TOI-4600 b and c: Two Long-period Giant Planets Orbiting an Early K Dwarf

TOI-4600 (TIC 232608943; V = 12.6) was observed by TESS Cameras 3 and 4 for 20 sectors to date. It was observed in only the Full Frame Image (FFI) observations from Sector 14 to Sector 19 (UT 2019 July 18 to UT 2019 December 24) and Sector 21 to Sector 26 (UT 2020 January 21 to UT 2020 July 4). It was then observed in 2 minute cadence in Sectors 40 and 41 (UT 2021 June 24 to UT 2021 August 20), Sectors 47–49 (UT 2021 December 30 to UT 2022 March 26), and Sectors 51–53 (UT 2022 April 22 to UT 2022 July 9). TESS observations were interrupted between each of the 13.7 day long orbits of the satellite when data were downloaded to Earth. The TESS Science Processing Operations Center (SPOC) pipeline (Jenkins et al. 2016) at NASA Ames Research Center calibrated the FFIs and processed the 2 minute data, producing two light curves per sector called Simple Aperture Photometry (SAP) and Presearch Data Conditioning SAP (PDCSAP; Smith et al. 2012; Stumpe et al. 2012, 2014), the latter of which is corrected for instrumental signatures, screened for outliers, and corrected for crowding effects. The TESS-SPOC pipeline (Caldwell et al. 2020) extracted photometry from the SPOC-calibrated FFIs.

TOI-4600 b was first identified as a planet candidate over a year prior to becoming a TESS Object of Interest (TOI 4600.01; Guerrero et al. 2021), by the TESS Single Transit Planet Candidate Working Group (TSTPC WG). The TSTPC WG focuses on searching light curves produced by the Quick-Look Pipeline (QLP; Huang et al. 2020) for single transit events, and validating and/or confirming those that are true planets, with the aim of increasing the yield of intermediate-to-long-period planets found by TESS (Villanueva & Dragomir 2019; Huang et al. 2020).

The TSTPC WG identified two ∼0.5% deep transits, one in Sector 16 and one in Sector 22 for TOI-4600 b. Further inspection of the QLP light curve revealed additional transits in Sectors 19 and 25, confirming the period to be 82.69 days (Figure 1). It also revealed an additional ∼1% deep transit in Sector 17, which we designate TOI-4600 c. 2 minute cadence data from the Extended Mission have revealed additional transits for both planets, including a transit in Sector 53 of nearly identical depth and duration as that of the transit of TOI-4600 c in Sector 17. We consider this transit in Sector 53 to also be of TOI-4600 c, allowing us to constrain the orbital period to only two possible values, as detailed in Section 3.2. The TESS apertures for TOI-4600 include two neighbors with ΔG < 8 (Figure 2), although both are 6 mag fainter than TOI-4600 and thus too faint to be the sources of the transits. We use vetting (Hedges 2021) to rule out centroid offsets for the transits of both TOI-4600 b and c, as detailed in Section 3.3.

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The TESS pixel scale is ∼21'' pixel⁻¹, and photometric apertures typically extend out to roughly 1', which generally results in multiple stars blending in the TESS aperture (Figure 2). We conducted ground-based photometric follow-up observations of TOI-4600 as part of the TESS Follow-up Observing Program (TFOP; Collins 2019) ³¹ Sub Group 1 to attempt to rule out or identify nearby eclipsing binaries (NEBs) as potential sources of the TESS detection, measure the transit-like event on target to confirm the depth and thus the TESS photometric deblending factor, and refine the TESS ephemeris. We used TESS Transit Finder, which is a customized version of the Tapir software package (Jensen 2013), to schedule our transit observations.

2.2.1. Las Cumbres Observatory

2.2.2. Wendelstein Observatory

2.2.3. Kotizarovci Observatory

2.2.4. Whitin Observatory

We obtained nine reconnaissance spectra of TOI-4600 using the Tillinghast Reflector Echelle Spectrograph (TRES) on the 1.5 m Tillinghast telescope at the Fred L. Whipple Observatory (FLWO) on Mt. Hopkins in Arizona (Fűrész 2008). The nine spectra span dates from 2021 May 31 to 2022 September 28, spanning the full orbit of TOI-4600 b and from phase 0 to 0.5 for TOI-4600 c, and range in signal-to-noise ratios from 21.9 to 35.4 (Table 1). TRES spectra were extracted using procedures outlined in Buchhave (2010) and a multiorder relative velocity analysis was then performed by cross-correlating, order by order, the strongest observed spectrum as a template, against all other spectra. We use the Stellar Parameter Classification (SPC) tool to derive spectroscopic parameters (T_eff, $\mathrm{log}\,g$, $v\,\sin \,i$, and [m/H]) for each spectrum (Buchhave et al. 2012). We use the spectroscopic parameters to determine the physical parameters of the host star, as described in Section 3.1. We also use the RVs to rule out brown dwarf and stellar mass companions as the sources of the transits, as detailed in Section 3.3.

