# Three low-mass companions around aged stars discovered by TESS

Zitao Lin,1 Tianjun Gan,1 Sharon X. Wang,1 Avi Shporer,2 Markus Rabus,3 George
Zhou,4 Angelica Psaridi,5 François Bouchy,5 Allyson Bieryla,6 David W.
Latham,3 Shude Mao,1,7 Keivan G. Stassun,8 Coel Hellier,9 Steve B. Howell,10
Carl Ziegler,11 César Briceño,12 Douglas A. Caldwell,10,13 Karen A. Collins,14
Jason L. Curtis,15 Jacqueline K. Faherty,16 Crystal L. Gnilka,10 Samuel K.
Grunblatt,17 Jon M. Jenkins,10 Marshall C. Johnson,18 Nicholas Law,19 Monika
Lendl,5 Colin Littlefield,10,20 Michael B. Lund,21 Mikkel N. Lund,22 Andrew W.
Mann,19 Scott McDermott,23 Lokesh Mishra,5 Dany Mounzer,5 Martin Paegert,24
Tyler Pritchard,25 George R. Ricker,2 Sara Seager,2,26,27 Gregor Srdoc,28
Jiaxin Tang,1 Stéphane Udry,5 Roland Vanderspek,2 David Watanabe,29 Joshua N.
Winn,30 and Jie Yu31
Affiliations are listed at the end of the paper E-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

We report the discovery of three transiting low-mass companions to aged stars:
a brown dwarf (TOI-2336b) and two objects near the hydrogen burning mass limit
(TOI-1608b and TOI-2521b). These three systems were first identified using
data from the Transiting Exoplanet Survey Satellite (TESS). TOI-2336b has a
radius of $1.05\pm 0.04\ R_{J}$, a mass of $69.9\pm 2.3\ M_{J}$ and an orbital
period of 7.71 days. TOI-1608b has a radius of $1.21\pm 0.06\ R_{J}$, a mass
of $90.7\pm 3.7\ M_{J}$ and an orbital period of 2.47 days. TOI-2521b has a
radius of $1.01\pm 0.04\ R_{J}$, a mass of $77.5\pm 3.3\ M_{J}$ and an orbital
period of 5.56 days. We found all these low-mass companions are inflated. We
fitted a relation between radius, mass and incident flux using the sample of
known transiting brown dwarfs and low-mass M dwarfs. We found a positive
correlation between the flux and the radius for brown dwarfs and for low-mass
stars that is weaker than the correlation observed for giant planets.

###### keywords:

stars: brown dwarfs – stars: low mass – techniques: photometric – techniques:
radial velocities

††pubyear: 2015††pagerange: Three low-mass companions around aged stars
discovered by TESS–B

##  1 Introduction

Brown dwarfs are objects with masses between those of giant planets and stars.
Typically, the lower mass limit is taken to be $\sim 13\ M_{J}$ (Spiegel et
al., 2011), the minimum mass to ignite deuterium fusion, and the upper mass
limit is taken to be $\sim 80\ M_{J}$ (Baraffe et al., 2002), the minimum mass
to ignite ordinary hydrogen fusion. Thousands of isolated brown dwarfs have
been discovered in this century (e.g., Henry et al., 2018). However, previous
works using radial velocity data have shown that, on relatively short orbital
periods, brown dwarf companions around solar-type stars are significantly
rarer compared to planetary and stellar companions, which is known as the
‘brown dwarf desert’ (e.g., Marcy & Butler, 2000; Grether & Lineweaver, 2006).
Similarly, the known transiting brown dwarfs are outnumbered by hot Jupiters
(e.g., the NASA Exoplanet Archive). Despite the similar radii of hot Jupiters
and brown dwarfs (Chen & Kipping, 2017), only about 30 transiting brown dwarfs
have been found, in contrast with the hundreds of transiting hot Jupiters.
This desert may result from two different formation mechanisms, i.e.,
gravitational instability and cloud fragmentation, dominating different mass
regimes for the formation of brown dwarfs (see discussion in Palle et al.,
2021). For example, Ma & Ge (2014) analyzed a catalog of 62 brown dwarf
companions and found that they are most depleted at $30\ M_{J}<M\sin i<55\
M_{J}$ and $P<100$ days. Using $42.5\ M_{J}$ as a threshold, they found that
the eccentricity distribution of lower-mass brown dwarfs is consistent with
that of massive planets, while the eccentricity distribution of higher mass
brown dwarfs is more consistent with that of stellar binaries. They suggested
that the brown dwarfs below $42.5\ M_{J}$ may form through disc gravitational
instability, while the brown dwarfs above $42.5\ M_{J}$ may form through
molecular cloud fragmentation like stars. However, the scope and robustness of
such statistical works is limited by the small size of the brown dwarf sample.
Therefore, we need more brown dwarf companions with precisely determined
parameters such as mass, radius, and age to analyze the origin of the ‘brown
dwarf desert’ and their formation mechanism.

NASA’s Transiting Exoplanet Survey Satellite (TESS) mission (Ricker et al.,
2015) is an all-sky transit survey, searching for planets around nearby bright
stars, which also provides an exciting opportunity to enlarge the sample of
transiting brown dwarfs. TESS has completed its Primary Mission (July 2018 -
July 2020), where it observed most of the sky, and its First Extended Mission
(July 2020 - September 2022) where it observed most of the sky a second time,
and it is now in its Second Extended Mission (started September 2022) where it
continues to re-observe the sky. Up to late 2022, TESS has successfully found
more than 10 transiting brown dwarfs (e.g., Grieves et al., 2021; Carmichael
et al., 2022; Psaridi et al., 2022). Light curves from TESS can provide us
with the radius of a transiting brown dwarf, its period, and its orbital
inclination. However, transit data alone are not sufficient to determine the
masses, which are important to distinguish brown dwarfs from hot Jupiters. In
addition, mass is a fundamental parameter for brown dwarf population studies
as in the works mentioned above. Therefore, we need to observe the spectra of
the host stars of transiting brown dwarfs to extract their radial velocities
(RVs) and derive the true mass (instead of the minimal mass $m\sin i$) and
other orbital parameters such as the eccentricity. Combining the information
from transit and RVs, the sample of transiting brown dwarfs is ideal for
population statistics and studies of their formation and evolution scenarios.

As different evolution models are developed for brown dwarfs and low-mass
stars (e.g., Baraffe et al., 2003; Phillips et al., 2020a; Marley et al.,
2021), the transiting brown dwarf sample can be used to test these models
(e.g., Carmichael et al., 2021). Compared with field brown dwarfs, the
transiting ones tend to be better characterized in terms of radius, mass, age
and even elemental abundances, thanks to the wealth of information provided by
their host stars, for example, assuming they were formed together and co-
evolved.

A special stage in the evolution of brown dwarfs in binary systems is when
their host stars evolve off the main sequence. Low-mass objects orbiting
evolved stars are poorly understood due to the challenges in their detection
and characterization largely caused by the large radii and stellar RV jitter
of the host stars (Yu et al., 2018; Tayar et al., 2019). It has been suggested
that planets around evolved stars should be different from planets around
main-sequence stars in many aspects due to their dynamical interactions with
the evolved host (Veras, 2016). For example, the increasing tidal interaction
may speed up the orbital decay and planet engulfment (Hut, 1981; Jackson et
al., 2008), which may induce a different eccentricity distribution in the
observed population of surviving hot Jupiters(Villaver et al., 2014; Grunblatt
et al., 2018). Additionally, due to the increasing luminosity of the evolved
host stars, close-in planets would receive more flux, which makes such systems
ideal for studies of planet inflation (e.g., Saunders et al., 2022; Grunblatt
et al., 2022) or re-inflation (e.g., Grunblatt et al., 2016). In contrast to
the numerous works on planets around evolved stars, few works discuss brown
dwarfs or very low-mass stars around evolved stars, probably due to the small
sample size of such systems. Do low-mass stars and brown dwarfs around evolved
stars undergo orbital or radius changes in a similar way as planets? Gathering
a sample of well-characterized transiting brown dwarfs would be the crucial
first step in addressing this question.

Here we report the discovery of a transiting brown dwarf, TOI-2336b, and two
transiting objects near the hydrogen burning mass limit, TOI-1608b and
TOI-2521b, around three aged stars using TESS data and ground-based follow-up
observations. In Section 2 we describe the data we used in this work. Section
3 presents the analyses to determine the properties of the host stars of these
three systems. Section 4 presents the joint analysis of the light curves and
the RV data and how we obtained the orbital parameters of the companions. In
Section 5, we analyze the radius inflation of brown dwarfs and low mass stars,
and we discuss the tidal evolution of these systems. We conclude our findings
in Section 6.

##  2 Observations

###  2.1 TESS photometry

TOI-1608 (TIC 138017750) was observed with the 2-minute cadence exposures in
TESS Sector 18 from 3rd November 2019 to 27th November 2019. The photometry
were extracted by the Science Processing Operations Center (SPOC; Jenkins et
al., 2016) pipeline. SPOC produces two kinds of light curves, using Simple
Aperture Photometry (SAP) or the Presearch Data Conditioning SAP (PDCSAP;
Smith et al., 2012; Stumpe et al., 2012; Stumpe et al., 2014). PDCSAP is
corrected for the instrumental systematic effects and dilution, and we used
PDCSAP light curves in our analysis. Eight transit signals were identified in
this light curve with a period of 2.47 days.

TOI-2336 (TIC 88902249) was observed with the 30-minute cadence exposures in
TESS Sector 11 from 23rd April 2019 to 20th May 2019, and it was re-observed
with the 10-minute cadence exposures in TESS Sector 38 from 29rd April 2021 to
26th May 2021 during the TESS First Extended Mission. Both light curves were
extracted by the TESS-SPOC pipeline (Caldwell et al., 2020) and we used the
PDCSAP in our analysis. Two transit signals were identified in the 30-minute
data and four transit signals were identified in the 10-minute data with a
period of 7.71 days.

TOI-2521 (TIC 72556406) was observed with the 30-minute cadence exposures in
TESS Sector 6 from 15rd December 2018 to 6th January 2019, and it was re-
observed with the 10-minute cadence exposures in TESS Sector 33 from 18th
December 2020 to 13th January 2021 during the TESS First Extended Mission.
Again we used the PDCSAP light curves extracted by the TESS-SPOC pipeline
(Caldwell et al., 2020) in our analysis. Four transit signals were identified
in the 30-minute data and five transit signals were identified in the
10-minute data with a period of 5.56 days.

We obtained all TESS data from the Mikulski Archive for Space Telescopes
(MAST) Portal111https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html.
We then detrended the TESS light curve to remove the remaining systematic
trends by fitting a Gaussian Process (GP) model after masking out the in-
transit data using the juliet package (Espinoza et al., 2019), which employs
the package celerite (Foreman-Mackey et al., 2017) to build the GP model. The
PDCSAP light curves with the best-fit GP models are shown in Figure 1.

