# Searching for Giant Exoplanets around M-dwarf Stars (GEMS) I: Survey
Motivation

Shubham Kanodia Earth and Planets Laboratory, Carnegie Institution for
Science, 5241 Broad Branch Road, NW, Washington, DC 20015, USA  Caleb I. Cañas
NASA Postdoctoral Fellow  NASA Goddard Space Flight Center, 8800 Greenbelt
Road, Greenbelt, MD 20771, USA  Suvrath Mahadevan Department of Astronomy &
Astrophysics, 525 Davey Laboratory, The Pennsylvania State University,
University Park, PA 16802, USA  Center for Exoplanets and Habitable Worlds,
525 Davey Laboratory, The Pennsylvania State University, University Park, PA
16802, USA  Eric B. Ford Department of Astronomy & Astrophysics, 525 Davey
Laboratory, The Pennsylvania State University, University Park, PA 16802, USA
Center for Exoplanets and Habitable Worlds, 525 Davey Laboratory, The
Pennsylvania State University, University Park, PA 16802, USA  Institute for
Computational and Data Sciences, The Pennsylvania State University, University
Park, PA, 16802, USA  Center for Astrostatistics, 525 Davey Laboratory, The
Pennsylvania State University, University Park, PA, 16802, USA  Ravit Helled
Center for Theoretical Astrophysics & Cosmology, University of Zurich,
Winterthurerstr. 190, CH-8057 Zurich, Switzerland  Dana E. Anderson Earth and
Planets Laboratory, Carnegie Institution for Science, 5241 Broad Branch Road,
NW, Washington, DC 20015, USA  Alan Boss Earth and Planets Laboratory,
Carnegie Institution for Science, 5241 Broad Branch Road, NW, Washington, DC
20015, USA  William D. Cochran McDonald Observatory and Department of
Astronomy, The University of Texas at Austin, USA  Center for Planetary
Systems Habitability, The University of Texas at Austin, USA  Megan Delamer
Department of Astronomy & Astrophysics, 525 Davey Laboratory, The Pennsylvania
State University, University Park, PA 16802, USA  Center for Exoplanets and
Habitable Worlds, 525 Davey Laboratory, The Pennsylvania State University,
University Park, PA 16802, USA  Te Han Department of Physics & Astronomy, The
University of California, Irvine, Irvine, CA 92697, USA  Jessica E. Libby-
Roberts Department of Astronomy & Astrophysics, 525 Davey Laboratory, The
Pennsylvania State University, University Park, PA 16802, USA  Center for
Exoplanets and Habitable Worlds, 525 Davey Laboratory, The Pennsylvania State
University, University Park, PA 16802, USA  Andrea S.J. Lin Department of
Astronomy & Astrophysics, 525 Davey Laboratory, The Pennsylvania State
University, University Park, PA 16802, USA  Center for Exoplanets and
Habitable Worlds, 525 Davey Laboratory, The Pennsylvania State University,
University Park, PA 16802, USA  Simon Müller Center for Theoretical
Astrophysics & Cosmology, University of Zurich, Winterthurerstr. 190, CH-8057
Zurich, Switzerland  Paul Robertson Department of Physics & Astronomy, The
University of California, Irvine, Irvine, CA 92697, USA  Guðmundur Stefánsson
NASA Sagan Fellow  Department of Astrophysical Sciences, Princeton University,
4 Ivy Lane, Princeton, NJ 08540, USA  Johanna Teske Earth and Planets
Laboratory, Carnegie Institution for Science, 5241 Broad Branch Road, NW,
Washington, DC 20015, USA Shubham Kanodia<EMAIL_ADDRESS>
(Received October 1, 2023; Revised January 26, 2024)

###### Abstract

Recent discoveries of transiting giant exoplanets around M-dwarf stars (GEMS),
aided by the all-sky coverage of TESS, are starting to stretch theories of
planet formation through the core-accretion scenario. Recent upper limits on
their occurrence suggest that they decrease with lower stellar masses, with
fewer GEMS around lower-mass stars compared to solar-type. In this paper, we
discuss existing GEMS both through confirmed planets, as well as
protoplanetary disk observations, and a combination of tests to reconcile
these with theoretical predictions. We then introduce the Searching for GEMS
survey, where we utilize multi-dimensional nonparameteric statistics to
simulate hypothetical survey scenarios to predict the required sample size of
transiting GEMS with mass measurements to robustly compare their bulk-density
with canonical hot-Jupiters orbiting FGK stars. Our Monte-Carlo simulations
predict that a robust comparison requires about 40 transiting GEMS (compared
to the existing sample of $\sim$ 15) with 5-$\sigma$ mass measurements.
Furthermore, we discuss the limitations of existing occurrence estimates for
GEMS, and provide a brief description of our planned systematic search to
improve the occurrence rate estimates for GEMS.

M-dwarfs, occurrence, giant planet

††journal: AAS Journals††software:  astropy (Robitaille et al., 2013; Astropy
Collaboration et al., 2018), ipython (Pérez & Granger, 2007), matplotlib
(Hunter, 2007), MRExo (Kanodia et al., 2019, 2023a), numpy (Oliphant, 2006),
pandas (McKinney, 2010), scipy (Oliphant, 2007; Virtanen et al., 2020), tess-
point (Burke et al., 2020),

##  1 Introduction

M-dwarfs are the most common type of star in the Galaxy (Reid & Gizis, 1997;
Henry et al., 2006; Reylé et al., 2021). The M-dwarf spectral type spans
almost an order of magnitude in mass ranging from $\sim$ 0.08 $M_{\odot}$ to
$\sim$ 0.6 $M_{\odot}$, and about 2600 K to 4000 K in effective temperature
(Pecaut & Mamajek, 2013). Compared to solar-type stars, these low mass stars
are expected to have correspondingly lower mass protoplanetary disks (Andrews
et al., 2013; Pascucci et al., 2016), and longer Keplerian orbital timescales
(at a fixed distance). The combination of these factors is theorized to make
it difficult to form giant planets around these stars in protoplanetary disks
(Class II) under the core-accretion formation paradigm. Under this paradigm,
traditionally it has been thought that a rocky heavy-element core of roughly
$\sim 10$ $M_{\oplus}$ must first form, which is then followed by runaway
gaseous accretion to rapidly accumulate a massive gaseous envelope111See
Helled et al. (2014); Dawson & Johnson (2018); Fortney et al. (2021); Helled &
Morbidelli (2021) for comprehensive reviews on giant planet formation.
(Mizuno, 1980; Pollack et al., 1996). Early studies showed that due to the
lower disk masses and longer orbital timescales, the formation of a
protoplanet massive enough to initiate runaway gaseous accretion would take
too long with respect to the lifetime of the gas (primarily H/He) in
protoplanetary disks (Laughlin et al., 2004; Ida & Lin, 2005). An alternative
rapid formation mechanism has been proposed in the form of gravitational
instability (GI; Boss, 1997, 2006), which takes place during the proto-stellar
phase (Class 0 or I disk) of massive disks when the star is still embedded in
a molecular cloud (Lada, 1987; Dauphas & Chaussidon, 2011).

While it is estimated that M-dwarfs host multiple small terrestrial planets on
average (Dressing & Charbonneau, 2015; Hardegree-Ullman et al., 2019; Hsu et
al., 2020), the occurrence of giant planets around these types of stars is
more uncertain due to their rarity (Endl et al., 2006; Johnson et al., 2010a;
Maldonado et al., 2019; Schlecker et al., 2022; Gan et al., 2023a; Bryant et
al., 2023). Attempts to understand the occurrence of giant exoplanets around
M-dwarf stars (GEMS) have traditionally been limited to radial velocity (RV)
surveys (Endl et al., 2006; Johnson et al., 2010a; Maldonado et al., 2019;
Sabotta et al., 2021; Schlecker et al., 2022; Pinamonti et al., 2022), since
M-dwarfs accounted for only a minor fraction of the target stars observed by
the Kepler mission (Borucki et al., 2010). This has recently started to change
with NASA’s Transiting Exoplanet Survey Satellite (TESS), and its all-sky
coverage that includes millions of bright M-dwarfs amenable to RV follow-up
(Ricker et al., 2014; Muirhead et al., 2018). Despite the low predicted
occurrence of GEMS, enough M-dwarf host stars have been observed within the
first few TESS cycles that attempts have been made to characterize the
occurrence rate of transiting GEMS (Gan et al., 2023a; Bryant et al., 2023).
However in subsequent sections, we discuss how these recent investigations are
just a first step and motivate a more detailed analysis and characterization
of the TESS detection sensitivity for GEMS, and subsequent estimation of their
occurrence.

