# Verification of Gaia DR3 Single-lined Spectroscopic Binary Solutions With
Three Transiting Low-mass Secondaries

Stephen P. Schmidt William H. Miller III Department of Physics and Astronomy, Johns Hopkins University, 3400 N Charles St, Baltimore, MD 21218, USA  Kevin C. Schlaufman William H. Miller III Department of Physics and Astronomy, Johns Hopkins University, 3400 N Charles St, Baltimore, MD 21218, USA  Carnegie Institution for Science Earth & Planets Laboratory, 5241 Broad Branch Road NW, Washington, DC 20015, USA  Keyi Ding (丁可怿) William H. Miller III Department of Physics and Astronomy, Johns Hopkins University, 3400 N Charles St, Baltimore, MD 21218, USA  Samuel K. Grunblatt William H. Miller III Department of Physics and Astronomy, Johns Hopkins University, 3400 N Charles St, Baltimore, MD 21218, USA  Theron Carmichael Institute for Astronomy, University of Edinburgh, Royal Observatory, Blackford Hill, Edinburgh, EH9 3HJ, UK  Allyson Bieryla Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA  Joseph E. Rodriguez Center for Data Intensive and Time Domain Astronomy, Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA  Jack Schulte Center for Data Intensive and Time Domain Astronomy, Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA  Noah Vowell Center for Data Intensive and Time Domain Astronomy, Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA  George Zhou Centre for Astrophysics, University of Southern Queensland, West Street, Toowoomba, QLD 4350, Australia  Samuel N. Quinn Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA  Samuel W. Yee Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA  Joshua N. Winn Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA  Joel D. Hartman Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA  David W. Latham Center for Astrophysics | Harvard & Smithsonian, 60 Garden St, Cambridge, MA 02138, USA  Douglas A. Caldwell SETI Institute, Moffett Field, Mountain View, CA, 94035, USA  NASA Ames Research Center, Moffett Field, CA 94035, US  M. M. Fausnaugh Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Department of Physics & Astronomy, Texas Tech University, Lubbock, TX, 79410-1051, USA  Christina Hedges University of Maryland, Baltimore County, 1000 Hilltop Cir, Baltimore, MD 21250, USA  NASA Goddard Space Flight Center, 8800 Greenbelt Rd, Greenbelt, MD 20771, USA  Jon M. Jenkins NASA Ames Research Center, Moffett Field, CA 94035, US  Hugh P. Osborn Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  NCCR/Planet-S, Universität Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland  S. Seager Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Department of Aeronautics and Astronautics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA Stephen Schmidt<EMAIL_ADDRESS>
(Received August 2, 2023; Revised October 5, 2023)

###### Abstract

While secondary mass inferences based on single-lined spectroscopic binary
(SB1) solutions are subject to $\sin{i}$ degeneracies, this degeneracy can be
lifted through the observations of eclipses. We combine the subset of Gaia
Data Release (DR) 3 SB1 solutions consistent with brown dwarf-mass secondaries
with the Transiting Exoplanet Survey Satellite (TESS) Object of Interest (TOI)
list to identify three candidate transiting brown dwarf systems. Ground-based
precision radial velocity follow-up observations confirm that TOI-2533.01 is a
transiting brown dwarf with
$M=72^{+3}_{-3}~{}M_{\text{Jup}}=0.069^{+0.003}_{-0.003}~{}M_{\odot}$ orbiting
TYC 2010-124-1 and that TOI-5427.01 is a transiting very low-mass star with
$M=93^{+2}_{-2}~{}M_{\text{Jup}}=0.088^{+0.002}_{-0.002}~{}M_{\odot}$ orbiting
UCAC4 515-012898. We validate TOI-1712.01 as a very low-mass star with
$M=82^{+7}_{-7}~{}M_{\text{Jup}}=0.079^{+0.007}_{-0.007}~{}M_{\odot}$
transiting the primary in the hierarchical triple system BD+45 1593. Even
after accounting for third light, TOI-1712.01 has radius nearly a factor of
two larger than predicted for isolated stars with similar properties. We
propose that the intense instellation experienced by TOI-1712.01 diminishes
the temperature gradient near its surface, suppresses convection, and leads to
its inflated radius. Our analyses verify Gaia DR3 SB1 solutions in the low
Doppler semiamplitude limit, thereby providing the foundation for future joint
analyses of Gaia radial velocities and Kepler, K2, TESS, and PLAnetary
Transits and Oscillations (PLATO) light curves for the characterization of
transiting massive brown dwarfs and very low-mass stars.

Brown dwarfs(185) — Eclipsing binary stars(444) — Substellar companion
stars(1648) — Spectroscopic binary stars(1557) — Stellar radii(1626) — Trinary
stars(1714) — Low mass stars(2050)

††journal: the Astronomical Journal††facilities: ADS, CDS, CTIO:1.5m (CHIRON),
CTIO:2MASS, ExoFOP, FLWO:1.5m (TRES), FLWO:2MASS, Gaia, GALEX, IRSA, MAST,
NEOWISE, TESS, WISE, WIYN (NEID)††software: astropy (Astropy Collaboration et
al., 2013, 2018, 2022), isochrones (Morton, 2015), juliet (Espinoza et al.,
2019), matplotlib (Hunter, 2007), numpy (Harris et al., 2020), pandas (Wes
McKinney, 2010; pandas development team, 2020), PyMultiNest (Buchner et al.,
2014), R (R Core Team, 2023), scipy (Virtanen et al., 2020; Jones et al.,
2001–), thejoker (Price-Whelan et al., 2017)

##  1 Introduction

While isochrones can be used to infer the masses and radii of isolated low-
mass stars and brown dwarfs, the theoretical models underlying those
isochrones have difficulties reproducing the measured masses and radii of such
objects (Kraus et al., 2011; Birkby et al., 2012). On the other hand, the
minimum masses of the secondary stars and brown dwarfs in single-lined
spectroscopic binary systems (SB1s) can be measured relative to the masses of
the primaries in such systems. If the photospheres of the primaries in such
systems are much, much hotter than the temperatures of the secondaries’
“night” sides and the secondaries have significantly smaller radii (both of
which apply for the lowest-mass stellar secondaries), then the eclipses of
such systems can be treated like the transits and occultations of exoplanet
systems. In that case, the depth of an eclipse of the primary by the secondary
can be used to measure the radius of the low-mass secondary relative to the
radius of the primary. The eclipsing nature of such a system lifts the
$\sin{i}$ degeneracy of an SB1 solution as well. If the primary is a Sun-like
star, then isochrone-inferred masses and radii can be both accurate and
precise. Consequently, a homogeneous analysis of a large sample of transiting
single-lined spectroscopic binaries would be a powerful way to infer accurate
and precise masses and radii for a large population of low-mass stars and
brown dwarfs.

Gaia Data Release (DR) 3’s non-single stars catalog (Gaia Collaboration et
al., 2023a; Babusiaux et al., 2023; Katz et al., 2023; Pourbaix et al., 2022)
presented a list of 181,327 single-lined spectroscopic binary solutions. This
collection of SB1s is almost two orders of magnitudes larger than previous
catalogs (e.g., Pourbaix et al., 2004). Of these solutions, 32,455 have Sun-
like primaries (F5V through K5V according to Pecaut & Mamajek, 2013) based on
absolute Gaia $G$-band magnitudes and effective temperatures $T_{\text{eff}}$
from Gaia DR3’s astrophysical parameters data product (Creevey et al., 2023;
Fouesneau et al., 2023). Of these, 2,490 secondaries have minimum masses in
the brown dwarf regime defined by an upper mass cutoff of 78.5 Jupiter masses
(Chabrier et al., 2023). Because of Gaia’s limited time baseline and Doppler
precision, these systems usually have short periods and relatively high
transit probabilities. These SB1 solutions have been validated down to the
brown dwarf mass regime (Gaia Collaboration et al., 2023b). The infrequent
sampling of light curves based on Gaia photometry leaves them insensitive to
all but the shortest-period eclipsing systems.

NASA’s Transiting Exoplanet Survey Satellite (TESS) mission has observed most
of the sky in search of exoplanets and other transiting objects (Ricker et
al., 2014). TESS light curves have been exhaustively searched for candidate
transiting planets (Guerrero et al., 2021). Now in its Second Extended
Mission, TESS data has led to the discovery of thousands of eclipsing binaries
(Prša et al., 2022) and over a dozen transiting very low-mass stars and brown
dwarfs (e.g., Grieves et al., 2021; Carmichael et al., 2022; Psaridi et al.,
2022; Lin et al., 2023; Vowell et al., 2023). The population of TESS-
discovered eclipsing or transiting systems is biased towards periods shorter
than about 10 days due to TESS’s observing strategy and the smaller transit
probabilities of systems with longer periods.

The combination of Gaia SB1 solutions and TESS light curves would be a
powerful resource for the accurate and precise characterization of low-mass
stars and brown dwarfs. In this article, we seek out systems with Gaia SB1
solutions consistent with brown dwarf-mass secondaries that have also been
observed by TESS to transit their primaries. We identify three such systems
for which the SB1 period is well matched by the TESS-inferred period: BD+45
1593 (TOI-1712), TYC 2010-124-1 (TOI-2533), and UCAC4 515-012898 (TOI-5427).
Our ground-based follow-up analysis subsequently reveals that BD+45 1593 Ab
(TOI-1712.01) is a very low-mass star transiting the primary in a hierarchical
triple, TYC 2010-124-1 b (TOI-2533.01) is a transiting brown dwarf, and UCAC4
515-012898 b (TOI-5427.01) is a transiting very low-mass star. We describe the
initial identification of these three objects as potential brown dwarfs in
Section 2. We describe our follow-up observations for the systems and our
subsequent analysis to confirm or validate the three objects in Section 3. We
discuss the implications of these discoveries in Section 4 and summarize our
conclusions in Section 5.