(a) TOI-1608

(b) TOI-2336

(c) TOI-2521

Figure 1: TESS light curves of TOI-1608, TOI-2336 and TOI-2521. Left panels:
The dots are the TESS PDCSAP data and the red dots highlight the transit
signals. The blue lines are the best-fit GP models for detrending. The sector
18 for TOI-1608 was observed with the 2-minute cadence exposures. The sector
11 for TOI-2336 and the sector 6 for TOI-2521 were observed with the 30-minute
cadence exposures. The sector 38 for TOI-2336 and the sector 33 for TOI-2521
were observed with the 10-minute cadence exposures.Right panels: The final
detrended TESS PDCSAP light curves.

###  2.2 Ground-Based photometry

####  2.2.1 LCOGT

We conducted ground-based follow-up observations of TOI-2336 using the Las
Cumbres Observatory Global Telescope (LCOGT222https://lco.global/) network
(Brown et al., 2013) on 1st July 2021. We used the TESS Transit Finder, which
is a customized version of the Tapir software package (Jensen, 2013), to
schedule the transit observation. The observation was taken using the 1.0-m
telescopes located at Cerro Tololo Interamerican Observatory (CTIO). The
images were acquired with the Sinistro cameras in the Pan-STARRS $z$-short
band ($z_{s}$) with an exposure time of 30 s. The photometric analysis was
then carried out using AstroImageJ (Collins et al., 2017). We show the ground
light curve in Figure 7.

####  2.2.2 WASP

WASP-South has an array of 8 cameras forming the South African station of the
WASP transit-search survey (Pollacco et al., 2006). The field of TOI-2336 was
observed during 2006, 2007 and 2008, and then again in 2011 and 2012. Within
each year, observations of the field every $\sim$ 15 min were obtained on each
clear night, over a span of $\sim$ 150 nights. During that time, WASP-South
was equipped with 200 mm, f/1.8 lenses, observing with a 400–700 nm bandpass,
and with a photometric extraction aperture of 48 arcsecs. A total of 32 000
photometric data points were obtained.

TOI-2336 was not selected as a WASP candidate, but with hindsight we find that
the standard WASP search algorithm (Collier Cameron et al., 2007) produces a
match to the TESS transit detection with a period of 7.712 days.

We also searched the WASP data for signs of a stellar rotational modulation,
using the methods from Maxted et al. (2011), but found only a 95% upper limit
of 1 mmag on any modulation in the range 1 d to 100 d.

###  2.3 Spectroscopic Observations

####  2.3.1 NRES

We acquired 16 spectra for TOI-1608 from 3rd December 2021 to 9th January 2022
at the McDonald Observatory, and 21 spectra for TOI-2336 between 22nd April
2021 and 19th July 2021 at South African Astronomical Observatory (SAAO) using
the LCOGT Network of Robotic Echelle Spectrographs (NRES; Siverd et al.,
2018). NRES has four sets of identical echelle spectograph units at four LCOGT
sites. It has a resolving power of $R\sim 53,000$, covering a wavelength range
from 3900 to 8600 Å. We reduced the spectra and extracted the RVs using the
CERES pipeline (Brahm et al., 2017).

####  2.3.2 CHIRON

We observed TOI-2336 and TOI-2521 with the CHIRON fiber-fed echelle
spectrograph at the SMARTS 1.5-m telescope located at Cerro Tololo Inter-
American Observatory, Chile (Tokovinin et al., 2013). CHIRON is a high
resolution echelle spectrograph with a spectral coverage of 4100 to 8700 Å.
Both targets were observed in the _fiber_ mode, with a spectral resolving
power of $R\sim$ 28,000. For TOI-2336 we obtained 12 spectra from May 7 to
August 5 2021 with an exposure time of 1200 s. For TOI-2521 we gathered 8 RVs,
from March 20 to March 30 2021, using an exposure time of 1800 s. We used the
spectra extracted via the official CHIRON pipeline as per Paredes et al.
(2021). Radial velocities were derived from a least-squares deconvolution
between the observation and a nonrotating synthetic template, generated using
the ATLAS9 atmosphere models (Castelli & Hubrig, 2004) at the spectral
parameters of the targets. The derived broadening profile is fitted with a
kernel accounting for the effects of rotational, macroturbulent, instrumental
broadening, and radial velocity shift. Spectroscopic atmosphere parameters
were derived by matching the observed CHIRON spectra against a library of
10,000 observed spectra previously classified via the Stellar Parameter
Classification (Buchhave et al., 2012). The library is convolved to the
instrument resolution of CHIRON and interpolated via a gradient-boosting
regressor. Uncertainties on the spectroscopic parameters are determined by the
spectrum-to-spectrum scatter of these values.

####  2.3.3 CORALIE

We used the high resolution spectrograph CORALIE to obtain spectra for
TOI-2336. CORALIE is installed at the Swiss 1.2-m Leonhard Euler Telescope at
La Silla Observatory, Chile (Queloz et al., 2001), and it has a resolution of
R = 60, 000 and is fed by a 2″ fibre. We acquired 16 RV measurements from
April 14 to June 08 2021 with exposure times ranging between 1200 s and 2400 s
and S/N between 15 and 33. The RV measurement of each epoch was extracted by
cross-correlating the spectrum with a binary G2 mask (Pepe et al., 2002). The
bisector-span (BIS), the contrast (depth) and the full width at half-maximum
(FWHM) were recorded using the standard CORALIE data reduction pipeline.

####  2.3.4 TRES

We observed TOI-1608 and TOI-2521 with the Tillinghast Reflector Echelle
Spectrograph (TRES; Fűrész, 2008). TRES is a fiber-fed echelle spectrograph
mounted on the 1.5m telescope at the Fred Lawrence Whipple Observatory (FLWO)
in Arizona. TRES covers a wavelength range of 3900 to 9100 Å and has a
resolving power of $R\sim$ 44,000. Two reconnaissance spectra of TOI-1608 were
obtained on December 31 2019 and January 4 2020 and similarly two
reconnaissance spectra of TOI-2521 were obtained on March 11 and March 19
2021. The spectra were extracted as described in Buchhave et al. (2010) and
were then used to derive stellar parameters using the Stellar Parameter
Classification (SPC; Buchhave et al., 2012) tool. SPC cross correlates the
observed spectrum against a grid of synthetic spectra derived from the Kurucz
atmospheric models (Kurucz, 1992). SPC was used to derive parameters $T_{\rm
eff}$, $\rm[m/H]$, and $v\sin i_{\ast}$ by fixing the $\log g_{\ast}$ from
isochrone fitting as described in section 3.2. We did not use the TRES spectra
for RV measurements as they include only two epochs for each of TOI-1608 and
TOI-2521.

###  2.4 High Angular Resolution Imaging

The presence of a close companion star can be the cause of a transit-like
event (i.e., a false positive), can cause incorrect stellar and exoplanet
parameters to be determined (Furlan & Howell, 2017, 2020), or can lead to non-
detections of small planets residing with the same exoplanetary system (Lester
et al., 2021). Given that nearly one-half of FGK stars are in binary or
multiple star systems (Matson et al., 2019), high-resolution imaging provides
crucial information toward our understanding of exoplanetary formation,
dynamics and evolution (Howell et al., 2021). We conducted high-resolution
imaging observations on the three systems and found no nearby stars within the
observational limits of each instrument. We summarize our imaging observations
and results in this section.

####  2.4.1 Gemini

TOI-1608 was observed on 2022 February 10 UT using the ‘Alopeke speckle
instrument on the Gemini North 8-m telescope and TOI-2336 was observed on 2022
May 05 UT using the Zorro instrument on the Gemini South 8-m telescope (Scott
et al., 2021; Howell & Furlan, 2022).

‘Alopeke and Zorro both provide simultaneous speckle imaging in two bands (562
nm and 832 nm) with output data products including a reconstructed image with
robust contrast limits on companion detections. Our observations reveal that
both TOI-1608 and TOI-2336 have no stellar companions to within the angular
and image contrast levels achieved.

For each star, three sets of 1000 x 0.06 second images were obtained and
processed in our standard reduction pipeline (Howell et al., 2011). Figure 2
shows our final contrast curves and the 832 nm reconstructed speckle image. We
find that both TOI-1608 and TOI-2336 have no stellar companions brighter than
5-9 magnitudes below that of the target star from the 8-m telescope
diffraction limit (20 mas) out to 1.2”. At the distance of TOI-1608 (d=100 pc)
these angular limits correspond to (sky-projected) spatial limits of 2 to 120
AU and for TOI-2336 (d=298 pc) the angular limits correspond to 6 to 358 AU.

Figure 2: We show the $5\sigma$ speckle imaging contrast curves in both
filters as a function of the angular separation from the diffraction limit out
to 1.2 arcsec. The inset shows the reconstructed 832 nm image with a 1 arcsec
scale bar. Both stars were found to have no close companions to within the
angular and contrast levels achieved.

####  2.4.2 SOAR

We searched for stellar companions to TOI-2336 and TOI-2521 with speckle
imaging on the 4.1-m Southern Astrophysical Research (SOAR) telescope
(Tokovinin, 2018) on 7 February 2021 UT and 1 October 2021, respectively. The
observations were performed in Cousins I-band, a similar visible bandpass as
TESS. Both observations were sensitive to an approximately 5-magnitude fainter
star at an angular distance of 0.25 arcsec from the target. More details of
the observations within the SOAR TESS survey are available in Ziegler et al.
(2020). The 5$\sigma$ detection sensitivity and speckle auto-correlation
functions from the observations are shown in Figure 3. No nearby stars were
detected within 3″of either TOI-2336 or TOI-2521 in the SOAR observations.

(a) TOI-2336

(b) TOI-2521

Figure 3: Speckle auto-correlation functions for TOI-2336 and TOI-2521
obtained in the $I$ band using SOAR. The $5\sigma$ contrast curves are shown
as the black points. The black solid line shows the linear fit of the data.