In this manuscript, we present the motivation for our Searching for GEMS
survey, where in Section 2 we present the existing GEMS in planet samples, and
protoplanetary disks. In Section 3, we discuss predictions from different
formation and population synthesis models, and observational tests to
distinguish between formation mechanisms. Next, in Section 4 we introduce our
Searching for GEMS survey, its requirements, motivations and provide a brief
outline, before summarizing our work in Section 5. Preliminary results and
trends seen in these existing samples will be discussed and evaluated in part
II of this work.

##  2 GEMS

Figure 1: Planetary mass plotted as a function of stellar mass for GEMS. The
transiting planets have true mass measurements, whereas $M_{p}$ sini is
plotted for the non-transiting ones, with the exception of GJ 463 b which has
a true mass measurement of $\sim 1140$ $M_{\oplus}$ (or $\sim$ 3.6 $M_{J}$)
from astrometry (Sozzetti, 2023).

###  2.1 As Exoplanets

In Figure 1 we show the transiting and non-transiting GEMS as queried from the
NASA Exoplanet Archive on 2023 June 7 (Akeson et al., 2013; NASA Exoplanet
Archive, 2023) resulting in 10 non-transiting (with $M_{p}$ sini $>$ 100
$M_{\oplus}$) and 18 transiting GEMS (with planetary radius $\gtrsim$ 8
$R_{\oplus}$) orbiting stars with Teff $\lesssim$ 4000 K, with precise masses
$>$ 3-$\sigma$ (Table 1). In addition, there have been some detections of GEMS
with microlensing (Suzuki et al., 2016), and direct imaging (Lannier et al.,
2016; Nielsen et al., 2019) which contribute to statistical analysis; this
will especially be true with hundreds of expected detections from Roman Space
Telescope (Penny et al., 2019). While the transiting GEMS have largely been
clustered around early M-dwarfs (M0-M2), there have been recent discoveries of
four mid M-dwarf GEMS — TOI-5205 b (Kanodia et al., 2023b), TOI-3235 b (Hobson
et al., 2023), TOI-519 b (Kagetani et al., 2023), and TOI-4860 b (Almenara et
al., 2023; Triaud et al., 2023). The mid M-dwarf GEMS, in addition to recent
RV detections around mid-to-late M-dwarfs (e.g. Morales et al., 2019;
Quirrenbach et al., 2022), defy expectations from population synthesis models
in protoplanetary disks and break the inferred mass-budget222An extreme
example of this was the discovery of a 5 $M_{J}$ planet around a 25 $M_{J}$
brown dwarf, which likely formed through gravitational instability or
fragmentation (Lodato et al., 2005)..

We also note that while most of these transiting GEMS fall in the canonical
hot-Jupiter parameter space based on their typical orbital periods of $<$ 10
days, it is erroneous to classify them as hot. Because of their cooler and
lower-mass M-dwarf hosts, the transiting GEMS are not ‘hot’, with equilibrium
temperatures $<$ 1000 K, and hence not expected to be inflated like hot-
Jupiters (Weiss et al., 2013; Dawson & Johnson, 2018; Thorngren & Fortney,
2018). Furthermore, their $a/R_{*}$ are $\gtrsim 10$, and hence these objects
may not be tidally locked or tidally circularized like hot-Jupiters. Thus,
transiting GEMS likely have different planetary properties (both bulk and
atmospheric) than hot-Jupiters, and should not be classified as such.

Table 1: Transiting Giant Exoplanets around M-dwarf Stars (GEMS) with $R_{p}$
$\gtrsim$ 8 $R_{\oplus}$, and Teff$<$ 4000 K to within 1-$\sigma$.

Planet Name | Pl. Mass | Pl. Radius | Orbital Period | Insolation | Teff | St. Mass | Distance | This | References
---|---|---|---|---|---|---|---|---|---
| $M_{\oplus}$ | $R_{\oplus}$ | d | $S_{\oplus}$ | K | $M_{\odot}$ | pc | Survey |
TOI-3984 A b | 44.00${}^{+8.73}_{-7.99}$ | 7.90${}^{+0.24}_{-0.24}$ | 4.3533 | 17.19 $\pm$ 1.98 | 3480 $\pm$ 90 | 0.49 $\pm$ 0.02 | 108 | $\bigstar$ | a
TOI-3629 b | 82.64 $\pm$ 6.36 | 8.29 $\pm$ 0.22 | 3.9366 | 41.44 $\pm$ 4.82 | 3870 $\pm$ 90 | 0.63 $\pm$ 0.02 | 130 | $\bigstar$ | b, i
TOI-4860 b‡ | 86.70 $\pm$ 1.90 | 8.58 $\pm$ 0.30 | 1.5228 | 38.64 $\pm$ 3.41 | 3255 $\pm$49 | 0.34 $\pm$ 0.01 | 80 | - | c
TOI-5344 b | 135.00 $\pm$ 17.00 | 9.80 $\pm$ 0.50 | 3.7926 | 35.58 $\pm$ 3.05 | 3757 $\pm$ 51 | 0.59${}^{+0.03}_{-0.02}$ | 136 | $\bigstar$ | d, i
HATS-75 b | 156.05 $\pm$ 12.40 | 9.91 $\pm$ 0.15 | 2.7887 | 58.29 $\pm$ 0.74 | 3790 $\pm$ 6 | 0.60 $\pm$ 0.01 | 195 | - | e
TOI-1899 b | 211.71${}^{+13.76}_{-13.54}$ | 11.11${}^{+0.36}_{-0.34}$ | 29.0903 | 3.06 $\pm$ 0.33 | 3909 $\pm$ 90 | 0.63${}^{+0.03}_{-0.03}$ | 128 | $\bigstar$ | f
HATS-6 b | 101.00 $\pm$ 22.00 | 11.19 $\pm$ 0.21 | 3.3253 | 42.73 $\pm$ 2.09 | 3724 $\pm$ 18 | 0.57${}^{+0.02}_{-0.03}$ | 169 | - | g
TOI-3714 b | 222.48 $\pm$ 9.53 | 11.32 $\pm$ 0.34 | 2.1548 | 60.02 $\pm$ 6.53 | 3660 $\pm$ 90 | 0.53 $\pm$ 0.02 | 113 | $\bigstar$ | b, i
TOI-3235 b | 211.35 $\pm$ 7.95 | 11.40 $\pm$ 0.49 | 2.5926 | 17.47 $\pm$ 2.19 | 3196 $\pm$ 67 | 0.39 $\pm$ 0.02 | 73 | - | h
TOI-519 b | 147.00 $\pm$ 27.00 | 11.55 $\pm$ 0.33 | 1.2652 | 52.98 $\pm$ 4.43 | 3322 $\pm$ 49 | 0.34 $\pm$ 0.01 | 115 | - | j, i
HATS-74 A b* | 464.03 $\pm$ 44.50 | 11.57 $\pm$ 0.23 | 1.7319 | 108.46 $\pm$ 4.08 | 3777 $\pm$ 10 | 0.60 $\pm$ 0.01 | 299 | - | e
TOI-5205 b | 342.70${}^{+17.70}_{-16.80}$ | 11.60 $\pm$ 0.30 | 1.6308 | 49.24 $\pm$ 4.24 | 3430 $\pm$ 54 | 0.39 $\pm$ 0.01 | 87 | $\bigstar$ | k
TOI-5293 A b | 170.45${}^{+21.84}_{-21.86}$ | 11.89${}^{+0.44}_{-0.41}$ | 2.9303 | 37.70 $\pm$ 4.16 | 3586 $\pm$ 90 | 0.54 $\pm$ 0.02 | 161 | $\bigstar$ | a
TOI-3757 b | 85.30${}^{+8.80}_{-8.70}$ | 12.00${}^{+0.40}_{-0.50}$ | 3.4388 | 54.81 $\pm$ 3.78 | 3913 $\pm$ 56 | 0.64 $\pm$ 0.02 | 181 | $\bigstar$ | l
Kepler-45 b* | 162.09 $\pm$ 22.25 | 12.11${}^{+0.34}_{-0.45}$ | 2.4552 | 85.88 $\pm$ 6.11 | 3950 $\pm$ 40 | 0.62${}^{+0.02}_{-0.02}$ | 382 | - | m
HATS-76 b* | 835.57 $\pm$ 28.29 | 12.09 $\pm$ 0.35 | 1.9416 | 131.80 $\pm$ 5.44 | 4016 $\pm$ 17 | 0.66 $\pm$ 0.02 | 386 | - | e
TOI-4201 b | 823.00${}^{+21.00}_{-20.00}$ | 13.11 $\pm$ 0.45 | 3.5819 | 52.83 $\pm$ 4.49 | 3920 $\pm$ 50 | 0.63 $\pm$ 0.02 | 188 | $\bigstar$ | n, o, i
NGTS-1 b*† | 258.08${}^{+20.98}_{-23.84}$ | 14.91${}^{+6.84}_{-3.70}$ | 2.6473 | 67.26 $\pm$ 19.55 | 3916 $\pm$ 67 | 0.62${}^{+0.02}_{-0.06}$ | 218 | - | p

\textsuperscript{$\bigstar$}\textsuperscript{$\bigstar$}footnotetext:  GEMS
discovered through this survey

**footnotetext: GEMS with distance $>$ 200 pc, and hence will not be included
in the occurrence rate sample. We note that there exists HATS-77 b around a
dwarf star with Teff of 4071 K, which is ostensibly an M-dwarf host (Jordán et
al., 2022). However, since it the system is $>$ 200 pc, it is not included in
the statistical sample either

$\textdaggerdbl$$\textdaggerdbl$footnotetext: TOI-4860 b was also confirmed in
a separate publication by Triaud et al. (2023).

