Two Intermediate-mass Transiting Brown Dwarfs from the TESS Mission

The light curves of TOI-569 come from the TESS mission, in sectors 7 and 8, and the Las Cumbres Observatory (LCOGT). For the TESS light curve for TOI-569, we use the Pre-search Data Conditioning Simple Aperture Photometry flux (PDCSAP; Smith et al. 2012; Stumpe et al. 2014) from the Mikulski Archive for Space Telescopes (MAST).²¹ The PDCSAP light curve has systematic effects removed, and with this, we then normalize the light curve with the lightkurve package in Python (the Lightkurve Collaboration et al. 2018). The light curves used for TOI-1406 come from the full-frame images (30 minutes cadence) from the TESS mission in sectors 4, 5, and 6. We use the lightkurve package to extract and normalize the light curve of TOI-1406. Coordinates and magnitudes for TOI-569 and TOI-1406 are given in Table 1.

For the light-curve extraction, we use circular apertures that are fixed to the target star's position for each sector. When the star moves slightly between sectors, the aperture is moved to follow it. The counts from each pixel within the aperture are summed, and the resulting light curve is detrended using the lightkurve package's built-in flattening tool, which we use to remove stellar rotational variability, when present, as well as scattered background light. The detrended TESS light curves are shown in Figure 1.

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We observed an ingress of TOI-569 continuously for 140 minutes on 2019 April 15 using 15 s exposures and a z-short band filter from the LCOGT (Brown et al. 2013) 1.0 m node at Cerro Tololo Inter-American Observatory. We used the TESS Transit Finder, which is a customized version of the Tapir software package (Jensen 2013), to schedule our transit observations. The 4096 × 4096 LCOGT SINISTRO cameras have an image scale of 0389 per pixel, resulting in a 26' × 26' field of view. The images were calibrated by the standard LCOGT BANZAI pipeline, and photometric data were extracted with the AstroImageJ software package (Collins et al. 2017) using a circular aperture with radius 58. The images have typical stellar point-spread-functions with a half-width-half-maximum of 1''. We detect a ∼3000 ppm ingress on target with apertures as small as 2'' in radius. Systematic effects start to dominate the light curve for smaller apertures. Thus, we confirm that the source of the TESS detection is within 3'' of the target star location and that the transit depth from the LCOGT partial transit is consistent with the TESS depth for all aperture radii we checked down to 2''. We did not obtain any ground-based photometric follow-up of TOI-1406.

2.1.1. Light Curve Modulation and the Orbital Period of the TOI-569 System

Previous to the transit detections of TOI-569b from TESS, the Wide Angle Search for Planets (WASP) found a 13-day modulation in the light curve of TOI-569. The phased light curve for WASP is shown in Figure 2. The WASP data were taken during the 2011–2012 seasons, with a 150-day span of coverage. The transits of TOI-569b are too shallow to be detected in the WASP light curve, even when phase folded to the ephemeris from TESS, but they can be seen in the TESS light curve in the top panel of Figure 2. The WASP and TESS light curves show a similar but not exactly equal modulation. The periodic peaks and dips in both light curves are likely from brightness variations due to star spots, which are known to vary over time. The unevenness in the peaks in the TESS light curves is likely from multiple different starspot configurations on different areas on the surface of the star, in addition to the evolution of spot brightness over time.

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The gaps in the TESS light curve occur during every other transit of TOI-569b in the sectors where the host star was observed. This is the reason why the initial orbital period was reported to be 13.12 days (twice the true orbital period of 6.56 days). We discovered that this was the case, as the orbital solution developed with RV follow-up using the instruments described in later sections. It seems coincidental that this erroneous orbital period of 13.12 days is nearly equal to the 13 days modulation in the WASP light curve. This made the BD appear to have an orbit synchronized with the rotation rate of the star, but this turned out not to be the case upon a more thorough investigation that accounts for the orbital solution derived from RVs. We note that the observation of the transit of TOI-569b with the LCOGT 1 m telescope did not occur at opposite parity to the transits detected by TESS, so we cannot use this to independently confirm the 6.56 days period of TOI-569b.

Table 1.  Coordinates and Magnitudes for TOI-569 and TOI-1406

	Description	TOI-569	TOI-1406	Source
αJ2000	Equatorial	07 40 24.67	05 28 30.71	1
δJ2000	Coordinates	−42 09 16.79	−48 24 32.64	1
T	TESS T	9.473 ± 0.006	11.427 ± 0.006	2
G	Gaia G	9.936 ± 0.001	11.759 ± 0.001	1
BT	Tycho BT	11.036 ± 0.048	13.352 ± 0.359	3
VT	Tycho VT	10.173 ± 0.032	12.074 ± 0.195	3
J	2MASS J	8.829 ± 0.020	10.929 ± 0.020	4
H	2MASS H	8.575 ± 0.060	10.787 ± 0.030	4
KS	2MASS KS	8.444 ± 0.020	10.675 ± 0.020	4
WISE1	WISE 3.4 μm	8.419 ± 0.023	10.649 ± 0.023	5
WISE2	WISE 4.6 μm	8.467 ± 0.020	10.692 ± 0.021	5
WISE3	WISE 12 μm	8.414 ± 0.021	10.651 ± 0.062	5
WISE4	WISE 22 μm	8.180 ± 0.188	⋯	5

Note. The values here make up the spectral energy distributions used to model the stars.

References. (1) Lindegren et al. (2018), corrected from the J2015 epoch, (2) Stassun et al. (2018b), (3) Høg et al. (2000), (4) Skrutskie et al. (2006), (5) Wright et al. (2010).

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Using a Lomb–Scargle periodogram analysis on both the TESS and WASP light curves separately (Figure 3), we see a peak frequency at roughly 13 days (12.91 days for TESS and 13.01 days for WASP). This may suggest that TOI-569 and its companion BD are in a 2:1 spin–orbit resonance.