##  3 Stellar Properties

Table 1: Stellar parameters for TOI-1608, TOI-2336, and TOI-2521

Parameter | TOI-1608 | TOI-2336 | TOI-2521 | Source
---|---|---|---|---
Main identifiers |  |  |  |
TIC | 138017750 | 88902249 | 72556406 |
GaiaID | 125548289270234880 | 6206457922905803392 | 3005862518856922752 |
Equatorial Coordinates |  |  |  |
RA | 3:23:12.28 | 15:20:55.30 | 6:14:06.99 |  Gaia DR3[1]
DEC | 33:04:41.25 | -33:41:32.87 | -8:03:44.31 |  Gaia DR3
Photometric properties |  |  |  |
TESS (mag) | $7.42812\pm 0.0064$ | $10.1757\pm 0.0061$ | $10.8218\pm 0.0065$ | TICv8[2]
Gaia (mag) | $7.8280\pm 0.0004$ | $10.5905\pm 0.0002$ | $11.2692\pm 0.0008$ |  Gaia DR3
Gaia BP (mag) | $8.1097\pm 0.0016$ | $10.8992\pm 0.0006$ | $11.6116\pm 0.0024$ |  Gaia DR3
Gaia RP (mag) | $7.3789\pm 0.0012$ | $10.1146\pm 0.0004$ | $10.7518\pm 0.0018$ |  Gaia DR3
$B$ (mag) | $8.519\pm 0.027$ | $11.134\pm 0.119$ | $12.076\pm 0.231$ | Hipparcos
$V$ (mag) | $7.96\pm 0.03$ | $10.756\pm 0.009$ | $11.263\pm 0.017$ | Hipparcos
$J$ (mag) | $6.885\pm 0.027$ | $9.61\pm 0.024$ | $10.161\pm 0.026$ | 2MASS[3]
$H$ (mag) | $6.636\pm 0.017$ | $9.386\pm 0.026$ | $9.839\pm 0.023$ | 2MASS
$K$ (mag) | $6.584\pm 0.029$ | $9.283\pm 0.019$ | $9.743\pm 0.019$ | 2MASS
WISE1 (mag) | $7.571\pm 0.402$ | $9.247\pm 0.023$ | $9.707\pm 0.023$ | WISE[4]
WISE2 (mag) | $6.554\pm 0.024$ | $9.247\pm 0.02$ | $9.726\pm 0.021$ | WISE
WISE3 (mag) | $6.604\pm 0.015$ | $9.195\pm 0.039$ | $9.688\pm 0.043$ | WISE
WISE4 (mag) | $6.52\pm 0.072$ | $8.49\pm 0.331$ | $9.193^{(a)}$ | WISE
Astrometric properties |  |  |  |
$\varpi$ (mas) | $9.87\pm 0.04$ | $3.37\pm 0.02$ | $2.99\pm 0.02$ |  Gaia DR3
$\mu_{\rm\alpha}\ ({\rm mas~{}yr^{-1}})$ | $109.30\pm 0.04$ | $6.22\pm 0.02$ | $-7.68\pm 0.03$ |  Gaia DR3
$\mu_{\rm\delta}\ ({\rm mas~{}yr^{-1}})$ | $-58.30\pm 0.03$ | $-18.97\pm 0.02$ | $-0.37\pm 0.03$ |  Gaia DR3
RV (km s-1) | $44.5\pm 2.8$ | $35.6\pm 1.5$ | $-32.5\pm 2.0$ |  Gaia DR3
Derived parameters |  |  |  |
Distance (pc) | $101.3\pm 0.4$ | $296.6\pm 1.7$ | $334.4\pm 2.3$ | Section 3.2
$M_{\ast}$ ($M_{\odot}$) | $1.31\pm 0.07$ | $1.40\pm 0.07$ | $0.95\pm 0.06$ | Section 3.2
$R_{\ast}$ ($R_{\odot}$) | $2.16\pm 0.08$ | $1.82\pm 0.06$ | $1.74\pm 0.06$ | Section 3.2
$\rho_{\ast}$ ($\rm g~{}cm^{-3}$) | $0.18\pm 0.03$ | $0.33\pm 0.04$ | $0.25\pm 0.03$ | Section 3.2
$\log g_{\ast}$ | $3.89\pm 0.04$ | $4.06\pm 0.04$ | $3.93\pm 0.04$ | Section 3.2
$L_{\ast}$ (L⊙) | $5.56\pm 0.26$ | $5.10\pm 0.21$ | $2.74\pm 0.12$ | Section 3.2
$T_{\rm eff}$ (K) | $6028\pm 82$ | $6433\pm 84$ | $5625\pm 74$ | Section 3.2
$\rm[Fe/H]$ (dex) | $0.09\pm 0.12$ | $0.09\pm 0.11$ | $-0.28\pm 0.13$ | Section 3.2
Age (Gyr) | $4.0\pm 0.8$ | $2.4\pm 0.5$ | $10.1\pm 1.1$ | Section 3.2
$P_{\rm rot}$ (days) | $2.7\pm 0.2$ | $4.2\pm 0.5$ | $6.9\pm 0.9$ | Section 3.4

  * 1

[1] Gaia Collaboration et al. (2021), [2] Stassun et al. (2018b, 2019), [3]
Cutri et al. (2003), [4] Wright et al. (2010).

  * 2

(a) The WISE catalog did not report an uncertainty for this $W4$ magnitude for
TOI-2521. Therefore, we did not use this magnitude in our analysis.

###  3.1 Spectral Energy Distribution

As an independent determination of the basic stellar parameters, we performed
an analysis of the broadband spectral energy distribution (SED) of each star
together with the Gaia EDR3 parallax (with no systematic offset applied; see,
e.g., Stassun & Torres, 2021), in order to determine an empirical measurement
of the stellar radius, following the procedures described in Stassun & Torres
(2016); Stassun et al. (2017); Stassun et al. (2018a). Depending on the
photometry available for each source, we pulled the $B_{T}V_{T}$ magnitudes
from Tycho-2, the $BVgri$ magnitudes from APASS, the Strömgren $uvby$
magnitudes from Paunzen (2015), the $JHK_{S}$ magnitudes from 2MASS, the W1–W4
magnitudes from WISE, the $GG_{\rm BP}G_{\rm RP}$ magnitudes from Gaia, and
the FUV and/or NUV fluxes from GALEX. All these magnitudes are shown in Table
1. Together, the available photometry spans the stellar SED over the
approximate wavelength range 0.2–22 $\mu$m (see Figure 4).

Figure 4: Spectral energy distribution of TOI-1608 (top), TOI-2336 (middle),
and TOI-2521 (bottom). Red symbols represent the observed photometric
measurements, where the horizontal bars represent the effective width of the
passband. Blue symbols are the model fluxes from the best-fit Kurucz
atmosphere model (black).

We performed fits to the photometry using Kurucz stellar atmosphere models,
with the free parameters being the effective temperature ($T_{\rm eff}$),
metallicity ([Fe/H]), and surface gravity ($\log g_{\ast}$). We also fit for
the extinction, $A_{V}$, limited to the full line-of-sight value from the
Galactic dust maps of Schlegel et al. (1998). The resulting fits shown in
Figure 4 have a reduced $\chi^{2}$ ranging from 1.2 to 1.5, and the best-fit
parameters are summarized in Table 2.

Integrating the model SED gives the bolometric flux at Earth, $F_{\rm bol}$.
Taking the $F_{\rm bol}$ together with the Gaia parallax directly gives the
luminosity, $L_{\rm bol}$. Similarly, the $F_{\rm bol}$ together with the
$T_{\rm eff}$ and the parallax gives the stellar radius, $R_{\ast}$. Moreover,
the stellar mass, $M_{\ast}$, can be estimated from the empirical eclipsing-
binary based relations of Torres et al. (2010). When available, the GALEX
photometry allows the activity index, $\log R^{\prime}_{\rm HK}$ to be
estimated from the empirical relations of Findeisen et al. (2011). All
quantities are summarized in Table 2.

Table 2: Summary of derived and empirical parameters determined from fits to the stellar spectral energy distributions. $T_{\rm eff}$, $[{\rm Fe/H}]$ and $\log g_{\ast}$ here are used as the input of isochrone fitting in Section 3.2 Parameter | TOI-1608 | TOI-2336 | TOI-2521
---|---|---|---
$A_{V}$ | $0.01\pm 0.01$ | $0.44\pm 0.03$ | $0.07\pm 0.07$
$T_{\rm eff}$ [K] | $5950\pm 100$ | $6550\pm 100$ | $5600\pm 100$
$[{\rm Fe/H}]$ | $0.1\pm 0.3$ | $0.0\pm 0.3$ | $-0.3\pm 0.3$
$\log g_{\ast}$ | $4.0\pm 0.5$ | $4.2\pm 0.3$ | $4.0\pm 0.3$
$F_{\rm bol}$ [10-9 erg s-1 cm-2] | $17.40\pm 0.20$ | $1.918\pm 0.045$ | $0.793\pm 0.018$
$L_{\rm bol}$ [L⊙] | $5.564\pm 0.068$ | $5.25\pm 0.13$ | $2.771\pm 0.068$
$R_{\ast}$ [R⊙] | $2.222\pm 0.076$ | $1.781\pm 0.059$ | $1.770\pm 0.068$
$M_{\ast}$ [M⊙] | $1.38\pm 0.08$ | $1.41\pm 0.08$ | $1.10\pm 0.07$
$\log R^{\prime}_{\rm HK}$ | $-4.52\pm 0.05$ | … | …

###  3.2 Isochrone fitting

In order to obtain more precise stellar parameters, we performed isochrone
fitting with the package isochrones (Morton, 2015) based on the results of the
SED fitting and the photometry. We pulled $T_{\rm eff}$, $[{\rm Fe/H}]$ and
$\log g$ from SED fitting, the TESS magnitudes from TICv8, the $JHK_{S}$
magnitudes from 2MASS, the W1–W3 magnitudes from WISE and the $GG_{\rm
BP}G_{\rm RP}$ magnitudes from Gaia DR3. We also used the Gaia DR3 parallax as
well as the stellar density from the transit model, which is described in
Section 4.

In the results of TOI-1608, we find bimodality in the posterior distribution
of $M_{\ast}$, $\rho_{\ast}$, $\log g_{\ast}$ and age, which is shown in
Figure 13. We compared the best-fit values of these two peaks with the SED
fitting results and the Gaia DR3 Final Luminosity Age Mass Estimator (FLAME)
results (Gaia Collaboration et al., 2016, 2021), and we found that the results
of the posterior peak with a lower mass estimate are more consistent with the
SED fitting and Gaia DR3. Therefore, we adopted the set of posterior
distributions with a lower mass to derive the final stellar parameters and
uncertainties from the isochrone fitting for TOI-1608.

Similarly for TOI-2521, we checked for consistency and find that the mass
estimates are different in these three sets of results. The mass values are
$1.10\pm 0.07\ M_{\ast}$, $0.95\pm 0.06\ M_{\ast}$ and $1.05\pm 0.04\
M_{\ast}$ from the SED fitting, isochrone fitting and Gaia DR3, respectively.
Relatively speaking, the mass derived from the SED fitting using the empirical
relation from Torres et al. (2010) is not as accurate as the other two
estimates because the Torres et al. (2010) relation probably does not work
very well for aged stars due to the limit of their stellar sample. We consider
the mass from our isochrone fitting as the more reliable one, as we
incorporated the results from the SED fitting and also used more photometric
data than Gaia DR3. Therefore, we adopted the results from our isochrone
fitting as the final stellar parameters for TOI-2521.

For TOI-2336, our results from the isochrone fit are consistent with the SED
fitting and Gaia DR3, so we adopted the isochrone fitting results as the final
stellar parameters to stay consistent with the other two targets. The final
stellar parameters we adopted in the following analyses are summarized in
Table 1.

