TESS Hunt for Young and Maturing Exoplanets (THYME). VII. Membership, Rotation, and Lithium in the Young Cluster Group-X and a New Young Exoplanet

TOI 2048 was observed in sectors 16 (2019 September 12 to 2019 October 6) and 23–24 (2020 March 19 to 2020 May 12) during TESS's second year of operations. Data from the Quick Look Pipeline (QLP; Fausnaugh et al. 2020; Huang et al. 2020a, 2020b) were used to initially identify the planet candidate, which was announced via the alerts as TOI 2048.01 (Guerrero et al. 2021).

We used 2-minute-cadence data produced by the Science Processing and Operations Center (SPOC; Jenkins et al. 2016) and 30-minute-cadence data produced by a custom pipeline operating on the full-frame images (FFIs). We used lightkurve (Lightkurve Collaboration et al. 2018) to access SPOC data. The SPOC pipeline produces two data products (Jenkins 2015; Jenkins et al. 2016). The simple aperture photometry (SAP data; Twicken et al. 2010; Morris et al. 2020) includes calibration, extraction, and background subtraction. The pre-search data conditioning simple aperture photometry (PDCSAP data; Stumpe et al. 2012, 2014; Smith et al. 2012) additionally removes instrumental systematics using cotrending basis vectors. A strong systematic arises from scattered light, which includes signals at 1 and 14 days.

The TESS data for TOI 2048 and Group-X members are strongly affected by systematics, and the PDCSAP corrections did impact the observed stellar rotation signal in some cases. The particular regime in which this matters is stars with rotation periods (P_⋆) between about 6 and 12 days, for which PDCSAP corrections can modify the rotational variability such that the apparent period is P_⋆/2. This is illustrated in Figure 1. Similar behavior was noted in one target from Magaudda et al. (2022); our investigation shows that this is prevalent, at least in some sectors. Additionally, the PDC data for TOI 2048 shown in Figure 1 display some abrupt jumps not seen in the SAP data. We therefore used SAP data instead of PDCSAP data for our analyses.

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The SAP data have not been corrected for crowding, but crowding is negligible for this source (0.5% in the most contaminated sector). Data from these sectors were also processed with the original version of the sky background algorithm, which can result in significant overcorrection of the background flux. We analyzed the target pixel files for this star and found that the bias in the transit depth is less than 0.6% in all three sectors.

For stars that were not observed in 2-minute-cadence mode, we extracted light curves from the TESS FFIs using a custom pipeline. We downloaded FFI cutouts around the locations of candidate Group-X members using the TESSCut ²⁹ service (Brasseur et al. 2019). Following Vanderburg et al. (2019), we extracted light curves from 20 different stationary photometric apertures and selected the one that resulted in the lowest photometric scatter. We then performed a systematics correction on the selected light curve by decorrelating the raw aperture photometry with the mean and standard deviation of the spacecraft quaternion time series within each exposure and the SPOC PDC cotrending basis vectors. Because the Group-X members are young and tend to have high amplitude and short-period stellar variability, we modeled the rotation signals with basis splines with breakpoints spaced every 0.3 days.

We observed three transits using the Las Cumbres Observatory (LCO; Brown et al. 2013) 1 m telescopes at the McDonald Observatory, Texas, USA, and Teide Observatory, Canary Islands. The observations on 2020 September 05 were obtained in the Pan-STARRS z-short band and those on 2021 March 03 and 2022 April 20 in Sloan $i^{\prime}$ band. The LCO BANZAI pipeline (McCully et al. 2018) was used the calibrate the data, and AstroImageJ (Collins et al. 2017) was used to extract the photometry using an aperture with radius 58. No sources within this distance were identified with our AO imaging or in Gaia data (Section 2.4). The 2021 March 03 observation suffered from an equipment problem that resulted in the dome or camera shutter rotating into the image throughout the observation and eventually obscuring the target star.