##  2 Candidate Brown Dwarf Identification

We first cross-match Gaia Data Release 3’s non-single stars product with the
TESS Object of Interest (TOI) catalog and the TESS eclipsing binary catalog
(Prša et al., 2022). We next focus on systems where the orbital period from
the Gaia SB1 solution is within 0.3% of the period found by TESS. We then use
the binary mass function

$f(M)=\frac{M_{2}^{3}\sin^{3}{i}}{\left(M_{1}+M_{2}\right)^{2}}=\frac{PK_{1}^{3}}{2\pi
G}\left(1-e^{2}\right)^{3/2},$ (1)

in the $M_{2}\ll M_{1}$ limit to calculate a preliminary minimum mass for each
secondary:

$\frac{M_{2}\sin{i}}{M_{\text{Jup}}}=\left(\frac{K_{1}}{28.4329~{}\text{ms^{-1}}}\right)\sqrt{1-e^{2}}\left(M_{1}\right)^{2/3}P^{1/3}.$
(2)

The inputs to Equation (2) are a primary mass $M_{1}$ in Solar masses from
Gaia’s Final Luminosity Age Mass Estimator (FLAME) and a system’s Gaia DR3 SB1
solution orbital parameters Doppler semiamplitude $K_{1}$ in meters per second
and period $P$ in years. In cases in which a FLAME-derived mass is
unavailable, we use the primary mass from the TESS Follow-up Observing Program
(TFOP) instead. For systems in which secondary mass is comparable to that of a
brown dwarf, we refine our preliminary mass estimates by calculating more
precise stellar parameters for the primary.

This procedure yields three systems with candidate brown dwarfs: BD+45 1593
(Gaia DR3 917823348536224768/TOI-1712), TYC 2010-124-1 (Gaia DR3
1259922196651254272/TOI-2533), and UCAC4 515-012898 (Gaia DR3
3341401062125577088/TOI-5427). We illustrate these systems in Figure 1 and
describe them in detail below. These SB1 solutions are based on Gaia $R\approx
11500$ Radial Velocity Spectrometer (RVS) spectra that cover the wavelength
range $846\text{~{}nm}\lesssim\lambda\lesssim 870\text{~{}nm}$ (Katz et al.,
2023). The SB1 solutions for BD+45 1593 used 18 “good” radial velocity (RV)
measurements, TYC 2010-124-1 used 47 “good” RV measurements, and UCAC4
515-012898 used 25 “good” RV measurements. We refer to these systems by their
TESS Object of Interest designations TOI-1712, TOI-2533, and TOI-5427 in the
context of analyses based on data sourced solely from TESS or TFOP. In all
other context, we refer to these systems by their Simbad designations BD+45
1593, TYC 2010-124-1, and UCAC4 515-012898.

Figure 1: Doppler curves for BD+45 1593, TYC 2010-124-1, and UCAC4 515-012898 based on their Gaia DR3 and ground-based SB1 solutions. We plot 2000 partially transparent blue lines to represent individual solutions sampled from normal distributions of the Gaia Doppler semiamplitude, eccentricity, and argument of periastron, with the standard deviation of each distribution representing the reported error on its respective parameter. Gaia’s reported uncertainties on each parameter are derived from the variance-covariance matrix of the solution, but the matrix itself is not reported; we therefore do not correct for it in this plot. We plot the individual precision ground-based radial velocity measurements and their uncertainties as light blue circles with black borders and error bars. We plot the Doppler curves inferred from these data as dashed black lines. The agreement between the Gaia and precision ground-based SB1 solutions demonstrates the fidelity of the Gaia SB1 solutions at low Doppler semiamplitude. Table 1: TESS Observations of the Three Objects  Sector | Corresponding Time Period | Cadence (Minutes)
---|---|---
TOI-1712 (TIC 67926921)
20 | December 24, 2019 - January 20, 2020 | 30
47 | December 30, 2021 - January 28, 2022 | 2, 10
TOI-2533 (TIC 418012030)
23 | March 19, 2020 - April 15, 2020 | 30
TOI-5427 (TIC 52420398)
6 | December 12, 2018 - January 6, 2019 | 30
43 | September 16, 2021 - October 12, 2021 | 10
44 | October 12, 2021 - November 6, 2021 | 10
45 | November 6, 2021 - December 2, 2021 | 10

TOI-1712 and TOI-5427 were discovered by the TESS Science Office, while
TOI-2533 was first discovered by Rafael Brahm. TOI-2533 then became a
Community TESS Object of Interest (CTOI), and later was adopted as a TOI by
the TESS Science Office. TOI-1712 and TOI-2533 were identified by the TESS
Quick-Look Pipeline (QLP \- Huang et al., 2020a, b), while TOI-5427 was
identified by the TESS QLP Faint-star Search (Kunimoto et al., 2022). We give
in Table 1 the details of the TESS observations of the three objects. For
stars like the primaries in these systems that were not allocated postage
stamps, TESS collected observations with a 30 minute cadence during its Prime
Mission and a 10/2 minute cadence during its First/Second Extended Mission.

Because of TESS’s large pixels, third light can make it difficult to interpret
transit depths directly as secondary-to-primary radius ratios. To investigate
the possibility of third light in these systems, we do two things. We first
confirm that the TESS contamination ratios111TESS contamination ratios are
given as the column contratio in the TESS Input Catalog (Stassun et al.,
2019). are small for each system: 0.00054 for TOI-1712, 0.039224 for TOI-2533,
and 0.263055 for TOI-5427. We next use the high spatial resolution Gaia DR3
source catalog and the plotting function within the giants pipeline (Saunders
et al., 2022; Grunblatt et al., 2022a) to show that there are no bright
sources within the TESS lightkurve-generated (Lightkurve Collaboration et al.,
2018) Full Frame Image (FFI) PSF of any of these three systems (Figure 2).

Figure 2: TESS Full-Frame Image (FFI) approximately $2048\times 2048$ pixel
postage stamps for TOI-1712, TOI-2533, and TOI-5427. TESS collected the FFI
for TOI-1712 during Sector 20, the FFI for TOI-2533 during Sector 23, and the
FFI for TOI-5427 during Sector 6. We center each postage stamp on the target
and mark each $G<15$ Gaia DR3 source with a white circle of radius inversely
proportional to the Gaia $G$ magnitude. These FFIs show that blending is not a
concern for our light curves of the objects.

We derive the fundamental and photospheric stellar parameters of BD+45 1593,
TYC 2010-124-1, and UCAC4 515-012898 using the isochrones (Morton, 2015)
package to execute with MultiNest (Feroz & Hobson, 2008; Feroz et al., 2009,
2019) a simultaneous Bayesian fit of the MIST isochrone grid (Paxton et al.,
2011, 2013, 2018, 2019; Jermyn et al., 2023; Dotter, 2016; Choi et al., 2016)
to a curated collection of data for each star. The stellar parameters derived
using this approach have been shown by Ding et al. (2024, in prep) to
reproduce the asteroseismology-inferred masses and radii of the Kepler LEGACY
Sample (Silva Aguirre et al., 2017). We fit (1) a zero point-corrected Gaia
DR3 parallax (Gaia Collaboration et al., 2021; Fabricius et al., 2021;
Lindegren et al., 2021a, b; Rowell et al., 2021; Torra et al., 2021), (2)
Galaxy Evolution Explorer (GALEX) GUVcat_AIS NUV and FUV (Bianchi et al.,
2017), Tycho-2 $B_{T}$ and $V_{T}$ (Høg et al., 2000), Gaia DR2 $G$ (Gaia
Collaboration et al., 2016, 2018; Arenou et al., 2018; Evans et al., 2018;
Riello et al., 2018), Two-micron All-sky Survey (2MASS) $JHK_{\text{s}}$
(Skrutskie et al., 2006), and/or Wide-field Infrared Survey Explorer AllWISE
$W1W2W3$ photometry (Wright et al., 2010; Cutri & et al., 2014), and (3) an
estimated extinction value $A_{V}$ based on three-dimensional reddening maps
(Lallement et al., 2022; Vergely et al., 2022). GALEX photometry is
unavailable for TYC 2010-124-1 and UCAC4 515-012898, and Tycho-2 photometry is
unavailable for UCAC4 515-012898. We use a uniform extinction prior bounded by
$A_{V}\pm 5\sigma_{A_{V}}$ and a distance prior proportional to volume bounded
by the Bailer-Jones et al. (2021) lower and upper bounds on geometric
distances. We report the photospheric stellar parameters in Table 3 and
compare the isochrone-derived magnitudes to each star’s observed magnitudes in
Figure 3.

As we will show in the next section, there is evidence that BD+45 1593 is a
double-lined spectroscopic binary. To determine the properties of both
luminous stars in the system, we use an isochrones binary star model that
requires both the primary and secondary to have the same age, composition, and
distance. We use the full set of inputs mentioned previously. In Section 3.1,
we will use the TESS $T$-band flux ratio so inferred to show that the dilution
of the transit in the BD+45 1593 system is small. The modeled photometry that
results from the isochrones is consistent with the observed photometry, and
this consistency supports the accuracy of the binary star model output (Figure
3).

Our isochrone analysis gives us several reasons to believe that the low-mass
object in the BD+45 1593 system orbits the most massive component BD+45 1593
A. We infer $T$-band magnitudes $9.95^{+0.05}_{-0.03}$ for the primary BD+45
1593 A and $13.90^{+2.47}_{-1.12}$ for the secondary BD+45 1593 B. Given that
the posterior-derived Gaia DR2 $G$ magnitudes for the two stars are
$10.30^{+0.05}_{-0.02}$ and $14.46^{+2.81}_{-1.25}$, we argue that this flux
difference would have left Gaia insensitive to the possible RV variation of
BD+45 1593 B. As we will show, the second set of lines in Tillinghast
Reflector Echelle Spectrograph reconnaissance spectra of BD+45 1593 do not
move on the time scale of months. On the other hand, the agreement of the Gaia
DR3 SB1 period and the transit-inferred TOI period suggests that the object of
interest in the system orbits BD+45 1593 A rather than BD+45 1593 B.
Therefore, we refer to the primary of this system as BD+45 1593 Aa and report
its parameters in Table 3.

We find that BD+45 1593 Aa is a subgiant with $M_{\ast}\approx
1.63^{+0.01}_{-0.02}~{}M_{\odot}$. It has been shown that the rotation period
of a subgiant is proportional to its mass and therefore age (van Saders &
Pinsonneault, 2013). Assuming an aligned orbit, our inferred mass for BD+45
1593 Aa is large enough that Gaia should be able to detect the rotational
broadening of its absorption lines. Indeed, the Gaia DR3 vbroad parameter for
BD+45 1593 is $30\pm 6$ km s-1. We assume that all of the line broadening
measured by the Gaia DR3 vbroad parameter is a result of rotation and use it
to calculate the rotation period of BD+45 1593 Aa. We divide the isochrone-
inferred circumference of BD+45 1593 Aa by its Gaia DR3 vbroad parameter,
resulting in a rotation period $P_{\text{rot}}\approx 5$ days. It is possible
that angular momentum exchange between the system’s revolution and the
rotation of BD+45 1593 Aa would lead to its faster rotation, though in that
case one would expect that the system’s orbital period $P=3.566276\pm
0.0000044$ days and the rotation period of the primary $P_{\text{rot}}\approx
5$ days would be synchronized. The agreement between our inferred rotation
period $P_{\text{rot}}\approx 5$ and the van Saders & Pinsonneault
(2013)-predicted slow-launch rotation period $P_{\text{slow}}\approx 6$ days
for a star of the mass, composition, and temperature of BD+45 1593 Aa supports
our inference that BD+45 1593 Aa is massive.