###  3.3 Spectral analysis

We analyzed CHIRON and TRES spectra to derive the stellar parameters. The
observations and analyses are described in Section 2.3.2 and 2.3.4, and the
results are shown in Table 3. $T_{\rm eff}$ from both CHIRON and TRES and
$[{\rm Fe/H}]$ from CHIRON are consistent with the results from other methods.
$[{\rm M/H}]$ from TRES is obtained by using all of the lines in the
wavelength region from $\sim 505\ nm$ to $\sim 536\ nm$. As iron lines
dominate in this range, we compare these results with $[{\rm Fe/H}]$ from
other methods and found they agree with each other. However, $\log g_{\ast}$
for TOI-2521 from CHIRON spectra is not consistent with isochrone fitting
because TOI-2521 is a fast rotating star, where the spectra would have a
relatively poor constraint on $\log g_{\ast}$. Thus, we only use the
spectroscopic results as a check and adopt the results from isochrone fitting
for further analyses.

Table 3: Summary of derived parameters determined from CHIRON and TRES spectra analysis. Parameter | TOI-1608 | TOI-2336 | TOI-2521
---|---|---|---
CHIRON |  |  |
$T_{\rm eff}$ (K) | … | $6551\pm 100$ | $5526\pm 50$
$[{\rm Fe/H}]$ (dex) | … | $-0.03\pm 0.10$ | $-0.41\pm 0.10$
$\log g_{\ast}$ | … | $4.16\pm 0.10$ | $3.70\pm 0.10$
$v\sin i$ (km s-1) | … | $29.7\pm 0.5$ | $15.6\pm 0.5$
TRES |  |  |
$T_{\rm eff}$ (K) | $6051\pm 56$ | … | $5604\pm 50$
$[{\rm M/H}]$ (dex) | $0.10\pm 0.08$ | … | $-0.21\pm 0.08$
$v\sin i$ (km s-1) | $48.3\pm 0.5$ | … | $15.2\pm 0.5$

###  3.4 Stellar rotation period

We masked in-transit data points in the TESS PDCSAP light curves (shown in the
left panels of Figure 1) and used Generalized Lomb-Scargle periodograms to
estimate the stellar rotation period of TOI-1608, TOI-2336, and TOI-2521. The
Lomb-Scargle periodograms are shown in Figure 5. We attributed the highest
peak in the Lomb-Scargle periodogram to the stellar rotation period. For each
peak, we used a half of its FWHM as the uncertainty. For TOI-2336 and
TOI-2521, we find the results obtained from different sectors are consistent.
We used the average of different sectors as the final rotation period and use
the higher uncertainty as the final error estimation. The final results are
listed in Table 1. Using these rotation period and $R_{\ast}$ derived in
Section 3.2, we can calculate the rotation speeds of TOI-1608, TOI2336, and
TOI-2521 to be $40.5\pm 3.5$ km s-1, $21.9\pm 3.1$ km s-1, and $12.8\pm 1.9$
km s-1, respectively. Compared with $v\sin i$ derived from the spectra, the
speeds of TOI-1608 and TOI-2336 are about $2\sigma$ smaller and the speed of
TOI-2521 is about $1\sigma$ smaller, which is unphysical (though still roughly
consistent statistically speaking). This may result from the unreliable $v\sin
i$ estimates from spectral analyses caused by the fast rotation and low $\log
g_{\ast}$ of these three targets, or it may result from our estimates of the
rotation period using the TESS light curves, which have a limited baseline.
More accurate measurements on $v\sin i$ and the rotation period in the future
would be able to put constraints on the projected stellar obliquity (Masuda &
Winn, 2020).

Figure 5: The generalized Lomb-Scargle periodogram from the TESS PDCSAP light
curves of TOI-1608, TOI-2336, and TOI-2521. The periods of the highest peaks
in each periodogram are marked as different color bars.

###  3.5 Asteroseismic analysis

No unambiguous solar-like oscillations could be detected for these three
stars. We calculated their expected frequencies of maximum power ($\nu_{\rm
max}$) using the spectroscopic $T_{\rm eff}$ and $\log g_{\ast}$ from Section
3.2 with the seismic scaling relation (Brown et al., 1991; Kjeldsen & Bedding,
1995; Lund et al., 2016), yielding $\nu_{\rm max}\approx 853\ {\rm\mu Hz}$ for
TOI-1608, $\nu_{\rm max}\approx 1221\ {\rm\mu Hz}$ for TOI-2336, and $\nu_{\rm
max}\approx 968\ {\rm\mu Hz}$ for TOI-2521. Since these values are all greater
than the Nyquist frequencies for 30-min (${\sim}278\ {\rm\mu Hz}$) and 10-min
(${\sim}833\ {\rm\mu Hz}$) cadence light curves, only 2-min cadence data are
suited for the detection of solar-like oscillations – which is only available
for 1 sector for TOI-1608. We searched for oscillation signals using the
lightkurve package (Lightkurve Collaboration et al., 2018) and from manual
inspection of the power density spectra of the light curves with the transit
signals masked out, but could not identify any solar-like oscillations.
Considering the TESS magnitudes of the stars (Campante et al., 2016), the fact
that they were observed with TESS cameras 1 or 2 where background scattered
light can be significant, and that clear rotational signals are evident in the
photometry (at least for TOIs 1608 and 2521), indicating stellar surface
activity that may suppress oscillation mode amplitudes (Chaplin et al., 2011),
the non-detection is not entirely unexpected. Future TESS observations
appended to the current coverage-limited light curves will be useful for
searching for the oscillations of these stars.

##  4 Analysis and results

###  4.1 Joint RV and transit analysis

We jointly fit the detrended TESS light curves and RVs by utilizing the juliet
package (Espinoza et al., 2019). For TOI-2336, we also include the LCOGT light
curves in the joint fitting. Juliet is a modeling tool for exoplanetary
systems. It employs the batman package (Kreidberg, 2015) to build transit
models and employs radvel (Fulton et al., 2018) to build RV models. It can fit
both transit and RV data simultaneously by using bayesian inference with
nested sampling with multiple choices of samplers. We used the sampler dynesty
(Speagle, 2020) in the following analyses.

In the modeling for the three systems, we set the priors based in a similar
fashion. We set wide uniform priors for the transit epoch ($T_{0,b}$) and the
orbital period ($P_{0}$) around the values from ExoFOP
333https://exofop.ipac.caltech.edu/tess/, which are from the analysis by the
TESS project. For the radius ratio ($p=R_{b}/R_{*}$) and the impact parameter
($b=a\cos i/R_{*}$), juliet uses the approach described in Espinoza (2018) to
fit the parametrizations $r_{1}$ and $r_{2}$ instead of fitting $b$ and $p$
directly. This parametrization allows us to efficiently sample the physically
plausible zone in the (b,p) plane (i.e., b<1+p) by only simply uniformly
sampling $r_{1}$ and $r_{2}$ between 0 and 1. Here we set uninformative
uniform priors between 0 and 1 for $r_{1}$ and $r_{2}$. For the orbital
eccentricity ($e$) and the argument of periapsis ($\omega_{b}$), we chose to
fit for the parameters $e\sin\omega_{b}$ and $e\cos\omega_{b}$. The
uninformative priors should be uniform priors between -1 and 1 for these two
parameters and juliet will keep $e<1$ when sampling. However, we found the
fitting would be stuck if we allow $e$ to be sampled very close to 1.
Therefore, we set uniform priors between -0.7 and 0.7 for $e\sin\omega_{b}$
and $e\cos\omega_{b}$, which constrains $e<0.99$. As such priors are
significantly wider than the resulting posteriors and changing the width down
to (-0.1, 0.1) does not affect our final results, we think adopting such
priors is essentially equivalent to using uninformative priors.

For the TESS light curves, because the TESS PDCSAP light curves have already
been corrected for the light dilution, we fixed the dilution factor D as 1.
However, the dilution correction may not be optimal due to the complex TESS
background flux like scattered light. To make sure these possible errors would
not influence our results, we adopted a Gaussian prior with a mean of 1 and a
variance of 0.1 to fit the data and obtained consistent results. For TOI-2336
and TOI-2521, we also compared the transit depths from fitting the light
curves of each individual sector independently, and the results are
consistent. In addition, the contamination ratios of these 3 systems given by
TICv8 on ExoFOP are all less than 0.05, indicating that the contamination is
very small, so any errors in the contamination correction of the TESS PDCSAP
light curves should be negligible. Therefore, we conclude that contamination
or scattered light should not be an issue in the TESS light curves or affect
our results, and we fixed the dilution factor as 1 in our reported results. We
set a Gaussian prior with a zero mean and a small variance for the mean out-
of-transit flux M because the light curves have been detrended and normalized.
We fitted an extra flux jitter term ($\sigma$) to account for additional noise
and set a wide log-uniform prior based on the photometric uncertainties of the
light curves. We adopted a quadratic limb-darkening law for the TESS light
curves and fitted the limb-darkening parameters $q_{1}$ and $q_{2}$ (Kipping,
2013) with uninformative priors between 0 and 1. For the LCOGT light curves of
TOI-2336, we set similar priors for the dilution factor, mean out-of-transit
flux, extra flux jitter term and limb-darkening parameters.

For the RV data, we fitted for the RV semi-amplitude ($K$) for each system
with a systemic velocity ($\mu$) for each instrument. We estimated rough
values for these parameters from the data through visual inspection and set
very wide and essentially uninformative uniform priors around these values. We
fitted an extra RV jitter term to account for any additional systematics for
each instrument. We set wide log-uniform priors for them according to the RV
precision of each instrument. The RVs of NRES for TOI-1608 on 9th January 2022
(BJD=2459588.661) and of CORALIE for TOI-2336 on 9th May 2021
(BJD=2459343.687687) were observed in transit. However, based on the equation
(1) in Triaud (2018), the predicted semi-amplitudes of the Rossiter-McLaughlin
effect (Rossiter, 1924; McLaughlin, 1924) of these systems are both less than
50 m/s, which is less than the uncertainties of our RVs. Therefore, these RVs
were not highly biased due to the transit and should not influence our RV
modeling for the orbits.

We summarize the prior settings and the best-fit values for each parameter in
our modeling in Table LABEL:tab_1608_joint_model, LABEL:tab_2336_joint_model
and LABEL:tab_2521_joint_model. We show the light curves with the best-fit
model in Figure 6 and 7, and show the RV data with the best-fit model in
Figure 8.

We combine the parameters from the joint RV and transit modeling with the
stellar parameters obtained in Section 3 to derive the physical parameters of
the companions. The results are shown in Table 7.

