# TOI 762 A b and TIC 46432937 b: Two Giant Planets Transiting M Dwarf Stars

Joel D. Hartman Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA  Daniel Bayliss Dept. of Physics, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK  Centre for Exoplanets and Habitability, University of Warwick, Gibbet Hill Road, Coventry CV4 7AL, UK  Rafael Brahm Millennium Institute of Astrophysics (MAS), Nuncio Monseñor Sótero Sanz 100, Providencia, Santiago, Chile  Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av. Diagonal las Torres 2640, Peñalolén, Santiago, Chile  Data Observatory Foundation, Santiago, Chile  Edward M. Bryant Mullard Space Science Laboratory, University College London, Holmbury St Mary, Dorking, RH5 6NT, UK  Andrés Jordán Millennium Institute of Astrophysics (MAS), Nuncio Monseñor Sótero Sanz 100, Providencia, Santiago, Chile  Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Av. Diagonal las Torres 2640, Peñalolén, Santiago, Chile  Data Observatory Foundation, Santiago, Chile  Gáspár Á. Bakos Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA  Melissa J. Hobson Observatoire de Genève, Département d’Astronomie, Université de Genève, Chemin Pegasi 51b, 1290 Versoix, Switzerland  Elyar Sedaghati European Southern Observatory (ESO), Av. Alonso de Córdova 3107, 763 0355 Vitacura, Santiago, Chile  Xavier Bonfils Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France  Marion Cointepas  Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France  Observatoire de Genève, Département d’Astronomie, Université de Genève, Chemin Pegasi 51b, 1290 Versoix, Switzerland  Jose Manuel Almenara Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France  Observatoire de Genève, Département d’Astronomie, Université de Genève, Chemin Pegasi 51b, 1290 Versoix, Switzerland  Khalid Barkaoui Astrobiology Research Unit, Université de Liège, Allée du 6 Août 19C, B-4000 Liège, Belgium  Department of Earth, Atmospheric and Planetary Science, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA  Instituto de Astrofísica de Canarias (IAC), Calle Vía Láctea s/n, 38200, La Laguna, Tenerife, Spain  Mathilde Timmermans  Astrobiology Research Unit, Université de Liège, Allée du 6 Août 19C, B-4000 Liège, Belgium  George Dransfield  School of Physics & Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK  Elsa Ducrot Paris Region Fellow, Marie Sklodowska-Curie Action  AIM, CEA, CNRS, Université Paris-Saclay, Université de Paris, F-91191 Gif-sur-Yvette, France  Sebastián Zúñiga-Fernández Astrobiology Research Unit, Université de Liège, Allée du 6 Août 19C, B-4000 Liège, Belgium  Matthew J. Hooton Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK  Peter Pihlmann Pedersen  Cavendish Laboratory, JJ Thomson Avenue, Cambridge CB3 0HE, UK  Francisco J. Pozuelos Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta de la Astronomía s/n, 18008 Granada, Spain  Amaury H. M. J. Triaud School of Physics & Astronomy, University of Birmingham, Edgbaston, Birmingham B15 2TT, UK  Michaël Gillon Astrobiology Research Unit, Université de Liège, Allée du 6 Août 19C, B-4000 Liège, Belgium  Emmanuel Jehin Space Sciences, Technologies and Astrophysics Research (STAR) Institute, Université de Liège, Allée du 6 Août 19C, B-4000 Liège, Belgium  William C. Waalkes Department of Physics and Astronomy, Dartmouth College, Hanover NH 03755, USA  Zachory K. Berta-Thompson Department of Astrophysical & Planetary Sciences, University of Colorado Boulder, Boulder CO 80309, USA  Steve B. Howell NASA Ames Research Center, Moffett Field, CA 94035, USA  Elise Furlan NASA Exoplanet Science Institute, Caltech/IPAC, Mail Code 100-22, 1200 E. California Blvd., Pasadena, CA 91125, USA  George R. Ricker Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Roland Vanderspek Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  Sara 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  Joshua N. Winn Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08544, USA  Jon M. Jenkins NASA Ames Research Center, Moffett Field, CA 94035, USA  David Rapetti NASA Ames Research Center, Moffett Field, CA 94035, USA  Research Institute for Advanced Computer Science, Universities Space Research Association, Washington, DC 20024, USA  Karen A. Collins Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  David Charbonneau Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA  Christopher J. Burke Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA  David R. Rodriguez  Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD, 21218, USA

###### Abstract

We present the discovery of TOI 762 A b and TIC 46432937 b, two giant planets
transiting M dwarf stars. Transits of both systems were first detected from
observations by the NASA TESS mission, and the transiting objects are
confirmed as planets through high-precision radial velocity (RV) observations
carried out with VLT/ESPRESSO. TOI 762 A b is a warm sub-Saturn with a mass of
$0.251\pm 0.042$ $M_{\rm J}$, a radius of $0.744\pm 0.017$ $R_{\rm J}$, and an
orbital period of $3.4717$ d. It transits a mid-M dwarf star with a mass of
$0.442\pm 0.025$ $M_{\sun}$ and a radius of $0.4250\pm 0.0091$ $R_{\sun}$. The
star TOI 762 A has a resolved binary star companion TOI 762 B that is
separated from TOI 762 A by 3$\farcs$2 ($\sim 319$ AU) and has an estimated
mass of $0.227\pm 0.010$ $M_{\sun}$. The planet TIC 46432937 b is a warm
Super-Jupiter with a mass of $3.20\pm 0.11$ $M_{\rm J}$ and radius of
$1.188\pm 0.030$ $R_{\rm J}$. The planet’s orbital period is $P=1.4404$ d, and
it undergoes grazing transits of its early M dwarf host star, which has a mass
of $0.563\pm 0.029$ $M_{\sun}$ and a radius of $0.5299\pm 0.0091$ $R_{\sun}$.
TIC 46432937 b is one of the highest mass planets found to date transiting an
M dwarf star. TIC 46432937 b is also a promising target for atmospheric
observations, having the highest Transmission Spectroscopy Metric or Emission
Spectroscopy Metric value of any known warm Super-Jupiter (mass greater than
3.0 $M_{\rm J}$, equilibrium temperature below 1000 K).

planetary systems — stars: individual ( TOI 762 A, TIC 46432937, ) techniques:
spectroscopic, photometric

††facilities: TESS, LCOGT, TRAPPIST, Gaia, ExTrA, SPECULOOS, VLT:ESPRESSO,
Exoplanet Archive, IRSA, ZTF, ASAS-SN††software: VARTOOLS (Hartman & Bakos,
2016), MWDUST (Bovy et al., 2016), Astropy (Astropy Collaboration et al.,
2013, 2018), ESPRESSO DRS pipeline v2.3.5 (Sosnowska et al., 2015; Modigliani
et al., 2020), EsoReflex (Freudling et al., 2013), AstroImageJ (Collins et
al., 2017)

##  1 Introduction

How do the properties of giant planets depend on the masses of their host
stars? This is a key open question and topic of current research in the field
of exoplanets. Efforts to address this question have been limited by the
relatively small number of low-mass and high-mass stars that have been
searched for planets compared to the much larger sample of Solar-mass stars
that have been searched to date by surveys like Kepler (Borucki et al., 2010)
and TESS (Ricker et al., 2015).

In order to expand the sample of giant planets known around low-mass stars, we
have been carrying out a program to follow-up candidate transiting giant
planets around M dwarf stars and late K dwarf stars using the ESPRESSO
instrument on the 8 m Very Large Telescope (VLT) at Paranal Observatory, in
Chile. This facility has proven to be very efficient at gathering high-
precision RV observations for faint stars, even down to $V\sim 16.5$. We have
so far published the confirmation of seven systems through this effort
(Hartman et al., 2020; Jordán et al., 2022; Hobson et al., 2023; Almenara et
al., 2023), and in this paper we publish the confirmation of two more such
objects.

The planets that we confirm in this paper were identified by the NASA TESS
mission, which has been carrying out a wide-field search for transiting
planets around bright stars since its launch in 2018\. Thanks to its all-sky
observing strategy, the mission has also been successful at discovering the
rare instances of transiting giant planets ($M_{p}>0.1$ $M_{\rm J}$, for
discussion purposes) around M dwarf stars. A total of 17 of the 20 transiting
giant planets that have been confirmed around M dwarf stars so far were either
first identified by TESS, or included TESS follow-up observations as part of
the discovery (Bakos et al., 2020; Cañas et al., 2020, 2022, 2023; Gan et al.,
2022, 2023; Hartman et al., 2023; Hobson et al., 2023; Jordán et al., 2022;
Kagetani et al., 2023; Kanodia et al., 2021, 2022, 2023; Triaud et al., 2023).
The three other known systems were discovered by Kepler (Johnson et al.,
2012), HATSouth (Hartman et al., 2015) and NGTS (Bayliss et al., 2018).

While TESS has been very successful at discovering giant planets around M
dwarfs, many of these objects were not identified through the mission’s
primary transit search effort operated by the Science Processing Operations
Center (SPOC; Jenkins et al., 2016) at NASA Ames Research Center (ARC), which
focuses on the 2 min cadence observations, but were instead identified through
special efforts to produce light curves for faint M dwarf stars from the Full
Frame Images (FFIs) and to search them for transit signals. One such effort is
the TESS Faint-star search (Kunimoto et al., 2022), which has identified some
3200 faint TESS Objects of Interest (TOIs) to date, including 128 candidate
giant planets transiting M dwarfs. Another example is Bryant et al. (2023) who
conducted a search for transiting giant planets around M dwarfs in 30 min
cadence TESS observations and identified 15 candidates, including 7 that were
not previously identified by other projects. Both of the planets that we
confirm in this paper were included in the Bryant et al. (2023) sample, and
one of these systems (TIC 46432937 b) was first identified by them.

In the following section we discuss the observations that are used to confirm
and characterize each planetary system. In Section 3 we describe the analysis
methods. In Section 4 we discuss the results.

##  2 Observations

###  2.1 Initial Photometric Detection

Both TOI 762 A b and TIC 46432937 b were first identified as transiting planet
candidates from observations gathered by the NASA TESS mission. Table 2
summarizes the TESS observations that are available for each system. Both
targets were identified as candidates by an independent transit search
performed by Bryant et al. (2023) with the aim of measuring the occurrence
rates of giant planets with low-mass host stars. For this search the Box-
fitting Least Squares (BLS; Kovács et al., 2002) algorithm was utilised to
search for giant planets transiting low-mass stars in light curves generated
from the 30 minute cadence Full-Frame-Images (FFIs) from Cycle 1 of the TESS
mission by the TESS-SPOC team (Caldwell et al., 2020). During Cycle 1 both TOI
762 A b and TIC 46432937 b were observed in a single sector: Sector 10 for TOI
762 A b (2019 Mar 29–2019 Apr 22) and Sector 6 for TIC 46432937 b (2018 Dec
15–2019 Jan 6). Following the BLS detection of these two candidates a number
of automated checks were performed to investigate whether the transit-like
signals could be the result of a number of different false positive scenarios,
such as eclipsing binaries or variable stars. See Bryant et al. (2023) for
more details on the analyses performed. For both TOI 762 A b and TIC 46432937
b we found no evidence that the transit-like signals were a result of a non-
planetary scenario. An initial transit fitting analysis was performed on both
candidates using the batman package (Kreidberg, 2015) to generate the transit
models and using emcee (Foreman-Mackey et al., 2013) to perform the MCMC
sampling. Both objects were identified as high likelihood giant planet
candidates by this analysis and included in the sample reported in Bryant et
al. (2023).