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Though the LCOGT data give us a sense of whether or not the transit signals for TOI-569 are within roughly 3'' of the target star, we may use speckle imaging to confirm whether or not there is contamination even closer to the target. For TOI-569, we used SOAR speckle imaging to look for other objects within the TESS aperture that would significantly contaminate the transit and RV signals we observe. Nearby stars that fall within the same TESS image profile as the target can cause photometric contamination or be the source of an astrophysical false positive, such as a background or nearby eclipsing binary star. We searched for nearby sources to TOI-569 with SOAR speckle imaging (Tokovinin 2018) on 2019 May 18, observing in a similar visible bandpass as TESS (the Cousins-I-band). Further details of the observations are available in Ziegler et al. (2020). We detected no nearby stars within 3'' of TOI-569. The 5σ detection sensitivity and the speckle auto-correlation function from the SOAR observation are plotted in Figure 4.

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We also use data from Gaia DR2 (Gaia Collaboration et al. 2018) to gather a census of nearby stars, finding that no stars brighter than G = 17.0 are within 25'' and only two stars with G = 13.7 and G = 14.9 are approximately 26'' from TOI-569, which has a brightness of G = 9.9 (Table 2). These other fainter stars also do not share the same proper motion as TOI-569, which indicates that they are not associated with TOI-569 and are more distant background stars.

We do not have any high-resolution imaging (as in, sub-3'' coverage) for TOI-1406, but using Gaia DR2 data, we find only three other stars within 30'' of TOI-1406. The brightest of these other stars has a magnitude of G = 15.8 and is 19'' from TOI-1406, which has a magnitude of G = 11.8. We also find that none of these other stars share the same proper motion as TOI-1406 from the Gaia DR2 data (Table 2).

To characterize the RVs and stellar atmospheric parameters of TOI-569 and TOI-1406, we obtained a series of spectroscopic observations using the CHIRON spectrograph on the 1.5 m SMARTS telescope (Tokovinin et al. 2013), located at Cerro Tololo Inter-American Observatory, Chile. CHIRON is a high-resolution echelle spectrograph that is fed via an image slicer and a fiber bundle. CHIRON achieves a spectral resolving power of λ/Δλ ≡ R ∼ 80,000 over the wavelength region 4100–8700 Å. The wavelength calibration is obtained via thorium-argon hollow-cathode lamp exposures that bracket each stellar spectrum.

To derive the stellar RVs, we performed a least-squares deconvolution (Donati et al. 1997) between the observed spectra and a non-rotating synthetic template generated via ATLAS9 atmospheric models (Castelli & Kurucz 2004) at the stellar atmospheric parameters of each target. We then model the stellar line profiles derived from the least-squares deconvolution via an analytic rotational broadening kernel, as per Gray (2005). This procedure follows that done in Zhou et al. (2019). The derived RVs for TOI-569 and TOI-1406 are listed in Table 3, and this series of RVs helped reveal the true orbital period of the BD. The stellar parameters derived from the spectra of TOI-569 are T_eff = 5669 ± 106 K, $\mathrm{log}g=4.11\pm 0.18$, $[\mathrm{Fe}/{\rm{H}}]=+0.23\pm 0.05$ dex, and $v\sin {I}_{\star }=5.33\pm 0.50$ km s⁻¹. For TOI-1406, we find T_eff = 6347 ± 186 K, $\mathrm{log}g=4.09\pm 0.15$, $[\mathrm{Fe}/{\rm{H}}]\,=-0.05\,\pm 0.11$ dex, and an approximate full width at half max for the line broadening profile of $\mathrm{FWHM}=12.91\pm 0.24$ km s⁻¹ with the CHIRON spectra. For TOI-569, we take care to account for the instrumental profile and macroturbulence to extract $v\sin {I}_{\star }$ from the FWHM approximation ($\mathrm{FWHM}=6.65\pm 0.5$ km s⁻¹) with CHIRON, as this is important in our analysis of the stellar inclination in Section 3.3.

Table 3.  Relative Radial Velocities, Bisector Span (V_span), and FWHM of TOI-569 from CHIRON, CORALIE, and FEROS and of TOI-1406 from CHIRON and ANU

