# A More Precise Mass for GJ 1214 b and the Frequency of Multi-Planet Systems
Around Mid-M Dwarfs

Ryan Cloutier Banting Fellow  Center for Astrophysics $|$ Harvard &
Smithsonian, 60 Garden Street, Cambridge, MA, 02138, USA  David Charbonneau
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA, 02138, USA  Drake Deming Department of Astronomy, University of
Maryland at College Park, College Park, MD, 20742, USA  Xavier Bonfils
Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France  Nicola
Astudillo-Defru Departamento de Matemática y Física Aplicadas, Universidad
Católica de la Santìsima Concepción, Alonso de Rivera 2850, Concepción, Chile
Ryan Cloutier<EMAIL_ADDRESS>
###### Abstract

We present an intensive effort to refine the mass and orbit of the enveloped
terrestrial planet GJ 1214 b using 165 radial velocity (RV) measurements taken
with the HARPS spectrograph over a period of ten years. We conduct a joint
analysis of the RVs with archival Spitzer/IRAC transits and measure a
planetary mass and radius of $8.17\pm 0.43$ $M_{\oplus}$ and
$2.742^{+0.050}_{-0.053}$ $R_{\oplus}$. Assuming GJ 1214 b is an Earth-like
core surrounded by a H/He envelope, we measure an envelope mass fraction of
$X_{\rm env}=$ $5.24^{+0.30}_{-0.29}$%. GJ 1214 b remains a prime target for
secondary eclipse observations of an enveloped terrestrial, the scheduling of
which benefits from our tight constraint on the orbital eccentricity of $<$
$0.063$ at 95% confidence, which narrows the secondary eclipse window to 2.8
hours. By combining GJ 1214 with other mid-M dwarf transiting systems with
intensive RV follow-up, we calculate the frequency of mid-M dwarf planetary
systems with multiple small planets and find that $90^{+5}_{-21}$% of mid-M
dwarfs with a known planet with mass $\in[1,10]$ $M_{\oplus}$ and orbital
period $\in[0.5,50]$ days, will host at least one additional planet. We rule
out additional planets around GJ 1214 down to 3 $M_{\oplus}$ within 10 days
such that GJ 1214 is a single-planet system within these limits, a result that
has a $44^{+9}_{-5}$% probability given the prevalence of multi-planet systems
around mid-M dwarfs. We also investigate mid-M dwarf RV systems and show that
the probability that all reported RV planet candidates are real planets is
$<12$% at 99% confidence, although this statistical argument is unable to
identify the probable false positives.

planets and satellites: composition, individual (GJ 1214 b) – stars: low-mass
– techniques: photometric, radial velocities

††facilities: ESO 3.6m/HARPS, Spitzer/IRAC, STELLA/WiFSIP††software: astropy
(Astropy Collaboration et al., 2013, 2018), batman (Kreidberg, 2015), emcee
(Foreman-Mackey et al., 2013), celerite (Foreman-Mackey et al., 2013), scipy
(Virtanen et al., 2020), TERRA (Anglada-Escudé & Butler, 2012).

##  1 Introduction

GJ 1214 remains a benchmark system for M dwarf planetary systems with small
planets in the super-Earth to sub-Neptune size regime (Charbonneau et al.,
2009). The accessibility of the GJ 1214 system for detailed characterization
has warranted measurements of spectroscopic stellar parameters (Anglada-Escudé
et al., 2013a), photometric variability studies (Berta et al., 2011; Narita et
al., 2013; Nascimbeni et al., 2015), and X-ray activity monitoring (Lalitha et
al., 2014). The planet GJ 1214 b itself has been subject to a multitude of
searches for transit timing variations (TTV) (Carter et al., 2011; Harpsøe et
al., 2013; Fraine et al., 2013; Gillon et al., 2014) and atmospheric
characterization efforts (Bean et al., 2011; Croll et al., 2011; Crossfield et
al., 2011; de Mooij et al., 2012; Fraine et al., 2013; Kreidberg et al.,
2014). The TTV searches have sought to 1) scrutinize the multiplicity of the
inner planetary system, and 2) to improve constraints on the mass of GJ 1214 b
if a second transiting planet was uncovered. These searches yielded null
results and consequently were unable to provide an independent mass
measurement of GJ 1214 b. Years of follow-up campaigns have generated the
consensus that GJ 1214 is an inactive mid-M dwarf that hosts a single known
planet, and given the radial velocity (RV) mass and radius of GJ 1214 b
($m_{p}=6.26\pm 0.91$ $M_{\oplus}$; Anglada-Escudé et al. 2013a, $2.85\pm 0.2$
$R_{\oplus}$; Harpsøe et al. 2013), the planet must be enveloped in H/He gas,
although high-altitude clouds hinder the detection of chemical species in the
deep atmosphere (Kreidberg et al., 2014).

The null results from the aforementioned TTV searches and the absence of
additional RV measurements analyzed since the planet’s discovery implies that
the mass of GJ 1214 b has not been refined since its discovery. Anglada-Escudé
et al. (2013a) did however conduct a stellar plus RV analysis using the same
RV data from the discovery paper (Charbonneau et al., 2009). Anglada-Escudé et
al. (2013a) performed an independent RV extraction using the TERRA template-
matching algorithm (Anglada-Escudé & Butler, 2012) and recovered a consistent
RV semiamplitude with an equivalent measurement uncertainty of $\pm 1.6$ m
s-1.

In this study, we analyze the ten years of intensive RV follow-up of GJ 1214
with the aim to improve our understanding of the mass and orbital solution of
GJ 1214 b and to search for additional planetary signals in the RV data. With
these RV data, GJ 1214 joins the set of mid-M dwarfs with sufficient RV
observations to be sensitive to the detection of additional sub-Neptune-mass
planets within the system. By analyzing the RV sensitivity of GJ 1214 along
with other mid-M dwarf planetary systems with comparable RV follow-up
datasets, we present the frequency of mid-M dwarf planetary systems with
multiple small planets with masses between 1-10 $M_{\oplus}$ and with orbital
periods between 0.5-50 days.

In Section 2 we report the adopted stellar parameters. In Section 3 we present
the RV and transit observations used in this study. In Section 4 we jointly
model the RV and transit data to measure the mass and orbit of GJ 1214 b. In
Section 5 we derive the RV sensitivity and place constraints on the presence
of additional planets orbiting GJ 1214. In Section 6 we compute the frequency
the multi-planet systems orbiting mid-M dwarfs. We conclude with a summary of
our results in Section 7.

##  2 Stellar Parameters

GJ 1214 is a metal-rich M4 dwarf ([Fe/H]$=0.29\pm 0.12$ dex; Newton et al.
2014) located at a distance of $14.631\pm 0.012$ pc (Lindegren et al., 2021;
Bailer-Jones et al., 2021). We use the star’s absolute $K_{s}$-band magnitude
($M_{K}=7.956\pm 0.020$) and metallicity to infer its stellar mass
($M_{s}=0.178\pm 0.010$ $M_{\odot}$) and radius ($R_{s}=0.215\pm 0.008$
$R_{\odot}$) using the empirical M dwarf mass-luminosity and radius-luminosity
relations from Benedict et al. (2016) and Mann et al. (2015), respectively. We
adopt the stellar effective temperature from Anglada-Escudé et al. (2013a),
which derived $T_{\text{eff}}$ via SED fitting to the absolute
$BVRIJHK_{s}$W1W2W3W4 magnitudes. The resulting value of $T_{\text{eff}}$
$=3250\pm 100$ K is in good agreement with the values derived from $K$-band
water absorption (Rojas-Ayala et al., 2012) and from empirical temperature-
color relations (Mann et al., 2015). The stellar luminosity, density, and
surface gravity are derived self-consistently from the aforementioned
parameters. The stellar parameters adopted throughout this work are summarized
in Table 1.

Table 1: GJ 1214 physical stellar parameters.

Parameter | Value | Ref
---|---|---
_GJ 1214, TIC 467929202,_
_Gaia DR3 4393265392168829056_
Spectral type | M4 | 1
Stellar mass, $M_{s}$ [$M_{\odot}$] | $0.178\pm 0.010$ | 2
Stellar radius, $R_{s}$ [$R_{\odot}$] | $0.215\pm 0.008$ | 2
Stellar density, $\rho_{s}$ [g cm-3] | $25.4^{+3.5}_{-3.0}$ | 2
Stellar luminosity, $L_{s}$ [L⊙] | $0.0046^{+0.0007}_{-0.0006}$ | 2
Effective temperature, $T_{\text{eff}}$ [K] | $3250\pm 100$ | 3
Surface gravity, $\log{g}$ [dex] | $5.026\pm 0.040$ | 2
Metallicity, [Fe/H] [dex] | $0.29\pm 0.12$ | 1

Note. — References: 1) Newton et al. 2014 2) this work 3) Anglada-Escudé et
al. 2013a.

##  3 Observations

###  3.1 HARPS Radial Velocities

GJ 1214 b was observed with the High Accuracy Radial velocity Planet Searcher
(HARPS; Mayor et al., 2003) échelle spectrograph mounted at the 3.6m ESO
telescope at La Silla Observatory, Chile. A total of 168 spectra were taken
between UT 2009 June 11 and UT 2019 September 2, although we exclude three
spectra with exceptionally low S/N (i.e. total S/N $<5$). Included in the
remaining set of 165 spectra are the 21 RV observations originally published
in the discovery paper of GJ 1214 b (Charbonneau et al., 2009) and
subsequently reanalyzed by Anglada-Escudé et al. (2013a). The observations
were obtained through five time allocations: ESO programs 183.C-0972,
283.C-5022 (P.I. Udry), and 1102.C-0339, 183.C-0437, 198.C-0838 (P.I.
Bonfils). The exposure time was set to 2400 seconds in each program. Due to
the low flux of the target spectrum, the simultaneous cailbration fiber was
set to on-sky rather than to the standard thorium argon lamp to avoid any
potential contamination of the lamp into the science fiber at the bluest
échelle orders. A majority of the spectra were also read in the HARPS slow
mode in order to minimize the read-out noise. In the first two observing
programs (22/165 observations), we obtained a median S/N at the center of the
order 66 ($\sim 650$ nm) of 18.8, which corresponds to a median RV uncertainty
of 2.35 m s-1. In the remaining ESO programs 1102.C-0339 (10/165
observations), 183.C-0437 (68/165 observations), and 198.C-0838 (65/165
observations), median S/N values at 650 nm of 13.9, 16.6, and 16.2,
respectively, were obtained. The corresponding median RV uncertainties were
3.81, 3.35, and 3.25 m s-1.