Figure 3: Comparison between the observed magnitudes for the three systems and
the isochrone-derived magnitudes of the combined binary system (in the case of
BD+45 1593) or the primaries (in the case of TYC 2010-124-1 and UCAC4
515-012898). For each filter, we plot filled circles to represent the observed
magnitudes and unfilled circles to represent the isochrone-inferred
magnitudes. We plot the relative transmission for each filter obtained from
the SVO Filter Profile Service (Rodrigo et al., 2012; Rodrigo & Solano, 2020)
above its photometric pair. There is a discontinuity on the wavelength axis at
the top of each plot between the 2MASS magnitudes and AllWISE magnitudes. The
Tycho-2 magnitudes for TYC 2010-124-1 are close to the catalog’s limit of
reliability (Høg et al., 2000). The $B_{T}$ magnitude difference corresponds
to $2.1~{}\sigma$, which is unsurprising given the number of comparisons in
our sample. However, the $V_{T}$ magnitude difference corresponds to
$3.2~{}\sigma$, which we attribute to underestimated random uncertainties or
systematic uncertainties at the faint end of the Tycho-2 photometric
calibration. The consistency and overall small offsets between the observed
and isochrone-inferred magnitudes demonstrates that our method of deriving the
host stars’ parameters is robust, especially when UV photometry is available.

##  3 Follow-up Observations and Analysis

###  3.1 Light Curve Analysis

The Box-fitting Least Squares (BLS - Kovács et al., 2002) algorithm-based
transit depths reported in the TOI catalog are only indicative of true transit
depths. We therefore reanalyze the TESS light curves of TOI-1712, TOI-2533,
and TOI-5427 with the juliet Python package (Espinoza et al., 2019) to obtain
more accurate and precise radius ratio posteriors for these three systems. We
retrieve the light curve data products originally reduced by the TESS Science
Processing Operations Center (SPOC - Jenkins et al., 2016) for the three
systems from the Mikulski Archive for Space Telescopes (MAST). For each
system, we use the available 30-minute cadence light curves produced from TESS
Full-Frame Images by the TESS-SPOC High Level Science Products (HLSP) project
(Caldwell et al., 2020). For TOI-1712, we additionally use the 2-minute
cadence Presearch Data Conditioned (PDC-SAP \- Stumpe et al., 2012, 2014;
Smith et al., 2012) light curve from Sector 47 produced by the SPOC. Likewise,
for TOI-5427 we additionally use 10-minute cadence light curves from Sectors
43, 44, and 45 also produced from TESS FFIs by the TESS-SPOC HLSP project.

We mask the transits in each light curve based on the ephemeris from the TOI
catalog and use the Gaussian Process (GP) modeling tool in juliet to model out
the stellar variability. We use a simple approximate Matern kernel for this
process. For the dilution factor we use a fixed prior at 1, for the mean out-
of-transit flux we use a Gaussian prior centered at 0 with a standard
deviation of 0.1, for the jitter term $\sigma_{w}$ and amplitude of the GP
$\sigma_{GP}$ we use logarithmic uniform priors between $10^{-6}$ and
$10^{6}$, and for the Matern time-scale of the GP $\rho_{GP}$ we use a
logarithmic uniform prior between $10^{-3}$ and $10^{3}$. We plot the systems’
original TESS light curves along with the Gaussian process stellar variability
fit in Figure 4.

After removing stellar variability, for each light curve we unmask the
transits and use the TOI ephemeris as a prior on the juliet fit to extract
inclination and radius ratio posteriors. We use similar priors as before,
except we use the Kipping (2013) quadratic limb-darkening parameterization
with q1 and q2 $\mathcal{U}(0,1)$ priors appropriate for TESS. We plot in
Figure 5 the detrended light curves for the systems with their transit fits,
while in Figure 6 we plot the phase-folded 30 minute-cadence light curves and
transit fit for each system. We report the inclinations and radius ratios in
Table 3.

Figure 4: TESS Science Processing Operations Center (SPOC)-reduced light
curves for TOI-1712, TOI-2533, and TOI-5427. In addition to the 30-minute
cadence data for each of the objects, we include two-minute cadence data for
TOI-1712 from TESS Sector 47 and ten-minute cadence data for TOI-5427 from
Sectors 43, 44, and 45. We plot in red on top of the data the juliet-derived
Gaussian process fits to the light curves. We next use these fits to detrend
the light curves before subsequently fitting the transits. Figure 5: Detrended
light curves for TOI-1712, TOI-2533, and TOI-5427. In addition to the
30-minute cadence data for each of the objects, we include two-minute cadence
data for TOI-1712 from TESS Sector 47 and ten-minute cadence data for TOI-5427
from Sectors 43, 44, and 45. We overlay the data with red lines representing
our juliet-derived transit fits. These fits demonstrates that the periods and
transit shape parameters reported in the TOI catalog accurately reflect the
data and are sufficient for comparison to external catalogs. Figure 6: Phased
light curves for the 30-minute cadence data of TOI-1712, TOI-2533, and
TOI-5427. We overlay the data with red lines representing the juliet-derived
transit fits. The relative flux axes are consistent across the three objects
to illustrate their differing transit depths.

###  3.2 Doppler Analysis

We present in Table 2 27 ground-based precision radial velocity measurements
we obtained for TOI-2533 and TOI-5427. We collected 11 spectra for TOI-2533
between March 5, 2021 and April 8, 2021 using the Tillinghast Reflector
Echelle Spectrograph (TRES - Fűrész, 2008) on the 1.5-m Tillinghast Reflector
telescope, situated on Mt. Hopkins in Arizona, USA. TRES has $R\approx 44000$
over a wavelength range approximately 3900 Å to 9100 Å. The exposure times
ranged from 2200 s to 3600 s with signal-to-noise per resolution element (S/N)
between 23 and 35. We visually reviewed each order to remove cosmic rays.
Depending on the S/N, we used 10-14 echelle orders within this wavelength
range for each spectrum to measure each order’s relative radial velocity. We
used the spectrum with highest S/N per resolution element as the reference
value for these relative radial velocities.

We also collected three spectra for TOI-1712 between March 7, 2020 and
February 3, 2023 using TRES. The exposure times ranged from 600 s to 2160 s
and with S/N between 35 and 67. These spectra revealed a cross-correlation
function indicative of a double-lined spectroscopic binary. The evolution of
the cross-correlation function suggests that (1) the primary shows some
orbital motion as seen by Gaia and (2) the secondary appears stationary.
Combined with the Gaia DR3 SB1 solution, this observation suggests that BD+45
1593 (TOI-1712) is a hierarchical triple system with an unresolved secondary
diluting the transit observed by TESS.

We collected 11 spectra of TOI-5427 between September 25, 2022 and October 7,
2022 using the CTIO High Resolution Spectrometer (CHIRON \- Tokovinin et al.,
2013) high-resolution spectrograph, installed in the 1.5-m telescope at the
Cerro Tololo International Observatory. We used an image slicer that produces
spectra with $R\approx 80000$. The exposure times ranged between 3000 s and
3650 s with S/N between 47 and 82 at 550 nm. We calibrated each science
spectrum with a thorium-argon (ThAr) spectrum taken immediately beforehand to
account for instrumental spectral drift and to compute a new wavelength
solution automatically from this calibration with the CHIRON pipeline (Paredes
et al., 2021). To extract the radial velocities from these CHIRON
observations, we used a least squares deconvolution (Donati et al., 1997)
approach where we deconvolved each spectrum against an ATLAS9 synthetic
template (Kurucz, 1992) following the technique described by Zhou et al.
(2020).

We collected five spectra of TOI-5427 between November 14, 2022 and February
11, 2023 using the NEID spectrograph (Schwab et al., 2016; Halverson et al.,
2016), a stabilized fiber-fed optical spectrograph on the WIYN 3.5-m telescope
at Kitt Peak National Observatory (KPNO). We used the high-resolution (HR)
mode that produces spectra with $R\approx 110000$ with exposure times of 480
s. The data were reduced using v1.2.0 of the standard NEID Data Reduction
Pipeline (NEID-DRP) 222https://neid.ipac.caltech.edu/docs/NEID-DRP, which
derives radial velocities by cross-correlating the observed spectra with a
weighted stellar mask (Baranne et al., 1996; Pepe et al., 2002). The median
radial velocity precision attained was approximately 26 m/s.

