11institutetext: Max-Planck-Institut für Astronomie, Königstuhl 17, 69117
Heidelberg, Germany
11email<EMAIL_ADDRESS>22institutetext: Department of
Astronomy/Steward Observatory, The University of Arizona, 933 North Cherry
Avenue, Tucson, AZ 85721, USA 33institutetext: Physikalisches Institut,
University of Bern, Gesellschaftsstrasse 6, 3012 Bern, Switzerland
44institutetext: Landessternwarte, Zentrum für Astronomie der Universität
Heidelberg, Königstuhl 12, 69117 Heidelberg, Germany 55institutetext: Lunar
and Planetary Laboratory, University of Arizona, 1629 E. University Blvd.,
Tucson, AZ 85721, USA 66institutetext: Universitäts-Sternwarte München,
Ludwig-Maximilians-Universität München, Scheinerstraße 1, 81679 München,
Germany 77institutetext: Centre for Earth Evolution and Dynamics, Department
of Geosciences, University of Oslo, Sem Sælands vei 2b 0315 Oslo, Norway
88institutetext: Institut für Astrophysik, Georg-August-Universität,
Friedrich-Hund-Platz 1, 37077 Göttingen, Germany

# RV-detected planets around M dwarfs: Challenges for core accretion models

M. Schlecker RV-detected planets around M dwarfs: Challenges for core
accretion modelsRV-detected planets around M dwarfs: Challenges for core
accretion modelsRV-detected planets around M dwarfs: Challenges for core
accretion modelsRV-detected planets around M dwarfs: Challenges for core
accretion models R. Burn RV-detected planets around M dwarfs: Challenges for
core accretion modelsRV-detected planets around M dwarfs: Challenges for core
accretion modelsRV-detected planets around M dwarfs: Challenges for core
accretion modelsRV-detected planets around M dwarfs: Challenges for core
accretion models S. Sabotta RV-detected planets around M dwarfs: Challenges
for core accretion modelsRV-detected planets around M dwarfs: Challenges for
core accretion models A. Seifert RV-detected planets around M dwarfs:
Challenges for core accretion modelsRV-detected planets around M dwarfs:
Challenges for core accretion models Th. Henning RV-detected planets around M
dwarfs: Challenges for core accretion modelsRV-detected planets around M
dwarfs: Challenges for core accretion models A. Emsenhuber RV-detected planets
around M dwarfs: Challenges for core accretion modelsRV-detected planets
around M dwarfs: Challenges for core accretion modelsRV-detected planets
around M dwarfs: Challenges for core accretion modelsRV-detected planets
around M dwarfs: Challenges for core accretion modelsRV-detected planets
around M dwarfs: Challenges for core accretion modelsRV-detected planets
around M dwarfs: Challenges for core accretion models C. Mordasini RV-detected
planets around M dwarfs: Challenges for core accretion modelsRV-detected
planets around M dwarfs: Challenges for core accretion models S. Reffert RV-
detected planets around M dwarfs: Challenges for core accretion modelsRV-
detected planets around M dwarfs: Challenges for core accretion models Y. Shan
RV-detected planets around M dwarfs: Challenges for core accretion modelsRV-
detected planets around M dwarfs: Challenges for core accretion modelsRV-
detected planets around M dwarfs: Challenges for core accretion modelsRV-
detected planets around M dwarfs: Challenges for core accretion models H.
Klahr RV-detected planets around M dwarfs: Challenges for core accretion
modelsRV-detected planets around M dwarfs: Challenges for core accretion
models

###### Abstract

Context. Planet formation is sensitive to the conditions in protoplanetary
disks, for which scaling laws as a function of stellar mass are known.

Aims. We aim to test whether the observed population of planets around low-
mass stars can be explained by these trends, or if separate formation channels
are needed.

Methods. We address this question by confronting a state-of-the-art planet
population synthesis model with a sample of planets around M dwarfs observed
by the HARPS and CARMENES radial velocity (RV) surveys. To account for
detection biases, we performed injection and retrieval experiments on the
actual RV data to produce synthetic observations of planets that we simulated
following the core accretion paradigm.

Results. These simulations robustly yield the previously reported high
occurrence of rocky planets around M dwarfs and generally agree with their
planetary mass function. In contrast, our simulations cannot reproduce a
population of giant planets around stars less massive than 0.5 solar masses.
This potentially indicates an alternative formation channel for giant planets
around the least massive stars that cannot be explained with current core
accretion theories. We further find a stellar mass dependency in the detection
rate of short-period planets. A lack of close-in planets around the earlier-
type stars ($M_{\star}\gtrsim$0.4\text{\,}\mathrm{M_{\odot}}$$) in our sample
remains unexplained by our model and indicates dissimilar planet migration
barriers in disks of different spectral subtypes.

Conclusions. Both discrepancies can be attributed to gaps in our understanding
of planet migration in nascent M dwarf systems. They underline the different
conditions around young stars of different spectral subtypes, and the
importance of taking these differences into account when studying planet
formation.

###### Key Words.:

stars: low-mass – techniques: radial velocities – planets and satellites:
formation – planets and satellites: gaseous planets – methods: statistical –
methods: numerical

##  1 Introduction

M dwarf stars are not only the most abundant stars and planet hosts in the
solar neighborhood (Reylé et al., 2021; Hsu et al., 2020), but also rewarding
study objects regarding some of the most pressing questions on planet
formation. They have been shown to host many small, potentially habitable
planets (e.g., Ribas et al., 2018; Bluhm et al., 2020; Ment et al., 2020;
Kemmer et al., 2020; Cloutier et al., 2021; Kossakowski et al., 2021), and
much has been learned about their properties from recent demographic studies
(e.g., Mulders et al., 2015a; Dressing & Charbonneau, 2015; Gibbs et al.,
2020; Hsu et al., 2020). In particular, the occurrence of small planets
appears to increase with decreasing stellar mass (Mulders et al., 2015b;
Hardegree-Ullman et al., 2019), and they might orbit closer to their hosts at
very low stellar masses (Sabotta et al., 2021). On the other hand, the
frequency of giant planets around M dwarfs seems to be low (Endl et al., 2006;
Johnson et al., 2010; Bonfils et al., 2013; Obermeier et al., 2016; Ghezzi et
al., 2018, but see Jordán et al., 2022). To understand how such trends come
about, it is crucial to confront observational results with predictions made
by planet formation theory. In this paper, we perform such a comparison using
a well-defined observational sample and thorough characterization of its
biases, as well as a planet population synthesis tailored to that sample.

On the observational side, the tightest constraints on exoplanet demographics
have been based on data from the _Kepler_ transit survey (Borucki et al.,
2010), which detects planets down to very low masses and radii. However,
_Kepler_ has observed only a few thousand stars of spectral type M (e.g.,
Gaidos et al., 2016), and transit observations do not provide information
about the mass of a planet. Instead of planetary radii, which are degenerate
for giant planets (e.g., Hatzes & Rauer, 2015), radial velocity (RV) surveys
constrain projected planet masses. For comparisons to planet formation models,
this is the more fundamental quantity due to its direct relationship to
planetary accretion processes. RV surveys also require less restrictive
geometrical configurations of detectable planetary systems, enabling more
complete observations. They thus cover a unique parameter space and provide
complementary input to exoplanet demographics. Spectroscopic surveys are able
to identify active stars (e.g., Tal-Or et al., 2018; Jeffers et al., 2018) and
binary stars (e.g., Baroch et al., 2018, 2021), whose implicit or explicit
exclusion from catalogs affect the planet statistics (Moe & Kratter, 2021).
This makes this detection technique ideal for the comparison with current
population synthesis models, which currently do not include stellar
multiplicity or activity.

One of the most comprehensive datasets currently available have been generated
by the High Accuracy Radial velocity Planet Searcher (HARPS, Mayor et al.,
2003) and the Calar Alto high-Resolution search for M dwarfs with Exoearths
with Near-infrared and optical Echelle Spectrographs (CARMENES, Quirrenbach et
al., 2010, 2013; Reiners et al., 2018b) surveys. Bonfils et al. (2013)
reported planet occurrence rates for 102 low-mass stars and 14 planets, for
which they took approximately 2000 spectra with the HARPS instrument. A more
recent study by Sabotta et al. (2021) determined occurrence rates using
$\sim$6500 CARMENES spectra of 71 M dwarfs that host 27 planets. Combining
these two surveys results in 9 confirmed planets around $147$ target stars in
the regime of $0.52.5\text{\,}\mathrm{R_{\oplus}}$ and
$0.510\text{\,}\mathrm{d}$ where _Kepler_ found 13 confirmed planets around
561 target stars (Hardegree-Ullman et al., 2019), thus the overall detection
rate is higher for the RV sample.

From the perspective of formation theory, the first fully operational planet
population syntheses focused on isolated gas giant planets around solar-type
stars (Ida & Lin, 2005; Alibert et al., 2011). As theoretical understanding,
computational resources, and algorithms improved, more and more physical
mechanisms where added to the models, among them planet migration (Ida & Lin,
2008; Dittkrist et al., 2014), long-term evolution of planet interiors
(Mordasini et al., 2012), atmospheric escape (Jin et al., 2014), pebble
accretion (Brügger et al., 2018; Bitsch et al., 2019), and disk chemistry
(Cridland et al., 2016; Thiabaud et al., 2014). A major advancement was to
simulate in the same disk multiple planets that gravitationally interact
(Thommes et al., 2008; Ida & Lin, 2010; Alibert et al., 2013), which is
particularly important for realistic modeling of systems including small
planets (Mordasini, 2018). Models including such an N-body component are
capable of computing systems that include the whole range of planetary masses.
Most population synthesis efforts have limited themselves to planetary systems
around stars with solar mass, but some studies took into account stellar mass
dependencies (Ida & Lin, 2005; Alibert et al., 2011; Liu et al., 2019; Miguel
et al., 2020; Burn et al., 2021). The planetesimal accretion-based formation
model of Emsenhuber et al. (2021a) used here includes all above mechanisms
except for pebble accretion, and its extension to low-mass stars presented in
Burn et al. (2021) allows to cover the full M dwarf mass range.

Direct comparison of its synthetic planets with observed samples is impeded by
various selection effects and detection biases, some of which can be
corrected. A common approach is to “de-bias” the observed sample, for example
by weighting individual planet discoveries according to their detection
probability (e.g., Cumming et al., 2008) or by injecting a test planet
population to obtain the necessary correction factors (e.g., Bonfils et al.,
2013). The drawback of such techniques is that slightly different assumptions
can affect the calculated result, typically an occurrence rate in planetary
parameter space (Hsu et al., 2020; Sabotta et al., 2021). A method that does
not require extrapolations beyond the actually observed sample is to apply a
detection bias to the synthetic planet population (e.g., Mulders et al.,
2019), hence this is our method of choice.

In this paper, we compile a combined planet sample from the HARPS (Mayor et
al., 2003) and CARMENES (Quirrenbach et al., 2010, 2013; Reiners et al.,
2018b) M dwarf surveys. Instead of applying a simple RV amplitude cut, we use
the actual RV measurements for each star to compute the detection bias and
apply it to our synthetic population to simulate its observation. We further
take into account the stellar mass distribution of the combined survey.
Finally, we confront the observed and biased synthetic planet populations in
minimum mass-period-stellar mass-space. The main purpose of our comparison of
detection rates, planetary masses, and orbital period distributions is to show
which of the CARMENES and HARPS planet detections our model can and cannot
explain.

