# TOI-3984 A b and TOI-5293 A b: two temperate gas giants transiting mid-M
dwarfs in wide binary systems

Caleb I. Cañas NASA Postdoctoral Fellow  NASA Goddard Space Flight Center,
8800 Greenbelt Road, Greenbelt, MD 20771, USA  Department of Astronomy &
Astrophysics, The Pennsylvania State University, 525 Davey Laboratory,
University Park, PA 16802, USA  Center for Exoplanets and Habitable Worlds,
The Pennsylvania State University, 525 Davey Laboratory, University Park, PA
16802, USA  Shubham Kanodia Earth and Planets Laboratory, Carnegie Institution
for Science, 5241 Broad Branch Road, NW, Washington, DC 20015, USA  Department
of Astronomy & Astrophysics, The Pennsylvania State University, 525 Davey
Laboratory, University Park, PA 16802, USA  Center for Exoplanets and
Habitable Worlds, The Pennsylvania State University, 525 Davey Laboratory,
University Park, PA 16802, USA  Jessica Libby-Roberts Department of Astronomy
& Astrophysics, The Pennsylvania State University, 525 Davey Laboratory,
University Park, PA 16802, USA  Center for Exoplanets and Habitable Worlds,
The Pennsylvania State University, 525 Davey Laboratory, University Park, PA
16802, USA  Andrea S.J. Lin Department of Astronomy & Astrophysics, The
Pennsylvania State University, 525 Davey Laboratory, University Park, PA
16802, USA  Center for Exoplanets and Habitable Worlds, The Pennsylvania State
University, 525 Davey Laboratory, University Park, PA 16802, USA  Maria
Schutte Homer L. Dodge Department of Physics and Astronomy, University of
Oklahoma, 440 W. Brooks Street, Norman, OK 73019, USA  Luke Powers Department
of Astronomy & Astrophysics, The Pennsylvania State University, 525 Davey
Laboratory, University Park, PA 16802, USA  Center for Exoplanets and
Habitable Worlds, The Pennsylvania State University, 525 Davey Laboratory,
University Park, PA 16802, USA  Sinclaire Jones Department of Astronomy, The
Ohio State University, 4055 McPherson Laboratory, Columbus, OH 43210, USA
Department of Astrophysical Sciences, Princeton University, 4 Ivy Lane,
Princeton, NJ 08540, USA  Andrew Monson Steward Observatory, The University of
Arizona, 933 N. Cherry Avenue, Tucson, AZ 85721, USA  Songhu Wang Department
of Astronomy, Indiana University, 727 East 3rd Street, Bloomington, IN
47405-7105, USA  Guðmundur Stefánsson NASA Sagan Fellow  Department of
Astrophysical Sciences, Princeton University, 4 Ivy Lane, Princeton, NJ 08540,
USA  William D. Cochran McDonald Observatory and Center for Planetary Systems
Habitability, The University of Texas at Austin, Austin, TX 78730, USA  Paul
Robertson Department of Physics & Astronomy, The University of California,
Irvine, Irvine, CA 92697, USA  Suvrath Mahadevan Department of Astronomy &
Astrophysics, The Pennsylvania State University, 525 Davey Laboratory,
University Park, PA 16802, USA  Center for Exoplanets and Habitable Worlds,
The Pennsylvania State University, 525 Davey Laboratory, University Park, PA
16802, USA  ETH Zurich, Institute for Particle Physics & Astrophysics, Zurich,
Switzerland  Adam F. Kowalski National Solar Observatory, University of
Colorado Boulder, 3665 Discovery Drive, Boulder, CO 80303, USA  Department of
Astrophysical and Planetary Sciences, University of Colorado, Boulder, 2000
Colorado Ave, CO 80305, USA  Laboratory for Atmospheric and Space Physics,
University of Colorado Boulder, 3665 Discovery Drive, Boulder, CO 80303, USA
John Wisniewski Department of Physics and Astronomy, George Mason University,
Fairfax, VA 22030, USA  Brock A. Parker Department of Physics & Astronomy,
University of Wyoming, Laramie, WY 82070, USA  Alexander Larsen Department of
Physics & Astronomy, University of Wyoming, Laramie, WY 82070, USA  Franklin
A.L. Chapman  Department of Physics & Astronomy, University of Wyoming,
Laramie, WY 82070, USA  Henry A. Kobulnicky Department of Physics & Astronomy,
University of Wyoming, Laramie, WY 82070, USA  Arvind F. Gupta Department of
Astronomy & Astrophysics, The Pennsylvania State University, 525 Davey
Laboratory, University Park, PA 16802, USA  Center for Exoplanets and
Habitable Worlds, The Pennsylvania State University, 525 Davey Laboratory,
University Park, PA 16802, USA  Mark E. Everett NSF’s National Optical-
Infrared Astronomy Research Laboratory, 950 N. Cherry Ave., Tucson, AZ 85719,
USA  Bryan Edward Penprase Soka University of America, Aliso Viejo CA, 92656,
USA  Gregory Zeimann Hobby Eberly Telescope, University of Texas at Austin,
Austin, TX, 78712, USA  Corey Beard Department of Physics & Astronomy, The
University of California, Irvine, Irvine, CA 92697, USA  Chad F. Bender
Steward Observatory, The University of Arizona, 933 N. Cherry Avenue, Tucson,
AZ 85721, USA  Knicole D. Colón NASA Goddard Space Flight Center, 8800
Greenbelt Road, Greenbelt, MD 20771, USA  Scott A. Diddams Electrical,
Computer & Energy Engineering, 425 UCB, University of Colorado, Boulder, CO
80309, USA  Department of Physics, 390 UCB, University of Colorado, Boulder,
CO 80309, USA  National Institute of Standards & Technology, 325 Broadway,
Boulder, CO 80305, USA  Connor Fredrick National Institute of Standards &
Technology, 325 Broadway, Boulder, CO 80305, USA  Department of Physics, 390
UCB, University of Colorado, Boulder, CO 80309, USA  Samuel Halverson Jet
Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove
Drive, Pasadena, California 91109  Joe P. Ninan Department of Astronomy and
Astrophysics, Tata Institute of Fundamental Research, Homi Bhabha Road,
Colaba, Mumbai 400005, India  Lawrence W. Ramsey Department of Astronomy &
Astrophysics, The Pennsylvania State University, 525 Davey Laboratory,
University Park, PA 16802, USA  Center for Exoplanets and Habitable Worlds,
The Pennsylvania State University, 525 Davey Laboratory, University Park, PA
16802, USA  Arpita Roy Space Telescope Science Institute, 3700 San Martin
Drive, Baltimore, MD 21218, USA  Department of Physics and Astronomy, Johns
Hopkins University, 3400 N Charles St, Baltimore, MD 21218, USA  Christian
Schwab School of Mathematical and Physical Sciences, Macquarie University,
Balaclava Road, North Ryde, NSW 2109, Australia

###### Abstract

We confirm the planetary nature of two gas giants discovered by TESS to
transit M dwarfs with stellar companions at wide separations. TOI-3984 A
($J=11.93$) is an M4 dwarf hosting a short-period ($4.353326\pm 0.000005$
days) gas giant ($M_{p}=0.14\pm 0.03~{}\mathrm{M_{J}}$ and $R_{p}=0.71\pm
0.02~{}\mathrm{R_{J}}$) with a wide separation white dwarf companion. TOI-5293
A ($J=12.47$) is an M3 dwarf hosting a short-period ($2.930289\pm 0.000004$
days) gas giant ($M_{p}=0.54\pm 0.07~{}\mathrm{M_{J}}$ and $R_{p}=1.06\pm
0.04~{}\mathrm{R_{J}}$) with a wide separation M dwarf companion. We
characterize both systems using a combination of ground-based and space-based
photometry, speckle imaging, and high-precision radial velocities from the
Habitable-zone Planet Finder and NEID spectrographs. TOI-3984 A b
($T_{eq}=563\pm 15$ K and $\mathrm{TSM}=138_{-27}^{+29}$) and TOI-5293 A b
($T_{eq}=675_{-30}^{+42}$ K and $\mathrm{TSM}=92\pm 14$) are two of the
coolest gas giants among the population of hot Jupiter-sized gas planets
orbiting M dwarfs and are favorable targets for atmospheric characterization
of temperate gas giants and three-dimensional obliquity measurements to probe
system architecture and migration scenarios.

††facilities: ARC (ARCTIC), Exoplanet Archive, Gaia, HET (HPF, LRS2), Las
Campanas Observatory (TMMT, LCRO), LAMOST, MAST, PO:1.2m (ZTF), PO:1.5m (ZTF),
TESS, WIYN (NESSI)††software:  astroquery (Ginsburg et al., 2019), astropy
(Astropy Collaboration et al., 2018), barycorrpy (Kanodia & Wright, 2018),
dynesty (Speagle, 2020), eleanor (Feinstein et al., 2019), EXOFASTv2 (Eastman
et al., 2019), GLS (Zechmeister & Kürster, 2009), HPF-SpecMatch, juliet
(Espinoza et al., 2019), lightkurve (Lightkurve Collaboration et al., 2018),
LRS2Multi, matplotlib (Hunter, 2007), numpy (van der Walt et al., 2011),
Panacea, pandas (McKinney, 2010), PyHammer (Roulston et al., 2020), scipy
(Virtanen et al., 2020), telfit (Gullikson et al., 2014), thejoker (Price-
Whelan et al., 2017), TLS (Hippke & Heller, 2019), WD_models (Cheng et al.,
2019)

##  1 Introduction

Hot Jupiters (defined as $P<10$ days and $R_{p}\gtrsim
8~{}\mathrm{R_{\oplus}}$ in this work) are a rare class of exoplanets with an
occurrence rate of $\lesssim 1\%$ (see Beleznay & Kunimoto, 2022, and
references therein) around Sun-like stars measured through radial velocity
(RV) surveys (e.g., Cumming et al., 2008; Mayor et al., 2011; Wright et al.,
2012) and photometric surveys (e.g., Howard et al., 2012; Petigura et al.,
2018; Zhou et al., 2019). While the exact formation process is unknown (see
the review by Dawson & Johnson, 2018), the most promising channels for origins
are in-situ formation (e.g., Boley et al., 2016; Batygin et al., 2016) or ex-
situ formation with gas disk migration (e.g., Lin et al., 1996) or high
eccentricity tidal migration (e.g., Rasio & Ford, 1996; Weidenschilling &
Marzari, 1996; Ford & Rasio, 2008; Petrovich, 2015a).

Less is known about the formation of hot Jupiters orbiting M dwarfs, which
should be difficult to form under the process of core-accretion (e.g.,
Laughlin et al., 2004; Ida & Lin, 2005; Kennedy & Kenyon, 2008). Surveys of M
dwarfs have revealed that: (i) small ($R_{p}<4\mathrm{~{}R_{\oplus}}$) planets
on short-period orbits ($P<200$ days) are more common around M dwarfs than
Sun-like stars (e.g., Dressing & Charbonneau, 2015; Mulders et al., 2015;
Hardegree-Ullman et al., 2019; Hsu et al., 2020) and (ii) low-mass planets
($1\mathrm{~{}M_{\oplus}}<M_{p}<10\mathrm{~{}M_{\oplus}}$) are more frequently
found orbiting later type M dwarfs on short-period orbits ($P<200$ days)
(e.g., Bonfils et al., 2013; Tuomi et al., 2014, 2019; Sabotta et al., 2021;
Pinamonti et al., 2022). The majority of the aforementioned surveys have not
detected any appreciable number of hot Jupiters and can only place upper
limits of $\sim 2\%$ on the intrinsic occurrence rate.

A tighter constraint on the occurrence of hot Jupiters orbiting M dwarfs has
been made possible through an analysis of M dwarfs observed in the primary
TESS mission (Ricker et al., 2015). Gan et al. (2023) report an occurrence
rate of $0.27\pm 0.09\%$ for hot Jupiters orbiting early M dwarfs from an
analysis of 60819 M dwarfs with $10.5<T<13.5$ spanning
$0.45~{}M_{\odot}<M_{\star}<0.65~{}M_{\odot}$. Bryant et al. (2023)
independently derived an occurrence rate of $0.194\pm 0.072\%$ from an
analysis of 91306 M dwarfs spanning
$0.088~{}M_{\odot}<M_{\star}<0.71~{}M_{\odot}$. At present, these values are
smaller than occurrence rates for hot Jupiters orbiting Sun-like stars,
although they are consistent within $1-3\sigma$. An analysis of a larger
sample of TESS M dwarfs is needed to refine the occurrence rate and determine
whether the M dwarf gas giant population is consistent with the population
orbiting Sun-like stars.

In this paper, we confirm the planetary nature of two gas giants transiting
the M dwarfs TOI-3984 A ($J=11.93$, $T=13.46$) and TOI-5293 A ($J=12.47$,
$T=13.98$). We characterize these system using space and ground-based
photometry, speckle imaging, and precision RVs obtained with the Habitable-
zone Planet Finder (HPF; Mahadevan et al., 2012, 2014) and NEID (Schwab et
al., 2016; Halverson et al., 2016) spectrographs. We derive stellar parameters
for the host stars with HPF spectra and jointly model the photometry and RVs
to confirm the planetary nature of TOI-3984 A b and TOI-5293 A b.

This paper is structured as follows: Section 2 presents the photometric,
imaging, and spectroscopic observations. The best estimates of the stellar
parameters and properties are presented in Section 3. The modeling and
analysis of the photometry and RVs are presented in Section 4. Section 5
provides further discussion of the nature of these planets and the feasibility
for future study. A summary of our key results is presented in Section 6.

##  2 Observations

###  2.1 TESS

TESS (Ricker et al., 2015) observed (i) TOI-3984 A (TIC 20182780, Gaia DR3
1291955578869575552) in long-cadence mode (30-min cadence) during Sectors
$23-24$ (2020 March 18 through 2020 May 13) and in short-cadence mode (2-min
cadence) during Sectors $50-51$ (2022 April 22 through 2022 May 18) and (ii)
TOI-5293 A (TIC 250111245, Gaia DR3 2640121486388076032) in long-cadence mode
during Sector 42 (2021 August 20 through 2021 September 16). Each star has one
planet candidate, TOI-3984.01 and TOI-5293.01, that was identified by the
“quick-look pipeline” (QLP111https://tess.mit.edu/qlp/; Huang et al., 2020a,
b) as part of a search for planet candidates orbiting stars with a TESS
magnitude of $T>12$ (Kunimoto et al., 2022).

We extract the long-cadence photometry from the TESS full-frame images using
eleanor (Feinstein et al., 2019) to process a cut-out of $31\times 31$ pixels
from the calibrated full-frame images centered on each target. The light
curves from eleanor222https://github.com/afeinstein20/eleanor  use an aperture
which minimizes the hour-binned combined differential photometric precision
(CDPP; see Jenkins et al., 2010) to ensure that sharp features on timescales
of a few hours, such as transits, are preserved. Figures 1 and 2 present the
apertures used to extract the TESS light curves for TOI-3984 A and TOI-5293 A,
respectively. In each figure, panel (a) presents the apertures used to derive
the light curve from the latest TESS sector. In panel (b), the region
contained in an 11x11 pixel subgrid and the photometric apertures from all
TESS sectors are overlaid on images from the Zwicky Transient Facility (ZTF;
Bellm et al., 2019; Graham et al., 2019; Masci et al., 2019).

TOI-5293 A has a nearby companion in the optimal aperture (TIC 2052711961,
$T=18.44$, $\Delta~{}\mathrm{G_{RP}}=2.6$) at a separation of $3.57\arcsec$.
The second release of the Pan-STARRS survey (PS1; Chambers et al., 2016;
Magnier et al., 2020) measures a PS1 $i^{\prime}=17.15$ for the nearby
companion, which is too faint to contribute to any significant dilution in the
photometry obtained with TESS or small ground-based telescopes described in
Sections 2.2.1, 2.2.4, and 2.2.5.

Figure 1: (a) The $11\times 11$ TESS target pixel cutout from Sector 51
centered on TOI-3984 A (marked as a star). Stars identified in Gaia DR3 with
magnitudes $\Delta\mathrm{G_{RP}}<4$ are marked with diamond stars for
reference. (b) Overlay of the region contained in an $11\times 11$ pixel grid
centered on TOI-3984 A from all sectors (green polygon), the region contained
within all SPOC (red polygon) and eleanor (blue polygon) photometric
apertures, and other comparably bright stars on a ZTF $zi$ image. Figure 2:
Identical to Figure 1 but for TOI-5293 A. (a) The $31\times 31$ TESS target
pixel cutout centered around TOI-5293 A (marked as a star). Stars identified
in Gaia EDR3 having magnitudes $\Delta\mathrm{G_{RP}}<4$ are marked with
diamond stars. (b) Overlay of the TESS $11\times 11$ pixel subgrid (green
polygon), the region contained in the photometric aperture (red polygon), and
other comparably bright stars on a ZTF $zi$ image.

The long-cadence light curves analyzed in this work are the CORR_FLUX values
produced by eleanor. The combined differential photometric precision is the
value minimized by eleanor when searching for the best aperture and is 5661
ppm for TOI-3984 A and 7089 ppm for TOI-5293 A. Observations where the
background flux exceeds a threshold (FLUX_BKG $>2.5\times$CORR_FLUX) or with
non-zero data quality flags (see a detailed description in Table 28 in
Tenenbaum & Jenkins, 2018) are excluded from further analysis.

Figure 3: (a) The median normalized TESS long-cadence light curve for TOI-3984
A derived with eleanor. The solid green line is the best-fitting Gaussian
process model used to detrend the light curve. The triangles indicate the
observed transits. (b) Identical to (a) but for short-cadence TESS data. (c)
and (d) are the phase-folded light curves for short and long cadence TESS
data. In (c) and (d), the best-fitting model from the joint fit to the
photometry and RVs is plotted as a dashed line while the shaded regions denote
the $1\sigma$ (darkest), $2\sigma$, and $3\sigma$ (lightest) extent of the
model posteriors. The modeling of the photometry and RVs is described in
detail in Section 4.

The short-cadence photometry of TOI-3984 A is obtained from the pre-search
data-conditioned simple aperture photometry (PDCSAP; Jenkins et al., 2016)
light curves available at the Mikulski Archive for Space Telescopes
(MAST)333https://archive.stsci.edu/prepds/tess-data-alerts/. The PDCSAP
photometry is corrected for instrumental systematics and dilution from other
objects contained within the aperture using algorithms developed for the
Kepler mission (see Stumpe et al., 2012; Smith et al., 2012). As with the
long-cadence photometry, observations with non-zero data quality flags are
excluded from further analysis. We perform no additional outlier rejection
beyond the data quality flags and application of a background threshold value
for long-cadence data. Figures 3 and 4 display the TESS light curves.

Figure 4: Identical to Figure 3, but for TOI-5293 A. (a) The median normalized
TESS light curve for TOI-5293 A derived with eleanor along with the best-
fitting Gaussian process model. The triangles indicate the observed transits.
(b)$-$(g) are the light curves for the TESS, RBO, LCRO, and TMMT plotted with
model posteriors (shaded regions) from the joint fit to the photometry and
RVs.

