiting signal originated from within less than 6'' of the target star. Gaia EDR3 (Lindegren et al. 2021) reports no additional stars within 10'' of TOI-1268.

We include two good-quality full transits that were taken simultaneously with TESS in our transit modeling (see Figure 1). The first transit was taken on UT 2020 January 26 by the Deep Sky West 0.5 m telescope near Rowe, NM, USA, in the $g^{\prime}$ band and detected an on-time transit in a 109 aperture. The second transit was taken on UT 2020 February 3 simultaneously in g_p and z_s filters with a 63 aperture from the Las Cumbres Observatory Global Telescope (LCOGT; Brown et al. 2013) 0.4 m network node at Haleakala Observatory. The light curves were reduced with AstroImageJ (Collins et al.2017).

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In addition, we obtained six transits of TOI-1268b with full or partial transit baselines from various ground-based facilities. These transit observations played an important role in confirming the transit to be on target and ruling out nearby eclipsing binaries. We do not include these observations in our transit model because of their partial transit baselines or additional complications due to meridian flip (i.e., telescope pointing crossing the meridian during the observation) that introduces systematic flux offset. We list the observations below and these data can be found on the ExoFOP website. ⁴⁷

• 1.  
  On UT 2020 January 9, an ingress was observed to be on target from the Kotizarovci Observatory 0.3 m Telescope, near Viskovo, Croatia, in a Baader R 610 nm longpass filter using a 108 aperture, marginally contaminated by a nearby star.
• 2.  
  On UT 2020 January 10, an egress was observed from the LCOGT 0.4 m telescope from Teide Observatory in the z_s filter using an uncontaminated 102 aperture.
• 3.  
  On UT 2020 March 6, the TRAPPIST-North team observed an on-time, almost full transit in the B band. The detection is complicated by a meridian flip at ingress and a strong increase in sky background as the nominal time of egress approached.
• 4.  
  On UT 2021 April 18, a partial transit was observed in the B band from the OAUV-TURIA1 (0.143 m) telescope near Valencia, Spain. The detection is complicated by a meridian flip during predicted ingress and strong residuals.
• 5.  
  Also on UT 2021 April 18, the same transit was observed in a 67 uncontaminated aperture in the B band from the Observatory de Ca l'Ou 0.4 m telescope near Barcelona, Spain.
• 6.  
  On UT 2021 April 27, a partial was observed in the g band using an uncontaminated 47 aperture from the Wellesley College Whitin Observatory CDK700 telescope near Wellesley, MA, USA.

The KELT Survey (Pepper et al. 2003, 2007) also monitored the star for over two years from BJD 2,455,976 to BJD 2,457,022 as part of its normal survey. The precision of the KELT photometry is not sufficient to detect the transit signals. However, the long-term monitoring from KELT was used to measure the stellar rotation period. The Lomb–Scargle periodogram (Lomb 1976; Scargle 1982; VanderPlas 2018) of the KELT light curve reveals that the star has a rotation period of 10.8 days (Figure 2). This detection helps to break the degeneracy between the rotation period determined from the TESS light curves, which shows two peaks at ∼5 days and ∼10 days.

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High-resolution imaging is required to detect nearby companions or background objects that cannot be resolved by seeing-limited photometry. We obtained both adaptive optics (AO) and speckle imaging of TOI-1268, as shown in Figure 3. On UT 2020 January 8, the PHARO instrument (Hayward et al. 2001) on Palomar 5 m collected AO images of TOI-1268 in the narrowband Br γ filter. No companions are identified down to a contrast of 5.481 magnitudes at 05. On UT 2021 February 2, the `Alopeke speckle instrument (Scott 2019) on Gemini North 8 m took simultaneous speckle imaging in the 832 and 562 nm bands. No companions are detected down to a contrast of 6.36 mag at 05.

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Although not shown in Figure 3, we obtained the following observations on the Sternberg Astronomical Institute (SAI) 2.5 m telescope located at Mt Shatdzhatmaz in the North Caucasus and on the Shane 3 m telescope at Lick Observatory in Mount Hamilton, CA, USA. On UT 2020 November 29, the Speckle Polarimeter on SAI 2.5 m obtained speckle imaging of TOI-1268 in I filter. On UT 2019 November 12, the ShARCS instrument (Kupke et al. 2012; Gavel et al. 2014) on Shane 3 m collected AO images of TOI-1268 in the Ks and J filters. The ShARCS data were reduced and analyzed using the open-source Python-based SImMER pipeline available on GitHub and described in previous publications (Hirsch et al. 2019; Savel et al. 2020). TOI-1268 appeared single in both observations.