The paper is organized as follows: First, we introduce the observed samples in
Sect. 2, followed by a brief description of the formation model and synthetic
planet population in Sect. 3. We continue with presenting statistical
comparisons of the observed and simulated samples in Sect. 4, and discuss the
most relevant differences in 5. In Sect. 6, we provide conclusions to our
findings.

##  2 Observed sample

We compiled a sample of confirmed exoplanets orbiting M dwarf stars. By
combining results from two of the largest RV surveys targeting low-mass stars,
the HARPS survey (Bonfils et al., 2013) and the CARMENES search for exoplanets
around M dwarfs (Quirrenbach et al., 2010), we obtained an observed sample of
$35$ planets around $147$ stars. In the following, we describe how we compiled
this sample.

###  2.1 The CARM70 sample

One of the most comprehensive searches for exoplanets around M dwarfs is the
CARMENES high-precision RV survey. The CARMENES instrument consists of two
independent Echelle spectrographs, one for visual wavelengths
$0.551.05\text{\,}\mathrm{\SIUnitSymbolMicro m}$ and one for near-Infrared
wavelengths $0.951.7\text{\,}\mathrm{\SIUnitSymbolMicro m}$ (Quirrenbach et
al., 2013). Both channels are fiber-fed from the Calar Alto 3.5 m telescope.
In its Guaranteed Time Observations (GTO), CARMENES targets a sample of $\sim
350$ stars whose spectral type distribution peaks at M4V (Reiners et al.,
2018b). This survey started beginning of 2016 and has since produced more than
18,000 spectra (Sabotta et al., 2021) and led to various exoplanet discoveries
(e.g., Sarkis et al., 2018; Ribas et al., 2018; Luque et al., 2018; Morales et
al., 2019; Zechmeister et al., 2019; Stock et al., 2020; Nowak et al., 2020;
Trifonov et al., 2021). The survey has already been defined with the goal to
perform a population-level, statistical analysis on the datasets it produces.
For a subset of the GTO stars, at least 50 RV measurements have been collected
and observations are considered complete (Sabotta et al., 2021). In addition
to the constraint on the RV measurements, all spectroscopic binaries (Baroch
et al., 2018) were excluded as well as a set of very active stars with large
RV scatter (Tal-Or et al., 2018). After accounting for duplicate stars in our
combined stellar sample (see Sect. 2.3), a subset of 70 stars, here termed
CARM70, remains. Several of the CARMENES discoveries were published in
combination with other instruments. To ensure a homogeneous analysis, Sabotta
et al. (2021) thus used only CARMENES data in their automated search for
planetary signals with a false alarm probability (FAP) of less than 1 % in the
Generalized Lomb-Scargle periodograms (GLS) (Zechmeister et al., 2009). There,
we used a conservative period cutoff at half the time baseline to make sure
that at least two orbits of the planet were observed. In this way, we
identified 27 planets in 22 planetary systems.

###  2.2 The HARPS M dwarf sample

The High Accuracy Radial velocity Planet Searcher (HARPS) consists of an
Echelle spectrograph fed from the ESO La Silla 3.6 m telescope with a second
fiber for a reference spectrum (Mayor et al., 2003). It observes stars in
different mass regimes, reaching from $0.1\text{\,}\mathrm{M_{\odot}}$
(Bonfils et al., 2013) up to intermediate masses (Leão et al., 2018). The
original M dwarf subsample of the HARPS search for exoplanets comprises a
volume-limited ($<11\,\mathrm{pc}$) target list, which was further cropped by
declination ($\delta<+20\deg$) and magnitude ($V>14\,\mathrm{mag}$). It
contains 102 stars covering a mass range of mostly
$0.10.6\text{\,}\mathrm{M_{\odot}}$ (Bonfils et al., 2013). Their stellar
masses were calculated from empirical mass-luminosity relations (Delfosse et
al., 2000). As in Bonfils et al. (2013), we excluded four stars with less than
four RV measurements from the sample and additionally one target with five
very low signal-to-noise observations (L 707-74). A scarcity of stars between
$\sim 0.35\text{\,}\mathrm{M_{\odot}}$ and $0.40\text{\,}\mathrm{M_{\odot}}$
in their sample has been shown to be a statistical fluctuation (Neves et al.,
2013). Unlike CARM70, the HARPS M dwarf sample was not filtered to exclude
very active stars. For this reason the stellar mass distribution of the HARPS
sample has a lower median stellar mass of $0.29\text{\,}\mathrm{M}_{\odot}$.
In the planet sample of Bonfils et al. (2013) there were 14 planets. Several
of those planets were published as a result of analyzing a combination of time
series with other instruments. Therefore, for our analysis, we retrieved again
those signals that were present in HARPS data only. After this analysis, we
excluded the two planets Gl 849b and Gl 832b from the sample, because the time
baseline of the time series did not cover two whole orbits (they were
published in combination with data from other instruments). GJ 876d was also
excluded from our analysis as it is below the detection limit of HARPS data
alone. Furthermore, we excluded GJ 581d as it is probably a false positive
(Robertson et al., 2014; Hatzes, 2016).

###  2.3 The combined M dwarf sample

$22$ stars occur in both the HARPS and the CARMENES target list. To avoid
duplicates while preserving sensitivity, we determined for each star the
number of observations it received for each instrument and kept in the sample
the dataset with the greater number of RV values. Some of these duplicate
stars host planets, in which case we considered only the planets detectable
with the retained dataset. No planets were lost from the sample due to this
rule. In total, the combined HARPS&CARM70 sample consists of $147$ target
stars and $35$ planets. All targets and duplicate stars in our sample are
listed in Table 2. Not only the mass distribution of the stellar sample is
relevant for demographic assessments, but also how intensely each target was
monitored. We show both the mass distributions of each stellar subsample and
the number of observations for each mass bin in Fig. 1111For the distribution
of the numbers of spectra in spectral subtype we refer to Fig. 10 in Sabotta
et al. (2021).. Some of the planets have new parameter constraints from
follow-up studies, in which case we used the updated values (see Table 3).

##  3 Synthetic planet population

We computed a synthetic planet population using the Bern global model of
planetary formation and evolution. Here, we briefly summarize the methodology,
which has been described extensively in the previous works by Emsenhuber et
al. (2021a), Emsenhuber et al. (2021b), Schlecker et al. (2021a), Burn et al.
(2021), Schlecker et al. (2021b), and Mishra et al. (2021).

###  3.1 Formation and evolution model

The goal of a global model is to include all relevant physical processes that
occur in the formation and evolution, albeit in a simplified fashion, so that
their interactions can be studied. The current version of the model is based
on several previous generations (Alibert et al., 2005; Mordasini et al.,
2009a; Alibert et al., 2013). Its formation part computes concurrently the
evolution of a protoplanetary disk consisting of gas and planetesimals, the
accretion of solids and gas by the protoplanets, gas-driven migration, and the
dynamical interactions between multiple protoplanets. The starting point of
the simulations is an already-formed disk consisting of a gaseous part with
mass $M_{\rm disk}$, a dust component (1%) used to calculate the opacity, and
a disk of planetesimals with mass $M_{\rm plts}$. The gas disk is modeled as
an axissymmetric 1D radial profile following Lüst (1952) and Lynden-Bell &
Pringle (1974), where the vertical structure is obtained following the
procedure of Nakamoto & Nakagawa (1994) with direct irradiation from the star
on the surface (Hueso & Guillot, 2005). Turbulent viscosity is described by an
$\alpha$ parametrization (Shakura & Sunyaev, 1973). In addition, both internal
(Clarke et al., 2001) and external (Matsuyama et al., 2003) photoevaporation
are included. The combination of stellar accretion and photoevaporation leads
to the dispersal of the gas disk after a few $\text{\,}\mathrm{Myr}$,
comparable to observed disk lifetimes (e.g., Haisch et al., 2001; Mamajek et
al., 2009; Ribas et al., 2015).

Planets are assumed to form according to the core accretion paradigm (Perri &
Cameron, 1974; Mizuno, 1980). At the start of each simulation, we inserted
multiple planetary seeds with an initial mass of
$0.01\text{\,}\mathrm{M}_{\oplus}$. Their cores grow by accretion of
planetesimals in the oligarchic regime (Ida & Makino, 1993; Inaba et al.,
2001; Fortier et al., 2013). Gas accretion rates are determined by the ability
of the planet to cool by radiating away energy and, for planets exceeding a
critical mass, by the gas supply from the disk. We computed nominal accretion
rates by solving the internal structure equations of both the solid planetary
core and the gaseous envelope Bodenheimer & Pollack (1986), where we assumed
an onion-like structure, the deposition of accreted solids at the core-
envelope boundary, and spherical symmetry. The maximum accretion rate that can
be provided by the gas disk is given by the Bondi accretion in two or three
dimensions, that is, the amount of gas that is intersected by the planet with
a gas capture radius larger (two-dimensional problem) or smaller (three-
dimensional problem) than the local disk scale height D’Angelo & Lubow (2008).
Protoplanets embedded in a gas disk will undergo planetary migration. We
account for type-I migration following Paardekooper et al., 2011 with a
modulation due the planet’s orbital characteristics from Coleman & Nelson,
2014. For very massive planets, we model type-II migration following Dittkrist
et al. (2014). Planet eccentricities and inclinations are damped by the gas
disk, following the prescription of Cresswell & Nelson (2008) in type-I
migration and a fraction of the migration timescale in the type-II regime.
Dynamical interactions between the protoplanets are tracked by means of the
mercury N-body package (Chambers, 1999), which includes a prescription of
collisions described in Emsenhuber et al. (2021a). Beyond the gas disk phase,
planetesimals accretion and orbital evolution are tracked until the end of the
formation stage at $20\text{\,}\mathrm{Myr}$. In the following evolution stage
the thermodynamical evolution is tracked until $10\text{\,}\mathrm{Gyr}$
(Mordasini et al., 2012) . This stage includes migration due to stellar tides
for close-in planets (Benítez-Llambay et al., 2011) and photoevaporation to
remove atmospheric material (Jin et al., 2014). While comprehensive, the Bern
model currently lacks prescriptions for pebble accretion and the formation of
planetesimals and planet seeds, both of which are currently in development
(Brügger et al., 2020; Voelkel et al., 2021a, b).