Table 2: Ground-based Radial Velocities  BJD | RV (km/s) |  $\sigma_{\text{RV}}$ (km/s) | Source
---|---|---|---
TOI-5427 (UCAC4 515-012898)
2459847.882200 | -44.961 | 0.023 | CHIRON
2459848.840030 | -48.368 | 0.110 | CHIRON
2459849.816380 | -38.673 | 0.033 | CHIRON
2459850.875790 | -28.476 | 0.095 | CHIRON
2459851.851630 | -30.545 | 0.052 | CHIRON
2459852.934530 | -42.431 | 0.038 | CHIRON
2459854.774330 | -42.442 | 0.069 | CHIRON
2459855.860460 | -30.129 | 0.059 | CHIRON
2459856.872970 | -28.896 | 0.036 | CHIRON
2459858.810920 | -47.816 | 0.051 | CHIRON
2459859.871500 | -44.323 | 0.065 | CHIRON
2459899.80424983 | -40.6134 | 0.0167 | NEID
2459898.84596773 | -30.1702 | 0.026 | NEID
2459897.87815973 | -29.9688 | 0.0304 | NEID
2459986.6052418 | -32.669 | 0.0262 | NEID
2459983.81974946 | -43.1964 | 0.0277 | NEID
TOI-2533 (TYC 2010-124-1)
2459278.939688 | 0.00000 | 0.09313 | TRES
2459295.873683 | -12.98947 | 0.06914 | TRES
2459296.893338 | -8.77543 | 0.09313 | TRES
2459298.965028 | -0.02706 | 0.07976 | TRES
2459299.935610 | 1.81353 | 0.10406 | TRES
2459300.924987 | -3.61235 | 0.14566 | TRES
2459301.855739 | -12.74344 | 0.09697 | TRES
2459307.836527 | -6.70705 | 0.10400 | TRES
2459309.818673 | -10.75052 | 0.07288 | TRES
2459311.900542 | -1.65152 | 0.07649 | TRES
2459312.931876 | 1.23157 | 0.08170 | TRES

To fit Keplerian orbits to these ground-based radial velocity data for
TOI-2533 and TOI-5427, we use the thejoker Python package (Price-Whelan et
al., 2017). We do not fit the TRES spectra for TOI-1712. For each system, we
use the default priors in thejoker: a log-uniform prior in period set to be
between the 1-$\sigma$ confidence interval provided by TESS, a Gaussian prior
in Doppler semiamplitude described by $\sigma_{K}=1$ km/s, and a Gaussian
prior in systemic velocity described by $\sigma_{v}=1$ km/s. We sample this
prior 200000 times and conduct rejection sampling to obtain a set of initial
parameters for use in a pymc3 (Salvatier et al., 2016) MCMC simulation. Using
10 chains, we generate a posterior distribution of 5000 samples with a burn-in
of 20000 samples. The resulting posterior distributions are both symmetric and
unimodal. As suggested by the thejoker
documentation333https://thejoker.readthedocs.io/en/latest/, this unimodality
suggests that these posterior samples are independent and represent a unique
solution without the need for convergence tests due to the MCMC sampling
method used by thejoker. We then take the 16th, 50th, and 84th percentiles of
the resulting Doppler semiamplitude and eccentricity posteriors to obtain the
values and uncertainties reported in Table 3.

Zero-point offsets between different precision radial velocity spectrographs
with amplitudes of a few m/s are common. We therefore verify that any zero-
point offsets have not affected our analysis of TOI-5427. We fit the radial
velocities from each instrument with thejoker using the same process described
above. We then compare the systemic velocity distributions between the two in
Figure 7 and obtain 16th, 50th, and 84th percentiles of the offset
distribution by subtracting the systemic velocity posterior from NEID from the
systemic velocity posterior from CHIRON. This results in an offset of
$-0.6^{+21.0}_{-20.5}$ m/s that is smaller than the uncertainties of the
individual radial velocity measurements. This supports our decision to analyze
both data sets without including zero-point offsets between them. We therefore
combine the data from CHIRON and NEID to simultaneously fit the ensemble of
radial velocities for TOI-5427.

Figure 7: Comparison between CHIRON and NEID velocity zero-point offsets for
TOI-5427. We plot distribution medians as solid vertical lines and the 16th
and 84th percentiles as dashed vertical lines. Left: Distributions of Monte
Carlo samples from the thejoker Doppler solution systemic velocities after
fitting to radial velocity measurements taken by CHIRON and NEID. Right:
Distribution of the differences between the two distributions in the left-hand
panel. The resulting median offset is much smaller than the individual radial
velocity uncertainties, suggesting that the data sets are consistent with each
other.

###  3.3 Determination of the Companions’ Properties

We solve the binary star mass function assuming that $\sin^{3}{i}\approx 1$
and re-parameterize Equation (1) as a cubic equation

$\left(\frac{M_{2}}{M_{1}}\right)^{3}-B\left(\frac{M_{2}}{M_{1}}\right)^{2}-2B\left(\frac{M_{2}}{M_{1}}\right)-B=0,$
(3)

where the coefficient $B$ is defined as

$B=\frac{1}{2\pi G}\frac{PK_{1}^{3}\left(1-e^{2}\right)^{3/2}}{M_{1}}.$ (4)

Taking the real root of Equation (3) results in a mass ratio that can then be
used to determine companion masses given primary masses. Using the
inclinations from each system’s 30-minute cadence light curve juliet transit
fit, we find that the $\sin{i}$ corrections for the companions result in
adjustments smaller than 0.1%. The inclination uncertainties are therefore
insignificant in comparison to our mass estimate uncertainties.

We use the 30-minute cadence light curve juliet-inferred radius ratio
posteriors to calculate the secondary radii $R_{2}$ in Jupiter radii. In the
TOI-1712 system, the flux from the eclipsing pair is diluted by an unresolved
secondary revealed by the TRES spectra. We correct the dilution in the
TOI-1712 light curve by incorporating into a juliet transit fit the TESS
$T$-band flux ratio from our isochrones binary star fit posterior. We first
take each point in the isochrones posterior and calculate the flux ratio of
the secondary to the primary

$F=\frac{f_{\text{B}}}{f_{\text{Aa}}}=10^{(T_{\text{Aa}}-T_{\text{B}})/2.5},$
(5)

where $T_{\text{Aa}}$ and $T_{\text{B}}$ are the TESS $T$-band magnitudes of
the primary and secondary. We next calculate the dilution $D$ in the light
curve for each point

$D=\frac{1}{1+F}.$ (6)

We then fit the 30-minute cadence light curve of TOI-1712 for each point using
the same procedure described in Section 3.1, but with the dilution prior set
to be the posterior point’s calculated dilution. We obtain the median radius
ratio from each point’s individual juliet posterior, and use these points as
our full radius ratio posterior for BD+45 1593. We then use the same procedure
as with TYC 2010-124-1 and UCAC4 515-012898 to obtain the companion radius for
BD+45 1593.

We use a Monte Carlo simulation to infer the mass and radius uncertainties for
these three systems. We generate normal distributions for each thejoker
Doppler analysis-derived (for TYC 2010-124-1 and UCAC4 515-012898) or
Gaia/TESS-derived (for BD+45 1593) parameter. The number of Monte Carlo trials
is equal to the lesser of the number of data points in the mass and radius
posterior distributions from isochrones and the number of data points in the
radius ratio posterior from juliet (about 3000 samples). For the posterior
with more data points, we randomly select a number of posterior points equal
to the number of data points in the smaller posterior without replacement. For
each set of points in the combined posteriors, we calculate a mass and radius
inference for the companion, resulting in distributions that we characterize
with their 16th, 50th, and 84th percentiles reported in Table 3. This approach
captures the covariance between the stellar parameters from the isochrone fit
in order to generate an accurate uncertainty for the companion mass and radius
inferences. A caveat is that our reported uncertainties only include random
uncertainties. The inclusion of systematic uncertainties would likely result
in larger overall uncertainties (e.g., Tayar et al., 2022).

We account for contamination in the TESS light curve for UCAC4 515-012898
(TOI-5427), where the TESS contamination ratio is 26.3055%. This level of
light curve contamination corresponds to a radius underestimate of 5.13%, so
we increase our inferred radius for UCAC4 515-012898 b (TOI-5427) by 5.13% to
account for this effect. BD+45 1593 (TOI-1712) and TYC 2010-124-1 (TOI-2533)
have TESS contamination ratios indicative of radii underestimates that are
smaller than our report uncertainties, so we do not increase uncertainties in
either case. We provide corner plots illustrating this process and visualizing
the covariances for each systems in Figure 8.

Figure 8: Corner plots showing unimodal posteriors for BD+45 1593 (left), TYC
2010-124-1 (center), and UCAC4 515-012898 (right). Table 3: Properties of the
Objects and their Host Stars

Parameter | BD+45 1593 | TYC 2010-124-1 | UCAC4 515-012898 | Source
---|---|---|---|---
Identifiers
TOI | 1712.01 | 2533.01 | 5427.01 | Guerrero et al. (2021)
TIC | 67926921 | 418012030 | 52420398 | Stassun et al. (2018)
Gaia | 917823348536224768 | 1259922196651254272 | 3341401062125577088 | Gaia DR3
2MASS | J08314698+4449564 | J14103331+2625241 | J05355241+1255340 | 2MASS
TYC | 2981-1719-1 | 2010-124-1 |  $\cdots$ | Tycho-2
Astrometry
$\alpha$ | 127.945701 | 212.638795 | 83.968375 | Gaia DR3
$\delta$ | 44.832313 | 26.423432 | 12.925942 | Gaia DR3
$\alpha_{\text{J2000}}$ | 08h31m46.970745s | 14h10m33.327483s | 05h35m52.409305s | Gaia DR3
$\delta_{\text{J2000}}$ | 44d49m56.431574s | 26d25m24.046501s | 12d55m33.913407s | Gaia DR3
$\pi$ (mas) | $1.7653\pm 0.0182$ | $2.7078\pm 0.0221$ | $2.7589\pm 0.0158$ | Gaia DR3
Photometry
FUV (mag) | $18.2406\pm 0.0591$ |  $\cdots$ |  $\cdots$ | GALEX
NUV (mag) | $13.9118\pm 0.004$ |  $\cdots$ |  $\cdots$ | GALEX
$B_{\text{T}}$ (mag) | $10.817\pm 0.045$ | $12.354\pm 0.16$ |  $\cdots$ | Tycho-2
$V_{\text{T}}$ (mag) |  $10.485\pm 0.04$7 | $11.745\pm 0.131$ |  $\cdots$ | Tycho-2
$G$ (mag) | $10.2738\pm 0.0017$ | $12.0092\pm 0.0018$ | $12.0787\pm 0.0017$ | Gaia DR2
$T$ (mag) | $9.9841\pm 0.0062$ | $11.6417\pm 0.0069$ | $11.659\pm 0.0061$ | TESS
$J$ (mag) | $9.541\pm 0.0182$ | $11.142\pm 0.0242$ | $11.062\pm 0.0201$ | 2MASS
$H$ (mag) | $9.374\pm 0.0156$ | $10.872\pm 0.0308$ | $10.753\pm 0.0176$ | 2MASS
$K_{s}$ (mag) | $9.353\pm 0.016$ | $10.8\pm 0.019$ | $10.68\pm 0.0172$ | 2MASS
$W1$ (mag) | $9.314\pm 0.023$ | $10.78\pm 0.024$ | $10.669\pm 0.023$ | AllWISE
$W2$ (mag) | $9.327\pm 0.021$ | $10.798\pm 0.022$ | $10.701\pm 0.022$ | AllWISE
$W3$ (mag) | $9.287\pm 0.037$ | $10.735\pm 0.07$ | $10.707\pm 0.096$ | AllWISE
$A_{V}$ (mag) | $0.029^{+0.001}_{-0.016}$ | $0.019^{+0.001}_{-0.021}$ | $0.055^{+0.039}_{-0.039}$ | See Footnote
Spectroscopy
vbroad (km s-1) | $30\pm 6$ |  $\cdots$ |  $\cdots$ | Gaia DR3
Single-lined Spectroscopic Binary Solution
$K_{1}$ (km s-1) | $7.69\pm 0.61$ | $7.39\pm 1.10$ | $7.86\pm 0.98$ | Gaia DR3
|  $\cdots$ | 7.575 $\pm$ 0.046 | 9.841 $\pm$ 0.018 | Ground-Based
$e$ | $0.090\pm 0.072$ | $0.195\pm 0.131$ | $0.28\pm 0.11$ | Gaia DR3
|  $\cdots$ | $0.2473\pm 0.0059$ | $0.0332\pm 0.0015$ | Ground-Based
$P$ (days) | $3.566276\pm 0.0000044$ | $6.6847\pm 0.0034$ | $5.2374183\pm 0.0000056$ | TESS
Light Curve Transit Fit
$i$ (∘) | $88.36^{+0.25}_{-0.25}$ | $88.01^{+1.34}_{-1.94}$ | $87.48^{+1.50}_{-1.65}$ | This work
$R_{2}$ / $R_{\ast}$ | $0.0564^{+0.0006}_{-0.0003}$ | $0.079^{+0.004}_{-0.003}$ | $0.097^{+0.002}_{-0.002}$ | This work
Physical Properties
T${}_{\text{eff}}$ (K) | $6860^{+40}_{-30}$ | $6180^{+60}_{-60}$ | $6160^{+70}_{-40}$ | This Work
log g | $3.67^{+0.03}_{-0.02}$ | $4.36^{+0.03}_{-0.03}$ | $4.34^{+0.03}_{-0.02}$ | This Work
$\left[\mathrm{Fe/H}\right]$ | $-0.2^{+0.1}_{-0.1}$ | $-0.3^{+0.2}_{-0.2}$ | $-0.2^{+0.1}_{-0.1}$ | This Work
$M_{\star}$ ($M_{\odot}$) | $1.63^{+0.01}_{-0.02}$ | $1.02^{+0.06}_{-0.07}$ | $1.07^{+0.06}_{-0.05}$ | This Work
$R_{\star}$ ($R_{\odot}$) | $3.10^{+0.07}_{-0.10}$ | $1.11^{+0.01}_{-0.01}$ | $1.16^{+0.01}_{-0.01}$ | This Work
Age (Gyr) | $1.62^{+0.05}_{-0.03}$ | $4.26^{+2.18}_{-1.74}$ | $3.96^{+1.24}_{-1.53}$ | This Work
$M_{2}$ ($M_{\text{Jup}}$) |  $82^{+7}_{-7}$ (∗) | $72^{+3}_{-3}$ | $93^{+3}_{-3}$ | This Work
$R_{2}$ ($R_{\text{Jup}}$) | $1.74^{+0.08}_{-0.07}$ | $0.85^{+0.04}_{-0.03}$ | $1.15^{+0.03}_{-0.03}$ | This Work