$\textdagger$$\textdagger$footnotetext: NGTS-1 b is not shown in the plots in
this manuscript since it has an imprecise radius to due its grazing transit.

††footnotetext: a) Cañas et al. (2023), b) Cañas et al. (2022), c) Almenara et
al. (2023), d) Han et al. (in prep.), e) Jordán et al. (2022), f) Cañas et al.
(2020); Lin et al. (2023), g) Hartman et al. (2015), h) Hobson et al. (2023),
i) Hartman et al. (2023), j) Kagetani et al. (2023), k) Kanodia et al.
(2023b), l) Kanodia et al. (2022), m) Johnson et al. (2012), n) Delamer et al.
(2023), o) Gan et al. (2023b), p) Bayliss et al. (2018)

###  2.2 As structure in Protoplanetary Disks

In addition to the direct detection of exoplanets, the high-resolution and
high-contrast imaging of disks has enabled the detection of structures and
gaps in Class II protoplanetary disks. While there are a number of physical
mechanisms that have been proposed to explain these structures, the presence
of young proto-planets has been the topic of recent investigations (Dong et
al., 2015). For example, van der Marel & Mulders (2021) identify potential
trends between the incidence of structured disks and stellar mass that can be
accounted for by massive Jovian-sized exoplanets. While they note a reduction
in the prevalence of structured (rings or transition) with lower stellar
masses, roughly 1 – 10 $\%$ of their $>$ 100 disks around such stars require
the presence of a $>$ 1 $M_{J}$ planet to explain the features present. These
trends were extended to very low mass stars (VLMS) and brown dwarfs using
high-angular resolution observations from ALMA, which suggested the presence
of substructures in a fraction of these disks (Pinilla, 2022). Zhang et al.
(2023) perform a similar analysis and compare the incidence of structures in
disks in the Taurus star-forming region to infer the occurrence of
protoplanets as a function of stellar mass and orbital-separation.

Additionally, Curone et al. (2022) performed a detailed analysis of the disk
CIDA 1 around a $\sim$ 0.2 $M_{\odot}$ star and the substructures present
therein, and indicate a planet $>$ 1.4 $M_{J}$ at $\sim$ 10 AU being
responsible for the observed morphology. Similarly Long et al. (2023) note the
presence of two gaps in the disk around an M3.5 star J04124068+2438157, which
they attribute to a Saturn-mass planet at $\sim$ 90 AU.

To summarize, there is observational evidence regarding the existence of GEMS
at large separations in young (Class II, i.e., $<$ 10 Myr) systems, and at
closer separations in mature stellar systems. This will be further bolstered
by the addition of hundreds of GEMS projected to be astrometrically detected
through Gaia DR4 (Sozzetti et al., 2014; Perryman et al., 2014).

##  3 Searching for GEMS Survey Motivation

###  3.1 Forming GEMS

####  3.1.1 Formation during protoplanetary phase

Figure 2: Green triangles show an estimate of the heavy-element content
($M_{Z}$) of GEMS from planetary interior models (Thorngren et al., 2016)
compared to the Class II disk dust mass ($M_{d}$) estimates as orange squares
(Manara et al., 2022) and the median (and 1-$\sigma$) trend seen in the Lupus
sample (Ansdell et al., 2016). The green arrows show the approximate expected
disk dust masses required to form the green triangles assuming 10% formation
efficiency (Liu et al., 2019b). The red line shows the range of dust masses
for Class 0 and I disks from Tychoniec et al. (2020). Takeaway: The formation
of GEMS necessitates disks with many 100s of $M_{\oplus}$ of heavy-elements.
This could be achieved by anomalously massive Class II disks with
underestimated dust masses, or Class 0/I disks.

Under the core-accretion paradigm (Mizuno, 1980) — ignoring migration and
transport of material across the disk — a massive solid core is formed by
accretion inside the protoplanet’s feeding zone (roughly a few Hill radii; Ida
& Lin, 2004a; Alibert et al., 2005). Once this zone is depleted, the planet
reaches isolation mass where its accretion slows down and takes place
primarily through the accretion of gas on to its envelope up until it reaches
crossover mass where the mass of solids is roughly equal to the mass of gas
(planetary mass of $\sim$ 30 $M_{\oplus}$; heavy-element mass of $\sim 10-20$
$M_{\oplus}$). If this threshold is reached when the disk still contains its
gas, it initiates exponential runaway gaseous accretion where a Neptune sized
planet accretes enough gas to become a gas giant. We note the caveats that
recent studies have suggested that the energy released during accretion of the
solid core might delay runaway gaseous accretion (Venturini & Helled, 2020;
Kessler & Alibert, 2023), such that the runaway phase only occurs much later
at a planet mass of $\sim 100$ $M_{\oplus}$ with a heavy-element mass of 20–30
$M_{\oplus}$. On the other hand, simulations have also shown the feasibility
of forming giant planets with lower mass cores of $\sim$ 4 $M_{\oplus}$ in
lower surface density disks, albeit over much longer times-scales (Movshovitz
et al., 2010). Therefore predictions for solid core mass are dependent on
models and initial conditions assumed. Yet GEMS, especially those around mid-
to-late M-dwarfs, continue to present a challenge for theories of planet
formation during the protoplanetary phase, i.e., core-accretion (Ida & Lin,
2005; Liu et al., 2019a; Miguel et al., 2020; Burn et al., 2021).

Summarizing briefly, there are two main reasons for this difficulty in forming
GEMS in the protoplanetary phase with core accretion. The first is the the
dust mass333Given the greater than order of magnitude uncertainties associated
with the estimates for these quantities, in this manuscript we use the terms
dust, heavy-elements, solids and metals interchangeably. budget of the disk
available to form giant planets. Based on estimates of exoplanet heavy-element
content ($M_{Z}$) and efficiency of planet formation, Kanodia et al. (2023b)
show how median Class II disks might not have enough dust mass to form GEMS.
In Figure 2 we compare the estimated heavy-element content ($M_{Z}$) for GEMS
from planetary interior models (Thorngren et al., 2016), with disk dust masses
($M_{d}$) and show that assuming an optimistic planet formation efficiency of
10% through pebble accretion (Liu et al., 2019b) necessitates disks with many
100s of $M_{\oplus}$ of heavy-elements. We note the caveat here that recent
giant planet interior models tend to predict a lower heavy-element content
when incorporating results from newer equations of state, results from the
Solar-system gas giants, and atmospheric metallicity estimates (Miguel et al.,
2022; Miguel & Vazan, 2023; Müller & Helled, 2021, 2022), which might
alleviate some of this mass deficit. The second is the timescale of formation
of a solid massive core to initiate runaway gaseous accretion: the crossover
mass must be attained before the gas in the disk is accreted on to the host
star. Laughlin et al. (2004) show that the lower disk masses (and subsequently
surface density) coupled with longer Keplerian timescales (also due to lower
host stellar masses) mean that planetary cores around M-dwarfs at 5 AU might
take too long to form relative to the lifetime of the disks.

Underestimated disk dust masses: During the protoplanetary phase, i.e. Class
II disks, the dust mass444See Miotello et al. (2022) for a review of disk dust
mass estimation. is typically estimated based on continuum flux measurements
of mm sized dust particles, either extrapolating from measurements at 850
$\mu$m (Hildebrand, 1983), or fitting SEDs to multi-wavelength data (Pinte et
al., 2008). These flux-to-mass estimates break down if the continuum emission
is optically thick (Eisner et al., 2018; Rilinger et al., 2023; Xin et al.,
2023), or in the presence of gaps or rings in the disk (Liu et al., 2022),
leading to underestimation of the dust masses by 3 – 10x.