###  3.2 The synthetic M dwarf planet population

The population synthesis approach randomizes initial conditions to sample the
parameter space. In a Monte Carlo fashion, $M_{\rm disk}$ and $M_{\rm plts}$,
as well as the external photoevaporation rate, the inner disk edge, and the
starting location of each seed were varied between each run of the model. For
the M dwarf planet population, termed NGM, we chose discrete stellar masses of
0.1, 0.3, 0.5, and $0.7\text{\,}\mathrm{M}_{\odot}$ (Burn et al., 2021). Some
of the disk parameters have been shown to scale with stellar mass, and we
aimed to scale our Monte Carlo parameter distributions accordingly. For the
disk mass, we extrapolated the observed relation between disk mass and stellar
mass found by Pascucci et al. (2016); Ansdell et al. (2017) to earlier times
and adopted a linear relationship between star and disk masses. This choice of
scaling is supported by observed stellar accretion rates (Alcalá et al.,
2017), which compare reasonably well to the model output from Burn et al.
(2021, see their Fig. 3). In order to account for the observed scatter in the
measured disk masses of nearby star forming regions, we distributed our gas
disk masses closely following the reported masses of Class-I objects in
Tychoniec et al. (2018), resulting in a log-normal distribution
$\mathcal{N}(\mu=-1.49,\,\sigma^{2}=0.123)$ (compare Emsenhuber et al., 2021b;
Schlecker et al., 2021a). The total planetesimal mass $M_{\rm plts}$ directly
relates the gas disk mass $M_{\rm disk}$ with a dust-to-gas ratio, which in
turn translates to a stellar metallicity under the assumption that stellar and
disk metallicity are the same (Emsenhuber et al., 2021b). We distributed this
parameter normally according to [Fe/H] measurements in the solar neighborhood
(Santos et al., 2003). We assumed that inner disk edges are located at the co-
rotation radius with the stellar spin (Günther, 2013). In agreement with
measured T Tauri rotation rates (Venuti et al., 2017), our inner edges are
distributed log-normally
($\mu=$4.74\text{\,}\mathrm{d}$,\sigma^{2}=0.3\,\mathrm{dex}$) in orbital
period and do not scale with stellar mass. At each simulation run, we injected
50 planetary seeds into the disk at random radial positions between the inner
edge and an orbital period of $253\text{\,}\mathrm{yr}$. The external
photoevaportion rate of the gas disk is chosen such that the resulting
lifetimes of the disks are similar for all stellar masses with a median
$\sim$$3.0\text{\,}\mathrm{M}\mathrm{y}\mathrm{r}$. More details on the
distributions of the Monte Carlo parameters can be found in Burn et al.
(2021).

Other parameters are fixed for each simulation run: the viscous $\alpha$
parameter was set to $2\text{\times}{10}^{-3}\text{\,}$, and the initial slope
of the planetesimal and gas disk are -1.5 and -0.9, respectively. This results
in a high concentration of solid mass close to the star, where locally,
planetesimals can make up to $10\text{\,}\mathrm{\char 37\relax}$ of the total
disk mass. Furthermore, we specified a planetesimal diameter of
$600\text{\,}\mathrm{m}$. This relatively low value leads to efficient damping
by gas drag and therefore lower relative velocities between the growing planet
and the planetesimals. A more detailed discussion on planetesimal sizes can be
found in Emsenhuber et al. (2021b). In total, the NGM population consists of
4996 simulated systems.

Table 1: Synthetic planet population NGM. Each of the simulated systems started with $N_{\mathrm{emb,ini}}=50$ planetary seeds. Five stellar masses and corresponding effective temperatures were sampled with different weights to match the observed distribution, and a total of 100,000 systems were drawn. Each system was assigned a random, isotropic inclination angle $i$. The population NG75 extends NGM to solar-type stars (see Emsenhuber et al., 2021b; Burn et al., 2021) but did not have to be sampled to match the $M_{\star}$ distribution of the observed sample. Table adapted from Burn et al. (2021). Name | $M_{\star}$ | $T_{\mathrm{eff}\mathrm{,5Gyr}}^{(a)}$ | $N_{\mathrm{emb,ini}}$ | Simulated Systems |  Sampling Weights |  Resampled Systems
---|---|---|---|---|---|---
NGM10 | $0.1\text{\,}\mathrm{M_{\odot}}$ | $2811\text{\,}\mathrm{K}$ | 50 | 1000 | 0.204 | 21392
NGM14 | $0.3\text{\,}\mathrm{M_{\odot}}$ | $3416\text{\,}\mathrm{K}$ | 50 | 997 | 0.368 | 34768
NGM11 | $0.5\text{\,}\mathrm{M_{\odot}}$ | $3682\text{\,}\mathrm{K}$ | 50 | 1000 | 0.388 | 39862
NGM12 | $0.7\text{\,}\mathrm{M_{\odot}}$ | $4430\text{\,}\mathrm{K}$ | 50 | 999 | 0.039 | 3978
NG75(b) | $1.0\text{\,}\mathrm{M_{\odot}}$ | $5731\text{\,}\mathrm{K}$ | 50 | 1000 | 0 | 0
(a) following Baraffe et al. (2015) |  |
(b) population also discussed in Emsenhuber et al. (2021b). |  |

Table 1 lists the simulation runs for each host star mass with the
corresponding stellar effective temperature at a simulation time of
$5\text{\,}\mathrm{Gyr}$, $T_{\mathrm{eff}\mathrm{,5Gyr}}$. The initial number
of planetary seeds per system $N_{\mathrm{emb,ini}}$ was always 50.

###  3.3 Observing the synthetic population

In order to make a comparison to observed data meaningful, we had to take into
account their selection effects and detection biases. We implemented this by
weighted resampling of the simulated population and by performing injection-
and-retrieval experiments with the observed data.

####  3.3.1 Stellar mass sampling

Figure 1: Observed and simulated stellar sample. Top: The combined observed
stellar mass distribution is composed of different distributions for the HARPS
(purple) and CARM70 samples (yellow), respectively. The host star masses in
the synthetic NGM population are discrete. By weighted resampling of its
systems according to the combined observed sample (red), we obtain a new
distribution that approximates it (blue). Bottom: Number of observations per
stellar mass bin with the same color code as the top figure. Every target star
is represented by a white box.

In contrast to the NGM population, the HARPS&CARM70 sample has a continuous
distribution of stellar masses, as shown in Fig. 1. We approximated this
distribution by weighted resampling of our synthetic population. For this
purpose, we first computed a histogram of the observed stellar mass sample
with the bin edges defined as the center between the discrete NGM stellar
masses. The normalized histogram counts then served as sampling weights, which
for HARPS&CARM70 amount to 0.204, 0.368, 0.388, 0.039, and 0 for NGM’s host
star masses $0.11.0\text{\,}\mathrm{M_{\odot}}$ (see Tab. 1); that is, the
$1.0\text{\,}\mathrm{M_{\odot}}$ population has no contribution. In total, we
sampled 100,000 systems with replacement. As the synthetic planets will be
compared to RV-detected exoplanets, for which only minimum masses $M\sin(i)$
are known, we assigned them random orbital inclination angles $i$. Here, we
assumed an isotropic distribution of orbit orientations and, for each system,
drew a $\sin(i)$ term from the distribution (Ho & Turner, 2011)

$f(\sin(i))=\frac{\sin(i)}{\sqrt{1-\sin(i)^{2}}}.$ (1)

Hence, despite our oversampling of the NGM population, no planet occurs more
than once with the exact same properties. The resulting $M_{\star}$
distribution approximates the one of the HARPS&CARM70 sample (compare Fig. 1).
While the oversampled population contains $5\text{\times}{10}^{6}\text{\,}$
planets, in the following we consider only the 2,433,170 planets that survived
the formation and evolution phase until an assumed observation time
$t_{\mathrm{obs}}=$5\text{\,}\mathrm{Gyr}$$.

####  3.3.2 Accounting for detection bias

Exoplanet surveys are affected by detection biases, distorting the
distributions in planetary parameter space. Some of these biases can be
characterized and corrected, which is commonly done for occurrence rate
studies (e.g., Cumming et al., 2008; Zechmeister et al., 2009; Mayor et al.,
2011; Bonfils et al., 2013; Sabotta et al., 2021; Fulton et al., 2021). To
obtain a synthetic planet population that we can reasonably compare to our
observed sample, we applied the same bias to the NGM planets. In this way, we
obtain an observable NGM population that contains all the planets that a
combined HARPS&CARM70 survey would have detected if it observed our synthetic
systems.

In Sabotta et al. (2021), we conducted injection-and-retrieval experiments on
the RV data for each of the stars in the CARM70 sample. The period and
$M\sin(i)$ of the injected planets were taken from a grid of log-uniform
distribution. For every grid point, we injected 50 test planets with random
phase angles. If the test planet appeared in the GLS periodogram with a FAP of
less than 1 %, we counted it as a recovery. The detection probability for this
grid point is then the number of recovered planets divided by the total number
of injected planets. From the final survey sensitivity map, we excluded active
periods that show up in the periodogram of the activity indicators. We further
dismissed periods that are longer than half the time baseline by setting the
detection probability to zero. At those periods we would not accept a real
planetary signal (see Sect. 2), and therefore we do not accept corresponding
test planets.

To make sure that the analysis is as self-consistent as possible, we computed
survey sensitivities for the HARPS survey (Bonfils et al., 2013) in the same
way using RV measurements and activity indicators produced with SERVAL
(Zechmeister et al., 2018) by Trifonov et al. (2020). To produce global
sensitivity maps for a combined HARPS&CARM70 survey, we averaged the detection
probability of each grid point for all targets of the combined sample. The
resulting maps for the four stellar mass bins are shown in Fig. 2.

Figure 2: Combined detection probabilities for the HARPS&CARM70 survey binned
to the stellar masses used in the simulations. The color code represents the
detection probability of a planet with the corresponding period, minimum mass,
and host star mass. The zero-detection threshold resides at slightly lower
minimum masses for lower-mass stars, and the detection map for
$0.7\text{\,}\mathrm{M_{\odot}}$ stars is patchy due to the small sample
there.

Each NGM planet received a detection probability depending on the period-
minimum mass bin in which it falls on this map. We then assigned each
synthetic planet a uniformly drawn random number from the interval $[0.0,1.0)$
and considered the planet detected if this number is lower than the planet’s
detection probability. Out of the 2,433,170 synthetic planets, only 20,021 are
detectable according to this procedure.

##  4 Results

###  4.1 Observed and theoretical planet detections

Occurrence rates are often computed from incomplete catalogs by characterizing
a detection bias and correcting for it (e.g., Cumming et al., 2008; Bonfils et
al., 2013; Burke et al., 2015; Obermeier et al., 2016; Lienhard et al., 2020;
Wittenmyer et al., 2020; Gibbs et al., 2020). This has the disadvantage of
having to extrapolate to domains of low sensitivity, leading to large
uncertainties (Foreman-Mackey et al., 2014). Instead, here we apply the
calculated bias of the combined HARPS&CARM70 survey to our synthetic
population and directly compare the observed sample with detectable synthetic
planets. We focus on comparing planetary detection rates $n$, which refer to
the number of simulated or observed planet detections per finite interval in a
parameter (e.g., the orbital period). Owing to the RV-detected exoplanet
sample, our parameter space of choice is the one spanned by the minimum mass
$M\sin(i)$ and the orbital period $P$. The corresponding detection rate
density in this parameter space can be expressed as

$\Gamma_{M,P}=\frac{\partial^{2}n}{\partial\log M\sin(i)\,\partial\log P}.$
(2)

Across all masses and periods, we find a total number of observed planets per
star of $0.24\pm 0.04$ in the HARPS&CARM70 sample and a total number of
observable synthetic planets per star of $0.201\pm 0.001$, where the
uncertainties are Poisson errors from the counting statistic. With $35$
planets in $26$ systems, the observed mean multiplicity of $1.35$ is in
agreement with the synthetic one of 1.38 (20,021 synthetic planets in 14,513
systems). The total observed and theoretical detection rates thus appear
consistent with each other. We note that the detection rate of giant planets
in the CARMENES sample could be overestimated since the survey is not yet
completed (see Sect. 5.5, Sabotta et al. (2021)).