Note. — We derive the secondary mass for TYC 2010-124-1 and UCAC4 515-012898
using ground-based radial velocity data from TRES and both CHIRON and NEID,
respectively. We use Lallement et al. (2022) and Vergely et al. (2022) for the
$A_{V}$ extinctions.

##  4 Discussion

Our synthesis of Gaia DR3 SB1 solutions and TESS transit discoveries described
in this article has revealed a new transiting brown dwarf and two very low-
mass eclipsing binary companions. We confirm that TOI-2533.01 is a
$72^{+3}_{-3}~{}M_{\text{Jup}}=0.069^{+0.003}_{-0.003}~{}M_{\odot}$ transiting
brown dwarf, placing it in a class of only a few dozen such objects. We
include in Table 4 an up-to-date list of known transiting brown dwarfs. We
further confirm that TOI-5427.01 with a mass of
$93^{+3}_{-3}~{}M_{\text{Jup}}=0.088^{+0.002}_{-0.002}~{}M_{\odot}$ is one of
the lowest-mass stars that models suggest is just above the hydrogen-burning
limit for its composition. We validate TOI-1712.01 as an inflated
$82^{+7}_{-7}~{}M_{\text{Jup}}=0.079^{+0.007}_{-0.007}~{}M_{\odot}$ very low-
mass star in a hierarchical triple system.

The Gaia DR3 SB1 solutions for brown dwarfs and very low-mass stars are
consistent with the ground-based precision radial velocity-based SB1
solutions. While the Gaia Doppler semiamplitude for TYC 2010-124-1 is in good
agreement with ground-based follow-up observations, it appears that Gaia has
underestimated by about 2 km/s the Doppler semiamplitude of UCAC4 515-012898.
The Gaia DR3 SB1 solutions also suggest a higher eccentricity for UCAC4
515-012898 and a slight phase shift for TYC 2010-124-1. We are unable to
further investigate this issue, as the individual Gaia radial velocity
measurements are not publicly available. For these two systems, the agreement
between the Gaia DR3 and ground-based precision radial velocity SB1 solutions
suggests that Gaia DR3 SB1 solutions are sufficiently precise to characterize
massive brown dwarfs and low-mass stars.

It seems possible that the quality of the Gaia DR3 SB1 solution is
proportional to the number of “good” radial velocity observations available by
the time of Gaia DR3. The brown dwarf TYC 2010-124-1 b has 47 “good” Gaia DR3
radial velocities, while the very low-mass stars BD+45 1593 Ab and UCAC4
515-012898 b only have 18 and 25, respectively. We suggest that the number of
“good” Gaia radial velocities may be a useful metric for assessing future Gaia
SB1 solutions. The fact that BD+45 1593 is an SB2 system may have affected
Gaia’s SB1 fit to its radial velocity variations. BD+45 1593 A is both hot and
rotating quickly, and all of these facts make it a challenging target for
Doppler observations.

We plot our inferred masses and radii for the three secondaries in Figure 9.
While we find that our mass and radius inferences for TYC 2010-124-1 b are
consistent with the Chabrier et al. (2023) theoretical models for brown
dwarfs, the radius of UCAC4 515-012898 b is about 6% larger than predicted by
the Baraffe et al. (2015) theoretical models for isolated stars of the same
mass, as is often the case for low-mass stars. In contrast, the radius of
BD+45 1593 Ab $R_{2}=1.74^{+0.08}_{-0.07}~{}R_{\text{Jup}}$, is more than 80%
larger than the radius $R\approx 1.0~{}R_{\text{Jup}}$ predicted by the
Baraffe et al. (2015) models for an isolated star with similar mass,
composition, and age. Because the MIST and PAdova TRieste Stellar Evolutionary
Code (PARSEC - Bressan et al., 2012; Nguyen et al., 2022) isochrone grids do
not extend to these objects’ mass ranges, we are unable to use them for
comparison. Given the precisions of our mass, age, composition, and radius
inferences, we find at over 9-$\sigma$ significance that BD+45 1593 Ab is
inflated. We note that this estimate accounts for dilution from BD+45 1593 B,
and in any case dilution typically would lead to an underestimated radius.

Figure 9: Comparisons between predicted and observed masses and radii. Top:
mass–radius relations assuming our isochrone-inferred system ages. We plot the
mass–radius relation at each system’s age as the solid line and indicate its
1-$\sigma$ uncertainty as the shaded region. Bottom: mass–radius relations
assuming a range of ages. Thicker lines refer to older ages. BD+45 1593 Ab has
a significantly larger radius than predicted by the Baraffe et al. (2015)
theoretical models. UCAC4 515-012898 b is slightly bigger than predicted by
the Baraffe et al. (2015) theoretical models as is frequently observed for
low-mass stars. TYC 2010-124-1 b is consistent with the Chabrier et al. (2023)
models. The models we use assume solar metallicity, while our systems have
slightly subsolar metallicities. However, these differences in metallicity are
too small to produce a radius offset as large as we observe for BD+45 1593 Ab.
Indeed, at the low-mass end of the MIST isochroe grid at $M_{\ast}\approx
0.11~{}M_{\odot}\approx 110~{}M_{\text{Jup}}$ a 0.2 dex decrease in
$[\text{Fe/H}]$ corresponds to no more than a 5% decrease in stellar radius.

Observations of radii about 10% larger than predicted by theoretical models
appear to be ubiquitous among K and M dwarfs (Kraus et al., 2011; Birkby et
al., 2012). This larger observed radii correspond to observed temperatures
typically 5% cooler than predicted by models, though this effect is poorly
studied due to the difficulty of determining late-type stellar temperatures
(Ribas et al., 2008; Morales et al., 2009; Torres, 2013). Magnetic fields are
the most-favored explanation for this “radius inflation problem” (and its
parallel “temperature suppression problem”), as they can increase the radii of
stars yet are rarely included in stellar models. Strong magnetic fields are
responsible for the significant starspot coverage of late-type dwarfs and may
inhibit convective energy transport (Feiden & Chaboyer, 2013). The net result
of these two physical mechanisms is an increase in the photospheric radii of K
and M dwarfs.

To empirically correct for radius underestimates, attempts have been made in
the past to relate magnetic activity with proxies like rotation (Somers &
Stassun, 2017; Jaehnig et al., 2019), X-ray emission (López-Morales, 2007),
and H$\alpha$ emission (Stassun et al., 2012). On the other hand, Mann et al.
(2015) found no correlation between radius discrepancy and either H$\alpha$ or
X-ray emission. While models have had some success including magnetic fields
in the context of X-ray emission (MacDonald & Mullan, 2014), the magnetic
field strengths required in this context might not be realistic (Feiden &
Chaboyer, 2014). More quantitative predictions from Browning et al. (2016)
provided an upper bound for the possible strength of these magnetic fields via
magnetohydrodynamic simulations, and MacDonald & Mullan (2015) provided a
constraint based on lithium abundance. While this remains an active area of
research, it appears that magnetic fields contribute to radius inflation in
low-mass stars without accounting for all aspects of the radius inflation
problem.