To achieve the RV uncertainties quoted above, we extracted each spectrum’s RV
using the TERRA pipeline (Anglada-Escudé & Butler, 2012). TERRA employs a
template-matching scheme which has been shown to produce an improved RV
measurement precision on M dwarfs compared to the cross-correlation function
(CCF) technique (e.g. Anglada-Escudé & Butler, 2012; Astudillo-Defru et al.,
2015), which is employed by the HARPS Data Reduction Software (DRS; Lovis &
Pepe, 2007). TERRA constructs an empirical, high S/N template spectrum by
coadding the individual spectra after shifting to the solar system barycentric
reference frame using the barycentric corrections calculated by the HARPS DRS.
Spectral regions with telluric absorption exceeding 1% are omitted from the RV
extraction process. Each spectrum’s RV is extracted via a linear least squares
fit to the master template in velocity space. In our analysis, we ignore the
bluest orders at low S/N and only consider spectral orders redward of order
number 139 (440-687 nm).

In June 2015 (BJD = 2457176.5), the HARPS fiber link was upgraded to improve
the instrument’s throughput, stability, and the uniformity of the emergent
illumination pattern (Lo Curto et al., 2015). These improvements enhanced the
RV stability of the instrument but produced a velocity offset that we account
for in the RV extraction by constructing two template spectra for the subsets
of data corresponding to pre and post-fiber upgrade (containing 90 and 75 out
of 165 measurements, respectively). The resulting raw RV time series is
provided in Table 2 and does not include corrections for the secular
acceleration of GJ 1214111The parallax and proper motion of GJ 1214 from Gaia
EDR3 produce an instantaneous secular acceleration at J2015.5 of
$\text{d}v_{r}/\text{d}t=0.3$ m/s/yr (Eq. 2; Zechmeister et al., 2009)., nor
for any Rossiter-induced contamination of in-transit measurements. We find
that six out of 165 of our RV measurements occur in-transit and may be
marginally affected by the Rossiter-McLaughlin (RM) effect. Due to the slow
rotation of GJ 1214, we can safely ignore this effect in our RV modeling as
the expected RM signal amplitude is neglible compared to our typical RV
uncertainties ($K_{\rm RM}\sim 1.1$ m s-1; Gaudi & Winn, 2007) and in practice
would appear smaller given that the exposure time is comparable to the transit
duration.

Table 2: The raw HARPS RV time series of GJ 1214 in the solar system
barycentric reference frame

Time | RV | $\sigma_{\text{RV}}$ | Fiber Upgrade
---|---|---|---
$[$BJD - 2,450,000$]$ | $[\text{m s}^{-1}]$ | $[\text{m s}^{-1}]$ | Status
4993.767485 | -8.239 | 2.275 | pre
7872.894813 | -7.473 | 3.670 | post

Note. — For conciseness, only a subset of rows are depicted here to illustrate
the table’s contents. The entirety of this table is provided in the arXiv
source code.

Figure 1: Long-term multi-color photometric monitoring of GJ 1214 with STELLA
photometry. The blue, green, and red markers depict the $BVI$ light curves,
respectively. The black curves depict the mean function of the quasi-periodic
Gaussian process while the surrounding shaded region depicts its $3\sigma$
uncertainty.

###  3.2 Spitzer/IRAC Transit Photometry

Fraine et al. (2013) and Gillon et al. (2014) presented data from a continuous
photometric monitoring program of GJ 1214 with the Spitzer Space Telescope.
The program’s strategy and motivation were to monitor GJ 1214 for 20.9 days to
characterize the infrared transit and secondary eclipse of GJ 1214 b, and to
search for additional transiting planets down to Martian size out to the inner
edge of the star’s habitable zone. These observations did not yield a
significant detection of the secondary eclipse depth, nor did they yield any
additional planet detections. However, these data provide us with high quality
transit light curves that we will use in our global analysis of GJ 1214 b.

The data were taken with Spitzer’s Infrared Array Camera (IRAC; Fazio et al.,
2004) as part of the Spitzer programs 542 (P.I. Desért) and 70049 (P.I.
Deming). The observations feature three and fourteen transit observations at
3.6 and 4.5 µm, respectively. Gillon et al. (2014) also presented six
secondary eclipse observations of GJ 1214 b that did not produce any
significant detection of the eclipse depth. As such, we elect to omit the
eclipse observations from our analysis and focus solely on the transit
observations.

The Spitzer photometry contains apparent variations in flux due to intra-pixel
sensitivity variations in the IRAC detectors. We corrected for this effect
using the pixel-level decorrelation (PLD) method (Deming et al., 2015),
implementing the procedure broadly as described by Garhart et al. (2020). We
performed aperture photometry on all of the Spitzer data at both 3.6 and 4.5
µm, using eleven circular numerical apertures. The apertures had constant
radii ranging from tightly centered on the point spread function (PSF), to
much broader than the PSF (i.e. 1.6 to 3.5 pixels), and were centered on the
stellar image using a 2-D Gaussian fit. We applied the PLD analysis to
segments of the subsequent photometry, each segment having a span of
approximately $\pm 3$ hours and centered on the predicted times of transit. In
addition to fitting the photometry using the pixel coefficients (Deming et
al., 2015), we also fit a quadratic baseline in time. There are two principal
differences between our procedure and the analysis method described by Garhart
et al. (2020). Firstly, we are fitting transits and not eclipses, so we
include quadratic limb darkening using the values calculated for the Spitzer
bandpasses (Claret et al., 2012). Secondly, at this phase we are interested
only in removing the instrumental systematic effects in the data, not yet
fitting to the transits. However, because the transits are intertwined with
the instrument effects, we include a preliminary transit fit simultaneously
with removing the intra-pixel systematic effects. We implement the transit
portion of the fitting process by fixing most orbital parameters to the values
given by Charbonneau et al. (2009), with the exceptions being the time of mid-
transit and the transit depth, which are left as free parameters. The PLD code
also varies the coefficients of the quadratic ramp, and the pixel
coefficients, whose optimal values over the broad bandwidth are chosen using a
Markov Chain Monte Carlo method (see Sections 3.4 of Garhart et al. (2020) and
Section 3.3 of Deming et al. (2015)). We then remove the instrumental portion
and retain the detrended transit light curve for the more detailed fitting
process described in Section 4.2.

###  3.3 Multi-color Photometric Monitoring with STELLA

Numerous teams have presented data from long-term photometric monitoring
campaigns of GJ 1214 and reported discrepant values of the stellar rotation
period ranging from 40 days to $\gtrsim 100$ days (Berta et al., 2011; Narita
et al., 2013; Nascimbeni et al., 2015). Here we retrieve on the precise long-
term photometry of GJ 1214 from Mallonn et al. (2018). These observations were
obtained between May 2012 and September 2016 with the WiFSIP imager mounted on
the robotic STELLA telescope at Teide Observatory in Tenerife, Spain
(Strassmeier et al., 2004). The observations were taken in the Johnson $BV$
and Cousin $I$ bands. The light curves clearly reveal photometric modulation
by star spots with 5.3% variability in the $BV$-bands and 1.3% in the $I$-band
(Figure 1). With these data, Mallonn et al. (2018) reported a stellar rotation
period of $125\pm 5$ days. This photometric variability arising from magnetic
activity will also induce an RV signal that we will model in our joint transit
plus RV analysis. We will use the STELLA photometry to train our activity
model as described in Section 4.1.

##  4 Data Analysis & Results

###  4.1 Training the RV Activity Model

GJ 1214 exhibits strong photometric variability at the level of 2-5% in the
$BVI$ bands. Whether induced by dark star spots or bright photospheric faculae
(Rackham et al., 2017), the magnetic activity will also be manifested in the
RVs. We opt to model the RV stellar activity with a semi-parametric Gaussian
process (GP), which models the temporal correlations in the RV data induced by
rotationally modulated stellar activity (e.g. Haywood et al., 2014).
Specifically, we will model the RV activity as a quasi-periodic GP given that
GJ 1214 clearly exhibits rotational modulation that is not strictly periodic
due to the finite lifetimes of active regions. The quasi-periodic covariance
kernel is

$k_{ijp}=a_{\rm GP,b}^{2}\exp{\left[-\frac{(t_{i}-t_{j})^{2}}{\lambda_{\rm
GP}^{2}}-\Gamma_{\rm GP}^{2}\sin^{2}{\left(\frac{\pi|t_{i}-t_{j}|}{P_{\rm
rot}}\right)}\right]}$ (1)

and is parameterized by the hyperparameters $a_{\rm GP,b}$, the covariance
amplitude in each of the $BVI$-bands (indexed by $b$), an exponential
timescale related to active region lifetimes $\lambda_{\rm GP}$, a coherence
parameter $\Gamma_{\rm GP}$, and the stellar rotation period $P_{\rm rot}$. To
first order, the origin of the photometric variability is common with the
origin of RV activity such that activity signals in each time series will show
similar temporal covariance. Consequently, we elect to train our GP activity
model on the photometry shown in Figure 1. However, we note that chromospheric
plages can induce a significant RV signal but with little to no photometric
signature, while near-polar active region configurations can have the opposite
effect of large photometric variations with a small RV counterpart. As such,
GP training on photometry in general may only provide a partially complete
picture of activity but any discrepancies for GJ 1214 are likely small due to
the star’s inactivity.