###  2.2 Ground-based photometry

We observed 6 transits of TOI-3984 A and 5 transits of TOI-5293 A using a
combination of five separate ground-based facilities. All ground-based light
curves were derived using AstroImageJ (Collins et al., 2017). Following the
methodology in Stefánsson et al. (2017), the estimated scintillation noise was
included in the flux uncertainty. The light curves are not detrended with any
external parameter (e.g., airmass or time). The observations are described in
detail below and summarized in Table 1. The ground-based photometry for
TOI-5293 A and TOI-3984 A are presented in Figures 4 and 5, respectively.

Table 1: Summary of ground-based photometry  Civil Night | Facility | Filter | Exposure Time | Aperture and sky annulus
---|---|---|---|---
|  |  | (s) | (pixels and ″)
TOI-3984 A:
2021-07-18 | RBO | Bessell I | 240 | 10/20/30 ($7.3/14.6/21.9\arcsec$)
2021-08-13 | RBO | Bessell I | 240 | 10/20/30 ($7.3/14.6/21.9\arcsec$)
2021-08-13 | Soka | Sloan $i^{\prime}$ | 300 | 10/15/20 ($7.0/10.5/14.0\arcsec$)
2022-02-03 | RBO | Bessell I | 240 | 10/20/30 ($7.3/14.6/21.9\arcsec$)
2022-03-23 | ARCTIC | Sloan $i^{\prime}$ | 45 | 20/30/40 ($9.1/13.7/18.2\arcsec$)
2022-05-10 | ARCTIC | Sloan $z^{\prime}$ | 30 | 12/20/30 ($5.5/9.1/13.7\arcsec$)
TOI-5293 A:
2022-07-27 | RBO | Bessell I | 240 | 10/20/30 ($7.3/14.6/21.9\arcsec$)
2022-09-12 | RBO | Bessell I | 240 | 10/20/30 ($7.3/14.6/21.9\arcsec$)
2022-10-23 | LCRO | Sloan $i^{\prime}$ | 420 | 6/10/15 ($4.6/7.7/11.6\arcsec$)
2022-02-03 | TMMT | Bessell I | 300 | 6/10/15 ($7.2/11.9/17.9\arcsec$)
2022-03-23 | TMMT | Bessell I | 180 | 6/10/15 ($7.2/11.9/17.9\arcsec$)

####  2.2.1 RBO 0.6 m telescope

The 0.6 m telescope at the Red Buttes Observatory (RBO) in Wyoming (Kasper et
al., 2016) is a f/8.43 Ritchey-Cretien constructed by DFM Engineering, Inc.
and equipped with an Apogee Alta F16M camera. We observed (i) TOI-3984 A on
the nights of 2021 July 18, 2021 August 13, and 2022 February 03 and (ii)
TOI-5293 A on the nights of 2022 July 27 and 2022 September 12. All
observations were moderately defocused, used an exposure time of 240s, and
operated with the Bessell I filter (Bessell, 1990) and the $2\times 2$ on-chip
binning mode, which provides a gain of 1.39 $\mathrm{e^{-}/ADU}$, a plate
scale of $0.731\arcsec/\mathrm{pixel}$, and a readout time of $\sim 2.4$s.

Figure 5: Identical to Figure 3, but displaying the ground-based photometry
for TOI-3984 A. (a)$-$(f) are the light curves from RBO, Soka, and ARCTIC
plotted with model posteriors (shaded regions) from the joint fit to the
photometry and RVs.

####  2.2.2 SUA Nieves 0.3 m telescope

The 0.3 m Planewave CDK14 telescope located at the Luis and Linda Nieves
Observatory444https://sites.soka.edu/SUO/about/ at Soka University of America
in Aliso Viejo, CA uses a FLI Proline 16803 camera with a $47\times 47\arcmin$
field-of-view and a quantum efficiency $>50\%$. We observed TOI-3984 A on the
night of 2021 August 13 in the Sloan $i^{\prime}$ filter using a 300s exposure
time and $1\times 1$ binning mode, which provides a plate scale of
$0.7\arcsec/\mathrm{pixel}$.

####  2.2.3 APO 3.5m telescope

We used the 3.5m Astrophysical Research Consortium (ARC) Telescope Imaging
Camera (ARCTIC; Huehnerhoff et al., 2016) on the ARC 3.5m Telescope at Apache
Point Observatory (APO) to obtain transits of TOI-3984 A on the nights of 2022
March 23 and 2022 May 10. The observation on 2022 March 23 were performed in
the Sloan $i^{\prime}$ filter using an engineered diffuser (Stefánsson et al.,
2017) with an exposure time of 45 s. ARCTIC was operated in the quad-amplifier
and fast readout modes using the $4\times 4$ on-chip binning mode to achieve a
gain of 2 $\mathrm{e^{-}/ADU}$, a plate scale of
$0.456\mathrm{\arcsec/pixel}$, and a readout time of 1.3 s. The observation on
2022 May 10 was performed in the Sloan $z^{\prime}$ filter slightly out of
focus with an exposure time of 30s. Hardware issues prevented the use of quad
mode and instead ARCTIC was operated in the single-amplifier and fast readout
modes using the $4\times 4$ on-chip binning mode with a readout time of 11 s.

####  2.2.4 LCO 0.3m TMMT telescope

The robotic Three-hundred MilliMeter Telescope (TMMT; Monson et al., 2017) at
Las Campanas Observatory (LCO) is a f/7.8 FRC300 from Takahashi on a German
equatorial AP1600 GTO mount with an Apogee Alta U42-D09 CCD Camera, FLI ATLAS
focuser, and Centerline filter wheel. We observed TOI-5293 A on the nights of
2022 October 23 and 2022 October 26. The observations were performed slightly
out of focus in the Bessell $I$ filter (Bessell, 1990) with an exposure time
of 300s and 180s, respectively, while operating in a 1x1 binning mode. In this
mode, TMMT has a gain of 1.35 $\mathrm{e^{-}/ADU}$, a plate scale of
$1.194\arcsec/\mathrm{pixel}$, and a readout time of $\sim 6$s.

####  2.2.5 LCRO 0.3m telescope

The 305 mm Las Campanas Remote Observatory
telescope555http://lcobot.duckdns.org/ (LCRO) at LCO observed TOI-5293 A on
the night of 2022 October 23. LCRO is an f/8 Maksutov-Cassegrain from Astro-
Physics on a German Equatorial AP1600 GTO mount with an FLI Proline 16803 CCD
Camera, FLI ATLAS focuser and Centerline filter wheel. The observations were
performed slightly out of focus in the Sloan $i^{\prime}$ filter with an
exposure time of 420s. We used the $1\times 1$ binning mode, providing a gain
of 1.52 $\mathrm{e^{-}/ADU}$, a plate scale of $0.773\arcsec/\mathrm{pixel}$,
and a readout time of 17s.

###  2.3 High-contrast imaging

NESSI (Scott et al., 2018) is a dual-channel speckle imager on the WIYN 3.5m
Telescope at Kitt Peak National Observatory (KPNO). Both TOI-3984 A and
TOI-5293 A were observed on 2022 April 18 using NESSI. The faintness of these
targets ($r^{\prime}>14$) prevented observations in the narrow filters that
NESSI traditionally uses, while hardware issues during the observing run only
allowed for observing with the redder channel in the Sloan $z^{\prime}$
filter. The images were reconstructed following the procedures described in
Howell et al. (2011).

Figures 6 and 7 display the $5\sigma$ contrast curve with insets of the
reconstructed NESSI speckle images. The NESSI data reveal no bright ($\Delta
z^{\prime}<3$) companions and no potential sources of dilution at separations
of $0.2-1.2\arcsec$ from the host stars. These angular limits correspond to
projected spatial limits of $22-130$ au for TOI-3984 A and $32-193$ au for
TOI-5293 A. AO imaging of TOI-3984 A (see Gan et al., 2023) obtained with the
Palomar High Angular Resolution Observer on the Palomar-5.1m telescope
(Hayward et al., 2001) also reveal it is an isolated star having a contrast of
$\Delta\textrm{mag}>6.1$ at distances $>0.5\arcsec{}$ ($>54$ au) from the star
in the near-infrared ($1-2.5$ µm).

Figure 6: The $5\sigma$ contrast curves for TOI-3984 A obtained from speckle
imaging with NESSI in the Sloan $z^{\prime}$ filter. The data reveal no bright
companions at separations of $0.2\arcsec{}-1.75\arcsec{}$. The inset is the
$4.7\arcsec{}\times 4.7\arcsec{}$ NESSI speckle image centered on TOI-3984 A
in the Sloan $z^{\prime}$ filter. Figure 7: Similar to Figure 6 but for
TOI-5293 A. The $5\sigma$ contrast curve for TOI-5293 A obtained from speckle
imaging using NESSI along with an inset displaying the $4.7\arcsec{}\times
4.7\arcsec{}$ reconstructed speckle image in the Sloan $z^{\prime}$ filter.

###  2.4 High-precision spectroscopy

####  2.4.1 HPF spectrograph

HPF (Mahadevan et al., 2012, 2014) is a high-resolution ($R\sim 55,000$),
fiber-fed (Kanodia et al., 2018), temperature stabilized (Stefánsson et al.,
2016), near-infrared ($\lambda\sim 8080-12780$ Å) spectrograph located on the
10m Hobby-Eberly Telescope at McDonald Observatory (HET; Ramsey et al., 1998;
Hill et al., 2021). Observations are executed in a queue by the HET resident
astronomers (Shetrone et al., 2007). We obtained (i) 35 visits of TOI-3984 A
between 2021 August 23 and 2022 May 10 and (ii) 16 visits of TOI-5293 A
between 2022 September 10 and 2022 December 01 with median signal-to-noise
ratios (S/N) per 1D extracted pixel at 1070nm of 38 and 35, respectively.

The HxRGproc tool666https://github.com/indiajoe/HxRGproc  (Ninan et al., 2018)
processed the raw HPF data and performed bias noise removal, nonlinearity
correction, cosmic-ray correction, and slope/flux and variance image
calculations. The 1D spectra were extracted following the procedures in Ninan
et al. (2018), Kaplan et al. (2019), and Metcalf et al. (2019). The wavelength
solution and drift correction were extrapolated using laser frequency comb
(LFC) frames obtained from routine calibrations, which enable wavelength
calibration on the order of $<30~{}\mathrm{cm~{}s^{-1}}$ (see Appendix A in
Stefánsson et al., 2020a).

The RVs were calculated using a modified version of the SpEctrum Radial
Velocity AnaLyser code (SERVAL; Zechmeister et al., 2018) optimized for HPF RV
extractions (see Metcalf et al. (2019) and Stefánsson et al. (2020a) for
details). SERVAL uses the template-matching technique to derive RVs (e.g.,
Anglada-Escudé & Butler, 2012) and creates a master template from the
observations by minimizing the $\chi^{2}$ statistic. The master template is
generated from the observed spectra after masking sky-emission lines and
telluric regions identified using a synthetic telluric-line mask generated
from telfit (Gullikson et al., 2014). The barycentric correction is calculated
using barycorrpy (Kanodia & Wright, 2018). Table 2 reports the HPF RVs, the
$1\sigma$ uncertainties, and the S/N per pixel for TOI-3984 A and TOI-5293 A.
Figure 8 presents the RVs for TOI-3984 A and Figure 9 presents the RVs for
TOI-5293 A.

####  2.4.2 NEID spectrograph

NEID (Schwab et al., 2016; Halverson et al., 2016) is an environmentally
stabilized (Stefánsson et al., 2016; Robertson et al., 2019), high-resolution
($R\sim 110,000$), fiber-fed (Kanodia et al., 2018) spectrograph with an
extended red wavelength coverage ($\lambda\sim 3800-9300$ Å) installed on the
WIYN 3.5m telescope at KPNO in Arizona. We obtained 6 visits of TOI-3984 A
between 2022 March 13 and 2022 April 09 in high-resolution mode with a median
S/N per 1D extracted pixel of 12 at 850nm. The NEID data were reduced using
the NEID Data Reduction Pipeline777https://neid.ipac.caltech.edu/docs/NEID-
DRP/ (DRP), and the Level-2 1D extracted spectra were retrieved from the NEID
Archive888https://neid.ipac.caltech.edu/. We measured the RVs using a modified
version of the SERVAL code (see Stefánsson et al., 2022) and extracted RVs
using the wavelength range from $5440-8920$ Å (order indices $61-104$) and the
inner most 3000 pixels of each order (similar to Cañas et al., 2022). Figure 8
presents the RVs while Table 2 reports the NEID RVs, the $1\sigma$
uncertainties, and the S/N per pixel.

Table 2: Radial velocities.

$\mathrm{BJD_{TDB}}$ | RV | $\sigma$ | S/Na | Instrument
---|---|---|---|---
| $(\mathrm{m~{}s^{-1}})$ | $(\mathrm{m~{}s^{-1}})$ |  |
TOI-3984 A:
2459449.620693 | 0.5 | 18.2 | 49 | HPF
2459451.620722 | 36.5 | 20.4 | 45 | HPF
2459452.603981 | 2.5 | 19.5 | 47 | HPF
2459574.028699 | $-78.2$ | 89.8 | 34 | HPF
2459576.029258 | $-83.2$ | 43.0 | 34 | HPF
2459578.027929 | $-45.0$ | 29.6 | 34 | HPF
2459584.013323 | 49.1 | 27.5 | 36 | HPF
2459597.974678 | 21.8 | 31.4 | 31 | HPF
2459598.978176 | $-18.4$ | 33.4 | 31 | HPF
2459602.968624 | $-18.0$ | 24.4 | 38 | HPF
2459604.966266 | 15.5 | 22.1 | 43 | HPF
2459605.946813 | 9.3 | 24.7 | 38 | HPF
2459606.947872 | $-50.6$ | 22.7 | 41 | HPF
2459623.909836 | $-56.8$ | 25.1 | 37 | HPF
2459624.911946 | $-41.7$ | 23.8 | 40 | HPF
2459626.894567 | 2.8 | 28.0 | 40 | HPF
2459629.887569 | $-9.2$ | 24.5 | 40 | HPF
2459649.837986 | 63.6 | 33.7 | 40 | HPF
2459651.825176 | $-35.1$ | 33.4 | 40 | HPF
2459658.810776 | $-24.1$ | 24.7 | 40 | HPF
2459677.769462 | 69.8 | 33.9 | 36 | HPF
2459677.984824 | $-10.6$ | 26.6 | 36 | HPF
2459678.752641 | 82.2 | 26.6 | 36 | HPF
2459681.740411 | $-5.2$ | 32.6 | 35 | HPF
2459681.974403 | 2.1 | 27.1 | 35 | HPF
2459682.749492 | $-37.4$ | 29.1 | 35 | HPF
2459683.749510 | 44.2 | 25.2 | 38 | HPF
2459683.967863 | 63.7 | 29.0 | 33 | HPF
2459684.983417 | 2.2 | 33.7 | 30 | HPF
2459685.974345 | $-33.1$ | 42.6 | 38 | HPF
2459686.729534 | 7.7 | 25.5 | 38 | HPF
2459687.723023 | 56.4 | 25.7 | 38 | HPF
2459687.962405 | 33.6 | 18.9 | 49 | HPF
2459691.942885 | 1.5 | 27.9 | 35 | HPF
2459709.673097 | 64.5 | 20.1 | 47 | HPF
2459651.891110 | $-60.1$ | 12.4 | 12 | NEID
2459656.839623 | $-57.8$ | 9.7 | 14 | NEID
2459663.895380 | $-76.5$ | 7.1 | 19 | NEID
2459664.861500 | $-71.9$ | 11.5 | 12 | NEID
2459671.889748 | $-68.6$ | 14.7 | 10 | NEID
2459678.892545 | $-29.9$ | 9.6 | 15 | NEID
TOI-5293 A:
2459846.735412 | 179.6 | 37.6 | 28 | HPF
2459852.721876 | 195.1 | 40.3 | 27 | HPF
2459853.718534 | $-3.3$ | 27.9 | 36 | HPF
2459856.709588 | $-113.7$ | 30.2 | 35 | HPF
2459856.779480 | $-104.9$ | 26.7 | 38 | HPF
2459864.686793 | 52.8 | 28.8 | 37 | HPF
2459865.683293 | $-98.5$ | 29.9 | 35 | HPF
2459873.660887 | 33.8 | 25.7 | 39 | HPF
2459877.648474 | $-82.7$ | 40.8 | 26 | HPF
2459879.639891 | 11.6 | 26.2 | 38 | HPF
2459880.642541 | $-109.4$ | 46.9 | 23 | HPF
2459882.640435 | 48.1 | 28.5 | 35 | HPF
2459885.630585 | $-10.3$ | 24.6 | 41 | HPF
2459890.687347 | 138.9 | 25.4 | 39 | HPF
2459893.611705 | 44.1 | 37.3 | 29 | HPF
2459907.568470 | 56.2 | 30.1 | 34 | HPF

aafootnotetext: The S/N is the median value per 1D extracted pixel at 1070nm
for HPF and 850nm for NEID. HPF observations use an exposure time of 1890s.
NEID observations use an exposure time of 1800s.

Figure 8: (a) shows the RVs for TOI-3984 A derived with modified versions of
SERVAL. (b) displays the phase-folded RVs plotted with model posteriors and
the residuals to the fit. In (a) and (b), the dashed line is the best-fitting
Keplerian model. The shaded regions denote the $1\sigma$ (darkest), $2\sigma$,
and $3\sigma$ (lightest) extent of the model posteriors. The modeling is
described in Section 4. Figure 9: Identical to Figure 8, but for TOI-5293 A.
(a) shows the HPF RVs. (b) displays the phase-folded RVs plotted with model
posteriors.

##  3 Stellar Parameters

###  3.1 Spectroscopic parameters

The stellar effective temperature ($T_{e}$), surface gravity ($\log
g_{\star}$), and metallicity ([Fe/H]) were calculated using the HPF-
SpecMatch999https://gummiks.github.io/hpfspecmatch  package (Stefánsson et
al., 2020a), which derives stellar parameters using the empirical template
matching methodology discussed in Yee et al. (2017). It uses a two-step
$\chi^{2}$ minimization to identify the five best-matching spectra from a
library of well-characterized stars and derives spectroscopic parameters using
a weighted, linear combination of the library stars. HPF-SpecMatch derives
$v\sin i_{\star}$ by broadening the library spectra using a linear limb
darkening law (Gray, 2005). The reported uncertainties are the standard
deviation of the residuals from a leave-one-out cross-validation procedure
applied to the entire spectral library.