As part of our standard process for validating transiting exoplanets to assess the possible contamination of bound or unbound companions on the derived planetary radii (Ciardi et al. 2015), we observed TOI-4600 with a combination of high-resolution resources including near-infrared (NIR) adaptive optics (AO) imaging at Palomar Observatory. While the optical observations tend to provide higher resolution, the NIR AO tend to provide better sensitivity, especially to lower-mass stars. The combination of the observations in multiple filters enables better characterization for any companions that may be detected. The Palomar Observatory observations of TOI-4600 were made with the PHARO instrument (Hayward et al. 2001) behind the natural guide star AO system P3K (Dekany et al. 2013) on 2021 June 19 and 2021 June 21 in a standard five-point quincunx dither pattern with steps of 5'' in the narrow-band Brγ filter (λ_o = 2.1686 μm; Δλ = 0.0326 μm). Each dither position was observed three times, offset in position from each other by 05 for a total of 15 frames, with an integration time of 29.7 s per frame, respectively for total on-source times of 445.5 s. PHARO has a pixel scale of 0025 pixel^–1 for a total field of view of ∼25''.

The AO data were processed and analyzed with a custom set of IDL tools. The science frames were flat fielded and sky subtracted. The flat fields were generated from a median average of dark-subtracted flats taken on sky. The flats were normalized such that the median value of the flats is unity. The sky frames were generated from the median average of the 15 dithered science frames; each science image was then sky subtracted and flat fielded. The reduced science frames were combined into a single combined image using an intrapixel interpolation that conserves flux, shifts the individual dithered frames by the appropriate fractional pixels, and median coadds the frames. The final resolutions of the combined dithers were determined from the FWHM of the point spread functions: 012 on 2022 June 19 and 010 on 2022 June 21.

The sensitivities of the final combined AO image were determined by injecting simulated sources azimuthally around the primary target every 20° at separations of integer multiples of the central source's FWHM (Furlan et al. 2017). The brightness of each injected source was scaled until standard aperture photometry detected it with 5σ significance. The resulting brightness of the injected sources relative to TOI-4600 set the contrast limits at that injection location. The final 5σ limit at each separation was determined from the average of all of the determined limits at that separation and the uncertainty on the limit was set by the rms dispersion of the azimuthal slices at a given radial distance. The final sensitivity curve for the Palomar data is shown in Figure 3. No additional stellar companions were detected on either night of observing.

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In addition to the high-resolution imaging, we have utilized Gaia to identify any wide stellar companions that may be bound members of the system. Typically, these stars are already in the TIC and their flux dilution to the transit has already been accounted for in the transit fits and associated derived parameters. Based upon similar parallaxes and proper motions (Mugrauer & Michel 2020, 2021), there are no additional widely separated companions identified by Gaia.

Additionally, the Gaia DR3 astrometry provides additional information on the possibility of inner companions that may have gone undetected by either Gaia or the high-resolution imaging. The Gaia renormalized unit weight error (RUWE) is a metric, similar to a reduced chi-square, where values that are ≲1.4 indicate that the Gaia astrometric solution is consistent with the star being single whereas RUWE values ≳1.4 may indicate excess astrometric noise, possibly caused the presence of an unseen companion (e.g., Ziegler et al. 2020). TOI-4600 has a Gaia DR3 RUWE value of ∼1 indicating that the astrometric fits are consistent with the single star model.