TOI 762 A b was independently identified as a candidate by the TESS Science
Processing Operations Center (SPOC; Jenkins et al., 2016) at NASA ARC. The
SPOC conducted a transit search of Sector 10 on 23 May 2019 with an adaptive,
noise-compensating matched filter (Jenkins, 2002; Jenkins et al., 2010, 2020),
producing a threshold crossing event (TCE) for which an initial limb-darkened
transit model was fitted (Li et al., 2019) and a suite of diagnostic tests
were conducted to help make or break the planetary nature of the signal
(Twicken et al., 2018). The transit signature was also detected in a search of
FFI data by the Quick Look Pipeline (QLP) at MIT (Huang et al., 2020a, b). The
TESS Science Office (TSO) reviewed the vetting information and issued an alert
on 27 February 2020 (Guerrero et al., 2021). The signal was repeatedly
recovered as additional observations were made in Sectors 36, 37, and 63, and
the transit signature passed all the diagnostic tests presented in the Data
Validation reports. The difference image centroiding test located the host
star within $3\farcs 18\pm 2\farcs 58$ of the source of the transit signal.

The light curves used in the analysis of each system are summarized in Table
2. For TOI 762 A we make use of the 2 min cadence TESS light curves corrected
for systematics by SPOC using the Presearch Data Conditioning Simple Aperture
Photometry (PDCSAP) method of Stumpe et al. (2012, 2014) and Smith et al.
(2012). For TIC 46432937 we use the TESS-SPOC 30 min and 10 min cadence light
curves, which were also corrected using the PDCSAP method. All of the TESS
light curves were obtained from the Mikulski Archive for Space Telescopes
(MAST) at the Space Telescope Science Institute (STScI).

###  2.2 High Contrast Imaging

TOI 762 A was observed with the Zorro speckle imager on the Gemini-South 8 m
telescope (Scott et al., 2021). Observations were obtained in the $832\pm 40$
nm and $562\pm 54$ nm band-passes on 2020 Jan 11, and these were processed to
generate reconstructed images following Howell et al. (2011). No companions to
TOI 762 A were resolved through these observations within the field of view of
Zorro, with contrast limits of $\Delta m_{832}>5.42$ and $\Delta m_{562}>4.9$
beyond 0$\farcs$5\. Figure 1 shows the 832 nm reconstructed image, and the
resulting $5\sigma$ magnitude contrast limits that we place on any companions
to TOI 762 A.

Table 1:  Sources within 10$\arcsec$ of TOI 762 A  Parameter | TOI 762 A | TOI 762 B |
---|---|---|---
GAIA DR3 ID | 5362352744504000256 | 5362352744496315264 | 5362352744496318336
$\Delta$R.A. (arcsec) | $\cdots$ | $-2.754\pm 0.098$ | $5.501\pm 0.092$
$\Delta$Dec. (arcsec) | $\cdots$ | $-1.62\pm 0.10$ | $4.003\pm 0.092$
$\mu_{\rm R.A.}$ ($\rm mas\,yr^{-1}$) | $-159.174\pm 0.020$ | $-157.37\pm 0.13$ | $-5.15\pm 0.12$
$\mu_{\rm Dec.}$ ($\rm mas\,yr^{-1}$) | $-24.780\pm 0.020$ | $-24.38\pm 0.12$ | $1.02\pm 0.11$
$\pi$ (mas) | $10.118\pm 0.023$ | $9.79\pm 0.14$ | $0.40\pm 0.13$
$G$ (mag) | $14.9297\pm 0.0028$ | $18.1928\pm 0.0031$ | $18.3444\pm 0.0030$
$BP-RP$ (mag)aaWe caution that the $BP$ and $RP$ photometry for each source appears to be contaminated by the other neighboring sources as indicated by the high values of phot_bp_rp_excess_factor. | $2.7548\pm 0.0059$ | $3.554\pm 0.093$ | $0.994\pm 0.020$
$M_{\star}$ ($M_{\sun}$) | $0.442\pm 0.025$ | $0.227\pm 0.010$ | $\cdots$

There are two resolved sources within 10″ of TOI 762 A that are listed in the
Gaia DR3 catalog (see also Table 1). Gaia DR3 5362352744496318336 is located
4$\farcs$7 to the northeast from TOI 762 A with $\Delta G=3.42$ mag, $\Delta
BP=2.29$ mag, and $\Delta RP=4.05$ mag relative to TOI 762 A. The parallax and
proper motion of this object differ significantly from the values for TOI 762
A indicating that the two sources are not physically associated. Gaia DR3
5362352744496315264 is located 3$\farcs$2 to the southwest of TOI 762 A and
does appear to be physically bound to TOI 762 A with $\pi=9.79\pm 0.14$ mas,
${\rm pmRA}=-157.37\pm 0.13$ mas yr-1 and ${\rm pmDE}=-24.38\pm 0.12$ mas yr-1
compared to $\pi=10.118\pm 0.023$ mas, ${\rm pmRA}=-159.174\pm 0.020$ mas yr-1
and ${\rm pmDE}=-24.780\pm 0.020$ mas yr-1 for TOI 762 A.

Gaia DR3 5362352744496315264 has $\Delta G=3.26$ mag, $\Delta BP=3.73$ mag and
$\Delta RP=2.93$ mag relative to TOI 762 A. This object has been identified as
a wide binary companion to TOI 762 A in both the SUPERWIDE catalog (Hartman &
Lépine, 2020) and the catalog of El-Badry et al. (2021). We refer to this star
as TOI 762 B. Assuming this is a bound companion, we estimate that it is a
late-M dwarf star with a mass of $0.227\pm 0.010$ $M_{\sun}$, at a projected
physical separation of $\sim 319$ AU from TOI 762 A. The mass estimate is
based on comparing the absolute $G$ magnitude of the source to version 1.2 of
the MIST theoretical stellar evolution models (Paxton et al., 2011, 2013,
2015; Choi et al., 2016; Dotter, 2016), assuming the age, distance,
metallicity and redenning to the source are the same as for TOI 762 A. The
inferred 2MASS and WISE magnitudes for the star are $J=14.057\pm 0.086$ mag,
$H=13.475\pm 0.076$ mag, $K_{S}=13.177\pm 0.085$ mag, $W_{1}=12.954\pm 0.095$
mag, $W_{2}=12.71\pm 0.10$ mag, $W_{3}=12.45\pm 0.11$ mag, and $W_{4}=12.33\pm
0.11$ mag. The measured relative proper motion difference between TOI 762 A
and TOI 762 B of $\Delta{\rm pm}=1.85\pm 0.13$ mas yr-1 is less than the value
of $2.9$ mas yr-1 that would be expected if the binary star system has a
circular, face-on orbit, and suggests that the orbit is inclined and/or
eccentric. We note that TOI 762 B is resolved from TOI 762 A in the
spectroscopic observations (Section 2.4) which reveal a RV variation for TOI
762 A that is in phase with the transit ephemeris. Based on this we conclude
that TOI 762 A is the source of the transit signal.

The closer of the two neighbors is within the photometric aperture of the
follow-up light curves for TOI 762 A, and would also have been unresolved in
the 2MASS and WISE photometry of this source. The source is also close enough
to be of concern for the Gaia $BP$ and $RP$ measurements. Using the relations
in Riello et al. (2021) for the expected value and scatter of
phot_bp_rp_excess_factor as functions of the $BP-RP$ color and $G$ magnitude
for an isolated source, we find that TOI 762 A has a value that is 3.3$\sigma$
greater than expected. This indicates that these measurements may be
contaminated, however correcting for this contamination may be difficult to do
accurately. For this reason we exclude the $BP$ and $RP$ photometry from the
analysis of this system. We account for the contamination in the follow-up
light curves as described in Section 3.3. To account for the contamination in
the 2MASS and WISE photometry we use MIST v 1.2 to estimate the apparent
magnitude of the neighbor in these band-passes given the absolute $G$
magnitude, and assuming the age, metallicity, distance, and redenning to the
source have the values that we determine for TOI 762 A. We then subtract the
flux contribution of the neighbor from the observed 2MASS and WISE magnitudes,
accounting for the uncertainty on the flux of the neighbor. These corrected
magnitudes are listed in Table 4, and are also the values that we include in
our analysis of the system (Section 3.3).

There are no nearby stars within 10″ of TIC 46432937 listed in the Gaia DR3
catalog. Ground-based high-spatial-resolution imaging is not available for
this target, so stellar companions within 1″ cannot be ruled out.

Figure 1:  $5\sigma$ magnitude contrast limits for any companions to TOI 762 A
based on Gemini/Zorro observations obtained in the 562 nm and 832 nm band-
passes. The inset shows the reconstructed Gemini/Zorro image of TOI 762 A in
the 832 nm band-pass.

###  2.3 Ground-based Photometric Follow-up

Ground-based photometric follow-up observations of TOI 762 A were obtained
through the TESS Follow-up Program (TFOP; Collins, 2019). Photometric follow-
up observations of TIC 46432937 were also obtained. The observations that have
been carried out for these systems are listed in Table 2. The data are shown
in Figures 2 and 5, and are made available in Table 3. Here we briefly discuss
each facility that was used.

####  2.3.1 LCOGT 1 m

A transit of TOI 762 A b was observed using a 1 m telescope at the Siding
Spring Observatory, Australia station of the Las Cumbres Observatory Global
Telescope (LCOGT) network (Brown et al., 2013). The observations were gathered
through an $I_{C}$ band-pass on 2019 June 28 using a SINISTRO imager. The
images have a pixel scale of 0$\farcs$389 px-1, with an estimated average
point-spread function (PSF) full-width at half-maximum (FWHM) of $\sim
2\arcsec$. The data were reduced to ensemble-corrected light curves using the
AstroImageJ package (Collins et al., 2017) with a photomeric aperture radius
of 14 pixels.

A separate photometric measurement was performed on the LCOGT images using a 2
pixel (0$\farcs$8) radius aperture to confirm that the transits are due to TOI
762 A rather than the neighbor TOI 762 B. This analysis revealed that TOI 762
A is indeed the source of the $\sim 30$ ppt transit events.

####  2.3.2 ExTrA 0.6 m

A total of six transits of TOI 762 A b and six transits of TIC 46432937 b were
monitored using the Exoplanets in Transits and their Atmospheres (ExTrA)
facility (Bonfils et al., 2015) at La Silla Observatory in Chile. Several of
the transits were simultaneously observed by more than one of the three 0.6 m
telescopes in the facility, and for TIC 46432937 b we made multi-band light
curves from the observations, leading to a total of 14 separate transit light
curves of TOI 762 A b and 64 separate light curves of TIC 46432937 b from this
facility. The facility performs spectro-photometric observations over a
wavelength range of 0.85 $\mu$m to 1.55 $\mu$m using a NIRvana 640 LN camera
which is fed with optical fibers from the three telescopes. Observations of
TOI 762 A were gathered with the 4″ diameter aperture fibers, except for on
the night of 2021 Mar 13 when the 8″ fibers were used. All transits of TIC
46432937 were observed with 8″ diameter aperture fibers. Both targets were
observed with the low resolution mode of the spectrograph ($R\sim 20$). Band-
pass-integrated, comparison-star-corrected light curves were produced
following the method of Cointepas et al. (2021). For TOI 762 A we made use of
light curves integrated over the full band-pass of the instrument, while for
TIC 46432937, where the grazing transits make the solution more sensitive to
limb darkening, we produced and analyzed separate $Z$, $Y$, $J$ and $H$-band
light curves from each transit observation.