The HPF spectral library contains 166 stars and spans the following parameter
space: $2700~{}\mathrm{K}<T_{e}<6000~{}\mathrm{K}$, $4.3<\log g_{\star}<5.3$,
and $-0.5<\mathrm{[Fe/H]}<0.5$. The library includes 87 M dwarfs ($T_{e}\leq
4000$ K) of which 40 are M dwarfs spanning $3300\mathrm{K}\leq T_{e}\leq
3700~{}\mathrm{K}$, $4.66<\log g_{\star}<4.97$, and
$-0.5<\mathrm{[Fe/H]}<0.4$. The spectral matching was performed using HPF
order index 5 ($8534-8645$ Å) due to minimal telluric contamination. TOI-3984
A is determined to have $T_{e}=3476\pm 88$ K, $\log g_{\star}=4.81\pm 0.05$,
and $\mathrm{[Fe/H]=0.18\pm 0.12}$. TOI-5293 A is determined to have
$T_{e}=3586\pm 88$ K, $\log g_{\star}=4.77\pm 0.05$, and
$\mathrm{[Fe/H]=-0.03\pm 0.12}$. The resolution limit of HPF ($R\sim 55,000$)
places a constraint of $v\sin i<2\mathrm{~{}km~{}s^{-1}}$ for both TOI-3984 A
and TOI-5293 A. Table 3.1 presents the derived spectroscopic parameters with
their uncertainties.

Table 3: Summary of stellar parameters.

Parameter | Description | TOI-3984 A | TOI-5293 A | Reference
---|---|---|---|---
Main identifiers: |
TIC |  $\cdots$ | 20182780 | 250111245 | TIC
Gaia DR3 |  $\cdots$ | 1291955578869575552 | 2640121486388076032 | Gaia DR3
Coordinates, proper motion, distance, maximum extinction, and spectral type: |
$\alpha_{\mathrm{J2016}}$ | Right Ascension (RA) | 15:05:20.91 | 23:43:18.88 | Gaia DR3
$\delta_{\mathrm{J2016}}$ | Declination (Dec) | 36:47:13.14 | -02:02:42.33 | Gaia DR3
$\mu_{\alpha}$ | Proper motion (RA, mas yr-1) | $-48.95\pm 0.02$ | $-17.13\pm 0.03$ | Gaia DR3
$\mu_{\delta}$ | Proper motion (Dec, mas yr-1) | $42.65\pm 0.02$ | $0.49\pm 0.02$ | Gaia DR3
$l$ | Galactic longitude | 60.26769 | $87.00510$ | Gaia DR3
$b$ | Galactic latitude | $60.17839$ | $-60.07625$ | Gaia DR3
$d$ | Geometric distance (pc) | $108.4_{-0.2}^{+0.3}$ | $160.8\pm 0.6$ | Bailer-Jones
$A_{V,max}$ | Maximum visual extinction | $0.04$ | $0.06$ | Green
Spectral type |  $\cdots$ | M$4\pm 0.5$ | M$3\pm 1$ | LAMOST/This worka
Broadband photometric magnitudes: |
$B$ | Johnson $B$ mag |  $\cdots$ | $17.4\pm 0.2$ | APASS
$V$ | Johnson $V$ mag |  $\cdots$ | $15.9\pm 0.2$ | APASS
$g^{\prime}$ | Pan-STARRS1 $g^{\prime}$ mag | $16.35\pm 0.02$ | $16.496\pm 0.007$ | PS1
$r^{\prime}$ | Pan-STARRS1 $r^{\prime}$ mag | $15.18\pm 0.01$ | $15.29\pm 0.01$ | PS1
$i^{\prime}$ | Pan-STARRS1 $i^{\prime}$ mag | $13.96\pm 0.01$ | $14.310\pm 0.008$ | PS1
$z^{\prime}$ | Pan-STARRS1 $z^{\prime}$ mag | $13.41\pm 0.02$ | $13.87\pm 0.01$ | PS1
$y^{\prime}$ | Pan-STARRS1 $y^{\prime}$ mag | $13.15\pm 0.02$ | $13.657\pm 0.008$ | PS1
$J$ |  $J$ mag | $11.93\pm 0.02$ | $12.47\pm 0.03$ | 2MASS
$H$ |  $H$ mag | $11.32\pm 0.02$ | $11.86\pm 0.03$ | 2MASS
$K_{s}$ |  $K_{s}$ mag | $11.05\pm 0.02$ | $11.64\pm 0.04$ | 2MASS
W1 | WISE1 mag | $10.94\pm 0.02$ | $11.37\pm 0.02$ | WISE
W2 | WISE2 mag | $10.85\pm 0.02$ | $11.30\pm 0.02$ | WISE
W3 | WISE3 mag | $10.69\pm 0.06$ | $11.2\pm 0.2$ | WISE
Spectroscopic parametersb: |
$T_{e}$ | Effective temperature (K) | $3476\pm 88$ | $3586\pm 88$ | This work
$\log g_{\star}$ | Surface gravity (cgs) | $4.81\pm 0.05$ | $4.77\pm 0.05$ | This work
$\mathrm{[Fe/H]}$ | Metallicity (dex) | $0.18\pm 0.12$ | $-0.03\pm 0.12$ | This work
$v\sin i_{\star}$ | Rotational broadening (km s-1) | $<2$ | $<2$ | This work
Model-dependent parameters from a stellar SED and isochrone fitc: |
$M_{\star}$ | Mass ($M_{\odot}$) | $0.49\pm 0.02$ | $0.54\pm 0.02$ | This work
$R_{\star}$ | Radius ($R_{\odot}$) | $0.47\pm 0.01$ | $0.52_{-0.01}^{+0.02}$ | This work
$\rho_{\star}$ | Density ($\mathrm{g~{}cm^{-3}}$) | $6.9\pm 0.5$ | $5.4\pm 0.4$ | This work
$A_{v}$ | Visual extinction (mag) | $0.02\pm 0.01$ | $0.04\pm 0.03$ | This work
Other stellar parameters: |
Rotation Period | Days | $50.0_{-2.7}^{+2.8}$ | $20.6_{-0.4}^{+0.3}$ | This work
Aged | Age (Gyrs) | $0.7-5.1$ | $0.7-5.1$ | This work
$RV$ | Systemic RV (km s-1) | $-5.77\pm 0.06$ | $8.0\pm 0.1$ | This work
$U,V,W$ | Barycentric Galactic velocities (km s-1) | $-33.01\pm 0.07,~{}-5.86\pm 0.03,~{}5.65\pm 0.06$ | $11.56\pm 0.06,~{}9.63\pm 0.06,~{}-3.32\pm 0.09$ | This work
$(U,V,W)_{\mathrm{LSR}}$ | Galactic velocities w.r.t. LSRe (km s-1) | $-21.9\pm 0.8,~{}6.4\pm 0.5,~{}12.9\pm 0.4$ | $22.7\pm 0.8,~{}21.9\pm 0.5,~{}3.9\pm 0.4$ | This work

aafootnotetext: Spectral type for TOI-3984 A is from LAMOST DR8. Spectral type
for TOI-5293 A is derived with PyHammer.

bbfootnotetext: Derived with the HPF-SpecMatch package.

ccfootnotetext: Derived with the EXOFASTv2 package using MIST isochrones.

ddfootnotetext: Ages based on the rotation period and corresponding age range
from Newton et al. (2016).

eefootnotetext: Calculated using the solar velocities from Schönrich et al.
(2010).

References. — TIC (Stassun et al., 2019), Gaia DR3 (Gaia Collaboration et al.,
2022), Bailer-Jones (Bailer-Jones et al., 2021), Green (Green et al., 2019),
LAMOST (Zhong et al., 2019), APASS (Henden et al., 2018), PS1 (Chambers et
al., 2016), 2MASS (Cutri et al., 2003), WISE (Wright et al., 2010)

###  3.2 Spectral classification

####  3.2.1 LAMOST

The Large Sky Area Multi-Object Fibre Spectroscopic Telescope (LAMOST) is a 4
m telescope equipped with 4000 fibers distributed over a 5° FOV that is
capable of acquiring spectra in the optical band (3700-9000Å) at $R\approx
1800$ (Cui et al., 2012). TOI-3984 A was observed as part of its spectroscopic
survey of the Galaxy (Deng et al., 2012; Yuan et al., 2015; Xiang et al.,
2017) while TOI-5293 A has not been observed. The data used in this work are
from the public DR8v2.0101010http://dr8.lamost.org/ release (Wang et al.,
2022).

The LAMOST stellar classification pipeline uses stellar templates to identify
molecular absorption features (e.g., CaH, TiO) typical for M-type stars
(Lépine et al., 2007) and has been shown to report the subclass of an M dwarf
with an accuracy of $\pm 0.5$ subtypes (Zhong et al., 2015). A successful
classification requires that a target have (i) a mean S/N$>5$, (ii) a best-
matching template that is an M type, and (iii) molecular band indices that are
located in the M-type stellar regime identified by Zhong et al. (2019)
($0<\mathrm{TiO5}<1.2$ and $0.6<\mathrm{CaH2+CaH3}<2.4$). LAMOST classifies
TOI-3984 A as an M$4\pm 0.5$ dwarf, which agrees with the derived parameters
in Section 3.1.

####  3.2.2 LRS2

The second generation Low Resolution Spectrograph (LRS2; Chonis et al., 2014,
2016) is a low-resolution, optical integral-field unit spectrograph on the
HET. LRS2 has broad wavelength coverage spread between two fiber-fed, dual-
channel spectrographs which simultaneously observe independent fields
separated by 100″. The blue spectrograph pair (LRS2-B) covers
$364\mathrm{nm}\leq\lambda\leq 467\mathrm{nm}$ and
$454\mathrm{nm}\leq\lambda\leq 700\mathrm{nm}$ with $R\sim 2500$ and $R\sim
1400$, respectively. The red spectrograph pair (LRS2-R) covers
$643\mathrm{nm}\leq\lambda\leq 845\mathrm{nm}$ and
$823\mathrm{nm}\leq\lambda\leq 1056\mathrm{nm}$ at $R\sim 2500$. TOI-5293 A
was observed with LRS2-R on 2022 November 01 and LRS2-B on 2022 December 11
using exposure times of 1800s.

The LRS2 data were processed using the automated pipeline,
Panacea111111https://github.com/grzeimann/Panacea , and the package
LRS2Multi121212https://github.com/grzeimann/LRS2Multi . We follow the
methodology outlined in Kanodia et al. (2023). Briefly, Panacea performs bias-
correction, wavelength calibration, fiber extraction, and an initial flux
calibration while LRS2Multi extracts spectra from Panacea products. The
stellar spectra are extracted from a 3.5″ aperture centered on TOI-5293 A
using sky-subtracted frames. The response correction was derived using
calibrated standard stars observed between 2021 May through 2022 May while the
telluric correction was constructed using telluric standard stars.

We use PyHammer (Kesseli et al., 2017; Roulston et al., 2020) to estimate the
spectral type of TOI-5293. PyHammer assigns a spectral type by measuring
spectral indices for various atomic and molecular lines and comparing the
values to those measured from observed stellar templates. The M dwarf
templates used by PyHammer are from the MaNGA Stellar Library
($362.2-1035.4~{}nm$ at $R\sim 1800$; Yan et al., 2019). PyHammer selects the
best-matching spectral type by minimizing the $\chi^{2}$ difference between
spectral indices of the stellar template and observed star. We estimate a
spectral type of M$3\pm 1$ from the LRS2 data, which agrees with the derived
parameters in Section 3.1.

###  3.3 Spectral energy distribution fitting

We modeled the spectral energy distribution (SED, see more details in Appendix
A) for each target using the EXOFASTv2 analysis package (Eastman et al., 2019)
to derive model-dependent stellar parameters. EXOFASTv2 calculates the
bolometric corrections for the SED fit by linearly interpolating the
precomputed bolometric
corrections131313http://waps.cfa.harvard.edu/MIST/model_grids.html#bolometric
in $\log g_{\star}$, $\mathrm{T_{e}}$, [Fe/H], and $A_{V}$ from the MIST model
grids (Dotter, 2016; Choi et al., 2016). Table 3.1 contains the stellar priors
and derived stellar parameters with their uncertainties. The model-dependent
mass and radius are (i) $0.49\pm 0.02~{}\mathrm{M_{\odot}}$ and $0.47\pm
0.01~{}\mathrm{R_{\odot}}$ for TOI-3984 A and (ii) $0.54\pm
0.02~{}\mathrm{M_{\odot}}$ and $0.52_{-0.01}^{+0.02}~{}\mathrm{R_{\odot}}$ for
TOI-5293 A.

###  3.4 Stellar rotation period

We search the ZTF photometry for a rotation period of any activity-induced
photometric modulations in TOI-3984 A and TOI-5293 A using the
GLS141414https://github.com/mzechmeister/GLS  package. We only consider
significant peaks in the periodogram calculated with GLS where the false alarm
probability (FAP), as calculated following Zechmeister & Kürster (2009), is
below a threshold of $0.1\%$. Data within transits were excised using the
duration and ephemeris from the QLP. Significant peaks (FAP$<0.1\%$) of $\sim
52$ were found in both the $zr$ and $zg$ photometry of TOI-3984 A while a
significant peak of $\sim 21$ days was seen only in the $zr$ photometry of
TOI-5293 A.

The measured rotation periods are $50.0_{-2.7}^{+2.8}$ days for TOI-3984 A and
$20.6_{-0.4}^{+0.3}$ days for TOI-5293 A (see a detailed description of the
measurement in Appendix B). These intermediate rotation periods suggest both
TOI-3984 A and TOI-5293 A most likely have ages between $0.7-5.1$ Gyr if we
adopt the classification scheme of Newton et al. (2016). This range is
consistent with the estimated rotation periods of $\sim 3.1$ Gyr for TOI-3293
A and $\sim 1.3$ Gyr for TOI-5293 A using the rotation period and age
relationship from Engle & Guinan (2018) for M2.5-M6 dwarfs. The age estimates
from the model-dependent SED fit of $7.9_{-5.0}^{+4.1}$ Gyr and
$7.7_{-4.9}^{+4.1}$ Gyr for TOI-3984 A and TOI-5293 A, respectively, are also
consistent with the age range from Newton et al. (2016).

###  3.5 Galactic kinematics

The UVW velocities are derived with galpy (Bovy, 2015) and provided with
respect to the local standard of rest from Schönrich et al. (2010) using the
Gaia DR3 astrometry and the systemic velocity derived from HPF. The values in
Table 3.1 are in a right-handed coordinate system (Johnson & Soderblom, 1987)
where UVW are positive in the directions of the Galactic center, Galactic
rotation, and the north Galactic pole, respectively. The BANYAN $\Sigma$
algorithm (Gagné et al., 2018), which uses sky positions, proper motions,
parallax, and RVs to constrain cluster membership probabilities, classifies
both TOI-3984 A and TOI-5293 A as field stars having no membership with known
nearby young associations. Both TOI-3984 A and TOI-5293 A are also classified
as a member of the thin disk ($\mathrm{P_{Thick}}/\mathrm{P_{Thin}}<0.02$)
following the kinematic selection criteria from Bensby et al. (2003).

##  4 Photometric and RV modeling

We use the juliet analysis package (Espinoza et al., 2019) to jointly model
the RVs with a standard Keplerian RV curve generated from the radvel (Fulton
et al., 2018) package and the light curves with a transit model generated from
the batman package (Kreidberg, 2015). juliet performs the parameter estimation
using dynesty (Speagle, 2020). The photometric model adopts a quadratic limb-
darkening law where the coefficients are sampled from uniform priors following
the parameterization in Kipping (2013a). For the long-cadence TESS photometry,
the transit model utilizes the supersampling option in batman with exposure
times of 30 minutes and a supersampling factor of 30.

The TESS photometric model includes a Gaussian process noise model identical
to that described in Section 3.4 to account for correlated noise. We do not
expect the period of the Gaussian process model to reflect the rotation
periods measured in Section 3.4 because the TESS PDCSAP and eleanor CORR_FLUX
are generated using algorithms known to attenuate long-period ($>10$ days)
signals (e.g., Van Cleve et al., 2016; Holcomb et al., 2022).

A dilution factor, $D$, is included in the TESS photometric model to account
for dilution from nearby stars in the large apertures (see Figures 1 and 2).
We assume the higher spatial resolution and smaller photometric apertures used
for the ground-based photometry result in negligible contamination from
neighboring stars, such that the dilution term is fixed to unity for all
ground-based transits. The fit uses a uniform prior on the TESS dilution term
of $0-2$ to account for potential over-compensation of the dilution term that
can occasionally occur with short-cadence PDCSAP TESS light curves in crowded
fields (e.g., Burt et al., 2020). Both the photometric and RV models include a
white-noise model parameterized as a jitter term that is added in quadrature
to the uncertainty of each data set.

Tables 4 and 5 provide the priors used for the fit along with the inferred
system parameters and the confidence intervals ($16\mathrm{th}-84\mathrm{th}$
percentile) for TOI-3984 A and TOI-5293 A, respectively. The fits suggest a
significant detection of the RV orbit for each system and we investigate this
by comparing the Bayesian Information Criterion (BIC; Schwarz, 1978; Liddle,
2007) and the Akaike Information Criterion (AIC; Akaike, 1974). We compute the
AIC and BIC of the best fit RV curve (9 free parameters for the RV curve: 5
orbital elements and one offset and jitter per instrument) and a flat line (4
free parameters: one offset and jitter per instrument) to determine the
significance of the detection. We use the likelihood function ($\mathcal{L}$)
employed by juliet (Equation 4 in Espinoza et al., 2019) for this comparison.
For TOI-3984 A, $\Delta\ln\mathcal{L}=13.5$, $\Delta\mathrm{AIC}=17.1$, and
$\Delta\mathrm{BIC}=8.4$ between a flat line and Keplerian model, which
provide support for the existence of a Keplerian orbit in the data. Similarly,
for TOI-5293 A, $\Delta\ln\mathcal{L}=46.3$, $\Delta\mathrm{AIC}=82.7$, and
$\Delta\mathrm{BIC}=78.2$ provide very strong evidence for a Keplerian orbit
in the data.

Figures 3–5, 8, and 9 display the model posteriors for each system. TOI-3984 A
b is a sub-Saturn ($M_{p}=0.14\pm 0.03~{}\mathrm{M_{J}}$ and $R_{p}=0.71\pm
0.02~{}\mathrm{R_{J}}$) on a nearly circular orbit with a period of
$4.353326\pm 0.000005$ days. TOI-5293 A b is a hot Jupiter ($M_{p}=0.54\pm
0.07~{}\mathrm{M_{J}}$ and $R_{p}=1.06\pm 0.04~{}\mathrm{R_{J}}$) on a nearly
circular orbit with a period of $2.930289\pm 0.000004$ days.

We estimate the timescales for circularization using the formalism of Jackson
et al. (2008), assuming that the tidal quality factor of each planet spans
$Q_{p}=10^{5}-10^{8}$, based on the observed range from Bonomo et al. (2017),
and the tidal quality factor of each host star is $Q_{\star}=10^{7}$, based on
the modeling of Gallet et al. (2017). The timescale for circularization spans
$0.1-10$ Gyr when using the orbit parameters derived from the joint fit. With
no precise age or $Q_{p}$ constraint, these systems may be able to retain a
small but non-zero eccentricity. Existing data are only sufficient to confirm
these are low-eccentricity planetary systems with $3\sigma$ upper limits of
$e<0.23$ for TOI-3984 A b and $e<0.38$ for TOI-5293 A b. Higher-precision
photometric observations of the occultation (e.g., Alonso, 2018) would be most
sensitive to measuring any non-zero eccentricity.