More fundamentally, the dust mass in mm sized particles in ALMA measurements
does not represent the true primordial mass budget, with formation already
underway locking up the dust in $>$ mm sized particles (Greaves & Rice, 2010;
Najita & Kenyon, 2014), as well as depletion due to radial drift (Appelgren et
al., 2023). If the accretion processes required to form a massive core (to
initiate runaway gaseous accretion) started earlier in the protoplanetary
stage, the true mass budget available for planet formation would be larger
than the 850 $\mu$m mass estimates.

Alleviating the timescale problem: The timescale problem could potentially be
circumvented by pebble accretion, which is a much faster process than
planetesimal accretion555Though population synthesis by Liu et al. (2019a)
still struggled to form giant planets around low-mass stars through pebble-
accretion. (Lambrechts & Johansen, 2012; Savvidou & Bitsch, 2023).
Furthermore, new theories suggest that some young disks could concentrate
solids in their mid-plane through self-gravitating spiral waves, thereby
hastening the formation of a core massive enough to reach the runaway stage
(Haghighipour & Boss, 2003; Baehr, 2023). Lastly, studies of large disk
clusters have shown that the low mass M-dwarf disks tend to have longer
lifetimes than those around more massive stars (Pfalzner et al., 2022), with
some extreme examples existing in the form of so-called ‘Peter-Pan’ disks
lasting many 10s of Myr (Flaherty et al., 2019; Silverberg et al., 2020;
Coleman & Haworth, 2020; Wilhelm & Portegies Zwart, 2022). Together these
factors might ease the timescale problem against the formation of GEMS.

Thus, to summarize, while it has traditionally been thought to have been
difficult to form GEMS through core-accretion during the protoplanetary disk
phase, this formation pathway cannot be ruled out given the dependence on
numerous poorly understood disk parameters.

####  3.1.2 Formation during protostellar phase

An alternative to core accretion is the formation of GEMS in the protostellar
phase, i.e., in Class 0, I disks (or protostars) through gravitational
instability (GI; Kuiper, 1951; Cameron, 1978; Boss, 1997). Under this
mechanism, massive disks can start fragmenting and forming dense self-
gravitating clumps as precursors to giant planets666See Durisen et al. (2007)
for a review of GI.. Gravitational collapse is prevented close to the star due
to the faster rotation period and thermal pressure (higher temperatures). Boss
(2006, 2011) show how GI can form giant planets around M-dwarfs with mass 0.1
and 0.5 $M_{\odot}$ at large separations, which is then followed by migration
induced by the spiral arms (Boss, 2023). Mercer & Stamatellos (2020) run
smoothed particle hydrodynamics (SPH) simulations to show that the GI
formation of GEMS necessitates disk-to-star mass ratios between $\sim 0.3$ to
0.6. Haworth et al. (2020) presented a range of SPH models for massive disks
around M-dwarf stars to show the difference between disk-to-star mass ratios
for disks that remain axisymmetric, produce spiral arms or fragment into
clumps. Boss & Kanodia (2023) perform GI-based population synthesis to
quantify the frequency with which GEMS can be formed around a range of stellar
masses ranging from 0.1 to 0.5 $M_{\odot}$ and disk-to-star mass ratios of
0.05 to 0.3. These mass ratios are consistent with those seen in Class 0/I
protostellar samples from VLA (Tychoniec et al., 2018, 2020; Xu, 2022;
Fiorellino et al., 2023).

###  3.2 Observationally distinguishing between formation mechanisms

In this subsection, we consider and establish a few potential mechanisms to
observationally distinguish between forming GEMS via core-accretion vs.
gravitational instability.

####  3.2.1 Stellar metallicity trends

Since the discovery of the first few gas giant exoplanets studies have shown
that the occurrence of these planets around FGK stars is positively correlated
with stellar metallicity (Gonzalez, 1997; Santos et al., 2001; Fischer &
Valenti, 2005; Wang & Fischer, 2015; Petigura et al., 2018; Narang et al.,
2018; Osborn & Bayliss, 2020). This positive metallicity trend agrees well
with predictions from the core-accretion model of formation (Ida & Lin, 2004b;
Matsumura et al., 2021), where higher metallicity stars correspond to higher
metallicity disks (higher solid surface density) and translates into faster
solid core formation777Here we simplistically assume uniform disk metallicity,
which is likely inaccurate given the possibility of dust settling, local
metallicity gradients and bumps.. This explanation of the metallicity trend is
further aided by observational evidence that higher metallicity disks tend to
live longer (Yasui et al., 2009, 2010), with the higher opacity potentially
shielding the disk against photoevaporation and reducing disk accretion (Yasui
et al., 2021). Overall, numerous simulations and observational studies have
shown that the core accretion paradigm of planet formation favours higher
metallicity host stars.

On the contrary, simulations suggest that lower metallicity molecular clouds
should favour gravitational fragmentation (Matsukoba et al., 2022; Elsender &
Bate, 2021), while GI is largely agnostic of the disk metallicity (Boss,
2002). Put simply, in the optically thick environment of the disk, higher
stellar metallicity leads to higher opacities (due to more free electrons in
the disk) that can slow down the cooling timescales. However, Boss (2002)
showed that at the typical separations at which GI occurs, the temperature
equilibrates much faster than the dynamical timescales and a factor of few
($\pm$ 0.5 dex) change in metallicity does not make an appreciable difference
in the efficiency of GI.

This difference in stellar metallicity dependence is indeed seen as one of the
cleaner ways to distinguish between the two mechanisms of giant planet
formation given a large enough sample, which also sidesteps the numerous
complications associated with other methods such as atmospheric chemistry
(Venturini & Helled, 2020; Mollière et al., 2022). This difference has been
noticed in the observed sample of giant planets and brown dwarfs orbiting FGK
stars around 4 – 10 $M_{J}$ (Santos et al., 2017; Schlaufman, 2018; Narang et
al., 2018), with lower mass giant planets ($\sim$ 1–4 $M_{J}$) preferring more
metal rich stars than the more massive giant planets ($>$ 10 $M_{J}$). This
feature is hypothesized to be explained by prevalence of core-accretion formed
planets below this transition, and GI formed objects (planets or brown dwarfs)
above it.

##### Complexities in M dwarf metallicity determination

While studies have attempted to extend the positive metallicity dependence of
giant planets to the M-dwarf spectral type (Johnson et al., 2010b; Maldonado
et al., 2019; Gan et al., 2022), with their cooler stellar atmospheres,
M-dwarfs are blanketed with molecular features that cause complexities in
metallicity determination. The molecular features complicate continuum
estimation (Pineda et al., 2013) and cause inaccuracies in line profile
measurement due to line blending, especially in the optical, where the spectra
is dominated by TiO as the dominant source of opacity (Kirkpatrick et al.,
1991; Jorgensen, 1994). Rains et al. (2021) discuss how the errors in the TiO
line lists (McKemmish et al., 2019) manifest in discrepancies in model
atmospheres. These issues are further exacerbated by low S/N observations in
the optical due to the low effective temperatures and luminosities of these
stars.

Recently, the introduction of high-resolution near-infrared spectrographs has
helped ameliorate this situation as metallicity determination has moved from
photometric relations (Bonfils et al., 2005; Schlaufman & Laughlin, 2010;
Neves et al., 2013) and spectral indices (Terrien et al., 2012; Mann et al.,
2013; Kuznetsov et al., 2019) to empirical spectral comparison methods (Yee et
al., 2017; Stefansson et al., 2020). With increased sophistication and
accuracy for stellar atmospheric models, spectral synthesis techniques (Bean
et al., 2006; Woolf & Wallerstein, 2005; Lindgren et al., 2016; Souto et al.,
2020, 2021, 2022; Tabernero et al., 2022; Iyer et al., 2022) and data driven
approaches (Antoniadis-Karnavas et al., 2020; Birky et al., 2020; Passegger et
al., 2020) have demonstrated a further improvement. Additionally, studies are
now moving towards elemental abundance measurements for M dwarfs (Veyette et
al., 2017; Souto et al., 2017; Maldonado et al., 2020; Ishikawa et al., 2021;
Shan et al., 2021), similar to those for FGK stars (Hinkel et al., 2016; Teske
et al., 2019; Wilson et al., 2021). Uniform high-resolution H-band
observations with APOGEE of all confirmed exoplanets and TOIs through SDSS V
(Kollmeier et al., 2017) may offer the opportunity to utilize existing
techniques that have been demonstrated to work well for M-dwarf metallicities.