####  2.3.3 TRAPPIST-South 0.6 m

Two transits of TOI 762 A b were observed using the southern 0.6 m TRAnsiting
Planets and PlanetesImals Small Telescope (TRAPPIST-South; Jehin et al., 2011;
Gillon et al., 2011; Barkaoui et al., 2019) at La Silla Observatory. The first
transit on 2023 February 2 was observed through an $I+z$ filter, while the
second transit on 2023 April 16 was observed through a $z^{\prime}$ filter.
Observations were obtained at a pixel scale of 0$\farcs$64\. For the first
transit the estimated PSF FWHM was 2$\farcs$6 and a photometric aperture of
3$\farcs$84 was used, while for the second transit the estimated PSF FWHM was
1$\farcs$4 and a photometric aperture of 4$\farcs$48 was used. Scheduling of
the observations was performed using the tools of Jensen (2013), while
ensemble-corrected light curves were derived from the observations following
the methods of Garcia et al. (2022).

####  2.3.4 SPECULOOS-South 1.0 m

A transit of TOI 762 A b was observed on 2023 April 16 using three of the 1.0
m telescopes in the Search for habitable Planets EClipsing ULtra-cOOl Stars
Southern observatory (SPECULOOS-South Delrez et al., 2018; Sebastian et al.,
2021; Burdanov et al., 2022) at Paranal Observatory in Chile. Observations
were gathered through $g^{\prime}$, $r^{\prime}$ and $z^{\prime}$ filters. The
observations had a pixel scale of 0$\farcs$35 and an estimated PSF FWHM of
1$\farcs$6\. An aperture of radius 2$\farcs$45 was used to extract the
photometry. Scheduling of the observations was performed following Sebastian
et al. (2021), while the data were reduced to ensemble-corrected light curves
following Murray et al. (2020) and Garcia et al. (2021, 2022).

###  2.4 Spectroscopic Observations

Time series spectroscopy was obtained for both TOI 762 A and TIC 46432937
using the ESPRESSO instrument (Pepe et al., 2021) on the 8 m Very Large
Telescope (VLT) at Paranal Observatory in Chile. We obtained five observations
of TOI 762 A in Period 104 between 2019 Dec 1 and 2019 Dec 24, and three
observations in Period 110 between 2022 Dec 25 and 2022 Dec 30. We allow for
an instrumental offset between the two periods in fitting the observations of
TOI 762 A. For TIC 46432937, we obtained eight observations all in Period 110,
between 2022 Nov 24 and 2023 Feb 22. Observations were gathered in HR mode
(using a single Unit Telescope, and a spectroscopic resolution of
$R\equiv\Delta\lambda/\lambda\sim 140,000$; the fiber aperture is 1″ in this
mode, and the wavelength coverage is from $3770$ Å to 7910 Å). A sky fiber was
placed 7″ from the target fiber, however there is no difference in the RVs
derived from the sky-subtracted spectra compared to the non-sky-subtracted
spectra due to the fact that moon contamination was minimal during the
observations. An exposure time of 1800 s was used for the observations. For
TOI 762 A the peak S/N varied from 23 to 30, while for TIC 46432937 the peak
S/N was between 41 and 64.

The data were reduced to Radial Velocity (RV) measurements in the Solar System
Barycentric frame using the ESPRESSO DRS pipeline (v2.3.5 Sosnowska et al.,
2015; Modigliani et al., 2020) in the EsoReflex environment (Freudling et al.,
2013). The ESPRESSO DRS calculates the RVs from individual spectra by
measuring the cross correlation function (CCF) for each slice separately,
using a template (stellar model) matching closest to the spectral type of the
star. It then adds all these CCFs for the different orders, ignoring those
orders that are severely affected by telluric contamination. It then fits a
Gaussian function to the final CCF, where the center of the Gaussian is the RV
and the width of the Gaussian represents the precision of the RV measurements.

Both systems show significant RV variations that are in phase with the transit
ephemerides, and of amplitudes that indicate the transiting components are of
planetary mass in both cases. The RV measurements for each system are shown in
Figures 2 and 4 and are listed in Table 5.

Table 2:  Summary of photometric observations  Instrument/Fieldaa For TESS data we list the Sector from which the observations are taken.  | Date(s) | # Imagesbb Excluding any outliers or other data not included in the modelling.  | Cadencecc The median time between consecutive images rounded to the nearest second. Due to factors such as weather, the day–night cycle, guiding and focus corrections the cadence is only approximately uniform over short timescales.  | Filter | Precisiondd The RMS of the residuals from the best-fit model. Note that in the case of TESS observations the transit may appear artificially shallower due to over-filtering and/or blending from unresolved neighbors. As a result the S/N of the transit may be less than what would be calculated from $R_{p}/R_{\star}$ and the RMS estimates given here.
---|---|---|---|---|---
|  |  | (sec) |  | (mmag)
TOI 762 A
TESS/Sector 10 | 2019 Mar–2019 Apr | 13,754 | 120 | $T$ | 11.7
TESS/Sector 36 | 2021 Mar–2021 Apr | 15,491 | 120 | $T$ | 13.3
TESS/Sector 37 | 2021 Apr | 15,061 | 120 | $T$ | 13.9
TESS/Sector 63 | 2023 Mar–2023 Apr | 17,456 | 120 | $T$ | 12.5
LCOGT 1.0 m | 2019 Jun 28 | 109 | 97 | $I_{C}$ | 3.1
ExTrA - tel2 | 2021 Mar 13 | 214 | 62 | 0.85–1.55 $\mu$m | 5.9
ExTrA - tel2 | 2021 Mar 27 | 168 | 62 | 0.85–1.55 $\mu$m | 4.7
ExTrA - tel3 | 2021 Mar 27 | 164 | 62 | 0.85–1.55 $\mu$m | 5.5
ExTrA - tel2 | 2021 Apr 24 | 159 | 62 | 0.85–1.55 $\mu$m | 5.7
ExTrA - tel3 | 2021 Apr 24 | 160 | 62 | 0.85–1.55 $\mu$m | 7.5
TRAPPIST-South | 2023 Feb 2 | 214 | 83 | I$+z$ | 3.5
ExTrA - tel1 | 2023 Apr 9 | 129 | 62 | 0.85–1.55 $\mu$m | 7.3
ExTrA - tel2 | 2023 Apr 9 | 131 | 62 | 0.85–1.55 $\mu$m | 4.8
ExTrA - tel3 | 2023 Apr 9 | 131 | 62 | 0.85–1.55 $\mu$m | 5.0
SPECULOOS-South | 2023 Apr 16 | 97 | 210 | $g^{\prime}$ | 3.2
SPECULOOS-South | 2023 Apr 16 | 169 | 85 | $r^{\prime}$ | 3.2
SPECULOOS-South | 2023 Apr 16 | 565 | 36 | $z^{\prime}$ | 2.9
ExTrA - tel1 | 2023 Apr 16 | 162 | 62 | 0.85–1.55 $\mu$m | 7.4
ExTrA - tel2 | 2023 Apr 16 | 167 | 62 | 0.85–1.55 $\mu$m | 4.4
ExTrA - tel3 | 2023 Apr 16 | 169 | 62 | 0.85–1.55 $\mu$m | 5.4
TRAPPIST-South | 2023 Apr 16 | 141 | 111 | $z^{\prime}$ | 3.0
ExTrA - tel1 | 2023 Apr 23 | 173 | 62 | 0.85–1.55 $\mu$m | 11.2
ExTrA - tel2 | 2023 Apr 23 | 175 | 62 | 0.85–1.55 $\mu$m | 6.1
ExTrA - tel3 | 2023 Apr 23 | 174 | 62 | 0.85–1.55 $\mu$m | 8.7
TIC 46432937
TESS/Sector 6 | 2018 Dec – 2019 Jan | 959 | 1800 | $T$ | 1.2
TESS/Sector 32 | 2020 Nov – 2020 Dec | 3467 | 600 | $T$ | 2.4
ExTrA - tel1 | 2023 Oct 10 | 141 | 62 | $Z$ | 7.7
ExTrA - tel1 | 2023 Oct 10 | 141 | 62 | $Y$ | 4.7
ExTrA - tel1 | 2023 Oct 10 | 140 | 62 | $J$ | 4.3
ExTrA - tel1 | 2023 Oct 10 | 141 | 62 | $H$ | 12.6
ExTrA - tel2 | 2023 Oct 10 | 140 | 62 | $Z$ | 7.0
ExTrA - tel2 | 2023 Oct 10 | 140 | 62 | $Y$ | 5.0
ExTrA - tel2 | 2023 Oct 10 | 140 | 62 | $J$ | 4.3
ExTrA - tel2 | 2023 Oct 10 | 141 | 62 | $H$ | 12.6
ExTrA - tel3 | 2023 Oct 10 | 141 | 62 | $Z$ | 7.1
ExTrA - tel3 | 2023 Oct 10 | 141 | 62 | $Y$ | 5.9
ExTrA - tel3 | 2023 Oct 10 | 141 | 62 | $J$ | 5.3
ExTrA - tel3 | 2023 Oct 10 | 140 | 62 | $H$ | 22.8
ExTrA - tel1 | 2023 Nov 5 | 165 | 62 | $Z$ | 9.9
ExTrA - tel1 | 2023 Nov 5 | 165 | 62 | $Y$ | 5.6
ExTrA - tel1 | 2023 Nov 5 | 165 | 62 | $J$ | 5.2
ExTrA - tel1 | 2023 Nov 5 | 163 | 62 | $H$ | 14.7
ExTrA - tel2 | 2023 Nov 5 | 163 | 62 | $Z$ | 7.5
ExTrA - tel2 | 2023 Nov 5 | 165 | 62 | $Y$ | 5.4
ExTrA - tel2 | 2023 Nov 5 | 165 | 62 | $J$ | 5.3
ExTrA - tel2 | 2023 Nov 5 | 165 | 62 | $H$ | 14.7
ExTrA - tel3 | 2023 Nov 5 | 165 | 62 | $Z$ | 8.9
ExTrA - tel3 | 2023 Nov 5 | 165 | 62 | $Y$ | 5.7
ExTrA - tel3 | 2023 Nov 5 | 164 | 62 | $J$ | 6.0
ExTrA - tel3 | 2023 Nov 5 | 165 | 62 | $H$ | 18.8
ExTrA - tel1 | 2023 Nov 18 | 269 | 62 | $Z$ | 8.0
ExTrA - tel1 | 2023 Nov 18 | 268 | 62 | $Y$ | 4.9
ExTrA - tel1 | 2023 Nov 18 | 269 | 62 | $J$ | 4.6
ExTrA - tel1 | 2023 Nov 18 | 270 | 62 | $H$ | 12.2
ExTrA - tel2 | 2023 Nov 18 | 269 | 62 | $Z$ | 7.6
ExTrA - tel2 | 2023 Nov 18 | 269 | 62 | $Y$ | 4.8
ExTrA - tel2 | 2023 Nov 18 | 268 | 62 | $J$ | 4.5
ExTrA - tel2 | 2023 Nov 18 | 270 | 62 | $H$ | 11.8
ExTrA - tel3 | 2023 Nov 18 | 269 | 62 | $Z$ | 8.6
ExTrA - tel3 | 2023 Nov 18 | 268 | 62 | $Y$ | 4.9
ExTrA - tel3 | 2023 Nov 18 | 269 | 62 | $J$ | 5.2
ExTrA - tel3 | 2023 Nov 18 | 269 | 62 | $H$ | 14.8
ExTrA - tel1 | 2023 Dec 1 | 221 | 62 | $Z$ | 8.1
ExTrA - tel1 | 2023 Dec 1 | 221 | 62 | $Y$ | 5.8
ExTrA - tel1 | 2023 Dec 1 | 220 | 62 | $J$ | 5.0
ExTrA - tel1 | 2023 Dec 1 | 220 | 62 | $H$ | 11.8
ExTrA - tel2 | 2023 Dec 1 | 219 | 62 | $Z$ | 8.0
ExTrA - tel2 | 2023 Dec 1 | 219 | 62 | $Y$ | 5.6
ExTrA - tel2 | 2023 Dec 1 | 220 | 62 | $J$ | 4.7
ExTrA - tel2 | 2023 Dec 1 | 220 | 62 | $H$ | 15.1
ExTrA - tel3 | 2023 Dec 1 | 220 | 62 | $Z$ | 9.0
ExTrA - tel3 | 2023 Dec 1 | 220 | 62 | $Y$ | 5.9
ExTrA - tel3 | 2023 Dec 1 | 220 | 62 | $J$ | 5.9
ExTrA - tel3 | 2023 Dec 1 | 221 | 62 | $H$ | 16.4
ExTrA - tel1 | 2023 Dec 14 | 312 | 62 | $Z$ | 8.3
ExTrA - tel1 | 2023 Dec 14 | 313 | 62 | $Y$ | 5.3
ExTrA - tel1 | 2023 Dec 14 | 314 | 62 | $J$ | 4.8
ExTrA - tel1 | 2023 Dec 14 | 313 | 62 | $H$ | 15.8
ExTrA - tel3 | 2023 Dec 14 | 314 | 62 | $Z$ | 8.2
ExTrA - tel3 | 2023 Dec 14 | 314 | 62 | $Y$ | 5.6
ExTrA - tel3 | 2023 Dec 14 | 311 | 62 | $J$ | 5.8
ExTrA - tel3 | 2023 Dec 14 | 313 | 62 | $H$ | 15.9
ExTrA - tel1 | 2024 Jan 6 | 141 | 62 | $Z$ | 8.6
ExTrA - tel1 | 2024 Jan 6 | 141 | 62 | $Y$ | 5.2
ExTrA - tel1 | 2024 Jan 6 | 141 | 62 | $J$ | 5.0
ExTrA - tel1 | 2024 Jan 6 | 140 | 62 | $H$ | 12.7
ExTrA - tel3 | 2024 Jan 6 | 141 | 62 | $Z$ | 8.7
ExTrA - tel3 | 2024 Jan 6 | 140 | 62 | $Y$ | 5.7
ExTrA - tel3 | 2024 Jan 6 | 141 | 62 | $J$ | 6.2
ExTrA - tel3 | 2024 Jan 6 | 140 | 62 | $H$ | 16.4
Table 3:  Light curve data for TOI 762 A and TIC 46432937.