${\mathrm{BJD}}_{\mathrm{TDB}}-2450000$	RV (m s−1)	σRV (m s−1)	Vspan (km s−1)	FWHM (km s−1)	Instrument	Target
8594.61690	75132.3	16.6	⋯	6.93	CHIRON	TOI-569
8606.50771	77754.5	20.7	⋯	6.67	CHIRON	TOI-569
8596.60327	66265.0	18.9	⋯	7.00	CHIRON	TOI-569
8595.59076	69864.3	16.4	⋯	6.67	CHIRON	TOI-569
8607.53323	75982.6	17.8	⋯	6.73	CHIRON	TOI-569
8611.59733	71966.8	24.9	⋯	6.65	CHIRON	TOI-569
8612.57581	76679.3	30.6	⋯	6.57	CHIRON	TOI-569
8649.44500	66193.1	24.2	⋯	6.63	CHIRON	TOI-569
8651.48774	74957.8	34.9	⋯	6.81	CHIRON	TOI-569
8654.46711	70453.9	27.2	⋯	6.40	CHIRON	TOI-569
8593.58587	79347.5	13.0	−0.030	10.55	CORALIE	TOI-569
8597.47058	68533.2	31.6	0.089	10.48	CORALIE	TOI-569
8599.52446	78333.6	16.8	−0.023	10.48	CORALIE	TOI-569
8602.58567	69230.9	13.3	0.001	10.46	CORALIE	TOI-569
8603.46595	67533.0	20.3	−0.012	10.47	CORALIE	TOI-569
8614.49353	75527.3	32.4	0.004	10.43	CORALIE	TOI-569
8615.48815	70140.4	29.5	0.060	10.66	CORALIE	TOI-569
8594.49170	77148.3	6.5	0.019	⋯	FEROS	TOI-569
8595.52217	71681.8	8.4	0.015	⋯	FEROS	TOI-569
8597.51010	68686.8	6.8	0.015	⋯	FEROS	TOI-569
8617.52568	70033.1	6.0	0.009	⋯	FEROS	TOI-569
8533.07797	−19568.3	465.0	⋯	⋯	ANU	TOI-1406
8534.98787	−17679.4	197.8	⋯	⋯	ANU	TOI-1406
8536.06364	−15764.1	260.2	⋯	⋯	ANU	TOI-1406
8537.96961	−12090.1	725.7	⋯	⋯	ANU	TOI-1406
8538.93516	−12135.5	282.9	⋯	⋯	ANU	TOI-1406
8561.89365	−13798.5	270.0	⋯	⋯	ANU	TOI-1406
8540.61381	−13631.0	69.2	⋯	12.91	CHIRON	TOI-1406
8541.60193	−15950.3	91.8	⋯	12.69	CHIRON	TOI-1406
8542.56709	−17888.9	45.1	⋯	13.07	CHIRON	TOI-1406
8544.52353	−19087.2	166.4	⋯	12.78	CHIRON	TOI-1406
8546.51779	−15936.1	91.8	⋯	12.81	CHIRON	TOI-1406
8562.57958	−15177.2	126.9	⋯	12.89	CHIRON	TOI-1406
8566.55925	−18101.1	97.7	⋯	12.77	CHIRON	TOI-1406
8567.59295	−16029.1	83.6	⋯	12.82	CHIRON	TOI-1406
8568.54388	−13973.3	124.3	⋯	13.36	CHIRON	TOI-1406
8569.57364	−12432.3	110.3	⋯	12.98	CHIRON	TOI-1406

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To help confirm TOI-1406b as a BD, we obtained six spectroscopic observations with the echelle spectrograph on the Australian National University (ANU) 2.3 m telescope, located at Siding Spring Observatory, Australia. The ANU 2.3 m echelle is a slit-fed spectrograph that yields a resolving power of R ∼ 23,000 over the wavelength region of 3700–6700 Å. Wavelength calibration was provided by bracketing thorium-argon lamp exposures, and the spectra were reduced as per Zhou et al. (2014). The RVs from each exposure were measured via the least-squares deconvolution technique, as described in Section 2.3. To derive T_eff, $\mathrm{log}g$, and $[\mathrm{Fe}/{\rm{H}}]$ for TOI-1406, we use SpecMatch-emp (Yee et al. 2017), which matches the input spectra to a library of stars with well-determined parameters derived with a variety of independent methods, such as interferometry, optical and NIR photometry, asteroseismology, and LTE analysis of high-resolution optical spectra. From the ANU spectra and SpecMatch-emp, we find T_eff = 6283 ± 110 K, $\mathrm{log}g=4.13\,\pm 0.12$, $[\mathrm{Fe}/{\rm{H}}]\,=-0.09\pm 0.09$ dex, and $\mathrm{FWHM}=15.0\,\pm 1.0$ km s⁻¹ for TOI-1406.

TOI-569 was observed with the CORALIE spectrograph on the Swiss 1.2 m Euler telescope at La Silla Observatories, Chile (Queloz et al. 2001), between 2019 April 19 and May 11. CORALIE has a resolving power of R ∼ 60,000 and is fed by two fibers: one 2'' diameter on-sky science fiber encompassing the star and another that can be connected to either a Fabry-Pérot etalon for simultaneous wavelength calibration or on-sky for background subtraction of the sky-flux. RVs were computed for each epoch by cross-correlating with a binary G2 mask (Pepe et al. 2002). Bisector-span, full-width half-max, and other line-profile diagnostics were computed as well, using the standard CORALIE data reduction software. Exposure times ranged from 450 to 1200 s.  We obtain internal error estimates of 13–32 m s⁻¹. The resulting velocities are plotted in Figure 5 and are listed in Table 3.

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The CORALIE spectra were shifted to the stellar rest frame and stacked while weighting the contribution from each spectrum with its mean flux to produce a high signal-to-noise spectrum for spectral characterization using SpecMatch-emp (Yee et al. 2017). We used the spectral region around the Mgb triplet (5100–5340 Å) to match our spectrum to the library spectra through χ² minimization. A weighted linear combination of the five best matching spectra was used to extract bulk stellar parameters: ${T}_{\mathrm{eff}}=5481\pm 110$ K, $\mathrm{log}g=4.08\pm 0.12$, and $[\mathrm{Fe}/{\rm{H}}]=+0.41\pm 0.09$ dex for TOI-569.

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TOI-569 was observed with the FEROS spectrograph (Kaufer & Pasquini 1998) mounted on the MPG 2.2 m telescope installed at the ESO La Silla Observatory. Four spectra were obtained between 2019 April 20 and May 14. Observations were performed with the simultaneous calibration mode, where a second fiber is illuminated with a thorium-argon lamp for tracking the instrumental drift in RV during the science exposure. The adopted exposure time was of 400 s, which produced spectra with a typical signal-to-noise ratio per resolution element of 90. FEROS data were processed with the ceres pipeline (Brahm et al. 2017a), which performs the optimal extraction of the raw data, the wavelength calibration, the instrumental drift correction, and the computation of precise RVs and bisector spans. The results are presented in Table 3. The four FEROS spectra were combined in order to measure the atmospheric parameters using the zaspe package (Brahm et al. 2017b), obtaining T_eff = 5669 ± 80 K, $\mathrm{log}g$ = 4.21 ± 0.12, $[\mathrm{Fe}/{\rm{H}}]=+0.28\,\pm 0.05$ dex, and an approximate $\mathrm{FWHM}=6.45\pm 0.30$ km s⁻¹ for TOI-569.

