11institutetext:  Univ. Grenoble Alpes, CNRS, IPAG, F-38000 Grenoble, France 22institutetext: Observatoire de Genève, Département d’Astronomie, Université de Genève, Chemin Pegasi 51, 1290 Versoix, Switzerland 33institutetext: Institute for Computational Science, University of Zurich, Winterthurerstr. 190, CH-8057 Zurich, Switzerland 44institutetext: Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Alonso de Rivera 2850, Concepción, Chile 55institutetext: Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 66institutetext: Department of Physics and Astronomy, University of Kansas, Lawrence, KS 66045 77institutetext: Astrophysics Group, Keele University, Staffordshire ST5 5BG, UK 88institutetext: Department of Physics, Engineering and Astronomy, Stephen F. Austin State University, 1936 North St, Nacogdoches, TX 75962, USA 99institutetext: SOAR Telescope/NSF’s NOIRLab, Casilla 603, La Serena, Chile 1010institutetext: George Mason University, 4400 University Drive, Fairfax, VA, 22030 USA 1111institutetext: International Center for Advanced Studies (ICAS) and ICIFI (CONICET), ECyT-UNSAM, Campus Miguelete, 25 de Mayo y Francia, (1650) Buenos Aires, Argentina 1212institutetext: Noqsi Aerospace Ltd., 15 Blanchard Avenue, Billerica, MA 01821, USA 1313institutetext: Department of Astronomy and Tsinghua Centre for Astrophysics, Tsinghua University, Beijing 100084, China 1414institutetext: Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 1515institutetext: NASA Ames Research Center, Moffett Field, CA 94035, USA 1616institutetext: Dept. of Physics & Astronomy, Swarthmore College, Swarthmore PA 19081, USA 1717institutetext: Department of Physics and Astronomy, The University of North Carolina at Chapel Hill, Chapel Hill, NC 27599-3255, USA 1818institutetext: Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD, 21218, USA 1919institutetext: Instituto de Astrofísica de Canarias (IAC), E-38200 La Laguna, Tenerife, Spain 2020institutetext: Dept. Astrofísica, Universidad de La Laguna (ULL), E-38206 La Laguna, Tenerife, Spain 2121institutetext: AAVSO, 5 Inverness Way, Hillsborough, CA 94010, USA 2222institutetext: Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 2323institutetext: Department of Aeronautics and Astronautics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA 2424institutetext: Perth Exoplanet Survey Telescope, Perth, Western Australia 2525institutetext: Curtin Institute of Radio Astronomy, Curtin University, Bentley, Western Australia 6102 2626institutetext: SETI Institute, Mountain View, CA 94043, USA 2727institutetext: Department of Astrophysical Sciences, Princeton University, NJ 08544, USA

# Atmosphere of the mini-Neptune transiting the bright M dwarf GJ 3090 can be
characterized

J.M. Almenara Atmosphere of the mini-Neptune transiting the bright M dwarf GJ
3090 can be characterizedAtmosphere of the mini-Neptune transiting the bright
M dwarf GJ 3090 can be characterized X. Bonfils Atmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere of the
mini-Neptune transiting the bright M dwarf GJ 3090 can be characterized J.F.
Otegi Atmosphere of the mini-Neptune transiting the bright M dwarf GJ 3090 can
be characterizedAtmosphere of the mini-Neptune transiting the bright M dwarf
GJ 3090 can be characterizedAtmosphere of the mini-Neptune transiting the
bright M dwarf GJ 3090 can be characterizedAtmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterized O. Attia Atmosphere
of the mini-Neptune transiting the bright M dwarf GJ 3090 can be
characterizedAtmosphere of the mini-Neptune transiting the bright M dwarf GJ
3090 can be characterized M. Turbet Atmosphere of the mini-Neptune transiting
the bright M dwarf GJ 3090 can be characterizedAtmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterized N. Astudillo-Defru
Atmosphere of the mini-Neptune transiting the bright M dwarf GJ 3090 can be
characterizedAtmosphere of the mini-Neptune transiting the bright M dwarf GJ
3090 can be characterized K.A. Collins Atmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere of the
mini-Neptune transiting the bright M dwarf GJ 3090 can be characterized A.S.
Polanski Atmosphere of the mini-Neptune transiting the bright M dwarf GJ 3090
can be characterizedAtmosphere of the mini-Neptune transiting the bright M
dwarf GJ 3090 can be characterized V. Bourrier Atmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere of the
mini-Neptune transiting the bright M dwarf GJ 3090 can be characterized C.
Hellier Atmosphere of the mini-Neptune transiting the bright M dwarf GJ 3090
can be characterizedAtmosphere of the mini-Neptune transiting the bright M
dwarf GJ 3090 can be characterized C. Ziegler Atmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere of the
mini-Neptune transiting the bright M dwarf GJ 3090 can be characterized F.
Bouchy Atmosphere of the mini-Neptune transiting the bright M dwarf GJ 3090
can be characterizedAtmosphere of the mini-Neptune transiting the bright M
dwarf GJ 3090 can be characterized C. Briceño Atmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere of the
mini-Neptune transiting the bright M dwarf GJ 3090 can be characterized D.
Charbonneau Atmosphere of the mini-Neptune transiting the bright M dwarf GJ
3090 can be characterizedAtmosphere of the mini-Neptune transiting the bright
M dwarf GJ 3090 can be characterized M. Cointepas Atmosphere of the mini-
Neptune transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere
of the mini-Neptune transiting the bright M dwarf GJ 3090 can be
characterizedAtmosphere of the mini-Neptune transiting the bright M dwarf GJ
3090 can be characterizedAtmosphere of the mini-Neptune transiting the bright
M dwarf GJ 3090 can be characterized K.I. Collins Atmosphere of the mini-
Neptune transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere
of the mini-Neptune transiting the bright M dwarf GJ 3090 can be characterized
I. Crossfield Atmosphere of the mini-Neptune transiting the bright M dwarf GJ
3090 can be characterizedAtmosphere of the mini-Neptune transiting the bright
M dwarf GJ 3090 can be characterized X. Delfosse Atmosphere of the mini-
Neptune transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere
of the mini-Neptune transiting the bright M dwarf GJ 3090 can be characterized
R.F. Díaz Atmosphere of the mini-Neptune transiting the bright M dwarf GJ 3090
can be characterizedAtmosphere of the mini-Neptune transiting the bright M
dwarf GJ 3090 can be characterized C. Dorn Atmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere of the
mini-Neptune transiting the bright M dwarf GJ 3090 can be characterized J.P.
Doty Atmosphere of the mini-Neptune transiting the bright M dwarf GJ 3090 can
be characterizedAtmosphere of the mini-Neptune transiting the bright M dwarf
GJ 3090 can be characterized T. Forveille Atmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere of the
mini-Neptune transiting the bright M dwarf GJ 3090 can be characterized G.
Gaisné Atmosphere of the mini-Neptune transiting the bright M dwarf GJ 3090
can be characterizedAtmosphere of the mini-Neptune transiting the bright M
dwarf GJ 3090 can be characterized T. Gan Atmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere of the
mini-Neptune transiting the bright M dwarf GJ 3090 can be characterized R.
Helled Atmosphere of the mini-Neptune transiting the bright M dwarf GJ 3090
can be characterizedAtmosphere of the mini-Neptune transiting the bright M
dwarf GJ 3090 can be characterized K. Hesse Atmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere of the
mini-Neptune transiting the bright M dwarf GJ 3090 can be characterized J.M.
Jenkins Atmosphere of the mini-Neptune transiting the bright M dwarf GJ 3090
can be characterizedAtmosphere of the mini-Neptune transiting the bright M
dwarf GJ 3090 can be characterized E.L.N. Jensen Atmosphere of the mini-
Neptune transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere
of the mini-Neptune transiting the bright M dwarf GJ 3090 can be characterized
D.W. Latham Atmosphere of the mini-Neptune transiting the bright M dwarf GJ
3090 can be characterizedAtmosphere of the mini-Neptune transiting the bright
M dwarf GJ 3090 can be characterized N. Law Atmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere of the
mini-Neptune transiting the bright M dwarf GJ 3090 can be characterized A.W.
Mann Atmosphere of the mini-Neptune transiting the bright M dwarf GJ 3090 can
be characterizedAtmosphere of the mini-Neptune transiting the bright M dwarf
GJ 3090 can be characterized S. Mao Atmosphere of the mini-Neptune transiting
the bright M dwarf GJ 3090 can be characterizedAtmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterized B. McLean
Atmosphere of the mini-Neptune transiting the bright M dwarf GJ 3090 can be
characterizedAtmosphere of the mini-Neptune transiting the bright M dwarf GJ
3090 can be characterized F. Murgas Atmosphere of the mini-Neptune transiting
the bright M dwarf GJ 3090 can be characterizedAtmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere of the
mini-Neptune transiting the bright M dwarf GJ 3090 can be
characterizedAtmosphere of the mini-Neptune transiting the bright M dwarf GJ
3090 can be characterized G. Myers Atmosphere of the mini-Neptune transiting
the bright M dwarf GJ 3090 can be characterizedAtmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterized S. Seager
Atmosphere of the mini-Neptune transiting the bright M dwarf GJ 3090 can be
characterizedAtmosphere of the mini-Neptune transiting the bright M dwarf GJ
3090 can be characterizedAtmosphere of the mini-Neptune transiting the bright
M dwarf GJ 3090 can be characterizedAtmosphere of the mini-Neptune transiting
the bright M dwarf GJ 3090 can be characterizedAtmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere of the
mini-Neptune transiting the bright M dwarf GJ 3090 can be characterized A.
Shporer Atmosphere of the mini-Neptune transiting the bright M dwarf GJ 3090
can be characterizedAtmosphere of the mini-Neptune transiting the bright M
dwarf GJ 3090 can be characterized T.G. Tan Atmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere of the
mini-Neptune transiting the bright M dwarf GJ 3090 can be
characterizedAtmosphere of the mini-Neptune transiting the bright M dwarf GJ
3090 can be characterizedAtmosphere of the mini-Neptune transiting the bright
M dwarf GJ 3090 can be characterized J.D. Twicken Atmosphere of the mini-
Neptune transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere
of the mini-Neptune transiting the bright M dwarf GJ 3090 can be
characterizedAtmosphere of the mini-Neptune transiting the bright M dwarf GJ
3090 can be characterizedAtmosphere of the mini-Neptune transiting the bright
M dwarf GJ 3090 can be characterized J. Winn Atmosphere of the mini-Neptune
transiting the bright M dwarf GJ 3090 can be characterizedAtmosphere of the
mini-Neptune transiting the bright M dwarf GJ 3090 can be characterized