Objectaa Either TOI 762 A or TIC 46432937.  | BJDbb Barycentric Julian Dates in this paper are reported on the Barycentric Dynamical Time (TDB) system.  | Magcc The out-of-transit level has been subtracted. For observations made with TESS these magnitudes have been corrected for trends prior to fitting the transit model. For observations made with follow-up instruments (anything other than “TESS” in the “Instrument” column), the magnitudes have been corrected for a quadratic trend in time fit simultaneously with the transit.  | $\sigma_{\rm Mag}$ | Mag(orig)dd Raw magnitude values without correction for the quadratic trend in time, or for trends correlated with the seeing. These are only reported for the follow-up observations.  | Filter | Instrument
---|---|---|---|---|---|---
TOI-762 | $2458595.24514$ | $0.01246$ | $0.01259$ | $\cdots$ | $T$ | TESS/Sec10
TOI-762 | $2458577.88699$ | $-0.00409$ | $0.01237$ | $\cdots$ | $T$ | TESS/Sec10
TOI-762 | $2458588.30220$ | $-0.01125$ | $0.01251$ | $\cdots$ | $T$ | TESS/Sec10
TOI-762 | $2458581.35921$ | $-0.01476$ | $0.01231$ | $\cdots$ | $T$ | TESS/Sec10
TOI-762 | $2458591.77437$ | $-0.02359$ | $0.01222$ | $\cdots$ | $T$ | TESS/Sec10
TOI-762 | $2458574.41615$ | $0.01873$ | $0.01261$ | $\cdots$ | $T$ | TESS/Sec10
TOI-762 | $2458595.24653$ | $-0.01866$ | $0.01228$ | $\cdots$ | $T$ | TESS/Sec10
TOI-762 | $2458577.88838$ | $-0.01855$ | $0.01225$ | $\cdots$ | $T$ | TESS/Sec10
TOI-762 | $2458588.30358$ | $-0.00049$ | $0.01260$ | $\cdots$ | $T$ | TESS/Sec10
TOI-762 | $2458581.36059$ | $-0.01694$ | $0.01228$ | $\cdots$ | $T$ | TESS/Sec10

Note. — This table is available in a machine-readable form in the online
journal. A portion is shown here for guidance regarding its form and content.

Figure 2:  Observations incorporated into the analysis of the transiting
planet system TOI 762 A. Additional light curves are shown in Fig. 3. Top:
Transit light curves with best-fitted model (maximum likelihood) overplotted.
The dates, filters and instruments used are indicated. The term $z-H$ used for
the ExTrA light curves refers to the full band-pass not the color. (Caption
continued on next page.)  Figure 2:  (Caption continued from previous page.)
For TESS we phase-fold the data, and plot the un-binned observations in grey,
with the phase-binned values overplotted in blue. The residuals for each light
curve are shown on the right-hand-side in the same order as the original light
curves. The error bars represent the photon and background shot noise, plus
the readout noise. Note that these uncertainties are scaled in the fitting
procedure to achieve a reduced $\chi^{2}$ of unity, but the uncertainties
shown in the plot have not been scaled. Bottom Left: High-precision RVs phased
with respect to the mid-transit time. The top panel shows the phased
measurements together with the best-fit (maximum likelihood) model with no
eccentricity. The gray-scale curves show 1000 models randomly selected from
the MCMC posterior distribution generated when the eccentricity is allowed to
vary. The center-of-mass velocity has been subtracted. The middle panel shows
the phase-folded velocity $O\\!-\\!C$ residuals. The error bars include the
estimated jitter, which is varied as a free parameter in the fitting. The
bottom panel shows the $O\\!-\\!C$ residuals as a function of time. Bottom
Right: Color-magnitude diagram (CMD) and spectral energy distribution (SED).
The top panel shows the absolute $G$ magnitude vs. the de-reddened $G-K_{S}$
color compared to theoretical isochrones (black lines) and stellar evolution
tracks (green lines) from the MIST models interpolated at the best-estimate
value for the metallicity of the host. The age of each isochrone is listed in
black in Gyr, while the mass of each evolution track is listed in green in
solar masses. The solid red lines show isochrones at higher and lower
metallicities than the best-estimate value, with the metallicity and age in
Gyr of each isochrone labelled on the plot. The filled blue circles show the
measured reddening- and distance-corrected values from Gaia DR3 and 2MASS,
while the blue lines indicate the $1\sigma$ and $2\sigma$ confidence regions,
including the estimated systematic errors in the photometry. The middle panel
shows the SED as measured via broadband photometry through the listed filters.
Here we plot the observed magnitudes without correcting for distance or
extinction. Overplotted are 200 model SEDs randomly selected from the MCMC
posterior distribution produced through the global analysis (gray lines). The
model makes use of the predicted absolute magnitudes in each bandpass from the
MIST isochrones, the distance to the system (constrained largely via Gaia DR3)
and extinction (constrained from the SED with a prior coming from the mwdust
3D Galactic extinction model). The bottom panel shows the $O\\!-\\!C$
residuals from the best-fit model SED. The errors listed in the catalogs for
the broad-band photometry measurements are shown with black lines, while the
errors including an assumed 0.02 mag systematic uncertainty, which is added in
quadrature to the listed uncertainties, are shown with red lines. These latter
uncertainties are what we use in the fit.  Figure 3:  Additional follow-up
light curves of TOI 762 A, shown as described in Fig. 2.

Figure 4:  Same as Figure 2, here we show the observations of TIC 46432937
together with our best-fit model. Additional follow-up light curves of TIC
46432937 are shown in Figure 5.

Figure 5:  Additional follow-up light curves of TIC 46432937, shown as
described in Fig. 2.

##  3 Analysis

###  3.1 Derivation of Stellar Atmospheric Parameters

We used the machine-learning based ODUSSEAS package (Antoniadis-Karnavas et
al., 2020) to measure the photospheric effective temperature $T_{\rm
eff\star}$ and metallicity $\rm[Fe/H]$ (assuming Solar-scaled abundances) of
both systems from the ESPRESSO spectra. The tool measures pseudo-equivalent
widths for thousands of lines and compares them to a training set of reference
M dwarf stars observed by HARPS. Although the HARPS spectra have lower
resolution than the ESPRESSO spectra, the tools has been shown to work
effectively on high resolution ESPRESSO spectra as well (Lillo-Box et al.,
2020). The analysis was performed using the “wide error” mode, taking into
consideration the intrinsic uncertainties of the reference parameters in the
machine learning process in addition to the output machine learning model
errors. We measure $T_{\rm eff\star}=3150\pm 67$ K and $\rm[Fe/H]=0.24\pm
0.10$ for TOI 762 A and $T_{\rm eff\star}=3535\pm 65$ K and $\rm[Fe/H]=0.03\pm
0.10$ for TIC 46432937.

###  3.2 Stellar Activity and Galactic Kinematics

We checked the TESS light curves of TOI 762 A for evidence of stellar
rotational variability or flaring events. No significant out-of-transit
variability is observed. A total of $6$ bright outliers ($>5\sigma$) are
present in the out-of-transit TESS light curves of TOI 762 A. In each case the
outlier is isolated to a single point, so if it is due to a flare, the flare
would have been shorter than 120 s in duration. It is worth noting though that
no $5\sigma$ faint outliers are detected. We also checked the publicly
available ASAS-SN (Hart et al., 2023; Shappee et al., 2014) light curve TOI
762 A and see no evidence for periodic variability or stellar flares.