3.1.1. MESA Isochrones and Stellar Tracks Analysis

As noted in Section 2.3, we use the SPC tool on the TRES spectra to derive initial values for the spectral parameters of the host star, obtaining T_eff = 5200 K, log g = 4.6, and [m/H] = 0.15. We then used the spectroscopic parameters along with the Gaia DR3 parallax and magnitudes (G, B_P , and R_P ), Two Micron All Sky Survey (2MASS) magnitudes (J, H, and K_S ), and Wide-field Infrared Survey Explorer (WISE) magnitudes (W₁, W₂, and W₃) to perform an isochrone fit in order to constrain further the spectroscopic parameters and derive the physical parameters of the host star. The spectroscopic parameters, parallax, and magnitudes are used as priors to determine the goodness of fit. We use the isochrone package (Morton 2015) to generate the isochrone models used to sample the stellar parameters and find the best-fit parameters by using a Markov Chain Monte Carlo (MCMC) routine using the emcee package (Foreman-Mackey et al. 2013). The routine consists of 40 independent walkers each taking 25,000 steps, of which the first 2000 are discarded as burn-in. emcee checks whether the chains have converged by determining if the total number of steps is at least 100 times the autocorrelation time, as is the case with the routine here. The fitted spectroscopic parameters and derived physical parameters, including stellar age, of the host star are reported in Table 2, and are consistent with a K1V star (Pecaut & Mamajek 2013). We inflate the uncertainties of the parameters derived from the isochrone modeling in order to correct for the underestimated uncertainties commonly seen with stellar evolutionary models, adding the systematic uncertainty floors from Tayar et al. (2022) in quadrature to the uncertainties from the isochrone modeling.

Table 2. System Information

Stellar Parameter	Value	Source	Planet Parameter	TOI-4600 b	TOI-4600 c
TIC	232608943	TIC v8 a	Fitted parameters
R.A.	17:13:48.09	Gaia DR3 b	q1,TESS	${0.31}_{-0.14}^{+0.24}$	⋯
Decl.	+64:33:58.22	Gaia DR3 b	q2,TESS	${0.34}_{-0.21}^{+0.33}$	⋯
μra (mas yr−1)	7.625 ± 0.012	Gaia DR3 b	Rp /R⋆	${0.0769}_{-0.0017}^{+0.0021}$	${0.1067}_{-0.0026}^{+0.0024}$
μdec (mas yr−1)	0.266 ± 0.016	Gaia DR3 b	(R⋆ + Rp )/a	${0.01161}_{-0.00052}^{+0.00060}$	${0.00362}_{-0.00015}^{+0.00017}$
Parallax (mas)	4.620 ± 0.011	Gaia DR3 b	$\cos i$	${0.0041}_{-0.0025}^{+0.0016}$	0.00169 ${}_{-0.00056}^{+0.00038}$
Epoch	2016.0	Gaia DR3 b	T0 (BJD – 2,457,000)	${2750.1421}_{-0.0019}^{+0.0020}$	2751.6008 ± 0.0020
B (mag)	13.487 ± 0.048	AAVSO DR9 c	P (days)	82.6869 ± 0.0003	${482.8191}_{-0.0017}^{+0.0018}$
V (mag)	12.577 ± 0.034	AAVSO DR9 c	$\sqrt{e}\,\cos \,\omega$	${0.01}_{-0.52}^{+0.55}$	${0.04}_{-0.55}^{+0.54}$
Gaia (mag)	12.43401 ± 0.00027	Gaia DR3 b	$\sqrt{e}\,\sin \,\omega$	−0.30${}_{-0.16}^{+0.20}$	${0.10}_{-0.22}^{+0.20}$
BP	12.92036 ± 0.00285	Gaia DR3 b	ln σGP,s	−6.92${}_{-0.37}^{+0.66}$	⋯
RP	11.82667 ± 0.00380	Gaia DR3 b	ln ρGP,s	−0.33${}_{-0.73}^{+0.82}$	⋯
TESS (mag)	11.8787 ± 0.006	TIC v8 a	offsetGP,s	${0.0007}_{-0.0007}^{+0.0005}$	⋯
J (mag)	11.075 ± 0.023	2MASS d	ln σGP,l	−6.48${}_{-0.30}^{+0.41}$	⋯
H (mag)	10.659 ± 0.021	2MASS d	ln ρGP,l	${0.59}_{-0.58}^{+0.68}$	⋯
KS (mag)	10.571 ± 0.018	2MASS d	offsetGP,l	0.001 ± 0.001	⋯
M⋆ (M⊙)	0.89 ± 0.05	This work	ln σTESS,s	−5.85 ± 0.02	⋯
R⋆ (R⊙)	0.81 ± 0.03	This work	ln σTESS,l	−7.10 ± 0.05	⋯
ρ⋆ (g cm−3)	2.36 ± 0.29	This work
L⋆ (L⊙)	0.42 ± 0.02	This work	Derived parameters
Teff (K)	5170 ± 120	This work	i (°)	${89.76}_{-0.10}^{+0.14}$	${89.90}_{-0.02}^{+0.03}$
[Fe/H]	0.16 ± 0.08	This work	a (au)	0.349 ± 0.021	1.152 ± 0.068
log g	4.57 ± 0.02	This work	b	${0.39}_{-0.24}^{+0.17}$	${0.46}_{-0.19}^{+0.10}$
Age (Gyr)	${2.3}_{-1.6}^{+2.8}$	This work	Ttot e (hr)	${7.54}_{-0.13}^{+0.14}$	${11.14}_{-0.19}^{+0.20}$
$v\,\sin i$ (km s−1)	2.5 ± 1.3	This work	Tfull f (hr)	${6.26}_{-0.20}^{+0.15}$	${8.47}_{-0.33}^{+0.30}$
			Rp (R⊕)	${6.80}_{-0.30}^{+0.31}$	${9.42}_{-0.41}^{+0.42}$
			Rp (RJ)	${0.607}_{-0.026}^{+0.028}$	0.841 ± 0.037
			e	${0.25}_{-0.17}^{+0.33}$	${0.21}_{-0.14}^{+0.29}$
			ω	${260}_{-57}^{+70}$	${142}_{-116}^{+148}$
			Teq g (K)	${347}_{-11}^{+12}$	191 ± 6