We report the detection of GJ 3090 b (TOI-177.01), a mini-Neptune on a 2.9-day
orbit transiting a bright (K = 7.3 mag) M2 dwarf located at 22 pc. The planet
was identified by the Transiting Exoplanet Survey Satellite and was confirmed
with the High Accuracy Radial velocity Planet Searcher radial velocities.
Seeing-limited photometry and speckle imaging rule out nearby eclipsing
binaries. Additional transits were observed with the LCOGT, Spitzer, and ExTrA
telescopes. We characterise the star to have a mass of 0.519 $\pm$ 0.013
$M_{\odot}$ and a radius of 0.516 $\pm$ 0.016 $R_{\odot}$. We modelled the
transit light curves and radial velocity measurements and obtained a planetary
mass of 3.34 $\pm$ 0.72 $\mathrm{M_{E}}$, a radius of 2.13 $\pm$ 0.11
$\mathrm{R_{E}}$, and a mean density of
1.89${}^{+0.52}_{-0.45}\leavevmode\nobreak\ {\rm g/cm^{3}}$. The low density
of the planet implies the presence of volatiles, and its radius and insolation
place it immediately above the radius valley at the lower end of the mini-
Neptune cluster. A coupled atmospheric and dynamical evolution analysis of the
planet is inconsistent with a pure H-He atmosphere and favours a heavy mean
molecular weight atmosphere. The transmission spectroscopy metric of
221${}^{+66}_{-46}$ means that GJ 3090 b is the second most favourable mini-
Neptune after GJ 1214 b whose atmosphere may be characterised. At almost half
the mass of GJ 1214 b, GJ 3090 b is an excellent probe of the edge of the
transition between super-Earths and mini-Neptunes. We identify an additional
signal in the radial velocity data that we attribute to a planet candidate
with an orbital period of 13 days and a mass of 17.1${}^{+8.9}_{-3.2}$
$\mathrm{M_{E}}$, whose transits are not detected.

###### Key Words.:

stars: individual: GJ~3090 – stars: planetary systems – techniques:
photometric – techniques: radial velocities

##  1 Introduction

The structure, bulk composition, and atmosphere of planets that are found in
transit can be characterised. To achieve precise measurements, planets that
transit bright and small stars are especially well suited.

Over the past two decades, $\sim$3,500 transiting planets have been
discovered111according to https://exoplanetarchive.ipac.caltech.edu as of
2021, October 12th and important statistical properties have emerged. Super-
Earths and mini-Neptunes, which are absent in our Solar System, have been
found to be frequent around other stars for a wide range of orbital periods
(Howard et al. 2012). After the seminal discovery of HD209458 b (Charbonneau
et al. 2000; Henry et al. 2000), many other planets have been detected on
short-period orbits with transit surveys ($P<10$ days), but a relative
scarcity is observed for hot Neptunes; this has been called the ’hot-Neptune
desert’ (Lecavelier Des Etangs 2007; Davis & Wheatley 2009; Szabó & Kiss 2011;
Beaugé & Nesvorný 2013; Lundkvist et al. 2016; Mazeh et al. 2016). Another
robust outcome of recent research is the bimodal radius distribution of
planets smaller than $\sim 4$ $\mathrm{R_{E}}$. The two modes seem to divide
rocky from volatile-rich planets with a transition at around $1.8$
$\mathrm{R_{E}}$ (Fulton et al. 2017; Zeng et al. 2019; Mousis et al. 2020)
that may depend on orbital period (Van Eylen et al. 2018; Petigura 2020) and
stellar mass (Cloutier & Menou 2020). The wish to interpret these demographics
has inspired much theoretical work. Models of photoevaporation and orbital
evolution have been shown to reproduce the radius valley (Owen & Wu 2017; Jin
& Mordasini 2018). Evaporation can also be powered by the intrinsic heat of
the planet that accumulates during the phase of accretion, which might play a
role for small exoplanets and for the formation of their bimodal distribution
(Ginzburg et al. 2018; Gupta & Schlichting 2019). Today, discovering new
planets near the edge of the valley and around stellar hosts of various masses
is particularly important for understanding where these mechanisms are
predominant. In particular, the precise characterisation of the atmospheric
composition of mini-Neptunes can reveal clues for their formation location and
migration and for their interior structure (Bean et al. 2021). Mini-Neptunes
around M dwarfs are generally favourable targets for atmospheric
characterisation through transit spectroscopy because of their large-scale
height and the ratio of the planet-to-star radius, although the final
amplitude of the signal also depends on the atmospheric composition and the
presence of clouds or hazes (Kreidberg et al. 2014).

We report here the confirmation of GJ 3090 b, a mini-Neptune around a bright M
dwarf (GJ 3090, HIP 6365) whose atmosphere can be characterised. GJ 3090 b was
first identified by the Transiting Exoplanet Survey Satellite (TESS; Ricker et
al. 2015), which is a space-based all-sky transit survey mission that focuses
on bright targets.

In Sect. 2 we describe the data we used to detect the candidate GJ 3090 b and
confirm its planetary nature. In Sect. 3 we characterise its host star, and in
Sect. 4 we model the radial velocity (RV) and transit data. Finally, in Sect.
5 we discuss the results of our work, including interior characterisation,
coupled atmospheric and dynamical evolution, and prospects for atmospheric
characterisation.

##  2 Observations

###  2.1 TESS

TESS observed GJ 3090 (TIC 262530407) in sectors 2, 3, and 29 with a two-
minute cadence for a total of 81 days and a time span of 760 days. A transit
signature with a 2.853-day period and 1.7 mmag depth was first identified in
the Science Processing Operations Center (SPOC; Jenkins et al. 2016)
transiting planet search (Jenkins 2002; Jenkins et al. 2010, 2020) of the
sector 2 light curve. The SPOC threshold crossing event was subsequently
promoted by the TESS Science Office to TESS object of interest (TOI; Guerrero
et al. 2021) status as TOI 177.01 based on the model fit and diagnostic test
results in the SPOC data validation report (Twicken et al. 2018; Li et al.
2019).

The SPOC simple aperture photometry (SAP; Twicken et al. 2010; Morris et al.
2020) and pre-search data conditioning SAP (PDCSAP, Stumpe et al. 2012, 2014;
Smith et al. 2012) light curves are shown in Fig. 1. The PDCSAP light curve is
the SAP light curve from which long-term trends have been removed. Twenty
complete transits were observed. The SAP light curve shows an amplitude
variability (peak to peak) of $\sim$1.5% that is compatible with the stellar
rotational modulation. To estimate the stellar rotation period, we removed the
transits from the SAP light curve, normalised each sector by the maximum flux
of a 0.5-day sliding mean, and modelled the resulting light curve with juliet
(Espinoza et al. 2019; Speagle 2020) and the quasi-periodic (QP) kernel
Gaussian process (GP) included in celerite (Foreman-Mackey et al. 2017). We
obtained a period of the QP kernel of 17.65 $\pm$ 0.48 days (Fig. 1).

Next, we prepared the TESS data for transit modelling as follows: We started
from the PDCSAP light curve, that include corrections for contamination in the
aperture of 1.5, 1.3, and 2.0% for sectors 2, 3, and 29 respectively, mainly
due to one contaminant at 28″ that is $\sim$4 magnitudes fainter than the
target in the Gaia G band (Fig. 22). We extracted the data spanning three
transit durations around the centre of each transit. Finally, we normalised
each transit with a straight line fitted to the out-of-transit parts of the
light curve. To account for the differences in stellar flux levels at the
times of each transit (Czesla et al. 2009), we corrected the normalised
transit curves using the GP model described above. To properly correct for the
stellar flux variability, it would be necessary to have observed the purely
photospheric stellar surface without any activity regions. This does not seem
to be the case here, as there is no global flat flux maximum. In any case,
this is a minor correction that changes the planet-to-star radius ratio by
only $\sim$0.2 $\sigma$.

Figure 1: TESS-normalised SAP (blue points) and PDCSAP (grey points, offset
for clarity). Transits of GJ 3090 b (orange) were excluded before fitting the
data with a QP kernel GP (the black line is the posterior median model). The
vertical green lines and bands are the median and 68.3% credible interval
transit window for a second planet candidate in the system (Sects. 4.1 and
4.2).

###  2.2 Seeing-limited photometry

We conducted ground-based photometric follow-up observations of GJ 3090 as
part of the TESS follow-up observing program222https://tess.mit.edu/followup
(TFOP; Collins 2019) to attempt to rule out or identify nearby eclipsing
binaries (NEBs) as potential sources of the TESS detection, measure the
transit-like event on target to confirm the depth and thus the TESS
photometric deblending factor, and refine the TESS ephemeris. We used the TESS
Transit Finder, which is a customised version of the Tapir software package
(Jensen 2013), to schedule our transit observations.

The observations are summarised in Table 1. Three observations searched for
and found no NEBs within $2.5\arcmin$ of the target star. One observation was
inconclusive, and two observations detected the event on-target, as indicated
in Table 1. The Las Cumbres Observatory Global Telescope (LCOGT; Brown et al.
2013) observations were calibrated with the standard LCOGT BANZAI pipeline
(McCully et al. 2018). All photometric data were extracted using AstroImageJ
(Collins et al. 2017), except for the Perth Exoplanet Survey Telescope
observation, which used a custom pipeline based on
C-Munipack333http://c-munipack.sourceforge.net.

Table 1: TFOP photometric follow-up observations.

Observatory / Location | UTC Date | Aperture (m) | Filter | Coverage | Result
---|---|---|---|---|---
Perth Exoplanet Survey Telescope, Perth, Australia | 2018-10-24 | 0.3 | Rc | Full | No NEB detected
Myers Observatory, Siding Spring, Australia | 2018-10-24 | 0.4 | Rc | Full | No NEB detected
LCOGT CTIO, Chile | 2018-11-01 | 0.4 | $i^{\prime}$ | Full | Inconclusive
LCOGT SAAO, South Africa | 2018-11-07 | 0.4 | $r^{\prime}$ | Full | On-target detection
Myers Observatory, Siding Spring, Australia | 2018-11-10 | 0.4 | Lum | Full | No NEB detected
LCOGT CTIO, Chile | 2018-11-19 | 1.0 | $r^{\prime}$ | Ingress | On-target detection

444The filter designation Lum indicates a luminance filter with central
wavelength 550 nm and bandwidth 300 nm. LCOGT indicates observations from the
Las Cumbres Observatory Global Telescope (Brown et al. 2013) 1.0 m and 0.4 m
network nodes at the Cerro Tololo Inter-American Observatory (CTIO) and at the
South Africa Astronomical Observatory (SAAO).