For TIC 46432937 a periodic signal is present in the out-of-transit TESS light
curves from Sectors 6 and 32. We find a period of $P=5.88\pm 0.54$ day with a
signal-to-noise ratio of $35.2$ as measured in the Generalized Lomb-Scargle
periodogram (Zechmeister & Kürster, 2009). The estimated uncertainty on the
period is the half-width at half-maximum of the peak in the periodogram. The
peak-to-peak variation in the phase-binned light curve is $\sim 1$ mmag. The
formal false alarm probability is vanishingly small, however this does not
account for the possibility of systematic variations in the light curve due to
uncorrected instrumental effects. We checked the Zwicky Transient Facility
(ZTF; Masci et al., 2019) DR20 light curve of TIC 46432937, but see no
evidence for periodic variability. However, the scatter in the ZTF light curve
of this source is $\sim 0.02$ mag, and the $\sim 1$ mmag amplitude signal seen
in TESS would not be detectable if it is present in the ZTF data. No
variability is detected in the publicly available ASAS-SN light curve of TIC
46432937 either, which also has too high a scatter to permit detection of a
signal comparable to that seen in the TESS data. No candidate flare events or
significant bright outliers are seen in the TESS light curve of TIC 46432937.
While bright outliers are seen in the ZTF and ASAS-SN light curves, the number
of faint outliers is comparable.

If the $P=5.88$ day signal corresponds to the rotation period of TIC 46432937,
the rotation period would be faster than the bulk of early M dwarf stars with
$0.9<G-RP<1.1$ seen in the Kepler sample (McQuillan et al., 2014). While this
may suggest a young age for TIC 46432937, existing gyrochronology relations
are not suitable for M dwarf stars (Popinchalk et al., 2021). It is also
unclear how a relatively massive $\sim 3$ $M_{\rm J}$ planet on a short
$P=1.44$ day orbit might affect the rotational evolution of an early M dwarf.
For these reasons we do not attempt to estimate an age for TIC 46432937 based
on the possible rotation period.

As an additional check on the ages of the systems, we used the systemic radial
velocities that we measured for each star with ESPRESSO, together with the
position, parallax, and proper motions from Gaia DR3 to compute the $U$, $V$,
and $W$ space velocities. We follow the convention that $U$ increases toward
the Galactic center, $V$ increases in the direction of Galactic rotation, and
$W$ increases toward the North Galactic Pole. These velocities are then
corrected to be relative to the local standard of rest (LSR) by adding the
Solar peculiar velocities of $(U_{\odot},$ $V_{\odot},$ $W_{\odot})$ $=$
$(11.10,$ $12.24,$ $7.25)$ $\rm km\,s^{-1}$ from Schönrich et al. (2010). This
yields $(U_{LSR},$ $V_{LSR},$ $W_{LSR})$ $=$ $(-35.56,$ $-58.28,$ $-22.96)$
$\rm km\,s^{-1}$ and $(-73.82,$ $-33.11,$ $-31.21)$ $\rm km\,s^{-1}$, for TOI
762 A and TIC 46432937, respectively. We use these velocities to compute the
relative probabilities of each star being a member of the Galactic thin and
thick disks, or of the Galactic halo, following Bensby et al. (2014). We find
that both objects have a 70% probability of being members of the thin disk, a
30% probability of being members of the thick disk, and negligible
probabilities of being members of the Galactic halo. While each object is more
likely than not to be in the thin disk, both stars exhibit fairly high space
motion compared to typical thin disk members (both stars have at least one
component of their space velocity that is different from the mean value for
thin disk stars by more than $2\sigma$), which is consistent with both objects
being older main sequence stars.

###  3.3 Joint Stellar and Planet Modeling

We carried out a joint analysis of the available observations to determine the
stellar and planetary parameters of each system following the method of
Hartman et al. (2019) and Bakos et al. (2020). For each object we performed a
simultaneous fit to all light curves, RV measurements, the observed Spectral-
Energy Distribution (SED) as traced by the available catalog broad-band
photometry, the spectroscopic $T_{\rm eff\star}$ and $\rm[Fe/H]$, and the
astrometric parallax from Gaia DR3 (Gaia Collaboration et al., 2022). The
catalog photometry, spectroscopic parameters, and parallax values that we
included in the fit for each system are listed in Table 4. The light curves
were modeled using the semi-analytic Mandel & Agol (2002) model with quadratic
limb darkening coefficients that were allowed to vary in the fit, but with
priors based on the stellar atmospheric parameters in the theoretical
tabulations by Claret et al. (2012, 2013); Claret (2018). For the TESS
observations of TIC 46432937, which have exposure times of 30 min or 10 min,
we integrated the model over the exposure time. The RVs were fit assuming the
star follows a Keplerian orbit around the system barycenter. The stellar
parameters and SED are constrained to follow the MIST stellar evolution model
while allowing for systematic errors in the stellar physical parameters
following the procedure of Hartman et al. (2023). We adopt systematic errors
of $0.08$ dex, 2.4%, 5% and 0.021 mag on the metallicity, effective
temperature, stellar mass, and bolometric magnitudes in the stellar evolution
models, respectively (Tayar et al., 2022). We impose a prior on the line of
sight extinction $A_{V}$ using the MWDUST 3D Galactic extinction model (Bovy
et al., 2016) and assume an $R_{V}=3.1$ extinction law. The fit is performed
twice for each system, first assuming a circular orbit for the planet, and
second allowing for a non-zero eccentricity. The fit is carried out through a
differential evolution Markov Chain Monte Carlo procedure (see Hartman et al.,
2019 for a full list of parameters and priors) using visual inspection to
confirm that the chains are converged and well-mixed, and to set the burn-in
period.

To account for dilution in the light curves of TOI 762 A due to the
neighboring star TOI 762 B, we include for each light curve a parameter that
specifies the fraction of the flux in the light curve that comes from the
transit hosting star when out of transit (with the remaining fraction assumed
to come from a constant source). This parameter is allowed to vary in the fit
independently for each light curve. We place a prior and uncertainty on each
of these dilution parameters by calculating the expected flux from TOI 762 B
that would contaminate the aperture, assuming that TOI 762 B has the same
distance, reddening, age, and metallicity as inferred for TOI 762 A, and using
the $G$ magnitude together with the MIST isochrones to infer the mass of TOI
762 B and its expected magnitude in various passbands. We note that the
resulting dilution values are all consistent with the priors, to within the
uncertainties. No systematic trend with wavelength is observed in the
residuals.

The observations are consistent with circular orbits for both systems. For TOI
762 A b we place a 95% confidence upper limit on the eccentricity of
$e<0.083$, while for TIC 46432937 b the upper limit is $e<0.009$. We therefore
adopt the parameters derived for each system assuming a circular orbit. Note
that in the joint fitting that we perform, the constraints on the
eccentricities of these systems come not just from the RVs, but also from the
combination of the transit observations and the theoretical stellar evolution
models that constrain the allowed combinations of stellar mass, radius, and
metallicity. For M dwarf host stars the constraints on mass and radius from
comparing the observations to stellar evolution models are much tighter than
for higher mass stars, which can lead to a much tighter constraint on the
eccentricity than might be allowed by the RVs alone (e.g., Hartman et al.,
2015). This appears to be the case for TOI 762 A b, for which the RVs alone
provide a much less stringent constraint on the eccentricity than is achieved
through our joint analysis of the data. For TIC 46432937 b on the other hand,
the eccentricity constraint appears to come primarily from the RVs. Here the
large semi-amplitude of the orbital variation caused by the massive planet and
the well-sampled phase curve enable a strong constraint on the orbital
eccentricity.

We find that TIC 46432937 b exhibits grazing transits which in some cases can
lead to a strong degeneracy between the impact parameter of the planet and the
planet-to-star radius ratio. In such cases it may only be possible to provide
a lower bound on the planetary radius (e.g., HATS-23 b; Bento et al., 2017).
We confirmed that the Markov Chains for TIC 46432937 are well-converged and
display a clear upper-limit on both the impact parameter and the planet-to-
star radius ratio, so we provide best-estimates for these parameters together
with two-sided uncertainties.

Figures 2–5 compare the best-fit models to the observational data. The adopted
stellar parameters are listed in Table 6, while the adopted planetary
parameters are listed in Table 7 and the limb darkening coefficients are
listed in Table 8. In both cases we list the parameters determined assuming
circular orbits.

###  3.4 Ruling Out Blended Stellar Eclipsing Binary Scenarios

A line-of-sight blend of three stars, including two that are eclipsing, is a
relatively common false positive that can mimic the photometric transit and
radial velocity signals produced by a transiting giant planet system (e.g.,
Torres et al., 2004). In order to rule out such an explanation for the
observations of TOI 762 A or TIC 46432937 we carried out a blend analysis
following the methods of Hartman et al. (2019). Here we model the photometric
and astrometric data as a blend of three or more stars using the MIST version
1.2 stellar evolution models to constrain the physical properties of the
stars. We consider both a hierarchical triple star system, where the two
fainter stars in the system are eclipsing, and a line-of-sight blend between
an eclipsing binary star system and a brighter, physically unrelated,
foreground star.

For TIC 46432937 the analysis was performed before the photometric follow-up
observations were available, and only the TESS light curves are included in
this case. Because the parameters that were measured for the system when using
only the TESS observations are fully consistent with the parameters found when
incorporating the follow-up light curves from ExTrA, and as we discuss below
we can already rule out blend scenarios with only the TESS data, we do not
expect the conclusions from the blend analysis to change when incorporating
the follow-up light curves. We therefore did not repeat the blend analysis for
this system after obtaining the light curves from ExTrA.

For the hierarchical triple system the parameters that we vary in the fit
include the times of two reference transit events, the age of the stellar
system, the masses of the three stellar companions, the distance to the
system, the metallicity of the system, the impact parameter of the eclipsing
pair, and the limb darkening coefficients of the primary star in the eclipsing
pair. For the line-of-sight blend we include all of these parameters, as well
as the age, metallicity and distance of the blended foreground star. We
compare both of these scenarios to a model consisting of a transiting planet
around a single star for which we vary the system age, metallicity and
distance, the mass and limb darkening coefficients of the host star, the
impact parameter of the system, and the planet-to-star radius ratio.

For TOI 762 we also account for the 3$\farcs$2 resolved stellar companents TOI
762 A and TOI 762 B in modelling the observations, and attempt to model the
brighter target TOI 762 A, for which RV variations consistent with a planetary
companion were detected, as a blended system of three stars. Thus for TOI 762
there are four stars considered in the blend scenarios that we investigate,
and we include the mass of the 3$\farcs$2 neighbor TOI 762 B as a parameter in
the fit, and assume that this neighbor has the same age, metallicity and
distance as the brightest star in the blended object TOI 762 A.

We find that for both TOI 762 and TIC 46432937 the transiting planet scenario
provides a significantly better fit to the photometric and astrometric data,
with lower $\chi^{2}$, than the blended stellar eclipsing binary systems that
we considered. For TOI 762 we find that the best-fit transiting planet model
has $\Delta\chi^{2}=-305$ compared to the best-fit blended eclipsing binary
model, while for TIC 46432937 we find $\Delta\chi^{2}=-287$. As the transiting
planet scenarios also require fewer free parameters, we can confidently rule
out a blended stellar eclipsing binary system for both objects. The best-fit
blend scenarios exhibit secondary eclipses that are ruled out by the TESS
observations, as well as distances and implied apparent magnitudes for the
blended objects that are inconsistent with the Gaia parallax and measured
apparent magnitudes. We therefore consider both objects to be confirmed
transiting planet systems.