Table 4: Prior settings and the best-fit values along with the 68% credibility
intervals in the final joint fit for TOI-1608. $\mathcal{N}$($\mu\ ,\
\sigma^{2}$) means a normal prior with mean $\mu$ and standard deviation
$\sigma$. $\mathcal{U}$(a , b) stands for a uniform prior ranging from a to b.
$\mathcal{J}$(a , b) stands for a Jeffrey’s prior ranging from a to b.

.  Parameter Prior Best-fit Description Planetary parameters $P_{b}$ (days)
$\mathcal{U}$ ($2.4$ , $2.6$) $2.47275^{+0.00004}_{-0.00004}$ Orbital period
of TOI-1608b. $T_{0,b}$ (BJD) $\mathcal{U}$ ($2458791$ , $2458793$)
$2458792.4599^{+0.0007}_{-0.0007}$ Mid-transit time of TOI-1608b. $r_{1,b}$
$\mathcal{U}$ (0 , 1) $0.952^{+0.003}_{-0.004}$ Parametrisation for p and b.
$r_{2,b}$ $\mathcal{U}$ (0 , 1) $0.057^{+0.002}_{-0.002}$ Parametrisation for
p and b. $e\sin\omega_{b}$ $\mathcal{U}$ (-0.7 , 0.7) $-0.02^{+0.03}_{-0.03}$
Parametrisation for $e$ and $\omega_{b}$. $e\cos\omega_{b}$ $\mathcal{U}$
(-0.7 , 0.7) $0.02^{+0.03}_{-0.03}$ Parametrisation for $e$ and $\omega_{b}$.
TESS photometry parameters $D_{\rm TESS-S18}$ Fixed $1$ TESS photometric
dilution factor. $M_{\rm TESS-S18}$ $\mathcal{N}$ (0 , $0.1^{2}$)
$0.0000004^{+0.000003}_{-0.000003}$ Mean out-of-transit flux of TESS
photometry. $\sigma_{\rm TESS-S18}$ (ppm) $\mathcal{J}$ ($0.1$ , $1000$)
$239^{+3}_{-3}$ TESS additive photometric jitter term. $q_{\rm 1,TESS-S18}$
$\mathcal{U}$ (0 , 1) $0.12^{+0.12}_{-0.08}$ Quadratic limb darkening
coefficient. $q_{\rm 2,TESS-S18}$ $\mathcal{U}$ (0 , 1) $0.48^{+0.30}_{-0.32}$
Quadratic limb darkening coefficient. Stellar parameters ${\rho}_{\ast}$ ($\rm
kg\ m^{-3}$) $\mathcal{J}$ ($10$ , $1000$) $224^{+27}_{-21}$ Stellar density.
RV parameters $K_{b}$ ($\rm m\ s^{-1}$) $\mathcal{U}$ ($0$ , $20000$)
$10649^{+232}_{-216}$ RV semi-amplitude of TOI-1608b. NRESRV parameters
$\rm\mu_{NRES}$ ($\rm m\ s^{-1}$) $\mathcal{U}$ ($70000$ , $100000$)
$82204^{+281}_{-283}$ Systemic velocity for NRES. $\rm\sigma_{NRES}$ ($\rm m\
s^{-1}$) $\mathcal{J}$ ($1$ , $10000$) $542^{+139}_{-103}$ Extra jitter term
for NRES.

Table 5: Similar to Table LABEL:tab_1608_joint_model but for TOI-2336

.  Parameter Prior Best-fit Description Planetary parameters $P_{b}$ (days)
$\mathcal{U}$ ($7.6$ , $7.8$) $7.711978^{+0.000013}_{-0.000013}$ Orbital
period of TOI-2336b. $T_{0,b}$ (BJD) $\mathcal{U}$ ($2459358$ , $2459360$)
$2459359.0493^{+0.0006}_{-0.0006}$ Mid-transit time of TOI-2336b. $r_{1,b}$
$\mathcal{U}$ (0 , 1) $0.874^{+0.013}_{-0.014}$ Parametrisation for p and b.
$r_{2,b}$ $\mathcal{U}$ (0 , 1) $0.0592^{+0.0008}_{-0.0007}$ Parametrisation
for p and b. $e\sin\omega_{b}$ $\mathcal{U}$ (-0.7 , 0.7)
$-0.007^{+0.008}_{-0.007}$ Parametrisation for $e$ and $\omega_{b}$.
$e\cos\omega_{b}$ $\mathcal{U}$ (-0.7 , 0.7) $0.006^{+0.003}_{-0.003}$
Parametrisation for $e$ and $\omega_{b}$. TESS photometry parameters $D_{\rm
TESS-S11}$ Fixed $1$ TESS photometric dilution factor. $D_{\rm TESS-S38}$
Fixed $1$ $M_{\rm TESS-S11}$ $\mathcal{N}$ (0 , $0.1^{2}$)
$0.0000009^{+0.000009}_{-0.000009}$ Mean out-of-transit flux of TESS
photometry. $M_{\rm TESS-S38}$ $\mathcal{N}$ (0 , $0.1^{2}$)
$-0.000001^{+0.000008}_{-0.000008}$ $\sigma_{\rm TESS-S11}$ (ppm)
$\mathcal{J}$ ($1$ , $10000$) $12^{+35}_{-10}$ TESS additive photometric
jitter term. $\sigma_{\rm TESS-S38}$ (ppm) $\mathcal{J}$ ($1$ , $10000$)
$117^{+28}_{-42}$ $q_{\rm 1,TESS-S11}$ $\mathcal{U}$ (0 , 1)
$0.19^{+0.12}_{-0.09}$ Quadratic limb darkening coefficient. $q_{\rm
1,TESS-S38}$ $\mathcal{U}$ (0 , 1) $0.12^{+0.07}_{-0.06}$ $q_{\rm 2,TESS-S11}$
$\mathcal{U}$ (0 , 1) $0.32^{+0.31}_{-0.20}$ Quadratic limb darkening
coefficient. $q_{\rm 2,TESS-S38}$ $\mathcal{U}$ (0 , 1) $0.56^{+0.27}_{-0.32}$
LCOGT photometry parameters $D_{\rm LCOGT}$ Fixed $1$ LCOGT photometric
dilution factor. $M_{\rm LCOGT}$ $\mathcal{N}$ (0 , $0.1^{2}$)
$0.00015^{+0.00009}_{-0.00009}$ Mean out-of-transit flux of LCOGT photometry.
$\sigma_{\rm LCOGT}$ (ppm) $\mathcal{J}$ ($1$ , $10000$) $1179^{+62}_{-62}$
LCOGT additive photometric jitter term. $q_{\rm 1,LCOGT}$ $\mathcal{U}$ (0 ,
1) $0.02^{+0.04}_{-0.02}$ Quadratic limb darkening coefficient. $q_{\rm
2,LCOGT}$ $\mathcal{U}$ (0 , 1) $0.54^{+0.28}_{-0.32}$ Quadratic limb
darkening coefficient. Stellar parameters ${\rho}_{\ast}$ ($\rm kg\ m^{-3}$)
$\mathcal{J}$ ($10$ , $1000$) $246^{+38}_{-31}$ Stellar density. RV parameters
$K_{b}$ ($\rm m\ s^{-1}$) $\mathcal{U}$ ($0$ , $20000$) $5549^{+37}_{-39}$ RV
semi-amplitude of TOI-2336b. NRESRV parameters $\rm\mu_{NRES}$ ($\rm m\
s^{-1}$) $\mathcal{U}$ ($20000$ , $40000$) $33707^{+327}_{-314}$ Systemic
velocity for NRES. $\rm\sigma_{NRES}$ ($\rm m\ s^{-1}$) $\mathcal{J}$ ($0.1$ ,
$10000$) $1586^{+264}_{-205}$ Extra jitter term for NRES. CHIRONRV parameters
$\rm\mu_{CHIRON}$ ($\rm m\ s^{-1}$) $\mathcal{U}$ ($20000$ , $40000$)
$30288^{+52}_{-50}$ Systemic velocity for CHIRON. $\rm\sigma_{CHIRON}$ ($\rm
m\ s^{-1}$) $\mathcal{J}$ ($0.1$ , $10000$) $4.7^{+27.3}_{-4.2}$ Extra jitter
term for CHIRON. CORALIERV parameters $\rm\mu_{CORALIE}$ ($\rm m\ s^{-1}$)
$\mathcal{U}$ ($20000$ , $40000$) $32432^{+36}_{-34}$ Systemic velocity for
CORALIE. $\rm\sigma_{CORALIE}$ ($\rm m\ s^{-1}$) $\mathcal{J}$ ($0.1$ ,
$10000$) $104^{+40}_{-33}$ Extra jitter term for CORALIE.

Table 6: Similar to Table LABEL:tab_1608_joint_model but for TOI-2521

.  Parameter Prior Best-fit Description Planetary parameters $P_{b}$ (days)
$\mathcal{U}$ ($5.5$ , $5.6$) $5.563060^{+0.000007}_{-0.000007}$ Orbital
period of TOI-2521b. $T_{0,b}$ (BJD) $\mathcal{U}$ ($2459226$ , $2459228$)
$2459227.2466^{+0.0007}_{-0.0007}$ Mid-transit time of TOI-2521b. $r_{1,b}$
$\mathcal{U}$ (0 , 1) $0.71^{+0.07}_{-0.13}$ Parametrisation for p and b.
$r_{2,b}$ $\mathcal{U}$ (0 , 1) $0.0598^{+0.0011}_{-0.0013}$ Parametrisation
for p and b. $e\sin\omega_{b}$ $\mathcal{U}$ (-0.7 , 0.7)
$0.003^{+0.007}_{-0.007}$ Parametrisation for $e$ and $\omega_{b}$.
$e\cos\omega_{b}$ $\mathcal{U}$ (-0.7 , 0.7) $0.0003^{+0.004}_{-0.004}$
Parametrisation for $e$ and $\omega_{b}$. TESS photometry parameters $D_{\rm
TESS-S6}$ Fixed $1$ TESS photometric dilution factor. $D_{\rm TESS-S33}$ Fixed
$1$ $M_{\rm TESS-S6}$ $\mathcal{N}$ (0 , $0.1^{2}$)
$-0.000004^{+0.000012}_{-0.000012}$ Mean out-of-transit flux of TESS
photometry. $M_{\rm TESS-S33}$ $\mathcal{N}$ (0 , $0.1^{2}$)
$-0.000004^{+0.000011}_{-0.000011}$ $\sigma_{\rm TESS-S6}$ (ppm) $\mathcal{J}$
($0.001$ , $10$) $0.25^{+2.38}_{-0.24}$ TESS additive photometric jitter term.
$\sigma_{\rm TESS-S33}$ (ppm) $\mathcal{J}$ ($0.001$ , $10$)
$0.12^{+1.99}_{-0.11}$ $q_{\rm 1,TESS-S6}$ $\mathcal{U}$ (0 , 1)
$0.15^{+0.11}_{-0.07}$ Quadratic limb darkening coefficient. $q_{\rm
1,TESS-S33}$ $\mathcal{U}$ (0 , 1) $0.16^{+0.11}_{-0.08}$ $q_{\rm 2,TESS-S6}$
$\mathcal{U}$ (0 , 1) $0.40^{+0.37}_{-0.28}$ Quadratic limb darkening
coefficient. $q_{\rm 2,TESS-S33}$ $\mathcal{U}$ (0 , 1) $0.31^{+0.27}_{-0.20}$
Stellar parameters ${\rho}_{\ast}$ ($\rm kg\ m^{-3}$) $\mathcal{J}$ ($10$ ,
$1000$) $269^{+115}_{-77}$ Stellar density. RV parameters $K_{b}$ ($\rm m\
s^{-1}$) $\mathcal{U}$ ($0$ , $20000$) $8725^{+73}_{-59}$ RV semi-amplitude of
TOI-2521b. CHIRONRV parameters $\rm\mu_{CHIRON}$ ($\rm m\ s^{-1}$)
$\mathcal{U}$ ($-40000$ , $-20000$) $-31334^{+49}_{-48}$ Systemic velocity for
CHIRON. $\rm\sigma_{CHIRON}$ ($\rm m\ s^{-1}$) $\mathcal{J}$ ($0.1$ , $1000$)
$113^{+96}_{-53}$ Extra jitter term for CHIRON.