We sample the posterior of the logarithmic GP hyperparameters using the emcee
Markov Chain Monte Carlo (MCMC) code (Foreman-Mackey et al., 2013) and employ
the celerite software package (Foreman-Mackey et al., 2017) to construct the
GP and evaluate its likelihood during the MCMC sampling. The mean GP model is
depicted in Figure 1 in each passband along with the associated $3\sigma$
uncertainty. The corresponding hyperparameters are listed in Table 3. We
recover a photometric stellar rotation period of $P_{\rm rot}$ $=$
$124.7^{+5.0}_{-4.8}$ days, which is consistent with the value reported by
Mallonn et al. (2018). In our forthcoming transit plus RV modeling efforts, we
will adopt the joint posterior of $\\{\ln{\lambda_{\rm GP}},\ln{\Gamma_{\rm
GP}},\ln{P_{\rm rot}}\\}$ as a prior on our RV activity model.

Table 3: Point estimates of the GP hyperparameters from training on the STELLA photometry Hyperparameter | Value
---|---
_Measured hyperparameters_
Log $B$-band covariance amplitude, $\ln{a_{\rm GP,B}}$ | $-0.81^{+0.31}_{-0.29}$
Log $V$-band covariance amplitude, $\ln{a_{\rm GP,V}}$ | $-0.59^{+0.28}_{-0.27}$
Log $I$-band covariance amplitude, $\ln{a_{\rm GP,I}}$ | $-0.68^{+0.43}_{-0.49}$
Log exponential timescale, $\ln{\lambda_{\rm GP}}$ [days] | $5.85\pm 0.21$
Log coherence, $\ln{\Gamma_{\rm GP}}$ | $-0.68+^{+0.01}_{-0.02}$
Log stellar rotation period, $\ln{P_{\rm rot}}$ [days] | $4.825^{+0.039}_{-0.041}$
_Derived hyperparameters_
Stellar rotation period, $P_{\rm rot}$ [days] | $124.7^{+5.0}_{-4.8}$

###  4.2 Global Transit + RV Model

Here we produce a global model of the available transit and RV observations.
Our primary goals are to refine the mass and orbital eccentricity of GJ 1214 b
to aid in the scheduling of any future secondary eclipse observations.

We fit the detrended Spitzer transits in both IRAC channels with an analytical
transit model (Mandel & Agol, 2002) computed using the batman software
(Kreidberg, 2015). Past TTVs searches have concluded that no significant TTV
signals persist in the system (Carter et al., 2011; Harpsøe et al., 2013;
Fraine et al., 2013; Gillon et al., 2014). As such, we assume a linear
ephemeris parameterized by the orbital period $P$ and the time of mid-transit
$T_{0}$. We also fit for the impact parameter $b$, orbital eccentricity $e$,
and the argument of periastron $\omega$. We mitigate the Lucy-Sweeney bias
towards non-zero eccentricities by sampling the parameters
$h=\sqrt{e}\cos{\omega}$ and $k=\sqrt{e}\sin{\omega}$ rather than sampling $e$
and $\omega$ directly (Eastman et al., 2013; Lucy & Sweeney, 1971).

Furthermore, rather than sampling the scaled semimajor axis $a/R_{s}$
directly, we sample the stellar mass $M_{s}$ and radius $R_{s}$ directly from
their respective measurement uncertainties, which we use together with $P$ to
ensure self-consistent values of $a/R_{s}$ with the aforementioned parameters.
Furthermore, by sampling $M_{s}$ and $R_{s}$, we can compute the corresponding
stellar density $\rho_{s}$, which is related to the transit parameters
$\\{P,e,\omega,a/R_{s}\\}$ according to

$\frac{\rho_{s}}{\rho_{s,\text{transit}}}=\frac{(1-e^{2})^{3/2}}{(1+e\sin{\omega})^{3}},$
(2)

(Moorhead et al., 2011; Dawson & Johnson, 2012) where

$\rho_{s,\text{transit}}=\frac{3\pi}{GP^{2}}\left(\frac{a}{R_{s}}\right)^{3},$
(3)

(Seager & Mallén-Ornelas, 2003) where $G$ is the gravitational constant. These
expressions are intended to ensure consistency between the transit and
stellar, which when considered jointly with the RV measurements, provide the
strongest possible constraints on the orbital eccentricity of GJ 1214 b. In
addition to the aforementioned transit parameters, we also fit separate sets
of the following wavelength-dependent parameters for the 3.6 and 4.5 µm
channels: the planet-star ratio $r_{p}/R_{s}$, the quadratic limb-darkening
parameters $u_{1}$ and $u_{2}$, and the flux baseline $F_{0}$.

Figure 2: The pre (circles) and post (squares) fiber-upgrade HARPS RVs of GJ
1214\. In descending order, the left column depicts the RV time series of the
raw RVs, the RV activity component, the GJ 1214 b planetary component, and the
RV residuals. The resulting residual rms values for the pre and post fiber-
upgrade RVs are 6.2 and 4.7 m s-1, respectively. The right column depicts the
corresponding GLS periodogram. The vertical bands highlight the orbital period
of GJ 1214 b ($P=1.58$ days), the stellar rotation period ($P_{\rm rot}$
$=124.7$ days) and its first harmonic ($P_{\rm rot}/2=62.4$ days). The dashed
horizontal lines in each periodogram panel depict the 1% FAP. Bottom panel:
the activity-corrected RVs phase-folded to the orbital solution of GJ 1214 b.
Figure 3: Phase-folded Spitzer/IRAC transit light curves of GJ 1214 b. Upper
panel: three 3.6 µm transits. Lower panel: fourteen 4.5 µm transits. Solid
curves depict the MAP transit model in each IRAC channel. White circle markers
depict the photometry binned for visualization purposes.

We first correct the raw RVs for the deterministic secular acceleration of GJ
1214 (i.e. 0.3 m/s/yr). We then fit the RVs with a Keplarian orbital solution
for GJ 1214 b plus an RV activity model trained on the STELLA photometry
(Section 4.1). In addition to the shared transit model parameters
$\\{P,T_{0},h,k\\}$, the Keplerian solution includes the RV semiamplitude $K$,
which we sample logarithmically in our MCMC. Our RV model also features
velocity offset $\gamma$ and logarithmic scalar jitter $\ln{s_{\rm RV}}$
parameters for the subsets of our HARPS RV time series corresponding to the
pre and post fiber-upgrade. Similarly to the GP training step, our RV activity
model is described by the hyperparameters $\\{\ln{a_{\rm RV}},\ln{\lambda_{\rm
GP}},\ln{\Gamma_{\rm GP}},\ln{P_{\rm rot}}\\}$, with the latter three being
shared with our training model on the STELLA photometry. Our full transit plus
RV model consists of 24 parameters:
$\\{M_{s},R_{s},P,T_{0},b,h,k,(r_{p}/R_{s})_{3.6},u_{1,3.6},u_{2,3.6},F_{0,3.6},$
$(r_{p}/R_{s})_{4.5},u_{1,4.5},u_{2,4.5},F_{0,4.5},\ln{K},\gamma_{\rm
pre},\gamma_{\rm post},\ln{s_{\rm pre}},$ $\ln{s_{\rm post}},\ln{a_{\rm
RV}},\ln{\lambda_{\rm GP}},\ln{\Gamma_{\rm GP}},\ln{P_{\rm rot}}\\}$. Our
adopted priors are outlined in Table 6.

Similarly to the training step, we sample the posterior of our transit plus RV
model parameters using the emcee MCMC code and use the celerite package to
construct the GP. We use the resulting maximum a-posteriori (MAP) GP
hyperparameters to construct the predictive GP distributions for the pre and
post fiber-upgrade time series. We adopt the mean functions of each GP
posterior as our best-fit activity models, which are highlighted in the second
row of Figure 2. The MAP values of the remaining parameters are reported in
Table 6. We use these parameters to compute the Keplerian orbital solution
shown in Figure 2 and the 3.6 and 4.5 µm transit models shown in Figure 3.

###  4.3 Resulting Physical Planetary Parameters

From our measured planetary parameters we refine the orbital and physical
parameters of GJ 1214 b. Similarly to Gillon et al. (2014), our Spitzer/IRAC
analysis reveals $r_{p}/R_{s}$ values in the 3.6 and 4.5 µm channels that are
consistent within $1\sigma$. We therefore average their marginalized
posteriors and measure a planetary radius of $r_{p}=$
$2.742^{+0.050}_{-0.053}$ $R_{\oplus}$. Our analysis of 165 HARPS RV
measurements provides the most detailed view of the mass of GJ 1214 b to date.
We measure a 27$\sigma$ RV semiamplitude of $14.36\pm 0.53$ m s-1, which
corresponds to a $19\sigma$ planetary mass measurement of $m_{p}=$ $8.17\pm
0.43$ $M_{\oplus}$. We note that with our RV dataset, GJ 1214 b’s mass
measurement precision is now limited by the uncertainty in the stellar mass
with only a small contribution from the RV semi-amplitude uncertainty. The
same remains true for the planet’s radius whose measurement uncertainty is
dominated by the uncertainty in the stellar radius Gillon et al. (2014).
Taking $r_{p}$ and $m_{p}$ together, we recover a $13\sigma$ detection of the
planet’s bulk density: $2.20^{+0.17}_{-0.16}$ g cm-3.

Figure 4: Mass–radius-insolation diagram for small planets with masses
measured to $\leq 33$% and orbiting mid-M dwarfs ($M_{s}\in[0.1,0.4]$
$M_{\odot}$). GJ 1214 b is depicted by the large square marker. The solid
curves depict internal structure models with mass fractions of 100% water,
100% silicate rock, 33% iron plus 67% rock (i.e. Earth-like), and 100% iron
(Zeng & Sasselov, 2013). The dashed curves depict models of Earth-like cores
enshrouded in solar metallicity H/He envelopes at the zero-albedo equilibrium
temperature of GJ 1214 b (i.e. 600 K) and with various envelope mass
fractions. The shaded region bounded by the dotted curve highlights the
forbidden region according to models of maximum collisional mantle stripping
by giant impacts (Marcus et al., 2010).