The eventual solution of the radius inference problem is important, as
accurate and precise radii for low-mass stars are required for the inference
of accurate and precise radii for the apparently common planets transiting
low-mass stars (Cassan et al., 2012). As low-mass stars are also the most
common type of star in the Galaxy (Bochanski et al., 2010), we expect that a
sizable portion of the Milky Way’s exoplanet population orbits low-mass stars.
Because of their small separations and high transit probabilities, habitable
zone terrestrial planets are easier to find orbiting low-mass stars than
solar-type stars. Low-mass stars are therefore the preferred target for
habitable transiting planet search programs (Gould et al., 2003) like MEarth
(Nutzman & Charbonneau, 2008), the TRAnsiting Planets and PlanetesImals Small
Telescope (TRAPPIST \- Jehin et al., 2011), the Next-Generation Transit Survey
(NGTS \- West et al., 2016), and TESS (Ricker et al., 2014). Unaccounted for,
the radius inflation problem would lead to underestimated radii for planets
transiting M dwarfs. An underestimated radius could cause a non-terrestrial,
uninhabitable planet to be mistakenly classified as a terrestrial, possibly
habitable planet.

We propose that the intense instellation experienced by BD+45 1593 Ab is
primarily responsible for its inflated radius. It has been suggested that in
binary systems with two very low-mass stars, instellation-driven radius
inflation is limited to 5% (Lucy, 2017). In this situation though, BD+45 1593
Ab is experiencing 500 times more flux than in situation investigated by Lucy
(2017). The increased surface temperature resulting from this instellation
could decrease the temperature gradient of BD+45 1593 Ab near its surface,
thereby limiting the ability of convection to transport energy in the same
manner as magnetic fields. This diminished efficiency of convective energy
transport would then lead to the inflated radius of BD+45 1593 Ab. Indeed, the
photospheric temperature of an isolated star with similar properties to BD+45
1593 Ab is expected to be $T_{\text{eff}}\approx 2300$ (Baraffe et al., 2015).
Give its instellation, the “day side” equilibrium temperature of BD+45 1593 Ab
could be has high as $T_{\text{eq}}\approx 4400$ K assuming a Bond albedo
$A_{\text{B}}=0$ and no redistribution of heat.

Tidal effects may also play a role in the inflated radius of BD+45 1593 Ab,
perhaps in a process analogous to the shear mixing that may occur in massive
binaries (Song et al., 2013). The fully convective nature of BD+45 1593 Ab
would make shear mixing inefficient, and its small eccentricity likely
minimizes the importance of tidal dissipation for its inflated radius. The
probable fast rotation of BD+45 1593 Ab resulting from its circular orbit,
short circularization time, and therefore likely spin–orbit synchronization
could contribute to its inflated radius, though the relationship between
rotation and radius inflation is unclear (Jackson et al., 2019). Consequently,
we conclude that instellation is the best candidate to explain the inflated
radius of BD+45 1593 Ab.

We searched the light curves of TOI-1712 and TOI-5427 for secondary eclipses
perfectly opposite the transits of TOI-1712.01 and TOI-5427.01 as expected for
circular orbits. We were unable to identify a secondary eclipse in either
case. This is not surprising, as the detection of a secondary eclipse would
have eliminated these objects from consideration for inclusion in the TOI
catalog. Given the precision of the TESS data for TOI-1712 and TOI-5427 and
the expected secondary eclipse durations of TOI-1712.01 and TOI-5427.01, we
would have been able to identify secondary eclipses above 0.003% for TOI-1712
and above 0.01% for TOI-5427. This lack of a secondary eclipse detection does
not constrain the “day side” temperature for UCAC4 515-012898 b, as we expect
its photospheric temperature to have been undetectable given our limits. On
the other hand, the non-detection of a secondary eclipse for TOI-1712.01
limits the “day side” temperature of BD+45 1593 Ab to less than 3200 K. The
“day side” of BD+45 1593 Ab could be as hot as 4400 K if its albedo is zero
and heat is not redistributed, so our non-detection of a secondary eclipse
implies that either its albedo is nonzero or the redistribution of heat is
efficient.

We compare TYC 2010-124-1 b to the Chabrier et al. (2023) brown dwarf
isochrones in Figure 9. Isolated brown dwarfs cool and contract to settle into
a typical mass range of between 0.7 and 1.4 Jupiter radii (Deleuil et al.,
2008; Šubjak et al., 2020; Carmichael et al., 2020) after at least 100 Myr.
Objects younger than this tend to be larger (David et al., 2019; Stassun et
al., 2006). According to isolated brown dwarf evolution models (e.g., Baraffe
et al., 2003; Saumon & Marley, 2008; Phillips et al., 2020; Marley et al.,
2021; Carmichael, 2023), the rate of this contraction should decrease over
time. However, the details of this process are unclear (Carmichael, 2023). We
find that the radius of TYC 2010-124-1 b is 9% bigger than the model
prediction for an isolated brown dwarf, but this offset is only significant at
the 1.9-$\sigma$ level.

The radius inflation we observe in BD+45 1593 Ab potentially parallels the
processes responsible for the inflation of some hot Jupiters (e.g., Grunblatt
et al., 2016, 2017, 2022b; Wittenmyer et al., 2022). Hot Jupiters orbiting
evolved stars are sometimes observed with larger radii than hot Jupiters
orbiting main sequence stars (Grunblatt et al., 2019). In this case, the
inflated radii of mature hot Jupiters orbiting evolved stars have been
“reinflated” due to rapid transport of energy deposited into their surface
layers by their evolved host star into their interiors (e.g., Lopez & Fortney,
2016; Thorngren & Fortney, 2018; Thorngren et al., 2021). Intense instellation
may therefore be capable of inflating radii from the giant planet regime up to
the very low-mass star regime.

Because theoretical models have struggled to explain the data that is
currently available, a much larger sample of massive brown dwarfs and very
low-mass stars with precisely inferred radii relative to their solar-type
primaries could lead to useful empirical radius calibrations. Our approach
that combines Gaia SB1 solutions with transit discoveries from Kepler, K2,
TESS, and the future PLAnetary Transits and Oscillations (PLATO) mission has
the potential to precisely characterize over 400 brown dwarfs and low-mass
stars. Such a catalog of radii precisely inferred relative to the radii of
their primaries could be the basis of future empirical brown dwarf and low-
mass star mass–radius relations. The masses inferred in this way using the
Doppler technique with a known inclination are more precise and likely more
accurate than other currently employed methods of mass inference such as
astrometry or atmospheric fitting444Different mass inference methods have
yielded inconsistent mass results for the first confirmed brown dwarf Gliese
229 B (Marley et al., 1996; Brandt et al., 2020). As transit surveys typically
focus on low-mass stars to find small-radius planets, these empirical stellar
radii could lead to better planet radii which would lead to better atmospheric
and interior structure inferences for terrestrial planets like those in the
TRAPPIST-1 system.