Figure 6: The TESS light curves near the transit of TOI-1608, TOI-2336, and
TOI-2521 with arbitrary offsets. The red line is the best-fit transit model
obtained by juliet. The gray dots are the TESS data. Figure 7: The LCOGT light
curves of TOI-2336 in the Pan-STARRS $z$-short ban ($z_{S}$). The red line is
the best-fit transit model obtained by juliet. The gray dots are the LCOGT
data and the blue dots are binned data. Figure 8: The RVs of TOI-1608,
TOI-2336 and TOI-2521. The black line is the best-fit model obtained by
juliet. The error bars are the quadrature sum of the instrument jitter term
and the measurement uncertainties for all RVs. The left row is the RVs as a
function of time and the right row is the phase-folded RVs. Table 7: Physical
parameters for TOI-1608b, TOI-2336b, and TOI-2521b.

Parameter | TOI-1608 | TOI-2336 | TOI-2521 | Description
---|---|---|---|---
$P$ (days) | $2.47275^{+0.00004}_{-0.00004}$ | $7.711978^{+0.000013}_{-0.000013}$ | $5.563060^{+0.000007}_{-0.000007}$ | Orbital period.
$R_{b}/R_{\ast}$ | $0.0574_{-0.0016}^{+0.0020}$ | $0.0592_{-0.0007}^{+0.0008}$ | $0.0598_{-0.0013}^{+0.0011}$ | Planet radius in units of stellar radius.
$R_{b}$ ($R_{J}$) | $1.21_{-0.06}^{+0.06}$ | $1.05_{-0.04}^{+0.04}$ | $1.01_{-0.04}^{+0.04}$ | Planet radius.
$M_{b}$ ($M_{J}$) | $90.7_{-3.7}^{+3.7}$ | $69.9_{-2.3}^{+2.3}$ | $77.5_{-3.3}^{+3.3}$ | Planet mass.
$\rho_{b}$ ($\rm g~{}cm^{-3}$) | $63.7_{-9.1}^{+10.3}$ | $75.0_{-7.9}^{+8.9}$ | $92.8_{-10.7}^{+12.3}$ | Planet density.
$b$ | $0.929_{-0.006}^{+0.004}$ | $0.810_{-0.021}^{+0.019}$ | $0.56_{-0.20}^{+0.11}$ | Impact parameter.
$a/R_{\ast}$ | $4.17_{-0.13}^{+0.16}$ | $9.18_{-0.40}^{+0.44}$ | $7.60_{-0.81}^{+0.96}$ | Semi-major axis in units of stellar radius.
$a$ (au) | $0.0419_{-0.0021}^{+0.0022}$ | $0.0777_{-0.0042}^{+0.0046}$ | $0.0615_{-0.0067}^{+0.0079}$ | Semi-major axis.
$i$ (deg) | $77.1_{-0.5}^{+0.5}$ | $84.9_{-0.4}^{+0.4}$ | $85.8_{-1.4}^{+1.8}$ | Inclination angle.
$e$ | $0.041_{-0.019}^{+0.024}$ | $0.010_{-0.005}^{+0.006}$ | $<0.035^{[1]}$ | Eccentricity.
$F$ ($F_{\oplus}$) | $3155_{-278}^{+289}$ | $846_{-86}^{+92}$ | $721_{-153}^{+185}$ | Insolation flux relative to the Earth.

  * 1

[1] Here we report the upper limits at the 99% confidence level.

##  5 Discussion

###  5.1 Mass-radius relation and Inflation

We compare TOI-1608b, TOI-2336b and TOI-2521b with the models from Baraffe et
al. (2015) and Phillips et al. (2020b) in the mass-radius diagram, which is
shown in Figure 9. For the following analysis, we also build a sample of
transiting companions with a mass between $13\ M_{J}$ and $150\ M_{J}$. We use
the sample compiled by Grieves et al. (2021) and add NGTS-19b (Acton et al.,
2021), TOI-1278 B (Artigau et al., 2021), TOI-2119b (Cañas et al., 2022;
Carmichael et al., 2022), TIC-320687387 B (Gill et al., 2022), TOI-629b,
TOI-1982b, and TOI-2543b (Psaridi et al., 2022). We call them the ‘previous
transiting BDs and low mass stars sample’ in the following text, and we also
plot this sample in Figure 9. According to the plotted model isochrones, all
of our three targets seem to be younger than $0.5\ Gyr$. This is much younger
than the ages we obtained in Section 3 for the primary stars. For TOI-1608 and
TOI-2336, if we adopt a very low age limit of $1\ Gyr$ for these systems, the
predicted radii from the model isochrones would be $1.06\ R_{J}$ and $0.90\
R_{J}$, respectively, both of which are $3\sigma$ smaller than the measured
values from our joint transit and RV analyses. For TOI-2521, if we adopt a
very low age limit of $5\ Gyr$ for the system, the predicted radius would be
$0.84\ R_{J}$, which is $4\sigma$ smaller than our measured value. Therefore,
TOI-1608b, TOI-2336b, and TOI-2521b should be all inflated.

Figure 9: Radius-mass diagram for the transiting brown dwarfs and low-mass
stars. The black dots are the ‘previous transiting BDs and low mass stars
sample’, and the red dots are the three targets in this work. The isochrones
of 0.1 Gyr, 0.5 Gry, 1 Gyr, 5 Gyr and 10 Gyr at solar metallicity from Baraffe
et al. (2015) and Phillips et al. (2020b) models are shown as lines with
different colors.

The radius inflation of hot Jupiters has been observed and characterized in
multiple previous studies (e.g., Bouchy et al., 2011; Weiss et al., 2013), and
the stellar incident flux clearly plays an important role in shaping the sizes
of these giant planets. Weiss et al. (2013) obtained an empirical relation for
the inflation of planets heavier than $150\ M_{\oplus}$ (0.47 $M_{J}$)

$\frac{R_{p}}{R_{\oplus}}=2.5\left(\frac{M_{p}}{M_{\oplus}}\right)^{-0.039}\left({\rm\frac{F}{erg\
s^{-1}\ cm^{-2}}}\right)^{0.094}.$ (1)

However, for brown dwarf companions, previous research concluded that the
stellar irradiation would not affect their radii as much as observed for the
hot Jupiters (e.g., Bouchy et al., 2011). To compare the dependence on the
incident flux for hot Jupiters, brown dwarfs, and low mass stars, we used the
‘previous transiting BDs and low mass stars sample’ and fitted a similar
power-low relation to that of Weiss et al. (2013) as

$\frac{R_{p}}{R_{J}}=C\left(\frac{M_{p}}{M_{J}}\right)^{\alpha}\left(\frac{F}{F_{\oplus}}\right)^{\beta}.$
(2)

We fitted this relation for brown dwarfs and low mass stars separately with a
cutoff at $80\ M_{J}$ (Baraffe et al., 2002) to distinguish them. For both
sets of samples, we calculated their time-averaged incident flux using the
equation

$F=\sigma_{\rm sb}T_{\rm
eff}^{4}\left(\frac{R_{*}}{a}\right)^{2}\sqrt{\frac{1}{1-e^{2}}},$ (3)

where $\sigma_{\rm sb}$ is the Stefan–Boltzmann constant. In the following
analyses we excluded from the samples objects with precision on the radius or
mass that is poorer than 30%. We also excluded RIK 72b because it is a very
young brown dwarf (David et al., 2019) and probably still in the process of
initial contraction after its formation. Finally, we have 35 brown dwarfs and
25 low mass stars in our fitting. We used emcee (Foreman-Mackey et al., 2013)
to perform a Markov chain Monte Carlo (MCMC) fitting for each sample. We set
uniform priors between -5 and 5 for $\alpha$ and $\beta$ and a log-uniform
prior between $10^{-5}$ and $10^{5}$ for $C$. The MCMC posteriors are shown in
Figure 11 and the best-fit values are listed in Table 8. The best-fit relation
for the brown dwarfs ($13\ M_{J}\sim 80\ M_{J}$) is

$\frac{R_{p}}{R_{J}}=1.11\left(\frac{M_{p}}{M_{J}}\right)^{-0.052}\left(\frac{F}{F_{\oplus}}\right)^{0.009},$
(4)

and the relation for the low mass stars ($80\ M_{J}\sim 150\ M_{J}$) is

$\frac{R_{p}}{R_{J}}=0.021\left(\frac{M_{p}}{M_{J}}\right)^{0.86}\left(\frac{F}{F_{\oplus}}\right)^{0.019}.$
(5)

In Figure 10, we show the sample of transiting low-mass companions with mass
around $13\ M_{J}\sim 150\ M_{J}$ as well as the sample of known hot Jupiters
with the published values for the radius and the incident flux with a
precision better than 50%. The empirical relations for different types of
objects are shown as different lines. Comparing these results, we find that
the mass-radius relations for the three kinds of objects are different. The
mass dependence for brown dwarfs and low mass stars from our fits is
consistent with Chen & Kipping (2017). Besides, the power-law indices for
incident flux in the mass-radius relation for brown dwarfs and low mass stars
are significantly lower than that for hot Jupiters, which confirms that brown
dwarfs are less affected by stellar irradiation (e.g., Bouchy et al., 2011).
However, the best-fit indices for incident flux for brown dwarfs and low mass
stars are not consistent with zero, which suggests that the effect on the size
of these objects by the stellar irradiation is perhaps not totally negligible.
Naively, one would speculate that objects with higher densities or higher
surface gravity should be more difficult to inflate via external irradiation.
Since brown dwarfs have higher densities than low mass stars on average
(Baraffe et al., 2003; Baraffe et al., 2008) (also see Fig. 8 in Persson et
al., 2019), it is consistent with the expectation that the brown dwarfs’ radii
have a weaker dependence on the incident flux than the low mass stars’ (a
smaller power-law index). Future studies with a larger and better
characterized sample of low-mass objects may reveal more on this potential
difference between these two populations. Finally, we caution that the radii
of brown dwarfs and low mass stars can change considerably as they age (as can
be seen in the isochrones in Figure 9). For example, since the radii decrease
with age, and irradiation is higher closer to the star where tidal effects are
also stronger, if tides are strong enough to cause orbital decay and
destruction of close-in objects within the main sequence lifetime of the host
stars, then this may result in closer-in brown dwarfs being systematically
younger and with larger radii than farther out objects. This may mimic a trend
with irradiation. However, due to the lack of samples with accurate ages, we
were unable to consider the dependence on age in this discussion. More samples
with accurate age estimates and a more comprehensive analysis are needed in
future works.