We include the refined mass and radius of GJ 1214 b in the mass-radius diagram
of small planets orbiting mid-M dwarfs in Figure 4. When compared to two-
layered, solid structure models (Zeng & Sasselov, 2013), we see that GJ 1214
b’s low bulk density is clearly inconsistent with a purely solid body and
therefore requires an extended H/He envelope to explain its mass and radius.
With the inclusion of a gaseous envelope component, we are unable to place
meaningful constraints on the mass fraction of materials that compose the
planet’s underlying structure. However, population synthesis models of small
close-in planets indicate that planets that form the sub-Neptune peak in the M
dwarf radius valley (Hirano et al., 2018; Cloutier & Menou, 2020) are likely
enveloped terrestrials222I.e. an Earth-like solid core surrounded by a H/He
envelope with an envelope mass fraction of up to a few percent. rather than
volatile-rich planets (Owen & Wu, 2017; Gupta & Schlichting, 2019; Rogers &
Owen, 2021). If we assume that GJ 1214 b is in fact an enveloped terrestrial,
we can calculate the envelope mass fraction $X_{\rm env}\equiv m_{\rm
env}/m_{p}$ required to explain its mass and radius. We perform this
calculation assuming that GJ 1214 b consists of an Earth-like core with a 33%
iron core mass fraction and hosts a H/He envelope with solar-metallicity
($\mu=2.35$). The thermal structure of the gaseous envelope is described by
the semi-analytical radiative-convective model from (Owen & Wu, 2017). We
sample GJ 1214 b’s joint $\\{m_{p},r_{p},T_{\rm eq}\\}$ posterior and
calculate the core and envelope masses required to match the mass and radius
of GJ 1214 b with an isothermal upper atmosphere at $T_{\text{eq}}$. We
measure an envelope mass fraction of $X_{\rm env}=$ $5.24^{+0.30}_{-0.29}$%.
We note that the uncertainty on $X_{\rm env}$ reported here is determined
solely by the uncertainties in the input parameters $\\{m_{p},r_{p},T_{\rm
eq}\\}$ and does not include uncertainties in the physical model.

###  4.4 Orbital Eccentricity Result

Our joint transit plus RV analysis, together with a-priori constraints on the
stellar density, produces a strong constraint on GJ 1214 b’s orbital
eccentricity. The marginalized posterior on the orbital eccentricity heavily
favors a circular orbital solution. We find that $e<$ $0.063$ and $<$ $0.100$
at 95% and 99% confidence, respectively. Although a circular orbit is expected
given the planet’s short circularization timescale of $<1$ Myr, precise
knowledge of the quantity $e\cos{\omega}$ has critical consequences for
scheduling secondary eclipse observations with facilities such as JWST. The
value of $e\cos{\omega}$ directly translates into an uncertainty window on the
eclipse center time according to

$\Delta t=\frac{2}{\pi}\,P\,e\cos{\omega}.$ (4)

Our stringent constraints on $e\cos{\omega}$ translate into an uncertainty on
the center time of eclipse of 2.8 hours at 95% confidence, or approximately
3.2 times the eclipse duration of 53 minutes.

##  5 Limits on Additional Planets

Continuous photometric monitoring of GJ 1214 with Spitzer revealed no new
transiting planets larger than Mars and interior to the inner HZ at 20.9 days
(Gillon et al., 2014). Furthermore, TTV searches have not yielded any
additional planetary signals (Carter et al., 2011; Harpsøe et al., 2013;
Fraine et al., 2013; Gillon et al., 2014). With our intensive RV time series
we are able to place meaningful constraints on additional planets orbiting GJ
1214, which need not be in a transiting configuration, nor locked into a mean-
motion resonance with GJ 1214 b where the TTV sensitivity is maximized. Here
we derive limits on the existence of additional planets around GJ 1214 via a
set of injection-recovery tests into our RV time series.

###  5.1 Injection-Recovery Methodology

We conduct a pair of Monte Carlo (MC) simulations to calculate the detection
sensitivity of planets as a function of planet mass and orbital period. The MC
simulations differ in their noisy RV time series into which we will inject
planetary signals.

_MCI_ : we sample the Gaussian distribution $\mathcal{N}(0,\sigma_{\rm eff})$
at the times of our HARPS pre and post-fiber upgrade window functions. The
effective RV noise $\sigma_{\rm eff}$ values are calculated from the
quadrature addition of the median RV uncertainties and the MAP scalar jitters
from our RV analysis ($\sigma_{\text{eff,pre}}=4.53$ m s-1,
$\sigma_{\text{eff,post}}=3.99$ m s-1). However in practice, RV uncertainties
are never purely Gaussian, often as a result of variable observing conditions
or imperfect corrections of instrumental drifts. To mimic these stochastic
effects, we add a unique noise term to the RV dispersion by sampling from
$\mathcal{U}(-1,1)$ m s-1, whose limits are generally appropriate for HARPS
observations. The MCI simulations should be interpreted as a best case
scenario in that the noisy RVs are uncontaminated by temporally-correlated
signals from stellar activity, which are implicitly assumed to have been
perfectly mitigated.

_MCII_ : alternatively, we construct noisy RVs by subtracting the MAP
Keplerian signal of GJ 1214 b from the pre and post fiber-upgrade RVs (see
second row in Figure 2). In this way, signals from the window function and
stellar activity are preserved. The MCII simulations can be interpreted as a
worst case scenario wherein stellar activity is left uncorrected and will
contribute to the rms of the RVs. This effect will be particularly detrimental
to the recovery of injected planetary signals close to $P_{\rm rot}$ (Newton
et al., 2016; Vanderburg et al., 2016). However undesirable the persistence of
stellar activity, and aliases, are for planet searches, these effects make the
MCII results a more realistic assessment of the detection sensitivity compared
to MCI.

In each of the $10^{5}$ MC realizations in MCI and MCII, we inject a single
synthetic Keplerian signal into the noisy HARPS RVs. We note that searching
for individual planets in our injection-recovery tests is a simplification as
blind searches for multiple RV planets are often conducted simultaneousy
rather than iteratively, such that the detection sensitivity is not
independent of the number of planetary signals. The masses and periods of our
synthetic planet population are sampled uniformly in log-space between 0.1-20
$M_{\oplus}$ and 0.5-200 days, respectively. We select these outer bounds with
the intention to cover the majority of the parameter space which is spanned by
known planets around mid-M dwarfs (see Section 6). Our adopted mass range
encapsulates planets with masses smaller than that of Neptune (i.e. 17
$M_{\oplus}$). The mass of GJ 1214 is sampled from its posterior
$\mathcal{N}(0.178,0.010)$ $M_{\odot}$. Due to the transiting nature of GJ
1214 b and the characteristically low dispersion in mutual inclinations among
Kepler multi-planet systems with $a/R_{s}>5$ (Dai et al., 2018)333Note that
for GJ 1214 b, $a/R_{s}=14.85\pm 0.16$., we focus on orbital inclinations that
are nearly co-planar with GJ 1214 b ($i_{b}=88.7\pm 0.1^{\circ}$) by sampling
inclinations from $\mathcal{N}(i_{b},\sigma_{i})$, where
$\sigma_{i}=2^{\circ}$ (Ballard & Johnson, 2016; Dai et al., 2018). Planets’
orbital phases are sampled from $\mathcal{U}(-\pi,\pi)$ and all orbits are
assumed to be circular.

We inject the synthetic planetary signal in each realization and attempt to
recover the injected signal based on two criteria in MCI plus one additional
criterion in MCII. Because the search for non-transiting planets lacks the
luxury of a-priori knowledge of the planet’s ephemeris, a non-transiting
planet’s signal in an RV time series must emerge as a prominent periodicity in
the data. As such, our first criterion for the successful recovery of an
injected planet is that the maximum GLS periodogram power within 10% of the
injected period must exceed a FAP of 1%. However, sporadic peaks in the GLS
periodogram can masquerade as a planetary signal. We safeguard against these
false positives by insisting that a six-parameter Keplerian model at the GLS
peak’s orbital period must be strongly favored over the null hypothesis (i.e.
a flat line with a velocity offset). For this model comparison, we calculate
the Bayesian Information Criteria (BIC) for the Keplerian model and for the
null model. The BIC $=-2\ln{\mathcal{L}}+\nu\ln{N}$, where $\mathcal{L}$ is
the likelihood of the RV data given the model, $\nu$ is the number of model
parameters ($\nu=6$ or 1 for the Keplerian and null models, respectively), and
$N=165$ is the number of RV measurements. Taken together, we claim the
successful recovery of an injected planet in MCI if 1) the GLS power of the
injected periodic signal has FAP $<1$%, and 2) the BIC of the null hypothesis
is greater than ten times the BIC of the Keplerian model.

Figure 5: RV detection sensitivity to planets orbiting GJ 1214 as a function
of planet mass and orbital period for two noise models. Left panel: the
results of the MCI simulation, which should be interpreted as a best case
scenario. Right panel: the results of the MCII simulation, which should be
interpreted as the worst case scenario (see Section 5.1 for details). The
solid contours highlight the 10% and 90% sensitivity levels. The thin dashed
lines represent lines of constant semiamplitude with illustrative values equal
to 1, 4, and 10 m s-1. The circle markers highlight GJ 1214 b. The vertical
shaded regions span the habitable zone of GJ 1214.

The noisy RVs in MCI do not contain any temporally correlated signals such
that there are no significant peaks in the GLS periodogram of the noisy RVs.
However, the MCII noisy RVs retain the stellar activity signals which can
produce GLS periodogram peaks at $P_{\rm rot}$, one of its low order
harmonics, or at spurious periods unrelated to stellar rotation (Nava et al.,
2020). We therefore apply a third condition to the successful recovery of an
injected Keplerian signal in MCII. Namely, the largest peak in the GLS
periodogram of the noisy RVs within 10% of the injected period must have FAP
$>1$%. This is to ensure that the injected signal is not confused with a
residual signal or an alias close to the its orbital frequency. Alternatively,
if the BIC of the null hypothesis is greater than $10^{3}$ times the BIC of
the Keplerian model, then we continue to accept the injected signal as a
detection because such RV signals have very high S/N and should not be
rejected by relatively minor aliases present in the window function.