The masses and radii of the BDs are derived using EXOFASTv2. A full description of EXOFASTv2 is given in Eastman et al. (2019). EXOFASTv2 uses the Monte Carlo-Markov Chain (MCMC) method. For each MCMC fit, we use N = 36 (N = 2 × n_parameters) walkers, or chains, and run for 50,000 steps, or links. To derive stellar parameters, EXOFASTv2 utilizes the MIST isochrone models (Paxton et al. 2015; Choi et al. 2016; Dotter 2016). 

The parameters for which we set priors and the types of priors we set for each (i.e., uniform ${ \mathcal U }[a,b]$ or Gaussian ${ \mathcal G }[a,b]$) are shown for clarity. We rely on our spectroscopic measurements of [Fe/H] and T_eff and parallax measurements from Gaia to define our Gaussian priors, which penalize the fit for straying beyond the width, b, away from the mean, a, of the parameter. We use an upper limit for the A_V extinction. See Table 3 of Eastman et al. (2019) for a detailed description of priors in EXOFASTv2. For the choice of priors for [Fe/H] and T_eff, we use the CHIRON values, since CHIRON has the highest spectral resolution R = 80,000 of the spectrographs we used (see Table 4). The resulting EXOFASTv2 values are consistent with the input values from CHIRON. The spectral energy distribution for each star is also taken into account with EXOFASTv2, as shown in Figure 8. The BD parameters are derived with the normalized TESS and LCOGT light curves and non-phase folded RVs into EXOFASTv2 as inputs. The phase folded light curves and RVs are shown in Figure 5 and Figure 6. The non-negligible BD mass is properly accounted for in EXOFASTv2, so no particularly special treatment is needed with regard to deriving the companion masses.

Table 4.  Spectroscopic Values for TOI-569 and TOI-1406 from CHIRON, ANU, CORALIE, and FEROS

TOI-569	CHIRON	CORALIE	FEROS
Teff (K)	5669 ± 106	5481 ± 110	5669 ± 80
$\mathrm{log}g$	4.11 ± 0.18	4.08 ± 0.12	4.21 ± 0.12
$[\mathrm{Fe}/{\rm{H}}]$ (dex)	+0.23 ± 0.05	+0.41 ± 0.09	+0.28 ± 0.05
FWHM (km s−1)	6.65 ± 0.16	10.50 ± 0.50	6.45 ± 0.30
R (resolution)	80,000	60,000	48,000
TOI-1406	ANU	CHIRON
Teff (K)	6283 ± 110	6347 ± 186	⋯
$\mathrm{log}g$	4.13 ± 0.12	4.09 ± 0.15	⋯
$[\mathrm{Fe}/{\rm{H}}]$ (dex)	−0.09 ± 0.09	−0.05 ± 0.11	⋯
FWHM (km s−1)	15.0 ± 1.0	12.9 ± 0.2	⋯
R (resolution)	23,000	80,000	⋯

Note. We use [Fe/H] and T_eff values from CHIRON as inputs to the global model described in Section 3.1.

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We see bimodality in the posterior distribution for the age (and correlated parameters) of TOI-569, so we present the two most probable solutions resulting from the bimodal posterior distributions with the absolute most probable solution taken as the final  adopted value. The most relevant bimodal posterior distributions are shown in Figure 7. The probability of the solution (for age and mass) we report here is 0.73, with the less likely solution having a probability of 0.27.

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The resulting stellar SED models from EXOFASTv2 for TOI-569 and TOI-1406 are shown in Figure 8. These follow the procedures outlined in Stassun & Torres (2016) and Stassun et al. (2017, 2018a).

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As an independent check on our EXOFASTv2 analysis, we also carried out an analysis with the pyaneti²² (Barragán et al. 2019) software. Using a Bayesian approach combined with MCMC sampling, we performed a joint analysis of the RV measurements and the TESS light curves, and modeled posterior distributions of the fitted parameters. The RV data were fitted with Keplerian orbits, and for each different instrumental set-up, an offset term for each systemic velocity is included. The non-negligible mass of the brown dwarf is properly taken into account in pyaneti (Barragán et al. 2019). The photometric data are modeled with the quadratic limb-darkening model of Mandel & Agol (2002).

We use uniform priors and fit for the BD-to-star radius ratio, the orbital period, the mid-transit time, the scaled orbital distance, the eccentricity, the argument of periastron, the impact parameter (b), and the Doppler semi-amplitude variation (K). The allowed ranges for the fit parameters for pyaneti are shown in Table 5.

Table 5.  Allowed Ranges for Fit Parameters from pyaneti

Parameter	TOI-569	TOI-1406
RBD/R⋆	[0, 0.1]	[0, 0.1]
Porb (days)	[6.5541, 6.5580]	[10.5721, 10.5762]
${T}_{0}-2458400$ (BJDTDB)	[96.858, 96.878]	[14.5061, 14.7061]
a/R⋆	[1.1, 12]	[1.1, 19]
$e\cos \omega$	[−1, 1]	[−1, 1]
$e\sin \omega$	[−1, 1]	[−1, 1]
Impact parameter b	[0, 1]	[0, 1]
Semi-amplitude K (km s−1)	[0, 15]	[0, 15]

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We used 500 independent chains and checked for convergence after every 5000 iterations. After convergence, a posterior distribution of 250,000 independent points for every parameter was computed from the last 500 iterations. We find the eccentricity to be consistent with zero for both BDs. We find a mass and radius of TOI-569b and TOI-1406b to be consistent within 1σ of the values from the EXOFASTv2 models.