Table 4: Literature Sample of Transiting Brown Dwarfs as of June 2023

Name | Simbad Name | $P$ | $M_{\text{BD}}$ | $R_{\text{BD}}$ | Eccentricity | $M_{\ast}$ | $R_{\ast}$ | $T_{\text{eff}}$ | [Fe/H] | Reference
---|---|---|---|---|---|---|---|---|---|---
|  | (days) | ($M_{\text{Jup}}$) | ($R_{\text{Jup}}$) |  | ($M_{\odot}$) | ($R_{\odot}$) | (K) |  |
TYC 2010-124-1 b | TYC 2010-124-1 | $6.6847^{+0.0034}_{-0.0034}$ | $72^{+3}_{-3}$ | $0.85^{+0.04}_{-0.03}$ | $0.2473^{+0.0059}_{-0.0059}$ | $1.02^{+0.06}_{-0.07}$ | $1.11^{+0.01}_{-0.01}$ | $6180^{+60}_{-60}$ | $-0.3^{+0.2}_{-0.2}$ | This Work
TOI-2119 b | StM 274 | 7.201 | $64.4^{+2.3}_{-2.2}$ | $1.08^{+0.03}_{-0.03}$ | $0.337^{+0.0019}_{-0.00064}$ | $0.525^{+0.02}_{-0.021}$ | $0.5^{+0.015}_{-0.015}$ | $3621^{+48}_{-46}$ | $0.055^{+0.084}_{-0.077}$ | 1
TOI-2521 b | TYC 5362-881-1 | 5.563 | $77.5^{+3.3}_{-3.3}$ | $1.01^{+0.04}_{-0.04}$ | $<0.035$ | $1.1^{+0.07}_{-0.07}$ | $1.77^{+0.068}_{-0.068}$ | $5600^{+100}_{-100}$ | $-0.3^{+0.3}_{-0.3}$ | 2
TOI-2336 b | TYC 7317-698-1 | 7.712 | $69.9^{+2.3}_{-2.3}$ | $1.05^{+0.04}_{-0.04}$ | $0.01^{+0.006}_{-0.005}$ | $1.41^{+0.08}_{-0.08}$ | $1.781^{+0.059}_{-0.059}$ | $6550^{+100}_{-100}$ | $0^{+0.3}_{-0.3}$ | 2
TOI-2543 b | TYC 229-654-1 | 7.543 | $67.62^{+3.45}_{-3.45}$ | $0.95^{+0.09}_{-0.09}$ | $0.009^{+0.003}_{-0.002}$ | $1.29^{+0.08}_{-0.08}$ | $1.86^{+0.15}_{-0.15}$ | $6060^{+82}_{-82}$ | $-0.28^{+0.1}_{-0.1}$ | 3
TOI-1982 b | BD-22 3669 | 17.172 | $65.85^{+2.75}_{-2.72}$ | $1.08^{+0.04}_{-0.04}$ | $0.272^{+0.014}_{-0.014}$ | $1.41^{+0.08}_{-0.08}$ | $1.51^{+0.05}_{-0.05}$ | $6325^{+110}_{-110}$ | $-0.1^{+0.09}_{-0.09}$ | 3
TOI-629 b | HD 44717 | 8.718 | $66.98^{+2.96}_{-2.95}$ | $1.11^{+0.05}_{-0.05}$ | $0.298^{+0.008}_{-0.008}$ | $2.16^{+0.13}_{-0.13}$ | $2.37^{+0.11}_{-0.11}$ | $9100^{+200}_{-200}$ | $0.1^{+0.15}_{-0.15}$ | 3
TOI-1994 b | HD 298656 | 4.03 | $22$ | $1.28$ | $0.29$ | $\cdots$ | $\cdots$ | $7550$ | $\cdots$ | 4
GPX-1 b | UCAC4 731-024744 | 1.75 | $19.7^{+1.6}_{-1.6}$ | $1.47^{+0.10}_{-0.10}$ | $\cdots$ | $1.68^{+0.10}_{-0.10}$ | $1.56^{+0.10}_{-0.10}$ | $7000^{+200}_{-200}$ | $\cdots$ | 5
TOI-746 b | TOI-746 | 10.98 | $82.2^{+4.9}_{-4.4}$ | $0.95^{+0.09}_{-0.06}$ | $0.199^{+0.003}_{-0.003}$ | $0.98^{+0.06}_{-0.06}$ | $0.957^{+0.051}_{-0.051}$ | $5593^{+215}_{-215}$ | $0.01^{+0.29}_{-0.29}$ | 6
TOI-587 b | HD 74162 | 8.04 | $81.1^{+7.1}_{-7.0}$ | $1.32^{+0.07}_{-0.06}$ | $0.051^{+0.049}_{-0.036}$ | $2.32^{+0.14}_{-0.14}$ | $2.031^{+0.092}_{-0.092}$ | $10400^{+300}_{-300}$ | $0.07^{+0.12}_{-0.12}$ | 6
TOI-148 b | TOI-148 | 4.87 | $77.1^{+5.8}_{-4.6}$ | $0.81^{+0.05}_{-0.06}$ | $0.005^{+0.006}_{-0.004}$ | $1.03^{+0.06}_{-0.06}$ | $1.192^{+0.068}_{-0.068}$ | $5836^{+286}_{-286}$ | $-0.28^{+0.28}_{-0.28}$ | 6
NGTS-19 b | NGTS-19 | 17.8397 | $69.5^{+5.7}_{-5.4}$ | $1.034^{+0.055}_{-0.053}$ | $0.3767^{+0.0061}_{-0.0061}$ | $0.807^{+0.038}_{-0.043}$ | $0.896^{+0.04}_{-0.035}$ | $4716^{+39}_{-28}$ | $0.11^{+0.074}_{-0.07}$ | 7
TOI-1278 B | TOI-1278 | 14.476 | $18.5^{+0.5}_{-0.5}$ | $1.09^{+0.24}_{-0.2}$ | $0.013^{+0.004}_{-0.004}$ | $0.55^{+0.02}_{-0.02}$ | $0.573^{+0.012}_{-0.012}$ | $3799^{+42}_{-42}$ | $-0.01^{+0.28}_{-0.28}$ | 8
TOI-811 b | TOI-811 | 25.166 | $59^{+13}_{-8.6}$ | $1.26^{+0.06}_{-0.06}$ | $0.509^{+0.075}_{-0.075}$ | $1.32^{+0.07}_{-0.07}$ | $1.27^{+0.09}_{-0.09}$ | $6107^{+77}_{-77}$ | $0.4^{+0.09}_{-0.09}$ | 9
TOI-852 b | TOI-852 | 4.946 | $53.7^{+1.4}_{-1.4}$ | $0.83^{+0.04}_{-0.04}$ | $0.004^{+0.004}_{-0.004}$ | $1.32^{+0.05}_{-0.05}$ | $1.72^{+0.04}_{-0.04}$ | $5768^{+84}_{-84}$ | $0.33^{+0.09}_{-0.09}$ | 9
HATS-70 b | HATS-70 | 1.888 | $12.9^{+1.8}_{-1.8}$ | $1.38^{+0.08}_{-0.08}$ | $<0.18$ | $1.78^{+0.12}_{-0.12}$ | $1.88^{+0.07}_{-0.07}$ | $7930^{+820}_{-820}$ | $0.04^{+0.11}_{-0.11}$ | 10
KELT-1 b | KELT-1 | 1.218 | $27.4^{+0.9}_{-0.9}$ | $1.12^{+0.04}_{-0.04}$ | $0.01^{+0.01}_{-0.01}$ | $1.34^{+0.06}_{-0.06}$ | $1.47^{+0.05}_{-0.05}$ | $6516^{+49}_{-49}$ | $0.05^{+0.08}_{-0.08}$ | 11
NLTT 41135 b | LP 563-38 | 2.889 | $33.7^{+2.8}_{-2.8}$ | $1.13^{+0.27}_{-0.27}$ | $<0.02$ | $0.19^{+0.03}_{-0.03}$ | $0.21^{+0.02}_{-0.02}$ | $3230^{+130}_{-130}$ | $-0.25^{+0.25}_{-0.25}$ | 12
LHS 6343 c | G 205-57 | 12.713 | $62.9^{+2.3}_{-2.3}$ | $0.83^{+0.02}_{-0.02}$ | $0.056^{+0.032}_{-0.032}$ | $0.37^{+0.01}_{-0.01}$ | $0.38^{+0.01}_{-0.01}$ | $\cdots$ | $0.02^{+0.19}_{-0.19}$ | 13
LP 261-75 b | LP 261-75 | 1.882 | $68.1^{+2.1}_{-2.1}$ | $0.9^{+0.02}_{-0.02}$ | $<0.007$ | $0.3^{+0.02}_{-0.02}$ | $0.31^{+0.01}_{-0.01}$ | $3100^{+50}_{-50}$ | $\cdots$ | 14
WASP-30 b | WASP-30 | 4.157 | $62.5^{+1.2}_{-1.2}$ | $0.95^{+0.03}_{-0.03}$ | 0 (adopted) | $1.25^{+0.04}_{-0.04}$ | $1.4^{+0.03}_{-0.03}$ | $6202^{+51}_{-51}$ | $0.08^{+0.1}_{-0.1}$ | 15
WASP-128 b | WASP-128 | 2.209 | $37.2^{+0.9}_{-0.9}$ | $0.94^{+0.02}_{-0.02}$ | $<0.007$ | $1.16^{+0.04}_{-0.04}$ | $1.15^{+0.02}_{-0.02}$ | $5950^{+50}_{-50}$ | $0.01^{+0.12}_{-0.12}$ | 16
CoRoT-3 b | CoRoT-3 | 4.257 | $21.7^{+1}_{-1}$ | $1.01^{+0.07}_{-0.07}$ | 0 (adopted) | $1.37^{+0.09}_{-0.09}$ | $1.56^{+0.09}_{-0.09}$ | $6740^{+140}_{-140}$ | $-0.02^{+0.06}_{-0.06}$ | 17
CoRoT-15 b | CoRoT-15 | 3.06 | $63.3^{+4.1}_{-4.1}$ | $1.12^{+0.3}_{-0.3}$ | 0 (adopted) | $1.32^{+0.12}_{-0.12}$ | $1.46^{+0.31}_{-0.31}$ | $6350^{+200}_{-200}$ | $0.1^{+0.2}_{-0.2}$ | 18
CoRoT-33 b | CoRoT-33 | 5.819 | $59^{+1.8}_{-1.8}$ | $1.1^{+0.53}_{-0.53}$ | $0.07^{+0.002}_{-0.002}$ | $0.86^{+0.04}_{-0.04}$ | $0.94^{+0.14}_{-0.14}$ | $5225^{+80}_{-80}$ | $0.44^{+0.1}_{-0.1}$ | 19
Kepler-39 b | Kepler-39 | 21.087 | $20.1^{+1.3}_{-1.3}$ | $1.24^{+0.1}_{-0.1}$ | $0.112^{+0.057}_{-0.057}$ | $1.29^{+0.07}_{-0.07}$ | $1.4^{+0.1}_{-0.1}$ | $6350^{+100}_{-100}$ | $0.1^{+0.14}_{-0.14}$ | 20
KOI-189 b | Kepler-486 | 30.36 | $78^{+3.4}_{-3.4}$ | $1^{+0.02}_{-0.02}$ | $0.275^{+0.004}_{-0.004}$ | $0.76^{+0.05}_{-0.05}$ | $0.73^{+0.02}_{-0.02}$ | $4952^{+40}_{-40}$ | $-0.07^{+0.12}_{-0.12}$ | 21
KOI-205 b | Kepler-492 | 11.72 | $39.9^{+1}_{-1}$ | $0.81^{+0.02}_{-0.02}$ | $<0.031$ | $0.92^{+0.03}_{-0.03}$ | $0.84^{+0.02}_{-0.02}$ | $5237^{+60}_{-60}$ | $0.14^{+0.12}_{-0.12}$ | 22
KOI-415 b | KOI-415 | 166.788 | $62.1^{+2.7}_{-2.7}$ | $0.79^{+0.12}_{-0.12}$ | $0.689^{+0.001}_{-0.001}$ | $0.94^{+0.06}_{-0.06}$ | $1.15^{+0.15}_{-0.15}$ | $5810^{+80}_{-80}$ | $-0.24^{+0.11}_{-0.11}$ | 23
EPIC 201702477 b | UCAC2 33047398 | 40.737 | $66.9^{+1.7}_{-1.7}$ | $0.76^{+0.07}_{-0.07}$ | $0.228^{+0.003}_{-0.003}$ | $0.87^{+0.03}_{-0.03}$ | $0.9^{+0.06}_{-0.06}$ | $5517^{+70}_{-70}$ | $-0.16^{+0.05}_{-0.05}$ | 24
EPIC 212036875 b | TYC 1400-1873-1 | 5.17 | $52.3^{+1.9}_{-1.9}$ | $0.87^{+0.02}_{-0.02}$ | $0.132^{+0.004}_{-0.004}$ | $1.29^{+0.07}_{-0.07}$ | $1.5^{+0.03}_{-0.03}$ | $6238^{+60}_{-60}$ | $0.01^{+0.1}_{-0.1}$ | 25
AD 3116 b | Cl* NGC 2632 HSHJ 430 | 1.983 | $54.2^{+4.3}_{-4.3}$ | $1.02^{+0.28}_{-0.28}$ | $0.146^{+0.024}_{-0.024}$ | $0.28^{+0.02}_{-0.02}$ | $0.29^{+0.08}_{-0.08}$ | $3200^{+200}_{-200}$ | $0.16^{+0.1}_{-0.1}$ | 26
CWW 89 Ab | UCAC4 366-166973 | 5.293 | $36.84^{+0.97}_{-0.97}$ | $0.94^{+0.03}_{-0.03}$ | $0.1929^{+0.002}_{-0.002}$ | $1.01^{+0.04}_{-0.04}$ | $1.01^{+0.03}_{-0.03}$ | $5850^{+85}_{-85}$ | $0.03^{+0.08}_{-0.08}$ | 27
RIK 72 b | UGCS J160339.22-185129.4 | 97.76 | $59.2^{+6.8}_{-6.8}$ | $3.1^{+0.31}_{-0.31}$ | $0.146^{+0.012}_{-0.012}$ | $0.44^{+0.04}_{-0.04}$ | $0.96^{+0.1}_{-0.1}$ | $3349^{+142}_{-142}$ | $\cdots$ | 28
TOI-503 b | BD+13 1880 | 3.677 | $53.7^{+1.2}_{-1.2}$ | $1.34^{+0.26}_{-0.15}$ | 0 (adopted) | $1.8^{+0.06}_{-0.06}$ | $1.7^{+0.05}_{-0.05}$ | $7650^{+160}_{-160}$ | $0.61^{+0.07}_{-0.07}$ | 29
TOI-569 b | CD-41 3255 | 6.556 | $63.8^{+1}_{-1}$ | $0.75^{+0.02}_{-0.02}$ | $<0.0035$ | $1.21^{+0.03}_{-0.03}$ | $1.48^{+0.03}_{-0.03}$ | $5705^{+76}_{-76}$ | $0.4^{+0.08}_{-0.08}$ | 30
TOI-1406 b | HD 274870 | 10.574 | $46^{+2.7}_{-2.7}$ | $0.86^{+0.02}_{-0.02}$ | $<0.039$ | $1.18^{+0.09}_{-0.09}$ | $1.35^{+0.03}_{-0.03}$ | $6290^{+100}_{-100}$ | $-0.08^{+0.09}_{-0.09}$ | 30
NGTS-7 Ab | NGTS-7 | 0.676 | $75.5^{+3}_{-13.7}$ | $1.38^{+0.13}_{-0.14}$ | 0 (adopted) | $0.48^{+0.13}_{-0.13}$ | $0.61^{+0.06}_{-0.06}$ | $3359^{+106}_{-106}$ | $\cdots$ | 31
2M0535-05 a | V* V2384 Ori | 9.779 | $56.7^{+4.8}_{-4.8}$ | $6.5^{+0.33}_{-0.33}$ | $0.323^{+0.006}_{-0.006}$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 32
2M0535-05 b | V* V2384 Ori | 9.779 | $35.6^{+2.8}_{-2.8}$ | $5.00^{+0.25}_{-0.25}$ | $0.323^{+0.006}_{-0.006}$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 32
2M1510 Aa | 2MASS J15104786-2818174 | 20.902 | $40^{+2.9}_{-2.9}$ | $1.53^{+0.15}_{-0.15}$ | $0.309^{+0.022}_{-0.022}$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 33
2M1510 Ab | 2MASS J15104761-2818234 | 20.902 | $39.9^{+2.9}_{-2.9}$ | $1.53^{+0.15}_{-0.15}$ | $0.309^{+0.022}_{-0.022}$ | $\cdots$ | $\cdots$ | $\cdots$ | $\cdots$ | 33
CoRoT-34 b | UCAC4 431-022970 | 2.119 | $71.4^{+8.9}_{-8.6}$ | $1.09^{+0.17}_{-0.16}$ | 0 (adopted) | $1.66^{+0.08}_{-0.15}$ | $1.85^{+0.29}_{-0.25}$ | $7820^{+160}_{-160}$ | $-0.2^{+0.2}_{-0.2}$ | 34
Kepler-503 b | Kepler-503 | 7.258 | $78.6^{+3.1}_{-3.1}$ | $0.96^{+0.06}_{-0.04}$ | $0.025^{+0.014}_{-0.012}$ | $1.154^{+0.047}_{-0.042}$ | $1.764^{+0.08}_{-0.068}$ | $5670^{+100}_{-110}$ | $0.169^{+0.046}_{-0.045}$ | 35
HIP 33609 b | HD 52470 | 39.472 | $68.0^{+7.4}_{-7.4}$ | $1.580^{+0.074}_{-0.070}$ | $0.560^{+0.029}_{-0.031}$ | $2.383^{+0.10}_{-0.095}$ | $1.863^{+0.087}_{-0.082}$ | $10400^{+800}_{-600}$ | $-0.01^{+0.19}_{-0.20}$ | 36
TOI-2336 b | TYC 7317-698-1 | 7.71 | $69.9^{+2.3}_{-2.3}$ | $1.05^{+0.04}_{-0.04}$ | $0.010^{+0.006}_{-0.005}$ | $1.40^{+0.07}_{-0.07}$ | $1.82^{+0.06}_{-0.06}$ | $6433^{+84}_{-84}$ | $0.09^{+0.11}_{-0.11}$ | 37