The 2020 September 05 data do not contain sufficient out-of-transit baseline to constrain the timing or depth of the transit itself, but they demonstrate that neighboring stars do not show clear eclipses that could be responsible for the signal associated with TOI 2048.01. The 2022 April 20 data firmly rule out nearby eclipsing binaries in stars within 25 of the target over a −2.1σ to + 2.7σ observing window relative to the TESS sector 24 QLP ephemeris. The light curves of all neighboring stars had rms less than 0.2 times the depth required in the respective star to produce the detection in the TESS aperture. The target starlight curve shows a tentative detection of a ∼1000 ppm ingress arriving roughly 24 minutes (0.5σ) late relative to the nominal TESS sector 24 QLP ephemeris. Given the ingress-only coverage, we consider the timing of the apparent ingress detection tentative and do not further consider it in the analyses in this paper. The 2021 March 03 data show an apparent short, deep event; the latter part of the 2021 March 03 data were not used owing to data issues. The apparent event is inconsistent with both the TESS transit and the data from 2020 September 05 and 2022 April 20 and coincides with a similar event seen on a neighboring star. We conclude that the event seen on 2021 March 03 is likely caused by the equipment problem and is not astrophysical. All ground-based follow-up light curves are available on ExoFOP. ³⁰

We obtained four spectra with the Tillinghast Reflector Echelle Spectrograph (TRES; Fűrész et al. 2008). TRES is on the 1.5 m Tillinghast Reflector at Fred Lawrence Whipple Observatory (FLWO), Arizona, USA. TRES has a resolution of around 44,000 and covers 3900–9100 Å. The signal-to-noise ratio (S/N) of our spectra is about 25. Spectra were timed to coincide with opposite quadratures of the candidate planetary signal. The spectra were obtained between 2020 August and 2021 September. The spectra were extracted as described in Buchhave et al. (2010). These data are available on ExoFOP (ExoFOP 2019). (See footnote 33.)

We obtained high-resolution visible spectra of 33 candidate members of Group-X (including TOI 2048) with the Tull coudé spectrograph on the 2.7 m Harlan J. Smith telescope at McDonald Observatory (Tull et al. 1995). The Tull spectrograph is a cross-dispersed echelle spectrograph that covers a wavelength range from ∼3400 to 10000 Å at a resolution of roughly 60,000. A total of 47 spectra were taken between 2020 July and 2021 May, with 11 targets having multiple observations taken (to increase S/N or investigate possible binarity). Exposure times ranged from 300 to 1500 s, chosen to achieve sufficient S/N to measure the equivalent width (EW) of the Li λ6708 spectral feature. Spectra were reduced using a custom Python implementation of standard reduction procedures. After bias subtraction, flat-field correction, and cosmic-ray rejection, we extract 1D spectra from the 2D echellograms using optimal extraction. We derive wavelength solutions using a series of ThAr lamp comparison observations taken throughout each observing night. Finally, we fit the continuum using an iterative B-spline to produce flux-normalized spectra. The median S/N in the spectral order that contains the Li λ6708 feature is 45.

We measured RVs from the Tull coudé spectra by computing spectral-line broadening functions (BFs) using the Python implementation saphires (Rucinski 1992; Tofflemire et al. 2019). The BF is computed from the deconvolution of an observed spectrum with a narrow-lined template and effectively reconstructs the average stellar absorption-line profile in velocity space. As templates, we used Phoenix spectra (Husser et al. 2013) that are matched to the stellar effective temperature estimated from the Gaia G_BP − G_RP color. We computed BFs for the 25 spectral orders with minimal telluric contamination, covering a wavelength range from 4730 to 8900 Å. We fit the BFs with a Gaussian to determine the peak location and thus the RV. We correct for barycentric motion using barycorrpy (Kanodia & Wright 2018), which implements the formalism of Wright & Eastman (2014). We then iteratively combine the RVs from the individual orders. At each iteration, we compute the mean, take the error in the mean as the uncertainty, and then reject orders with RVs > 3σ from the average and recomputing the combined RV. No exposure uses fewer than 22 of the 25 orders. For the 11 objects with multiple observations, we adopt the weighted average and error as the RV and its uncertainty.

Two targets (Gaia EDR3 1601557162529801856 and Gaia EDR3 1615954442661853056) have double-peaked BFs, which indicates that they are SB2s, and were each observed twice. For the former, we compute the weighted average and error for each exposure using the individually fit RV peaks to get the systemic RV, and then we adopt the average systemic RV across both observations. For the latter, we adopt the RV computed from the second observation as the two SB2 components are overlapping, which should provide the systemic RV.

The RVs from our high-resolution optical spectra are used as part of our Li measurements (Section 4.5) but are not included in our assessment of Group-X membership. They are included in Table 7.