####  3.2.2 Planet atmospheric composition

Studies have suggested that the metallicity and composition of the atmospheres
of giant planets could hold clues to their formation and evolution history,
ranging from the mechanisms and location of formation, and the subsequent
migration if applicable (Helled & Bodenheimer, 2010; Öberg et al., 2011;
Madhusudhan et al., 2014; Knierim et al., 2022; Dash et al., 2022, and
numerous others). Further to the above discussions, studies have suggested
potential differences in planetary atmospheres formed through core-accretion
vs GI (Hobbs et al., 2022). However, this initial picture is likely to be
complicated by location of formation and subsequent planetary accretion
(Helled et al., 2014). Mollière et al. (2022) list the numerous pitfalls and
challenges in trying to invert the formation history of individual planets
based on present-day atmospheres. Therefore, extreme caution must be exercised
in attempts to connect the present-day atmospheric observations with formation
and evolution history. Instead it is possible that planetary mass –
atmospheric (and bulk) metallicity trends across a homogeneous sample of
planets, sampled across planetary mass and stellar mass space, present a
better alternative (Thorngren et al., 2016; Teske et al., 2019; Welbanks et
al., 2019).

We can now begin to test the limits of these current theories by atmospheric
characterization of carefully selected samples across a larger stellar mass
range with JWST and ARIEL in the future (Tinetti et al., 2016).

##  4 Searching for GEMS Survey Outline

Considering the theoretical considerations (and limitations) presented above,
and the observational tests to discriminate between the formation of GEMS in
the protoplanetary (through core-accretion) vs. protostellar (through
gravitational instability), we ask the following questions:

  1. 1.

Mass-Radius+ (M-R+) relations: Does the difficulty in forming GEMS in this
mass starved regime through core-accretion manifest as a systematic difference
in their bulk-properties as a function of stellar mass, i.e., characterizing
the sample of giant planets in 3D (planet mass, planet radius, stellar mass)?

  2. 2.

Occurrence Rates: Do GEMS have a lower occurrence compared to similar giant
planets in short orbital periods (the canonical hot Jupiters) around FGK
stars? In particular, the M-dwarf spectral type also allows comparison of the
giant planet occurrence rate as a function of spectral sub-type.

To address these questions, we have started the Searching for GEMS survey,
which is currently ongoing, and entails the follow-up and confirmation of new
GEMS planet candidates in addition to performing demographical analysis on
TESS light-curves to provide improved constraints on the occurrence of short-
period GEMS. This transit survey will also contribute GEMS to answer the
question: Does the metallicity transition in the planetary mass – stellar
metallicity plane between core-accretion and GI (Section 3.2.1) also depend on
the host-stellar mass? To answer this we will extend the 2D planetary mass –
stellar metallicity plane to a third dimension to include stellar mass.

###  4.1 Requirements for M-R+ relations

In order to answer the question posed for M-R+ relations we generate a set of
simulated catalogues of GEMS with mass measurements based on different
assumptions, and assess our ability to recover the input trends as a function
of catalogue size.

Figure 3: The 2D joint probability distribution — f(m, r)FGK — for $\sim$ 350
giant planets orbiting FGK stars shown in the background, where the black
points are the real measurements of FGK planets, the red dots are an example
simulation of 50 GEMS obtained via rejection sampling of the underlying FGK
PDF, and the underlying colour is their probability. Each of these simulated
GEMS sample of planets is compared with the FGK sample to estimate the minimum
sample size required for the survey as described in Section 4.1.1.

Figure 4:  Figure 5:

Figure 6: The solid line represents the median metric value, while the
different shaded regions represent the 16-84%, 5-95%, and 1-99% percentile
regions respectively. The vertical black line is the minimum sample size of
transiting GEMS required based on the comparing the distributions. Left:
Welch’s t-test suggesting the need for about 20 GEMS if we solely wish to
compare the median values of the two distributions assuming normality. Right:
The Earth-mover distance, which suggests the need for about 35 GEMS to
distinguish between the two distributions agnostic of Gaussian assumptions.
Takeaway: Identifying the minimum sample size required to distinguish between
a sample of giant planets around FGK stars and GEMS based on their M-R
distribution when the GEMS are about 60% in mass of the FGK planets, we
require about 35 GEMS if we do not assume an underlying normal distribution
and 20 GEMS if we do.

####  4.1.1 Mass-Radius (M-R)

We start off by trying to recover and quantify differences in the 2D mass-
radius (M-R) distribution of giant planets between a sample of FGK stars (
$\sim$ 0.6 $M_{\odot}$ – 1.5 $M_{\odot}$) and a simulated sample of GEMS
($\lesssim$ 0.6 $M_{\odot}$). Our FGK sample consists of $\sim 350$ transiting
giant planets (15 $M_{\oplus}$ $\gtrsim$ $R_{p}$ $\gtrsim$ 8 $M_{\oplus}$)
orbiting stars ranging from $\sim$ 0.6 $M_{\odot}$ to 1.5 $M_{\odot}$ with
planetary masses known to better than 3-$\sigma$ (Akeson et al., 2013; NASA
Exoplanet Archive, 2023). The steps followed are as follows:

  1. 1.

Obtain joint mass-radius distribution — f(m, r)FGK — for FGK giant planet
sample: We fit the FGK sample using the updated non-parametric inference tool
MRExo (Kanodia et al., 2019, 2023a), which utilizes beta-density (or
normalized Bernstein) polynomials (Ning et al., 2018) to jointly fit their
masses and radii and obtain the 2D probability density function (PDF) — f(m,
r) — where m, r signify the planetary mass and radius respectively. We use the
cross-validation functionality within this toolkit to optimize for the number
of degrees (for the beta density functions) to be 48 in each dimension
(Kanodia et al., 2019, 2023a).

  2. 2.

Simulated M-dwarf sample: We perform rejection sampling in 2D to obtain a
simulated M-dwarf sample with n planets in mass-radius space following the
same distribution as the FGK sample, to which we ascribe 10-$\sigma$ and
5-$\sigma$ errors in radius and mass respectively. We then use MRExo to fit
the joint M-R distribution — f(m, r)M=FGK — for this simulated sample. We
repeat this step 100x times to obtain a distribution of PDFs for the simulated
M-dwarf catalogue across 6 steps spanning a range of GEMS sample sizes from 15
to 150 (Figure 3). We simulate the entire M-dwarf catalogue from scratch
(instead of training it on the existing transiting GEMS sample) to avoid any
potential biases that could be incurred from this limited sample size, and
estimate the ideal sample-size in an agnostic manner.

  3. 3.

Obtain the conditional distribution of planetary mass for a given radius —
$f(m|r)$ — for real FGK and simulated M-dwarf samples: The 2D joint
distribution — f(m, r) — can then be conditioned on Jupiter’s radius ($R_{J}$
or 11.2 $R_{\oplus}$) to obtain a PDF for the inferred mass of Jovian-sized
objects — $f(m|r=1R_{J})$. The ability to compare $f(m|r=1R_{J})_{FGK}$ and
$f(m|r=1R_{J})_{M=FGK}$ in this simulated case where we have the same joint
M-R distribution for the two (i.e., the conditional distributions should be
similar), represents our ‘best-case scenario’ in being able to distinguish
between two the inferred PDFs for two samples of a given size.

  4. 4.

Metrics to compare the two 1-D PDFs: We then perform rejection sampling in 1-D
of the normalized PDFs — $f(m|r=1R_{J})$ — to obtain a histogram of 1000
simulated planetary masses to compare across the M-dwarf and FGK samples. We
adopt the Earth-mover distance (EMD; Rubner et al., 1998; Ramdas et al.,
2015), which is a metric to quantify the difference between two distributions
without any assumptions of normality or similar variance. In other words, the
EMD can be used to compare two distributions, and reduces to zero when they
are completely identical. The EMD is identical to the Wasserstein distance
(Kantorovich, 1960; Ramdas et al., 2015) in the case of PDFs and is
implemented in the scipy python package (Oliphant, 2007; Virtanen et al.,
2020). We do not adopt the Kullback-Leibler (KL) divergence since that assumes
comparison of a sample with a known distribution (Kullback & Leibler, 1951),
and is asymmetric. For the purposes of comparing just the mean of the two PDFs
(while assuming normality), we adopt Welch’s t-test (Welch, 1947; Ruxton,
2006) implemented in scipy through the ttest_ind function that allows for
unequal variances. This is in lieu of the two-sample Kolmogorov-Smirnov (K-S;
Kolmogorov, 1933; Smirnov, 1948) and Anderson-Darling (AD; Anderson & Darling,
1952; Scholz & Stephens, 1987) tests.

  5. 5.