Due to the exhaustive nature of computing the GLS periodogram and its
corresponding FAP in $2\times 10^{5}$ realizations via bootstrapping with
replacement, we adopt an analytical framework to approximate the FAP as a
function of the normalized GLS power $p$ (Cumming, 2004). For a time series
containing $N$ measurements and spanning $T$ days, the FAP is well-
approximated by

$\text{FAP}(p_{0})=1-[1-\text{Pr}(p>p_{0})]^{M},$ (5)

where $M=\Delta f/\delta f$ is the number of sampled independent frequencies
and is calculated from the ratio of the frequency range $\Delta f=f_{2}-f_{1}$
and the frequency resolution $\delta f=1/T$. For a GLS periodogram normalized
by the sample variance, such that the GLS power is bounded between 0 and 1
(Zechmeister & Kürster, 2009), the probability term in Eq. 5 becomes
$\text{Pr}(p>p_{0})=(1-p_{0})^{(N-3)/2}$ (Cumming et al., 1999), where $p_{0}$
is the power whose FAP we are interested in. Adopting this analytical
formalism significantly decreases the computation time of the MC simulations
and enables the mass-period space to be sampled more efficiently.

###  5.2 RV Detection Sensitivity

We calculate the RV detection sensitivity in each MC simulation as the ratio
of the number of recovered planets to the number of injected planets as a
function of mass and period. Our results are shown in Figure 5.

The results from MCI indicate that the sensitivity is a smooth function of RV
semiamplitude. Injected planetary signals with $K\gtrsim 4$ m s-1 are detected
$>90$% of the time. The sensitivity contours derived from MCII are broadly
similar with the 90% contour being well-aligned with $K=4$ m s-1 as in MCI.
However in MCII, residual low FAP frequencies in the HARPS RVs after the
removal of GJ 1214 b disrupt the smoothness of the map. In particular,
residual GLS peaks introduce structure in the sensitivity map, which can in
principle inhibit the detection of planetary signals whose orbital periods are
close to low FAP peaks (or aliases thereof) in the window function. However,
the persistent periodicities seen in the GJ 1214 RV residuals at 1, 16, and
125 days have FAPs $>1$% such that they are not confused with injected signals
with similar orbital periods. But because these periodicities do exhibit a
slightly enhanced GLS power, the GLS power of injected planets near these
frequencies is artificialy enhanced. This effect manifests itself as a
moderate improvement to the sensitivity in the second panel of Figure 5. We
caution that these enhancements are an undesirable outcome of a periodogram-
based recovery process and should not be taken at face value in a planet
search.

The residual signals at 1 and 125 can be easily explained by the daily alias
and the rotation period of GJ 1214, respectively. However, the signal at
approximately 16 days is not trivially explained by a stellar rotation
harmonic or by an alias from the HARPS window function. The FAP of the 16-day
signal in the MCII RV residuals (i.e. without removing a GP) exceeds 1% such
that the evidence for a periodic signal at 16 days is marginal. Furthermore,
the GLS periodogram of the RV residuals following the removal of GJ 1214 b and
the GP activity model does not show any significant power at 16 days (see the
fourth row in Figure 2). As a check, we conducted a separate RV analysis that
included a second Keplerian signal with a narrow uniform prior on its orbital
period centered on 16 days. The resulting marginalized posterior on the
Keplerian semiamplitude is consistent with zero and has an upper limit of 2.45
m s-1 at 95% confidence (i.e. corresponding to $m_{p}\sin{i}$ $<3.03$
$M_{\oplus}$). Together these tests indicate that there is no strong evidence
for a second planet near 16 days but continued RV follow-up will help to
establish the nature of this signal.

Modulo the detailed structure in the MCII detection sensitivity, the results
are broadly consistent with MCI. The similarity between each MC simulation’s
detection sensitivity is depicted in Figure 6 as a function of $K$. The two MC
simulation curves are in close agreement for all values of $K$ with the MCII
sensitivity exhibiting a slight detriment relative to MCI up to $K\geq 5$ m
s-1, at which point all planetary signals within 200 days are detectable.
Focusing on the more realistic results from MCII and taking the 90% contour to
mark the boundary between detection and non-detection, we are able to rule out
additional planets around GJ 1214 above 4 $M_{\oplus}$ and within 10 days. By
adopting the empirical recent Venus and early Mars limits of the inner and
outer habitable zone (HZ) from Kopparapu et al. (2013), we can rule out
planets down to 5 $M_{\oplus}$ within the HZ. The most massive Super-Earths
have masses $\lesssim 7$ $M_{\oplus}$ (LHS 1140 b; Ment et al. 2019, TOI-1235
b; Cloutier et al. 2020b), such that we are only able to rule out the most
massive HZ rocky planets.

Figure 6: RV detection sensitivity to planets orbiting GJ 1214 as a function
of RV semiamplitude for two noise models. The gold and blue curves depict the
RV detection sensitivity from the MCI and MCII simulations, respectively. The
errorbars illustrate the error on the mean detection sensitivity in each
semiamplitude bin. The vertical solid line highlights the semiamplitude of GJ
1214 b.

Within our sensitivity limits, GJ 1214 continues to be a system with only one
known planet. In the next section, we will argue that this result is somewhat
surprising. We will show that multi-planet systems around mid-M dwarfs are
common and that GJ 1214 belongs to a small subset of mid-M dwarf systems that
appear to be singles given the current RV data available.

##  6 The Frequency of Multi-Planet Systems Around Mid-M Dwarfs

The Kepler transit survey revealed that multi-planet systems around early M
dwarfs are ubiquitous (e.g. Dressing & Charbonneau, 2015; Ballard & Johnson,
2016; Gaidos et al., 2016; Hsu et al., 2020). Studies focused on later type M
dwarfs have shown preliminary evidence of a trend of increasing small planet
multiplicity with decreasing stellar mass. Although the current inferences are
largely limited by poor counting statistics around mid-to-late M dwarfs
(Muirhead et al., 2015; Hardegree-Ullman et al., 2019). If it is true that
mid-M dwarf planetary systems frequently host multiple planets, then the
confirmation of even a single transiting planet around a mid-M dwarf would
establish a strong prior on the existence of additional planets. Indeed, a
number of apparently single transiting mid-M dwarf systems have revealed
additional planets in RV follow-up campaigns (GJ 357c,d; Luque et al. 2019, GJ
1132c; Bonfils et al. 2018b, GJ 3473c; Kemmer et al. 2020, LHS 1140c; Ment et
al. 2019). Here we aim to quantify this phenomenon by calculating the
frequency of mid-M dwarf planetary systems that host multiple planets.

###  6.1 Calculating the Frequency of Mid-M Dwarf Planetary Systems with
Multiple Planets

Let $f$ denote the fraction of mid-M dwarf planetary systems that host two or
more planets over some range of planet masses and orbital periods. That is,
among the sample of mid-M dwarfs that host at least one known planet, $f$
denotes the fraction of those planetary systems that host multiple planets. To
calculate $f$, we will consider $N$ mid-M dwarfs that host at least one known
planet (see Section 6.1.1), which we index by $i=1,\dots,N$. For each mid-M
dwarf in our sample, we define $p_{i}\equiv$ the RV detection sensitivity to
additional planets over the range of planet masses $\in[m_{p,\rm min},m_{p,\rm
max}]$ and orbital periods $\in[P_{\rm min},P_{\rm max}]$. We will compute the
$p_{i}$ values in Section 6.1.2. Because each planetary system contains at
least one confirmed planet, the probability of detecting an additional planet
proceeds as a Bernoulli process with probability $fp_{i}$. We denote the
results of the $N$ Bernoulli experiments by the set of binary values
$\\{k_{i}\\}$ wherein $k_{i}=1$ if the $i^{\rm th}$ system contains multiple
planets, with at least one planet having $m_{p}\in[m_{p,\rm min},m_{p,\rm
max}]$ and $P\in[P_{\rm min},P_{\rm max}]$. Otherwise, $k_{i}=0$. For example,
in Section 5.2 we showed that GJ 1214 appears to be a single-planet system
such that $k_{\rm GJ1214}=0$, whereas for the two-planet system LHS 1140,
$k_{\rm LHS1140}=1$. The likelihood of measuring $\\{k_{i}\\}$ from $N$
systems is given by

$\mathcal{L}(\\{k_{i}\\}|f,\\{p_{i}\\})=\prod_{i=1}^{N}\>(fp_{i})^{k_{i}}\>(1-fp_{i})^{1-k_{i}}.$
(6)

Assuming that all values of $f$ are equally probable (i.e. $\text{Pr}(f)=1$),
the normalized posterior of $f$ follows from Bayes theorem:

$\text{Pr}(f|\\{k_{i}\\},\\{p_{i}\\})=\frac{\mathcal{L}(\\{k_{i}\\}|f,\\{p_{i}\\})}{\int_{0}^{1}\mathcal{L}(\\{k_{i}\\}|f,\\{p_{i}\\})df},$
(7)

which we can use to calculate $f$ and its corresponding confidence intervals.

####  6.1.1 Sample of Mid-M Dwarf Systems

We proceed with measuring $f$ by compiling the set of mid-M dwarfs with

  1. 1.

$M_{s}\in[0.1,0.4]$ $M_{\odot}$, and

  2. 2.

that host at least one confirmed planet with an RV mass measurement.

Because our second condition requires systems to contain at least one
confirmed planet with an RV mass measurement, we only consider transiting
planetary systems because the absolute masses of non-transiting planets
uncovered in RV datasets are unmeasurable. The existence of a mass detection
also (largely) implies that sufficient RV follow-up was conducted and a robust
RV detection sensitivity has been achieved. This will enable meaningful
constraints on $f$ to be produced. We note that our second criterion excludes
systems with TTV masses only. This is desirable given the difficulty in
accessing the TTV sensitivity to additional planets; a function which itself
is not smooth with orbital separation due to the dependence of the TTV
amplitude on the proximity of planet pairs to low-order mean motion
resonances. We identify $12$ transiting planetary systems based on our
criteria and list them in Table 4.