We obtained 14 spectra with TRES on the 1.5 m telescope at the Fred Lawrence Whipple Observatory, from UT 2019 December 10 to UT 2020 December 27. TRES has a resolving power of R ≈ 44,000, and covers a wavelength range from 385 to 906 nm. The spectra were extracted following Buchhave et al. (2010), and radial velocities were measured using a cross-correlation analysis against a template spectrum generated from a median combination of all TRES observed spectra (Quinn et al. 2012). We also make use of the TRES spectra to measure the atmospheric parameters of the host star via the Stellar Classifications Pipeline (SPC; Buchhave et al. 2012, 2014), finding an effective temperature of T_eff = 5288 ± 50 K, surface gravity $\mathrm{log}g$ of 4.62 ± 0.10, and bulk metallicity [M/H] of +0.16 ± 0.08. The projected broadening width v_broadening = 4.1 ± 0.5 km s⁻¹. The v_broadening here does not correct for macroturbulence, so the stellar rotational velocity $v\sin {i}_{\star }$ must be smaller than the reported value.

TOI-1268 exhibits significant photometric variability due to its youth, and as such we also expect significant jitter in the radial velocities (RVs). The TRES RVs exhibit scatter at the 50 m s⁻¹ level with a typical RV precision at ∼30 m s⁻¹. The Lomb–Scargle periodogram of the TRES RVs detects neither the stellar rotation period nor the planetary orbital period due to sparse observations and entangled stellar activity and planetary signals.

We observed one transit of TOI-1268b with the extremely high precision NEID spectrograph (Halverson et al. 2016; Schwab et al. 2016) on the 3.5 m WIYN telescope at the Kitt Peak National Observatory (KPNO) in Tucson, AZ, USA. NEID is a fiber-fed (Kanodia et al. 2018), actively environmentally stabilized spectrograph (Stefansson et al. 2016; Robertson et al. 2019) with a resolution of R ≈ 110,000 and a wavelength coverage of 380 nm to 930 nm. The observation was taken on UT 2021 May 4 during the transit of TOI-1268b, and covered about 1.5 hr baseline before the transit. We used an exposure time of 8 minutes for each observation, and in total obtained 37 spectra. The spectra were extracted and radial velocities were reduced by the NEID standard data reduction pipeline NEID-DRP v1.1.2, ⁴⁸ which derives cross-correlation-based RVs (we used CCFRVMOD data produced by the pipeline), and separately by the SERVAL pipeline (Zechmeister et al. 2018), which derives RVs using the reconstructed stellar template from observations (see Section 3.1 in Stefansson et al. 2021 for the NEID customization). The two pipelines derive similar RVs that are consistent within 1σ uncertainties except a few data points. The achieved median, photon-limited RV precision for both pipelines is ∼5.8 m s⁻¹. Reduced RVs are presented in Figure 5.

To directly measure the Doppler shadow cast by the planet on the spectroscopic line profiles of the star, we perform a least-squares deconvolution (Donati et al. 1997) between the NEID spectra and a synthetic nonrotating spectral template. The synthetic template is generated using a set of ATLAS-9 atmosphere models (Castelli & Kurucz 2004) at the stellar atmosphere parameters of TOI-1268. The line profiles are computed for each order of an observation, and weighted-average combined into a single line profile per epoch. Section 4.2 describes the modeling of the line profiles to retrieve the planetary orbital obliquity, and Figure 5 shows the tomographic shadow of the planetary transit.

We use astroARIADNE ⁴⁹ to model the spectra energy distribution (SED) of the star. We use the three Gaia band magnitudes, three 2MASS band magnitudes, and the four WISE band magnitudes in the modeling, and use the Gaia parallax, T_eff, and [M/H] derived from the TRES spectra as our priors. The uncertainties of the photometry bands are inflated following methods described in EXOFASTv2 (Eastman et al. 2013, 2019). We use the PHOENIX models and MIST isochrones in the SED modeling. The best-fitted stellar parameters and their uncertainties are R_⋆ = 0.86 ± 0.02 R_⊙, M_⋆ = 0.9 ± 0.13 M_⊙, T_eff = 5257 ± 40 K, $\mathrm{log}g=4.52\pm 0.07$, and [M/H] = +0.17 ± 0.06.