We observed TOI 2048 on 2021 January 23 UT with the Speckle Polarimeter (Safonov et al. 2017) on the 2.5 m telescope at the Caucasian Observatory of Sternberg Astronomical Institute (SAI) of Lomonosov Moscow State University. SPP uses Electron Multiplying CCD Andor iXon 897 as a detector. The atmospheric dispersion compensator allowed observation of this relatively faint target through the wide-band I_c filter. The power spectrum was estimated from 4000 frames with 30 ms exposure. The detector has a pixel scale of 20.6 mas pixel⁻¹, and the angular resolution was 89 mas. We did not detect any stellar companions brighter than ΔI_C = 4 and 5.8 at ρ = 025 and 10, respectively, where ρ is the separation between the source and the potential companion.

We observed TOI 2048 on 2021 March 29 using the ShARCS camera on the Shane 3 m telescope at Lick Observatory, California, USA (Kupke et al. 2012; Gavel et al. 2014; McGurk et al. 2014). Observations were taken with the Shane adaptive optics system in natural guide star mode. We collected two sequences of observations, one with a K_s filter (λ₀ = 2.150 μm, Δλ = 0.320 μm) and one with a J filter (λ₀ = 1.238 μm, Δλ = 0.271 μm). A more detailed description of the observing strategy and reduction procedure can be found in Savel et al. (2020). We find no nearby stellar companions within our detection limits.

We observed TOI 2048 with PHARO (Hayward et al. 2001) at Palomar Observatory, California, USA, on 2021 February 23, behind the natural guide star AO system P3K (Dekany et al. 2013). PHARO has a pixel scale of 0025 pixel⁻¹ for a total FOV of ∼25''. We used the narrowband Brγ filter (λ_o =2.1686 μm, Δλ = 0.0326 μm). We used a standard five-point dither with steps of 5'', revisiting each position with an offset of 05 three times. The total integration time was 148 s.

We processed the AO data with a custom set of IDL tools that performed flat-fielding, sky subtraction, and dark subtraction. The images were combined, producing a combined image with a point-spread function (PSF) FWHM of 0098. No stellar companions were found within the 5σ detection limits, as determined in injection and recovery tests.

We observed TOI 2048 and two calibrator stars (HIP 77903 and TYC 3877-725-1) on 2020 June 30 at Keck Observatory using the NIRC2 adaptive optics camera and natural guide star adaptive optics. We observed each star with standard imaging, coronagraphic imaging, and nonredundant aperture masking in the $K^{\prime}$ filter (λ = 2.124 μm). All observations used the narrow camera, which has a pixel scale of 9.971 mas pixel⁻¹ (Service et al. 2016) and an FOV of 10''. For sensitivity at wide separations, we obtained some images with the partially transmissive 06-diameter coronagraph. For sensitivity to companions near and inside the diffraction limit, we also obtained interferograms using a nine-hole aperture mask placed in the reimaged pupil plane.

All images were dark-subtracted with mode-matched dark frames, linearized, flat-fielded, and screened for cosmic rays and known hot/dead pixels. Additionally, spatially coherent EM interference noise (which manifests as "stripe noise" and which otherwise dominates the faint-source flux limit) was subtracted from each quadrant using the median of the other three quadrants.

We estimated companion detection limits following the procedures of Kraus et al. (2016). To summarize, we create two residual maps tailored to identify wide-separation candidate companions amid read noise and scattered light and to identify close-in candidate companions amid speckle noise. The former is generally more sensitive at separations ≳05. We estimated the source detection limits as a function of projected separation as the contrast that would correspond to a +6σ outlier at any given radius.

We found no candidate detections above this limit for TOI 2048 or either calibrator, and hence no candidate companions within the NIRC2 FOV around each star (ρ ≲ 5''). We summarize the NIRC2 observations and detection limits in Table 1 and plot the contrast curve in Figure 2.