Table 4: Summary of known mid-M dwarfaaIn this study, mid-M dwarfs are defined as having a stellar mass $\in[0.1,0.4]$ $M_{\odot}$. transiting planetary systems with RV follow-up Star name | $M_{s}$ | Number | Average RV | Ref.
---|---|---|---|---
| [$M_{\odot}$] | of known | Sensitivity, |
|  | planets |  $p_{i}$ [%]bbAverage RV sensitivity from MCII for the planets with $m_{p}\in[1,10]$ $M_{\oplus}$ and $P\in[0.5,50]$ days that follow log-uniform distributions versus $m_{p}$ and $P$. |
GJ 1214 | $0.178$ | 1 | 65.8 | this work
LHS 1140 | $0.179$ | $\geq 2$ | 75.7 |  Me19,Li20
GJ 1132 | $0.191$ | 2 | 73.4 | Bo18b
LHS 1478 | $0.235$ | 1 | 52.5 | So21
LTT 1445A | $0.257$ | 2 | 66.2 | Wi21 submitted
L 98-59 | $0.313$ | 3 | 57.0 | Cl19
GJ 486 | $0.323$ | 1 | 94.8 | Tr21
GJ 3473 | $0.371$ | 2 | 43.3 | Ke20
GJ 357 | $0.380$ | 2 | 54.7 | Lu19
GJ 1252 | $0.381$ | 1 | 14.1 | Sh20
LTT 3780 | $0.399$ | 2 | 60.0 |  Cl20,No20
TOI-270 | $0.400$ | 3 | 65.2 | Va21

One notable system is excluded from our sample on the basis of having TTV
masses only without substantial RV follow-up (K2-146; Hamann et al., 2019).
Other notable exclusions include the nearby system LHS 3844 (Vanderspek et
al., 2019) and the multi-transiting system around LP 791-18 (Crossfield et
al., 2019), both of which lack published RV time series.

####  6.1.2 RV Detection Sensitivity

We compute each system’s RV detection sensitivity $p_{i}$ identically to the
procedure described in Section 5.1 as applied to our GJ 1214 RV data (i.e. via
injection-recovery tests in the RV time series). We retrieve each system’s RV
time series from its literature source listed in Table 4. For the MCI
simulations, we construct the noisy RVs from the quadrature addition of the
median RV uncertainty and the scalar jitter for each unique spectrograph. In
the MCII simulations, we subtract the orbtial signal for each known planet
using the Keplerian parameters reported in the reference. The planet
injections and subsequent recoveries proceed identically as with GJ 1214. The
resulting sensitivity maps for each mid-M dwarf are displayed in Appendix B.

####  6.1.3 The Frequency of Multi-Planet Systems Around Mid-M Dwarfs

After tabulating $\\{k_{i}\\}$ and calculating $\\{p_{i}\\}$ among the mid-M
dwarfs in our sample, we compute the posterior of $f$ using Eq. 7. Due to the
relatively small number of mid-M dwarf systems with at least one known
transiting planet, we consider a wide range of planetary parameters bounded by
$m_{p}\in[1,10]$ $M_{\oplus}$ and $P\in[0.5,50]$ days. However, because the
typical RV detection sensitivity varies widely across such a wide range of
parameters (c.f. Figure 5), our results as discussed below only apply under
the assumption that planets are distributed uniformly as a function of
$\log{m_{p}}$ and $\log{P}$, both of which are implicitly assumed based on our
sampling of injected planets. Of the $12$ systems in our sample, eight host a
multi-planet system with at least one planet within the aforementioned mass
and period bounds. The resulting $f$ posteriors from MCI and MCII are depicted
in Figure 7. The MAP values of $f$, along with confidence intervals bounded by
the 16th and 84th percentiles are $86^{+6}_{-20}$% and $90^{+5}_{-21}$% for
MCI and MCII, respectively. The 95% lower limits on $f$ are $>57$% and $>59$%
for MCI and MCII, respectively. The consistency of $f$ between both MC
simulations highlights the insensitivity of our results to the exact method of
constructing the noisy RVs.

The consensus from our calculations is that the most likely fraction of mid-M
dwarf planetary systems that host multiple planets less massive than 10
$M_{\oplus}$ and within 50 days is approximately 90%. To rephrase, the
discovery of a single planet within these bounds around a mid-M dwarf
establishes a strong prior that favors the existence of at least one
additional planet in the system.

Figure 7: The posterior probability of $f$: the fraction of mid-M dwarf
planetary systems that host multiple planets with masses $m_{p}\in[1,10]$
$M_{\oplus}$ and orbital periods $P\in[0.5,50]$ days. The analytical $f$
posterior is calculated using Eq. 7 given the mid-M dwarfs with multiple
planets $\\{k_{i}\\}$ and their respective RV detection sensitivities
$\\{p_{i}\\}$. The solid gold curve depicts the result using the detection
sensitivities from MCI whereas the solid blue curve depicts the result from
MCII (see Section 5.1). As a check, the dashed curves depict the $f$ posterior
from each MC when RV planetary systems are included in the calculation (see
Section 6.1.4).

####  6.1.4 Including RV Systems When Calculating $f$

Our previous calculation of $f$ focused solely on transiting planets whose
absolute masses are measured with an RV detection. Here we relax this
requirement by equating minimum planet mass $m_{p}\sin{i}$ to planet mass,
which allows us to include $15$ additional mid-M dwarf planetary systems
(Table 5). We construct our RV planet sample by retrieving planets with
$m_{p}\sin{i}$ $\in[1,10]$ $M_{\oplus}$, $P\in[0.5,50]$ days, and whose
dispositions have not been directly contested in the literature. Our sample of
RV planets is compared to the transiting planet sample in $P$-$m_{p}$ space in
Figure 8. Most of the RV planets have minimum masses and periods that are
distinct from the transiting planet sample. RV planets often have longer
orbital periods because of the geometric selection effect of decreasing
transit probability with increasing orbital period. Many of the RV planets are
also less massive than the transiting planets because their host stars tend to
be brighter, which often results in more intensive RV follow-up and with
higher S/N measurements, both of which facilitate the detection of low
amplitude signals.

Table 5: Summary of known mid-M dwarfaaIn this study, mid-M dwarfs are defined as having a stellar mass $\in[0.1,0.4]$ $M_{\odot}$. RV planetary systems Star name | $M_{s}$ | Number | Average RV | Ref.
---|---|---|---|---
| [$M_{\odot}$] | of known | Sensitivity, |
|  | planets |  $p_{i}$ [%]bbAverage RV sensitivity from MCII for the planets with $m_{p}\sin{i}$ $\in[1,10]$ $M_{\oplus}$ and $P\in[0.5,50]$ days that follow log-uniform distributions versus $m_{p}$ and $P$. |
Prox Cen | $0.122$ | $\geq 1$ | 47.0 |  An16,Su20
GJ 1061 | $0.123$ | 3 | 74.3 | Dr20
YZ Ceti | $0.133$ | 3 | 86.1 | As17a
GJ 3323 | $0.166$ | 2 | 76.2 | As17b
GJ 1265 | $0.168$ | 1 | 59.0 | Lu18
Ross 128 | $0.173$ | 1 | 81.7 | Bo18a
GJ 3779 | $0.264$ | 1 | 50.4 | Lu18
Wolf 1061 | $0.302$ | 3 | 71.5 | As17b
GJ 581 | $0.311$ | 3 | 72.5 | Tr18
GJ 273 | $0.314$ | 2 | 22.0 | As17b
GJ 625 | $0.320$ | 1 | 69.7 | Su17
GJ 667C | $0.330$ | $\geq 2$ | 57.1 | An13
GJ 876 | $0.350$ | 4 | 72.0 | Tr18
GJ 251 | $0.360$ | 1 | 64.3 | St20
GJ 411 | $0.386$ | 1 | 71.2 | Dí19
Figure 8: Minimum masses and orbital periods of planets orbiting mid-M dwarfs
and with an RV mass detection. The circles depict 19 transiting planets in
$12$ systems. The squares depict four RV planets in three transiting planetary
systems. The triangles depict the 31 RV planets in $15$ systems. The shaded
rectangle outlines the range of planetary parameters over which we calculate
$f$ in this work.

We wish to include the set of confirmed RV planets into our calculation of
$f$. We proceed by computing each system’s detection sensitivity $\\{p_{i}\\}$
via injection-recovery in a manner similar to the transiting planetary
systems. The only difference being that we modify the distribution from which
the injected planet’s inclinations are drawn. Because the orbital inclinations
$i$ of RV planets are unconstrained, we draw $i$ from the distribution that is
uniform in $\cos{i}$: i.e. $i$ drawn from $\arccos{\mathcal{U}(-1,1)}$. In
general, this geometric effect is detrimental to the RV detection sensitivity
because it allows for the possibility of nearly face-on orbits for which the
RV semiamplitude goes to zero given its linear dependence on $\sin{i}$. The
resulting sensitivity maps for each mid-M dwarf in our RV sample are included
in Appendix B.

By combining the $15$ RV systems with the $12$ transiting systems, we expand
the sets $\\{k_{i}\\}$ and $\\{p_{i}\\}$. We note that tentative evidence for
additional planet candidates orbiting Proxima Centauri at 5.15 days (Suárez
Mascareño et al., 2020) and at 5.2 years (Damasso et al., 2020) have been
independently reported. However, due to the present uncertainties regarding
the nature of these signals, here we will treat Proxima Centauri as a single-
planet system (i.e. $k_{\rm Prox}=0$). We note that the resulting value of $f$
is largely insensitive to any one system and that if instead we set $k_{\rm
Prox}=1$, the resulting value of $f$ will only increase by 3%. With $8/15=53$%
of RV systems hosting multiple planets over the range of interest, the sample
of RV systems exhibits a slightly lower fraction of known multi-planet systems
compared to the transiting systems ($8/12=68$%). However despite having
fractionally fewer multi-planet systems and systematically larger detection
sensitivities, we find that the $f$ values obtained when RV systems are
included in the calculation are consistent with those obtained from transiting
systems only (Figure 7). Explicitly, including RV planets in MCI we measure
$75^{+9}_{-12}$%, compared to $86^{+6}_{-20}$% with transiting systems alone.
Similarly in MCII, we measure $85^{+8}_{-15}$%, compared to $90^{+5}_{-21}$%
with transiting systems alone. Therefore the following statement continues to
be robust: the existence of one known planet orbiting a mid-M dwarf
establishes a strong prior that favors the existence of at least one
additional planet in the system.