As discussed in Sections 2.1 and 2.3, the star exhibits a clear rotation signature in both the TESS and KELT photometry. We use a Gaussian Process (GP) model with a rotation kernel to infer the rotation period of TOI-1268. The rotation kernel is composed of two damped harmonic oscillators with the characteristic frequencies of 1/P and 2/P to model stellar variability at the rotation period itself and at harmonics. Five free parameters in the rotation term are the rotation period P, two quality factors Q₀ and dQ describing the damping timescales of each oscillator, and σ and f describing the amplitudes of each oscillator. We apply the kernel to the KELT, TESS, and TRES observations using the celerite2 package (Foreman-Mackey et al. 2017; Foreman-Mackey 2018). Transits are masked from TESS light curves. We run Markov Chain Monte Carlo (MCMC) to sample posteriors using the PyMC package (Salvatier et al. 2016). We sample four chains, each with 10,000 burn-in steps, 3000 draws, and use a target acceptance rate of 0.95. We assess the MCMC convergence using the Gelman–Rubin diagnostic ($\hat{R}\lt 1.01$ for convergence) and find all the inferred parameters have $\hat{R}\leqslant 1.001$. In Figure 2, we present the flux and RV variations predicted by GP models in blue curves and draw from the posteriors in light blue curves. The GP models perform well on predicting flux variations on TRES and TESS light curves, whereas perform they poorly on TRES RVs due to the sparse sampling on TRES data and complication from planetary signal. We test the GP models with and without TRES RVs and find similar rotation period posteriors. The inferred rotation period for TOI-1268 is P_rot = 10.84 ± 0.07 days. Combining the stellar rotation period and radius, the equatorial rotational velocity of TOI-1268 is v_rot =2π R_⋆/P_rot = 4.02 ± 0.10 km s⁻¹.

The inclination of the star i_⋆ can be inferred from the projected rotational velocity of the host star and its equatorial rotational velocity (Masuda & Winn 2020). Since we only know the projected broadening width (v_broadening = 4.1 ± 0.5 km s⁻¹), the true projected rotational velocity could be smaller and this difference could lead to an overestimation of the inclination. We apply priors on P_rot and R_⋆ from stellar fits, as well as a uniform prior on $\cos {i}_{\star }$, and infer i_⋆ = 76° ± 10° using the MCMC.

TOI-1268 does not belong to any known association based on a search of the BANYAN Σ catalog (Gagné et al. 2018). We used the Comove package ⁵⁰ (Tofflemire et al. 2021) to query Gaia EDR3 (Lindegren et al. 2021) and search for associations within 50 pc, and did not find any clear clustering in velocity space. We identified 95 candidate stars that are brighter than TESS magnitude of 13.5, and could be associated with TOI-1268, and use the Discrete Fourier Transform (DFT) clean algorithm in VARTOOLS (Hartman & Bakos 2016) to measure their rotation periods with TESS FFI light curves. The relation between the rotation period and effective temperature in comparison to clusters with well-determined age are shown in Figure 4(a). The effective temperatures were obtained from the TIC-v8 catalog (Stassun et al. 2019). While some of the candidates have rotation periods consistent with being young (dark gray crosses), many do not (light gray crosses). A full vetting of candidates would be required to use them to further refine the age of TOI-1268. From the rotation period only, TOI-1268's age is most likely to be between the 120 Myr Pleiades (colored in blue) and the 2.5 Gyr NGC 6819 (colored in purple).

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For K-type stars like TOI-1268, lithium is expected to be depleted when the star is older than Praesepe/Hyades ages (e.g., Boesgaard et al. 2016; Cummings et al. 2017). However, Li I 6707.8 nm is clearly detected in both the TRES and NEID spectra. The equivalent width (EW) is measured to be 0.069 ± 0.011 Å from the TRES data. In Figure 2(b), we compare the Li abundance of TOI-1268 to Pleiades (120 Myr), Group X (250 Myr; Netwon et al. 2022, in preparation), and Praesepe (670 Myr) clusters. The Li measurements for Pleiades and Praseape are obtained from Zhou et al. (2021), where the spectra were obtained as part of the long-term radial-velocity surveys on the TRES spectrograph by Quinn et al. (2012) and Quinn et al. (2014). The Li abundance of TOI-1268 is richer than Praesepe and in agreement with Pleiades. Combining Li and stellar rotation period information, TOI-1268's age is likely between Pleiades and Praesepe clusters, i.e., 120–670 Myr.