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We fit the SED by comparing spectral templates to the observed photometry, following the method detailed in Mann et al. (2016). The templates cover 0.4–2.3 μm, with gaps in regions of high telluric contamination (primarily H₂O bands in the NIR). To fill these gaps and extend the spectrum past the limits, we included BT-SETTL CIFIST atmospheric models (Baraffe et al. 2015) in the fit, which also provided an estimate of T_eff. To estimate the bolometric flux (F_bol), we took the integral of the resulting absolutely calibrated spectrum. We combined F_bol with the Gaia parallax to determine the total stellar luminosity (L_⋆). With T_eff and L_⋆, we calculated R_⋆ from the Stefan–Boltzmann relation. Reddening is expected to be small but potentially nonzero for a star at this distance, so we include extinction as part of the fit. To account for variability in the star, we added (in quadrature) 0.02 mag to the errors of all optical photometry. The resulting fit included six free parameters: the spectral template, A_V , three BT-SETTL model parameters ($\mathrm{log}\,g$, T_eff, and [M/H]), and a scaling factor that matches the model to the photometry. This scale factor is equivalent to ${({R}_{\star }/D)}^{2}$, which provides a separate estimate of R_⋆ based on the infrared flux method (IRFM; Blackwell & Shallis 1977). We show an example fit in Figure 3.

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The resulting fit yielded E(B − V) = 0.11 ± 0.07, T_eff =5185 ± 60K, F_bol = (9.64 ± 0.022) × 10⁻¹⁰ (erg cm⁻² s⁻¹), L_⋆ = 0.41 ± 0.04 L_⊙, and R_⋆ = 0.79 ± 0.04 R_⊙. The errors account for uncertainty due to template choice, measurement uncertainties in the photometry and parallax, and uncertainties in the filter zero-points and profiles (Mann & von Braun 2015). The IRFM fit gave a radius of 0.75 ± 0.02 R_⊙, consistent with the value from the Stefan–Boltzmann relation. We adopt the former as the more conservative radius for our analysis.

The TRES spectra were also used to derive stellar parameters using the Stellar Parameter Classification tool (SPC; Buchhave et al. 2010). SPC cross-correlates an observed spectrum against a grid of synthetic spectra based on the Kurucz atmospheric model (Kurucz 1992). SPC values were consistent with those derived above and additionally indicate near-solar metallicity ([m/H] = 0.02 ± 0.08).

To determine M_⋆, we interpolated the observed photometry onto a grid of solar-metallicity models from the Dartmouth Stellar Evolution Program (DSEP; Dotter et al. 2008). We fit for age, A_V , and M_⋆ within a Markov Chain Monte Carlo (MCMC) framework. An additional parameter (f, in mag) captured underestimated uncertainties in the data or models. For computational efficiency, we used a two-step interpolator to find the nearest age in the model grid and then performed linear interpolation in mass to obtain stellar parameters and synthetic photometry. This nearest-age approach can generate errors from the grid spacing, so we pre-interpolated the grid in age and mass using the isochrones package (Morton 2015). This also let us use a grid uniform in age. To redden model photometry, we used synphot (Lim 2020). We applied Gaussian priors on age (300 ± 50 Myr, based on the rotation analysis in Section 4.3) and T_eff (5185 ± 60 K, based on our SED fit). Other parameters had uniform priors.

This analysis yielded M_⋆ = 0.844 ± 0.017 M_⊙ and R_⋆ =0.742 ± 0.011 R_⊙. Repeating the analysis with the PARSEC isochrones (Bressan et al. 2012; Chen et al. 2015) yielded a mass of M_⋆ = 0.824 ± 0.013 M_⊙ and R_⋆ = 0.735 ± 0.009 R_⊙. The values are consistent with those derived using DSEP models, and both radii are consistent with our SED fit. The errors were likely underestimated, however, as model systematics dominate and we did not account for effects of metallicity or activity. For the mass, we adopted M_⋆ = 0.83 ± 0.03 M_⊙, with the larger errors more reflective of model disagreements seen in young eclipsing binaries (e.g., David et al. 2016; Kraus et al. 2017). We adopted the more empirical R_⋆ from the previous section.

We modeled the stellar rotation signal in the TESS photometry using Gaussian processes. As discussed in Section 2.1.1, we used the SAP light-curve data for this analysis. We masked the transits using a preliminary transit fit before proceeding.

We used the exoplanet package (Foreman-Mackey et al. 2021) for our model and pymc3 (Salvatier et al. 2016) for optimization. The stellar rotation capabilities for exoplanet are built on the Gaussian process package celerite2 (Foreman-Mackey et al. 2017; Foreman-Mackey 2018). We set up the model ³² with a nonperiodic component to describe trends in the data and a quasi-periodic component to describe the stellar rotation signal.