Figure 3: Observed and biased synthetic planets in minimum mass-period space.
Markers show confirmed planets from the HARPS and CARM70 surveys,
respectively. The color code shows the frequency of simulated planets in this
parameter space after applying the detection bias of the combined HARPS&CARM70
survey. Low-mass planets are well reproduced by the model, but intermediate-
mass planets appear more common than theoretically predicted.

Figure 3 shows the distribution of biased simulated and observed planets in
minimum mass-period space. The bulk of the unbiased theoretical occurrence
density lies at roughly Earth-mass planets at orbital periods of a few hundred
days (Burn et al., 2021). After applying the observational bias, the peak
density is shifted to the super-Earth mass regime with periods of a few days.
Super-Neptunes and giant planets
($M\sin(i)\gtrsim$20\text{\,}\mathrm{M_{\oplus}}$$) are expected to be much
rarer. The model anticipates detections to be most likely at
$M\sin(i)\gtrsim$200\text{\,}\mathrm{M_{\oplus}}$$ with orbits around $\sim
100\text{\,}\mathrm{d}$. It produces the fewest planets in the
$20200\text{\,}\mathrm{M_{\oplus}}$ mass range, especially in orbits beyond
$\sim 20\text{\,}\mathrm{d}$. However, such a dearth of Neptune to Saturn-mass
planets at intermediate orbits is not obvious in the sample of detected CARM70
and HARPS planets. Although these are subject to selection effects (see Sect.
5.5), it is noticeable that the mass distribution here is rather continuous
and does not reveal multiple modes. Concerning the agreement of simulated and
observed planets, low-mass planets up to around
$10\text{\,}\mathrm{M_{\oplus}}$ are well reproduced by the model, whereas
intermediate-mass planets occur in the observed sample but are hardly present
in the synthetic population.

###  4.2 Planet detections as a function of host star mass

Figure 4: As Fig. 3, but partitioned according to host star mass. In the
biased synthetic population, the domination by low-mass planets on short to
intermediate orbits is similar for all stellar masses, but giant planets occur
only around stars with masses $0.5\text{\,}\mathrm{M_{\odot}}$ and higher. In
contrast, the subgiant and giant planets in the observed sample occur most
frequently around less massive stars. GJ 3512b, the giant planet in the
$0.1\text{\,}\mathrm{M_{\odot}}$ panel, is particularly at odds with
theoretical predictions. No simulated planets with
$M\sin(i)\gtrsim$10\text{\,}\mathrm{M_{\oplus}}$$ occur in this stellar mass
bin.

To identify trends with planet host mass, we explore planet detections in the
four stellar mass bins defined above (Fig. 4). The synthetic population shows
a clear trend of increasing giant planet detection rate with host star mass.
In general, giant planets are rare and occur only in the stellar mass bins
$0.5\text{\,}\mathrm{M_{\odot}}$ and $0.7\text{\,}\mathrm{M_{\odot}}$.
However, the giant planets in the HARPS&CARM70 sample are in stark contrast to
this trend. The observed giant planet closest to a synthetic counterpart is GJ
876b, which orbits its host with a period of $61\text{\,}\mathrm{d}$ and has a
projected mass of $M\sin(i)\approx$761\text{\,}\mathrm{M_{\oplus}}$$ (Marcy et
al., 2001; Rivera et al., 2005; Trifonov et al., 2018). With a stellar mass of
$0.37\text{\,}\mathrm{M_{\odot}}$, the system just barely ended up in the
stellar mass bin with zero detection rate density in the giant domain and is
in fact relatively close to the few synthetic giant planets in the
$0.5\text{\,}\mathrm{M_{\odot}}$ population.

There are four discovered planets on intermediate and large orbits
($P=$101000\text{\,}\mathrm{d}$$) with projected masses
$M\sin(i)=$20200\text{\,}\mathrm{M_{\oplus}}$$, where NGM shows a deep valley
in the detection rates: GJ 1148b,c (Haghighipour et al., 2010; Trifonov et
al., 2018), HD 147379b (Reiners et al., 2018a), and GJ 3512b (Morales et al.,
2019). While none of them would have been expected based on our simulations,
the $147\text{\,}\mathrm{M_{\oplus}}$ giant GJ 3512b is particularly difficult
to reconcile with theoretical predictions: it orbits a late (M5.5) M dwarf
with a very low mass of $0.123\pm 0.009\text{\,}\mathrm{M_{\odot}}$ (Morales
et al., 2019). Our theoretical model produces no giant planets in this stellar
mass regime, and its existence remains a challenge for planet formation
theories based on core accretion (Liu et al., 2020; Burn et al., 2021; Schib
et al., 2021, but also see Kurtovic et al., 2021).

On the other hand, low-mass planets
($M\sin(i)\lesssim$30\text{\,}\mathrm{M_{\oplus}}$$) are well represented by
our synthetic systems. Populations from all stellar mass bins contribute to
the synthetic planet detections in this domain, where in the unbiased
population the maximum occurence rate density is invariably at orbits $\sim
1001000\text{\,}\mathrm{d}$ and terrestrial planet masses of a few
$\text{\,}\mathrm{M_{\oplus}}$. In the biased population presented here, this
peak is shifted toward periods on the order of a few days. The observed low-
mass planets in CARM70 and HARPS lie all in the domain where significant
synthetic detection rate density exists.

Figure 5: Simulated and observed planet detections as a function of host star
mass. In each stellar mass bin, vertical lines denote
$68\text{\,}\mathrm{\char 37\relax}$ confidence intervals based on the
binomial distribution (not visible for the synthetic population), and the
counts are normalized to the number of stars in that bin. The synthetic
population features discrete $M_{\star}$ values and shows a linear increase of
detections with increasing stellar mass. The observed sample shows no clear
trend.

If we normalize the planet detections by the number of stars of a given host
star mass range (see Fig. 5), the simulated detections increase linearly with
stellar mass. This can be directly attributed to the scaling of disk masses
(see Sect. 3.2). In contrast, the observed detections vary across the
considered range without a clear trend. While the total numbers of detections
in both samples agree, significant deviations exist when they are partitioned
by stellar mass. This is particularly true for giant planets.

###  4.3 Planetary mass function

By marginalizing over the orbital period axis, a minimum mass distribution of
the samples can be obtained.

Figure 6: Minimum mass distribution of synthetic and observed planets. Thick
vertical lines show $68\text{\,}\mathrm{\char 37\relax}$ confidence intervals
based on the binomial distribution. Thin blue lines show standard deviations
of 2000 bootstrapped synthetic samples, where each sample had the size of the
observed sample. Top: Total normalized counts. The distribution of the biased
synthetic population appears bimodal in $M\sin(i)$. The observed sample shows
a peak in the terrestrial planet regime of a few Earth masses and a continuous
downward slope without a valley. At low masses, the theoretical and observed
distributions agree. The formation model underproduces planets
$\gtrsim$30\text{\,}\mathrm{M_{\oplus}}$$ and features a “sub-Saturn valley.”
Center and bottom: Normalized counts for late
($<0.4\,$\text{\,}\mathrm{M_{\odot}}$$) and early
($>0.4\,$\text{\,}\mathrm{M_{\odot}}$$) M dwarfs separately. At minimum masses
beyond $\sim$30\text{\,}\mathrm{M_{\oplus}}$$, theory and observations
disagree: while observed subgiant and giant planets occur mostly around stars
with masses $\sim$0.3\text{\,}\mathrm{M_{\odot}}$$, the formation model
produces such planets only around more massive stars.

Figure 6 shows normalized histograms in $M\sin(i)$ for the observed and
simulated planet samples. In the domain of terrestrial planets and super-
Earths up to $\sim$30\text{\,}\mathrm{M_{\oplus}}$$, the observed and
theoretical distributions match well. Beyond that, the model predicts fewer
planets than observed and shows a few very massive planets
($>$1000\text{\,}\mathrm{M_{\oplus}}$$) that are not observed. To quantify the
difference between the observed and theoretical planetary mass functions, we
performed a two-sample Anderson-Darling test (Anderson & Darling, 1952), which
is based on a nonparametric and distribution-free test statistic $A^{2}$.
Using the implementation in
Scipy222https://docs.scipy.org/doc/scipy/reference/generated/scipy.stats.anderson_ksamp.html
and critical values from Scholz & Stephens (2007), we find that the null
hypothesis that the two samples stem from the same distribution can only be
rejected at an estimated $25\text{\,}\mathrm{\char 37\relax}$ significance
level ($A^{2}=0.97$). We cannot exclude an identical underlying base
distribution. If we divide the samples into early
($>0.4\,$\text{\,}\mathrm{M_{\odot}}$$) and late
($<0.4\,$\text{\,}\mathrm{M_{\odot}}$$) host
stars333$0.4\,$\text{\,}\mathrm{M_{\odot}}$$ is centered between two of our
simulated, discrete stellar masses and close to the threshold mass where stars
become fully convective (Cifuentes et al., 2020)., the agreement remains only
for the early subsample. Due to the large difference in giant planet detection
rates, the observed and simulated samples for later spectral subtypes are
different at the $\leavevmode\nobreak\ 0.1\text{\,}\mathrm{\char 37\relax}$
level.

Figure 6 further shows an apparent bimodality of the synthetic $M\sin(i)$
distribution for early host stars. We tested the significance of such a
bimodality by applying Hartigan’s dip test (Hartigan & Hartigan, 1985), which
tests the null hypothesis of a unimodal distribution, on the different
synthetic subsamples. For the overall synthetic sample, the resulting p-value
of $7\text{\times}{10}^{-3}\text{\,}$ suggests a multimodal distribution.
Distinguishing between early and late stars, the dip test suggests a
multimodal distribution for the early sample
($p_{\mathrm{early}}=$1\text{\times}{10}^{-3}\text{\,}$$) but not for the late
sample ($p_{\mathrm{late}}=0.9$). This is because at stellar masses below
$0.4$ $\text{\,}\mathrm{M_{\odot}}$ no giant planets occur and the
distribution features a single slope. The minimum mass distribution of the
observed sample is overall more flat and shows an almost continuous negative
slope. Consequently, Hartigan’s dip test suggests a unimodal distribution
($p_{\mathrm{obs}}=0.8$).

###  4.4 Orbital period distributions

One of the most directly accessible planetary properties is the orbital
period.

Figure 7:  Period distribution of observed and synthetic planets. As in Fig.
6, thick error bars denote $68\text{\,}\mathrm{\char 37\relax}$ confidence
intervals based on the binomial distribution, and thin lines show the standard
deviations of bootstrapped simulated samples. Top: Total normalized counts.
The distributions are comparable over most of the range, except for a peak at
$\sim 1030\text{\,}\mathrm{d}$ that only occurs in the observed sample. Center
and bottom: Normalized counts for early
($>0.4\,$\text{\,}\mathrm{M_{\odot}}$$) and late
($<0.4\,$\text{\,}\mathrm{M_{\odot}}$$) M dwarfs separately, both showing a
high detection rate at a few tens of days in the observed sample.
Significantly fewer short-period planets are found around higher-mass stars,
which is in disagreement with our model.