###  2.3 Spitzer

GJ 3090 was observed with Spitzer on 2019 April 4, using director’s
discretionary time (Crossfield et al. 2018). A single transit was observed
using the 4.5 $\mu$m channel (IRAC2; Fazio et al. 2004) in subarray mode with
an integration time of 2 seconds. The transit observation spanned 18 hr 37
min, totalling 711 frames with short observations taken before and after
transit to check for bad pixels. Peak-up mode was used to place the star as
close as possible to the well-characterised sweet spot of the detector.

###  2.4 ExTrA

The ExTrA facility (Bonfils et al. 2015), located at La Silla observatory,
consists of a near-infrared (0.85 to 1.55 $\mu$m) multi-object spectrograph
fed by three 60 cm telescopes. Five fibre positioners intercept the light from
a target and four comparison stars at the focal plane of each telescope. We
observed three transits of GJ 3090 b on nights UTC 2019 December 14, 2020
January 3, and 2020 November 29. On the first and second nights, we observed
simultaneously with two telescopes, whereas on the third night, we observed
with three telescopes. We used the fibres with 8″ apertures, the higher
resolution mode of the spectrograph (R$\sim$200), and 60-second exposures. The
comparison stars were 2MASS J01235337-4659196, 2MASS J01235446-4647126, 2MASS
J01241189-4633588, and 2MASS J01224245-4617544. The ExTrA data were analysed
using custom data reduction software.

###  2.5 WASP

WASP-South was a wide-field array of eight cameras forming the southern
station of the WASP transit-search survey (Pollacco et al. 2006). The field of
GJ 3090 was observed over a span of 200 nights in 2006, and then over
170-night spans in 2010 and 2011. During that time, WASP-South was equipped
with 200 mm f/1.8 lenses, observing with a 400–700 nm passband, and with a
photometric extraction aperture of 48 arcsec. GJ 3090 was also observed over
180-night spans in 2012, 2013, and 2014 each, when WASP-South was equipped
with 85-mm f/1.2 lenses using an SDSS-$r$ filter. In total, 71 000 photometric
observations were obtained, with a typical cadence of 12 minutes. GJ 3090 is
the brightest star in the extraction apertures. The next brightest star is 4
magnitudes fainter and also dominates the TESS contamination correction. We
searched each season of data for modulations due to stellar rotation using the
methods from Maxted et al. (2011). We show the results in Fig. 2.

A clear rotational modulation is seen at a period of 18.2 $\pm$ 0.4 days. The
error is estimated from the dispersion between different datasets and is
dominated by phase shifts of the modulation. In most years, the modulation has
an amplitude of 12 to 18 mmag. In 2011, the modulation is weaker and the
dominant power is at the first harmonic, and in 2014, the modulation is below
our detection threshold.

Figure 2: Periodograms of the WASP-South data for GJ 3090 (left) and the
resulting folds on the 18.2 d rotational period (right). The dotted lines mark
the estimated 1% likelihood false-alarm level.

###  2.6 SOAR speckle imaging

High-angular resolution imaging can identify nearby sources that contaminate
the TESS photometry, resulting in an underestimated planetary radius or in
astrophysical false positives such as background eclipsing binaries. We
searched for stellar companions to GJ 3090 with speckle imaging on the 4.1 m
Southern Astrophysical Research (SOAR) telescope (Tokovinin 2018) on UT 2018
October 21, observing in Cousins I band, similar to the TESS bandpass. More
details of the observation are available in Ziegler et al. (2020). The
5$\sigma$ detection sensitivity and speckle autocorrelation functions from the
observations are shown in Fig. 3. The SOAR observations detect no additional
stars within 3″ of GJ 3090.

Figure 3: I-band 5$\sigma$ contrast curve from SOAR speckle imaging of GJ 3090
(TIC 262530407). The inset depicts the corresponding speckle autocorrelation
function.

###  2.7 HARPS

We obtained differential RVs of GJ 3090 with HARPS (Mayor et al. 2003), the
optical velocimeter installed at the ESO 3.6 m telescope at La Silla
Observatory. Under the ESO program 1102.C-0339(A), we collected 55 spectra
over a time span of 326 days, with an exposure time of 900 seconds each (one
spectrum taken on 2019 January 24 has a 1800-second exposure time), slow
readout mode (104 kHz), and without simultaneous wavelength calibration. The
signal-to-noise ratio at 650 nm ranges between 20 and 71, with a mean value of
42. The spectra were reduced with the HARPS Data Reduction
Software555http://www.eso.org/sci/facilities/lasilla/instruments/harps/doc/DRS.pdf.
We used the template-matching algorithm described in Astudillo-Defru et al.
(2017b) to compute the RVs, which have a median RV precision of 2.0 m/s
(dominated by photon noise) and a dispersion of 8.5 m/s. The RVs, cross-
correlation function (CCF) characteristics, and activity indicators are listed
in Table 6. We measured $\log R^{\prime}_{HK}=-4.416\pm 0.048$ in the average
of the HARPS spectra, from which the Astudillo-Defru et al. (2017a) activity-
rotation relation estimates a rotation period of $P_{\rm rot}=17.0\pm 1.3$
days that excellently agrees with the results presented in Sects. 2.1 and 2.5.

##  3 Stellar parameters

We derived the mass and radius of GJ 3090 from empirical relations based on
luminosity. We used the Gaia-corrected (Gaia Collaboration et al. 2016, 2021;
Lindegren et al. 2021) parallax determination (44.555 $\pm$ 0.027 mas) to
compute the distance (22.444 $\pm$ 0.013 pc) and an absolute magnitude of
$M_{K_{s}}\leavevmode\nobreak\ =\leavevmode\nobreak\ 5.538\leavevmode\nobreak\
\pm\leavevmode\nobreak\ 0.026$. We then used the empirical relations of Mann
et al. (2019) and Mann et al. (2015) to derive a mass of
$M_{\star}\leavevmode\nobreak\ =\leavevmode\nobreak\ 0.519\leavevmode\nobreak\
\pm\leavevmode\nobreak\ 0.013\leavevmode\nobreak\ \hbox{$M_{\odot}$}$ and a
radius of $R_{\star}\leavevmode\nobreak\ =\leavevmode\nobreak\
0.517\leavevmode\nobreak\ \pm\leavevmode\nobreak\ 0.016\leavevmode\nobreak\
\hbox{$R_{\odot}$}$, respectively, which were used as priors for Sect. 4.3. We
derived the stellar radius independently of the spectral energy distribution
(SED) that we constructed using the magnitudes from Gaia (Riello et al. 2021),
the 2-Micron All-Sky Survey (2MASS; Skrutskie et al. 2006; Cutri et al. 2003),
and the Wide-field Infrared Survey Explorer (WISE; Wright et al. 2010; Cutri &
et al. 2013). The measurements are listed in Table 2. We modelled these
magnitude measurements using the procedure described in Díaz et al. (2014),
with the PHOENIX/BT-Settl (Allard et al. 2012) stellar atmosphere models. We
used informative priors for the effective temperature
($T_{\mathrm{eff}}=3556\pm 70$ K, which corresponds to an M2 spectral type),
and metallicity ($[\rm{Fe/H}]=-0.06\pm 0.12$ dex) derived from the co-added
HARPS spectra (which we analysed with SpecMatch-Emp; Yee et al. 2017), and for
the distance from Gaia. We used non-informative priors for the rest of the
parameters. We used an additive jitter (Gregory 2005) for each set of
photometric bands (Gaia, 2MASS, and WISE). The parameters, priors, and
posteriors are listed in Table 7. The maximum a posteriori (MAP) model is
shown in Fig 4. The derived radius
($R_{\star}=0.531^{+0.016}_{-0.012}\leavevmode\nobreak\ \hbox{$R_{\odot}$}$)
is compatible with the radius computed above using an empirical radius-
luminosity relation.

We used the stellar mass and the HARPS stellar rotation period to derive a
gyrochronological age, neglecting the influence of the planets. Using the
formulation of Barnes (2010) and Barnes & Kim (2010) with initial periods P0
between 0.12 and 3.4 d, the age is 1.02${}^{+0.23}_{-0.15}$ Gyr. We added a
10% systematic error to the statistical error (Meibom et al. 2015). This value
agrees with 1.07 $\pm$ 0.22 Gyr that was determined based on the rotational
period alone using the Engle & Guinan (2011) relation. This was obtained from
a sample of M dwarfs.

Table 2: Astrometry, photometry, and stellar parameters for GJ 3090.

Parameter | Value | Source
---|---|---
Designations | CD-47 399 | Thome (1900)
| HIP 6365 | Turon et al. (1993)
| TIC 262530407 |  Stassun et al. (2019)
RA (ICRS, J2000) |  01h 21m 45.39s | Gaia EDR3
Dec (ICRS, J2000) |  $-46^{\rm o}$ 42’ 51.8” | Gaia EDR3
$\mu$ RA [mas yr-1] |  -111.089 $\pm$ 0.020 | Gaia EDR3
$\mu$ Dec [mas yr-1] |  -79.954 $\pm$ 0.025 | Gaia EDR3
Parallax [mas] |  44.555 $\pm$ 0.027 |  Gaia EDR3a
Distance [pc] |  22.444 $\pm$ 0.013 |  Sect. 3
Gaia-BP |  11.6698 $\pm$ 0.0032 | Gaia EDR3
Gaia-G |  10.5567 $\pm$ 0.0028 | Gaia EDR3
Gaia-RP |  9.5053 $\pm$ 0.0029 | Gaia EDR3
2MASS-J |  8.168 $\pm$ 0.021 | 2MASS
2MASS-H |  7.536 $\pm$ 0.036 | 2MASS
2MASS-Ks |  7.294 $\pm$ 0.026 | 2MASS
WISE-W1 |  7.053 $\pm$ 0.038 | WISE
WISE-W2 |  7.113 $\pm$ 0.019 | WISE
WISE-W3 |  7.045 $\pm$ 0.016 | WISE
WISE-W4 |  6.941 $\pm$ 0.082 | WISE
Mass, $M_{\star}$ [$M_{\odot}$] |  0.519 $\pm$ 0.013 |  Mann et al. (2019), Sect. 4.3
Radius, $R_{\star}$ [$R_{\odot}$] |  0.516 $\pm$ 0.016 |  Mann et al. (2015), Sect. 4.3
Effective temperature, $T_{\rm eff}$ [K] | $3556\pm 70$ | SpecMatch-Emp
Metallicity, [Fe/H] [dex] | $-0.06\pm 0.12$ | SpecMatch-Emp
Luminosity [L⊙] |  0.0455${}^{+0.0018}_{-0.0016}$ | SED
$v$ sin $i_{\star}$ [ km s-1] | $\leq 1.468\pm 0.050$ |  $R_{\star}$, Prot
666a Corrected according to Lindegren et al. (2021).