The quasi-periodic term ³³ is a mixture of two damped simple harmonic oscillators at P_⋆ and P_⋆/2. This mixture is characterized by the rotation period (P_⋆, sampled as $\mathrm{log}{P}_{\star }$), the amplitudes of two oscillators, and the quality factors of the two oscillators that characterized how damped they are. The amplitudes are characterized by the amplitude of the signal at P_⋆ (σ, sampled as $\mathrm{log}\sigma$) and the fractional amplitude of the signal at P_⋆/2 (f, such that the amplitude is f σ). The quality factors are characterized by the quality factor Q (Q, such that the quality factor of signal at P_⋆/2 is Q₂ = 1/2 + Q), and the difference in the quality factors between the signals (δ Q, such that the quality factor of signal at P_⋆ is Q₁ = 1/2 + Q + δ Q). These were sampled as $\mathrm{log}Q$ and $\mathrm{log}\delta Q$. Priors are listed in Table 4.

Table 4. Stellar Rotation Modeling Priors and Fits

Parameter	Prior			Fitted Value
	μ	σ	Limits
$\mathrm{log}{P}_{\star }/d$	8	1	⋯	${2.076}_{-0.025}^{+0.010}$
$\mathrm{log}a$	⋯	⋯	[−20,5]	${1.29}_{-0.13}^{+0.18}$
f	⋯	⋯	[0,1]	${0.65}_{-0.26}^{+0.22}$
$\mathrm{log}{Q}_{0}$	0	5	⋯	${1.3}_{-0.39}^{+0.44}$
$\mathrm{log}{\rm{\Delta }}Q$	0	5	⋯	${2.7}_{-3.6}^{+3.2}$
μflux (ppt)	0	0.5	⋯	0.17 ± 0.19

Note. Parameters that use Gaussian priors list the mean (μ) and standard deviation σ, while those that use uniform priors list the limits. The best-fitting values are the median and 68% confidence intervals of the posterior distribution.

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We first fit the data using maximum likelihood and clipped 5σ outliers from the model (with the standard deviation calculated as 1.48 times the median absolute deviation). We then refit the data using MCMC. We used 4000 tuning steps, used 4000 draws, and ran six chains. The Gelman–Rubin statistic ($\hat{R}$) was 1.0 for all parameters, and there were no divergences in the chains after tuning (which would indicate regions of parameter space that may not have effectively been explored), indicating that the MCMC converged.

Figure 4 shows our fit to the data for TESS sectors 23 and 24. Table 4 includes our best-fit values and errors. We measure a stellar rotation period of ${7.97}_{-0.19}^{+0.08}$ days.

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We measure the projected rotational velocity ($v\sin {i}_{\star }$) from the first three TRES spectra by fitting the co-added BFs from each epoch with a broadened absorption-line model (Gray 2008). The model is broadened by the TRES spectral resolution, rotational broadening, and macroturbulent velocity (v_mac). The computed BF profile is not sufficiently broad with respect to the TRES instrumental resolution to independently constrain the rotational and v_mac broadening components. As such, we estimate v_mac using the T_eff- and $\mathrm{log}g$-dependent relation derived by Doyle et al. (2014). For TOI 2048, this corresponds to 2.3 km s⁻¹. Adopting this value, we fit an average $v\sin {i}_{\star }$ value of 4.8 km s⁻¹ from the three epochs. Adding the standard deviation of our $v\sin {i}_{\star }$ measurements with the uncertainty of the v_mac relation in quadrature, we determine an uncertainty of 0.7 km s⁻¹.

The Doyle et al. (2014) v_mac relation is derived using a sample of field-age stars, and there may be an age dependence missing in the relationship that we have not accounted for. To validate our adopted value v_mac above, we perform the same measurement on the single Tull coudé spectrum we obtained (Section 2.3.2), finding a consistent $v\sin {i}_{\star }$ value. Due to the difference in their spectral resolutions, obtaining a consistent $v\sin {i}_{\star }$ measurement between these instruments provides independent confirmation that the adopted v_mac value is appropriate for TOI 2048.

We used the photometric rotation period, stellar radius, and $v\sin i$ to measure the inclination of the stellar spin axis to the line of sight. We sampled the posterior of $\cos {i}_{* }$ as in Newton et al. (2019), following the formalism from Masuda & Winn (2020) and using the Goodman & Weare (2010) affine-invariant MCMC sampler. We used the values and errors in Table 2 to set Gaussian priors on all parameters and imposed $0\leqslant \cos {i}_{* }\leqslant 1$. We took the median and 68% confidence intervals of the $\cos {i}_{* }$ posterior as the best value and error. Under the convention that i_⋆ < 90°, $i\,=\,{71}_{-14}^{+12}$. This is suggestive that the stellar rotation and the planetary orbital axes are misaligned; however, the uncertainties are large, and the 2σ confidence interval extends to 89°.