The period distributions of the synthetic and observed planets agree well,
except for a peak at a few tens of days that occurs only in the observed
sample (Fig. 7). Orbital periods of the biased synthetic population resemble a
power law with a single slope, and a common origin of the distributions can be
excluded on the $0.1\text{\,}\mathrm{\char 37\relax}$ level ($A^{2}=7.85$).
The apparent log-linear slope is partly shaped by the period-sensitive
completeness function, which masks a slight drop in occurrence for planets
with periods shorter than $\sim 10\text{\,}\mathrm{d}$.

The observed sample shows a dearth of such short-period planets around stars
of higher masses. This feature is not reproduced in our model with its current
prescription for planet migration and the innermost parts of the
protoplanetary disk (see discussion in Sect. 5.4). inline]multimodal period
dist in single vs. multis: (Schlecker et al., 2021b)

##  5 Discussion

The overall number of planet detections per star in the observed HARPS&CARM70
sample and in the biased synthetic NGM population are in good agreement. In
$M\sin(i)$-$P$-$M_{\star}$ space, some discrepancies occur that we discuss
here.

###  5.1 A missing sub-Saturn valley in the planetary mass function

The simulated and observed minimum mass distributions agree for small planets
up to $\sim 30\text{\,}\mathrm{M_{\oplus}}$. Beyond that, their shapes
deviate: while the minimum mass distribution recovered from the discovered
planets follows a smooth power law, the simulated planets show a significant
bimodality. The valley between about $30\text{\,}\mathrm{M_{\oplus}}$ and
$200\text{\,}\mathrm{M_{\oplus}}$ is also present in the unbiased synthetic
population (Burn et al., 2021). Testing the existence or nonexistence of this
demographic feature in M dwarf planetary systems may provide clues about the
gas accretion process in the core accretion paradigm for different stellar
types.

In the classical picture, the valley separates all planets that attained solid
cores massive enough to enter runaway gas accretion and became giant planets
from those that did not (Mizuno et al., 1978; Pollack et al., 1996). Due to
the short duration of the runaway phase, only few planets retain intermediate
masses (Mordasini et al., 2009b; Emsenhuber et al., 2021b, Emsenhuber et al.,
in prep.). Three-dimensional hydrodynamical simulations of planetary gas
accretion have challenged this prediction by proposing significantly lower
accretion rates (Szulágyi et al., 2014; Moldenhauer et al., 2021).

Preliminary observational evidence for the predicted valley has been provided
by Mayor et al. (2011), who computed bias-corrected occurrence rates and the
mass distribution for the HARPS RV survey, albeit for mostly solar-type stars.
They reported a decrease of their bias-corrected mass distribution “between a
few Earth masses and $\sim$40\text{\,}\mathrm{M_{\oplus}}$$.” In contrast, the
valley appears absent in the MOA-II microlensing survey (Suzuki et al., 2016,
2018). Furthermore, the significance of the Mayor et al. (2011)-dip has
recently been questioned (Bennett et al., 2021).

Future expansion of the observational sample will shed light on the existence,
strength, and physical origin of a “sub-Saturn valley.” Planets on
intermediate orbits of $\sim 10100\text{\,}\mathrm{d}$ period (e.g., Espinoza
et al., 2016; Kipping et al., 2019; Brahm et al., 2020; Schlecker et al.,
2020; Hobson et al., 2021) are suitable study objects, as they are less
affected by direct interaction with the host star than their hotter siblings
(Thorngren et al., 2016). The absence of a significant valley in our sample
already indicates that it is not as pronounced as our canonical gas accretion
formalism predicts.

###  5.2 Excess of giant planets around late M dwarfs

The stellar mass dependence of giant planet detections with
$M\sin(i)>$100\text{\,}\mathrm{M_{\oplus}}$$ in the synthetic NGM and observed
samples are at odds with each other. While the formation model generally
produces giant planets only around earlier stars, our RV-detected giant
planets orbit only host stars with masses lower than
$0.5\text{\,}\mathrm{M_{\odot}}$. This is despite a significantly higher
survey sensitivity around earlier stars (Sabotta et al., 2021).

Further constraints on the occurrence of giant planets around stars over a
wide range of masses are provided by the California Legacy Survey (CLS,
Rosenthal et al., 2021; Fulton et al., 2021), which combined and extended
previous surveys using Keck and HIRES data (Cumming et al., 2008; Howard et
al., 2010; Hirsch et al., 2021).

Figure 8: Giant planet detections as a function of stellar mass. From the
HARPS and CARMENES planets presented here (red), the California Legacy Survey
(Rosenthal et al., 2021, gray), and our synthetic NGM planets (blue), we
include detections with RV semi-amplitude
$K>$10\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$,
$M\sin(i)>$100\text{\,}\mathrm{M}_{\oplus}$$, and
$P<$1000\text{\,}\mathrm{d}$$. We removed planets around close spectroscopic
binaries ($P<$20\text{\,}\mathrm{y}\mathrm{r}$$) reported in Rosenthal et al.
(2021). In addition, we show the fit of Ghezzi et al. (2018) for metallicities
consistent with the assumptions made to draw the synthetic planets. Shaded
areas and vertical lines denote $68\text{\,}\mathrm{\char 37\relax}$
confidence intervals, respectively. We note that the HARPS&CARM70 sample is
limited to M dwarfs $\lesssim$0.7\text{\,}\mathrm{M}_{\odot}$$. The observed
giant planets around very low-mass stars indicate a possible break of the
monotonic stellar mass trend at around $\sim$$0.5\text{\,}\mathrm{M}_{\odot}$,
which theoretical predictions fail to reproduce. If type-I migration is
inhibited by a factor $\times 10$ in the simulations (dashed blue line), giant
planets occur also in systems with lower stellar masses.

We plot its giant planet detections as a function of the stellar host mass in
Fig. 8 together with the combined HARPS&CARM70 sample and our synthetic
systems, where we included analogous populations for
$1.0\text{\,}\mathrm{M_{\odot}}$ and $1.5\text{\,}\mathrm{M_{\odot}}$ stellar
hosts (Emsenhuber et al., 2021b). We further add the linear giant planet
occurrence trend found in Ghezzi et al. (2018). For the latter, we use the
stellar mass-metallicity fit of their most recent planetary sample at
$\mathrm{[Fe/H]}=-0.02$, which is the mean metallicity in the synthetic sample
of Burn et al. (2021). The fit deviates from the CLS sample for larger stellar
masses due to enhanced metallicities in the latter. Thus, the derived, almost
linear dependency ($\propto M_{\star}^{1.05}$) by Ghezzi et al. (2018) is
better suited for comparison to the synthetic data than the full CLS sample.
Similar to previous findings (Kennedy & Kenyon, 2008), our synthetic giant
planet occurrences show a trend in stellar mass that is significantly steeper
than the observed one. This suggests that the gas accretion process needs to
be revised in the model (Emsenhuber et al., in prep).

Instead of changing the formation process itself, the proposed stellar mass
scaling of the initial disk mass, which controls the number and masses of
planetary cores (e.g., Kennedy & Kenyon, 2008), may be questioned. It is based
on millimeter dust continuum emission, which may be affected by unknown
opacities, conversion of solid material into larger particles, or optical
depth effects (e.g., Molyarova et al., 2017; Pascucci et al., 2016). A
shallower stellar mass-scaling could resolve the discrepancy, however,
realistic stellar accretion rates should be maintained. In our nominal
scaling, these are consistent with observed data by Alcalá et al. (2017) (see
Burn et al., 2021), although slightly shallower. Indeed, other authors matched
observed accretion rates with a steeper than linear slope: Alibert et al.
(2011) found a good match to observational data (Muzerolle et al., 2003; Natta
et al., 2004) for a slope of $1.2$. Similarly, the scaling of Manara et al.
(2012) used in Liu et al. (2020) results in a steeper relationship. Thus, a
shallower than linear slope for the disk mass as a function of stellar mass is
disfavored by accretion rate observations and not a viable option to enhance
the giant planet formation rate around low-mass stars.

Giant planets occur more frequently around stars of higher metallicity
(Gonzalez, 1997; Santos et al., 2001, 2004; Fischer & Valenti, 2005; Buchhave
et al., 2018). Since for a fixed disk mass, metallicity controls the solid
budget of the disk, larger metallicities promote the formation and growth of
planetesimals and ultimately of planetary cores able to reach the runaway
regime. It is thus reasonable to suspect that the giant planets in our sample
have formed in particularly metal-rich environments that are not represented
by the metallicity distribution of the synthetic systems. The reported values
for the giant planet hosts mentioned above are $0.05\pm 0.2$ dex or $-0.11\pm
0.10$ for Gl 876 b (Correia et al., 2010; Marfil et al., 2021), $0.36\pm 0.2$
for Gl 317 (Anglada-Escudé et al., 2012), $0.13\pm 0.16$ or $-0.17\pm 0.10$
for GJ 1148 (Passegger et al., 2018; Marfil et al., 2021) and $-0.07\pm 0.16$
or $-0.12\pm 0.16$ for GJ 3512 (Morales et al., 2019; Marfil et al., 2021),
leaving only Gl 317 with an enhanced metallicity. Although the uncertainties
are large, the other targets are consistent with solar metallicity. Recent
discoveries in the literature like the sub-Neptune around TOI-2406
($M_{\star}=$0.16\text{\,}\mathrm{M}_{\odot}$$, [Fe/H] = $-0.38\pm
0.07\text{\,}$, Wells et al. (2021)) and the giant planet around a late (M5.0
V) M dwarf with subsolar metallicity (Quirrenbach et al., 2022) complement the
pattern. A high metal content alone thus cannot explain the existence of giant
planets around very low-mass stars.

Already Johnson et al. (2010) reported an elevated giant planet occurrence at
stellar masses below $0.5\text{\,}\mathrm{M_{\odot}}$, and both CARMENES and
CLS find a slight excess of giant planets around stars with masses
$\sim$0.4\text{\,}\mathrm{M}_{\odot}$$ compared to the linear trend in (Ghezzi
et al., 2018): Gl 876b and c (J22532-142, Marcy et al., 1998; Delfosse et al.,
1998; Marcy et al., 2001; Rivera et al., 2005; Millholland et al., 2018;
Trifonov et al., 2018) are included in both the CLS and the CARMENES sample.
Furthermore, Gl 317 (Johnson et al., 2007; Anglada-Escudé et al., 2012) and
HIP 57050 (GJ 1148, Ross 1003, see also Trifonov et al., 2020) both host a
planet with mass $>100\text{\,}\mathrm{M_{\oplus}}$ and are included in CLS
but not in the CARM70 sample (the latter is reported with a mass
$M\sin(i)$$<100\text{\,}\mathrm{M_{\oplus}}$ in CARMENES).

Stellar masses below $\sim 0.3\text{\,}\mathrm{M_{\odot}}$ have not been
thoroughly explored by RV surveys, and the currently small sample does not
allow us to draw a definite conclusion about a deviation from the linear trend
at very low stellar masses. However, Quirrenbach et al. (2022) recently
reported the discovery of another giant planet around a late (M5.0 V) M dwarf.
Although not part of our sample, the roughly Saturn-mass planet provides
additional evidence for the existence of giant planets around stars of all
masses.