We use BAFFLES (Stanford-Moore et al. 2020), a package that uses empirically determined relations to compute age posteriors for field stars from measurements of $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }$ calcium emission or lithium equivalent width absorption and B − V color to estimate the age of TOI-1268. From the TRES spectra, we measure $\mathrm{log}{R}_{\mathrm{HK}}^{{\prime} }\,=\,$−4.3 ± 0.19. Since the B − V color from the catalog has relatively large error bars, we use MIST isochrones and the best-fitted SED of TOI-1268 to derive a more accurate B − V color of 0.83 ± 0.03. The ages independently estimated from the calcium emission and lithium lines are consistent with each other. The calcium age posterior gives 130 Myr–1.4 Gyr in the 1σ credible interval (CI). The lithium age posterior gives 220–500 Myr in the 1σ CI. The combined posterior estimates the age of TOI-1268 is 190–370 Myr in the 1σ CI (or 76–600 Myr in the 2σ CI).

Using the above information, we conclude that the rotation, lithium abundance, and activity index all give consistent ages, and confirm the youth of TOI-1268.

We use a quadratic limb-darkening transit model (Mandel & Agol 2002; Kipping 2013) plus a rotational Gaussian Process kernel (Foreman-Mackey et al. 2017; Foreman-Mackey 2018) to model the transit light curves and the rotational modulation introduced by stellar activity. We perform the light-curve fit using the TESS 2 minute cadence data only and also the TESS data jointly with two ground-based transits described in Section 2.2. To reduce the computational time, we trim the TESS light curves to roughly 3 times the transit duration before the ingress and after the egress. No transit-timing variations on TOI-1268b are detected in a preliminary light-curve fit. Because of that, we directly model the orbital period P and the reference mid-transit time T_C . Free parameters in our model include $\left\{{\rho }_{\mathrm{circ}},b,{r}_{{\rm{p}}}/{r}_{\star },P,{T}_{C}\right\}$, the quadratic limb-darkening parameter $\left\{{u}_{0},{u}_{1}\right\}$, and GP parameters for the rotational kernel (see Section 3). We take the GP parameters derived from the out-of-transit TESS data as priors. Here we model ρ_circ, the stellar density of the host star assuming zero eccentricity, and later compare it to the ρ_⋆ from isochrone fitting to infer the planet's eccentricity e and argument of periapse ω. To jointly model the TESS and ground-based light curves, we use an independent pair of limb-darkening parameters for each filter and separate GP models for TESS and ground-based transits due to different cadences.

In Figure 1, we present the detrended TESS and ground-based transit light curves from a joint fit. The orbital period and transit ephemeris of the planet are tightly constrained. A summary of planetary parameters can be found in Table 1.

Table 1. Median Values and 68% Credible Intervals for the Stellar and Planetary Parameters of the TOI-1268 (TIC-142394656) System