Astronomers may calculate the angle at which a star is inclined to the line of sight, I_⋆, in order to learn about the relative alignment between this angle and the orbital inclination angle, i, of a transiting or eclipsing object. Using the spectroscopic $v\sin {I}_{\star }$ measurement from CHIRON (the highest resolution spectrograph we used with R ∼ 80,000) and the P_rot results from the Lomb–Scargle periodogram analysis, we calculate the inclination of the rotational axis of TOI-569 to be ${I}_{\star }={65.58}_{-8.55}^{+17.75\circ}$ (1σ uncertainties). This is traditionally done by taking

$$\begin{eqnarray}&&{I}_{\star }={\sin }^{-1}\left(\displaystyle \frac{v\sin {I}_{\star }}{{V}_{\mathrm{rot}}}\right),\end{eqnarray} \tag{ 1 }$$

where $v\sin {I}_{\star }=5.33\pm 0.50$ km s⁻¹ and ${V}_{\mathrm{rot}}=2\pi {R}_{\star }/{P}_{\mathrm{rot}}\,=5.80\pm 0.58$ km s⁻¹. However, this traditional technique neglects the fact that the prior on $v\sin {I}_{\star }$ and the prior on V_rot depend on each other and are therefore covariant. Masuda & Winn (2020) provide guidance on how to properly address this flaw with the traditional technique. We show our results of I_⋆ for TOI-569 using the formulation described by Masuda & Winn (2020) in Figure 9. We use the MCMC distribution to calculate the 1σ uncertainties as the 16th and 84th percentiles of the distribution, with the mean being the peak of the analytic distribution. We use the peak of the distribution because the distribution is skewed, and so the median would bias I_⋆ to higher values. This method neglects the effects of differential rotation in the star, which may change the P_rot we report here by up to 15%, depending on the latitude of the star spots (Quinn & White 2016).

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The orbital inclination of TOI-569b is $i={85.37}_{-0.11}^{+0.13\circ}$. Given the probability distribution of the stellar inclination ${I}_{\star }={65.58}_{-8.55}^{+17.75\circ}$ of TOI-569 (see Figure 9), we argue that this system is marginally misaligned and that alignment cannot be ruled out.

When only $v\sin {I}_{\star }$ is known, as is the case with TOI-1406, we use the following equation to place an upper limit on P_rot (5.3 days), which is much shorter than the orbital period of TOI-1406b (10.6 days), meaning that the system is not synchronized:

$$\begin{eqnarray}&&{P}_{\mathrm{rot}}\leqslant \displaystyle \frac{2\pi {R}_{\star }}{v\sin {I}_{\star }}.\end{eqnarray} \tag{ 2 }$$

Over time, tidal interactions between a host star and any companions affect their orbits. Generally, the orbits of the companions and their host stars first begin to circularize according to what is known as the circularization timescale. Next, the orbital period of the companion synchronizes with their host star's rotation (the synchronization timescale). Finally, the system experiences a spin–orbit co-alignment (Mazeh 2008). These timescales are influenced by the mass, radius, separation, and tidal quality factor Q of both the host star and companion of a system.  Here, we restrict our discussion to the circularization timescales for different values of the tidal quality factors, Q_⋆ and Q_BD, that may be most appropriate for the TOI-569 and TOI-1406 systems.

Following the formalism from Jackson et al. (2008), the equations for the orbital circularization timescale for a close-in companion are

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\tau }_{\mathrm{circ},\star }}=\displaystyle \frac{171}{16}\sqrt{\tfrac{G}{{M}_{\star }}}\displaystyle \frac{{R}_{\star }^{5}{M}_{\mathrm{BD}}}{{Q}_{\star }}{a}^{-\tfrac{13}{2}}\end{eqnarray} \tag{ 3 }$$

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\tau }_{\mathrm{circ},\mathrm{BD}}}=\displaystyle \frac{63}{4}\displaystyle \frac{\sqrt{{{GM}}_{\star }^{3}}{R}_{\mathrm{BD}}^{5}}{{Q}_{\mathrm{BD}}{M}_{\mathrm{BD}}}{a}^{-\tfrac{13}{2}}\end{eqnarray} \tag{ 4 }$$

$$\begin{eqnarray}&&\displaystyle \frac{1}{{\tau }_{{\rm{e}}}}=\displaystyle \frac{1}{{\tau }_{\mathrm{circ},\star }}+\displaystyle \frac{1}{{\tau }_{\mathrm{circ},\mathrm{BD}}},\end{eqnarray} \tag{ 5 }$$

where ${\tau }_{{\rm{e}}}$ is the circularization timescale, a is the semimajor axis, M_⋆ is the stellar mass, R_⋆ is the stellar radius, M_BD is the BD mass, R_BD is the BD radius, Q_⋆ is the tidal quality factor for the star, and Q_BD is the tidal quality factor for the BD. Equation (5) is a prediction on how long it takes for the orbital eccentricity of an object to decrease by an exponential factor (the relationship ${\tau }_{{\rm{e}}}\propto {dt}\propto -{de}/e$), based on the tides raised on the star and BD.

Use of this equation comes with a number of assumptions that we reiterate here from Jackson et al. (2008): (1) the BD is in a short orbital period (10 days or less), (2) the orbital eccentricity e is small (though for companions in the planetary mass range, higher-order terms may be important to account for higher e in the past), (3) the BD's orbital period P_orb is smaller than the host star's rotation period P_rot, and (4) Q_⋆ is independent of the tidal forcing frequency. Admittedly, Equation (5) and these assumptions cater to hot Jupiters and not the type of more massive BDs in this study. 