Table 4: System parameters for TOI-3984 A

Parameter | Units | Prior | Value
---|---|---|---
Ground-based photometry parameters |  |  | RBO (07-19) | RBO (08-14) | Soka | RBO (02-04) | ARCTIC (03-23) | ARCTIC (05-10)
Linear limb-darkening coefficienta | $q_{1}$ | $\mathcal{U}(0,1)$ | $0.2\pm 0.1$ | $0.2\pm 0.1$ | $0.09_{-0.02}^{+0.03}$ | $0.2\pm 0.1$ | $0.09_{-0.02}^{+0.03}$ | $0.05_{-0.02}^{+0.03}$
Quadratic limb-darkening coefficienta | $q_{2}$ | $\mathcal{U}(0,1)$ | $0.5\pm 0.2$ | $0.5\pm 0.2$ | $0.8_{-0.2}^{+0.1}$ | $0.5\pm 0.2$ | $0.8_{-0.2}^{+0.1}$ | $0.6\pm 0.2$
Photometric jitter |  $\sigma_{phot}$ (ppm) | $\mathcal{J}(10^{-6},10^{4})$ | $0.0001_{-0.0001}^{+0.0261}$ | $0.3_{-0.3}^{+20.4}$ | $0.0001_{-0.0001}^{+0.0045}$ | $0.02_{-0.02}^{+2.35}$ | $0.3_{-0.3}^{+24.1}$ | $989_{-9}^{+7}$
TESS photometry parameters |  |  | Long-cadence | Short-cadence
Linear limb-darkening coefficienta | $q_{1}$ | $\mathcal{U}(0,1)$ | $0.24_{-0.09}^{+0.13}$ | $0.24_{-0.09}^{+0.13}$
Quadratic limb-darkening coefficienta | $q_{2}$ | $\mathcal{U}(0,1)$ | $0.6_{-0.3}^{+0.2}$ | $0.6_{-0.3}^{+0.2}$
Photometric jitter |  $\sigma_{phot}$ (ppm) | $\mathcal{J}(10^{-6},10^{3})$ | $0.01_{-0.01}^{+2.56}$ | $10_{-10}^{+80}$
Dilution factor | $D$ | $\mathcal{U}(0,2)$ | $0.89_{-0.03}^{+0.04}$ | $0.87_{-0.03}^{+0.04}$
TESS Gaussian process hyperparameter:
$B$ | Amplitude (ppm) | $\mathcal{J}(10^{-4},10^{12})$ | $37_{-7}^{+9}$ | $10_{-4}^{+779}$
$C$ | Additive factor | $\mathcal{J}(10^{-3},10^{3})$ | $0.02_{-0.01}^{+0.26}$ | $1.4_{-1.3}^{+244.2}$
$L$ | Length scale (days) | $\mathcal{J}(10^{-3},10^{3})$ | $1.7_{-0.4}^{+0.6}$ | $1.8_{-0.8}^{+133.6}$
$P_{GP}$ | Period (days) | $\mathcal{J}(1.0,100)$ | $2.7_{-0.5}^{+1.2}$ | $2.7_{-0.5}^{+1.2}$
RV parameters |  |  | HPF | NEID
Systemic velocity | $\gamma~{}\mathrm{(m~{}s^{-1})}$ | $\mathcal{U}(-10^{3},10^{3})$ | $2.4_{-3.8}^{+4.0}$ | $-57.7_{-10.4}^{+10.7}$
RV Jitter | $\sigma_{RV}~{}\mathrm{(m~{}s^{-1})}$ | $\mathcal{J}(10^{-3},10^{3})$ | $3.1_{-2.8}^{+8.0}$ | $29.0_{-9.8}^{+14.0}$
Orbital parameters:
Orbital period |  $P$ (days) | $\mathcal{N}(4.35,0.1)$ | $4.353326\pm 0.000005$
Time of mid-transit |  $T_{C}$ (BJDTDB) | $\mathcal{N}(2459715.02,0.1)$ | $2459715.02268\pm 0.00009$
$\sqrt{e}\cos\omega$ |  | $\mathcal{U}(-1,1)$ | $-0.07_{-0.17}^{+0.23}$
$\sqrt{e}\sin\omega$ |  | $\mathcal{U}(-1,1)$ | $-0.05_{-0.10}^{+0.09}$
Semi-amplitude velocity | $K~{}\mathrm{(m~{}s^{-1})}$ | $\mathcal{U}(0,10^{3})$ | $27.7_{-5.0}^{+5.4}$
Scaled radius | $R_{p}/R_{\star}$ | $\mathcal{U}(0,1)$ | $0.1558_{-0.0008}^{+0.0009}$
Impact parameter | $b$ | $\mathcal{U}(0,1)$ | $0.18_{-0.07}^{+0.06}$
Scaled semi-major axis | $a/R_{\star}$ | $\mathcal{J}(1,100)$ | $19.1_{-0.4}^{+0.5}$
Derived parameters:
Eccentricity | $e$ |  $\cdots$ |  $0.04_{-0.03}^{+0.05}$, $3\sigma<0.23$
Argument of periastron |  $\omega$ (degrees) |  $\cdots$ | $-54_{-108}^{+189}$
Orbital inclination |  $i$ (degrees) |  $\cdots$ | $89.5\pm 0.2$
Transit duration |  $T_{14}$ (hours) |  $\cdots$ | $2.01\pm 0.01$
Mass |  $M_{p}$ ($\mathrm{M_{J}}$ / $\mathrm{M_{\oplus}}$) |  $\cdots$ |  $0.14\pm 0.03$ / $44.0_{-8.0}^{+8.7}$
Radius |  $R_{p}$ ($\mathrm{R_{J}}$ / $\mathrm{R_{\oplus}}$) |  $\cdots$ |  $0.71\pm 0.02$ / $7.9\pm 0.24$
Surface gravity |  $\log g_{p}$ (cgs) |  $\cdots$ | $2.84\pm 0.08$
Density |  $\rho_{p}$ ($\mathrm{g~{}cm}^{-3}$) |  $\cdots$ | $0.49_{-0.10}^{+0.11}$
Semi-major axis |  $a$ (au) |  $\cdots$ | $0.041_{-0.001}^{+0.002}$
Average Incident flux |  $\langle F\rangle$ ($\mathrm{10^{8}\ erg~{}s^{-1}~{}cm^{-2}}$ / $S_{\oplus}$) |  $\cdots$ |  $0.23_{-0.02}^{+0.03}$ / $16.7\pm 1.8$
Equilibrium temperatureb |  $T_{eq}$ (K) |  $\cdots$ | $563\pm 15$

aafootnotetext: Using the $q1$ and $q2$ parameterization from Kipping (2013a).

bbfootnotetext: The planet is assumed to be a black body and we ignore heat
redistribution.

Table 5: System parameters for TOI-5293 A

Parameter | Units | Prior | Value
---|---|---|---
Ground-based photometry parameters |  |  | RBO (07-28) | RBO (09-13) | LCRO | TMMT (10-24) | TMMT (10-27)
Linear limb-darkening coefficienta | $q_{1}$ | $\mathcal{U}(0,1)$ | $0.4_{-0.2}^{+0.3}$ | $0.4_{-0.2}^{+0.3}$ | $0.5\pm 0.2$ | $0.3_{-0.2}^{+0.3}$ | $0.3_{-0.2}^{+0.3}$
Quadratic limb-darkening coefficienta | $q_{2}$ | $\mathcal{U}(0,1)$ | $0.3_{-0.2}^{+0.3}$ | $0.3_{-0.2}^{+0.3}$ | $0.5\pm 0.3$ | $0.3\pm 0.2$ | $0.3\pm 0.2$
Photometric jitter |  $\sigma_{phot}$ (ppm) | $\mathcal{J}(10^{-6},10^{3})$ | $0.007_{-0.007}^{+2.846}$ | $0.04_{-0.04}^{+12.96}$ | $0.1_{-0.1}^{+34.6}$ | $0.05_{-0.05}^{+18.11}$ | $0.2_{-0.2}^{+40.7}$
TESS photometry parameters:
Linear limb-darkening coefficienta | $q_{1}$ | $\mathcal{U}(0,1)$ | $0.4\pm 0.2$
Quadratic limb-darkening coefficienta | $q_{2}$ | $\mathcal{U}(0,1)$ | $0.3\pm 0.2$
Photometric jitter |  $\sigma_{phot}$ (ppm) | $\mathcal{J}(10^{-6},10^{3})$ | $0.2_{-0.2}^{+37.5}$
Dilution factor | $D$ | $\mathcal{U}(0,2)$ | $0.94_{-0.03}^{+0.04}$
TESS Gaussian process hyperparameter:
$B$ | Amplitude | $\mathcal{J}(10^{-10},10^{6})$ | $0.0010_{-0.0005}^{+0.0015}$
$C$ | Additive factor | $\mathcal{J}(10^{-3},10^{3})$ | $0.2_{-0.2}^{+5.5}$
$L$ | Length scale (days) | $\mathcal{J}(10^{-3},10^{3})$ | $40_{-22}^{+63}$
$P_{GP}$ | Period (days) | $\mathcal{J}(1.0,100)$ | $16_{-5}^{+10}$
RV parameters:
Systemic velocity | $\gamma~{}\mathrm{(m~{}s^{-1})}$ | $\mathcal{U}(-10^{3},10^{3})$ | $21.3_{-11.4}^{+10.8}$
RV Jitter | $\sigma_{RV}~{}\mathrm{(m~{}s^{-1})}$ | $\mathcal{J}(10^{-3},10^{3})$ | $27.4_{-9.8}^{+12.3}$
Orbital parameters:
Orbital period |  $P$ (days) | $\mathcal{N}(2.9,0.1)$ | $2.930289\pm 0.000004$
Time of mid-transit |  $T_{0}$ (BJDTDB) | $\mathcal{N}(2459448.9,0.1)$ | $2459448.9148\pm 0.0004$
$\sqrt{e}\cos\omega_{\star}$ |  | $\mathcal{U}(-1,1)$ | $-0.07_{-0.16}^{+0.17}$
$\sqrt{e}\sin\omega_{\star}$ |  | $\mathcal{U}(-1,1)$ | $-0.17\pm 0.22$
Semi-amplitude velocity | $K~{}\mathrm{(m~{}s^{-1})}$ | $\mathcal{U}(0,10^{3})$ | $115.6\pm 14.5$
Scaled radius | $R_{p}/R_{\star}$ | $\mathcal{U}(0,1)$ | $0.210_{-0.004}^{+0.005}$
Impact parameter | $b$ | $\mathcal{U}(0,1)$ | $0.32_{-0.14}^{+0.12}$
Scaled semi-major axis | $a/R_{\star}$ | $\mathcal{J}(1,100)$ | $14.1_{-1.1}^{+1.6}$
Derived parameters:
Eccentricity | $e$ |  $\cdots$ |  $0.08_{-0.06}^{+0.11}$, $3\sigma<0.38$
Argument of periastron |  $\omega_{\star}$ (degrees) |  $\cdots$ | $-92_{-45}^{+161}$
Orbital inclination |  $i$ (degrees) |  $\cdots$ | $88.8_{-0.6}^{+0.5}$
Transit duration |  $T_{14}$ (hours) |  $\cdots$ | $1.94_{-0.04}^{+0.05}$
Mass |  $M_{p}$ ($\mathrm{M_{J}}$ / $\mathrm{M_{\oplus}}$) |  $\cdots$ |  $0.54\pm 0.07$ / $170.4_{-21.9}^{+21.8}$
Radius |  $R_{p}$ ($\mathrm{R_{J}}$ / $\mathrm{R_{\oplus}}$) |  $\cdots$ |  $1.06\pm 0.04$ / $11.9\pm 0.4$
Surface gravity |  $\log g_{p}$ (cgs) |  $\cdots$ | $3.11_{-0.11}^{+0.12}$
Density |  $\rho_{p}$ ($\mathrm{g~{}cm}^{-3}$) |  $\cdots$ | $0.56\pm 0.09$
Semi-major axis |  $a$ (au) |  $\cdots$ | $0.034_{-0.003}^{+0.004}$
Average Incident flux |  $\langle F\rangle$ ($\mathrm{10^{8}\ erg~{}s^{-1}~{}cm^{-2}}$ / $S_{\oplus}$) |  $\cdots$ |  $0.47_{-0.09}^{+0.12}$ / $34.6_{-6.2}^{+8.6}$
Equilibrium temperatureb |  $T_{eq}$ (K) |  $\cdots$ | $675_{-30}^{+42}$

aafootnotetext: Using the $q1$ and $q2$ parameterization from Kipping (2013a).

bbfootnotetext: The planet is assumed to be a black body and we ignore heat
redistribution.

##  5 Discussion

###  5.1 The dynamical implication of wide separation companions

Hot Jupiters are difficult to form in-situ at their observed separations from
host stars (e.g., Dawson & Johnson, 2018; Anderson et al., 2020; Poon et al.,
2021) either through gravitational instability (e.g., Boss, 1997; Durisen et
al., 2007) or core accretion (e.g., Perri & Cameron, 1974; Pollack et al.,
1996; Chabrier et al., 2014). These systems are hypothesized to have migrated
from larger distances via the loss of angular momentum due to gravitational
interactions with the circumstellar gas disk (e.g., Goldreich & Tremaine,
1980; Lin & Papaloizou, 1986; Lin et al., 1996; Ida & Lin, 2008; Baruteau et
al., 2014) or with other massive companions (e.g., Rasio & Ford, 1996;
Weidenschilling & Marzari, 1996; Wu & Murray, 2003; Fabrycky & Tremaine, 2007;
Petrovich, 2015a, b). Gas disk migration predominantly results in circular
planetary orbits which are well aligned with the spin axis of the host star
while high eccentricity tidal migration preferentially results in planets on
misaligned and eccentric orbits that may have large-separation companions.

We search for wide companions using a list of spatially resolved binary stars
from an analysis of proper motions in Gaia EDR3 (El-Badry et al., 2021). Wide
separation binary systems in El-Badry et al. (2021) are flagged as having
either a main sequence or white dwarf companion using the location of the
companion on the Gaia color-absolute magnitude diagram (El-Badry & Rix, 2018).
TOI-3984 A has a white dwarf companion, Gaia DR3 1291955574574621056 (TIC
1101522311), at a projected distance of 3.27″ or a projected separation of 356
au. TOI-5293 A has an M dwarf companion, Gaia DR3 2640121482094497024 (TIC
2052711961), at a projected distance of 3.57″ or a projected separation of 579
au. The companions have a negligible probability ($<4\times 10^{-8}$, see a
detailed description in Appendix A of El-Badry et al., 2021) of being a false
detection as a result of chance alignment with a background source having
spurious parallax and proper motion measurements. The NEID and HPF spectra are
not contaminated by either companion as they lie outside the HPF fiber ($\sim
1.7\arcsec$ on-sky; Kanodia et al., 2018) and the NEID HR fiber ($\sim
0.9\arcsec$ on-sky; Schwab et al., 2016).

We use the phot_bp_rp_excess_factor to determine if any of the measured blue
or red Gaia photometry of either bound companion is blended or contaminated
(Evans et al., 2018b; Riello et al., 2021). The value of
phot_bp_rp_excess_factor in the Gaia DR3 archive is known to have a strong
color dependence and the Gaia documentation suggests using a corrected excess
factor that compares the reported value to the expected excess factor at a
given color as derived from sources with good quality photometry (see the
discussion in Section 6 of Riello et al., 2021). We calculate a corrected
phot_bp_rp_excess_factor151515using Table 2 and Equation 6 from Riello et al.
(2021) of 0.20 and 0.15 for TOI-3984 B and TOI-5923 B, respectively. The
deviation from zero reveals that both companions have some contamination in
their Gaia colors. We adopt these colors as nominal values to qualitatively
characterize the wide separation companions but note that photometry and
spectroscopy of each target is required to provide robust companion
parameters.

The Gaia General Stellar Parameterizer from Photometry algorithm (GSP-Phot;
Andrae, R. et al., 2022) confirms that (i) the companion to TOI-3984 A is most
likely a white dwarf based on its location on the Gaia color-magnitude diagram
and (ii) the companion to TOI-5293 A is a main sequence M dwarf. TOI-5293 B is
reported to be a mid M dwarf with a radius of
$R_{\star}=0.26_{-0.08}^{+0.10}~{}\mathrm{R_{\odot}}$ (Creevey & Lebreton,
2022; Creevey, O. et al., 2022) with a $\log g=4.72_{-0.14}^{+0.16}$ and
$\mathrm{T_{e}}=3041_{-41}^{+280}$ K derived by the Gaia DR3 Multiple Star
Classifier161616https://gea.esac.esa.int/archive/documentation/GDR3/Data_analysis/chap_cu8par/sec_cu8par_apsis/ssec_cu8par_apsis_msc.html
(MSC) analysis of Gaia BP/RP spectra.

Neither GSP-Phot nor MSC provide stellar parameters for TOI-3984 B and we use
the WD_models171717https://github.com/SihaoCheng/WD_models  package from Cheng
et al. (2019) to derive a photometric age and mass. We assume the atmosphere
is composed of hydrogen, due to the prevalence of DA white dwarfs in the 100
pc SDSS sample ($\sim 65\%$ of the sample; Kilic et al., 2020; Kepler et al.,
2021) and adopt the cooling models of Bédard et al. (2020). The estimated mass
for the white dwarf companion is $M_{WD}\sim 0.75~{}\mathrm{M_{\odot}}$ with a
cooling age of $\sim 2.9$ Gyr (see Appendix C) and its progenitor star has a
mass in the range $1.6-4.5~{}\mathrm{M_{\odot}}$ when using the MIST semi-
empirical white dwarf initial-final mass relationship from Cummings et al.
(2018). We note the cooling age is consistent with the age estimate of
TOI-3984 A from the rotation period in Section 3.4 ($0.7-5.1$ Gyr).

Various studies have been performed on the hot Jupiter population orbiting
Sun-like stars to test the significance of multi-body interactions (e.g., Wang
et al., 2014; Knutson et al., 2014; Ngo et al., 2015, 2016; Evans et al.,
2018a; Ziegler et al., 2018; Marzari & Thebault, 2019; Fontanive et al., 2019;
Belokurov et al., 2020; Hwang et al., 2020; Fontanive & Bardalez Gagliuffi,
2021; Ziegler et al., 2021). Hot Jupiters orbiting Sun-like stars have been
reported to have a high wide binary fraction, relative to field stars. Ngo et
al. (2016) measured that $47\pm 7\%$ of systems with a hot Jupiter have
resolved stellar companions between separations of $50-2000$ au while
Fontanive et al. (2019) determined that $79.0^{+13.2}_{-14.7}\%$ of systems
with a massive substellar object have a wide companion between 20-10000 au.
Moe & Kratter (2021) reassessed these claims by accounting for biases and
selection effects in the sample to conclude that wide separation companions
($50-2000$ au) to hosts of hot Jupiters do not enhance the formation of hot
Jupiters at a statistically significant level and that the larger wide binary
fraction of hot Jupiters is a result of suppressed hot Jupiter formation in
close binaries. This non-enrichment in wide separation companions has been
observed in the hot Jupiter population orbiting Sun-like stars with Gaia
(Hwang et al., 2020).