Parameter	Units	Values
Stellar Parameters
M*...	Mass (M ⊙ )...	0.90 ± 0.13
R*...	Radius (R ⊙ )...	0.86 ± 0.02
ρ*...	Density (cgs)...	1.98 ± 0.33
$\mathrm{log}g$...	Surface gravity (cgs)...	4.52 ± 0.07
Teff...	Effective temperature (K)...	5257 ± 40
[M/H]...	Bulk metallicity (dex)...	+0.17 ± 0.06
Prot...	Stellar rotation period (day)...	10.84 ± 0.07
vrot...	Equatorial velocity (km s−1)...	4.02 ± 0.10
G...	Gaia G magnitude (EDR3; Lindegren et al. 2021)...	10.690 ± 0.001
BP...	Gaia BP magnitude (EDR3; Lindegren et al. 2021)...	11.131 ± 0.002
RP...	Gaia RP magnitude (EDR3; Lindegren et al. 2021)...	10.089 ± 0.001
J...	2MASS J magnitude (Skrutskie et al. 2006)...	9.40 ± 0.02
H...	2MASS H magnitude (Skrutskie et al. 2006)...	9.034 ± 0.023
KS...	2MASS KS magnitude (Skrutskie et al. 2006)...	8.911 ± 0.014
WISE1...	WISE1 magnitude (Wright et al. 2010)...	8.886 ± 0.023
WISE2...	WISE2 magnitude (Wright et al. 2010)...	8.941 ± 0.019
WISE3...	WISE3 magnitude (Wright et al. 2010)...	8.846 ± 0.026
WISE4...	WISE4 magnitude (Wright et al. 2010)...	8.878 ± 0.411
Planetary Parameters (transit+RM-effect joint model)
P...	Period (days)...	${8.157728}_{-0.000005}^{+0.000005}$
TC ...	Mid-transit time (BJD)...	${{\rm{2,458,703.5895}}}_{-0.0002}^{+0.0003}$
Rp /R⋆ ...	Planet–star radius ratio ...	${0.089}_{-0.001}^{+0.001}$
Rp ...	Radius (RJup) ...	${0.747}_{-0.018}^{+0.018}$
		With NEID-DRP RVs	With SERVAL RVs	With DT
ρcirc	Stellar density assuming the planet has a circular orbit (cgs)...	${1.437}_{-0.099}^{+0.051}$	${1.407}_{-0.094}^{+0.064}$	${1.450}_{-0.049}^{+0.033}$
a/R⋆...	Planet–star separation...	${17.164}_{-0.403}^{+0.199}$	${17.043}_{-0.387}^{+0.256}$	${17.215}_{-0.196}^{+0.131}$
a...	Semimajor axis (au)...	${0.0684}_{-0.0021}^{+0.0019}$	${0.0680}_{-0.0021}^{+0.0020}$	${0.0688}_{-0.0018}^{+0.0017}$
b...	Transit impact parameter ...	${0.191}_{-0.130}^{+0.149}$	${0.246}_{-0.128}^{+0.121}$	${0.135}_{-0.049}^{+0.065}$
i...	Inclination (°)...	${89.469}_{-0.413}^{+0.361}$	${89.318}_{-0.337}^{+0.354}$	${89.552}_{-0.222}^{+0.165}$
∣λ∣...	Projected stellar obliquity (°)...	${13.623}_{-9.572}^{+14.145}$	${24.873}_{-13.134}^{+12.963}$	${39.997}_{-9.904}^{+7.223}$
$v\sin {i}_{\star }$...	Rotational line broadening (km s−1)...	${4.472}_{-0.392}^{+0.454}$	${4.303}_{-0.454}^{+0.553}$	${4.183}_{-0.231}^{+0.306}$
e...	Eccentricity...	${0.13}_{-0.13}^{+0.24}$	${0.13}_{-0.13}^{+0.27}$	${0.12}_{-0.12}^{+0.24}$
ω...	Argument of periapse (°)...	${210}_{-19}^{+25},{329}_{-25}^{+21}$	${212}_{-20}^{+25},{328}_{-26}^{+19}$	${210}_{-19}^{+24},{329}_{-27}^{+20}$
σRV...	Radial-velocity jitter (m s−1)...	${3.473}_{-2.243}^{+1.791}$	${2.255}_{-1.683}^{+1.861}$	⋯
vmacro...	Macroturbulence of the host star (km s−1)...	⋯	⋯	${2.020}_{-0.131}^{+0.134}$
u0,TESS ...	Quadratic limb-darkening coefficient 0...	${0.235}_{-0.074}^{+0.072}$	${0.236}_{-0.077}^{+0.076}$	${0.239}_{-0.074}^{+0.071}$
u1,TESS ...	Quadratic limb-darkening coefficient 1...	${0.473}_{-0.142}^{+0.150}$	${0.462}_{-0.146}^{+0.154}$	${0.478}_{-0.137}^{+0.151}$

Note. Due to the asymmetric and bimodal shapes of the eccentricity and argument of periapse posteriors, instead of reporting their medians and 68% credible intervals, we report their 68% highest posterior density intervals.

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We use the NEID RM-effect signals to infer the stellar obliquity from two separate approaches: (1) model the RV anomalies reduced by the NEID-DRP v1.1.2 and the SERVAL pipelines (see Section 2.6 for the description of the pipelines), and (2) directly model the planetary shadow extracted from the spectra using the planet's transit (i.e., DT; Collier Cameron et al. 2010). For both approaches, we jointly model the RM-effect signals with the TESS transit light curves. Doing so allows one to solve the complex covariances between the impact parameter b, the projected stellar obliquity λ, and the projected stellar rotation velocity $v\sin {i}_{\star }$.