With these considerations in mind, we calculate τ_e for TOI-569b and TOI-1406b for a range of Q_⋆ and Q_BD for each system in Table 6.  The choice to adopt a Q_BD as low as 10^4.5 comes from Beatty et al. (2018), who directly constrain Q_BD for CWW 89Ab, a M_BD = 39 ${M}_{{\rm{J}}}$ BD. The choice to adopt a Q_⋆ as low as 10⁵ comes from studies of circularization of binary stars (Meibom & Mathieu 2005; Milliman et al. 2014). For the bimodal posterior distributions of TOI-569, we only use the most probable M_⋆, R_⋆, M_BD, R_BD, and a.  

Table 6.  Circularization Timescales for Different Values of Q_⋆ and Q_BD, with Stellar Rotational Period and BD Orbital Period also Shown

Object Name and Age	Q⋆	QBD	τe (Gyr)
TOI-569	107	106	${8.0}_{-1.2}^{+0.8}$
${4.70}_{-1.30}^{+1.30}$ Gyr	107	104.5	${7.5}_{-1.1}^{+0.7}$
	106	104.5	${0.80}_{-0.12}^{+0.08}$
	105	104.5	${0.08}_{-0.01}^{+0.01}$
TOI-1406	107	106	${127}_{-24.1}^{+26.2}$
${3.20}_{-1.60}^{+2.20}$ Gyr	107	104.5	${91.5}_{-17.0}^{+18.4}$
	106	104.5	${12.4}_{-2.3}^{+2.5}$
	105	104.5	${1.3}_{-0.2}^{+0.3}$
	Prot (days)	Porb (days)	$v\sin {I}_{\star }$ (km s−1)
TOI-569	12.9	6.6	5.3 ± 0.5
	Prot (days)	Porb (days)	$\mathrm{FWHM}$ (km s−1)
TOI-1406	≤5.3	10.6	12.9 ± 0.2

Note. We quote the rotation period measured from the periodogram of the TESS light curve for TOI-569. An upper limit on the rotation period of TOI-1406 is calculated (Equation (2)) using R_⋆ from EXOFASTv2 and FWHM as an approximation for $v\sin {I}_{\star }$. Since we do not directly have the inclination of the star's rotation axis for TOI-1406, the rotation period listed for this star is an upper limit. The $v\sin {I}_{\star }$ and FWHM values are from CHIRON.  These show 1σ uncertainties.

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We highlight the tidal theory here to show that for these BDs, it is difficult to pin down the timescale over which tidal interactions influence their orbits. Though both BDs have circular orbits, we may only conclude that TOI-1406b likely underwent a low-eccentricity migration, unless tidal dissipation was extremely efficient. The circularization timescales for TOI-569 may be short enough such that tidal interactions alone would have circularized the orbit of the BD over the system's age, thus making it difficult to tell whether or not the BD formed in a circular orbit. 

The mass distribution for the current population of transiting BDs is shown in Figure 10, with the effective temperature of the host star indicated by the colors of the points. From all published studies of transiting BDs to date, there is no obvious preference for a particular type of star to host transiting BDs. Interestingly, we see that six transiting BDs (roughly 20% of the transiting BD population, see Table 7) are hosted by an M dwarf star. This is in contrast to hot Jupiters, where only a small percentage of the hot Jupiter population is found transiting M dwarf stars (e.g., Kepler-45b Johnson et al. 2012, HATS-6b Hartman et al. 2015, WASP-80b Triaud et al. 2015, NGTS-1b Bayliss et al. 2018).  By placing the transiting BD population in the context of eclipsing low-mass stars and hot Jupiters, we may also explore the idea that the scarcity of transiting BDs stems from them, spanning the space between the tail ends of the distributions for companions that form like giant planets versus companions that form like low-mass stars. However, more transiting BD discoveries are needed for such studies to yield meaningful results.

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Here we will discuss how we use transiting BDs with well-determined masses, radii, and ages to test the substellar isochrones from Baraffe et al. (2003; for irradiated BDs) and Saumon & Marley (2008; for non-irradiated BDs). The Baraffe et al. (2003) models use the same input physics that Chabrier & Baraffe (1997) used for main-sequence stars. These are scaled appropriately in Baraffe et al. (2003) for low-mass stars and substellar objects down to $\lessapprox 1{M}_{{\rm{J}}}$. The way we test these models is by having independent measurements of a transiting BD's age, which comes from the age of its host star (assuming the BD is the same age as its host star). We prioritize the use of stellar ages obtained through studies of clusters, asteroseismology, and gyrochronology, but with Gaia DR2, we are able to reliably determine stellar properties to derive accurate ages with stellar isochrone models such as MIST. This increases the number of transiting BDs for which we have reliable and independently determined ages for comparison to substellar isochrones. This is important because we only know the companion as well as we know the host star, and with Gaia DR2, we can now examine the host star with greater precision than ever before.

The mass–radius diagram for transiting BDs is shown in Figure 11. All of the BDs on this diagram are necessarily transiting because it is through the transit method that we can measure the radius. However, even though a transit provides some measure of the radius, the measurement is not always very precise (i.e., as precise as the measurement of the stellar radius). This is important because the radius of a BD changes drastically with its age (see Baraffe et al. 2003). Substellar isochrones are challenging to test at ages beyond a few Gyr because the rate at which BD radii contract significantly decelerates, resulting in transiting BD radii approaching an asymptotic limit for the oldest systems (see the differences between the 3 and 10 Gyr substellar isochrones in Figure 11).