It also allows a simple estimate: For 66 stars of spectral type M4.0 V or
later, which roughly corresponds to our lowest-mass bin, CARMENES has already
collected 30 or more RV measurements. This rather arbitrary threshold serves
as a reasonable reference value above which CARMENES would detect gas giants
in a wide range of periods. Two of these stars have been shown to host giant
planets (Morales et al., 2019; Quirrenbach et al., 2022). We can thus estimate
a lower limit on the giant planet occurrence rate of
$2/66\approx$3\text{\,}\mathrm{\char 37\relax}$$ in this stellar mass bin.
This assumes a generic completeness of $100\text{\,}\mathrm{\char 37\relax}$ –
the actual occurrence rate is likely higher when realistic detection
sensitivities are taken into account. This crude estimate hints at a possible
break of the linear occurrence rate trend reported before. However, no
conclusions in this regard should be drawn ahead of a thorough sensitivity and
occurrence rate assessment upon completion of the survey.

Discoveries of gas giants through gravitational microlensing support the
existence of massive planets around low-mass stars (e.g., Udalski et al.,
2005; Dong et al., 2009; Bennett et al., 2020; Bhattacharya et al., 2021; Han
et al., 2021). These discoveries reinforce the puzzling disagreement of the
new CARMENES results with theoretical predictions of giant planet occurrence
as a function of stellar mass. This raises the question of whether planets
with extreme planet-to-star mass ratios may form within the core accretion
framework, or, as hypothesized in Morales et al. (2019), via a disk
instability scenario (Cameron, 1978).

###  5.3 Potential of core accretion to produce giant planets around low-mass
stars

There is a substantial body of literature that provides predictions about the
existence of gas giant planets around low-mass stars. From the point of view
of disk and exoplanet observations, Manara et al. (2018) found that the
combined solid masses of planetary systems frequently exceed those of the most
massive dust disks
($$30\text{\,}\mathrm{M_{\oplus}}$\approx${10}^{-4}\text{\,}\mathrm{M_{\odot}}$$
for M dwarfs), leading them to speculate about a dedicated formation pathway
for massive giant planets around very low-mass stars. It should be noted that
the total mass and distribution of planetesimals may strongly deviate from
what is implied by the gas and dust distribution in disks (Lenz et al., 2019;
Voelkel et al., 2020). At the times of observation, a large fraction of the
dust might already be hidden in planetesimals (Gerbig et al., 2019).

Miguel et al. (2020) performed a population synthesis of planetary systems
around (very) low-mass stars
($M_{\star}=$0.050.25\text{\,}\mathrm{M_{\odot}}$$) based on a semi-analytical
model assuming classical planetesimal accretion. Their model was originally
designed to study circumplanetary disks and features a low-viscosity gas disk
model as well as type-I and type-II migration (Miguel & Ida, 2016). They find
efficient planet formation only in sufficiently massive disks
($\gtrsim${10}^{-2}\text{\,}\mathrm{M_{\odot}}$$). Even under these
conditions, their model fails to form any planets more massive than
$5\text{\,}\mathrm{M_{\oplus}}$.

Using a pebble accretion based model, Liu et al. (2019) model gas accretion
onto planetary cores that have reached pebble isolation mass (Lambrechts et
al., 2014). Their gas accretion method is optimistic (reduced opacity by a
factor of $\sim 20-100$, no leftover thermal energy due to prior solid
accretion) and therefore useful as an upper limit for giant planet formation
in the pebble accretion scenario (see also Brügger et al., 2020). In this case
and for varied disk properties, they found a minimum stellar mass for giant
planet formation of $\sim 0.3\text{\,}\mathrm{M}_{\odot}$.

In good agreement, Mulders et al. (2021) also use a pebble drift and accretion
model to simulate planet growth around stars with different masses. They find
that no giant planets form at stellar host masses
$\lesssim$0.3\text{\,}\mathrm{M_{\odot}}$$.

Adams et al. (2021) explore the planetary mass function of planets formed
around host stars of different masses with a semi-empirical approach. They
find a steeper planetary mass function for low-mass stars, suggesting a low
probability of producing Jovian planets around stars with
$M_{\star}\lesssim$0.5\text{\,}\mathrm{M_{\odot}}$$.

Zawadzki et al. (2021) used N-body calculations to simulate the formation of
planets around $0.2\text{\,}\mathrm{M_{\odot}}$ stars. Under the assumptions
of early planetesimal formation (Lenz et al., 2019) and including type-I
migration, they find efficient growth of planetary cores through early
collisions of planetary seeds. Their setup assumes a rather high solid disk
mass of $\sim$2\text{\times}{10}^{-2}\text{\,}\mathrm{M_{\odot}}$$ at the
start of their simulations. While they do not model gas accretion onto
planets, many of these cores grow to super-Earths, some of which in the mass
range where runaway accretion could be triggered.

Finally, Burn et al. (2021) discuss modifications to the model used here that
would enable the formation of giant planets around very low-mass stars. The
dashed line in Fig. 8 shows the fraction of giant planets in simulations with
type-I migration velocities inhibited by a factor $\times 10$. For
computational cost considerations, we initialized these otherwise identical
simulations with only 20 planetary cores per disk, which is adequate for
modeling gas giant systems (Emsenhuber et al., 2021b). Suppressing planet
migration clearly has a strong effect on giant planet formation: In the
modified simulations, planetary cores are able to reach masses beyond $\sim
10\text{\,}\mathrm{M_{\oplus}}$ even around late M stars without rapidly
migrating into the star, enabling giant planet formation.

Even though there is no obvious evidence justifying such tuning of the
migration scheme, it does mimic “planet traps” caused by inverted gas pressure
gradients to some degree. Their existence is indicated by numerous
observations of disk substructures (e.g., Andrews et al., 2018), which are
suspected to be also common around very low-mass stars (Kurtovic et al., 2021;
Pinilla et al., 2021). This way out of the giant planet conundrum still
requires relatively high initial solid disk masses
$M_{\mathrm{solid,0}}\gtrsim$66\text{\,}\mathrm{M_{\oplus}}$\approx$2\text{\times}{10}^{-4}\text{\,}\mathrm{M_{\odot}}$$.
The efficiency of planet core formation could be further enhanced if large
amounts of planetesimals are concentrated at intermediate orbital distances,
possibly at the water iceline (Dra̧żkowska & Alibert, 2017).

Giant planet formation could be further facilitated if planet cores accrete
not only planetesimals but also mm to cm-sized pebbles (Klahr & Bodenheimer,
2006; Ormel & Klahr, 2010; Lambrechts & Johansen, 2012; Voelkel et al., 2020),
but sufficiently high pebble fluxes are needed in order for growth to outweigh
rapid inward migration (Bitsch et al., 2019).

We conclude that common theories of planet formation via core accretion do not
predict planets like the gas giants around late M stars described here. Giant
planet growth via direct collapse in a gravitationally instable disk remains
an alternative scenario to form these planets, provided that the mass infall
rate onto the disk at any one time exceeded its turbulence-driven accretion
rate (Boss, 1997). However, planets formed via disk instability are thought to
form at large orbital radii and to be massive with $M\sim
10\,\mathrm{M_{Jup}}$ (e.g., Adams & Benz, 1992; Kratter et al., 2010). It is
thus difficult to reconcile the planets considered here with this scenario.

A larger observational sample and systematic parameter studies of various
formation models will be required to reveal the physical mechanisms
responsible for this enigmatic subpopulation of planets. Stronger constraints
can be expected from future microlensing surveys, which predominantly probe
low-mass stars (Gould et al., 2010; Zang et al., 2021; Hwang et al., 2021).
The high angular resolutions achieved by extremely large telescopes will allow
one to resolve and measure masses of the hosts of essentially all microlensing
planets discovered to date. Furthermore, astrometric data from Gaia Data
Releases 3 and 4 will enable much better statistics on a sample of several
thousand nearby M dwarfs (Sozzetti et al., 2014; Perryman et al., 2014). For
these stars, giant planets on short and intermediate orbits can then be
characterized, which will greatly enhance the sample size for precise
occurrence rate studies of giant planets around low-mass stars.

###  5.4 A stellar-mass dependent drop in the period distribution

The present-day picture of planet-disk interactions generally leads to
migration of planets toward the star. Therefore, if and where they are stopped
can in principle be constrained by the orbital period distribution of
exoplanets. Our sample provides new constraints to this topic, which was
previously predominantly explored with transit surveys. Mulders et al. (2015a)
found a stellar mass dependent drop in occurrence rates of inner planets from
the _Kepler_ mission and determined a semi-major axis break point that scales
as $\propto M_{\star}^{1/3}$. The innermost planet of a given system appears
to be located at a preferred orbital period $\simeq$$10\text{\,}\mathrm{d}$
(Mulders et al., 2018). As drivers of such a drop, migration traps (Plavchan &
Bilinski, 2013) or the removal of inner planets are being discussed.

It seems possible to utilize trends with stellar host mass to pinpoint which
process determines the location of the innermost planets. Mulders et al.
(2015a) favored planetary tides or the stellar corotation radius to match
their $\propto M_{\star}^{1/3}$ scaling. They further considered a removal of
planets by stellar tides or dust sublimation in passive or viscously heated
disks. Here, we briefly discuss new insights gained on the possible origins of
this scaling in light of our sample.

From Fig. 7 it becomes apparent that the Bern model fails to correctly predict
the stopping mechanism: synthetic planets commonly migrate closer to the star
than their observed counterparts. The model includes an inner edge set at
corotation radii based on observed stellar rotation rates (Venuti et al.,
2017, see the discussion in Burn et al., 2021). The drop of the gas surface
density at this location causes a type-I outward migration zone (Masset et
al., 2006), which should foster a local pile-up of planets (Mulders et al.,
2019). However, we observe only little change of the innermost periods in
simulations with increased inner disk edge position (see Appendix B). As
indicated in Schlecker et al. (2021b), we find that the presence of planets
within the corotation radius can be attributed to N-body interactions with
planets further out: Often, an inner planet gets locked in a mean-motion
resonance with an outer migrating planet that pushes it further in. This model
outcome is in agreement with the result of Ataiee & Kley (2021), who use 2D
hydrodynamic models instead of migration rate prescriptions to address this
issue. They found that commonly an inner edge is not able to stop migration of
resonant chains of planets. This is in agreement with the resonant chain-
shaped period distribution of planets around solar-mass stars in Carrera et
al. (2019). Thus, the inner edge of the disk is only an efficient migration
trap for solitary planets. With our multiseed setup, such a scenario is rare.

A more efficient migration trap could be the inner edge of a nonionized dead
zone (Gammie, 1996). Due to a change in ionization rate, the viscosity is
expected to increase, which in turn leads to a lowered surface density and a
pressure bump in a steady-state disk. The typical temperature at which this
happens is $\sim$$1000\text{\,}\mathrm{K}$ (Flock et al., 2016, 2017). More
detailed models were presented by Mohanty et al. (2018) and Jankovic et al.
(2021): They find pressure bumps outside the dust sublimation front.
Therefore, these pressure bumps can cause the first trap that resonant convoys
of migrating planets would encounter. According to Ataiee & Kley (2021), the
dead zone inner edge is efficient in halting migration of planets in resonant
chains. The scaling of the orbital period of the pressure bump found by
Mohanty et al. (2018) is $\propto M_{\star}^{3q/4}$, where $q$ is the exponent
with which the stellar accretion rate depends on stellar mass $q\sim 1.8$
(Alcalá et al., 2017). Such a steep dependency of a trap mechanism on stellar
mass is required to explain the tentative evidence from our sample shown in
the right panel of Fig. 7. While the scaling found by Mohanty et al. (2018)
looks promising to explain the trend, the absolute values of the found
pressure bump locations are too large with orbital periods of
$\sim$$60\text{\,}\mathrm{d}$ for solar-mass stars. However, the work of
Mohanty et al. (2018) does not include dust evolution, which would change the
outcome.