##  4 Discussion

Figure 6:  Left: Planet radius vs. mass. The two new planet discoveries are
indicated (we abbreviate TIC 46432937 b as 4643). Small grey points show all
transiting planets with measured masses and radii around K and earlier type
stars from the NASA Exoplanet Archive, while red points show transiting
planets with masses and radii around M dwarfs. Right: Planet mass vs. host
star mass for transiting planets around sub-solar-mass stars. The red points
in this case indicate giant planets transiting M dwarfs, while the grey points
indicate other planets.

Figure 7:  Top Left: Transmission Spectroscopy Metric (TSM) vs. planet mass.
TOI 762 A b and TIC 46432937 b are indicated. Small gray points show all
transiting planets with measured masses $>0.1$ $M_{\rm J}$ and radii $>0.5$
$R_{\rm J}$ from the NASA Exoplanet Archive, while small black points denote
planets with M dwarf host stars. For all panels we restrict the plots to
planets for which the plotted quantities have uncertainties of less than 30%.
Top Right: TSM vs. planet Teq computed assuming zero albedo and full
redistribution of heat. The symbols are the same as in the left plot. Bottom
Left: Emission Spectroscopy Metric (ESM) vs. planet mass. Bottom Right: ESM
vs. planet Teq. TIC 46432937 b has the highest value of TSM or ESM for a
transiting warm Super-Jupiter with $M_{p}>3.0$ $M_{\rm J}$ and $T_{\rm
eq}<1000$ K.

In this paper we presented the discovery of two transiting giant planets that
orbit M dwarf stars. Fig. 6 compares these two new systems to other confirmed
transiting planet systems listed in the NASA Exoplanet Archive
(2024)111Accessed on 2024-01-17 at 12:40. We find that with
$M_{\star}=0.442\pm 0.025$ $M_{\sun}$, TOI 762 A is one of the lowest mass
stars known to host a transiting giant planet (taking $M_{p}>0.1$ $M_{\rm J}$
as our definition of such a planet; it is also one of the lowest mass stars
known to host a giant planet in general), while with $M_{p}=3.20\pm 0.11$
$M_{\rm J}$, TIC 46432937 b is one of the highest mass planets known to
transit an M dwarf star. Stars of lower mass than TOI 762 A with giant planets
include TOI-4860 ($0.336$ $M_{\sun}$; Triaud et al., 2023), TOI-519 ($0.335$
$M_{\sun}$; Kagetani et al., 2023), TOI-5205 ($0.392$ $M_{\sun}$; Kanodia et
al., 2023), and TOI-3235 ($0.370$ $M_{\sun}$; Hobson et al., 2023). The only
transiting object orbiting an M dwarf star listed on the NASA Exoplanet
Archive with a higher mass than TIC 46432937 b is TOI-1278 b (Artigau et al.,
2021), but with a mass of $18.5\pm 0.5$ $M_{\rm J}$, this object may be a
brown dwarf rather than a planet. The next highest mass planet is TOI-4201 b
($2.48$ $M_{\rm J}$; Gan et al., 2023; Hartman et al., 2023; Delamer et al.,
2023).

##### Stellar Companion to TOI 762 A

We find that TOI 762 A has a resolved stellar companion TOI 762 B. The angular
separation of 3$\farcs$2 between the two stars corresponds to a current
projected physical separation of 319 AU. We estimate that TOI 762 B has a mass
of $0.227\pm 0.010$ $M_{\sun}$, which is 45% the mass of TOI 762 A. The
eccentricity and orbital period of TOI 762 B are not known.

If the eccentricity is sufficiently high, then TOI 762 B might have induced
high-eccentricity migration for the planet TOI 762 A b via the Kozai-Lidov
mechanism (Kozai, 1962; Lidov, 1962; Naoz, 2016). To get a rough sense for the
time-scale of this mechanism we use eq. 7 of Kiseleva et al. (1998) assuming
that the current projected physical separation corresponds to the semi-major
axis of the orbit of the binary star system. We find a time-scale of $\sim
100$ Myr if TOI 762 B has a very high eccentricity of 0.95, or $\sim 3.5$ Gyr
if the eccentricity is close to zero.

Four other M dwarfs that host transiting giant planets also have known wide
stellar binary companions (TOI 3984 A, Cañas et al., 2023; TOI 5293 A, Cañas
et al., 2023; TOI 3714, Cañas et al., 2022; and HATS-74 A, Jordán et al.,
2022). Thus, approximately 20% of the sample of M dwarfs with transiting giant
planets are known to have resolved stellar binary companions. The overall
stellar multiplicity fraction for M dwarf stars is estimated to be $46\pm 5$%
(Susemiehl & Meyer, 2022), however, a meaningful comparison to the rate for
giant planet host stars will require a careful correction for observational
completeness. Such a study will likely require a much larger sample of giant
planet-hosting M dwarfs to enable a statistically significant result.

Ngo et al. (2016) has compared the stellar multiplicity rate for FGK stars
that host hot Jupiters to the rate for field FGK stars. They find that the
fraction of hot Jupiter systems with stellar companions between 50 and 2000 AU
is approximately 2.9 times the field star companion fraction. But, they also
find that in the majority of cases the stellar binary companions could not
drive high-eccentricity migration through the Kozai-Lidov effect. They
conclude that for FGK systems, binarity is likely correlated with the
formation of hot Jupiters, but the binary companions themselves do not
physically drive their migration.

##### High mass of TIC 46432937 b

The high mass of TIC 46432937 b given its low stellar host mass of $0.563\pm
0.029$ $M_{\sun}$ poses a challenge for theories of planet formation and
evolution. Gan et al. (2023) carried out interior structure modelling of the
planet TOI-4201 b, which has a similar radius to TIC 46432937 b, similar host
star metallicity and similar planet equilibrium temperature, but somewhat
lower planet mass and somewhat larger host star mass. They find that TOI-4201
b requires a low planet bulk metal content which appears to be at odds with
the high metallicity inferred for the host star. Given the higher planet mass
of TIC 46432937 b, the comparably high metallicity of its host star, and the
comparable planet radius and equilibrium temperature, we anticipate that a
similar analysis would result in the same conclusion for TIC 46432937 b as Gan
et al. (2023) arrived at for TOI-4201 b—i.e., that the planet
mass/radius/equilibrium temperature requires a low planet bulk metal content
that is at odds with the high host star metallicity. Gan et al. (2023)
considered a variety of potential heating sources that might explain the
discrepancy, including tidal heating (Leconte et al., 2010), a gas giant
merger (Li et al., 2010; Liu et al., 2015), and a more quiescent process
involving an embryo being captured by a gas giant during inward migration (Lin
et al., 1996). The fact that two super-Jupiter-mass planets have now been
found around M dwarf stars with somewhat high radii, suggesting low bulk metal
content, indicates that if there is an additional heating mechanism, it may be
common for short-period giant planets around M dwarf stars. With only two
planets, however, we cannot draw any definite conclusions at this point.

##### Grazing Transits of TIC 46432937 b

We find that TIC 46432937 b exhibits grazing transit events. Two other giant
planets around M dwarfs have been found with grazing transits: NGTS 1 b
(Bayliss et al., 2018), and the super-massive planet or brown dwarf TOI 1278 B
(Artigau et al., 2021). The relatively large values of $R_{p}/R_{\star}$ for
giant planets orbiting M dwarfs enhances the probability of finding these
systems in grazing configurations compared to giant planets transiting Sun-
like stars. The deep V-shaped transit events exhibited by these systems are
often considered the hallmark of stellar eclipsing binaries, so it is
important to keep in mind that transiting planets can have light curves of
this form as well when searching for giant planets transiting M dwarf stars.

Because grazing transit events do not have second or third contact points (the
times when a planet starts and stops being fully in front of its host star,
respectively), there is less information content in a grazing transit than in
a full transit event. This can lead to degeneracies between the impact
parameter and the planet-to-star radius ratio, resulting in larger
uncertainties on the planetary radius and other parameters for these systems.
High-precision observations, such as from TESS, can help break this degeneracy
and enable an accurate measurement of the planetary radius in spite of the
grazing transits. We find that this is the case for TIC 46432937 b, for which
we find that the radius is measured to 2.5% uncertainty despite the grazing
events.

One minor advantage of a grazing system is that secular variations in the
orbits of these planets, due for example to the presence of exterior planets
in the system, may be detectable with higher signal-to-noise than for full
transits (Ribas et al., 2008). An example of this is K2-146 c, a planet whose
orbit has been seen to vary between grazing and fully transiting
configurations (Hamann et al., 2019). While no timing variations have been
detected yet for TIC 46432937 b, the grazing transits make this an interesting
target for continued transit timing observations going forward.

##### Prospects for Atmospheric Characterization

Transiting giant planets orbiting M dwarf stars can be useful objects for
atmospheric characterization due to their relatively deep transits. We
computed the Transmission Spectroscopy Metric and Emission Spectroscopy Metric
(TSM and ESM, respectively; Kempton et al., 2018) for the two new planetary
systems presented here, finding values of ${\rm TSM}=101\pm 21$, and ${\rm
ESM}=18.3\pm 1.5$ for TOI 762 A b and ${\rm TSM}=52.3\pm 4.8$, and ${\rm
ESM}=152\pm 11$ for TIC 46432937 b. These are compared to other transiting
giant planets from the NASA Exoplanet Archive in Figure 7. While neither of
the newly discovered planets has an especially high TSM value compared to the
overall sample of transiting giant planets, their values are both greater than
the median value of ${\rm TSM}=47.4$ for the full sample of confirmed warm
transiting giant planets with equilibrium temperatures $T_{\rm eq}<1000$ K.
When comparing against planets of $M_{p}>3.0$ $M_{\rm J}$, the TSM of TIC
46432937 b does stand out as being greater than the value for all but four of
these planets (i.e., it has a greater TSM value than 92% of all known
transiting planets with $M_{p}>3.0$ $M_{\rm J}$). These four planets (HAT-P-70
b, ${\rm TSM}=57.8$, Zhou et al., 2019; MASCARA-4 b, ${\rm TSM}=107$, Dorval
et al., 2020; TOI-1431 b, ${\rm TSM}=109$, Addison et al., 2021; and HIP 65 A
b, ${\rm TSM}=780$, Nielsen et al., 2020) all have equilibrium temperatures
greater than $1400$ K, which is significantly higher than that of TIC 46432937
b. The next highest TSM value for a planet with $M_{p}>3.0$ $M_{\rm J}$ and
$T_{\rm eq}<1000$ K is 13.1 for the planet HAT-P-20 b (Bakos et al., 2011),
which is a factor of four smaller than the value for TIC 46432937 b. TIC
46432937 b also has a high value of ESM for a giant planet with $T_{\rm
eq}<1000$ K. The only giant planet that has $T_{\rm eq}<1000$ K and a higher
ESM value than TIC 46432937 b is WASP-80 b (Triaud et al., 2013), with ${\rm
ESM}=216$. WASP-80 b also has a significantly lower mass of $0.538\pm 0.035$
$M_{\rm J}$. Thus TIC 46432937 b presents a particularly good opportunity for
studying the atmosphere of a warm, high-mass planet. The frequent transits of
this $P=1.4404$ d system should also facilitate scheduling such observations.