###  6.2 Is the Architecture of the GJ 1214 System Unique Among Mid-M Dwarfs?

In Section 6.1.3 we calculated that $90^{+5}_{-21}$% of mid-M dwarf planetary
systems host multiple planets when at least one planet has $m_{p}\in[1,10]$
$M_{\oplus}$ and $P\in[0.5,50]$ days. Given that GJ 1214 hosts one such
planet, and that multi-planet mid-M dwarf systems are extremely common, it may
appear contradictory that GJ 1214 does not show evidence for additional
planets (Section 5.2). The cause of this apparent discrepancy must be due to
either the GJ 1214 system being a bona-fide single-planet system, or that
additional planets are present but are undetected due to our imperfect RV
sensitivity.

Given GJ 1214’s detection sensitivity over the parameter range of interest
(Figure 5), we find the probability that GJ 1214 is a single-planet system to
be $44^{+9}_{-5}$. This value is consistent with 50% such that there is no
strong evidence that GJ 1214 is a bona-fide single-planet system. For example,
if intensive RV characterization were to continue to achieve a detection
sensitivity that is on par with that of the mid-M dwarf transiting system with
the highest RV sensitivity to-date ($p_{\rm GJ486}=90$%; Trifonov et al.,
2021), then we would measure the probability that GJ 1214 is a single-planet
system to be $15^{+21}_{-1}$% (if it remained a single-planet system). We
estimate that reaching this goal would require an additional 120 hours of
observing time with HARPS (Cloutier et al., 2018). But using an alternative
spectrograph like MAROON-X, which is optimized for M dwarfs (Seifahrt et al.,
2018), the observing time may be signficiantly reduced to about 15
hours.444MAROON-X ETC. We emphasize however that this calculation is
approximate as it does not consider telescope scheduling, overheads, and the
unknown effects of effects of stellar activity at red optical wavelengths.

Figure 9: The probability of detecting the either one (i.e. $k_{i}=0$) or
multiple planets (i.e. $k_{i}=1$) in each individual mid-M dwarf system with
at least one known small planet candidate. Each system’s probability is
calculated from the fraction of mid-M dwarf planetary systems with multi-
planet systems $f=$ $90^{+5}_{-21}$% and each system’s RV detection
sensitivity to planets with $m_{p}\in[1,10]$ $M_{\oplus}$ and $P\in[0.5,50]$
days; $p_{i}$. The uncertainty on $f$ produces the errorbars on each data
point. The number of known planet candidates in each system is annotated next
to its data point.

###  6.3 Are All Small RV Planets Detected Around Mid-M Dwarfs Real?

In general, small planets discovered in RV datasets are more difficult to
confirm than transiting planets. This is due to the typically lower S/N of RV
signals relative to transit signals and the strong impacts that stellar
activity and uneven sampling can have on the recovery of RV signals. As a
result, small RV planets have a greater probability of being a false positive
than their transiting counterparts (e.g. Robertson et al., 2014; Robertson &
Mahadevan, 2014; Rajpaul et al., 2016; Lubin et al., 2021).

Using the value of $f$ measured with transiting planets only (i.e.
$90^{+5}_{-21}$%), we can calculate the probability that all of the RV planets
in our sample are real planets. Among the $15$ RV systems in our sample,
$\sum_{i}\,k_{i}=8$ implying that single-planet RV systems may be more common
than expected based on the results from the transiting systems where the false
positive probability is negligible. Given that the RV detection sensitivities
for RV systems are often superior to the transiting planetary systems, we
would expect to detect multiple planets in closer to 90% of the RV systems, if
those multi-planet systems existed. The fact that only eight out of $15$ (53%)
of RV systems appear to be singles presents a discrepancy with the results
from the ‘clean’ transiting planet sample. We find that the probability of
finding only eight out of $15$ multi-planet mid-M dwarf RV systems to be $<$
$12$% at 99% confidence. That is, mid-M dwarf planetary systems with multiple
planets are common and because most of the RV systems with at least one known
planet have typically high RV detection sensitivity, it is improbable that
such a small fraction of the RV systems would appear to be singles. This
result, coupled with the modest false positive rate for small RV planet
candidates, suggests that it is highly unlikely that all of the small RV
planet candidates in our sample are real planets because the majority of RV
systems should appear to have either zero, or at least two planets.

We note that this statistical argument is unable to identify which RV planet
candidates are most likely to be false positives. We can however investigate
the probability of detecting $k_{i}$ planets in each individual system given
each $p_{i}$ and $f=$ $90^{+5}_{-21}$%; i.e. Pr$(k_{i}|f,p_{i})$. Figure 9
depicts these probabilities for the $15$ RV systems using the set of
$\\{p_{i}\\}$ from MCII. The system most likely to produce its observed result
is YZ Ceti ($72^{+9}_{-13}$%), which hosts three known planet candidates and
has the largest detection sensitivity among the RV systems ($p_{\rm
YZCeti}=86$%; Astudillo-Defru et al., 2017b). Conversely, GJ 273 has the
lowest probability of having its two small planet candidates detected
($18^{+2}_{-3}$%). We caution that this does not imply that the GJ 273 planet
candidates are the most likely false positives among our sample of RV systems.
Instead, the low detection sensitivity around GJ 273 is likely the result of
residual activity cycles that are left corrected in MCII and therefore inhibit
the detection of planetary signals.

##  7 Summary

In this paper we presented a joint transit plus RV analysis of the enveloped
terrestrial GJ 1214 b using archival Spitzer photometry and ten years of RV
follow-up with the HARPS spectrograph. Given the RV sensitivity of our data,
we do not detect any additional planets around GJ 1214. We place the
architecture of the GJ 1214 system in the context of mid-M dwarf systems by
calculating the frequency of mid-M dwarf planetary systems that host multiple
planets. Our main findings are summarized below.

  * •

Using 165 HARPS RV measurements, we refine the mass of GJ 1214 b to be
$8.17\pm 0.43$ $M_{\oplus}$. If the composition of GJ 1214 b resembles an
Earth-like interior with 33% Fe and 67% MgSiO3 mass fractions and surrounded
by an extended H/He envelope, the envelope mass fraction needed to explain its
mass and radius ($2.742^{+0.050}_{-0.053}$ $R_{\oplus}$) is $X_{\rm env}=$
$5.24^{+0.30}_{-0.29}$%.

  * •

We constrain the planet’s orbital eccentricity to be $<$ $0.063$ at 95%
confidence, which narrows its secondary eclipse window to 2.8 hours at 95%
confidence.

  * •

We compute the RV detection sensitivity to additional planets around GJ 1214
and are able to rule out planets more massive than 3 $M_{\oplus}$ within 10
days and planets more massive than 5 $M_{\oplus}$ within the HZ.

  * •

We compute the RV detection sensitivities of a set of $12$ mid-M dwarf
transiting systems (including GJ 1214) and measure the frequency of mid-M
dwarf planetary systems that host multiple planets with $m_{p}\in[1,10]$
$M_{\oplus}$ and $P\in[0.5,50]$ days to be $f=$ $90^{+5}_{-21}$%.

  * •

Given $f$, there is a $44^{+9}_{-5}$ chance that GJ 1214 is a bona-fide
single-planet system. This value is consistent with 50% because the RVs
presented in this paper are not very constraining and a second planet around
GJ 1214 may be present but is beyond our current RV detection capabilities.

  * •

Also, given $f$ and the typically high detection sensitivities of $15$ RV
systems around mid-M dwarfs, the probability that all of the RV planets within
those systems are real is just $<$ $12$% at 99% confidence.

R.C. acknowledges support from the Banting Postdoctoral Fellowship program,
administered by the Government of Canada, and from the National Aeronautics
and Space Administration in support of the TESS science mission. This material
is based upon work supported by the National Aeronautics and Space
Administration under grant No. 80NSSC18K0476 issued through the XRP Program.
N.A.D. acknowledges the support of FONDECYT project 3180063. Based on
observations collected at the European Southern Observatory under ESO
programmes 183.C-0972, 283.C-5022, 1102.C-0339, 183.C-0437, and 198.C-0838.
Based on data obtained from the ESO Science Archive Facility under request
number 578670. This work is based in part on archival data obtained with the
Spitzer Space Telescope, which was operated by the Jet Propulsion Laboratory,
California Institute of Technology under a contract with NASA.