A related effect that is included in the Bern model is radial migration due to
stellar tides (following e.g., Benítez-Llambay et al., 2011). Similar to the
consideration of Mulders et al. (2015b), they lead to the removal of the
planets closest to their host stars. In the simulations, tidal effects are at
least partially responsible for shaping the period distribution at close
orbits. Despite that, the observed distribution is not reproduced, which
indicates that either tides are stronger in reality or that we are missing
another important mechanism.

The modeled stellar tides in Burn et al. (2021) include the dependency on
stellar mass and radius (determined using the stellar evolution tracks by
Baraffe et al., 2015). Currently missing is the dependency on the ratio
between the quadrupolar hydrostatic Love number and the tidal dissipation
quality factor $Q_{\star}$ (Gallet et al., 2017). These parameters depend on
the interior structure and dynamics of the star, which vary with stellar mass.
A detailed analysis taking this into account was done by Strugarek et al.
(2017), who find that tides can become efficient for solar-mass stars and
orbital periods below $20\text{\,}\mathrm{d}$. However, the dynamical tide and
therefore a large portion of the effect is suppressed for fully convective M
dwarfs. Therefore, fewer planets would be removed at small orbital periods.
This could explain the different distribution of planetary orbital periods for
stars with $M_{\star}<$0.4\text{\,}\mathrm{M}_{\odot}$$.

Another effect leading to the same outcome as tides are magnetic planet-star
interactions. Strugarek et al. (2017) find that they can dominate over tides
in some regimes, but the order of magnitude is comparable to tides. For low
stellar masses, magnetic effects are stronger than tides but are not efficient
enough to lead to migration of planets on orbits with periods larger than a
day.

Overall, our M dwarf sample provides tentative evidence for a steep scaling of
a migration trap with stellar mass that could be caused by a magnetically
induced pressure bump. Alternatively, efficient tidal migration could remove
the innermost planets around larger stars but cease to be efficient for fully
convective small stars. These findings should be seen as further motivation
for a model revision of the planetary orbital migration and trapping
mechanisms already mentioned in Emsenhuber et al. (2021a) and Schlecker et al.
(2021a).

###  5.5 Caveats and limitations

Radial velocity surveys are not free from biases, and the CARMENES and HARPS
surveys are no exception. At the same time, no theoretical model can fully
reflect all physical mechanisms relevant for planet formation, and
simplifications have to be made. In the following, we list a number of
limitations of our study that could affect our conclusions.

####  5.5.1 Selection effects

Spectroscopic binaries have been removed from the HARPS and CARMENES samples,
respectively. Moe & Kratter (2021) showed that this may lead to overestimated
giant planet occurrence rates for G-type stars. Reported close binary
fractions are relatively constant across the M dwarf mass range (Offner et
al., 2022). Thus, the removal of binary stars from the samples can not lead to
the apparent overabundance of giant planets around late M dwarfs. Since all
our simulated planet hosts are single stars, this selection effect does not
impair our comparison of observed and synthetic populations.

One potential bias is the intensified observation of targets that already show
a tentative signal. Such a signal shows earlier for more massive planets, and
these planets are thus more likely to be eventually detected. The CARM70
sample was selected out of a larger sample with about 340 targets based on the
number of observations its stars received. Thus, “human intervention bias” may
lead to an overprediction of more massive observed planets up to a factor of
five, assuming there are no giant planets left in the rest of the sample (see
also Sabotta et al., 2021). Since we have no indication of an enhancement of
this effect for the least massive stars, it should not affect our conclusions.
The analysis of the full CARMENES sample will result in more robust occurrence
rates for giant planets.

####  5.5.2 Single-planet approximation during biasing

The NGM population consists of multiplanet systems where most systems maintain
several planets each. In practice, their detectability not only depends on the
individual planet properties but also on the combination of planets that
occupy a system: The measured RV signal of a multiplanet system is a
combination of contributions from the individual planets, and a successful
disentanglement of these signals depends on their shape. Nevertheless, when
applying the detection bias to synthetic systems, we treat each planet as
isolated and assign its detectability based on the survey sensitivity at its
orbital period and minimum mass. Injection-and-retrieval tests of each
synthetic system might lead to a more realistic bias but would be
computationally expensive and are not expected to have a significant impact on
the statistical result.

####  5.5.3 Simplifications of the formation model

Due to computational limits, there remain relevant model parameters, such as
the viscous $\alpha$ parameter or the planetesimal size, that are not included
in our parameter search. While the chosen values result in a good fit for the
solar mass case, there might be additional trends with stellar mass that need
to be explored in the future. In addition, our model does not yet include the
evolution of solids in the disk and the consistent formation of planetesimals
and seeds (Burn et al., 2021). We note that the popular theoretical
approximation of two grain sizes presented in Birnstiel et al. (2012) was
derived for solar-type stars and its application to an order of magnitude
lower stellar masses is not straightforward.

##  6 Conclusions

We have compared synthetic planet populations computed with a core accretion
formation model with a sample of planets around low-mass stars discovered by
the HARPS and CARMENES RV surveys. To correct for completeness, we performed
injection tests on the actual RV time series of the surveys. Instead of
extrapolating beyond the observed planet sample, we biased the population of
simulated systems according to the detection limits of the observed sample. We
then statistically compared the actual and synthetic surveys in
$M\sin(i)$-$P$-$M_{\star}$ space. Our main findings are:

  * •

Theory and observations are in agreement for short-period rocky planets, which
form the largest population. Their observed detection rates, planetary mass
function, and orbital period distribution are consistent with our simulations.

  * •

Observed detections of giant planets around late
($<0.5\text{\,}\mathrm{M_{\odot}}$) M dwarfs might be indicative of a break in
the giant planet occurrence as a function of host star mass. The existence of
these planets cannot be reconciled with our model, although the discrepancy is
reduced when planet migration is suppressed.

  * •

The observed and synthetic planetary mass functions diverge for intermediate
masses ($30200\text{\,}\mathrm{M_{\oplus}}$). More sophisticated treatments of
planetary gas accretion, in particular those that take into account 3D
effects, are possibly needed for convergence.

  * •

The observed orbital period distribution depends on the stellar mass with a
paucity of very short-period planets around stars
$\gtrsim$0.4\text{\,}\mathrm{M}_{\odot}$$. While the model reproduces the
distribution for less massive stars, it fails to remove planets or halt
planetary migration efficiently enough for earlier M dwarfs. Candidate
mechanisms to produce the observed trend are planet trapping due to a dead
zone inner edge or stellar tides.

Both the inability to explain the existence of giant planets around very low-
mass stars and the difference in the period distributions suggest that state-
of-the-art planet formation models are still missing a complete picture of
planet migration, or rather of the local disk conditions to which it is highly
sensitive. Constraints on the abundance of substructures in disks around very
low-mass stars through high angular resolution observations will be
particularly insightful here.

With $147$ stars and $35$ planets, the statistical power of our sample is
still limited. It will improve upon the completion of the CARMENES GTO survey,
and beyond that by including the imminent results from astrometry and
microlensing campaigns. On the theoretical side, future model improvements
will allow one to study the physical mechanisms responsible for the
discrepancies presented in this paper.

Our findings underscore the different conditions in protoplanetary disks
around different stellar spectral types, which has a measurable impact on the
outcome of planet formation. M dwarfs are not just small Suns.

###### Acknowledgements.

We wish to thank Pedro Amado, Hans Baehr, Jose Caballero, Mario Flock, Andrew
Gould, Diana Kossakowski, Nicolas Kurtovic, Kaitlin Kratter, Rafael Luque,
Ilaria Pascucci, Ansgar Reiners, and Andreas Quirrenbach for stimulating
discussions. We also thank the anonymous referee for their insightful comments
that improved the manuscript. This work was supported by the Deutsche
Forschungsgemeinschaft (DFG) via the DFG Research Unit FOR2544 “Blue Planets
around Red Stars” (RE 2694/8-1), as well as through the DFG priority programs
SPP 1992: “Exoplanet Diversity” (KL 1469/17-1) and SPP 1833 “Building a
Habitable Earth” (KL 1469/13-(1-2)). This material is based upon work
supported by the National Aeronautics and Space Administration under Agreement
No. 80NSSC21K0593 for the program “Alien Earths.” The results reported herein
benefitted from collaborations and/or information exchange within NASA’s Nexus
for Exoplanet System Science (NExSS) research coordination network sponsored
by NASA’s Science Mission Directorate. This research has made use of the
SIMBAD database, operated at CDS, Strasbourg, France. This research has made
use of the VizieR catalogue access tool, CDS, Strasbourg, France
(DOI:10.26093/cds/vizier). The original description of the VizieR service was
published in 2000, A&AS 143, 23.

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##  Appendix A Planet and stellar sample

Table 2: Observed stars used in this study. Adapted from Sabotta et al. (2021)
and Bonfils et al. (2013).