For the RV fits, our model includes parameters mentioned in the transit model (Section 4.1), and also the λ, $v\sin {i}_{\star }$, and an RV jitter term σ_RV as free parameters. We place uniform priors on these parameters. ⁵¹ The RV anomaly due to the transit is modeled using the starry package (Luger et al. 2019), which takes the analytical expression of the radial velocity of a stellar disk (Short et al. 2018) and converts the polynomials to spherical harmonic coefficients. The calculated RVs do not account for macroturbulence or instrumental broadening. We incorporate starry into exoplanet, and build and sample the joint model using the PyMC. We also add a quadratic trend to model the baseline RV trend introduced by either the planet's orbit or stellar activity. In total, we sample four chains, each with 20,000 tuning steps and 5000 draws. A target acceptance rate of 0.95 was used. All four chains are confirmed to be converged and the inferred parameters have $\hat{R}\leqslant 1.001$. The quadratic coefficients are consistent with zeros.

For the DT fit, similar to the RV fits, we incorporate the DT data into exoplanet, and build and sample the joint model using the PyMC. At each observing time t, we calculate the planet's position on the stellar disk, assuming the star rotates as a rigid body, and identify the stellar velocity channels being blocked by the planet, v_t . To model the planetary shadow, we use a Gaussian distribution that centers at v_t and has a standard deviation of $\sqrt{{v}_{\mathrm{res}}^{2}+{v}_{\mathrm{macro}}^{2}}$, where ${v}_{\mathrm{res}}$ is the velocity resolution set by the spectrograph resolution and v_macro is the macroturbulence velocity determined by the host star. The planetary shadow is further scaled with the photometric flux at time t and normalized by the total stellar velocity flux over the planetary velocity flux. We sum up the likelihoods of the DT signals at all observing times, and infer the planet's orbital orientation, along with its TESS transit light curves. The PyMC setup and the convergence test are the same as the ones described in the RV fits.

In Figures 5(a) and (b), we present the NEID RVs reduced by the NEID-DRP v1.1.2 and SERVAL pipelines, and their corresponding RM-effect models and uncertainties. In Figure 5(c), we show the DT data (left panel), the best-fit model (middle panel), and the residual of the data after subtracting the model (right). In all three inference models, the projected stellar obliquity ∣λ∣ posteriors extend from 0° to 60° (3σ CIs). A polar or retrograde solution of TOI-1268 system can be ruled out. However, the differences in ∣λ∣ posteriors from three models are still noticeable. In Table 1, we summarize the fitting parameters. The NEID-DRP RVs suggest an aligned system with $| \lambda | ={14}_{-10}^{+14\circ }$, whereas the SERVAL RVs and DT data suggest a slightly misaligned system (i.e., $| \lambda | ={25}_{-13}^{+13\circ }$ for SERVAL and $| {\lambda }_{\mathrm{DT}}| ={40}_{-10}^{+7\circ }$ for DT). A possible explanation for the high stellar obliquity inferred by the DT model is the stellar obliquity and impact parameter degeneracy: low impact parameters (b ∼ 0.1) correlate to high stellar obliquities (λ ∼ 40°), and slightly higher impact parameters (b ∼ 0.2) correlate to lower stellar obliquities (λ ∼ 20°). Since a low impact parameter solution is suggested by the DT model (b = 0.135 + 0.065 − 0.049), we derive the high stellar obliquity solution. The DT inferred impact parameter is still consistent with the one from transit-only fit (b = 0.2 ± 0.1). Breaking the degeneracy between the impact parameter and stellar obliquity will be the key step toward precise stellar obliquity measurement in future observations.

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Finally, we use the inclination of the host star (i_⋆ = 76° ± 10°) and the projected stellar obliquity to estimate the true stellar obliquity ψ of TOI-1268. Using the spherical law of cosines, $\cos \psi =\cos | \lambda | \sin i\sin {i}_{\star }+\cos i\cos {i}_{\star }$, where i is the orbital inclination, we get ψ = 227 ± 105 for NEID-DRP RVs, ψ = 304 ± 111 for SERVAL RVs, and ψ = 40° ± 10° for the DT signal.