Figure 10: The incident flux received by the companions vs. their radii. Dots with different colors are the ‘previous transiting BDs and low mass stars sample’, and the color scale represents the mass of these companions. The dots with red outlines are the three targets in this work and the gray dots in the background are hot Jupiters. Here we only show brown dwarfs and low mass stars with mass and radius measured with a precision better than 30% and hot Jupiters with radius and incident flux with a precision better than 50%. Equations 4 and 5 are plotted in solid lines for two different selected masses in each mass range, and the dashed line presents the Weiss et al. (2013) relation for hot Jupiters with $1\ M_{J}$. Table 8: MCMC fitting results of Equation 2 for brown dwarfs and low mass stars. Parameter | Brown dwarfs | Low mass stars
---|---|---
$\alpha$ | $-0.052_{-0.021}^{+0.021}$ | $0.860_{-0.026}^{+0.025}$
$\beta$ | $0.009_{-0.003}^{+0.003}$ | $0.019_{-0.004}^{+0.004}$
$C$ | $1.11_{-0.10}^{+0.11}$ | $0.021_{-0.002}^{+0.002}$

(a) Brown dwarfs

(b) Low mass stars

Figure 11: The posterior distribution of the MCMC fitting of Equation 2 for
brown dwarfs and low mass stars. $C$ is the normalization parameter, $\alpha$
is the power-law index for mass, and $\beta$ is the power-law index for
incident flux.

###  5.2 Tidal effects

Tidal effects are important in the evolution of close binary systems. There
are several tidal effects to consider for binary systems with a relatively
high mass ratio like the three cases here: tidal circularization, orbital
decay, and spin-orbit synchronization. We discuss each effect below.

The tidal circularization effect is where the tidal force damps the
eccentricity of the orbit. We examine the orbital circularization timescale

$\tau_{e}=\frac{e}{|de/dt|}$ (6)

of the three targets by using the equation from Jackson et al. (2008):

$\frac{1}{e}\frac{de}{dt}=-\left[\frac{63}{4}(GM_{\ast}^{3})^{1/2}\frac{R_{p}^{5}}{Q_{p}M_{p}}+\frac{171}{16}(G/M_{\ast})^{1/2}\frac{R_{\ast}^{5}M_{p}}{Q_{\ast}}\right]a^{-13/2},$
(7)

where $Q_{p}$ is the modified quality factor of the companion and $Q_{*}$ is
the modified quality factor of the host star. The quality factor is impossible
to measure precisely. For brown dwarf companions, this value is usually
adopted as $Q_{p}=Q_{\ast}=10^{6}$ at face value. Under this assumption, the
orbital circularization timescales are 2.13 Myr, 373 Myr, and 41.3 Myr for
TOI-1608, TOI-2336, and TOI-2521, respectively. Compared to the stellar ages
derived in Section 3, the orbital circularization time-scales are much
smaller, which means it is very likely that the orbits have been circularized.
This is consistent with the measured low eccentricities of these three
systems.

Dissipative tidal interactions can also change the orbital period and the
rotation period of the host star and may cause spin-orbit synchronization
(Goldreich & Soter, 1966). Comparing the stellar rotation periods and the
orbital periods of these three targets, we find that, for TOI-1608 and
TOI-2521, these two periods are very similar, which indicates that both
TOI-1608 and TOI-2521 may be spin-orbit synchronized. For TOI-2336, the
rotation period is $7\sigma$ smaller than the orbital period. It is not likely
to be spin-orbit synchronized.

Finally, when the orbital period is shorter than the stellar rotation period,
the semi-major axis of the companion’s orbit will shrink and the companion
will experience orbital tidal decay. According to the the equation in Patra et
al. (2017) derived from Goldreich & Soter (1966), the tidal decay timescale is

$\tau_{P}=\frac{P}{|dP/dt|}=\frac{2Q_{\ast}}{27\pi}\left(\frac{M_{\ast}}{M_{p}}\right)\left(\frac{a}{R_{\ast}}\right)^{5}P,$
(8)

where $P$ is the orbital period. For evolved host stars, the decay will become
faster as the radius of the host star increases. As the orbital periods of
TOI-1608 and TOI-2521 are shorter than their stellar rotation period, we
examine their orbital decay time scales. Using equation 8 and adopting
$Q_{\ast}=10^{6}$, we obtained 3.05 Myr and 73.2 Myr for TOI-1608 and TOI-2521
respectively. The orbital decay timescale of TOI-1608b is similar to that of
WASP-12b, whose orbital decay has been reported (Maciejewski et al., 2016;
Patra et al., 2017; Yee et al., 2020; Wong et al., 2022). Based on equation 8,
we calculated the decay rate of the orbital period to be $\sim 70\ {\rm ms\
yr^{-1}}$, which only induces a difference of $\sim 4.6$ ms in our TESS
observation. As the uncertainty on the orbital period in our analysis is 3.5
s, the orbital decay of TOI-1608b is not detectable in the TESS data gathered
so far. Nevertheless, the orbital decay of WASP-12b was measured after being
observed for a decade. We may have a chance to observe their orbital decay in
future observations (e.g., in TESS Extended Missions). However, as mentioned
above, whether TOI-1608 is spin-orbit synchronized is unclear, especially
given the uncertainty in the stellar rotation period. Given the relatively
large mass of TOI-1608b, the system might have reached spin-orbit
synchronization, and magnetic braking could then cause further orbital decay
(e.g., Barker & Ogilvie, 2009; Li & Winn, 2016; Benbakoura et al., 2019).

###  5.3 Eccentricity

In section 5.2, we showed the low eccentricities of TOI-1608b, TOI-2336b and
TOI-2521b are consistent with their relatively short tidal circularization
timescales. Here we put these three system in a broader context by comparing
their eccentricities with previous statistical results. In Ma & Ge (2014),
they analyzed brown dwarf companions and found different eccentricity
distributions for brown dwarfs below and above $42.5\ M_{J}$. The eccentricity
distribution of low mass brown dwarf companions is consistent with that of
massive planets, while the high mass brown dwarf companions show more
diversity in their eccentricity distribution, more consistent with binaries.
We show the eccentricities of the ‘previous transiting BDs and low mass stars
sample’ as well as our targets in Figure 12. More diverse eccentricities can
be seen in the region above $42.5\ M_{J}$. All our targets are in the high
mass region, but all have very low eccentricities in contrast to the majority
of the high-mass brown dwarfs. We can see that companions with smaller periods
tend to have lower eccentricities. This trend was also found by Ma & Ge
(2014), and they suggested that this is caused by tidal circularization. All
of our targets have relatively small orbital periods and thus are consistent
with their observed trend.

As all our targets are companions orbiting aged stars, we also compared the
eccentricities with the sample of hot Jupiters from Grunblatt et al. (2018).
They analyzed a sample of hot Jupiters around 419 dwarf stars and 136 giant
stars with planetary radii $R_{p}>0.4\ R_{J}$ and orbital periods between 4.5
and 30 days. They found that for the giant planets with short orbital periods,
the ones orbiting evolved stars tend to have higher eccentricities (with a
median eccentricity of $e\approx 0.152$) than the giant planets orbiting main-
sequence dwarfs (with a median eccentricity of $e\approx 0.056$). They suggest
that this may be caused by the increasing tidal effect of the evolved stars as
their radii increase post-main-sequence: the planets with shorter tidal
circularization timescales would also have relatively short tidal decay
timescales and thus be engulfed. As a result, we would observe more planets
with high eccentricities. All of our targets have eccentricities that are
significantly lower than the median eccentricity of $e\approx 0.152$ of the
evolved-star sample reported by Grunblatt et al. (2018). This is not
consistent with the trend in Grunblatt et al. (2018). As we discussed in
Section 5.2, orbital decay happens when the orbital period is shorter than the
stellar rotation period and thus may not happen in our targets. Perhaps this
mismatch shows the systematic difference between the post-main-sequence
evolution of systems with brown dwarf or low-mass star companions and hot
Jupiters. However, there is currently a small sample of brown dwarfs (or
planets) around evolved stars, and more detections are needed to reveal the
detailed statistical properties of this population in comparison with the
planetary population.

Figure 12: The eccentricities of transiting companions with a mass between
$13\ M_{J}$ and $150\ M_{J}$ as a function of mass. The dots with different
colors are the ‘previous transiting BDs and low mass stars sample’ and the
highlighted red dots are our targets. The dots are sized by the radii of the
companions and the color shows their orbital periods. The dashed line is the
mass threshold at $42.5\ M_{J}$ proposed by Ma & Ge (2014) separating two
different brown dwarf populations. The three companions confirmed in this work
are circled in red.

##  6 Summary

In this paper, we confirmed one brown dwarf, TOI-2336b, and two objects near
the hydrogen burning mass limit, TOI-1608b and TOI-2521b, transiting three
aged stars. We analyzed space and ground photometry as well as RVs from
ground-based high-resolution spectra. TOI-2336b has a radius of $1.05\pm 0.04\
R_{J}$, a mass of $69.9\pm 2.3\ M_{J}$ and an orbital period of 7.71 days.
TOI-1608b has a radius of $1.21\pm 0.06\ R_{J}$, a mass of $90.7\pm 3.7\
M_{J}$ and an orbital period of 2.47 days. TOI-2521b has a radius of $1.01\pm
0.04\ R_{J}$, a mass of $77.5\pm 3.3\ M_{J}$ and an orbital period of 5.56
days. Their host stars are F (TOI-2336, TOI-1608) and G (TOI-2521) type
subgiant stars.

We found that all three companions have inflated radii via comparison to the
evolution models for brown dwarfs and low mass stars. Combining data from the
literature, we fitted a relationship between radius, mass, and incident flux
for brown dwarfs and low mass stars, and we found that they are different
between these two populations and also differ from the relationship of hot
Jupiters. We found weak but statistically significant positive correlations
between radius and flux for both brown dwarfs and low mass stars, which
indicates that these companions may also be inflated by the irradiation from
host stars as hot Jupiters, though to a much lesser degree.