Figure 4: SED of GJ 3090. Top panel: MAP PHOENIX/BT-Settl interpolated
synthetic spectrum (solid line). Red circles show photometric observations,
and grey open circles are the result of integrating the synthetic spectrum in
the observed bandpasses. Bottom panel: Residuals of the MAP model (the jitter
has been added quadratically to the error bars of the data).

##  4 Analysis

###  4.1 Radial velocity

Figure 5 shows the generalised Lomb–Scargle periodogram (GLS, Zechmeister &
Kürster 2009) of the HARPS RVs. Its highest peak is at 12.7 days, with some
power at the stellar rotation period ($P_{\rm rot}$) and at its first harmonic
($P_{\rm rot}/2$). The main peak of the periodogram of the measured activity
indicators is at the stellar rotation period, except for SHK. For the
observations closest in time, the photometric modulation at the stellar
rotation period and the RV and activity indicators vary in anti-phase (Fig.
6).

Figure 5: GLS periodogram of the HARPS velocities for GJ 3090. The horizontal
lines represents 10, 1, and 0.1% false-alarm levels, from bottom to top,
respectively. Vertical lines mark the period of the transiting planet (green),
the highest peak of the periodogram (orange), and the stellar rotation period
estimated in Sect. 2.5 (red). Figure 6: TESS photometry and HARPS RV and
activity indicators folded at the stellar rotational period. To compute the
phase, we used the median period of the QPC from the adopted RV modelling
(17.733 days) and one arbitrary epoch (BJD 2,458,452.3). The TESS photometry
was better folded with a slightly different period at the expense of the RVs.
We favoured the general case in which the photometric variability observed by
TESS about two years apart could originate from different active regions at
different longitudes of the stellar surface. From top to bottom: Times of the
TESS (red, green, and blue, depending on the sector) and HARPS observations
(at times indicated by the colour of the circles) shifted by the median and
scaled with the standard deviation. TESS SAP light curve in which the transits
of GJ 3090 b are masked are coloured by sector. HARPS RV. The MAP Keplerians
of the adopted RV modelling are subtracted. HARPS H$\alpha$, H$\beta$,
H$\gamma$, NaD, and SHK activity indicators (their median is marked by the
horizontal line). The panels on the right show the GLS periodogram (in
frequency space) of the quantity shown at the left. Vertical lines mark the
period of planet b (green), planet c (orange), and $P_{\rm rot}$, $P_{\rm
rot}/2$, and $P_{\rm rot}/3$ (red). The HARPS observations started 45 days
after the end of TESS sector 3 and ended 311 days before the beginning of
sector 29.

To evaluate the significance of the periodicities in the RV series, we
iteratively added sine waves describing planets on circular orbits to the
model and estimated the evidence. We used juliet (Espinoza et al. 2019),
radvel (Fulton et al. 2018), and dynesty (Speagle 2020) to model the RV data,
and we used a GP regression model to account for the stellar activity signal
in the residuals. We first tried the QP kernel GP implemented in celerite
(Foreman-Mackey et al. 2017), with a prior on the period from Sect. 2.5. After
adding GJ 3090 b (with a prior on its period and time of conjunction from the
transiting TESS candidate), we searched for a second planet within a period
range between 3 and 20 days. The posterior orbital period of this second
signal peaks at half the stellar rotation period (Fig. 7).

We therefore elected to change the kernel function used in our GP model and
instead used the quasi-periodic with cosine (QPC) kernel (Perger et al. 2021),
which we implemented in george (Ambikasaran et al. 2015). With this new GP
model, we searched again for a second planet. As expected, the main peak of
the posterior probability for the orbital period now no longer is at half the
rotational period of the star, and the distribution presents several peaks
around 13 days. The log-Bayes factor ($\mathcal{Z}_{n}/\mathcal{Z}_{n-1}$,
with $\mathcal{Z}_{n}$ the evidence of the model with $n$ planets, and
$\mathcal{Z}_{n-1}$ the evidence of the model with $n-1$ planets) for this
model with respect to the one-planet model is 5.79 $\pm$ 0.23, which can be
interpreted as very strong evidence (Kass & Raftery 1995) in favour of this
model (as we assume equal probability for the $n$ and $n-1$ models, the Bayes
factor is equivalent to the odds ratio).

The marginal posterior distribution for the orbital period of the second
planet presents a number of peaks. The largest peak lies at 12.7 days, and the
remaining peaks are compatible with aliases (see Fig. 23). We nevertheless
estimated the contribution of each peak to the total evidence value. The peak
at 12.7 days holds the largest evidence, and the remaining peaks have log-
Bayes factors with respect to the main peak of -3.06 $\pm$ 0.49 for the 12.1
days, -2.83 $\pm$ 0.49 for the 13.4 days and -2.12 $\pm$ 0.49 for the 14.2
days. We adopted the 12.7-day period for the second planet.

We searched for a third planet within a period range between 2 (at less than
two days, multiple aliases appear at and around 1.0 and 1.5 days, see Fig. 5)
and 2.7 days, 3 to 12 days, and 13.1 to 100 days and found log-Bayes factors
with respect to the two-planet model of -2.39 $\pm$ 0.28, -0.71 $\pm$ 0.28,
and -0.33 $\pm$ 0.28, respectively. We conclude that the current data do not
favour a model with a third planet. The marginal posterior distribution for
the orbital period for the third planet peak again at half the stellar
rotation period despite the use of the QPC kernel.

If instead of the described scheme we start by modelling the 12.7-day signal
and then search for a second signal, the marginal posterior distribution of
the trial period in the 2 to 12 days range peaks at 2.881 days (an alias of
the 2.853 days period of GJ 3090 b, which is also present in the periodogram).
We also tested that the model with the QPC kernel and a planet with the priors
from the transiting TESS candidate is favoured, $\ln(\mathcal{Z}_{\rm
QPC+2.85^{d}}/\mathcal{Z}_{\rm QPC})=7.43\pm 0.31$, against a model with only
the QPC kernel (Fig. 7).

Figure 7: Search for planets in the HARPS data. Lower panel: Marginal
posterior of the trial planet period. Top panel: Bayes factor for the main
peaks of the posterior (error bars). The bands are the labelled limits from
Kass & Raftery (1995).

In order to study the influence of the eccentricity on the RV semi-amplitudes,
we finally modelled the RV with two planets on eccentric orbits. We used non-
informative priors for all the parameters except for the period and time of
conjunction of GJ 3090 b. The parameter priors, posterior medians, and 68.3%
credible intervals (CIs) are shown in Table 8. The period of the QPC kernel
parameter is well constrained although it has an uniform prior in this
analysis. This is even the most precise of all the fully compatible stellar
rotation period estimates presented in this section (Fig. 8). The parameter
$h_{2}$ of the QPC kernel is poorly constrained, even though it is key for the
$P_{\rm rot}/2$ signal.

Figure 8: Estimates of the stellar rotation period. The distributions are
normalised by their maximum for visibility purposes. Error bars at the top of
the distributions represent the median and the 68.3% CI. The distributions are
assumed normal for the periods derived from WASP-South and $\log
R^{\prime}_{HK}$.

We investigated how the RV semi-amplitude of GJ 3090 b changed with the model
assumption. We summarise the results in Fig. 9. The RV semi-amplitude and its
uncertainty can change by up to 40% and 86% (relative change when the adopted
value is taken as reference), respectively.

Figure 9: RV semi-amplitude due to GJ 3090 b from different models. Our
preferred model is shown in black.

###  4.2 Keplerian modelling

We first adjusted a Keplerian model to the TESS, Spitzer, ExTrA, and LCOGT
transit photometry and the HARPS RVs using juliet (Espinoza et al. 2019).
juliet uses batman (Kreidberg 2015) for the transit model and radvel (Fulton
et al. 2018) to model the RVs. For the transit photometry, a GP was used to
model the residuals, with the approximate Matern or QP kernels (depending on
the data set) included in celerite (Foreman-Mackey et al. 2017). We
oversampled the TESS and LCOGT models by a factor 5 (Kipping 2010) because the
averaging times of both data sets are longer than one minute. For the RV, we
used the QPC kernel (Perger et al. 2021) with george (Ambikasaran et al.
2015). We used a normal prior for the stellar density (5.29 $\pm$ 0.53
$\mathrm{g\;cm^{-3}}$) from the stellar mass and radius derived in Sect. 3 and
non-informative priors for the remaining parameters. We sampled from the
posterior with dynesty (Speagle 2020). In Table 3 we list the prior, the
median, and the 68% CI of the marginal distributions of the inferred system
parameters. We used the stellar mass and radius derived in Sect. 3 to compute
the planetary masses and the radius of planet b. Figure 24 shows the data sets
and the model from this analysis.