We used periodograms to identify rotation periods in every star from the Friends, Tang et al. (2019), or Fürnkranz et al. (2019) lists with TESS-SAP or FFI pipeline data. We used the Lomb–Scargle periodogram (Lomb 1976; Scargle 1982) from astropy (Astropy Collaboration 2018) as implemented in starspot. ³⁸ We considered rotation periods out to a maximum of 18 days and selected the highest peak in the periodogram as the candidate period. Some light curves still displayed significant long-term trends. We did not remove these systematics; however, if the highest peak in the periodogram is a result of the long-term trend and a clear candidate signal is seen in the periodogram at a shorter period, we adjusted the maximum period searched to zero in on the candidate signal. Additionally, TIC 229668649's periodogram peaked just beyond 18 days, and we extended the search to 20 days for this star. Based on visual inspection of the periodogram, the light curve, the phased light curve, and the autocorrelation function, we identified stars with photometric modulation.

As discussed in Section 2.1.1, the PDCSAP pipeline impacts a subset of the stellar rotation signals seen in the stars we analyzed, motivating our use of SAP data over PDCSAP data. The majority of stars with periods above 6 days (G_BP −G_RP > 1) were affected by this systematic, such that when using PDCSAP data there were two apparent gyrochrone sequences for the cluster with a break at G_BP − G_RP ∼ 1. Using rotation periods derived from SAP data or our FFI data, the Group-X candidate members follow a single gyrochrone.

Using TESS data and without the quality checks described below, we detect periods in 133 stars and candidate periods in an additional 30. We also detected two eclipsing binaries (TIC 441702640 and TIC 165407465) and a star with δ Scuti pulsations (TIC 137834492; Section 4.7).

Rotation period detection with TESS fell off significantly for T > 14; since the Friends include a significant number of stars fainter than this, the following discussion is limited to brighter stars. We detected secure periods in the TESS data alone for 61% ± 9% of Friends, 70% ± 8% of Tang et al. candidate members, and 68 ± 9% of Fürnkranz et al. candidate members. In comparison, the fraction of stars with detected rotation periods of P_rot < 18 days in the Kepler field—which we generally expect to have better sensitivity, especially to longer periods—is 14% (McQuillan et al. 2014) or 16% (Santos et al.2021).

Considering the Friends, we saw a strong trend in period detection as a function of tangential velocity offset: the detection rate in Friends with ${\rm{\Delta }}{v}_{\tan }\lt 2$ km s⁻¹ was 77% ±13% and drops off to 42% ± 10% at larger ${\rm{\Delta }}{v}_{\tan }$. This is expected because TOI 2048 lies at one extreme of Group-X. A one-third detection rate is still higher than one would expect for a field population, and the detection rate did not continue to fall off at the largest ${\rm{\Delta }}{v}_{\tan }$, where stars are less likely to be members. We identified a candidate small coeval association distinct from, but nearby to, Group-X, which is contributing to period detections at larger ${\rm{\Delta }}{v}_{\tan }$. We call this candidate association MELANGE-2, and we discuss it further in Appendix.

The $21^{\prime\prime}$ pixel size means that multiple Group-X members occasionally fall on the same pixel. We searched for neighboring members that are relatively bright (G_neighbor <G_target + 1) and within two TESS pixels and flagged them and the neighbors as blended. We then inspected each blend case: (a) when the stars were similar in color and brightness, we rejected the period for both stars; (b) when the brighter star is an A or early F star and the fainter star is solar-type, we rejected the period for the early-type star and assigned it to the solar-type star; and finally, (c) when the brighter star is solar-type and significantly brighter than the other, we rejected the period for the fainter star and assigned it to the brighter solar-type star.