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Table 9.  MIST Median Values and 68% Confidence Interval for TOI-569, Created Using EXOFASTv2 Commit Number 65aa674

Parameter	Units	Priors	Most-likely Values	Less-likely Values
Stellar Parameters:			Prob. = 0.73	Prob. = 0.27
M*	Mass (${M}_{\odot }$)	⋯	${\bf{1.21}}\pm {\bf{0.05}}$	${1.10}_{-0.05}^{+0.03}$
R*	Radius (${R}_{\odot }$)	⋯	${\bf{1.48}}\pm {\bf{0.03}}$	1.48 ± 0.03
L*	Luminosity (${L}_{\odot }$)	⋯	${\bf{2}}.{{\bf{15}}}_{-{\bf{0.12}}}^{+{\bf{0.15}}}$	2.09 ± 0.1
${\rho }_{* }$	Density (cgs)	⋯	${\bf{0.54}}\pm {\bf{0.04}}$	0.48 ± 0.03
$\mathrm{log}g$	Surface gravity (cgs)	⋯	${\bf{4.19}}\pm {\bf{0.03}}$	4.14 ± 0.02
Teff	Effective temperature (K)	${ \mathcal G }[5699,106]$	${{\bf{5768}}}_{-{\bf{92}}}^{+{\bf{110}}}$	${5720}_{-85}^{+94}$
$[\mathrm{Fe}/{\rm{H}}]$	Metallicity (dex)	${ \mathcal G }[+0.23,0.10]$	$+{\bf{0}}.{{\bf{29}}}_{-{\bf{0.08}}}^{+{\bf{0.09}}}$	+0.23 ± 0.09
Age	Age (Gyr)	⋯	${\bf{4.70}}\pm {\bf{1.3}}$	${7.50}_{-1.20}^{+1.80}$
AV	V-band extinction (mag)	${ \mathcal U }[0,1.1749]$	${\bf{0}}.{{\bf{067}}}_{-{\bf{0.049}}}^{+{\bf{0.083}}}$	${0.052}_{-0.038}^{+0.074}$
${\sigma }_{\mathrm{SED}}$	SED photometry error scaling	⋯	${\bf{2}}.{{\bf{99}}}_{-{\bf{0.71}}}^{+{\bf{1.10}}}$	${3.08}_{-0.78}^{+1.40}$
ϖ	Parallax (mas)	${ \mathcal G }[6.3723,0.0306]$	${\bf{6.374}}\pm {\bf{0.031}}$	${6.374}_{-0.030}^{+0.031}$
d	Distance (pc)	⋯	${\bf{156.88}}\pm {\bf{0.75}}$	156.83 ± 0.75
$v\sin {I}_{\star }$	Projected equatorial velocity (km s−1)	Not modeled	${\bf{5.30}}\pm {\bf{0.50}}$	5.30 ± 0.50
Brown Dwarf Parameters:
P	Period (days)	⋯	${\bf{6}}.{{\bf{55604}}}_{-{\bf{0.00015}}}^{+{\bf{0.00016}}}$	${6.55603}_{-0.00015}^{+0.00016}$
MP	Mass (${M}_{{\rm{J}}}$)	⋯	${\bf{64}}.{{\bf{1}}}_{-{\bf{1.4}}}^{+{\bf{1.9}}}$	${59.6}_{-1.7}^{+1.1}$
RP	Radius (${R}_{{\rm{J}}}$)	⋯	${\bf{0.75}}\pm {\bf{0.02}}$	0.76 ± 0.02
TC	Time of conjunction (${\mathrm{BJD}}_{\mathrm{TDB}}$)	⋯	${\bf{2458523}}.{{\bf{09192}}}_{-{\bf{0.00069}}}^{+{\bf{0.00070}}}$	2458523.09199 ± 0.00070
a	Semimajor axis (au)	⋯	${\bf{0.07428}}\pm {\bf{0.00059}}$	${0.07207}_{-0.00069}^{+0.00047}$
i	Orbital inclination (°)	⋯	${\bf{85}}.{{\bf{37}}}_{-{\bf{0.11}}}^{+{\bf{0.13}}}$	${85.15}_{-0.12}^{+0.13}$
e	Eccentricity	⋯	${\bf{0}}.{{\bf{0017}}}_{-{\bf{0.0012}}}^{+{\bf{0.0018}}}$	${0.0017}_{-0.0012}^{+0.0018}$
$e\cos {\omega }_{* }$		⋯	${\bf{0}}.{{\bf{0002}}}_{-{\bf{0.0011}}}^{+{\bf{0.0015}}}$	${0.0002}_{-0.0011}^{+0.0015}$
$e\sin {\omega }_{* }$		⋯	${\bf{0}}.{{\bf{0005}}}_{-{\bf{0.0012}}}^{+{\bf{0.0022}}}$	${0.0005}_{-0.0012}^{+0.0022}$
Teq	Equilibrium temperature (K)	⋯	${{\bf{1227}}}_{-{\bf{12}}}^{+{\bf{13}}}$	${1237}_{-13}^{+12}$
K	RV semi-amplitude (m s−1)	⋯	${\bf{5884}}\pm {\bf{17}}$	${5884}_{-17}^{+18}$
log K	Log of RV semi-amplitude	⋯	${\bf{3.7697}}\pm {\bf{0.0013}}$	3.7697 ± 0.0013
RP/R*	Radius of planet in stellar radii	⋯	${\bf{0}}.{{\bf{05217}}}_{-{\bf{0.00091}}}^{+{\bf{0.00094}}}$	${0.05258}_{-0.00096}^{+0.0010}$
a/R*	Semimajor axis in stellar radii	⋯	${\bf{10}}.{{\bf{81}}}_{-{\bf{0.21}}}^{+{\bf{0.22}}}$	${10.44}_{-0.20}^{+0.21}$
δ	Transit depth (fraction)	⋯	${\bf{0}}.{{\bf{002721}}}_{-{\bf{0.000096}}}^{+{\bf{0.00010}}}$	${0.002765}_{-0.000100}^{+0.00011}$
τ	Ingress/egress transit duration (days)	⋯	${\bf{0}}.{{\bf{0214}}}_{-{\bf{0.0013}}}^{+{\bf{0.0015}}}$	${0.0230}_{-0.0013}^{+0.0014}$
b	Transit impact parameter	⋯	${\bf{0}}.{{\bf{8739}}}_{-{\bf{0.0084}}}^{+{\bf{0.0082}}}$	${0.8820}_{-0.0075}^{+0.0068}$
${\rho }_{P}$	Density (cgs)	⋯	${{\bf{187}}}_{-{\bf{16}}}^{+{\bf{17}}}$	${169}_{-15}^{+16}$
log gP	Surface gravity	⋯	${\bf{5}}.{{\bf{444}}}_{-{\bf{0.031}}}^{+{\bf{0.029}}}$	${5.412}_{-0.026}^{+0.027}$
${M}_{P}\sin i$	Minimum mass (${M}_{{\rm{J}}}$)	⋯	${\bf{63.6}}\pm {\bf{1.0}}$	${59.9}_{-0.9}^{+0.8}$
${M}_{P}/{M}_{* }$	Mass ratio	⋯	${\bf{0}}.{{\bf{05043}}}_{-{\bf{0.00050}}}^{+{\bf{0.00097}}}$	${0.05195}_{-0.00039}^{+0.00045}$
Wavelength Parameters:		I-band	TESS band
u1	Linear limb-darkening coeff	0.292 ± 0.050	0.335 ± 0.049
u2	Quadratic limb-darkening coeff	0.260 ± 0.050	0.269 ± 0.049
RV Parameters:		CHIRON	CORALIE	FEROS
${\gamma }_{\mathrm{rel}}$	Relative RV offset (m s−1)	${71964}_{-16}^{+14}$	${73413}_{-18}^{+17}$	${73402}_{-55}^{+63}$
${\sigma }_{J}$	RV jitter (m s−1)	${37}_{-17}^{+23}$	${37}_{-16}^{+27}$	${70}_{-70}^{+210}$
${\sigma }_{J}^{2}$	RV jitter variance	${1420}_{-1000}^{+2300}$	${1390}_{-920}^{+2800}$	${5000}_{-7100}^{+76000}$
Transit Parameters:		LCOGT UT 2019-04-15 (I-band)	TESS
${\sigma }^{2}$	Added variance	${0.00000202}_{-0.00000026}^{+0.00000030}$	${0.0000001792}_{-0.0000000062}^{+0.0000000063}$
F0	Baseline flux	0.99981 ± 0.00013	1.0000143 ± 0.0000053