Table 3: Inferred system parameters.

|  | Keplerian | Keplerian | Dynamical | Dynamical | Stable
---|---|---|---|---|---|---
Parameter | Units | Prior | Median and 68.3% CI | Prior | Median and 68.3% CI | Median and 68.3% CI
_Star_ |  |  |  |  |  |
Mass, $M_{\star}$ |  [$M_{\odot}$] |  |  | $N(0.519,0.013)$ | $0.520\pm 0.013$ | $0.519\pm 0.013$
Radius, $R_{\star}$ |  [$\mathcal{R}^{\rm N}_{\odot}$] |  |  | $N(0.517,0.016)$ | $0.517\pm 0.016$ | $0.516\pm 0.016$
Mean density, $\rho_{\star}$ |  [$\mathrm{g\;cm^{-3}}$] | $N(5.29,0.53)$ | $5.36\pm 0.53$ |  | $5.30^{+0.52}_{-0.45}$ | $5.32^{+0.52}_{-0.47}$
Surface gravity, $\log g$ | [cgs] |  |  |  | $4.727\pm 0.028$ | $4.727\pm 0.029$
Kipping (2013) $q_{1}$ TESS |  | $U(0,1)$ | $0.21^{+0.23}_{-0.12}$ | $U(0,1)$ | $0.33^{+0.29}_{-0.18}$ | $0.34^{+0.28}_{-0.18}$
Kipping (2013) $q_{2}$ TESS |  | $U(0,1)$ | $0.22^{+0.39}_{-0.17}$ | $U(0,1)$ | $0.18^{+0.31}_{-0.14}$ |  $0.18^{+0.30}_{-0.13}$
_Planet b_ |  |  |  |  |  |
Semi-major axis, $a$ | [au] |  | $0.0317\pm 0.0014$ |  | $0.03165\pm 0.00027$ | $0.03165\pm 0.00027$
Eccentricity, $e$ |  |  | $0.18\pm 0.12$ |  | $0.16\pm 0.11$ |  ¡ 0.32†, ($0.15\pm 0.11$)
Argument of pericentre, $\omega$ | [°] |  | $146^{+68}_{-53}$ |  | $142^{+74}_{-57}$ | $137^{+73}_{-57}$
Inclination, $i$ | [°] |  | $87.09^{+0.98}_{-0.29}$ | $S(0,90)$ | $87.13^{+0.80}_{-0.29}$ | $87.14^{+0.79}_{-0.30}$
Longitude of the ascending node, $\Omega$ | [°] |  |  |  |  180 (fixed at $t_{\mathrm{ref}}$) |  180 (fixed at $t_{\mathrm{ref}}$)
Mean anomaly, $M_{0}$ | [°] |  |  |  | $327^{+40}_{-73}$ | $331^{+41}_{-69}$
$\sqrt{e}\cos{\omega}$ |  | $U(-1,1)$ | $-0.25^{+0.26}_{-0.17}$ | $U(-1,1)$ | $-0.21^{+0.26}_{-0.18}$ | $-0.19^{+0.26}_{-0.19}$
$\sqrt{e}\sin{\omega}$ |  | $U(-1,1)$ | $0.14\pm 0.32$ | $U(-1,1)$ | $0.17^{+0.27}_{-0.34}$ | $0.17^{+0.27}_{-0.31}$
Mass ratio, $M_{\mathrm{p}}/M_{\star}$ |  |  |  | $U(0,1)$ | $(19.3\pm 4.2)\times 10^{-6}$ | $(19.4\pm 4.1)\times 10^{-6}$
Radius ratio, $R_{\mathrm{p}}/R_{\star}$ |  |  | $0.0384^{+0.0011}_{-0.0015}$ | $U(0,1)$ | $0.0379^{+0.0012}_{-0.0014}$ | $0.0379^{+0.0011}_{-0.0014}$
Scaled semi-major axis, $a/R_{\star}$ |  |  | $13.22\pm 0.45$ |  | $13.16\pm 0.42$ | $13.18\pm 0.42$
Impact parameter, $b$ |  |  | $0.650^{+0.093}_{-0.32}$ |  | $0.633^{+0.099}_{-0.27}$ | $0.631^{+0.093}_{-0.26}$
Transit duration, $T_{14}$ | [h] |  | $1.270\pm 0.022$ |  | $1.281\pm 0.025$ | $1.281\pm 0.024$
Espinoza (2018) $r_{1}$ |  | $U(0,1)$ | $0.766^{+0.062}_{-0.21}$ |  |  |
Espinoza (2018) $r_{2}$ |  | $U(0,1)$ | $0.0384^{+0.0011}_{-0.0015}$ |  |  |
T0 ($T_{0}^{\prime}$) - 2 450 000 |  [BJDTDB] | $U(8370.417,8370.420)$ | $8370.41849\pm 0.00030$ | $U(7370,9370)$ | $8370.41847\pm 0.00034$ | $8370.41849\pm 0.00034$
Orbital period, P ($P^{\prime}$) | [d] | $U(2.85309,2.85312)$ | $2.8531054\pm 0.0000023$ | $U(0,1000)$ | $2.853134^{+0.000066}_{-0.000047}$ | $2.853136^{+0.000064}_{-0.000038}$
RV semi-amplitude, K ($K^{\prime}$) |  [m s-1] | $U(0,8)$ | $2.36\pm 0.52$ |  | $2.37\pm 0.51$ | $2.38\pm 0.51$
Mass, $M_{\mathrm{p}}$ |  [$\mathrm{M_{E}}$ ] |  | $3.31\pm 0.73$ |  | $3.33\pm 0.72$ | $3.34\pm 0.72$
Radius, $R_{\mathrm{p}}$ |  [$\mathcal{R}^{\rm N}_{e\rm E}$] |  | $2.16\pm 0.10$ |  | $2.13\pm 0.11$ | $2.13\pm 0.11$
Mean density, $\rho_{\mathrm{p}}$ |  [$\mathrm{g\;cm^{-3}}$] |  | $1.80^{+0.50}_{-0.42}$ |  | $1.87^{+0.54}_{-0.44}$ | $1.89^{+0.52}_{-0.45}$
Surface gravity, $\log$ $g_{\mathrm{p}}$ | [cgs] |  | $2.841^{+0.095}_{-0.11}$ |  | $2.853^{+0.098}_{-0.11}$ | $2.857^{+0.096}_{-0.11}$
Equilibrium temperature, Teq | [K] |  | $692\pm 18$ |  | $693\pm 17$ |  $693\pm 18$
_Planet c_ |  |  |  |  |  |
Semi-major axis, $a$ | [au] |  | $0.0860\pm 0.0039$ |  | $0.08578\pm 0.00075$ | $0.08575\pm 0.00074$
Eccentricity, $e$ |  |  | $0.23^{+0.12}_{-0.13}$ |  | $0.17\pm 0.11$ |  ¡ 0.31†, ($0.16\pm 0.11$)
Argument of pericentre, $\omega$ | [°] |  | $357^{+0.31}_{-0.37}$ |  | $0^{+44}_{-53}$ | $-1^{+45}_{-57}$
Inclination, $i$ | [°] |  |  | $S(0,180)$ | $85\pm 55$ | $86\pm 46$
Longitude of the ascending node, $\Omega$ | [°] |  |  | $U(90,270)$ | $157^{+70}_{-44}$ | $157^{+50}_{-39}$
Mean anomaly, $M_{0}$ | [°] |  |  |  | $68\pm 47$ | $68\pm 51$
$\sqrt{e}\cos{\omega}$ |  | $U(-1,1)$ | $0.42^{+0.14}_{-0.26}$ | $U(-1,1)$ | $0.33^{+0.15}_{-0.32}$ | $0.32^{+0.15}_{-0.33}$
$\sqrt{e}\sin{\omega}$ |  | $U(-1,1)$ | $-0.02\pm 0.22$ | $U(-1,1)$ | $0.00\pm 0.23$ | $0.00\pm 0.22$
Mass ratio, $M_{\mathrm{p}}/M_{\star}$ |  |  |  | $U(0,1)$ | $(10.8^{+8.5}_{-2.5})\times 10^{-5}$ | $(9.9^{+5.1}_{-1.9})\times 10^{-5}$
Scaled semi-major axis, $a/R_{\star}$ |  |  | $35.8\pm 1.2$ |  | $35.7\pm 1.1$ | $35.7\pm 1.1$
Impact parameter, $b$ |  |  |  |  | $22.2^{+9.4}_{-14}$ | $19^{+10}_{-13}$
T0 ($T_{0}^{\prime}$) - 2 450 000 |  [BJDTDB] | $U(8367,8376)$ | $8370.39^{+1.3}_{-0.90}$ | $U(7370,9370)$ | $8370.92^{+1.1}_{-0.90}$ | $8370.96^{+1.2}_{-0.90}$
Orbital period, P ($P^{\prime}$) | [d] | $U(12.5,12.9)$ | $12.736^{+0.023}_{-0.030}$ | $U(0,1000)$ | $12.730^{+0.025}_{-0.031}$ | $12.729^{+0.025}_{-0.031}$
RV semi-amplitude, K ($K^{\prime}$) |  [m s-1] | $U(0,20)$ | $6.23\pm 0.75$ |  | $6.05\pm 0.69$ | $6.01\pm 0.69$
Mass, $M_{\mathrm{p}}$ |  [$\mathrm{M_{E}}$ ] |  |  |  | $18.6^{+15}_{-4.4}$ | $17.1^{+8.9}_{-3.2}$
$M_{\mathrm{p}}\sin{i}$ |  [$\mathrm{M_{E}}$ ] |  | $14.3\pm 1.5$ |  | $14.0\pm 1.6$ | $13.9\pm 1.6$
Equilibrium temperature, Teq | [K] |  | $420\pm 11$ |  | $421\pm 11$ |  $421\pm 11$
Mutual inclination, $i_{\mathrm{rel}}$ | [°] |  |  |  | $66^{+16}_{-27}$ |  $59^{+16}_{-26}$

777The table lists the prior, posterior median, and 68.3% CI for the
parameters of the Keplerian model (Sect. 4.2) and of the dynamical model,
without and with the stability constraint (Sect. 4.3). For the dynamical
model, the Jacobi orbital elements are given for the reference time
$t_{\mathrm{ref}}\leavevmode\nobreak\ =\leavevmode\nobreak\
2\;458\;370.418461$ BJDTDB. † Upper limit, 95% confidence. The planetary
equilibrium temperature is computed with $T_{\mathrm{eff}}=3556\pm 70$ K and
for zero albedo and full day-night heat redistribution. T0 is the time of
conjunction. $T^{\prime}_{0}\equiv
t_{\mathrm{ref}}-\frac{P^{\prime}}{2\pi}\left(M_{0}-E+e\sin{E}\right)$ with
$E=2\arctan{\left\\{\sqrt{\frac{1-e}{1+e}}\tan{\left[\frac{1}{2}\left(\frac{\pi}{2}-\omega\right)\right]}\right\\}}$,
$P^{\prime}\equiv\sqrt{\frac{4\pi^{2}a^{3}}{\mathcal{G}M_{\star}}}$,
$K^{\prime}\equiv\frac{M_{p}\sin{i}}{M_{\star}^{2/3}\sqrt{1-e^{2}}}\left(\frac{2\pi\mathcal{G}}{P^{\prime}}\right)^{1/3}$.
CODATA 2018: $\mathcal{G}$ = 6.674 30$\;\times 10^{-11}$
$\rm{m^{3}\;kg^{-1}\;s^{-2}}$. IAU 2012: au = 149 597 870 700 m . IAU 2015:
$\mathcal{R}^{\rm N}_{\odot}$= 6.957$\;\times 10^{8}$ m, $\mathcal{(GM)}^{\rm
N}_{\odot}$= 1.327 124 4$\;\times 10^{20}$ $\rm{m^{3}\;s^{-2}}$,
$\mathcal{R}^{\rm N}_{e\rm E}$ = 6.378 1$\;\times 10^{6}$ m,
$\mathcal{(GM)}^{\rm N}_{\rm E}$= 3.986 004$\;\times 10^{14}$
$\rm{m^{3}\;s^{-2}}$, $M_{\odot}$ = $\mathcal{(GM)}^{\rm
N}_{\odot}$/$\mathcal{G}$, $\mathrm{M_{E}}$ = $\mathcal{(GM)}^{\rm N}_{\rm
E}$/$\mathcal{G}$, $k^{2}$ = $\mathcal{(GM)}^{\rm
N}_{\odot}$$\;(86\;400\leavevmode\nobreak\ \rm{s})^{2}$/$\rm{au}^{3}$.
$N(\mu,\sigma)$: Normal distribution with mean $\mu$, and standard deviation
$\sigma$. $U(a,b)$: A uniform distribution defined between a lower $a$ and
upper $b$ limit. $S(a,b)$: A Sinusoidal distribution defined between a lower
$a$ and upper $b$ limit. $J(a,b)$: Jeffreys (or log-uniform) distribution
defined between a lower $a$ and upper $b$ limit.