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##  5 Conclusion

We characterized three transiting systems: one brown dwarf and two low-mass
stars in the TYC 2010-124-1, UCAC4 515-012898, and BD+45 1593 systems
identified by TESS as TOI-2533.01, TOI-5427.01, and TOI-1712.01. We showed
that the synthesis of Gaia DR3 SB1 solutions and TESS TOI discoveries offers a
promising path to the large-scale, homogeneous characterization of low-mass
stars and massive brown dwarfs. Gaia DR3 SB1 solutions and TESS transits
indicated that the masses of the transiting objects in all three systems were
consistent with the brown dwarf mass range. Ground-based precision radial
velocity measurements confirmed that TOI-2533.01 is a transiting brown dwarf
with $M=72^{+3}_{-3}~{}M_{\text{Jup}}=0.069^{+0.003}_{-0.003}~{}M_{\odot}$ and
that TOI-5427.01 is a transiting very low-mass star with
$M=93^{+2}_{-2}~{}M_{\text{Jup}}=0.088^{+0.002}_{-0.002}~{}M_{\odot}$. We
validated TOI-1712.01 as a very low-mass star with
$M=82^{+7}_{-7}~{}M_{\text{Jup}}=0.079^{+0.007}_{-0.007}~{}M_{\odot}$
transiting a massive subgiant primary in the hierarchical triple system BD+45
1593. The star BD+45 1593 Ab has a radius nearly 80% larger than predicted by
theoretical models for an isolated star with a similar masses, solar
composition, and age equal to that of the primary BD+45 1593 Aa. While the
observation of very-low mass stars with radii larger than predicted by
theoretical models seems to be ubiquitous and often attributed to magnetic
fields, radius inflation as large as we observe in BD+45 1593 Ab cannot be
attributed to the effect of magnetic fields alone. We suggest that the intense
instellation experienced by BD+45 1593 Ab heats its photosphere to beyond the
photospheric temperature expected for an isolated star with similar
parameters. This increased photospheric temperature diminishes the temperature
gradient in the outer parts of BD+45 1593 Ab, suppresses convection, and leads
to its inflated radius. Future studies of systems with Gaia SB1 solutions and
Kepler, K2, TESS, or PLATO transit detections have the potential to provide
hundreds of precise radius inferences for brown dwarfs and very-low mass
stars. These latter data would be critical for the calibration of empirical
mass–radius relations for very low-mass stars that would be very useful for
the calculation of the radii of planets transiting very low-mass stars like
those in the TRAPPIST-1 system.

## Acknowledgments

We thank the anonymous referee for many very helpful comments. We thank Zafar
Rustamkulov for helpful suggestions about visualizing our results. We also
thank Daniel Thorngren for helpful insight about our findings. This material
is based upon work supported by the National Science Foundation under grant
number 2009415 and by the TESS General Investigator program under NASA grant
80NSSC20K0059. Samuel N. Quinn acknowledges support from the TESS mission via
subaward s3449 from MIT. This work was supported by a NASA WIYN PI Data Award,
administered by the NASA Exoplanet Science Institute. This work has made use
of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This paper includes data
collected with the TESS mission, obtained from the MAST data archive at the
Space Telescope Science Institute (STScI). Funding for the TESS mission is
provided by the NASA Explorer Program. STScI is operated by the Association of
Universities for Research in Astronomy, Inc., under NASA contract NAS 5–26555.
We acknowledge the use of public TESS data from pipelines at the TESS Science
Office and at the TESS Science Processing Operations Center. This research has
made use of the Exoplanet Follow-up Observation Program (ExoFOP; DOI:
10.26134/ExoFOP5) website, which is operated by the California Institute of
Technology, under contract with the National Aeronautics and Space
Administration under the Exoplanet Exploration Program. Resources supporting
this work were provided by the NASA High-End Computing (HEC) Program through
the NASA Advanced Supercomputing (NAS) Division at Ames Research Center for
the production of the SPOC data products. This paper contains data taken with
the NEID instrument, which was funded by the NASA-NSF Exoplanet Observational
Research (NN-EXPLORE) partnership and built by Pennsylvania State University.
NEID is installed on the WIYN telescope, which is operated by the National
Optical Astronomy Observatory, and the NEID archive is operated by the NASA
Exoplanet Science Institute at the California Institute of Technology. NN-
EXPLORE is managed by the Jet Propulsion Laboratory, California Institute of
Technology under contract with the National Aeronautics and Space
Administration. Data presented herein were obtained at the WIYN Observatory
from telescope time allocated to NN-EXPLORE through the scientific partnership
of the National Aeronautics and Space Administration, the National Science
Foundation, and NOIRLab. The authors are honored to be permitted to conduct
astronomical research on Iolkam Du’ag (Kitt Peak), a mountain with particular
significance to the Tohono O’odham. This publication makes use of data
products from the Two Micron All Sky Survey, which is a joint project of the
University of Massachusetts and the Infrared Processing and Analysis
Center/California Institute of Technology, funded by the National Aeronautics
and Space Administration and the National Science Foundation. This publication
makes use of data products from the Wide-field Infrared Survey Explorer, which
is a joint project of the University of California, Los Angeles, and the Jet
Propulsion Laboratory/California Institute of Technology, and NEOWISE, which
is a project of the Jet Propulsion Laboratory/California Institute of
Technology. WISE and NEOWISE are funded by the National Aeronautics and Space
Administration. Some/all of the data presented in this paper were obtained
from the Mikulski Archive for Space Telescopes (MAST). STScI is operated by
the Association of Universities for Research in Astronomy, Inc., under NASA
contract NAS5-26555. Support for MAST for non-HST data is provided by the NASA
Office of Space Science via grant NNX13AC07G and by other grants and
contracts. This research has made use of the Exoplanet Follow-up Observation
Program website, which is operated by the California Institute of Technology,
under contract with the National Aeronautics and Space Administration under
the Exoplanet Exploration Program. This research has made use of the Spanish
Virtual Observatory (https://svo.cab.inta-csic.es) project funded by
MCIN/AEI/10.13039/501100011033/ through grant PID2020-112949GB-I00. This
research has made use of the SIMBAD database (Wenger et al., 2000), operated
at CDS, Strasbourg, France. This research made use of Lightkurve, a Python
package for Kepler and TESS data analysis (Lightkurve Collaboration et al.,
2018). This work made use of the IPython package (Perez & Granger, 2007). This
research has made use of NASA’s Astrophysics Data System.

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