Name |  Mass (M⊙) | Name |  Mass (M⊙) | Name |  Mass (M⊙) | Name |  Mass (M⊙)
---|---|---|---|---|---|---|---
BD-055715a | 0.46 | Gl402 | 0.26 | HIP31292 | 0.31 |  KarmnJ17378+185a | 0.42
BD-073856 | 0.61 | Gl413.1 | 0.46 | HIP31293 | 0.43 |  KarmnJ17578+046a | 0.17
BD+053409a | 0.51 | Gl433 | 0.47 | KarmnJ00051+457 | 0.50 | KarmnJ18174+483 | 0.58
BD+092636 | 0.53 | Gl438 | 0.33 |  KarmnJ00067-075a | 0.11 |  KarmnJ19169+051Na | 0.45
BD+63869 | 0.55 | Gl447 | 0.17 | KarmnJ00183+440 | 0.34 | KarmnJ19346+045 | 0.55
CD-44863 | 0.22 | Gl465 | 0.34 | KarmnJ01025+716 | 0.47 | KarmnJ20305+654 | 0.36
G108-21 | 0.23 | Gl479 | 0.43 | KarmnJ01026+623 | 0.50 | KarmnJ21164+025 | 0.35
G192-013 | 0.25 | Gl480.1 | 0.18 |  KarmnJ01125-169a | 0.15 | KarmnJ21348+515 | 0.46
G264-018A | 0.53 | Gl514 | 0.53 | KarmnJ02222+478 | 0.55 | KarmnJ21466+668 | 0.25
GJ1061 | 0.12 | Gl526 | 0.5 |  KarmnJ02362+068a | 0.25 |  KarmnJ22021+014a | 0.57
GJ1065 | 0.19 | Gl536 | 0.52 | KarmnJ02442+255 | 0.34 | KarmnJ22114+409 | 0.15
GJ1068 | 0.13 | Gl551 | 0.12 | KarmnJ02530+168 | 0.09 | KarmnJ22115+184 | 0.54
GJ1123 | 0.21 | Gl555 | 0.28 |  KarmnJ03133+047a | 0.16 | KarmnJ22252+594 | 0.34
GJ1125 | 0.29 | Gl569 | 0.49 | KarmnJ03463+262 | 0.57 |  KarmnJ22532-142a | 0.32
GJ1224 | 0.14 | Gl581 | 0.3 |  KarmnJ04153-076a | 0.27 |  KarmnJ23216+172a | 0.39
GJ1232 | 0.20 | Gl588 | 0.47 | KarmnJ04290+219 | 0.64 | KarmnJ23351-023 | 0.13
GJ1236 | 0.22 | Gl618.1 | 0.39 | KarmnJ04376+528 | 0.54 | KarmnJ23381-162 | 0.35
GJ1256 | 0.19 | Gl643 | 0.21 | KarmnJ04588+498 | 0.58 | KarmnJ23419+441 | 0.15
GJ2066 | 0.46 | Gl667 | 0.3 | KarmnJ06103+821 | 0.40 | LHS1134 | 0.2
Gl1 | 0.39 | Gl674 | 0.35 |  KarmnJ06105-218a | 0.52 | LHS1481 | 0.17
Gl12 | 0.22 | Gl680 | 0.47 | KarmnJ06548+332 | 0.34 | LHS1513 | 0.09
Gl145 | 0.32 | Gl682 | 0.27 | KarmnJ08413+594 | 0.12 | LHS1723 | 0.17
Gl176 | 0.50 | Gl693 | 0.26 | KarmnJ09143+526 | 0.59 | LHS1731 | 0.27
Gl191 | 0.27 | Gl729 | 0.17 | KarmnJ09144+526 | 0.59 | LHS1935 | 0.29
Gl203 | 0.19 | Gl754 | 0.18 |  KarmnJ10122-037a | 0.53 | LHS288 | 0.1
Gl205b | 0.60 | Gl803 | 0.75 |  KarmnJ10289+008a | 0.41 | LHS337 | 0.15
Gl213 | 0.22 | Gl832 | 0.45 | KarmnJ10482-113 | 0.12 | LHS3583 | 0.4
Gl250 | 0.45 | Gl87 | 0.45 |  KarmnJ10564+070a | 0.15 | LHS3746 | 0.24
Gl273 | 0.29 | Gl877 | 0.43 | KarmnJ11033+359 | 0.34 | LP771-95 | 0.24
Gl285 | 0.31 | Gl880 | 0.58 | KarmnJ11054+435 | 0.35 | LP816-60 | 0.23
Gl299 | 0.14 | Gl887 | 0.47 | KarmnJ11417+427 | 0.35 |  LP819-052a | 0.18
Gl300 | 0.26 | Gl908 | 0.42 | KarmnJ11421+267 | 0.41 | LTT9759 | 0.54
Gl341 | 0.55 |  HD165222a | 0.44 | KarmnJ11511+352 | 0.45 | NLTT56083 | 0.29
Gl357 | 0.33 | HD168442 | 0.59 | KarmnJ12123+544S | 0.57 | Ross1020 | 0.26
Gl358 | 0.42 | HD199305 | 0.51 | KarmnJ14257+236W | 0.60 | Ross104 | 0.36
Gl367 | 0.49 | HD260655 | 0.45 | KarmnJ16167+672S | 0.60 |  Wolf437a | 0.31
Gl388 | 0.42 | HD97101B | 0.54 |  KarmnJ16303-126a | 0.29 |  |

(a) Duplicate star removed from the HARPS sample
(b) Duplicate star removed from the CARMENES sample

Table 3: Observed planets used in this study. Adapted from Sabotta et al.
(2021) and Bonfils et al. (2013).

Simbad | Karmn. |  | $P$ | $M\sin(i)$ | Ref. | Ref.
---|---|---|---|---|---|---
Identifier | ID |  | (d) |  ($\mathrm{M}_{\oplus}$) | Discovery | Param.
YZ Cet | J01125-169 | c | 3.060 | $1.14^{+0.11}_{-0.10}$ | Ast17 | Sto20a
YZ Cet | J01125-169 | d | 4.656 | $1.09^{+0.12}_{-0.12}$ | Ast17 | Sto20a
Teegarden's Star | J02530+168 | b | 4.910 | $1.05^{+0.13}_{-0.12}$ | Zec19 | Zec19
Teegarden's Star | J02530+168 | c | 11.41 | $1.11^{+0.16}_{-0.15}$ | Zec19 | Zec19
CD Cet | J03133+047 | b | 2.291 | $3.95^{+0.42}_{-0.43}$ | Bau20 | Bau20
HD 285968 | J04429+189 | b | 8.78 | $9.06^{+1.54}_{-0.7}$ | For09 | Tri18
HD 265866 | J06548+332 | b | 14.24 | $4.00^{+0.40}_{-0.40}$ | Sto20b | Sto20b
G 234-45 | J08413+594 | b | 203.6 | $147^{+7.0}_{-7.0}$ | Mor19 | Mor19
HD 79211 | J09144+526 | b | 24.45 | $10.3^{+1.5}_{-1.4}$ | Gon20 | Gon20
HD 95735 | J11033+359 | b | 12.95 | $2.69^{+0.25}_{-0.25}$ | Dia19 | Sto20b
Ross 1003 | J11417+427 | b | 41.38 | $96.6^{+1.3}_{-1.0}$ | Hag10 | Tri20
Ross 1003 | J11417+427 | c | 532.02 | $72.1^{+0.3}_{-7.0}$ | Tri18 | Tri20
Ross 905 | J11421+267 | b | 2.644 | $21.4^{+0.20}_{-0.21}$ | But04 | Tri18
HD 238090 | J12123+544S | b | 13.67 | $6.89^{+0.92}_{-0.95}$ | Sto20b | Sto20b
Wolf 437a | J12479+097 | b | 1.467 | $2.82^{+0.11}_{-0.12}$ | Tri21 | Tri21
Ross 1020 | J13229+244 | b | 3.023 | $8.0^{+0.5}_{-0.5}$ | Luq18 | Luq18
BD-07 4003 | J15194-077 | b | 5.37 | $15.20^{+0.22}_{-0.27}$ | Bon05 | Tri18
BD-07 4003 | J15194-077 | c | 12.92 | $5.65^{+0.39}_{-0.24}$ | Udr07 | Tri18
BD-07 4003 | J15194-077 | e | 3.15 | $1.66^{+0.24}_{-0.16}$ | May09 | Tri18
HD 147379 | J16167+672S | b | 86.54 | $24.7^{+1.8}_{-2.4}$ | Rei18 | Rei18
BD-12 4523 | J16303-126 | b | 1.26 | $1.92^{+0.37}_{-0.37}$ | Wri16 | Sab21
BD-12 4523 | J16303-126 | c | 17.87 | $4.15^{+0.37}_{-0.37}$ | Wri16 | Wri16
BD+18 3421 | J17378+185 | b | 15.53 | $6.24^{+0.58}_{-0.59}$ | Lal19 | Lal19
HD 180617 | J19169+051N | b | 105.9 | $12.2^{+1.0}_{-1.4}$ | Kam18 | Kam18
LSPM J2116+0234 | J21164+025 | b | 14.44 | $13.3^{+1.0}_{-1.1}$ | Lal19 | Lal19
G 264-12 | J21466+668 | b | 2.305 | $2.50^{+0.29}_{-0.30}$ | Ama21 | Ama21
G 264-12 | J21466+668 | c | 8.052 | $3.75^{+0.48}_{-0.47}$ | Ama21 | Ama21
L 788-37 | J22137-176 | b | 3.651 | $7.4^{+0.5}_{-0.5}$ | Luq18 | Luq18
G 232-70 | J22252+594 | b | 13.35 | $16.6^{+0.94}_{-0.95}$ | Nag19 | Nag19
BD-15 6290 | J22532-142 | b | 61.08 | $761^{+1.0}_{-1.0}$ | Del98 | Tri18
BD-15 6290 | J22532-142 | c | 30.13 | $242^{+0.7}_{-0.7}$ | Mar01 | Tri18
CD-31 9113 |  | b | 7.37 | 5.49 | Bon13 | Del13
CD-46 11540 |  | b | 4.69 | 11.39 | Bon07 | Boi11
HD 156384C |  | b | 7.20 | $5.6^{+1.4}_{-1.3}$ | Bon13 | Ang13
HD 156384C |  | c | 28.14 | $3.8^{+1.5}_{-1.2}$ | Bon13 | Ang13

Ama21: Amado et al. 2021; Ang13: Anglada-Escudé et al. 2013; Ast17: Astudillo-
Defru et al. 2017; Bau20: Bauer et al. 2020; Boi11: Boisse et al. 2011; Bon05:
Bonfils et al. 2005; Bon07: Bonfils et al. 2007; Bon13: Bonfils et al. 2013;
But04: Butler et al. 2004; Del98: Delfosse et al. 1998; Del13: Delfosse et al.
2013; Dia19: Díaz et al. 2019; For09: Forveille et al. 2009; Gon20: González-
Álvarez et al. 2020; Hag10: Haghighipour et al. 2010; Kam18: Kaminski et al.
2018; Lal19: Lalitha et al. 2019; Luq18: Luque et al. 2018; Mar01: Marcy et
al. 2001; May09: Mayor et al. 2009; Mor19: Morales et al. 2019; Nag19: Nagel
et al. 2019; Rei18: Reiners et al. 2018b; Riv05: Rivera et al. 2005; Sto20a:
Stock et al. 2020; Sto20b: Stock et al. 2020; Tri18: Trifonov et al. 2018;
Tri20: Trifonov et al. 2020 ; Tri21: Trifonov et al. 2021; Udr07: Udry et al.
2007; Wri16: Wright et al. 2016; Zec19: Zechmeister et al. 2019.
$a$$a$footnotetext: For the transiting planet GJ 486b (J12479+097), its
determined mass is listed instead of its minimum mass.

##  Appendix B Influence of the inner disk edge on orbital periods

In Sect. 4.4 we showed that our model produces synthetic detection rates that
decrease approximately log-linearly with orbital period. This shape is caused
by the combination of the underlying synthetic period distribution and
applying the detection bias to the synthetic sample, which favors short-period
planets over more distant ones. The relative occurrence rates differ from the
_Kepler_ sample, which peaks at $\sim 10\text{\,}\mathrm{d}$ and declines for
shorter orbital periods (Howard et al. 2012; Mulders et al. 2015a; Carrera et
al. 2019). We observed a similar departure from a linear trend at the shortest
orbital periods in the HARPS&CARM70 sample for the more massive stars.

This raises the question if the continuous decline in the model is merely
defined by a numerical inner disk edge that is too close to the star. To
explore this, we examine the period distribution of a comparison population
with inner edges at periods $>$7.3\text{\,}\mathrm{d}$$ instead of the nominal
distribution with a median of $4.74\text{\,}\mathrm{d}$ (Fig. 9). Differences
between the populations are visible, but no clear deviation from the original
trend appears. In particular, the increased distance of the disk edge is not
sufficient to overcome the effect of the detection bias and to cause a break
in the distribution with a drop at small periods. An extended cavity hence
fails to fully explain the observed lack of inner planets around the more
massive stars in our sample.

Figure 9: Dependence of the orbital period distribution on the position of the
inner disk edge. For each stellar host mass bin, we show the nominal biased
synthetic population and a comparison sample with inner disk edges greater
than $7.3\text{\,}\mathrm{d}$ (dashed gray lines), which is the 68th
percentile of the distribution. While differences occur, there is no clear
deviation from the approximate linear trend.