The small population of hot Jupiters orbiting M dwarfs precludes an extensive
study of stellar companions. Only two additional M dwarfs in the existing
population of 10 transiting hot Jupiters ($P<10$ days,
$R_{p}>8~{}\mathrm{R_{\oplus}}$), HATS-74A (Jordán et al., 2022) and TOI-3714
(Cañas et al., 2022), are known to have wide separation companions. HATS-74A
has an M dwarf companion at a separation of 238 au while TOI-3714 has a white
dwarf companion, potentially on an eccentric orbit, at a projected separation
of 302 au.

The separation of the main sequence companions may be too large to strongly
impact the formation of hot Jupiters, as some studies suggest only companions
at separations of $\lesssim 200$ au can impact the formation process (e.g.,
Kraus et al., 2016; Moe & Kratter, 2021; Ziegler et al., 2021). The white
dwarf progenitor in the TOI-3984 A system, however, would have been both more
massive and much closer if we assume adiabatic mass loss (see Nordhaus et al.,
2010; Nordhaus & Spiegel, 2013) and during the progenitor’s main sequence
lifetime, secular effects such as Kozai-Lidov cycles (Kozai, 1962; Lidov,
1962; Naoz, 2016) could have induced high-eccentricity tidal migration of
TOI-3984 A b that brought it to its observed location.

Gaia DR3 is able to constrain the eccentricity of resolved wide binaries
(e.g., Tokovinin, 2020; Hwang et al., 2022) using the angle between the
separation vector and the relative velocity vector ($v-r$ angle). Hwang et al.
(2022) estimate the eccentricity of the wide binary sample identified by El-
Badry et al. (2021) under the assumption that a wide companion has a random
orbital orientation. The inferred eccentricities calculated by Hwang et al.
(2022) are $e=0.64^{+0.18}_{-0.26}$ and $e=0.77_{-0.24}^{+0.16}$ for TOI-3984
B and TOI-5293 B, respectively.

Similar to the white dwarf companion to TOI-3714, the high-eccentricity of
TOI-3984 B is consistent with the scenario in which the progenitor star was
initially on a smaller orbit, potentially inducing the migration of TOI-3984 A
b, before later widening and becoming eccentric due to mass loss as it evolved
into a white dwarf. While TOI-5293 B is also eccentric, the large projected
separation ($579$ au) would result in a $>5$ Gyr timescale for the Kozai-Lidov
cycles (Equation 7 from Kiseleva et al., 1998) which would be too long to
affect the migration of TOI-5293 A b, particularly given the most likely age
of $0.7-5.1$ Gyr derived in Section 3.4.

###  5.2 Constraints on additional planetary companions

There are 15 confirmed planetary systems hosting a close-in gas giant ($P<100$
days and $M>0.6~{}\mathrm{M_{J}}$) with interior companions across all
spectral types (see Table A1 in Sha et al., 2023). For the transiting hot
Jupiter sample ($P<10$ days), there are only 6 confirmed gas giants with
nearby companions: WASP-47 b (Becker et al., 2015), Kepler-730 b (Zhu et al.,
2018; Cañas et al., 2019), TOI-1130 c (Huang et al., 2020c), WASP-148 b (Wang
et al., 2022), WASP-132 b (Hord et al., 2022), and TOI-2000 (Sha et al.,
2023). This low planetary multiplicity rate for hot Jupiters orbiting Sun-like
stars has been statistically confirmed with multiple ground and space-based
transiting samples (e.g., Steffen et al., 2012; Huang et al., 2016;
Maciejewski, 2020; Hord et al., 2021; Wang et al., 2021; Zhu & Dong, 2021). A
search for transit timing variations in the Kepler sample by Wu et al. (2023)
also determined that a minimum of $\sim 13\%$ of hot Jupiters orbiting Sun-
like stars have nearby companions and must have a quiescent formation history.
The scarcity of short-period companions to hot Jupiters is consistent with the
hypothesis that high-eccentricity migration is responsible for most of these
systems, as the inward migration of a gas giant is a dynamically “hot”
processes that would destabilize interior planets and leave an isolated gas
giant (e.g., Mustill et al., 2015; Dawson & Johnson, 2018).

We search for companions to TOI-3984 A and TOI-5923 using the available TESS
photometry and HPF RVs. We used the transit least squares algorithm (TLS;
Hippke & Heller, 2019) on the photometry to search for additional transiting
planets after subtracting the best-fitting transit model for each planet. For
this search, we only searched for candidate signals (depths $>1$ ppm) between
$1-13$ days. The maximum radius of a candidate signal was $\sim
5\mathrm{~{}R_{\oplus}}$ for TOI-3984 A and $\sim 7\mathrm{~{}R_{\oplus}}$ for
TOI-5293 A. TLS identified no candidates where the test statistic was above
the suggested threshold of 7 (corresponding to a false positive rate of
$1\%$). The current TESS photometry is only sufficient to exclude the
existence of additional short-period transiting gas giant companions.

To further constrain the existence of non-transiting companions, we analyzed
the residuals to the HPF RVs using thejoker (Price-Whelan et al., 2017) to
perform a rejection sampling analysis. Orbits were sampled using a log-uniform
prior for the period (between 1 day and twice the HPF RV baseline), the Beta
distribution from Kipping (2013b) as a prior for the eccentricity, and a
uniform prior for the argument of pericenter and the orbital phase. We
considered a total of $2^{28}$ samples using thejoker and had a total
acceptance rate of $<0.07\%$ for both systems. The surviving samples place a
$3\sigma$ upper limit on any coplanar ($\sin i\sim 1$) companions of
$M<2.4~{}\mathrm{M_{J}}$ ($K<\mathrm{150~{}m~{}s^{-1}}$) within 1 au ($P<520$
days) for TOI-3984 A181818For comparison, $M<0.5~{}\mathrm{M_{J}}$ within 0.3
au and $M<0.6~{}\mathrm{M_{J}}$ ($K<100\mathrm{~{}m~{}s^{-1}}$) within 0.3 au
($P<82$ days) for TOI-5293 A. The existing photometry and RVs reject the
existence of nearby, coplanar massive planetary companions. Future
observations with TESS and additional RVs are needed to provide robust
constraints on the existence of small and low-mass transiting or non-
transiting planetary companions.

###  5.3 Comparison to the M dwarf planet population

A comparison of the TOI-3984 A and TOI-5293 A systems to the planetary
mass–radius, stellar $T_{e}-\log g_{\star}$, and the period–insolation flux
distributions of known M dwarf systems hosting planets with
$R_{p}>4~{}\mathrm{R_{\oplus}}$ is shown in Figure 10. Among the population of
short-period gas giants ($P<10$ days and $R_{p}\gtrsim
8~{}\mathrm{R_{\oplus}}$) orbiting M dwarfs, TOI-3984 A b is a sub-Saturn with
the smallest mass ($\sim 0.47\mathrm{~{}M_{Saturn}}$) and radius ($\sim
0.88\mathrm{~{}R_{Saturn}}$) while TOI-5293 A has a radius and mass consistent
at $1\sigma$ to the median values of the existing M dwarf gas giant population
($1.02~{}\mathrm{R_{J}}$ and $0.50~{}\mathrm{M_{J}}$). The mid M dwarf nature
of the host stars makes these two planets some of the coolest transiting M
dwarf gas giants with an insolation flux of $S=16.7\pm 1.8~{}S_{\oplus}$ for
TOI-3984 A and $S=34.6_{-6.2}^{+8.6}~{}S_{\oplus}$ for TOI-5293 A, much lower
than the typical transiting M dwarf gas giant (median value of $S\sim
56~{}S_{\oplus}$).

There are two additional hot Jupiters orbiting mid M dwarfs (M3-M5,
$3400\mathrm{~{}K}<T_{e}<3600\mathrm{~{}K}$): HATS-71 b (Bakos et al., 2020)
and TOI-5205 b (Kanodia et al., 2023) while the rest orbit early M dwarfs
(M0-M2). The frequency of such hot Jupiters orbiting M dwarfs is theoretically
predicted to be intrinsically low in the framework of core accretion because
the low surface density of an M dwarf protoplanetary disk would impede growth
of cores and the onset of runaway gas accretion (e.g., Laughlin et al., 2004;
Ida & Lin, 2005; Kennedy & Kenyon, 2008; Burn et al., 2021). Constraints from
radial velocity surveys (e.g., Endl et al., 2006; Sabotta et al., 2021;
Pinamonti et al., 2022) and photometric surveys (e.g., Kovács et al., 2013;
Obermeier et al., 2016) have constrained the frequency of short-period gas
giants orbiting M dwarfs to $\lesssim 2\%$. Two independent searches for hot
Jupiters orbiting M dwarfs in TESS by Gan et al. (2023) and (Bryant et al.,
2023) have yielded an occurrence rate of $0.27\pm 0.09\%$ and $0.193\pm
0.072\%$, respectively.

The measured occurrence rates for M dwarfs are smaller to recent values
measured from TESS for Sun-like stars of $0.71\pm 0.31\%$ (Zhou et al., 2019)
and $0.98\pm 0.36\%$ (Beleznay & Kunimoto, 2022). For comparison, the
occurrence rate of hot Jupiters orbiting Sun-like stars has consistently been
estimated to be $\sim 1\%$ from various RV surveys such as the California
Planet Survey ($1.2\pm 0.38\%$; Wright et al., 2012), the HARPS/CORALIE survey
($0.9\pm 0.4\%$; Mayor et al., 2011), and the Anglo-Australian Planet Search
($0.84^{+0.7}_{-0.2}\%$; Wittenmyer et al., 2020). Results from the California
Legacy Survey (Zhu, 2022) suggest a higher occurrence rate of $2.8\pm 0.8\%$,
but Zhu (2022) warn that the differences in the stellar population (such as
binarity and metallicity) may result in a discrepancy from the canonical $\sim
1\%$ occurrence rate for Sun-like stars. The current results for M dwarfs from
Gan et al. (2023) and Bryant et al. (2023) suggest hot Jupiters are rarer as
companions to M dwarfs, but these values are also within $1-3\sigma$ of most
occurrence rates for Sun-like stars.

The apparent dearth of hot Jupiters orbiting M dwarfs is expected if they
predominantly form via core accretion. Giant planet formation must occur
before depletion of the gas disk. The low mass and surface density of
protoplanetary disks around M dwarfs (e.g., Andrews et al., 2013; Mohanty et
al., 2013; Stamatellos & Herczeg, 2015; Ansdell et al., 2017; Manara et al.,
2018) stymie this process with a lower mass supply and longer timescales of
planetesimal formation (Laughlin et al., 2004). While the preference for hot
Jupiters to orbit early M dwarfs may be an observational bias or a result of a
small population size, the efficiency and prevalence of gas giants should
increase when orbiting more massive M dwarfs.

Figure 10: (a) TOI-3984 A b (star) and TOI-5293 A b (diamond star) on the
mass-radius diagram for transiting M dwarf exoplanets with mass measurements
and $R_{p}>4~{}R_{\oplus}$. All previously known hot Jupiters ($P<10$ days and
$R_{P}\geq 8~{}\mathrm{R_{\oplus}}$) transiting M dwarfs are marked as
pentagons. (b) TOI-3984 A and TOI-5293 A on an effective temperature $-$
surface gravity diagram. (c) the insolation flux and radius for these planets.
The data were compiled from the NASA Exoplanet Archive (Akeson et al., 2013)
on 2023 May 03.

###  5.4 Comparison to planetary models

These planets are unlikely to exhibit any radius inflation due to stellar
flux-driven mechanisms. Studies of the Kepler population of transiting hot
Jupiters (e.g., Thorngren & Fortney, 2018; Thorngren et al., 2021) show that
inflated radii are only evident in when $T_{eq}>1000$ K or the incident flux
$>2\times 10^{8}\mathrm{~{}erg~{}s^{-1}~{}cm^{-2}}$. These values serve as the
threshold for the apparent radius anomaly of hot Jupiters, where some planets
have radii much larger than expected for a pure hydrogen and helium Jovian
analogue (see Fortney et al., 2021, and references therein).

The mass and radii for TOI-3984 A and TOI-5293 A are within $2-3\sigma$ of the
predicted values from models for gas giants between $1-5$ Gyrs by Baraffe et
al. (2008), which assume a gaseous hydrogen and helium envelope with a
distribution of heavy elements, and Fortney et al. (2007), which assume a
solar metallicity hydrogen and helium atmosphere with a heavy element core
composed of a $50/50$ mixture of ice and rock. We compared TOI-3984 A b and
TOI-5293 A b to the predicted mass and radii for a solar metallicity
atmosphere and note agreement in the mass and radius within $2-3\sigma$
regardless of age. For TOI-3984 A b, the models of Baraffe et al. (2008)
suggest a large heavy element fraction ($Z=M_{\mathrm{env}}/M_{p}\approx
0.3-0.5$) while the Fortney et al. (2007) models similarly suggest a core
$0.25-0.55$ times the planetary mass. It may be possible the apparent upper
limit for $Z$ is large because we ignore the effects of tidal heating. Through
a population study of sub-Saturns, Millholland et al. (2020) determined that
ignoring radius inflation due to tides driven by a non-zero eccentricity or
obliquity would result in ($Z\gtrsim 0.5$). The younger models favor a larger
$Z$, but the poor constraint on age and eccentricity preclude a detailed study
on tidal heating in TOI-3984 A. In comparison, the mass and radius of TOI-5293
A are typical of a canonical Jupiter and are consistent with models from
Fortney et al. (2007) having no core or the Baraffe et al. (2008) models with
a negligible heavy element fraction of $Z=2\%$.

###  5.5 Future Characterization

####  5.5.1 Stellar Obliquity

Measurements of the projected obliquity ($\lambda$), or the angle between a
star’s spin axis and the orbit normal of a companion, have been useful to
constrain the physical processes responsible for the observed planetary
architectures (see the reviews by Triaud (2018) and Albrecht et al. (2022)).
Studies of obliquities of hot Jupiters orbiting Sun-like stars have revealed
an obliquity distribution that is consistent with a formation process
involving high-eccentricity migration and tidal damping (e.g., Albrecht et
al., 2012; Rice et al., 2022).

Obliquity measurements for M dwarf systems are rarer due to the faintness of
the host stars and only a handful of M dwarf systems have obliquity
measurements: GJ 436 (Bourrier et al., 2018, 2022), K2-25 (Stefánsson et al.,
2020b), GJ 3740 (Stefánsson et al., 2022), TRAPPIST-1 (Hirano et al., 2020a;
Brady et al., 2023), and AU Mic (Hirano et al., 2020b; Palle et al., 2020;
Addison et al., 2021). Of these, GJ 436 b and GJ 3470 b are observed to be on
polar orbits, while the other systems are observed to be on well-aligned
orbits. Due to the rarity and relative faintness of M dwarfs hosting gas
giants, there exists no measurement of $\lambda$ for such a system and it is
not known if high-eccentricity migration coupled with tidal damping is also
the dominant formation pathway for these planets.

Both TOI-3984 A and TOI-5293 A have a measured rotation period (see Section
3.4), which allows for constraints on the three-dimensional obliquity, $\psi$
(e.g., Stefánsson et al., 2022; Frazier et al., 2023). A measurement of $\psi$
for exoplanets across all spectral types has revealed these planets do not
span the full range of $\psi$ but tend to be either polar or well-aligned,
which may be a remnant of formation and not due to tidal evolution (Albrecht
et al., 2021; Spalding & Winn, 2022). Measurements of $\psi$ for M dwarf gas
giants are required to investigate if the distribution is also bimodal for
this population.

We estimate the amplitude of the RV anomaly due to the Rossiter-Mclaughlin
effect (Equation 1 in Triaud, 2018) for both stars using the derived transit
and stellar parameters. We derive an equatorial velocity of $v_{eq}=0.47\pm
0.03~{}\mathrm{km~{}s^{-1}}$ and RM amplitude of $\sim
8~{}\mathrm{m~{}s^{-1}}$ for TOI-3984 A. For TOI-5293 A,
$v_{eq}=1.28_{-0.04}^{+0.06}~{}\mathrm{km~{}s^{-1}}$ with an RM amplitude of
$\sim 36~{}\mathrm{m~{}s^{-1}}$. The precision to detect these signals around
faint ($V=16-17$) M dwarfs can be achieved using current high-resolution
spectrometers on large telescopes, such as MAROON-X (Seifahrt et al., 2016) or
KPF (Gibson et al., 2016).

####  5.5.2 Transmission Spectroscopy

Understanding the atmospheric composition of an exoplanet is important to
determine its bulk planetary composition and provide an constraints to
internal structure models (see the reviews by Madhusudhan, 2019; Fortney et
al., 2020) which can then be linked to formation and evolutionary processes.
Theoretical studies have predicted that key volatile molecules, such as
$\mathrm{H_{2}O}$, $\mathrm{CH_{4}}$, and $\mathrm{CO}$ should be present in
the hydrogen-dominated atmospheres of hot Jupiters at high temperatures (e.g.,
Burrows & Sharp, 1999; Moses et al., 2011; Madhusudhan, 2012). Current and
future atmospheric missions, such as JWST (Greene et al., 2016) and ARIEL
(Tinetti et al., 2018), will characterize such gas giants over a wide range of
temperatures to adequately sample a wide range of transitions in atmospheric
chemistry (Madhusudhan et al., 2011; Mollière et al., 2015; Fortney et al.,
2020).

Beyond the detection of individual species, atmospheric abundances can be used
to estimate the C/N/O ratios, which in turn are thought to trace where within
the protoplanetary disk a planet forms (Öberg et al., 2011; Madhusudhan, 2012;
Öberg & Wordsworth, 2019; Turrini et al., 2021; Hobbs et al., 2022). Current
attempts to invert atmospheric composition to formation and migration
processes have had limited success and more detailed characterization of
exoplanet atmospheres would help inform planet formation models (e.g., Dash et
al., 2022; Mollière et al., 2022).