Note. The most likely values (probability of 0.73) and the ones we report for this system are shown in boldface. Here, ${ \mathcal U }$[a, b] is the uniform prior bounded between a and b, and ${ \mathcal G }[a,b]$ is a Gaussian prior of mean a and width b. We show $v\sin {I}_{\star }$ (measured from CHIRON) only for convenient reference; EXOFASTv2 does not model $v\sin {I}_{\star }$.

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Taking a more critical look at Figure 11, we notice some interesting features. Among the most noticeable are the large (>10%) uncertainties on the radii of no fewer than six transiting BDs (AD 3116b, CoRoT-15b, CoRoT-33b, NGTS-7b, NLTT 41135b, and TOI-503b). These are the least informative data points in the substellar mass–radius diagram, especially for objects younger than 1 Gyr, as the radius of a transiting BD changes rapidly at ages less than 1 Gyr. Notice also how the substellar isochrone models are mostly horizontal from roughly 20 to 70 ${M}_{{\rm{J}}}$. This means that testing the age of the isochrones is less sensitive to the precision on the mass than the precision on the radius of a transiting BD.

Interestingly, the oldest substellar isochrones are traced fairly well by a handful of transiting BDs. This suggests that the oldest substellar isochrones accurately predict the radii of transiting BDs that approach this asymptotic limit. Future works to improve on the Baraffe et al. (2003; COND03) and Saumon & Marley (2008; SM08) models must consider the effects of metallicity for transiting BDs, as this may be key to more finely distinguishing the older substellar isochrones from each other, especially in the asymptotic radius regime of 20 to 70 ${M}_{{\rm{J}}}$. The COND03 models do not explore a variety of different metallicity values, as the SM08 models do, but the SM08 models do not consider the effects of irradiation like the COND03 models. Additionally, improvements may be made to BDs less massive than 20 ${M}_{{\rm{J}}}$, as the input physical prescription from Baraffe et al. (2003) and Chabrier & Baraffe (1997) may not describe these BDs as well as they do more massive objects.

4.2.1. TOI-569b and TOI-1406b in the Mass–Radius Diagram

TOI-569b and TOI-1406b are two newly discovered BDs that transit their host stars in nearly circular orbits. TOI-569 appears to be a slightly evolved G dwarf star with strong photometric modulation interpreted as evolving star spots on the surface of the star. We use the TESS and WASP light curves to extract an estimate for the rotation period of this star to be 13 days and determine that the star's rotational axis is marginally misaligned with the orbital inclination of TOI-569b. In contrast, TOI-1406 is an F star on the main sequence, and with no noticeable photometric modulation over the sectors, the star was observed by TESS. By comparing the ages of each system to a range of plausible circularization timescales, we find that we are not able to convincingly determine the orbital history of TOI-569 and that we can at least rule out significant high-eccentricity orbital evolution followed by tidal circularization for TOI-1406.

We demonstrate here how stellar isochrones can be used to test substellar isochrones. This is done by leveraging Gaia DR2 for precise stellar parameters, which translate into better estimates of masses, radii, and ages, of transiting BD. 

Ultimately, we find TOI-569b and TOI-1406b to be special in that they contribute new measurements to the still sparsely populated mass–radius diagram for transiting BDs. In addition to providing some of the first examples of a test of the COND03 and SM08 models against stellar isochrones, these systems also offer themselves as new data to examine circularization models. As we build the population of transiting BDs, we will refine the predictive power of substellar isochrones and potentially turn them into tools that are useful for estimating the ages of transiting BDs.