Table 4: Continuation Table 3. |  | Keplerian | Keplerian | Dynamical | Dynamical | Stable
---|---|---|---|---|---|---
Parameter | Units | Prior | Median and 68.3% CI | Prior | Median and 68.3% CI | Median and 68.3% CI
_HARPS_ |  |  |  |  |  |
Systemic velocity, $\gamma$ |  [ km s-1] | $U(17.385,17.435)$ | $17.4095\pm 0.0046$ | $U(-100,100)$ | $17.4095\pm 0.0043$ | $17.4095\pm 0.0041$
Jitter |  [m s-1] | $J(0.01,4)$ | $0.101^{+0.41}_{-0.080}$ | $U(0,1000)$ | $0.66^{+0.67}_{-0.45}$ | $0.68^{+0.65}_{-0.46}$
QPC $h_{1}$ |  [m s-1] | $J(0.01,30)$ | $9.5^{+3.1}_{-2.2}$ | $J(0.01,30)$ | $9.1^{+3.2}_{-2.3}$ | $9.0^{+3.0}_{-2.2}$
QPC $h_{2}$ |  [m s-1] | $J(0.01,30)$ | $0.52^{+3.8}_{-0.49}$ | $J(0.01,30)$ | $0.55^{+3.6}_{-0.51}$ | $0.59^{+3.7}_{-0.55}$
QPC $P$ | [d] | $U(17,19)$ | $17.729^{+0.17}_{-0.036}$ | $U(17,19)$ | $17.732^{+0.17}_{-0.038}$ | $17.731^{+0.15}_{-0.038}$
QPC $\lambda$ | [d] | $U(1,2000)$ | $640^{+410}_{-350}$ | $U(1,2000)$ | $600^{+410}_{-320}$ |  $610^{+390}_{-320}$
_TESS_ |  |  |  |  |  |
Relative flux | [Relative flux] |  |  | $U(0.9,1.1)$ | $1.000003\pm 0.000023$ | $1.000002\pm 0.000022$
Jitter | [ppm] |  |  | $J(1,10000)$ | $9.0^{+33}_{-7.0}$ | $9.4^{+33}_{-7.3}$
Timescale of the GP | [days] |  |  | $J(0.001,1000)$ | $1.9^{+190}_{-1.9}$ | $1.8^{+180}_{-1.7}$
Amplitude of the GP | [ppm] |  |  | $J(1,10^{6})$ | $6.9^{+23}_{-5.1}$ | $6.6^{+22}_{-4.8}$

In the next section we use a self-consistent Newtonian model that takes the
gravitational interactions between the planets into account. Even though we
detect no interactions, accounting for them allows us to exclude combinations
of system parameters that would result in detectable interactions and
constrain the true mass of planet c.

###  4.3 Dynamical modelling

We modelled the TESS photometry and HARPS RV measurements and accounted for
the gravitational interactions between the adopted components of the system
using a photodynamical model. The TESS transits are the most numerous of the
transit observations and dominate the determination of the planet-to-star
radius ratio. For simplicity, we did not model the transit observations with
other instruments. The planet positions and velocities in time were obtained
through an n-body integration. The sky-projected positions were used to
compute the light curve (Mandel & Agol 2002) using a quadratic limb-darkening
law (Manduca et al. 1977) that we parametrised following Kipping (2013). The
light-time effect (Irwin 1952) was included in the model. To account for the
integration time, the model was oversampled by a factor 3 and was then binned
back to match the cadence of the data points (Kipping 2010). The line-of-sight
projected velocity of the star issued from the n-body integration was used to
model the RV measurements. We used the n-body code REBOUND (Rein & Liu 2012)
with the WHFast integrator (Rein & Tamayo 2015) and an integration step of
0.005 days, which resulted in a model error smaller than 1 ppm for the
photometric model. The model was parametrised using the stellar mass and
radius, planet-to-star mass ratios, planet b-to-star radius ratio, and Jacobi
orbital elements (Table 3) at reference time,
$t_{\mathrm{ref}}\leavevmode\nobreak\ =\leavevmode\nobreak\
2\;458\;370.418461$ BJDTDB. Due to the symmetry of the problem, we fixed the
longitude of the ascending node of the interior planet $\Omega_{\mathrm{b}}$
at $t_{\mathrm{ref}}$ to 180°, and we limited its inclination to $i_{c}<90$°.
We used a GP regression model with an approximate Matern kernel (celerite,
Foreman-Mackey et al. 2017) for the model of the error terms of the transit
light curves, and the QPC kernel (Perger et al. 2021; Ambikasaran et al. 2015)
for the RVs. For the photometric dataset, we added a transit normalisation
factor and a jitter parameter. For the RV, we added a systemic RV and a jitter
parameter. In total, the model has 28 free parameters. We used normal priors
for the stellar mass and radius from Sect. 3, non-informative sinusoidal
priors for the orbital inclinations (uniform in $\cos{i}$), and non-
informative uniform prior distributions for the rest of the parameters. The
joint posterior distribution was sampled using the emcee algorithm (Goodman &
Weare 2010; Foreman-Mackey et al. 2013) with 200 walkers with starting points
based on the results of Sect. 4.2. We ran 1.2${\times 10^{6}}$ steps of the
emcee algorithm and used the last 200 000 steps for the final inference. In
Table 3 we list the prior, the median, and the 68% CI of the inferred marginal
distributions of the system parameters.

We used the mean exponential growth of nearby orbits (MEGNO; Cincotta & Simó
2000; Cincotta et al. 2003), a chaos indicator that is implemented in REBOUND,
to assess the stability of the system. The period ratio of the planets is
4.461 $\pm$ 0.011, close to the 9:2 seventh-order mean motion resonance. For
each sample of the posterior, we ran REBOUND with the WHFast integrator for
104 orbits of planet c with time steps of 0.1 days and computed the MEGNO
value. We followed Hadden & Lithwick (2018) and considered stable the
solutions with a MEGNO value between 0 and 2.3 (Hadden & Lithwick 2018 ran the
simulation for 3000 orbits of the outer planet, with period $P^{\prime}$, and
considered regular trajectories, $\frac{3000P^{\prime}}{{\rm
MEGNO}}>1300P^{\prime}$, which corresponds to $0<{\rm MEGNO}\lesssim 2.3$).
This represents 77% of the posterior samples. The requirement of stability
removes some extreme values of mutual inclination and planet c mass from the
posterior (Fig. 10). We added a column to Table 3 based on the stable samples
of the posterior, which we adopt as our preferred values. We were only able to
determine upper limits for the eccentricities of planet b and c of 0.32 and
0.31, respectively, at 95% CI. The mutual inclination between the planets b
and c is poorly constrained by the observations.

Figure 10: Posterior distribution comparison for the planet eccentricities,
the mutual inclination between the planet orbits, and the mass of planet c
before (dynamical) and after (stable) the stability constraint. For the
eccentricities, the posteriors for the RV analysis (Sect. 4.1) and Keplerian
modelling (Sect. 4.2) are also shown.

The one- and two-dimensional projections of the posterior sample are shown in
Fig. 25. The MAP model is plotted in Figs. 11 and 12. The posteriors of the
transit-timing variations (TTVs) are shown in Fig. 26, along with individually
determined TESS transit times using juliet.

Figure 11: Transits of GJ 3090 b observed with TESS (blue points) and the MAP
model (black line). Each panel is centred at the linear ephemeris from Sect.
4.2 (indicated by the vertical grey lines). Each panel is labelled with the
epoch; zero is the transit closest to $t_{\mathrm{ref}}$. In the lower part of
each panel, the residuals after subtracting the MAP model are shown. Figure
12: HARPS RVs of GJ 3090 (green error bars) and the MAP model (black line). In
the lower panel, the residuals after subtracting the MAP model are shown.

##  5 Results and discussion

The GJ 3090 system is composed of an M2-dwarf star orbited by a transiting
mini-Neptune, GJ 3090 b, with a period of 2.9 days (Fig. 13). We found
evidence for an outer Neptune-mass planet candidate in the RV data whose
transits are not detected in TESS data (Fig. 1). Below we investigate the
transiting planet GJ 3090 b further.

Figure 13: Planet mass vs orbital semi-major axis in logarithmic scale. Error
bars mark the position of planet b (blue) and planet c (orange). Grey dots are
planets listed in the NASA Exoplanet Archive accessed through DACE API
(dace.unige.ch). A Gaussian kernel density estimate is shown in different
intensities of green. Solar System planets (black points, from left to right:
Venus, Earth, Jupiter, Saturn, Uranus, and Neptune) are taken from NASA.

###  5.1 Characterisation of the interior

With a mass of 3.34 $\pm$ 0.72 $\mathrm{M_{E}}$ and a radius of 2.13 $\pm$
0.11 $\mathrm{R_{E}}$, GJ 3090 b lies at the upper edge of the transition
between the populations of super-Earths and mini-Neptunes. Figure 14 shows M-R
curves corresponding to planets at $T_{\rm eq}=693$ K for compositions of pure
iron, an Earth-like composition (with a core mass fraction of 0.33), and
compositions of pure water. For reference, we also show exoplanets with
accurate and reliable mass and radius determinations (Otegi et al. 2020a).
Figure 14 shows that planets with masses up to nearly 4 $\mathrm{M_{E}}$ tend
to follow the Earth-like composition curve, while more massive planets have a
wide diversity in radius and density. GJ 3090 b is located in the upper part
of the exoplanet envelope that starts at around 4 $\mathrm{M_{E}}$. Most of
the exoplanets around M dwarfs are located in the upper part of the envelope,
which may be due to the observational bias. These planets have a lower
incoming irradiation that allows them to keep most of the H-He atmospheres. GJ
3090 b sits well above the Earth-composition curve, implying that a volatile-
envelope accounts for a significant fraction of its radius.