Notes.

^a Stassun et al. (2018b). ^b Gaia Collaboration et al. (2021). ^c Henden et al. (2016). ^d Cutri et al. (2003). ^e From first to last (fourth) contacts. ^f From second to third contacts. ^g Assuming an albedo of 0.3 and emissivity of 1.

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3.1.2. SED Analysis

As an independent determination of the basic stellar parameters, we performed an analysis of the broadband spectral energy distribution (SED) of the star together with the Gaia DR3 parallax (with no systematic offset applied; see, e.g., Stassun & Torres 2021), in order to determine an empirical measurement of the stellar radius, following the procedures described in Stassun & Torres (2016) and Stassun et al. (2017, 2018a). We pulled the JHK_S magnitudes from 2MASS, the W1−W4 magnitudes from WISE, and the G, G_BP, and G_RP magnitudes from Gaia. We also used the UVW1 and U-band measurements from the Swift satellite. Together, the available photometry spans the full stellar SED over the wavelength range 0.3–20 μm (see Figure 4).

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We performed a fit using Kurucz stellar atmosphere models, with the free parameters being the effective temperature (T_eff), surface gravity ($\mathrm{log}g$), and metallicity ([Fe/H]). The remaining free parameter is the extinction A_V , which we limited to the maximum line-of-sight value from the Galactic dust maps of Schlegel et al. (1998). The resulting fit (Figure 4) has a reduced χ² of 1.3 with best-fit A_V = 0.04 ± 0.04, T_eff = 5075 ± 75 K, $\mathrm{log}g=4.5\pm 0.5$, and [Fe/H] = −0.2 ± 0.3. Integrating the model SED gives the bolometric flux at Earth, F_bol = 2.791 ± 0.066 × 10⁻¹⁰ erg s⁻¹ cm⁻². Taking the F_bol and T_eff together with the Gaia parallax, gives the stellar radius, R_⋆ = 0.827 ± 0.026 R_⊙. In addition, we can estimate the stellar mass from the empirical relations of Torres et al. (2010), giving M_⋆ = 0.81 ± 0.10 M_⊙. These parameters are consistent with those derived using the isochrone models at the 1σ level.

The second transit of TOI-4600 c observed in Sector 53 meant the possible periods were reduced to a finite number of aliases of the separation between the two transits. As an initial check that the two transit events are caused by the same planet, we model each event independently using allesfitter (Günther & Daylan 2021; see Section 3.4 for a more detailed description) in order to obtain estimates for the planet radius and orbital period, assuming a circular orbit. We obtain a planet radius of ${9.42}_{-0.29}^{+0.37}\,{R}_{\oplus }$ and ${9.53}_{-0.25}^{+0.28}\,{R}_{\oplus }$ for the first and second transits, respectively. For the orbital periods, we obtain estimates of ${360}_{-100}^{+170}$ days and ${324}_{-58}^{+84}$ days, respectively. We find these values agree with each other to within 1σ and therefore we safely assume from this point on that these two events are caused by the same source and hence planetary candidate.