We then repeat steps 2 and 3, where we reduce the masses for the M-dwarf
planet sample to 60% and 80% of the FGK sample to quantify the ability to
distinguish between the M-dwarf and FGK samples as a function of M-dwarf
sample size.

  6. 6.

Finally we calculate the EMD and t-statistic for comparing histograms of the
M-dwarf and FGK samples, and pick the optimum sample size where the 5th
percentile of our metric comparing our test distribution with the FGK sample
($f(m|r=1R_{J})_{M=80\%FGK}$, $f(m|r=1R_{J})_{FGK}$) intersects with the 50th
percentile (median) of the metric distribution from our best-case scenario
with similar distributions ($f(m|r=1R_{J})_{M=FGK}$, $f(m|r=1R_{J})_{FGK}$).

Based on the criterion defined above, we are able to distinguish between mean
of the M-dwarf and FGK planetary M-R distributions when the M-dwarf
distribution is reduced to 80% and 60% (i.e., a 20% and 40% difference) with
about 100 and 20 GEMS respectively using Welch’s t-test. Using the more
agnostic and model-independent EMD, we require 35–40 GEMS to distinguish
between the two distributions when the M-dwarf masses are offset to 60%, and ¿
150 for the 80% case (Figure 6). Thus limiting the comparison between the two
samples to just the M-R plane would require about 35 GEMS with mass
measurements at the 5-$\sigma$ level to ascertain a 40% difference in mass.

####  4.1.2 Mass-Radius-Stellar Mass (MRStM)

Instead of restricting our analysis to just the two M-R dimensions, we can
incorporate additional information about these systems in the form of their
stellar mass to jointly fit f(m, r, stm), where stm refers to the continuous
variable spanning stellar mass using MRExo. Using the sample defined above, we
perform a fit in 3D where the cross-validation method optimizes for the number
of degrees to be 41 in each dimension. Conditioning on planetary radius and
stellar mass to predict the planetary mass — $f(m|r=1R_{J},stm)$ — across a
range of stellar masses, we obtain a nominal linear trend between the
expectation values of the PDFs and stellar mass as shown in Figure 7.

Figure 7: The scatter plot shows the expectation value — f(m— r, stm) — for
the Jovian sized objects orbiting different stellar masses, whereas the orange
line shows the linear fit, which we then extrapolate to lower stellar masses
for M-dwarfs.

Figure 8:  Figure 9:

Figure 10: Left: Similar to Figure 6, we plot the EMD while comparing
$f(m,r|stm=0.75~{}$$M_{\odot}$) with $f(m,r|stm=1.0~{}$$M_{\odot}$) in orange,
and then $f(m,r|stm=0.5~{}$$M_{\odot}$) with $f(m,r|stm=1.0~{}$$M_{\odot}$) in
red. Since the K-dwarf sample is fixed in sample-size, it does not show any
dependence on M-dwarf sample-size, whereas the red region converges. Right: To
establish convergence of the EMD as a function of sample-size (i.e., minimum
sample-size beyond which the improvement in EMD is marginal), we consider the
distance between the central 68-th percentile of the EMD distribution as a
function of sample-size and the sample-size where its fractional change goes
below 1%. Takeaway: The EMD metric to ascertain the stellar dependence for
$f(m|r=1R_{J},stm)$ starts to converge for a GEMS sample of $\sim$ 40.

Similar to the methodology from the 2D simulation, we generate synthetic
M-dwarf GEMS samples and compare the predicted masses for these planets with
those around more massive stars using the EMD metric. We do not investigate
the uncertainties associated with the expectation values here (Figure 7),
since the significance of this trend (both statistical and physical) will be
discussed in a follow-up paper. We first generate a synthetic M-dwarf
catalogue where the 2D joint $f(m,r)$ distribution is same as that for a solar
mass star, i.e., $f(m,r|stm=1.0~{}$$M_{\odot}$). For this synthetic catalogue
of planets, we ascribe stellar masses using a half-normal distribution
$\equiv$ 0.6 - $|\mathcal{N}$(0, 0.1)$|$, to avoid an undue bias towards mid-
to-late M-dwarfs or include K-dwarfs. Then the mass for each M-dwarf planet is
scaled based on an extrapolation of the linear trend shown in Figure 7, which
is further normalized to be $\sim$ 1 for Jupiters around solar-mass stars.
This way we ascribe a stellar mass dependence to the simulated GEMS, and
generate 100 such catalogues for each M-dwarf sample size ranging from 15 to
150 and append these to the original FGK sample. This dependence is partly
motivated by the existing sample, and will be explored in further detail (both
empirically and theoretically) in part II of this work. After performing a 3D
fit — f(m, r, stm) — for each catalogue of planets, we compare the
$f(m,r|stm=0.75~{}$$M_{\odot}$) with $f(m,r|stm=1.0~{}$$M_{\odot}$) as a
control that should not show any sample-size dependence (since fixed FGK
sample), and then $f(m,r|stm=0.5~{}$$M_{\odot}$) with
$f(m,r|stm=1.0~{}$$M_{\odot}$) to show the convergence of the EMD with
increasing the M-dwarf sample size (Figure 10).

To establish convergence, we consider the M-dwarf sample size where the
fractional change of the central 68th percentile (EMD84% \- EMD16%) of the EMD
distribution with sample-size goes below 1%, which is at $\sim 40$. To test
the impact of the assumed slope on the inferred sample-size, we also compare
the EMD for two $f(m|r=1R_{J})$ distributions for stellar masses of 0.55
$M_{\odot}$ and 0.6 $M_{\odot}$, i.e., with more similar distributions, and
find that we attain a similar 1% convergence criterion with a GEMS sample of
about 40 – 50.

These simulations help us assert that we should be able to distinguish between
different stellar-mass – planetary mass dependencies for Jovians with a sample
size of about 40 – 50 planets, which will help answer the first question
regarding M-R+ relations — whether the difficulty in forming GEMS through
core-accretion in a mass starved environment manifests as a systematic
difference in their bulk-density as a function of stellar mass.

###  4.2 Limitations of Existing Transiting GEMS Occurrence Estimates

Gan et al. (2023a) have estimated the occurrence of short-period (0.8 – 10
days) transiting giant planets (7 $R_{\oplus}$$\leq R_{p}\leq$ 2 $R_{J}$)
around $\sim$ 60,000 early M-dwarfs with Teff between 2900 K and 4000 K, and
stellar masses between 0.45 $M_{\odot}$ and 0.65 $M_{\odot}$and $10.5\leq
T_{mag}\leq 13.5$. as 0.27 $\pm$ 0.09 %. Similarly, Bryant et al. (2023)
suggest an occurrence rate of 0.194 $\pm$ 0.072 % for planets with (0.6
$R_{J}$ $\leq R_{p}\leq$ 2.0 $R_{J}$) orbiting $\sim$ 90,000 nearby low-mass
stars with $M_{*}\leq$ 0.71 $M_{\odot}$. These investigations have already
started to provide insight into the occurrence of transiting GEMS, however we
note three main limitations with these existing studies that our survey design
aims to improve upon:

  1. 1.

Jovian-sized objects can range in mass from 0.3 $M_{J}$ to $\sim$ 100 $M_{J}$,
i.e, Saturn-massed to late M-dwarfs. It is not possible to confirm the
planetary nature of transiting Jovian-sized objects, with statistical
validation alone (i.e., without dynamical confirmation with RVs or TTVs).
Therefore, the transiting objects used in these two studies to estimate
occurrence rates are not necessarily all planets and likely contaminated by
astrophysical false-positives such as brown-dwarfs and eclipsing binaries
(EBs). This is indeed seen for TOI-5375 B that was identified as a planet
candidate by Gan et al. (2023a), but confirmed to be an EB by Lambert et al.
(2023). These studies have attempted to account for this by assigning a false
positive probability (FPP) for the candidates discovered through their
pipelines using statistical validation tools (Bryant et al., 2023) or
estimating this based on the prevalence of brown-dwarfs and EBs in the
literature (Gan et al., 2023a). However, literature samples of brown-dwarfs
and giant planets are heterogeneous and not drawn from M-dwarf hosts,
additionally, as pointed out by Bryant et al. (2023), the statistical
techniques are limited in their ability to distinguish between giant planets,
brown-dwarfs and very low-mass stars. Without more relevant FPP estimates
(from M-dwarf host, short orbital period and homogeneous samples), or
spectroscopic validation, these existing estimates should be considered
carefully (and with the appropriate caveats) while comparing across samples
and surveys.

  2. 2.