We also measured rotation periods using data from ZTF (Bellm et al. 2019, 2019). Following the Curtis et al. (2020) procedure developed for archival Palomar Transient Factory data, we downloaded 8' × 8' ZTF r-band image cutouts from the NASA/IPAC Infrared Science Archive (IRSA) and used 6'' circular aperture photometry to extract light curves for each target and a selection of nearby reference stars, which are used to refine the target light curves. Rotation periods were determined using a similar Lomb–Scargle approach as was applied to the TESS data, except that the maximum period searched was set to 100 days (possible given ZTF's longer seasonal baseline). In all, we measured periods for 78 stars from ZTF data. Of these, 68 have periods or candidate periods also measured from TESS, and the periods agreed for all but five stars; in these cases, the discrepancies were due to multiple periods being detected in TESS or because the stars are eclipsing binaries. Checking TESS periods with ZTF is important because the short 27-day baseline for a single TESS sector, with a large midsector data gap, makes those light curves susceptible to half-period harmonic detection, especially as periods exceed 10 days. The remaining ZTF rotators missed by TESS are typically fainter and/or more slowly rotating. Combining TESS and ZTF thus allows for a more complete rotational census for a coeval population such as Group-X.

Periods for stars measured only from TESS (88 stars) or ZTF (18 stars) were adopted as the final period. We merged the TESS and ZTF periods by averaging stars with periods from both surveys that agreed to within 15%. In the few conflicting cases, we reviewed the light curves for all TESS sectors and ZTF seasons and decided what the most likely period was (e.g., when one survey found 1/2 the period of the other, we opted for the double period because the 1/2 period case is often caused by a temporary symmetric distribution of starspots around the stellar surface).

With MOLUSC, we explore the range of possible bound companions to the planet host that could produce the observed transit. To briefly summarize, MOLUSC simulates realistic stellar companions following a physically motivated distribution of physical parameters (period, mass ratio, etc.). For each companion, it compares the expected signal to the observed high-resolution imaging, RVs, and limits from Gaia imaging and astrometry. Companions are eliminated if they should have been recovered in the data, and they survive if not.

For TOI 2048, we generated one million companions, using all available constraints from Section 2 and the stellar parameters from Section 3.1. Of the generated companions, 8.5% survive the comparison. Most are stars that are on wide orbits and happen to be behind the target at the moment of observations, brown dwarfs that are below our imaging detection thresholds, or close-in objects on face-on orbits that elude the velocity constraints.

None of the surviving companions transit the host at the detected period, ruling out the possibility of an eclipsing binary. A hierarchical binary cannot be ruled out, but it is highly disfavored. Assuming that the deepest possible eclipse is 50%, none of the companions below ≃ 0.3 M_⊙ (q ≃ 0.37) can reproduce the transit. Only about 10% of the surviving companions pass this criterion (0.85% of the original generated sample).

With TRICERATOPS, we statistically validate TOI 2048 b. TRICERATOPS leverages Gaia DR2 and the TIC to consider false-positive probabilities for nearby stars, many of which are blended with the target star owing to TESS's large pixel size. TRICERATOPS considers contamination of the target star aperture from stars within 10 pixels of the target that are listed in the TIC. The false-positive scenarios that are modeled as possible origins of the putative transit signal include both bound stellar companions and background stars, as well as both eclipsing binaries and transiting planets other than the planet under consideration. We used the phase-folded 2-minute-cadence data detrended with the Gaussian process stellar variability model, which we bin for computational ease.

Background eclipsing binaries represented the most likely false positive for our data, so the limits on background sources from AO imaging were an important constraint. Our primary constraint was the NIRC2 contrast curve. The ΔK = 8.3 quoted at 2'' in Table 1 applies out to 5'', since from 2'' to 5'' background noise is the limiting magnitude. Beyond 5'', Gaia provided strong constraints. As there are no stars within 44'' of TOI 2048 in Gaia, we adopted ΔK = 15 as the contrast limit from 5'' to 25''. At separations below 015, we approximated the RUWE limits as ΔK = 0 at 003 and ΔK = 3 at 008.

We used TRICERATOPS to simulate the possible scenarios that could produce the observed transit, applying the composite K-band contrast curve as a constraint. We created 10⁶ realizations and calculated the false-positive probability (FPP) and nearby false-positive probability (NFPP). The NFPP describes the probability that a false-positive scenario arises owing to a known nearby star. We then repeated the experiment 10 times and took the mean and standard deviation of the resulting rates. NFPP is 0. The FPP from 10 trials is 0.0011 ± 0.0001.

Giacalone et al. (2021) established criteria of FPP < 0.015 and NFPP < 10⁻³; thus, we consider TOI 2048 b statistically validated.