In order to assess the potential orbital periods for TOI-4600 c, we first fitted the available TESS transits jointly using the MonoTools method outlined in Osborn et al. (2022). This fits the impact parameter, transit duration, and radius ratio in a way that is agnostic of the exoplanet period using the exoplanet fitting package (Foreman-Mackey et al. 2021a). When combined with stellar parameters such as the bulk density (in this case from the TIC v8 catalog; Stassun et al. 2019), the transit model allows us to derive an instantaneous transverse planetary velocity and therefore, for each allowed period alias, to compute a marginalized probability distribution. This uses a combination of the model likelihood, and priors from a combination of window function and occurrence rate (P⁻²; Kipping 2018), a geometric transit probability (p_g ∝ a⁻¹ ∼ P^−2/3), and an eccentricity prior (Van Eylen et al. 2019) applied using the eccentricity distribution implied by the ratio of the transverse planetary velocity to the circular velocity at that period alias. The stability of each possible orbit with respect to other planets with known periods in the system is also considered by suppressing the probabilities of any orbit which passes inside the Hill radii of inner planets.

Through MonoTools, we used the TESS-SPOC HLSP FFI light curves (S14–26), the 2 minute SPOC light curves (S40–41 and S47–49), and light curves extracted from the 10 minute TICA FFIs (S51–53) to analyze the system. We flattened the data using a basis spline set by a 1.2 day knot distance and with identified transits from planets b & c masked. We then clipped the TESS photometry to windows around each transit with 4.5 transit durations, thereby improving the speed of sampling. Sampling was performed using the PyMC3 implementation of Hamiltonian Monte Carlo (Salvatier 2016) using four chains and producing 2000 unique samples, with low Rubin–Gelman statistics ($\hat{r}$) ensuring that the chains were well mixed. The resulting log probabilities are then marginalized using the sum of the probability across all period aliases, and the final log probabilities were calculated by summing the probabilities across all samples (assuming equal weights for each independent model draw) and period aliases. Using MonoTools, we find again that the two transits are related and we find that there are only two periods permitted as a result of the vast data coverage—482.82 and 965.64 days—with potential transits at all other periods being ruled out by the TESS observations. Monotools assigns these two possible periods with probabilities of 99.97% and 0.03%, respectively, assuming the eccentricity distribution of multiplanet systems from Van Eylen et al. (2019). We note that when assuming a more general eccentricity distribution from Kipping (2013a), the 482.82 day alias is preferred by a factor of only ∼6.5. As a result of this analysis, we hereafter assume an orbital period of 482.82 days for TOI-4600 c. This period can be definitively confirmed with an additional transit observation, with the next predicted transit of only the 482.82 day alias occurring on UT 2023 October 16. Given the previous ground-based observations of the smaller inner planet, the predicted transit of TOI-4600 c should be readily detected.

We are able to rule out the various false-positive scenarios using the original TESS photometry and subsequent ground-based photometry, spectroscopy, and imaging. We analyze the in- and out-of-transit photocenters, or centroids, using vetting (Hedges 2021) to rule out NEBs as the source of the transits for both planets. vetting calculates a p-value for each TESS sector to determine whether a transit is on target or not. For TOI-4600 c, the p-values for both sectors are greater than 0.05, indicating the transits are on target. For TOI-4600 b, all sectors except Sector 19 have p-values greater than 0.05. The transit in that sector is contaminated by scattered light causing a false offset. We are able to rule out stellar masses for both planets using the TRES spectra. We fit the RVs using the radvel package (Fulton et al. 2018) in order to obtain mass limits for both planets. We assume circular orbits for both planets and fix the period and epoch to the best-fit values listed in Table 2 for both. We fit only the RV semiamplitudes of both planets, a jitter term for TRES, and a relative offset term. We use an MCMC routine consisting of 4 × 10⁶ steps with uniform priors for all parameters. We obtain 3σ upper mass limits of 3.02 M_J and 9.27 M_J for TOI-4600 b and c, respectively.