##  Appendix A GJ 1214 Global Transit + RV Model Results

Table 6: Point estimates of the GJ 1214 model parameters Parameter | Prior | Value
---|---|---
_Stellar parameters_
Stellar mass, $M_{s}$ [$M_{\odot}$] | $\mathcal{N}(0.178,0.010)$ | $0.178\pm 0.010$
Stellar radius, $R_{s}$ [$R_{\odot}$] | $\mathcal{N}(0.215,0.008)$ | $0.215\pm 0.008$
_RV parameters_
Log covariance amplitude, [m s-1] | $\mathcal{U}(-5,5)$ | $1.03^{+0.33}_{-0.35}$
$\ln{a_{\rm RV}}$ |  |
$a_{\rm RV}$, [m s-1] | $-$ | $2.79^{+1.08}_{-0.82}$
Log exponential timescale, $\ln{\lambda_{\rm GP}}$ [days] | Pr($\ln{\lambda_{\rm GP}}|\mathbf{f_{\rm STELLA}}$) | $5.86^{+0.23}_{-0.30}$
Log coherence, $\ln{\Gamma_{\rm GP}}$ | Pr($\ln{\Gamma_{\rm GP}}|\mathbf{f_{\rm STELLA}}$) | $-0.69\pm 0.01$
Log rotation period, $\ln{P_{\rm rot}}$ [days] | Pr($\ln{P_{\rm rot}}|\mathbf{f_{\rm STELLA}}$) | $4.835^{+0.038}_{-0.040}$
Log jitter, $\ln{s_{\rm RV,pre}}$ [m s-1] | $\mathcal{U}(-5,5)$ | $1.26^{+0.15}_{-0.16}$
$s_{\rm RV,pre}$, [m s-1] | $-$ | $3.52^{+0.56}_{-0.50}$
Log jitter, $\ln{s_{\rm RV,post}}$ [m s-1] | $\mathcal{U}(-5,5)$ | $0.81^{+0.28}_{-0.31}$
$s_{\rm RV,post}$, [m s-1] | $-$ | $2.25^{+0.76}_{-0.80}$
Velocity offset, $\gamma_{\rm RV,pre}$ [m s-1] | $\mathcal{U}(-\inf,\inf)$ | $-3.09^{+1.65}_{-1.66}$
Velocity offset, $\gamma_{\rm RV,post}$ [m s-1] | $\mathcal{U}(-\inf,\inf)$ | $-0.61^{+1.76}_{-1.90}$
_Transit parameters_
Baseline flux, $f_{0,3.6}$ | $\mathcal{U}(-\inf,\inf)$ | $2.9^{+4.6}_{-4.3}\times 10^{-5}$
Baseline flux, $f_{0,4.5}$ | $\mathcal{U}(-\inf,\inf)$ | $2.1^{+2.2}_{-2.1}\times 10^{-5}$
Limb darkening coefficient, $u_{1,3.6}$ | $\mathcal{N}(-0.0210,0.0052)$ | $-0.0210\pm 0.0052$
Limb darkening coefficient, $u_{2,3.6}$ | $\mathcal{N}(0.1852,0.0050)$ | $0.1852\pm 0.0050$
Limb darkening coefficient, $u_{1,4.5}$ | $\mathcal{N}(-0.0046,0.0050)$ | $-0.0046\pm 0.0050$
Limb darkening coefficient, $u_{2,4.5}$ | $\mathcal{N}(0.1976,0.0030)$ | $0.1976\pm 0.0030$
_GJ 1214 b parameters (measured)_
Orbital period, $P$ [days] | $\mathcal{U}(1,2)$ | $1.58040433\pm 0.00000013$
Time of mid-transit, | $\mathcal{U}(5700,5702)$ | $5701.413328^{+0.000066}_{-0.000059}$
$T_{0}$ [BJD - 2,450,000] |  |
Planet-to-star radius ratio, $(r_{p}/R_{s})_{3.6}$ | $\mathcal{U}(0.07,0.15)$ | $0.11608\pm 0.00034$
Planet-to-star radius ratio, $(r_{p}/R_{s})_{4.5}$ | $\mathcal{U}(0.07,0.15)$ | $0.11677\pm 0.00017$
Impact parameter, $b$ | $\mathcal{U}(0,1)$ | $0.325\pm 0.025$
Log RV semiamplitude, $\ln{K}$ | $\mathcal{U}(-1,4)$ | $2.665^{+0.036}_{-0.037}$
$h=\sqrt{e}\cos{\omega}$ | $\mathcal{U}(-1,1)$ | $-0.015^{+0.123}_{-0.135}$
$k=\sqrt{e}\sin{\omega}$ | $\mathcal{U}(-1,1)$ | $-0.008^{+0.057}_{-0.053}$
_GJ 1214 b parameters (derived)_
Planet radius, $r_{p,3.6}$ [$R_{\oplus}$] | - | $2.733^{+0.050}_{-0.052}$
Planet radius, $r_{p,4.5}$ [$R_{\oplus}$] | - | $2.749^{+0.050}_{-0.054}$
RV semiamplitude, $K$ [m s-1] | - | $14.36\pm 0.53$
Planet mass, $m_{p}$ [$M_{\oplus}$] | - | $8.17\pm 0.43$
Bulk density, $\rho_{p}$ [g cm-3] | - | $2.20^{+0.17}_{-0.16}$
Surface gravity, $g_{p}$ [m s-2] | - | $10.65^{+0.71}_{-0.67}$
Escape velocity, $v_{\rm esc}$ [km s-1] | - | $19.31^{+0.53}_{-0.54}$
Scaled semimajor axis, $a/R_{s}$ | - | $14.85\pm 0.16$
Inclination, $i$ [deg] | - | $88.7\pm 0.1$
Eccentricity, $e$ | - |  $<$ $0.063$aa95% upper limit.
Semimajor axis, $a$ [au] | - | $0.01490\pm 0.00026$
Insolation, $F$ [F⊕] | - | $21.0^{+2.7}_{-2.5}$
Equilibrium temperature, $T_{\rm eq}$ [K]bbAssuming uniform heat redistribution and zero albedo. | - | $596\pm 19$
Envelope mass fraction, $X_{\rm env}$ [%]ccAssuming an Earth-like solid core with a 33% iron core mass fraction. | - | $5.24^{+0.30}_{-0.29}$

##  Appendix B Mid-M Dwarf RV Detection Sensitivity

Figure 10: Raw and synthetic RVs time series, GLS periodograms, and
sensitivity maps for GJ 1214. Top row: raw RV data from HARPS with 90 pre
(squares) and 75 (circles) post-fiber upgrade measurements. Second row: an
example of a noisy synthetic RV dataset from the Monte-Carlo simulation MCI
wherein the noisy RVs are constructed from Gaussian noise with a dispersion
equal to the rms of the RV residuals. Third row: noisy synthetic RV data from
the Monte-Carlo simulation MCII wherein the noisy RVs are constructed from the
RV residuals after the removal of GJ 1214 b only (i.e. no GP component is
removed). Right column: each RV time series’ corresponding GLS periodogram.
The vertical shaded regions highlight the orbital period of the planet GJ 1214
b ($P=1.58$ days). The horizontal dashed lines depict the approximate 1% FAP
levels. Lower left panel: the RV detection sensitivity computed from MCI as a
function of planet mass and orbital period. Lower right panel: the RV
detection sensitivity from MCII. GJ 1214 b is depicted as a white circle
marker in both panels. The solid contours highlight the 10% and 90%
sensitivity levels. The dashed lines represent lines of constant semiamplitude
with illustrative values equal to 1 and 10 m s-1. Figure 11: Similar to Figure
10 but for the LHS 1140 transiting system using RV data from HARPS (Ment et
al., 2019). Figure 12: Similar to Figure 10 but for the GJ 1132 transiting
system using RV data from HARPS (Bonfils et al., 2018b). Figure 13: Similar to
Figure 10 but for the LHS 1478 transiting system using RV data from CARMENES
and IRD (Soto et al., 2021). Figure 14: Similar to Figure 10 but for the LTT
1445 A transiting system using RV data from ESPRESSO, HARPS, HIRES, MAROON-X,
and PFS (Winters et al. 2021 submitted). Figure 15: Similar to Figure 10 but
for the L 98-59 transiting system using RV data from HARPS (Cloutier et al.,
2019). Figure 16: Similar to Figure 10 but for the GJ 486 transiting system
using RV data from CARMENES, MAROON-X blue, and MAROON-X red (Trifonov et al.,
2021). Figure 17: Similar to Figure 10 but for the GJ 3473 transiting system
using RV data from CARMENES, HARPS, and IRD (Kemmer et al., 2020). Figure 18:
Similar to Figure 10 but for the GJ 357 transiting system using RV data from
CARMENES, HARPS, HIRES, PFS (with two distinct iodine calibrations), and UVES
(Luque et al., 2019). Figure 19: Similar to Figure 10 but for the GJ 1252
transiting system using RV data from HARPS (Shporer et al., 2020). Figure 20:
Similar to Figure 10 but for the LTT 3780 transiting system using RV data from
CARMENES (Nowak et al., 2020), HARPS, and HARPS-N (Cloutier et al., 2020a).
Figure 21: Similar to Figure 10 but for the TOI-270 transiting system using RV
data from ESPRESSO and HARPS (Van Eylen et al., 2021). Figure 22: Similar to
Figure 10 but for the Proxima Centauri RV system using RV data from ESPRESSO
(Suárez Mascareño et al., 2020), HARPS and UVES (Anglada-Escudé et al., 2016).
Figure 23: Similar to Figure 10 but for the GJ 1061 RV system using RV data
from HARPS (Dreizler et al., 2020). Figure 24: Similar to Figure 10 but for
the YZ Ceti RV system using RV data from HARPS (Astudillo-Defru et al.,
2017b). Figure 25: Similar to Figure 10 but for the GJ 3323 RV system using RV
data from HARPS (Astudillo-Defru et al., 2017a). Figure 26: Similar to Figure
10 but for the GJ 1265 RV system using RV data from CARMENES and HARPS (Luque
et al., 2018). Figure 27: Similar to Figure 10 but for the Ross 128 RV system
using RV data from HARPS (Bonfils et al., 2018a). Figure 28: Similar to Figure
10 but for the GJ 3779 RV system using RV data from CARMENES (Luque et al.,
2018). Figure 29: Similar to Figure 10 but for the Wolf 1061 RV system using
RV data from HARPS (Astudillo-Defru et al., 2017a). Figure 30: Similar to
Figure 10 but for the GJ 581 RV system using RV data from HARPS (Trifonov et
al., 2018). Figure 31: Similar to Figure 10 but for the GJ 273 RV system using
RV data from HARPS (Astudillo-Defru et al., 2017a). Figure 32: Similar to
Figure 10 but for the GJ 625 RV system using RV data from HARPS (Suárez
Mascareño et al., 2017). Figure 33: Similar to Figure 10 but for the GJ 667C
RV system using RV data from HARPS, HIRES, and PFS (Anglada-Escudé et al.,
2013b).

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