The input sample of M-dwarfs for both studies conflates mid-to-late K-dwarfs
with early M-dwarfs. For example, Bryant et al. (2023) have a Teff and stellar
mass upper limit of 4500 K and 0.75 $R_{\odot}$ respectively, which includes
mid-to-late K-dwarfs. Given the potentially positive correlation between
stellar-mass and the occurrence of GEMS (Johnson et al., 2010a; Zhou et al.,
2019; Bryant et al., 2023), this choice for the input stellar sample biases
the occurrence rate to higher values. Bryant et al. (2023) account for this by
dividing their stellar sample into different stellar mass bins.

  3. 3.

These studies utilize a sample of $<$ 100,000 M-dwarfs analyzing light-curves
for bright(-er) M-dwarfs as reduced by the Science Processing Operations
Center (SPOC; Jenkins et al., 2016; Caldwell et al., 2020) or Quick Look
Pipeline (QLP; Huang et al., 2020; Kunimoto et al., 2021) efforts. In
particular, the Gan et al. (2023a) sample is (mostly) magnitude limited, but
confounded by the selection function of the QLP sample, and consists of 60,000
(primarily) early M-dwarfs. Conversely, the Bryant et al. (2023) sample is
volume limited to 100 pc, but also includes cuts based on magnitude, which
precludes it from being volume complete across the entire M-dwarf spectral
type888For example, TRAPPIST-1 at 100 pc has a $T$ magnitude of $\sim$ 18 mag,
which is fainter than their magnitude cut of 16 mag.. Furthermore, the
distance and magnitude cut-offs chosen for the above studies do not include
most of the confirmed planetary transiting GEMS that have been discovered thus
far and are beyond 100 pc (Table 1).

Despite these limitations, these existing occurrence estimates already
demonstrate the lower occurrence of transiting GEMS compared to FGK hot-
Jupiters. We will attempt to tackle these limitations through our systematic
candidate search and analysis that we describe in the survey work plan.

###  4.3 Survey Design

While the methodology and preliminary results from this ongoing survey will be
described in upcoming manuscripts, we briefly describe the sample selection,
work plan and survey status here.

####  4.3.1 Sample Selection

Figure 11: (a) Cumulative distribution function (CDF) of the distance. Figure
12: (b) Color - absolute magnitude diagram. Figure 13: (c) Histogram of
Stellar Radius. Figure 14: (d) Histogram of Stellar Mass. Figure 15: (e)
Histogram of Effective Temperature. Figure 16: (f) CDF of J magnitude. Figure
17: (g) CDF of K magnitude. Figure 18: (h) CDF of TIC contamination ratio.

Figure 19: The $\sim 1.0$ million stars observed by TESS through Cycle 5 and
contained in our cuts in Section 4.3. Previous studies have used an upper
limit of 100 pc, which represents a small fraction ($<20$%) of our sample. Our
sample spans the full range of color and magnitude for M0-M9 . For reference,
we indicate GJ 488 (M0V; square) and TRAPPIST-1 (M8V; triangle) in the CMD to
approximately denote the two ends of the M-dwarf spectral type. The stellar
radii, masses and Teff are calculated from $M_{K_{s}}$ using relations from
Mann et al. (2015, 2019), whereas the TICv8.2 contamination ratio is from
Stassun et al. (2018).

We adopt a volume limited sample spanning 200 pc, which covers most of the
existing confirmed planets. Since confirming the planetary nature of
transiting Jovian objects requires resource-intensive observations, this
approach capitalizes on existing observations while providing a large sample
for more precise occurrence estimates. This is to help overcome the
limitations of existing efforts described above, primary among which is the
need for spectroscopic validation of the planet candidates utilized in
occurrence rate estimates, in the absence of which, the need for more informed
and relevant false positive estimates. To do this, we start off with the Gaia
DR3 catalogue (Gaia Collaboration et al., 2023), to which we apply the
following cuts:  (i) a parallax of $\varpi>5\mathrm{~{}mas}$ (corresponding to
an upper limit of 200 pc),  (ii) a parallax error of
$\sigma_{\varpi}<1\mathrm{~{}mas}$, and  (iii) null quality flags indicating
the object is not a quasi-stellar object or galaxy (not in the Gaia DR3
qso_candidates or galaxy_candidates tables), not a binary star (not identified
in Gaia DR3 as an astrometric, spectroscopic, or EB), and not a duplicated
source.

The source identifier in each Gaia data release is not guaranteed to be the
same and we obtain the Gaia DR2 source identifier using the dr2_neighbourhood
table provided with the Gaia DR3 release. We reject sources without matching
dr2_source_id and dr3_source_id because this can arise from a spurious
detection in either DR2 or DR3 (Torra, F. et al., 2021) or erroneous proper
motions from DR2 and double stars that are a resolved pair in Gaia DR3
(Fabricius et al., 2021). We cross-match the Gaia objects with TICv82 using
the Gaia DR2 source identifier (Stassun et al., 2019) and retain stars
observed through TESS Cycle 5 using the high-precision TESS pointing
tool999https://github.com/tessgi/tess-point (Burke et al., 2020). To select
the M dwarfs in the sample, we place constraints on the 2MASS colors and
magnitudes (Table 7; Cifuentes et al., 2020) keeping stars with  (i)
$0.8<J-K<1.25$and  (ii) $4.7<M_{K_{s}}<10.1$ , resulting in our final sample
of $\sim 1$ million stars, the properties of which are depicted in Figure 19.
Splitting this along the spectral sub-type using $M_{K_{s}}$ cut-offs from
Cifuentes et al. (2020), we note that about 21% of our sample are early
M-dwarfs (¡ M2.5V; $M_{K_{s}}$ ¡ 6), 30% as mid M-dwarfs (M2.5V – M4; 7.1 ¿
$M_{K_{s}}$ ¿ 6), with the rest being late M-dwarfs. While we denote GJ 488
and TRAPPIST-1 as example M0/M9s (Figure 19), these do not set our CMD cut-
offs due to the intrinsic astrophysical scatter ($\sim$ 0.2 mag) in the colour
of early M-dwarfs as seen in Figure 14 and A2 from Cifuentes et al. (2020). We
use the CMD to define our sample instead of derived stellar parameters which
often suffer from inaccuracies due to systematics in internal structure models
and evolutionary tracks (Kesseli et al., 2018; Dieterich et al., 2021;
Passegger et al., 2022).

####  4.3.2 Work Plan

Given their faintness, publicly available light curves such as from SPOC or
QLP do not exist for our entire sample. Instead we will extract light curves
from the full-frame image (FFI) images using the public package TESS-Gaia
Light Curves (TGLC; Han & Brandt, 2023) for all our targets, which performs a
contamination correction for the TESS light curves based on their Gaia
positions, magnitude and colour. A detailed characterization of the
photometric performance of this reduction routine for our sample as a function
of stellar properties, will be included in a future manuscript upon completion
of said search.Then, we will search for transiting GEMS candidates around our
M-dwarf sample using the box-least squares algorithm (Kovács et al., 2002) to
identify candidates and a combination of publicly available tools to vet these
candidates for astrophysical false positives (e.g., DAVE and TRICERATOPS;
Kostov et al., 2019; Giacalone et al., 2021). The candidates that survive
vetting will be observed from ground-based facilities to obtain transits (to
rule out background EBs), high-contrast imaging (to constrain dilution) and
spectra (to rule out EBs and brown-dwarfs). This sample of well-vetted planets
will be used to inform the occurrence rates for GEMS.

To estimate the number of true GEMS we will obtain here, we have assumed an
occurrence rate of 0.1 – 0.2% alongside a transit probability of 5 – 10%; the
occurrence rate is from a 0.4 % estimate for AFG stars from Zhou et al.
(2019), which is then scaled by a stellar mass dependent ratio. Additionally,
this estimate also agrees with recent upper limits from Gan et al. (2023a) and
Bryant et al. (2023). We note that since we already have $\sim 15$ confirmed
transiting GEMS (Table 1), we can already place a lower-limit on the
occurrence of short-period transiting GEMS at $\sim 0.03$% (after accounting
for transit probability). While we do not have good priors on the
astrophysical false-positives (brown-dwarfs, EBs) around M-dwarfs at these
orbital periods, we assume a 1:1 ratio between bonafide planets and false
positives here.

Based on these estimates, we expect to discover about $\sim$ 100 transiting
GEMS, the spectroscopically validated sub-sample (roughly half) of which will
enable accurate (due to spectroscopic mass upper limits) and precise (due to
the $>$ 10x larger input sample-size) occurrences. The large sample-size will
also enable estimating the occurrence rate as a function of stellar mass
across the M-dwarf spectral sub-type. To convert our detection efficiency into
occurrence rates, we will follow previous studies by performing injection &
recovery tests to quantify the search sensitivity (Dressing & Charbonneau,
2015; Gan et al., 2023a; Bryant et al., 2023; Ment & Charbonneau, 2023). The
parameters, methodology, and results from these tests will be published along
with the final occurrence rates.