In the context of hot Jupiters orbiting Sun-like stars, studies with HST have
revealed a diverse sample containing both cloudy and clear planets with
depleted water abundance relative to predictions from the Solar System (e.g.,
Sing et al., 2016; Pinhas et al., 2019; Welbanks et al., 2019). No studies
have been performed on M dwarf hot Jupiters to date. TOI-3984 A b and TOI-5293
A b have the precision on mass and radius ($>5\sigma$) required for detailed
atmospheric analysis (Batalha et al., 2019). The large transmission
spectroscopy metric (TSM; Kempton et al., 2018) of both TOI-3984 A b
($\mathrm{TSM}=138_{-27}^{+29}$) and TOI-5293 A b ($\mathrm{TSM}=92\pm 14$)
make these favorable targets among the M dwarf gas giant population to be
observed with JWST, as shown in Figure 11. While these planets do not have the
highest TSM among the M dwarf gas giants (TOI-3757 has a $\mathrm{TSM}=180\pm
30$), they are unique in the population due to their cooler equilibrium
temperature. Almost all of the corresponding hot Jupiters orbiting Sun-like
stars are hotter than TOI-3984 A b and TOI-5293 A b or have lower TSMs.

Figure 11: A comparison of the transmission spectroscopic metric (TSM) from
Kempton et al. (2018) for TOI-3984 A and TOI-5293 A to the existing M dwarf
gas giant population ($P<10$ days and $R_{p}>8\mathrm{~{}R_{J}}$). TOI-3984 A
is the coolest object in this sample and provides an opportunity to
characterize the chemistry in a gas giant with $T_{eq}<600$ K. TOI-5293 A b
overlaps with TOI-3629 b in equilibrium temperature and TSM but is much larger
in radius and provides an opportunity to study a temperate gas giant with
$R>1~{}\mathrm{R_{J}}$. The TSM value for hot Jupiters orbiting Sun-like stars
are included for reference. The data were compiled from the NASA Exoplanet
Archive on 2023 May 03.

An important factor in understanding exoplanetary atmospheres is the
prevalence of clouds and hazes, which may form in the hot Jupiters through
condensation chemistry or photochemical processes (e.g., Sudarsky et al.,
2003; Helling et al., 2008; Marley et al., 2013). The presence of clouds or
hazes can impact atmospheric processes and weaken or mask spectral features
(e.g., Sing et al., 2016; Sing, 2018). Numerous theoretical studies have
predicted the ubiquity of clouds in atmospheres at all temperatures (Marley &
Robinson, 2015; Gao et al., 2021). Disequilibrium processes are thought to be
more efficient in the atmospheres cooler exoplanets like TOI-3984 A b and
TOI-5293 A b (e.g., Moses et al., 2011) and the production of aerosols may be
enhanced due to the higher percentage of total UV energy from M dwarfs
compared to Sun-like stars (e.g., Liang et al., 2004; Line et al., 2010;
Youngblood et al., 2016; Melbourne et al., 2020). Transmission spectra with
JWST using NIRSpec would provide ample wavelength coverage needed to make
comparisons to predictions from cloud and haze models (e.g., Kawashima et al.,
2019; Mai & Line, 2019) and the effects of higher UV radiation environment of
early M dwarfs on atmospheric chemistry (e.g., Pineda et al., 2021).

##  6 Summary

We present and characterize two gas giants orbiting mid M dwarfs. TOI-3984 A b
is a short-period sub-Saturn ($M_{p}=0.14\pm 0.03~{}\mathrm{M_{J}}$ and
$R_{p}=0.71\pm 0.02~{}\mathrm{R_{J}}$) on a $P=4.353326\pm 0.000005$ day
orbit. TOI-5293 A b is a hot Jupiter ($M_{p}=0.54\pm 0.07~{}\mathrm{M_{J}}$
and $R_{p}=1.06\pm 0.04~{}\mathrm{R_{J}}$) on a $P=2.930289\pm 0.000004$ day
orbit. Both systems have measured rotation periods between 10 and 70 days and
probably ages between 0.71 and 5.1 Gyrs. They are in wide separation binary
systems, with TOI-3984 A having a white dwarf companion at a projected
separation of 356 au and TOI-5293 A having a later type M dwarf companion at a
projected separation of 579 au. For TOI-3984 A, the age range estimated from
the rotation period is in agreement with the nominal cooling age of its white
dwarf companion ($\sim 2.9$ Gyr), which may have once been close enough to
impact the migration of TOI-3984 A b. The companion to TOI-5293 A is too far
to impact the migration of its planet. Existing photometric and RV data are
sufficient to reject the presence of additional massive, close-period
planetary companions in these systems. TOI-3984 A b has the smallest mass and
radius among the current sample of M dwarf gas giants while TOI-5293 A b is a
typical M dwarf gas giant and consistent with the median mass and radius of
the existing population. These two planets are, however, much cooler than the
typical M dwarf gas giants due to the mid M dwarf nature of the host stars.
TOI-3984 A b and TOI-5293 A b are bright enough to facilitate observations
during transit to (i) further our understanding of their dynamical history
with a measurement of the projected and three dimensional obliquities
($\lambda$ and $\psi$) and (ii) explore the atmospheric chemistry of temperate
gas giants.

## Acknowledgments

We thank the anonymous referee for valuable feedback which has improved the
quality of this manuscript. CIC acknowledges support by NASA Headquarters
through an appointment to the NASA Postdoctoral Program at the Goddard Space
Flight Center, administered by USRA through a contract with NASA and the NASA
Earth and Space Science Fellowship Program through grant 80NSSC18K1114. SK
acknowledges research support from Carnegie Institution of Science through the
Carnegie Fellowship. GS acknowledges support provided by NASA through the NASA
Hubble Fellowship grant HST-HF2-51519.001-A awarded by STScI, which is
operated by AURA, for NASA, under contract NAS5-26555. The Center for
Exoplanets and Habitable Worlds is supported by the Pennsylvania State
University and the Eberly College of Science.

The computations for this research were performed on the Pennsylvania State
University’s Institute for Computational and Data Sciences’ Roar
supercomputer. This content is solely the responsibility of the authors and
does not necessarily represent the views of the Institute for Computational
and Data Sciences.

We acknowledge support from NSF grants AST 1006676, AST 1126413, AST 1310875,
AST 1310885, AST 2009554, AST 2009889, AST 2108512, AST 2108801 and the NASA
Astrobiology Institute (NNA09DA76A) in our pursuit of precision RVs in the
near-infrared. We acknowledge support from the Heising-Simons Foundation via
grant 2017-0494.

Some of these results are based on observations obtained with the Apache Point
Observatory 3.5m telescope, which is owned and operated by the Astrophysical
Research Consortium. We acknowledge support from NSF grants AST 1907622, AST
1909506, AST 1909682, AST 1910954 and the Research Corporation in connection
with precision diffuser-assisted photometry. We wish to thank the APO 3.5m
telescope operators in their assistance in obtaining these data.

These results are based on observations obtained with HPF and LRS2 on the HET.
The HET is a joint project of the University of Texas at Austin, the
Pennsylvania State University, Ludwig-Maximilians-Universität München, and
Georg-August Universität Gottingen. The HET is named in honor of its principal
benefactors, William P. Hobby and Robert E. Eberly. The HET collaboration
acknowledges the support and resources from the Texas Advanced Computing
Center. We are grateful to the HET Resident Astronomers and Telescope
Operators for their valuable assistance in gathering our HPF data. LRS2 was
developed and funded by the University of Texas at Austin McDonald Observatory
and Department of Astronomy and by Pennsylvania State University. We thank the
Leibniz-Institut für Astrophysik Potsdam (AIP) and the Institut für
Astrophysik Göttingen (IAG) for their contributions to the construction of the
integral field units. We would like to acknowledge that the HET is built on
Indigenous land. Moreover, we would like to acknowledge and pay our respects
to the Carrizo & Comecrudo, Coahuiltecan, Caddo, Tonkawa, Comanche, Lipan
Apache, Alabama-Coushatta, Kickapoo, Tigua Pueblo, and all the American Indian
and Indigenous Peoples and communities who have been or have become a part of
these lands and territories in Texas, here on Turtle Island.

Some of the data were obtained by the NEID spectrograph built by the
Pennsylvania State University and operated at the WIYN Observatory by NOIRLab,
which is managed by the Association of Universities for Research in Astronomy
(AURA) under a cooperative agreement with the NSF, and operated under the NN-
EXPLORE partnership of NASA and the NSF. WIYN is a joint facility of the
University of Wisconsin-Madison, Indiana University, NSF’s NOIRLab, the
Pennsylvania State University, Purdue University, University of California-
Irvine, and the University of Missouri. Observations with NEID were obtained
under proposal 2022A-763446 (PI: S. Wang). NEID results utilize the Data
Reduction Pipeline operated by NExScI and developed under subcontract 1644767
between JPL and the University of Arizona. This work was performed for the Jet
Propulsion Laboratory, California Institute of Technology, sponsored by the
United States Government under the Prime Contract 80NM0018D0004 between
Caltech and NASA. We thank the NEID Queue Observers and WIYN Observing
Associates for their skillful execution of the NEID observations. We thank
Zade Arnold, Joe Davis, Michelle Edwards, John Ehret, Tina Juan, Brian
Pisarek, Aaron Rowe, Fred Wortman, the Eastern Area Incident Management Team,
and all of the firefighters and air support crew who fought the recent
Contreras fire to save KPNO. The authors are honored to be permitted to
conduct astronomical research on Iolkam Du’ag (Kitt Peak), a mountain with
particular significance to the Tohono O’odham.

Some of the observations in this paper made use of the NN-EXPLORE Exoplanet
and Stellar Speckle Imager (NESSI). NESSI was funded by the NASA Exoplanet
Exploration Program and the NASA Ames Research Center. NESSI was built at the
Ames Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and
Emmett Quigley.

This paper is based on observations obtained from the Las Campanas Remote
Observatory through a partnership between Carnegie Observatories, the Astro-
Physics Corporation, Howard Hedlund, Michael Long, Dave Jurasevich, and SSC
Observatories.

Some of the data presented in this paper were obtained from MAST at STScI. The
specific observations analyzed can be accessed via
https://doi.org/10.17909/1xcp-2f18 (catalog DOI: 10.17909/1xcp-2f18). Support
for MAST for non-HST data is provided by the NASA Office of Space Science via
grant NNX09AF08G and by other grants and contracts. This work includes data
collected by the TESS mission, which are publicly available from MAST. Funding
for the TESS mission is provided by the NASA Science Mission directorate. This
research made use of the (i) NASA Exoplanet Archive, which is operated by
Caltech, under contract with NASA under the Exoplanet Exploration Program,
(ii) SIMBAD database, operated at CDS, Strasbourg, France, (iii) NASA’s
Astrophysics Data System Bibliographic Services, (iv) NASA/IPAC Infrared
Science Archive, which is funded by NASA and operated by the California
Institute of Technology, and (v) data from 2MASS, a joint project of the
University of Massachusetts and IPAC at Caltech, funded by NASA and the NSF.

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.

Some of the observations in this paper made use of the Guoshoujing Telescope
(LAMOST), a National Major Scientific Project built by the Chinese Academy of
Sciences. Funding for the project has been provided by the National
Development and Reform Commission. LAMOST is operated and managed by the
National Astronomical Observatories, Chinese Academy of Sciences.

Some of the observations in this paper were obtained with the Samuel Oschin
Telescope 48-inch and the 60-inch Telescope at the Palomar Observatory as part
of the ZTF project. ZTF is supported by the NSF under Grant No. AST-2034437
and a collaboration including Caltech, IPAC, the Weizmann Institute for
Science, the Oskar Klein Center at Stockholm University, the University of
Maryland, Deutsches Elektronen-Synchrotron and Humboldt University, the TANGO
Consortium of Taiwan, the University of Wisconsin at Milwaukee, Trinity
College Dublin, Lawrence Livermore National Laboratories, and IN2P3, France.
Operations are conducted by COO, IPAC, and UW.

##  Appendix A SED Fit to Broadband Photometry

The SED fits with EXOFASTv2 use Gaussian priors on the (i) broadband
photometry listed in Table 3.1, (ii) $\log g_{\star}$, $T_{e}$, and [Fe/H]
derived from HPF-SpecMatch, and (iii) the geometric distance calculated from
Bailer-Jones et al. (2021) for each respective star. We apply an upper limit
to the visual extinction based on estimates of Galactic dust (Green et al.,
2019) calculated at the distance determined by Bailer-Jones et al. (2021). The
$R_{v}=3.1$ reddening law from Fitzpatrick (1999) is used to convert the
extinction from Green et al. (2019) to a visual magnitude extinction. The
stellar parameters derived using the Gaia DR3 parallax are identical to the
values derived with the Bailer-Jones et al. (2021) distance priors. This is
expected because Bailer-Jones et al. (2021) note that the inverse parallax
provides a good distance estimate for stars with positive parallaxes and a
ratio between the parallax error to the parallax of
$\sigma_{\varpi}/\varpi<0.1$. Figures 12 and 13 present the SED fits for
TOI-3984 A and TOI-5293 A, respectively.

Figure 12: The SED fit to TOI-3984 A. Red points are the measurements from
broadband photometry while blue points represent the MIST model flux. The
horizontal bars represent the width for each filter. A NextGen BT-SETTL model
(Allard et al., 2012) is overlaid for reference as a grey line and smoothed
for clarity as a black line. The BT-SETTL model is not used as part of the SED
fit. Figure 13: Identical to Figure 12 but for TOI-5293 A.

##  Appendix B Measuring the rotation period

###  B.1 TESS

We do not search for photometric modulation using the TESS PDCSAP flux or
eleanor CORR_FLUX because the algorithms that produce these light curves are
known to attenuate and distort long-period ($>10$ day) astrophysical signals,
such as starspot-induced photometric variability (see Gilliland et al., 2015;
Van Cleve et al., 2016; Feinstein et al., 2019; Holcomb et al., 2022). We use
the TESS Systematics-insensitive Periodogram package (TESS-SIP; Hedges et al.,
2020) to simultaneously detrend systematics in the uncorrected flux and create
a Lomb-Scargle periodogram. TESS-SIP uses a linear model consisting of
regressors (for this search, 3 principal components and a mean offset) and a
sinusoid component. We limit the search to periods between $1-60$ days and
analyzed all available long-cadence data after excising the transits and bad
quality data (non-zero TESS quality flags) for each star. No significant
period for TOI-3984 A and TOI-5293 A were recovered.

###  B.2 ZTF

TOI-3984 A and TOI-5293 A were observed with ZTF as part of a survey of the
TESS northern sectors (van Roestel et al., 2019) and the photometry is
publicly available under DR15191919https://www.ztf.caltech.edu/ztf-public-
releases.html. ZTF has a plate scale of $1.012\arcsec{}\mathrm{~{}pixel^{-1}}$
(Yao et al., 2019) and all exposures for these objects are 30s. We use the
constraints from the ZTF Science Data System Explanatory
Supplement202020https://web.ipac.caltech.edu/staff/fmasci/ztf/ztf_pipelines_deliverables.pdf
(ZDS) and reject observations with (i) non-zero catflag values (see §13.6 in
ZDS), (ii) values of $\chi<0.5$ and $\chi>1.5$, where $\chi$ is the rms of the
residuals to the PSF fit on the source performed by the ZTF pipeline, (iii)
values of $|\mathtt{sharp}|\geq 0.5$, where sharp is the difference of the
observed and model squared PSF FWHM, and (iv) airmass $>1.8$. TOI-3984 A has
(i) 471 observations in the $zg$ filter between 2018 March 25 and 2022 July 22
with a median cadence and precision of 1 day and $\sim 1.3\%$ and (ii) 498
observations in the $zr$ filter between 2018 March 21 and 2022 July 17 with a
median cadence and precision of 1 day and $\sim 1.1\%$, respectively. TOI-5293
A has (i) 197 observations in the $zg$ filter between 2018 June 26 and 2022
November 06 with a median cadence and precision of 3 days and $\sim 1.5\%$ and
(ii) 181 observations in the $zr$ filter between 2018 June 29 and 2022 October
26 with a median cadence and precision of 3 days and $\sim 1.1\%$,
respectively.

To derive the rotation period, we modeled the ZTF photometry with the juliet
analysis package (Espinoza et al., 2019) which performs the parameter
estimation using the dynamic nested-sampling algorithm dynesty (Speagle,
2020). The photometric model consists of a Gaussian process noise model with
the approximate quasi-periodic covariance function from Foreman-Mackey et al.
(2017). This kernel has allowed for computationally efficient inference of
stellar rotation periods in large datasets that are not uniformly sampled
(e.g., Angus et al., 2018).

Figure 14: (a) displays the ZTF photometry for TOI-3984 A in each filter along
with the best-fitting Gaussian process model for reference. (b) and (c) are
the GLS periodograms for the $zr$ and $zg$ photometry from ZTF. A false-alarm
probability of 0.01% (following Zechmeister & Kürster, 2009) is shown for
reference. (d) presents the ground-based ZTF photometry from panel (a) phased
to the derived rotation period. The large black points represent 4-day bins of
the phased photometry. (e) presents the posterior distribution of the rotation
period from the Gaussian process model. The derived rotation period is
$50.0_{-2.7}^{+2.8}$ days. Figure 15: Identical to Figure 14 but for TOI-5293
A. (a) displays the ZTF photometry along with the best-fitting Gaussian
process model for reference. (b) presents the GLS periodogram for the ZTF
data. (c) presents the phased ground-based ZTF photometry with large black
points representing 2-day bins. (d) presents the posterior distribution of the
rotation period. The derived rotation period is $20.6_{-0.4}^{+0.3}$ days.

We interpret the periodicity of the Gaussian process kernel as the stellar
rotation period and force the same kernel periodicity for both ZTF filters of
the same star. We include a simple white-noise model $\sigma_{\mathrm{phot}}$
in the form of a jitter term that is added in quadrature to the error bars of
the photometry for each filter. The fit for each star uses a uniform prior on
the Gaussian process period of $1.1-100$ days, where the upper limit reflects
the upper limits from other photometric surveys of mid M dwarfs (McQuillan et
al., 2013; Newton et al., 2018). The priors and posterior values for the fits
to ZTF photometry are listed in Table 6. The ZTF data and best fitting
Gaussian process kernel are presented in Figure 14 and 15.

Table 6: ZTF photometric modeling  Parameter | Units | Prior | TOI-3984 A | TOI-5293 A
---|---|---|---|---
$zg$ Gaussian process parameters:
$B$ | Amplitude (ppm) | $\mathcal{J}(10^{-4},10^{12})$ | $470_{-60}^{+70}$ |  $\cdots$
$C$ | Additive factor | $\mathcal{J}(10^{-3},10^{3})$ | $0.8_{-0.8}^{+100}$ |  $\cdots$
$L$ | Length scale (days) | $\mathcal{J}(10^{-3},10^{3})$ | $4.6_{-0.9}^{+1.2}$ |  $\cdots$
Photometric jitter |  $\sigma_{phot}$ (ppm) | $\mathcal{J}(10^{-6},10^{3})$ | $0.03_{-0.03}^{+45.07}$ |  $\cdots$
$zr$ Gaussian process parameters:
$B$ | Amplitude (ppm) | $\mathcal{J}(10^{-4},10^{12})$ | $200_{-40}^{+60}$ | $160_{-50}^{+60}$
$C$ | Additive factor | $\mathcal{J}(10^{-3},10^{3})$ | $0.02_{-0.02}^{+0.21}$ | $0.03_{-0.02}^{+0.3}$
$L$ | Length scale (days) | $\mathcal{J}(10^{-3},10^{3})$ | $60_{-20}^{+40}$ | $120_{-70}^{+200}$
Photometric jitter |  $\sigma_{phot}$ (ppm) | $\mathcal{J}(10^{-6},10^{3})$ | $0.03_{-0.03}^{+43.36}$ | $0.05_{-0.05}^{+42.17}$
Shared Gaussian process parameters:
$P_{GP}$ | Period (days) | $\mathcal{J}(1.1,100)$ | $50.0_{-2.7}^{+2.8}$ | $20.6_{-0.4}^{+0.3}$

##  Appendix C White dwarf companion to TOI-3984 A

We use the nominal Gaia DR3 colors to predict the mass and cooling age using
the WD_models package and the models from Bédard et al. (2020) for a hydrogen
dominated atmosphere. Figure 16 displays the location of TOI-3984 B (TIC
1101522311) on the Gaia color-magnitude diagram (data from Gentile Fusillo et
al., 2021) along with the best-matching model used to derive a photometric age
and mass from its location.