We modelled the interior of GJ 3090 b as the superposition of a pure-iron
core, a silicate mantle, a pure-water layer, and a H-He atmosphere. The models
follow the basic model structure of Dorn et al. (2017), and the equation of
state (EOS) of the iron core is taken from Hakim et al. (2018), and the EOS of
the silicate-mantle is taken from Connolly (2009). For water, we used the AQUA
EOS from Haldemann et al. (2020), and for the H-He envelope, the SCVH EOS
(Saumon et al. 1995) with an assumed protosolar composition. Precise
characterisation of the internal planetary structure is very challenging
because different compositions can lead to identical mass and radius Rogers &
Seager (2010); Lopez & Fortney (2014); Dorn et al. (2015, 2017); Lozovsky et
al. (2018); Otegi et al. (2020b); Valencia et al. (2013). However, Fig. 14
shows that most super-Earths follow the Earth-like composition line. In
addition, chemical abundance ratios in solar metallicity stars have been found
to be homogeneous within 10% in the solar neighbourhood Bedell et al. (2018),
implying that exoplanets may exhibit less compositional diversity than
previously expected. It is therefore reasonable to assume an Earth-like solid
interior. The low density of the planet indicates the presence of volatiles,
which could be water, H-He, or a combination of these two. When an Earth-like
solid interior is assumed, the water mass fraction needed to match the
observed mass and radius is 55${}^{+18}_{-16}$%. On the other hand, in a
water-free scenario, the required H-He mass fraction would be
4.9${}^{+0.7}_{-1.3}$%. These two values are upper limits on the water and
H-He content because the planet may contain a mix of water vapour and H-He.

Alternatively, the degeneracy can also be partly broken without assumptions on
the planet abundances by using a generalised Bayesian inference method with a
nested-sampling scheme. This method allows quantifying the degeneracy and
correlation of the structural parameters of the planet and to estimate the
most likely region in the parameter space. Figure 15 shows ternary diagrams of
the inferred composition of GJ 3090 b. The ternary diagram illustrates the
composition degeneracy for exoplanets with measured mass and radius. We found
a median H-He mass fraction of about 2%, which is a lower bound because metal-
enriched H-He atmospheres are more compressed and therefore increase the
planetary H-He mass fraction. Formation models indeed suggest that mini-
Neptunes likely form through envelope enrichment Venturini & Helled (2017). We
found a very strong degeneracy between the core, the silicate mantle, and the
water layer, which prevents accurate estimates of the masses of these
constituents. Interior models cannot distinguish between water and H-He as the
source of the low-density material, and therefore we also ran three-layer
models without the H2O layer and without the H-He envelope. Table 5 lists the
inferred mass fractions of the core, mantle, water layer, and H-He for
different models. In the water-free case, we found that the planet is 4.3%
H-He, 45% iron, and 50.7% rock by mass, which sets maximum limits for the
atmospheric and core mass. However, a model without H-He would require 57%
water. This qualitatively agrees with previous calculations that took into
account that water must be mainly in vapour and supercritical states for
planets receiving an insolation above the runaway greenhouse limit, as applies
for GJ 3090 b (Turbet et al. 2020; Mousis et al. 2020; Aguichine et al. 2021).
A more self-consistent estimate of the water content would be obtained by
considering interactions with the rocky interior (Bower et al. 2019; Dorn &
Lichtenberg 2021). We note that the water and H-He mass fractions obtained in
the three-layer models are compatible with the results obtained assuming a
Earth-like solid interior. An iron-poor formation would be another possible
scenario, but a volatile (either H2/He or H2O-dominated) layer is always
required. An iron-free model gives a median atmospheric mass fraction of 0.9%,
a mantle of 84%, and a water layer of 15%. Because any iron added to this
model would increase the H-He mass fraction and decrease the mantle mass
fraction, these values set a minimum and maximum limit, respectively. We found
a strong degeneracy between the core and mantle mass fractions that might be
reduced with further constraints such as the stellar abundances.

---
Figure 14: Mass-radius diagram of exoplanets with an accurate mass and radius
determination from the updated version of Otegi et al. (2020a, accessible on
the Data & Analysis Center for Exoplanet
DACE999https://dace.unige.ch/exoplanets/). Exoplanets around M dwarfs are
highlighted in black. The curves are the composition lines of iron, Earth-
like, and pure water planets (at T${}_{\rm eq}=693$ K).
---
Figure 15: Ternary diagrams of the inferred internal composition of GJ 3090 b. We show the parameter space covered by the posterior distributions in mass (left) and radius (right). Table 5: Inferred interior structure properties of GJ 3090 b. Constituent | 4-layer | No H-He | No H2O | No Fe
---|---|---|---|---
$M_{\rm core}/M_{\rm total}$ | 0.44${}^{+0.14}_{-0.23}$ | 0.18${}^{+0.09}_{-0.07}$ | 0.45${}^{+0.21}_{-0.20}$ | -
$M_{\rm mantle}/M_{\rm total}$ | 0.37${}^{+0.21}_{-0.18}$ | 0.24${}^{+0.14}_{-0.11}$ | 0.51${}^{+0.23}_{-0.22}$ | 0.84${}^{+0.04}_{-0.07}$
$M_{\rm water}/M_{\rm total}$ | 0.17${}^{+0.14}_{-0.11}$ | 0.57${}^{+0.12}_{-0.13}$ | - | 0.15${}^{+0.06}_{-0.04}$
$M_{\rm atm}/M_{\rm total}$ | 0.019${}^{+0.006}_{-0.007}$ | - | 0.043${}^{+0.009}_{-0.008}$ | 0.009${}^{+0.005}_{-0.004}$

###  5.2 Coupled atmospheric and dynamical evolution

We study in this section the possible history of the atmosphere of planet b
with the JADE code (Attia et al. 2021). We refer to Attia et al. (2021) for
details, but briefly summarise the main ingredients here. The JADE code
simulates the evolution of a planet over secular (gigayears) timescales under
the coupled influence of complex dynamical and atmospheric mechanisms. The
planet is modelled as composed of an iron core, a silicate mantle, and a H/He
gaseous envelope, whose varying properties are self-consistently derived at
each time step as the orbit, stellar irradiation, and inner planetary heating
all evolve. The atmosphere easily erodes due to XUV-induced photoevaporation,
and JADE calculates this mass-loss rate using analytical formulae adjusted to
detailed simulations of upper atmospheric structures (Salz et al. 2016). This
provides a better agreement with observed mass-loss rates than the commonly
used energy-limited approximation (e.g. Watson et al. 1981; Lammer et al.
2003; Erkaev et al. 2007).

Figure 16: Evolution of three representative simulations of GJ 3090 b from the
exploration of Sect. 5.2, represented by three different colours. Top:
Evolution of the mass. Bottom: Evolution of the radius. The inset on the left
zooms into the first 10 Myr of each simulation. The star marks tidal
disruption. The two black points are the observed mass and radius of GJ 3090 b
at the inferred age of the system.

We first investigated the sole impact of photoevaporation by conducting purely
atmospheric simulations at a fixed orbit corresponding to the present-day
values of Table 3. We used protosolar abundances (Asplund et al. 2021) for the
atmospheric composition, and we accounted for the evolution of the star. The
stellar irradiation, which we separated into three spectral bands (bolometric,
X, and EUV), was analytically computed at each time step using the bolometric
luminosity of Table 7 and the model of Jackson et al. (2012). We simulated the
atmospheric evolution of planet b for a wide range of initial planet masses
above the currently observed mass $M_{\rm pl}=3.34$ $\mathrm{M_{E}}$ starting
at $t=0$ from the dissipation of the protoplanetary disc.

Within this framework, we find no initial configuration that is compatible
with the current bulk properties of the planet at the inferred age of the
system ($1.02^{+0.23}_{-0.15}$ Gyr, Sect. 3). In Fig. 16 we present the mass
and radius evolution of three typical simulations that are representative of
the outcomes of the exploration. The grey simulation represents a planet that
is too massive to be affected by photoevaporation, showing thus nearly
constant bulk properties during 2 Gyr. The orange simulation represents an
intermediate-mass planet that shows a substantial inflation of the planetary
radius when evaporation causes the mass to drop to $\sim 5$ $\mathrm{M_{E}}$,
which is typical for hot H/He-dominated low-mass atmospheres (e.g. Jin et al.
2014; Zeng et al. 2019; Gao & Zhang 2020). Due to this inflation, the
atmosphere extends to the Roche limit of the star (e.g. Gu et al. 2003;
Jackson et al. 2017) and is hence affected by tidal disruption. In this case,
we ended the simulation, considering that the atmosphere is completely
depleted. Finally, the blue simulation shows a low-mass planet whose
atmosphere is entirely exhausted after a few million years due to
photoevaporation. We highlight that only intermediate-mass planets (e.g. the
orange simulation) are tidally disrupted because they have lower densities.
For a given orbit, the Roche radius definition indeed provides a lower limit
on the density below which stellar tidal forces exceed the planetary
gravitational self-attraction.

Figure 17: Corner plot of the age, initial planetary mass, initial semi-major
axis, and initial eccentricity of the simulated GJ 3090 b from the exploration
of Sect. 5.2. Full vertical lines indicate the PDF median values, and dashed
vertical lines indicate the $1\sigma$ highest density intervals.

We then conducted fully coupled atmospheric and dynamical simulations by
running a grid of models that covered a broad range of initial masses, semi-
major axes, and eccentricities. These models also included planetary and
stellar tides, planetary and stellar spin evolution, and general relativity,
which are thought to be the most important dynamical forces on close-in orbits
(e.g. Eggleton & Kiseleva-Eggleton 2001; Mardling & Lin 2002). The purpose of
this exploration was to assess whether forming the planet farther away, thus
escaping the intense irradiation from the young star (e.g. Owen & Wu 2017;
Rogers & Owen 2021) before migrating only recently (Bourrier et al. 2018;
Attia et al. 2021), circumvents the issue of radius inflation. To estimate the
most likely region in the parameter space, we again employed a Bayesian
inference framework (Attia et al. in prep.), using emcee (Foreman-Mackey et
al. 2013) constrained by the values of Table 3 and with non-informative
priors. The jump parameters were the initial conditions and the age of the
system, and the probabilities were constructed by comparing the outcomes of
the simulations to the observations. The resulting corner plot (Foreman-Mackey
2016) is shown in Fig. 17. Again, no appropriate set of initial conditions is
found. The most compatible configurations are low-mass planets that entirely
erode after a few million years, leaving a bare core that is not consistent
with the bulk properties of planet b. Planets forming with an intermediate
mass are less preferred because they are still inflated and subsequently
tidally disrupted (as in Fig. 16), even though they only recently migrated
close to the star.