We use triceratops (Giacalone & Dressing 2020; Giacalone et al. 2021) to calculate false-positive probabilities (FPP) and nearby false-positive probabilities (NFPP) for each candidate in the system, including the contrast curve from our high-resolution imaging in order to provide additional constraints. False-positive scenarios include an eclipsing binary on target, on a background star, or an unseen companion as well as a transiting planet on a background star or unseen companion. Nearby false-positive scenarios include a transiting planet or eclipsing binary on a nearby star. We calculate FPP values of 0.0107 and 0.0205 for b and c, respectively. Due to the lack of sufficiently bright nearby neighbors, the NFPP values for both of the candidates default to 0. While the individual FPPs of both TOI-4600 b and c exceed the nominal maximum value of 0.015 stated by Giacalone et al. (2021) as the threshold to be considered statistically validated, multiplanet system candidates have been found to have much lower false-positive rates than single-planet system candidates. For the Kepler mission, the false-positive rate of multiplanet system candidates was ∼25 times lower than that of single-planet system candidates (Lissauer et al. 2012, 2014). For TESS this so-called "multiplicity boost" is ∼20 (Guerrero et al. 2021), meaning the FPP values of both TOI-4600 b and c are well below the previously mentioned 0.015 threshold and the often-used 0.01 threshold (Rowe et al. 2014; Montet et al. 2015; Heller et al. 2019; Castro González et al. 2020).

The FPP values of both planets are dominated by the secondary transit planet (STP) scenario, where the transits originate from a transiting planet orbiting an unresolved bound companion. However, triceratops only uses the contrast curve to determine possible bound companions in its analysis of the STP scenario. In order to incorporate additional data, we use Multi-Observational Limits on Unseen Stellar Companions (MOLUSC; Wood et al. 2021) in order to generate a sample of potential companions consistent with the combination of the contrast curve, RV data, Gaia astrometry (in the form of the RUWE), and Gaia imaging. Of the 5 million stars we generated using MOLUSC, only ∼10% were consistent with the aforementioned data. We then reran triceratops using this sample of plausible bound companions and recalculated the FPPs, finding that the STP scenario is now a negligible component of the FPP. We obtain an FPP of (3.7 ± 6.8) × 10⁻⁵ for b and (1.1 ± 3.9) × 10⁻⁷ for c, making both statistically validated planets.

In order to derive the orbital parameters and radii of the two planets, we performed model fitting using the publicly available allesfitter package (Günther & Daylan 2021). We used MCMC sampling using the emcee package to explore the parameter space and determine the best-fit values from the medians of the posteriors for the following parameters:

• 1.  
  quadratic stellar limb-darkening parameters q₁ and q₂, using the transformation from Kipping (2013b), with uniform priors from 0 to 1;
• 2.  
  radius ratio, R_p /R_⋆, where p denotes the individual planets, with uniform priors from 0 to 1;
• 3.  
  sum of radii divided by the orbital semimajor axis, (R_⋆ + R_p )/a, with uniform priors from 0 to 1;
• 4.  
  cosine of the orbital inclination, $\cos i$, with uniform priors from 0 to 1;
• 5.  
  orbital period, P, with uniform priors centered on the estimate of 82.69 days from MonoTools with a 1 day range;
• 6.  
  transit epoch, T₀, with uniform priors centered on the estimate of 482.82 days from MonoTools with a 1 day range;
• 7.  
  eccentricity parameters $\sqrt{e}\,\cos \,\omega$ and $\sqrt{e}\,\sin \,\omega$, each with uniform priors from −1 to 1, where e is the orbital eccentricity and ω the argument of periastron;
• 8.  
  the hyperparameters σ_GP and ρ_GP and offset_GP for a Matérn 3/2 kernel used to model the red noise for the 2 and 30 minute TESS data individually (denoted s and l, respectively); and
• 9.  
  white noise scaling terms for the 2 and 30 minute TESS data, σ_TESS,s and σ_TESS,l , respectively.

We initialized the MCMC with 200 walkers, performing two preliminary runs of 1000 steps per walker to obtain higher-likelihood initial guesses for the nominal run of 40,000 steps per walker. We then discarded the first 10,000 steps for each chain as a burn-in phase before thinning the chains by a factor of 100 and calculating the final posterior distributions. The values and uncertainties of the fitted and derived parameters listed in Table 2 are defined as the median values and 68% confidence intervals of the posterior distributions, respectively. The best-fit transit model light curves for the planets are shown in Figure 1.