Figure 16: The nominal position of TIC 1101522311, the white dwarf companion
to TOI-3984 A, on the color-magnitude diagram for white dwarfs identified in
Gaia EDR3 by Gentile Fusillo et al. (2021). Contours for fixed masses from
Bédard et al. (2020) are plotted for reference. The best-matching cooling
track from the models is shown with a dashed line.

## References

  * Addison et al. (2021) Addison, B. C., Horner, J., Wittenmyer, R. A., et al. 2021, AJ, 162, 137, doi: 10.3847/1538-3881/ac1685
  * Akaike (1974) Akaike, H. 1974, IEEE Transactions on Automatic Control, 19, 716
  * Akeson et al. (2013) Akeson, R. L., Chen, X., Ciardi, D., et al. 2013, PASP, 125, 989, doi: 10.1086/672273
  * Albrecht et al. (2012) Albrecht, S., Winn, J. N., Johnson, J. A., et al. 2012, ApJ, 757, 18, doi: 10.1088/0004-637X/757/1/18
  * Albrecht et al. (2022) Albrecht, S. H., Dawson, R. I., & Winn, J. N. 2022, PASP, 134, 082001, doi: 10.1088/1538-3873/ac6c09
  * Albrecht et al. (2021) Albrecht, S. H., Marcussen, M. L., Winn, J. N., Dawson, R. I., & Knudstrup, E. 2021, ApJ, 916, L1, doi: 10.3847/2041-8213/ac0f03
  * Allard et al. (2012) Allard, F., Homeier, D., & Freytag, B. 2012, Philosophical Transactions of the Royal Society of London Series A, 370, 2765, doi: 10.1098/rsta.2011.0269
  * Alonso (2018) Alonso, R. 2018, in Handbook of Exoplanets, ed. H. J. Deeg & J. A. Belmonte, 40, doi: 10.1007/978-3-319-55333-7_40
  * Anderson et al. (2020) Anderson, K. R., Lai, D., & Pu, B. 2020, MNRAS, 491, 1369, doi: 10.1093/mnras/stz3119
  * Andrae, R. et al. (2022) Andrae, R., Fouesneau, M., Sordo, R., et al. 2022, A&A, doi: 10.1051/0004-6361/202243462
  * Andrews et al. (2013) Andrews, S. M., Rosenfeld, K. A., Kraus, A. L., & Wilner, D. J. 2013, ApJ, 771, 129, doi: 10.1088/0004-637X/771/2/129
  * Anglada-Escudé & Butler (2012) Anglada-Escudé, G., & Butler, R. P. 2012, ApJS, 200, 15, doi: 10.1088/0067-0049/200/2/15
  * Angus et al. (2018) Angus, R., Morton, T., Aigrain, S., Foreman-Mackey, D., & Rajpaul, V. 2018, MNRAS, 474, 2094, doi: 10.1093/mnras/stx2109
  * Ansdell et al. (2017) Ansdell, M., Williams, J. P., Manara, C. F., et al. 2017, AJ, 153, 240, doi: 10.3847/1538-3881/aa69c0
  * Astropy Collaboration et al. (2018) Astropy Collaboration, Price-Whelan, A. M., Sipőcz, B. M., et al. 2018, AJ, 156, 123, doi: 10.3847/1538-3881/aabc4f
  * Bailer-Jones et al. (2021) Bailer-Jones, C. A. L., Rybizki, J., Fouesneau, M., Demleitner, M., & Andrae, R. 2021, AJ, 161, 147, doi: 10.3847/1538-3881/abd806
  * Bakos et al. (2020) Bakos, G. Á., Bayliss, D., Bento, J., et al. 2020, AJ, 159, 267, doi: 10.3847/1538-3881/ab8ad1
  * Baraffe et al. (2008) Baraffe, I., Chabrier, G., & Barman, T. 2008, A&A, 482, 315, doi: 10.1051/0004-6361:20079321
  * Baruteau et al. (2014) Baruteau, C., Crida, A., Paardekooper, S. J., et al. 2014, in Protostars and Planets VI, ed. H. Beuther, R. S. Klessen, C. P. Dullemond, & T. Henning, 667, doi: 10.2458/azu_uapress_9780816531240-ch029
  * Batalha et al. (2019) Batalha, N. E., Lewis, T., Fortney, J. J., et al. 2019, ApJ, 885, L25, doi: 10.3847/2041-8213/ab4909
  * Batygin et al. (2016) Batygin, K., Bodenheimer, P. H., & Laughlin, G. P. 2016, ApJ, 829, 114, doi: 10.3847/0004-637X/829/2/114
  * Becker et al. (2015) Becker, J. C., Vanderburg, A., Adams, F. C., Rappaport, S. A., & Schwengeler, H. M. 2015, ApJ, 812, L18, doi: 10.1088/2041-8205/812/2/L18
  * Bédard et al. (2020) Bédard, A., Bergeron, P., Brassard, P., & Fontaine, G. 2020, ApJ, 901, 93, doi: 10.3847/1538-4357/abafbe
  * Beleznay & Kunimoto (2022) Beleznay, M., & Kunimoto, M. 2022, MNRAS, 516, 75, doi: 10.1093/mnras/stac2179
  * Bellm et al. (2019) Bellm, E. C., Kulkarni, S. R., Graham, M. J., et al. 2019, PASP, 131, 018002, doi: 10.1088/1538-3873/aaecbe
  * Belokurov et al. (2020) Belokurov, V., Penoyre, Z., Oh, S., et al. 2020, MNRAS, 496, 1922, doi: 10.1093/mnras/staa1522
  * Bensby et al. (2003) Bensby, T., Feltzing, S., & Lundström, I. 2003, A&A, 410, 527, doi: 10.1051/0004-6361:20031213
  * Bessell (1990) Bessell, M. S. 1990, PASP, 102, 1181, doi: 10.1086/132749
  * Boley et al. (2016) Boley, A. C., Granados Contreras, A. P., & Gladman, B. 2016, ApJ, 817, L17, doi: 10.3847/2041-8205/817/2/L17
  * Bonfils et al. (2013) Bonfils, X., Delfosse, X., Udry, S., et al. 2013, A&A, 549, A109, doi: 10.1051/0004-6361/201014704
  * Bonomo et al. (2017) Bonomo, A. S., Desidera, S., Benatti, S., et al. 2017, A&A, 602, A107, doi: 10.1051/0004-6361/201629882
  * Boss (1997) Boss, A. P. 1997, Science, 276, 1836, doi: 10.1126/science.276.5320.1836
  * Bourrier et al. (2018) Bourrier, V., Lovis, C., Beust, H., et al. 2018, Nature, 553, 477, doi: 10.1038/nature24677
  * Bourrier et al. (2022) Bourrier, V., Zapatero Osorio, M. R., Allart, R., et al. 2022, A&A, 663, A160, doi: 10.1051/0004-6361/202142559
  * Bovy (2015) Bovy, J. 2015, ApJS, 216, 29, doi: 10.1088/0067-0049/216/2/29
  * Brady et al. (2023) Brady, M., Bean, J. L., Seifahrt, A., et al. 2023, AJ, 165, 129, doi: 10.3847/1538-3881/acb5f7
  * Bryant et al. (2023) Bryant, E. M., Bayliss, D., & Van Eylen, V. 2023, MNRAS, 521, 3663, doi: 10.1093/mnras/stad626
  * Burn et al. (2021) Burn, R., Schlecker, M., Mordasini, C., et al. 2021, A&A, 656, A72, doi: 10.1051/0004-6361/202140390
  * Burrows & Sharp (1999) Burrows, A., & Sharp, C. M. 1999, ApJ, 512, 843, doi: 10.1086/306811
  * Burt et al. (2020) Burt, J. A., Nielsen, L. D., Quinn, S. N., et al. 2020, AJ, 160, 153, doi: 10.3847/1538-3881/abac0c
  * Cañas et al. (2019) Cañas, C. I., Wang, S., Mahadevan, S., et al. 2019, ApJ, 870, L17, doi: 10.3847/2041-8213/aafa1e
  * Cañas et al. (2022) Cañas, C. I., Kanodia, S., Bender, C. F., et al. 2022, AJ, 164, 50, doi: 10.3847/1538-3881/ac7804
  * Chabrier et al. (2014) Chabrier, G., Johansen, A., Janson, M., & Rafikov, R. 2014, in Protostars and Planets VI, ed. H. Beuther, R. S. Klessen, C. P. Dullemond, & T. Henning, 619, doi: 10.2458/azu_uapress_9780816531240-ch027
  * Chambers et al. (2016) Chambers, K. C., Magnier, E. A., Metcalfe, N., et al. 2016, arXiv e-prints, arXiv:1612.05560.  https://arxiv.org/abs/1612.05560
  * Cheng et al. (2019) Cheng, S., Cummings, J. D., & Ménard, B. 2019, ApJ, 886, 100, doi: 10.3847/1538-4357/ab4989
  * Choi et al. (2016) Choi, J., Dotter, A., Conroy, C., et al. 2016, ApJ, 823, 102, doi: 10.3847/0004-637X/823/2/102
  * Chonis et al. (2014) Chonis, T. S., Hill, G. J., Lee, H., Tuttle, S. E., & Vattiat, B. L. 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9147, Ground-based and Airborne Instrumentation for Astronomy V, ed. S. K. Ramsay, I. S. McLean, & H. Takami, 91470A, doi: 10.1117/12.2056005
  * Chonis et al. (2016) Chonis, T. S., Hill, G. J., Lee, H., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9908, Ground-based and Airborne Instrumentation for Astronomy VI, ed. C. J. Evans, L. Simard, & H. Takami, 99084C, doi: 10.1117/12.2232209
  * Collins et al. (2017) Collins, K. A., Kielkopf, J. F., Stassun, K. G., & Hessman, F. V. 2017, AJ, 153, 77, doi: 10.3847/1538-3881/153/2/77
  * Creevey & Lebreton (2022) Creevey, O. L., & Lebreton, Y. 2022, Masses and ages from FLAME in DR3, Technical note, Observatoire de la Cote d’Azur, Nice.  https://dms.cosmos.esa.int/COSMOS/doc_fetch.php?id=1612899
  * Creevey, O. et al. (2022) Creevey, O., Sordo, R., Pailler, F., & et al. 2022, A&A, doi: 10.1051/0004-6361/202243688
  * Cui et al. (2012) Cui, X.-Q., Zhao, Y.-H., Chu, Y.-Q., et al. 2012, Research in Astronomy and Astrophysics, 12, 1197, doi: 10.1088/1674-4527/12/9/003
  * Cumming et al. (2008) Cumming, A., Butler, R. P., Marcy, G. W., et al. 2008, PASP, 120, 531, doi: 10.1086/588487
  * Cummings et al. (2018) Cummings, J. D., Kalirai, J. S., Tremblay, P. E., Ramirez-Ruiz, E., & Choi, J. 2018, ApJ, 866, 21, doi: 10.3847/1538-4357/aadfd6
  * Cutri et al. (2003) Cutri, R. M., Skrutskie, M. F., van Dyk, S., et al. 2003, VizieR Online Data Catalog, 2246
  * Dash et al. (2022) Dash, S., Majumdar, L., Willacy, K., et al. 2022, ApJ, 932, 20, doi: 10.3847/1538-4357/ac67f0
  * Dawson & Johnson (2018) Dawson, R. I., & Johnson, J. A. 2018, ARA&A, 56, 175, doi: 10.1146/annurev-astro-081817-051853
  * Deng et al. (2012) Deng, L.-C., Newberg, H. J., Liu, C., et al. 2012, Research in Astronomy and Astrophysics, 12, 735, doi: 10.1088/1674-4527/12/7/003
  * Dotter (2016) Dotter, A. 2016, ApJS, 222, 8, doi: 10.3847/0067-0049/222/1/8
  * Dressing & Charbonneau (2015) Dressing, C. D., & Charbonneau, D. 2015, ApJ, 807, 45, doi: 10.1088/0004-637X/807/1/45
  * Durisen et al. (2007) Durisen, R. H., Boss, A. P., Mayer, L., et al. 2007, in Protostars and Planets V, ed. B. Reipurth, D. Jewitt, & K. Keil, 607.  https://arxiv.org/abs/astro-ph/0603179
  * Eastman et al. (2019) Eastman, J. D., Rodriguez, J. E., Agol, E., et al. 2019, arXiv e-prints, arXiv:1907.09480.  https://arxiv.org/abs/1907.09480
  * El-Badry & Rix (2018) El-Badry, K., & Rix, H.-W. 2018, MNRAS, 480, 4884, doi: 10.1093/mnras/sty2186
  * El-Badry et al. (2021) El-Badry, K., Rix, H.-W., & Heintz, T. M. 2021, MNRAS, 506, 2269, doi: 10.1093/mnras/stab323
  * Endl et al. (2006) Endl, M., Cochran, W. D., Kürster, M., et al. 2006, ApJ, 649, 436, doi: 10.1086/506465
  * Engle & Guinan (2018) Engle, S. G., & Guinan, E. F. 2018, Research Notes of the American Astronomical Society, 2, 34, doi: 10.3847/2515-5172/aab1f8
  * Espinoza et al. (2019) Espinoza, N., Kossakowski, D., & Brahm, R. 2019, MNRAS, 490, 2262, doi: 10.1093/mnras/stz2688
  * Evans et al. (2018a) Evans, D. F., Southworth, J., Smalley, B., et al. 2018a, A&A, 610, A20, doi: 10.1051/0004-6361/201731855
  * Evans et al. (2018b) Evans, D. W., Riello, M., De Angeli, F., et al. 2018b, A&A, 616, A4, doi: 10.1051/0004-6361/201832756
  * Fabrycky & Tremaine (2007) Fabrycky, D., & Tremaine, S. 2007, ApJ, 669, 1298, doi: 10.1086/521702
  * Feinstein et al. (2019) Feinstein, A. D., Montet, B. T., Foreman-Mackey, D., et al. 2019, PASP, 131, 094502, doi: 10.1088/1538-3873/ab291c
  * Fitzpatrick (1999) Fitzpatrick, E. L. 1999, PASP, 111, 63, doi: 10.1086/316293
  * Fontanive & Bardalez Gagliuffi (2021) Fontanive, C., & Bardalez Gagliuffi, D. 2021, Frontiers in Astronomy and Space Sciences, 8, 16, doi: 10.3389/fspas.2021.625250
  * Fontanive et al. (2019) Fontanive, C., Rice, K., Bonavita, M., et al. 2019, MNRAS, 485, 4967, doi: 10.1093/mnras/stz671
  * Ford & Rasio (2008) Ford, E. B., & Rasio, F. A. 2008, ApJ, 686, 621, doi: 10.1086/590926
  * Foreman-Mackey et al. (2017) Foreman-Mackey, D., Agol, E., Ambikasaran, S., & Angus, R. 2017, AJ, 154, 220, doi: 10.3847/1538-3881/aa9332
  * Fortney et al. (2021) Fortney, J. J., Dawson, R. I., & Komacek, T. D. 2021, JGRE, 126, e06629, doi: 10.1029/2020JE006629
  * Fortney et al. (2007) Fortney, J. J., Marley, M. S., & Barnes, J. W. 2007, ApJ, 659, 1661, doi: 10.1086/512120
  * Fortney et al. (2020) Fortney, J. J., Visscher, C., Marley, M. S., et al. 2020, AJ, 160, 288, doi: 10.3847/1538-3881/abc5bd
  * Frazier et al. (2023) Frazier, R. C., Stefánsson, G., Mahadevan, S., et al. 2023, ApJ, 944, L41, doi: 10.3847/2041-8213/acba18
  * Fulton et al. (2018) Fulton, B. J., Petigura, E. A., Blunt, S., & Sinukoff, E. 2018, PASP, 130, 044504, doi: 10.1088/1538-3873/aaaaa8
  * Gagné et al. (2018) Gagné, J., Mamajek, E. E., Malo, L., et al. 2018, ApJ, 856, 23, doi: 10.3847/1538-4357/aaae09
  * Gaia Collaboration et al. (2022) Gaia Collaboration, Vallenari, A., Brown, A.G.A., Prusti, T., & et al. 2022, A&A, doi: 10.1051/0004-6361/202243940
  * Gallet et al. (2017) Gallet, F., Bolmont, E., Mathis, S., Charbonnel, C., & Amard, L. 2017, A&A, 604, A112, doi: 10.1051/0004-6361/201730661
  * Gan et al. (2023) Gan, T., Wang, S. X., Wang, S., et al. 2023, AJ, 165, 17, doi: 10.3847/1538-3881/ac9b12
  * Gao et al. (2021) Gao, P., Wakeford, H. R., Moran, S. E., & Parmentier, V. 2021, Journal of Geophysical Research (Planets), 126, e06655, doi: 10.1029/2020JE006655
  * Gentile Fusillo et al. (2021) Gentile Fusillo, N. P., Tremblay, P. E., Cukanovaite, E., et al. 2021, MNRAS, 508, 3877, doi: 10.1093/mnras/stab2672
  * Gibson et al. (2016) Gibson, S. R., Howard, A. W., Marcy, G. W., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9908, Ground-based and Airborne Instrumentation for Astronomy VI, ed. C. J. Evans, L. Simard, & H. Takami, 990870, doi: 10.1117/12.2233334
  * Gilliland et al. (2015) Gilliland, R. L., Chaplin, W. J., Jenkins, J. M., Ramsey, L. W., & Smith, J. C. 2015, AJ, 150, 133, doi: 10.1088/0004-6256/150/4/133
  * Ginsburg et al. (2019) Ginsburg, A., Sipőcz, B. M., Brasseur, C. E., et al. 2019, AJ, 157, 98, doi: 10.3847/1538-3881/aafc33
  * Goldreich & Tremaine (1980) Goldreich, P., & Tremaine, S. 1980, ApJ, 241, 425, doi: 10.1086/158356
  * Graham et al. (2019) Graham, M. J., Kulkarni, S. R., Bellm, E. C., et al. 2019, PASP, 131, 078001, doi: 10.1088/1538-3873/ab006c
  * Gray (2005) Gray, D. F. 2005, The Observation and Analysis of Stellar Photospheres
  * Green et al. (2019) Green, G. M., Schlafly, E., Zucker, C., Speagle, J. S., & Finkbeiner, D. 2019, ApJ, 887, 93, doi: 10.3847/1538-4357/ab5362
  * Greene et al. (2016) Greene, T. P., Line, M. R., Montero, C., et al. 2016, ApJ, 817, 17, doi: 10.3847/0004-637X/817/1/17