We thank the anonymous referee for their careful reading this paper, and
helpful comments that have improved the quality of the work. JH and GB
acknowledge funding from NASA grant 80NSSC22K0315. AJ and RB acknowledge
support from ANID – Millennium Science Initiative – ICN12_009. R.B.
acknowledges support from FONDECYT Project 11200751. AJ acknowledges support
from FONDECYT project 1210718. We acknowledge funding from the European
Research Council under the ERC Grant Agreement n. 337591-ExTrA. 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 made use of
data collected by the TESS mission and are publicly available from the
Mikulski Archive for Space Telescopes (MAST) operated by the Space Telescope
Science Institute (STScI). Funding for the TESS mission is provided by NASA’s
Science Mission Directorate. The specific observations from MAST analyzed in
this paper can be accessed from https://doi.org/10.17909/e3cp-9907 (catalog
DOI). We acknowledge the use of public TESS data from pipelines at the TESS
Science Office and at the TESS Science Processing Operations Center. 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
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.
The contributions at the Mullard Space Science Laboratory by EMB have been
supported by STFC through the consolidated grant ST/W001136/1. The
postdoctoral fellowship of KB is funded by F.R.S.-FNRS grant T.0109.20 and by
the Francqui Foundation. This publication benefits from the support of the
French Community of Belgium in the context of the FRIA Doctoral Grant awarded
to MT. MG is F.R.S.-FNRS Research Director and EJ is F.R.S.-FNRS Senior
Research Associate. F.J.P acknowledges financial support from the grant
CEX2021-001131-S funded by MCIN/AEI/ 10.13039/501100011033 and through
projects PID2019-109522GB-C52 and PID2022-137241NB-C43. Based on data
collected by the SPECULOOS-South Observatory at the ESO Paranal Observatory in
Chile.The ULiege’s contribution to SPECULOOS has received funding from the
European Research Council under the European Union’s Seventh Framework
Programme (FP/2007-2013) (grant Agreement n∘ 336480/SPECULOOS), from the
Balzan Prize and Francqui Foundations, from the Belgian Scientific Research
Foundation (F.R.S.-FNRS; grant n∘ T.0109.20), from the University of Liege,
and from the ARC grant for Concerted Research Actions financed by the
Wallonia-Brussels Federation. This work is supported by a grant from the
Simons Foundation (PI Queloz, grant number 327127). Based on data collected by
the TRAPPIST-South telescope at the ESO La Silla Observatory. TRAPPIST is
funded by the Belgian Fund for Scientific Research (Fond National de la
Recherche Scientifique, FNRS) under the grant PDR T.0120.21, with the
participation of the Swiss National Science Fundation (SNF). DR was supported
by NASA under award number NNA16BD14C for NASA Academic Mission Services.

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Table 4:  Astrometric, Spectroscopic and Photometric parameters for TOI 762 A and TIC 46432937  | TOI 762 A | TIC 46432937 |
---|---|---|---
Parameter | Value | Value | Source
Astrometric properties and cross-identifications
2MASS-ID | 11041818-4749169 | 05352856-1435504 |
TIC-ID | 178709444 | 46432937 |
TOI-ID | 762 | $\cdots$ |
GAIA DR2-ID | 5362352744504000256 | 2984391358868786816 |
R.A. (J2000) | $11^{\mathrm{h}}04^{\mathrm{m}}18.1831{\mathrm{s}}$ | $05^{\mathrm{h}}35^{\mathrm{m}}28.5693{\mathrm{s}}$ | GAIA DR3
Dec. (J2000) | $-47{\arcdeg}49{\arcmin}17.0030{\arcsec}$ | $-14{\arcdeg}35{\arcmin}50.4600{\arcsec}$ | GAIA DR3
$\mu_{\rm R.A.}$ ($\rm mas\,yr^{-1}$) | $-159.174\pm 0.020$ | $-13.365\pm 0.013$ | GAIA DR3
$\mu_{\rm Dec.}$ ($\rm mas\,yr^{-1}$) | $-24.780\pm 0.020$ | $36.962\pm 0.012$ | GAIA DR3
parallax (mas) | $10.118\pm 0.023$ | $11.031\pm 0.013$ | GAIA DR3
Spectroscopic properties
$T_{\rm eff\star}$ (K) | $3150\pm 67$ | $3535\pm 65$ | see Section 3.1
$\rm[Fe/H]$ | $0.24\pm 0.10$ | $0.03\pm 0.10$ | see Section 3.1
$\gamma_{\rm RV}$ ($\rm m\,s^{-1}$) | $48680.1\pm 4.3$ | $102279.1\pm 2.5$ | ESPRESSO
Photometric propertiesaa We only include in the table catalog magnitudes that
were included in our analysis of each system.
$G$ (mag)bb The listed uncertainties for the Gaia DR3 photometry are taken from the catalog. For the analysis we assume an additional systematic uncertainty of 0.02 mag for all bandpasses.  | $14.9297\pm 0.0028$ | $13.4172\pm 0.0028$ | GAIA DR3
$BP$ (mag)bb The listed uncertainties for the Gaia DR3 photometry are taken from the catalog. For the analysis we assume an additional systematic uncertainty of 0.02 mag for all bandpasses.  | $\cdots$ | $14.4962\pm 0.0033$ | GAIA DR3
$RP$ (mag)bb The listed uncertainties for the Gaia DR3 photometry are taken from the catalog. For the analysis we assume an additional systematic uncertainty of 0.02 mag for all bandpasses.  | $\cdots$ | $12.3785\pm 0.0039$ | GAIA DR3
$B$ (mag) | $\cdots$ | $15.707\pm 0.050$ | APASScc From APASS DR6 as listed in the UCAC 4 catalog (Zacharias et al., 2013). Although these measurements are also available for TOI 762 A, we do not include them in the analysis or list them here because the degree to which these measurements are contaminated by flux from the 3$\farcs$2 and 4$\farcs$7 neighbors is difficult to determine.
$V$ (mag) | $\cdots$ | $14.310\pm 0.020$ | APASScc From APASS DR6 as listed in the UCAC 4 catalog (Zacharias et al., 2013). Although these measurements are also available for TOI 762 A, we do not include them in the analysis or list them here because the degree to which these measurements are contaminated by flux from the 3$\farcs$2 and 4$\farcs$7 neighbors is difficult to determine.
$g$ (mag) | $\cdots$ | $15.040\pm 0.060$ | APASScc From APASS DR6 as listed in the UCAC 4 catalog (Zacharias et al., 2013). Although these measurements are also available for TOI 762 A, we do not include them in the analysis or list them here because the degree to which these measurements are contaminated by flux from the 3$\farcs$2 and 4$\farcs$7 neighbors is difficult to determine.
$r$ (mag) | $\cdots$ | $13.740\pm 0.020$ | APASScc From APASS DR6 as listed in the UCAC 4 catalog (Zacharias et al., 2013). Although these measurements are also available for TOI 762 A, we do not include them in the analysis or list them here because the degree to which these measurements are contaminated by flux from the 3$\farcs$2 and 4$\farcs$7 neighbors is difficult to determine.
$i$ (mag) | $\cdots$ | $12.792\pm 0.040$ | APASScc From APASS DR6 as listed in the UCAC 4 catalog (Zacharias et al., 2013). Although these measurements are also available for TOI 762 A, we do not include them in the analysis or list them here because the degree to which these measurements are contaminated by flux from the 3$\farcs$2 and 4$\farcs$7 neighbors is difficult to determine.
$J$ (mag)dd From the 2MASS catalog (Skrutskie et al., 2006). For TOI 762 A we subtracted an estimate for the flux contribution from TOI 762 B.  | $12.099\pm 0.080$ | $11.011\pm 0.022$ | 2MASS
$H$ (mag)dd From the 2MASS catalog (Skrutskie et al., 2006). For TOI 762 A we subtracted an estimate for the flux contribution from TOI 762 B.  | $11.444\pm 0.077$ | $10.427\pm 0.023$ | 2MASS
$K_{s}$ (mag)dd From the 2MASS catalog (Skrutskie et al., 2006). For TOI 762 A we subtracted an estimate for the flux contribution from TOI 762 B.  | $11.187\pm 0.082$ | $10.195\pm 0.020$ | 2MASS
$W1$ (mag)ee From the 2021 Feb 16 ALLWISE Data release of the WISE mission (Cutri et al., 2021). For TOI 762 A we subtracted an estimate for the flux contribution from TOI 762 B.  | $10.991\pm 0.088$ | $10.114\pm 0.023$ | WISE
$W2$ (mag)ee From the 2021 Feb 16 ALLWISE Data release of the WISE mission (Cutri et al., 2021). For TOI 762 A we subtracted an estimate for the flux contribution from TOI 762 B.  | $10.890\pm 0.094$ | $10.063\pm 0.020$ | WISE
$W3$ (mag)ee From the 2021 Feb 16 ALLWISE Data release of the WISE mission (Cutri et al., 2021). For TOI 762 A we subtracted an estimate for the flux contribution from TOI 762 B.  | $10.71\pm 0.12$ | $9.910\pm 0.055$ | WISE

Table 5:  ESPRESSO radial velocities for TOI 762 A and TIC 46432937.  System | BJD | RVaa The zero-point of these velocities is arbitrary. An overall offset $\gamma_{\rm rel}$ fitted to the orbit has been subtracted for each system.  |  $\sigma_{\rm RV}$bb Internal errors excluding the component of astrophysical jitter allowed to vary in the fit.  | Phase | Instrumentcc For TOI 762 A we distinguish between the ESPRESSO observations obtained during P104 and those obtained during P110 for which we allow independent zero-point offsets in the fit.
---|---|---|---|---|---
| (2,450,000$+$) | ($\rm m\,s^{-1}$) | ($\rm m\,s^{-1}$) | ($\rm m\,s^{-1}$) |
TOI-762 | $8818.82472$ | $47.93$ | $11.30$ | $0.901$ | ESPRESSO/P104
TOI-762 | $8822.81705$ | $-17.07$ | $11.50$ | $0.051$ | ESPRESSO/P104
TOI-762 | $8836.78307$ | $-29.07$ | $8.70$ | $0.074$ | ESPRESSO/P104
TOI-762 | $8839.83846$ | $19.93$ | $5.80$ | $0.954$ | ESPRESSO/P104
TOI-762 | $8841.81658$ | $4.93$ | $6.40$ | $0.524$ | ESPRESSO/P104
TOI-762 | $9938.80622$ | $3.43$ | $7.14$ | $0.506$ | ESPRESSO/P110
TOI-762 | $9942.74132$ | $39.13$ | $10.60$ | $0.639$ | ESPRESSO/P110
TOI-762 | $9943.75858$ | $40.13$ | $8.42$ | $0.932$ | ESPRESSO/P110
TIC-46432937 | $9907.75857$ | $-423.38$ | $1.95$ | $0.086$ | ESPRESSO
TIC-46432937 | $9909.74414$ | $-187.38$ | $2.49$ | $0.464$ | ESPRESSO
TIC-46432937 | $9928.64212$ | $417.62$ | $3.81$ | $0.584$ | ESPRESSO
TIC-46432937 | $9930.76066$ | $-277.38$ | $1.94$ | $0.054$ | ESPRESSO
TIC-46432937 | $9936.72836$ | $-797.38$ | $1.81$ | $0.197$ | ESPRESSO
TIC-46432937 | $9995.60029$ | $-336.38$ | $3.53$ | $0.068$ | ESPRESSO
TIC-46432937 | $9996.56963$ | $836.62$ | $2.08$ | $0.741$ | ESPRESSO
TIC-46432937 | $9997.63070$ | $-122.38$ | $1.97$ | $0.478$ | ESPRESSO
Table 6:  Adopted derived stellar parameters for TOI 762 A and TIC 46432937.