We analyzed the tidal interactions in these three systems. We found that their
relatively small eccentricities are consistent with their short orbital
circularization timescales. We also compared their stellar rotation periods
and orbital periods and found that TOI-1608 and TOI-2521 may be approaching
spin-orbit synchronization. Additionally, we found a short orbital decay
timescale of 3.05 Myr for TOI-1608, which may be detectable in future
observations if it has not reached spin-orbit synchronization yet.

We compared the eccentricities of these three systems with previous
statistical works. Compared with Ma & Ge (2014), they do not follow the trend
that high mass brown dwarf companions show more diversity in their
eccentricity distribution than low mass brown dwarf. Instead, they are
consistent with the trend that companions with smaller periods tend to have
lower eccentricities. This is in contrast with the hot Jupiter sample in
Grunblatt et al. (2018), where short-period giant planets orbiting evolved
stars tend to have larger eccentricities than the ones orbiting dwarfs. A
larger sample of transiting brown dwarfs and low mass stars will reveal more
on the similarities and differences between these populations and the gas
giant planets and provide more insights about their formation and evolution.

## Affiliations

1Department of Astronomy, Tsinghua University, Beijing 100084, China
2Department of Physics and Kavli Institute for Astrophysics and Space
Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
3Departamento de Matemática y Física Aplicadas, Facultad de Ingeniería,
Universidad Católica de la Santísima Concepción, Alonso de Rivera 2850,
Concepción, Chile
4Center for Astrophysics, Harvard & Smithsonian, 60 Garden Street,Cambridge,
MA 02138, USA
5Observatoire de Genève, Université de Genève, Chemin Pegasi, 51,1290 Versoix,
Switzerland
6Harvard University, Cambridge, MA 02138
7National Astronomical Observatories, Chinese Academy of Sciences, 20A Datun
Road, Chaoyang District, Beijing 100012, China
8Department of Physics and Astronomy, Vanderbilt University, Nashville, TN
37235, USA
9Astrophysics Group, Keele University, Staffordshire, ST5 5BG, UK
10NASA Ames Research Center, Moffett Field, CA 94035, USA
11Department of Physics, Engineering and Astronomy, Stephen F. Austin State
University, 1936 North St, Nacogdoches, TX 75962, USA
12Cerro Tololo Inter-American Observatory, Casilla 603, La Serena, Chile
13SETI Institute, Mountain View, CA 94043, USA
14Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA
15Department of Astronomy, Columbia University, 550 West 120th Street, New
York, NY, USA
16Department of Astrophysics, American Museum of Natural History, Central Park
West at 79th St., New York, NY 10024, USA
17Department of Physics and Astronomy, Johns Hopkins University, 3400 N
Charles St, Baltimore, MD 21218, USA
18Department of Astronomy, The Ohio State University, 4055 McPherson
Laboratory, 140 West 18th Avenue, Columbus, OH 43210 USA
19Department of Physics and Astronomy, The University of North Carolina at
Chapel Hill, Chapel Hill, NC 27599-3255, USA
20Bay Area Environmental Research Institute, Moffett Field, CA 94035, USA
21NASA Exoplanet Science Institute, Caltech/IPAC, Mail Code 100-22,1200 E.
California Blvd., Pasadena, CA 91125, USA
22Stellar Astrophysics Centre, Department of Physics and Astronomy, Aarhus
University, Ny Munkegade 120, DK-8000 Aarhus C, Denmark
23Proto-Logic LLC, 1718 Euclid Street NW, Washington, DC 20009, USA
24Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA
02138, USA
25NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771,
USA
26Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA
27Department of Aeronautics and Astronautics, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA
28Kotizarovci Observatory, Sarsoni 90, 51216 Viskovo, Croatia
29Planetary Discoveries in Fredericksburg, VA 22405, USA
30Department of Astrophysical Sciences, Princeton University, Princeton, NJ
08544, USA
31Max Planck Institute for Solar System Research, Justus-von-Liebig-Weg 3,
D-37077 Gottingen, Germany

## Acknowledgements

We thank Doug Lin for his suggestions on this work, especially on the tidal
effects. We also thank Theron W. Carmichael for useful suggestions. T.G. and
S.M. acknowledge the support from the National Science Foundation of China
(Grant No. 12133005). Funding for the TESS mission is provided by NASA’s
Science Mission Directorate. 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.
This paper includes data collected by the TESS mission that are publicly
available from the Mikulski Archive for Space Telescopes (MAST). 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 research has made use
of the NASA Exoplanet Archive, which is operated by the California Institute
of Technology, under contract with the National Aeronautics and Space
Administration under the Exoplanet Exploration Program. This work made use of
Astropy:444http://www.astropy.org a community-developed core Python package
and an ecosystem of tools and resources for astronomy (Astropy Collaboration
et al., 2013, 2018, 2022). We thank the Swiss National Science Foundation
(SNSF) and the Geneva University for their continuous support to our planet
search programs. This work has been carried out within the framework of the
NCCR PlanetS supported by the Swiss National Science Foundation under grants
51NF40_182901 and 51NF40_205606. ML acknowledges support of the Swiss National
Science Foundation under grant number PCEFP2_194576. This work makes use of
observations from the LCOGT network. Part of the LCOGT telescope time was
granted by NOIRLab through the Mid-Scale Innovations Program (MSIP). MSIP is
funded by NSF. 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 research made use of Lightkurve, a Python package for Kepler and TESS
data analysis (Lightkurve Collaboration et al., 2018).

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##  Appendix A The posterior distribution of isochrone fitting for TOI-1608

Figure 13: The posterior distribution of isochrone fitting for TOI-1608. The
bimodal solution can be seen in the distribution of $M_{\ast}$, $\rho_{\ast}$,
$\log g_{\ast}$ and age.

##  Appendix B RV measurements

Table 9: RV measurements of TOI-1608. BJDTDB | RV ($\rm m\ s^{-1}$) |  $\rm\sigma_{\rm RV}$ ($\rm m\ s^{-1}$) | Instrument
---|---|---|---
2459551.809 | 77505.8 | 111.0 | NRES
2459554.611 | 72125.0 | 115.4 | NRES
2459555.828 | 92426.6 | 113.9 | NRES
2459556.669 | 80122.9 | 117.2 | NRES
2459569.599 | 72143.9 | 117.0 | NRES
2459570.815 | 93377.1 | 136.1 | NRES
2459571.662 | 74946.0 | 112.5 | NRES
2459573.657 | 86689.7 | 317.6 | NRES
2459574.790 | 74478.1 | 111.6 | NRES
2459575.762 | 92945.6 | 119.0 | NRES
2459586.564 | 74329.9 | 128.5 | NRES
2459586.624 | 73127.7 | 135.1 | NRES
2459586.691 | 72150.8 | 99.8 | NRES
2459586.757 | 72454.5 | 140.4 | NRES
2459588.610 | 84335.5 | 147.5 | NRES
2459588.661 | 82298.2 | 101.6 | NRES
Table 10: RV measurements of TOI-2336. BJDTDB | RV ($\rm m\ s^{-1}$) |  $\rm\sigma_{\rm RV}$ ($\rm m\ s^{-1}$) | Instrument
---|---|---|---
2459327.48570 | 36031.7 | 146.3 | NRES
2459328.38208 | 33814.8 | 130.3 | NRES
2459329.55779 | 25601.2 | 205.3 | NRES
2459336.62242 | 34946.6 | 169.7 | NRES
2459341.32044 | 39963.4 | 127.5 | NRES
2459343.48120 | 35564.8 | 113.3 | NRES
2459344.56582 | 30027.7 | 129.6 | NRES
2459346.54750 | 30188.4 | 132.2 | NRES
2459347.49433 | 34602.6 | 102.0 | NRES
2459348.57917 | 37950.9 | 112.0 | NRES
2459371.34618 | 36932.0 | 97.0 | NRES
2459376.46575 | 29548.0 | 119.3 | NRES
2459377.29886 | 31467.1 | 142.8 | NRES
2459379.50367 | 38767.5 | 134.0 | NRES
2459406.40547 | 28286.2 | 134.2 | NRES
2459407.41335 | 26729.2 | 134.1 | NRES
2459410.41467 | 38226.1 | 132.9 | NRES
2459411.40852 | 37089.3 | 110.6 | NRES
2459412.37610 | 35113.3 | 128.6 | NRES
2459414.22659 | 28683.6 | 113.1 | NRES
2459415.38816 | 27329.2 | 113.1 | NRES
2459342.68168 | 34035 | 219 | CHIRON
2459345.72623 | 24826 | 164 | CHIRON
2459349.70953 | 35637 | 180 | CHIRON
2459357.62661 | 35468 | 206 | CHIRON
2459359.65638 | 27702 | 198 | CHIRON
2459371.64418 | 34388 | 128 | CHIRON
2459387.65515 | 35618 | 178 | CHIRON
2459399.56554 | 24837 | 215 | CHIRON
2459407.61081 | 24916 | 186 | CHIRON
2459417.57493 | 33182 | 191 | CHIRON
2459424.53740 | 30008 | 214 | CHIRON
2459432.50942 | 31233 | 205 | CHIRON
2459318.877767 | 37598.76 | 121.12 | CORALIE
2459321.804744 | 27564.77 | 111.03 | CORALIE
2459328.784617 | 29880.55 | 119.24 | CORALIE
2459331.872147 | 31218.65 | 99.95 | CORALIE
2459335.705030 | 33463.34 | 109.05 | CORALIE
2459343.687687 | 32204.50 | 125.93 | CORALIE
2459346.774496 | 29472.14 | 99.30 | CORALIE
2459351.752823 | 30681.14 | 120.70 | CORALIE
2459356.753344 | 37656.42 | 123.56 | CORALIE
2459357.723789 | 37554.35 | 77.79 | CORALIE
2459358.643619 | 34352.13 | 99.49 | CORALIE
2459361.597369 | 27438.35 | 65.13 | CORALIE
2459363.736155 | 36053.07 | 107.73 | CORALIE
2459365.700203 | 36685.84 | 69.86 | CORALIE
2459368.608494 | 27031.54 | 78.39 | CORALIE
2459373.599021 | 35976.49 | 64.58 | CORALIE
Table 11: RV measurements of TOI-2521. BJDTDB | RV ($\rm m\ s^{-1}$) |  $\rm\sigma_{\rm RV}$ ($\rm m\ s^{-1}$) | Instrument
---|---|---|---
2459294.52983 | -36161 | 53 | CHIRON
2459295.53896 | -40116 | 87 | CHIRON
2459296.53688 | -33715 | 53 | CHIRON
2459297.55627 | -24716 | 49 | CHIRON
2459298.51469 | -23135 | 112 | CHIRON
2459301.52687 | -38223 | 97 | CHIRON
2459303.52923 | -22904 | 66 | CHIRON
2459304.50997 | -25694 | 86 | CHIRON