  * Gullikson et al. (2014) Gullikson, K., Dodson-Robinson, S., & Kraus, A. 2014, AJ, 148, 53, doi: 10.1088/0004-6256/148/3/53
  * Halverson et al. (2016) Halverson, S., Terrien, R., Mahadevan, S., et al. 2016, Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9908, A comprehensive radial velocity error budget for next generation Doppler spectrometers, 99086P, doi: 10.1117/12.2232761
  * Hardegree-Ullman et al. (2019) Hardegree-Ullman, K. K., Cushing, M. C., Muirhead, P. S., & Christiansen, J. L. 2019, AJ, 158, 75, doi: 10.3847/1538-3881/ab21d2
  * Hayward et al. (2001) Hayward, T. L., Brandl, B., Pirger, B., et al. 2001, PASP, 113, 105, doi: 10.1086/317969
  * Hedges et al. (2020) Hedges, C., Angus, R., Barentsen, G., et al. 2020, Research Notes of the American Astronomical Society, 4, 220, doi: 10.3847/2515-5172/abd106
  * Helling et al. (2008) Helling, C., Woitke, P., & Thi, W. F. 2008, A&A, 485, 547, doi: 10.1051/0004-6361:20078220
  * Henden et al. (2018) Henden, A. A., Levine, S., Terrell, D., et al. 2018, in American Astronomical Society Meeting Abstracts, Vol. 232, American Astronomical Society Meeting Abstracts #232, 223.06
  * Hill et al. (2021) Hill, G. J., Lee, H., MacQueen, P. J., et al. 2021, AJ, 162, 298, doi: 10.3847/1538-3881/ac2c02
  * Hippke & Heller (2019) Hippke, M., & Heller, R. 2019, A&A, 623, A39, doi: 10.1051/0004-6361/201834672
  * Hirano et al. (2020a) Hirano, T., Gaidos, E., Winn, J. N., et al. 2020a, ApJ, 890, L27, doi: 10.3847/2041-8213/ab74dc
  * Hirano et al. (2020b) Hirano, T., Krishnamurthy, V., Gaidos, E., et al. 2020b, ApJ, 899, L13, doi: 10.3847/2041-8213/aba6eb
  * Hobbs et al. (2022) Hobbs, R., Shorttle, O., & Madhusudhan, N. 2022, MNRAS, 516, 1032, doi: 10.1093/mnras/stac2106
  * Holcomb et al. (2022) Holcomb, R. J., Robertson, P., Hartigan, P., Oelkers, R. J., & Robinson, C. 2022, ApJ, 936, 138, doi: 10.3847/1538-4357/ac8990
  * Hord et al. (2021) Hord, B. J., Colón, K. D., Kostov, V., et al. 2021, AJ, 162, 263, doi: 10.3847/1538-3881/ac2602
  * Hord et al. (2022) Hord, B. J., Colón, K. D., Berger, T. A., et al. 2022, AJ, 164, 13, doi: 10.3847/1538-3881/ac6f57
  * Howard et al. (2012) Howard, A. W., Marcy, G. W., Bryson, S. T., et al. 2012, ApJS, 201, 15, doi: 10.1088/0067-0049/201/2/15
  * Howell et al. (2011) Howell, S. B., Everett, M. E., Sherry, W., Horch, E., & Ciardi, D. R. 2011, AJ, 142, 19, doi: 10.1088/0004-6256/142/1/19
  * Hsu et al. (2020) Hsu, D. C., Ford, E. B., & Terrien, R. 2020, MNRAS, 498, 2249, doi: 10.1093/mnras/staa2391
  * Huang et al. (2016) Huang, C., Wu, Y., & Triaud, A. H. M. J. 2016, ApJ, 825, 98, doi: 10.3847/0004-637X/825/2/98
  * Huang et al. (2020a) Huang, C. X., Vanderburg, A., Pál, A., et al. 2020a, Research Notes of the American Astronomical Society, 4, 204, doi: 10.3847/2515-5172/abca2e
  * Huang et al. (2020b) —. 2020b, Research Notes of the American Astronomical Society, 4, 206, doi: 10.3847/2515-5172/abca2d
  * Huang et al. (2020c) Huang, C. X., Quinn, S. N., Vanderburg, A., et al. 2020c, ApJ, 892, L7, doi: 10.3847/2041-8213/ab7302
  * Huehnerhoff et al. (2016) Huehnerhoff, J., Ketzeback, W., Bradley, A., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9908, Ground-based and Airborne Instrumentation for Astronomy VI, ed. C. J. Evans, L. Simard, & H. Takami, 99085H, doi: 10.1117/12.2234214
  * Hunter (2007) Hunter, J. D. 2007, Computing In Science & Engineering, 9, 90, doi: 10.1109/MCSE.2007.55
  * Hwang et al. (2020) Hwang, H.-C., Hamer, J. H., Zakamska, N. L., & Schlaufman, K. C. 2020, MNRAS, 497, 2250, doi: 10.1093/mnras/staa2124
  * Hwang et al. (2022) Hwang, H.-C., Ting, Y.-S., & Zakamska, N. L. 2022, MNRAS, 512, 3383, doi: 10.1093/mnras/stac675
  * Ida & Lin (2005) Ida, S., & Lin, D. N. C. 2005, ApJ, 626, 1045, doi: 10.1086/429953
  * Ida & Lin (2008) —. 2008, ApJ, 673, 487, doi: 10.1086/523754
  * Jackson et al. (2008) Jackson, B., Greenberg, R., & Barnes, R. 2008, ApJ, 678, 1396, doi: 10.1086/529187
  * Jenkins et al. (2010) Jenkins, J. M., Caldwell, D. A., Chandrasekaran, H., et al. 2010, ApJ, 713, L87, doi: 10.1088/2041-8205/713/2/L87
  * Jenkins et al. (2016) Jenkins, J. M., Twicken, J. D., McCauliff, S., et al. 2016, in Proc. SPIE, Vol. 9913, Software and Cyberinfrastructure for Astronomy IV, 99133E, doi: 10.1117/12.2233418
  * Johnson & Soderblom (1987) Johnson, D. R. H., & Soderblom, D. R. 1987, AJ, 93, 864, doi: 10.1086/114370
  * Jordán et al. (2022) Jordán, A., Hartman, J. D., Bayliss, D., et al. 2022, AJ, 163, 125, doi: 10.3847/1538-3881/ac4a77
  * Kanodia & Wright (2018) Kanodia, S., & Wright, J. 2018, Research Notes of the American Astronomical Society, 2, 4, doi: 10.3847/2515-5172/aaa4b7
  * Kanodia et al. (2018) Kanodia, S., Mahadevan, S., Ramsey, L. W., et al. 2018, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10702, Ground-based and Airborne Instrumentation for Astronomy VII, ed. C. J. Evans, L. Simard, & H. Takami, 107026Q, doi: 10.1117/12.2313491
  * Kanodia et al. (2023) Kanodia, S., Mahadevan, S., Libby-Roberts, J., et al. 2023, AJ, 165, 120, doi: 10.3847/1538-3881/acabce
  * Kaplan et al. (2019) Kaplan, K. F., Bender, C. F., Terrien, R. C., et al. 2019, in Astronomical Society of the Pacific Conference Series, Vol. 523, Astronomical Data Analysis Software and Systems XXVII, ed. P. J. Teuben, M. W. Pound, B. A. Thomas, & E. M. Warner, 567
  * Kasper et al. (2016) Kasper, D. H., Ellis, T. G., Yeigh, R. R., et al. 2016, PASP, 128, 105005, doi: 10.1088/1538-3873/128/968/105005
  * Kawashima et al. (2019) Kawashima, Y., Hu, R., & Ikoma, M. 2019, ApJ, 876, L5, doi: 10.3847/2041-8213/ab16f6
  * Kempton et al. (2018) Kempton, E. M. R., Bean, J. L., Louie, D. R., et al. 2018, PASP, 130, 114401, doi: 10.1088/1538-3873/aadf6f
  * Kennedy & Kenyon (2008) Kennedy, G. M., & Kenyon, S. J. 2008, ApJ, 673, 502, doi: 10.1086/524130
  * Kepler et al. (2021) Kepler, S. O., Koester, D., Pelisoli, I., Romero, A. D., & Ourique, G. 2021, MNRAS, 507, 4646, doi: 10.1093/mnras/stab2411
  * Kesseli et al. (2017) Kesseli, A. Y., West, A. A., Veyette, M., et al. 2017, ApJS, 230, 16, doi: 10.3847/1538-4365/aa656d
  * Kilic et al. (2020) Kilic, M., Bergeron, P., Kosakowski, A., et al. 2020, ApJ, 898, 84, doi: 10.3847/1538-4357/ab9b8d
  * Kipping (2013a) Kipping, D. M. 2013a, MNRAS, 435, 2152, doi: 10.1093/mnras/stt1435
  * Kipping (2013b) —. 2013b, MNRAS, 434, L51, doi: 10.1093/mnrasl/slt075
  * Kiseleva et al. (1998) Kiseleva, L. G., Eggleton, P. P., & Mikkola, S. 1998, MNRAS, 300, 292, doi: 10.1046/j.1365-8711.1998.01903.x
  * Knutson et al. (2014) Knutson, H. A., Fulton, B. J., Montet, B. T., et al. 2014, ApJ, 785, 126, doi: 10.1088/0004-637X/785/2/126
  * Kovács et al. (2013) Kovács, G., Hodgkin, S., Sipőcz, B., et al. 2013, in European Physical Journal Web of Conferences, Vol. 47, European Physical Journal Web of Conferences, 01002, doi: 10.1051/epjconf/20134701002
  * Kozai (1962) Kozai, Y. 1962, AJ, 67, 591, doi: 10.1086/108790
  * Kraus et al. (2016) Kraus, A. L., Ireland, M. J., Huber, D., Mann, A. W., & Dupuy, T. J. 2016, AJ, 152, 8, doi: 10.3847/0004-6256/152/1/8
  * Kreidberg (2015) Kreidberg, L. 2015, batman: BAsic Transit Model cAlculatioN in Python, Astrophysics Source Code Library.  http://ascl.net/1510.002
  * Kunimoto et al. (2022) Kunimoto, M., Daylan, T., Guerrero, N., et al. 2022, ApJS, 259, 33, doi: 10.3847/1538-4365/ac5688
  * Laughlin et al. (2004) Laughlin, G., Bodenheimer, P., & Adams, F. C. 2004, ApJ, 612, L73, doi: 10.1086/424384
  * Lépine et al. (2007) Lépine, S., Rich, R. M., & Shara, M. M. 2007, ApJ, 669, 1235, doi: 10.1086/521614
  * Liang et al. (2004) Liang, M.-C., Seager, S., Parkinson, C. D., Lee, A. Y. T., & Yung, Y. L. 2004, ApJ, 605, L61, doi: 10.1086/392509
  * Liddle (2007) Liddle, A. R. 2007, MNRAS, 377, L74, doi: 10.1111/j.1745-3933.2007.00306.x
  * Lidov (1962) Lidov, M. L. 1962, Planet. Space Sci., 9, 719, doi: 10.1016/0032-0633(62)90129-0
  * Lightkurve Collaboration et al. (2018) Lightkurve Collaboration, Cardoso, J. V. d. M. a., Hedges, C., et al. 2018, Lightkurve: Kepler and TESS time series analysis in Python.  http://ascl.net/1812.013
  * Lin et al. (1996) Lin, D. N. C., Bodenheimer, P., & Richardson, D. C. 1996, Nature, 380, 606, doi: 10.1038/380606a0
  * Lin & Papaloizou (1986) Lin, D. N. C., & Papaloizou, J. 1986, ApJ, 309, 846, doi: 10.1086/164653
  * Line et al. (2010) Line, M. R., Liang, M. C., & Yung, Y. L. 2010, ApJ, 717, 496, doi: 10.1088/0004-637X/717/1/496
  * Maciejewski (2020) Maciejewski, G. 2020, Acta Astron., 70, 181, doi: 10.32023/0001-5237/70.3.2
  * Madhusudhan (2012) Madhusudhan, N. 2012, ApJ, 758, 36, doi: 10.1088/0004-637X/758/1/36
  * Madhusudhan (2019) —. 2019, ARA&A, 57, 617, doi: 10.1146/annurev-astro-081817-051846
  * Madhusudhan et al. (2011) Madhusudhan, N., Mousis, O., Johnson, T. V., & Lunine, J. I. 2011, ApJ, 743, 191, doi: 10.1088/0004-637X/743/2/191
  * Magnier et al. (2020) Magnier, E. A., Schlafly, E. F., Finkbeiner, D. P., et al. 2020, ApJS, 251, 6, doi: 10.3847/1538-4365/abb82a
  * Mahadevan et al. (2012) Mahadevan, S., Ramsey, L., Bender, C., et al. 2012, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8446, Proc. SPIE, 84461S, doi: 10.1117/12.926102
  * Mahadevan et al. (2014) Mahadevan, S., Ramsey, L. W., Terrien, R., et al. 2014, in Proc. SPIE, Vol. 9147, Ground-based and Airborne Instrumentation for Astronomy V, 91471G, doi: 10.1117/12.2056417
  * Mai & Line (2019) Mai, C., & Line, M. R. 2019, ApJ, 883, 144, doi: 10.3847/1538-4357/ab3e6d
  * Manara et al. (2018) Manara, C. F., Morbidelli, A., & Guillot, T. 2018, A&A, 618, L3, doi: 10.1051/0004-6361/201834076
  * Marley et al. (2013) Marley, M. S., Ackerman, A. S., Cuzzi, J. N., & Kitzmann, D. 2013, Clouds and Hazes in Exoplanet Atmospheres, ed. S. J. Mackwell, A. A. Simon-Miller, J. W. Harder, & M. A. Bullock, 367–392, doi: 10.2458/azu_uapress_9780816530595-ch015
  * Marley & Robinson (2015) Marley, M. S., & Robinson, T. D. 2015, ARA&A, 53, 279, doi: 10.1146/annurev-astro-082214-122522
  * Marzari & Thebault (2019) Marzari, F., & Thebault, P. 2019, Galaxies, 7, 84, doi: 10.3390/galaxies7040084
  * Masci et al. (2019) Masci, F. J., Laher, R. R., Rusholme, B., et al. 2019, PASP, 131, 018003, doi: 10.1088/1538-3873/aae8ac
  * Mayor et al. (2011) Mayor, M., Marmier, M., Lovis, C., et al. 2011, arXiv e-prints, arXiv:1109.2497.  https://arxiv.org/abs/1109.2497
  * McKinney (2010) McKinney, W. 2010, in Proceedings of the 9th Python in Science Conference, ed. S. van der Walt & J. Millman, 51 – 56
  * McQuillan et al. (2013) McQuillan, A., Aigrain, S., & Mazeh, T. 2013, MNRAS, 432, 1203, doi: 10.1093/mnras/stt536
  * Melbourne et al. (2020) Melbourne, K., Youngblood, A., France, K., et al. 2020, AJ, 160, 269, doi: 10.3847/1538-3881/abbf5c
  * Metcalf et al. (2019) Metcalf, A., Anderson, T., Bender, C., et al. 2019, Optica, 6, 233, doi: 10.1364/OPTICA.6.000233
  * Millholland et al. (2020) Millholland, S., Petigura, E., & Batygin, K. 2020, ApJ, 897, 7, doi: 10.3847/1538-4357/ab959c
  * Moe & Kratter (2021) Moe, M., & Kratter, K. M. 2021, MNRAS, 507, 3593, doi: 10.1093/mnras/stab2328
  * Mohanty et al. (2013) Mohanty, S., Greaves, J., Mortlock, D., et al. 2013, ApJ, 773, 168, doi: 10.1088/0004-637X/773/2/168
  * Mollière et al. (2015) Mollière, P., van Boekel, R., Dullemond, C., Henning, T., & Mordasini, C. 2015, ApJ, 813, 47, doi: 10.1088/0004-637X/813/1/47
  * Mollière et al. (2022) Mollière, P., Molyarova, T., Bitsch, B., et al. 2022, ApJ, 934, 74, doi: 10.3847/1538-4357/ac6a56
  * Monson et al. (2017) Monson, A. J., Beaton, R. L., Scowcroft, V., et al. 2017, AJ, 153, 96, doi: 10.3847/1538-3881/153/3/96
  * Moses et al. (2011) Moses, J. I., Visscher, C., Fortney, J. J., et al. 2011, ApJ, 737, 15, doi: 10.1088/0004-637X/737/1/15
  * Mulders et al. (2015) Mulders, G. D., Pascucci, I., & Apai, D. 2015, ApJ, 798, 112, doi: 10.1088/0004-637X/798/2/112
  * Mustill et al. (2015) Mustill, A. J., Davies, M. B., & Johansen, A. 2015, ApJ, 808, 14, doi: 10.1088/0004-637X/808/1/14
  * Naoz (2016) Naoz, S. 2016, ARA&A, 54, 441, doi: 10.1146/annurev-astro-081915-023315
  * Newton et al. (2016) Newton, E. R., Irwin, J., Charbonneau, D., et al. 2016, ApJ, 821, 93, doi: 10.3847/0004-637X/821/2/93
  * Newton et al. (2018) Newton, E. R., Mondrik, N., Irwin, J., Winters, J. G., & Charbonneau, D. 2018, AJ, 156, 217, doi: 10.3847/1538-3881/aad73b
  * Ngo et al. (2015) Ngo, H., Knutson, H. A., Hinkley, S., et al. 2015, ApJ, 800, 138, doi: 10.1088/0004-637X/800/2/138
  * Ngo et al. (2016) —. 2016, ApJ, 827, 8, doi: 10.3847/0004-637X/827/1/8
  * Ninan et al. (2018) Ninan, J. P., Bender, C. F., Mahadevan, S., et al. 2018, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 10709, Proc. SPIE, 107092U, doi: 10.1117/12.2312787
  * Nordhaus & Spiegel (2013) Nordhaus, J., & Spiegel, D. S. 2013, MNRAS, 432, 500, doi: 10.1093/mnras/stt569
  * Nordhaus et al. (2010) Nordhaus, J., Spiegel, D. S., Ibgui, L., Goodman, J., & Burrows, A. 2010, MNRAS, 408, 631, doi: 10.1111/j.1365-2966.2010.17155.x
  * Öberg et al. (2011) Öberg, K. I., Murray-Clay, R., & Bergin, E. A. 2011, ApJ, 743, L16, doi: 10.1088/2041-8205/743/1/L16