Despite using TGLC which performs a dilution correction based on Gaia
magnitudes, to estimate the yield from our sample we assume a TICv8.2
contamination ratio (Stassun et al., 2018) cut-off101010Where the flux from
contaminating background objects is equal to the flux from the target object.
of 1.0, beyond which the field is likely to be too crowded for follow-up.

The bulk of our spectroscopic follow-up will be performed with the Habitable-
zone Planet Finder (HPF; Mahadevan et al., 2012, 2014) on the 10-m Hobby
Eberly Telescope (Ramsey et al., 1998), NEID on the 3.5-m WIYN telescope
(Halverson et al., 2016; Schwab et al., 2016), alongside MAROON-X on the 8-m
Gemini-N (Seifahrt et al., 2022), and the Planet Finding Spectrograph (PFS) on
the 6.5-m Magellan Clay telescope (Crane et al., 2006, 2008, 2010). Based on
the existing observations and the exposure time calculator for
HPF111111https://psuastro.github.io/HPF/Exposure-Times/, we estimate being
able to perform spectroscopic validation for faint M-dwarfs going down to $J$
$<15$, which combined with the declination limits for the telescope of
approximately -10∘ to 72∘, and the contamination ratio cut-off defined above,
covers about half our sample of a million M-dwarfs. In addition to this, we
expect to be able to observe an additional fraction of stars with NEID,
MAROON-X, which despite being optical/red-optical instruments operate on
conventional telescopes with more lenient declination limits. Similarly PFS
operates in the Southern hemisphere, but uses the iodine gas-cell technique
for RV determination (Butler et al., 1996), making it less suitable for
following-up faint, red M-dwarfs. Given the vagaries of telescope time
allocation, we conservatively assume that using a combination of HPF, NEID,
MAROON-X and PFS, at least half the stars in our 200 pc sample will be
amenable to spectroscopic validation. Therefore, any planet candidates
discovered in this half of the sample should be available for validation and
potential confirmation. The faintness limit of HPF allows us to be almost
volume limited (covers $>$ 90% of the sample) in the region of the sky
accessible to HET, and derive empirical FPP for our sample, which we can then
apply to the rest of the candidates that might be too faint for follow-up from
the other facilities.

A subset of the validated planets ($\sim 40$ as motivated in previous
sections, of which 15 have already been confirmed) will be selected for more
intensive follow-up to obtain planetary mass measurements, which can then be
utilized for future investigations that will be discussed in our follow-up
paper. Through ongoing observations of objects classified as TOIs or CTOIs
that also form part of our 200 pc sample, the community has confirmed $\sim
15$ GEMS so far, in addition to which we have also identified over 20
astrophysical false positives that will be discussed in a follow-up paper
discussing preliminary results and trends from our survey.

####  4.3.3 Survey Status

The ongoing survey has currently led to the discovery and confirmation of nine
of the transiting GEMS indicated in Table 1, with five additional planet
confirmation manuscripts currently in preparation. The demographics analysis
has been initiated, with the sample selection (as explained above) complete,
and vetting procedure being finalized and trained based on a test run on a 100
pc sub-sample. Results from the 100 pc sample will be published as an
intermediate paper, which will also serve as a comparison with previous works
such as those by Bryant et al. (2023). Candidates from these searches are
simultaneously being validated and followed-up from ground-based telescopes to
confirm their planetary nature, and estimate their stellar and planetary
properties. After finalizing the candidate search and vetting procedure, we
will also perform injection and recovery tests to quantify our detection
sensitivity, and go from candidate detection, to measured occurrences. Upon
completion of the survey and confirmation of planet candidates, we will
perform the statistical analysis outlined in previous sections to quantify the
properties of the observed transiting GEMS and compare them with similar
transiting giant planets around more massive stars.

###  4.4 Requirements for planetary mass – stellar metallicity investigations

Extending the planetary mass – stellar metallicity plane into the stellar mass
dimension for GEMS requires not just additional planet discoveries, but also
precise stellar metallicities for M-dwarfs. Furthermore, it will be important
to determine the metallicities across the sample in a homogeneous manner to
avoid methodology dependent metallicity offsets (Passegger et al., 2022).

As can be seen in Figure 1, the current sample of GEMS typically range in mass
from around 0.3 – 3 $M_{J}$. These will be further augmented through our
follow-up of transiting GEMS, and also ongoing uninformed RV surveys with new
instruments such as HPF (Mahadevan et al., 2014), NEID (Halverson et al.,
2016; Gupta et al., 2021), ESPRESSO (Pepe et al., 2021), CARMENES (Sabotta et
al., 2021), SPIROU (Donati et al., 2020), NIRPS (Wildi et al., 2017), IRD
(Kotani et al., 2018), etc. Most importantly, Gaia astrometric detections
(primarily expected in DR4 in
2025–2026121212https://www.cosmos.esa.int/web/gaia/release) are expected to
add hundreds of GEMS and brown-dwarfs at intermediate orbital separations
($\sim$ AU) with mass measurements (Casertano et al., 2008; Sozzetti et al.,
2014; Perryman et al., 2014). An early example of this is GJ 463 b, for which
a true mass was determined using a combination of RVs and Gaia astrometry
(Endl et al., 2022; Sozzetti, 2023). This would offer a homogeneous and well-
characterized sample at intermediate-long periods, that is ripe for population
studies.

##  5 Summary

In this manuscript we discuss the small but growing sample of transiting giant
exoplanets around M-dwarf stars (GEMS), and the challenges posed to their
formation by existing theories of planet formation by core-accretion and
gravitational instability. We motivate the Searching for GEMS survey to find,
and characterize short-period transiting GEMS from TESS and ground-based
facilities. We utilize multi-dimensional nonparametric statistics in the
publicly available package MRExo to predict the required sample-size to
robustly confirm the tentative trends (between the mass of Jovian-sized
planets and host stellar mass) seen in the data to be about 40 confirmed
transiting planets with mass measurements. Lastly, we discuss the limitations
of existing occurrence rates estimates for these GEMS, and highlight the
stellar sample of $\sim$ 1 million M-dwarfs over a volume of 200 pc that we
will use to characterize their occurrence.

##  6 Acknowledgement

We thank the anonymous referee for the valuable feedback which has improved
the quality of this manuscript.

S.K. acknowledges and appreciates discussions with Kevin Schlaufman regarding
target selection and the Gaia queries, and also Theodora for proof-reading,
and Annie Clark for providing a suitable background. GS acknowledges support
provided by NASA through the NASA Hubble Fellowship grant HST-HF2-51519.001-A
awarded by the Space Telescope Science Institute, which is operated by the
Association of Universities for Research in Astronomy, Inc., for NASA, under
contract NAS5-26555. CIC acknowledges support from an appointment to the NASA
Postdoctoral Program at the Goddard Space Flight Center, administered by the
ORAU through a contract with NASA.

These results are based on observations obtained with the Habitable-zone
Planet Finder Spectrograph on the HET. We acknowledge support from NSF grants
AST-1006676, AST-1126413, AST-1310885, AST-1310875, ATI 2009889, ATI-2009982,
AST-2108512, AST-2108801 and the NASA Astrobiology Institute (NNA09DA76A) in
the pursuit of precision radial velocities in the NIR. The HPF team also
acknowledges support from the Heising-Simons Foundation via grant 2017-0494.
The Low Resolution Spectrograph 2 (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. The
Hobby-Eberly Telescope 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 thank the Resident astronomers and Telescope Operators at
the HET for the skillful execution of our observations with HPF. 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.

We acknowledge support from NSF grants AST-1910954, AST-1907622, AST-1909506,
AST-1909682 for the ultra-precise photometry effort.

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.
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. Data presented herein were obtained at the WIYN Observatory
from telescope time allocated to NN-EXPLORE through the scientific partnership
of NASA, the NSF, and NOIRLab.

Data presented herein were obtained at the WIYN Observatory from telescope
time allocated to NN-EXPLORE through the scientific partnership of the
National Aeronautics and Space Administration, the National Science
Foundation, and NOIRLab. This work was supported by a NASA WIYN PI Data Award,
administered by the NASA Exoplanet Science Institute. These results are based
on observations obtained with NEID on the WIYN 3.5 m telescope at KPNO, NSF’s
NOIRLab under proposal 2022B-785506 (PI: S. Kanodia), managed by the
Association of Universities for Research in Astronomy (AURA) under a
cooperative agreement with the NSF. 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.

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