| TOI 762 A | TIC 46432937
---|---|---
Parameter | Value | Value
$M_{\star}$ ($M_{\sun}$) | $0.442\pm 0.025$ | $0.563\pm 0.029$
$R_{\star}$ ($R_{\sun}$) | $0.4250\pm 0.0091$ | $0.5299\pm 0.0091$
$\log{g_{\star}}$ (cgs) | $4.827\pm 0.010$ | $4.740\pm 0.010$
$\rho_{\star}$ ($\rm g\,cm^{-3}$) | $8.11\pm 0.20$ | $5.34\pm 0.11$
$L_{\star}$ ($L_{\sun}$) | $0.0185\pm 0.0011$ | $0.0412\pm 0.0031$
$T_{\rm eff\star}$ (K) | $3266\pm 36$ | $3572\pm 57$
$\rm[Fe/H]$ | $0.357\pm 0.085$ | $0.323\pm 0.081$
Age (Gyr) | $9.3\pm 5.5$ | $7.4\pm 5.1$
$A_{V}$ (mag) | $0.1740\pm 0.0052$ | $0.023\pm 0.012$
Distance (pc) | $98.78\pm 0.22$ | $90.64\pm 0.10$

Note. — The listed parameters are those determined through the joint
differential evolution Markov Chain analysis, including systematic errors in
the stellar evolution models, described in Section 3.3. For all systems the RV
observations are consistent with a circular orbit, and we assume a fixed
circular orbit in generating the parameters listed here.

Table 7: Adopted orbital and planetary parameters for TOI 762 A b, and TIC
46432937 b

| TOI 762 A b | TIC 46432937 b
---|---|---
Parameter | Value | Value
Light curve parameters
$P$ (days) | $3.47168260\pm 0.00000072$ | $1.440445270\pm 0.000000087$
$T_{c}$ (${\rm BJD\\_{}TDB}$) aa Times are in Barycentric Julian Date calculated on the Barycentric Dynamical Time (TDB) system. $T_{c}$: Reference epoch of mid transit that minimizes the correlation with the orbital period. $T_{14}$: total transit duration, time between first to last contact; $T_{12}=T_{34}$: ingress/egress time, time between first and second, or third and fourth contact.  | $2459850.258470\pm 0.000094$ | $2459952.288940\pm 0.000039$
$T_{14}$ (days) aa Times are in Barycentric Julian Date calculated on the Barycentric Dynamical Time (TDB) system. $T_{c}$: Reference epoch of mid transit that minimizes the correlation with the orbital period. $T_{14}$: total transit duration, time between first to last contact; $T_{12}=T_{34}$: ingress/egress time, time between first and second, or third and fourth contact.  | $0.05789\pm 0.00031$ | $0.04981\pm 0.00026$
$T_{12}=T_{34}$ (days) aa Times are in Barycentric Julian Date calculated on the Barycentric Dynamical Time (TDB) system. $T_{c}$: Reference epoch of mid transit that minimizes the correlation with the orbital period. $T_{14}$: total transit duration, time between first to last contact; $T_{12}=T_{34}$: ingress/egress time, time between first and second, or third and fourth contact.  | $0.01877\pm 0.00048$ | $\cdots$
$\phi_{\rm occ}$ (phase) bb Inferred timing of occultation events calculated from the fit where the eccentricity is allowed to vary. Occultations have not been observed for either system. Times are in Barycentric Julian Date calculated on the Barycentric Dynamical Time (TDB) system. $\phi_{\rm occ}$: orbital phase of the occultation. Phase 0 refers to the time of mid transit. $T_{c,{\rm occ}}$: Reference epoch of mid occultation. $T_{14,{\rm occ}}$: total occultation duration.  | $0.513\pm 0.018$ | $0.50064\pm 0.00093$
$T_{c,{\rm occ}}$ (${\rm BJD\\_{}TDB}$) bb Inferred timing of occultation events calculated from the fit where the eccentricity is allowed to vary. Occultations have not been observed for either system. Times are in Barycentric Julian Date calculated on the Barycentric Dynamical Time (TDB) system. $\phi_{\rm occ}$: orbital phase of the occultation. Phase 0 refers to the time of mid transit. $T_{c,{\rm occ}}$: Reference epoch of mid occultation. $T_{14,{\rm occ}}$: total occultation duration.  | $2459876.341\pm 0.062$ | $2460009.1874\pm 0.0013$
$T_{14,{\rm occ}}$ (days) bb Inferred timing of occultation events calculated from the fit where the eccentricity is allowed to vary. Occultations have not been observed for either system. Times are in Barycentric Julian Date calculated on the Barycentric Dynamical Time (TDB) system. $\phi_{\rm occ}$: orbital phase of the occultation. Phase 0 refers to the time of mid transit. $T_{c,{\rm occ}}$: Reference epoch of mid occultation. $T_{14,{\rm occ}}$: total occultation duration.  | $0.05759\pm 0.00067$ | $0.05237\pm 0.00073$
$a/R_{\star}$ | $17.29\pm 0.14$ | $8.381\pm 0.056$
$\zeta/R_{\star}$ cc Reciprocal of the half duration of the transit used as a jump parameter in our MCMC analysis in place of $a/R_{\star}$. It is related to $a/R_{\star}$ by the expression $\zeta/R_{\star}=a/R_{\star}(2\pi(1+e\sin\omega))/(P\sqrt{1-b^{2}}\sqrt{1-e^{2}})$ (Bakos et al., 2010).  |  $47.79\pm 0.30$0 |  $64.73_{-0.89}^{+2.37}$0
$R_{p}/R_{\star}$ | $0.17985\pm 0.00095$ | $0.2301\pm 0.0037$
$b^{2}$ | $0.5710_{-0.0101}^{+0.0086}$ | $0.681_{-0.012}^{+0.026}$
$b\equiv a\cos i/R_{\star}$ | $0.7556_{-0.0067}^{+0.0057}$ | $0.8250_{-0.0073}^{+0.0157}$
$i$ (deg) |  $87.500\pm 0.039$0 |  $84.350_{-0.140}^{+0.090}$0
RV parameters
$K$ ($\rm m\,s^{-1}$) |  $58.1\pm 9.3$00 |  $837.1\pm 4.4$00
$e$ dd The 95% confidence upper limit on the eccentricity determined when $\sqrt{e}\cos\omega$ and $\sqrt{e}\sin\omega$ are allowed to vary in the fit.  | $<0.083$ | $<0.009$
RV jitter ESPRESSO 1ee Term added in quadrature to the formal RV uncertainties for each instrument. This is treated as a free parameter in the fitting routine. The two values listed for TOI 762 A b are for the ESPRESSO P104 and P110 data, respectively.  ($\rm m\,s^{-1}$) | $0.1\pm 4.1$ | $5.3\pm 2.5$
RV jitter ESPRESSO 2ee Term added in quadrature to the formal RV uncertainties for each instrument. This is treated as a free parameter in the fitting routine. The two values listed for TOI 762 A b are for the ESPRESSO P104 and P110 data, respectively.  ($\rm m\,s^{-1}$) | $0.0\pm 2.9$ | $\cdots$
Planetary parameters
$M_{p}$ ($M_{\rm J}$) | $0.251\pm 0.042$ | $3.20\pm 0.11$
$R_{p}$ ($R_{\rm J}$) | $0.744\pm 0.017$ | $1.188\pm 0.030$
$C(M_{p},R_{p})$ ff Correlation coefficient between the planetary mass $M_{p}$ and radius $R_{p}$ estimated from the posterior parameter distribution.  | $0.17$ | $0.53$
$\rho_{p}$ ($\rm g\,cm^{-3}$) | $0.76\pm 0.13$ | $2.37\pm 0.15$
$\log g_{p}$ (cgs) | $3.053\pm 0.072$ | $3.751_{-0.023}^{+0.016}$
$a$ (AU) | $0.03418\pm 0.00065$ | $0.02065\pm 0.00035$
$T_{\rm eq}$ (K) | $555.4\pm 6.4$ | $872\pm 14$
$\Theta$ gg The Safronov number is given by $\Theta=\frac{1}{2}(V_{\rm esc}/V_{\rm orb})^{2}=(a/R_{p})(M_{p}/M_{\star})$ (see Hansen & Barman, 2007).  | $0.0523\pm 0.0085$ | $0.1951\pm 0.0051$
$\log_{10}\langle F\rangle$ (cgs) hh Incoming flux per unit surface area, averaged over the orbit.  | $7.334\pm 0.020$ | $8.119\pm 0.028$

Note. — For both systems we adopt a model in which the orbit is assumed to be
circular. Except where noted otherwise, the listed parameters are calculated
assuming circular orbits. See the discussion in Section 3.3.

Table 8: Adopted limb darkening coefficients for TOI 762 A b, and TIC 46432937
b

| TOI 762 A b | TIC 46432937 b
---|---|---
Parameter | Value | Value
$c_{1},g$ | $0.65\pm 0.14$ | $\cdots$
$c_{2},g$ | $0.09\pm 0.16$ | $\cdots$
$c_{1},r$ | $0.44\pm 0.14$ | $\cdots$
$c_{2},r$ | $0.30\pm 0.17$ | $\cdots$
$c_{1},i$ | $0.23\pm 0.12$ | $\cdots$
$c_{2},i$ | $0.13\pm 0.15$ | $\cdots$
$c_{1},zs$ | $0.13\pm 0.11$ | $\cdots$
$c_{2},zs$ | $0.10\pm 0.12$ | $\cdots$
$c_{1},I+z$ | $0.37\pm 0.15$ | $\cdots$
$c_{2},I+z$ | $0.28_{-0.19}^{+0.14}$ | $\cdots$
$c_{1},Z$ | $\cdots$ | $0.143\pm 0.096$
$c_{2},Z$ | $\cdots$ | $0.15\pm 0.14$
$c_{1},Y$ | $\cdots$ | $0.108_{-0.071}^{+0.097}$
$c_{2},Y$ | $\cdots$ | $0.06\pm 0.12$
$c_{1},J$ | $\cdots$ | $0.142\pm 0.088$
$c_{2},J$ | $\cdots$ | $0.09\pm 0.13$
$c_{1},H$ | $\cdots$ | $0.20\pm 0.11$
$c_{2},H$ | $\cdots$ | $0.08\pm 0.15$
$c_{1},z$–$H$ | $0.050_{-0.037}^{+0.066}$ | $\cdots$
$c_{2},z$–$H$ | $-0.000\pm 0.071$ | $\cdots$
$c_{1},T$ | $0.132\pm 0.091$ | $0.20\pm 0.10$
$c_{2},T$ | $0.02\pm 0.13$ | $0.17\pm 0.14$

Note. — For all systems we adopt a model in which the orbit is assumed to be
circular. See the discussion in Section 3.3. The values listed are for a
quadratic law. The limb darkening parameters were directly varied in the fit,
using the tabulations from Claret et al. (2012, 2013); Claret (2018) to place
Gaussian prior constraints on their values, assuming a prior uncertainty of
$0.2$ for each coefficient.