This shows that whether planet b migrated earlyon or underwent a later, more
complex migration for example due to interactions with planet c, it would
always have lost a light H/He atmosphere in a few million years after arriving
at its current location. Because the current density is also not consistent
with an atmosphere-free rock-iron core, our simulations combined with the
results of Sect. 5.1 suggest that a different atmospheric composition makes it
more resilient to tidal disruption and photoevaporation. Particularly a water-
enriched (i.e. high-metallicity), possibly even water-dominated atmosphere
would allow for a wider range of initial masses with compact atmospheres
(Lopez 2017) because of the increased resilience of water to atmospheric
escape processes. Alternatively, silicate vapour in the lower parts of the
atmosphere might also result in a higher metallicity (Misener & Schlichting
2022), especially for such a close-in target. Future atmospheric
characterisation of the target would help to answer this question.

###  5.3 Prospects for atmospheric characterisation

Because it is very close to us (22 pc) and orbits a relatively low-mass M2V
star, GJ 3090 b is one of the best mini-Neptune-sized planets for a
characterisation of its atmosphere to date. The transmission spectroscopy
metric (TSM, Kempton et al. 2018) of GJ 3090 b is $221^{+66}_{-46}$, which
means that only one other of the known mini-Neptunes has an atmosphere that
can be better characterised through transit spectroscopy (Fig. 18). Below, we
further evaluate and discuss the prospects for characterising the true nature
of the atmosphere of GJ 3090 b.

Figure 18: Transmission spectroscopy metric for planets with radii between 1
and 4 $\mathrm{R_{E}}$ and T${}_{\rm eq}<1000$ K. Data are taken from Guo et
al. (2020), updated with recent results (Delrez et al. 2021; Burt et al. 2021;
Weiss et al. 2021; Kemmer et al. 2020; Nowak et al. 2020; Trifonov et al.
2021; Soto et al. 2021; Cointepas et al. 2021; Winters et al. 2021). Grey
areas are below the cutoffs that Kempton et al. (2018) suggested for follow-up
efforts. The planets above these cutoffs are labelled (T-1 is the abbreviation
for Trappist-1). The symbol with error bars is GJ 3090 b. The colour indicates
the equilibrium temperature of the planet, computed for zero albedo and full
day-night heat redistribution.

As shown in Sect. 5.1, theoretical mass-radius relations for planets predict
that GJ 3090 b has a volatile-rich envelope that is dominated in mass by
either hydrogen and helium, or water. Moreover and as shown in Sect. 5.2,
stability to atmospheric escape requirements suggest that the atmosphere of GJ
3090 b is likely to have super-solar metallicity. It should therefore have a
high water-mass fraction. A high metallicity is consistent with the empirical
mass-metallicity trend of Solar System planets (see Wakeford & Dalba 2020 and
references therein). In-depth analyses of HST/WFC3 transit spectra of some of
the least massive mini-Neptunes (e.g. GJ 1214 b and K2-18 b; see Kreidberg et
al. 2014; Benneke et al. 2019; Tsiaras et al. 2019) provide tentative evidence
that their metallicity is super-solar (Charnay et al. 2015; Gao & Benneke
2018; Charnay et al. 2021), which may – at least for the population of mini-
Neptunes most comparable in mass to GJ 3090 b – support this empirical trend.

Figure 19: Synthetic transit spectra of GJ 3090 b for different (hypothetical)
atmospheric compositions. (a) Transit spectra for cloud-free atmospheres (30x
solar metallicity, 1000x solar metallicity, and pure water vapour). (b)
Transit spectra for cloudy atmospheres with 1000x solar metallicity with cloud
layers at different altitudes (10-2, 10-3, 10-4 bar, and cloud-free). (c-d)
Two levels of zoom into the water-absorption lines (in the 1.4 $\mu$m water-
absorption band) at high spectral resolution for a 1000x solar metallicity
atmosphere with a cloud top at 10-3 bar. The spectral ranges of TESS,
HST/WFC3, JWST/NIRSPEC, and JWST/MIRI have been added for reference.

To further evaluate the feasibility of characterising the atmosphere of GJ
3090 b, we computed synthetic transit spectra (summarised in Fig. 19) from
visible to infrared wavelengths (0.5-16 $\mu$m) at low and high spectral
resolutions for different atmospheric scenarios (low metallicity, high
metallicity, pure water vapour, cloud-free, and cloudy). To construct these
spectra, we proceeded as follows. (1) Following Villanueva et al. (2018), we
calculated the pressure-temperature profiles using the prescriptions of
Parmentier & Guillot (2014) and assuming chemical equilibrium for the
atmospheric composition (Mbarek & Kempton 2016; Kempton et al. 2017). (2)
Transmission spectra were computed with petitRADTRANS (Mollière et al. 2019)
using the temperature, pressure, and mixing ratio vertical profiles. For
simplicity, we used 1D atmospheric models. Future works should revisit these
estimates with 3D atmospheric models, first to more realistically simulate the
position of clouds (see e.g. Charnay et al. 2015, 2021), and then to account
for the 3D transit geometry, which is particularly relevant for low-mass
highly irradiated planets with a potentially low mean molecular weight
atmosphere (Caldas et al. 2019; Pluriel et al. 2020; MacDonald & Lewis 2022;
Wardenier et al. 2022).

As expected from previous studies (Morley et al. 2013, 2015; Charnay et al.
2015; Greene et al. 2016; Mollière et al. 2017; Kawashima & Ikoma 2018), the
amplitude of the transmission spectra is highly dependent on metallicity (for
the explored range of metallicity, the higher the metallicity, the higher the
average molecular weight of the atmosphere, the lower the scale height, and
thus the lower the amplitude of the transit spectroscopy signal) and on the
presence and altitude of the cloud (top) layer. The simulated transit depth
amplitudes range from $\sim$ 200 ppm (30$\times$ solar metallicity, cloud-
free) down to 30 ppm (1000$\times$ solar metallicity, cloudy atmosphere).
Based on previous observations (Kreidberg et al. 2014) and models (Charnay et
al. 2015; Gao et al. 2020) of irradiated mini-Neptunes, clouds are very likely
to be present in the atmosphere of GJ 3090 b, and thus are likely to mute the
transmission spectrum.

Scaling from GJ 1214 b observations (Kreidberg et al. 2014), we estimate that
a $\sim$ 25 hours HST program using the G141 grism of the WFC3-IR instrument
would provide a $\sim$ 50 ppm transit depth uncertainty (around 1.4 $\mu$m,
for R=70), which is similar to that of Kreidberg et al. (2014). For a given
atmospheric composition, however, the expected amplitude of the near-infrared
transit spectroscopy signal is lower for GJ 3090 b than for GJ 1214 b by a
factor of $\sim$ 7\. This factor is mostly justified by its radius, which is
1.33 times smaller. Such an HST/WFC3-IR observation program would be able to
characterise an atmosphere of low to moderate metallicity, and only in the
absence of high-altitude clouds and/or hazes.

Figure 20: Synthetic transit spectra of GJ 3090 b. The estimated uncertainties
are calculated with the JWST ETC PandExo (Batalha et al. 2017) using NIRISS
(SOSS; blue) with the NIRSPec (395M grism; green) and with MIRI (LRS; red)
instrument modes of the JWST. This calculation assumes a 1000x solar
metallicity atmosphere (with a cloud top at 1 millibar). The initial modelled
transit spectrum is plotted in grey.

To make a step further in characterising the atmosphere of GJ 3090 b, JWST
(Gardner et al. 2006) observations are a very promising path. Using the JWST
ETC PandExo (Batalha et al. 2017), we evaluated the detectability of the
atmosphere of GJ 3090 b with NIRISS, NIRSPec, and MIRI-LRS for the range of
previously discussed atmospheric scenarios (shown in Fig. 19). Figure 20 shows
an illustrative result for a 1000x solar metallicity atmosphere with a cloud
top at 1 millibar. We find that NIRISS observations are the most promising for
this range of simulated atmospheres. GJ 3090 is too bright to use NIRSPec-
prism, and its rapid brightness decrease in the mid-infrared (compared to a
later-type star such as GJ 1214) makes MIRI observations expensive. A
reconnaissance observation program of two transits with NIRISS single-object
slitless spectroscopy (SOSS) – fewer than 10 hours of JWST time, after taking
overheads into account – could indeed characterise the atmosphere of GJ 3090 b
even if its metallicity is high and it has high clouds. The spectral coverage
of transit spectra could be extended to infrared wavelengths with
NIRSPec-G395H/M observations, but this would require two to three times more
transits to give a similar signal-to-noise ratio. A combination of transit
spectra obtained with NIRISS (SOSS) and NIRSPec (G395H/M) has indeed been
shown to be the optimal strategy for bright targets such as GJ 3090 (Batalha &
Line 2017; Batalha et al. 2018). None of the high-metallicity scenarios can be
characterised with MIRI-LRS in transit spectroscopy with a reasonable number
of observation hours. However, and given the expected equilibrium temperature
of GJ 3090 b, thermal emission might be searched for with MIRI (in LRS mode,
or alternatively, using a combination of filters) in a handful of secondary
eclipses.

GJ 3090 b is highly complementary to the benchmark GJ 1214 b (Charbonneau et
al. 2009; Cloutier et al. 2021). It is almost half as massive and falls in a
different region of the planetary radius-insolation diagram (Fig. 21),
immediately above the radius valley (Fulton et al. 2017; Cloutier & Menou
2020). Therefore, GJ 3090 b it is an excellent probe of the edge of the
transition between super-Earths and mini-Neptunes.

Figure 21: Planet radius vs insolation flux in Earth units and logarithmic
scale. Error bars mark the position of GJ 3090 b (green) and GJ 1214 b (blue).
The size of the symbols is proportional to the planet mass. Grey dots are
planets listed in the NASA Exoplanet Archive
(https://exoplanetarchive.ipac.caltech.edu/) with radius uncertainties smaller
than 20%. A Gaussian kernel density estimate is shown in different background
shades of grey. A subset of planets whose stars have a
$T_{\mathrm{eff}}<4000\leavevmode\nobreak\ K$ is shown as red dots, and its
Gaussian kernel density estimate is shown in different background shades of
red. The histogram of the entire sample (grey) and the subset (red) is shown
right of the panel. The black line shows the scaling relation for the radius
valley predicted by the photoevaporation model of Lopez & Rice (2018). The two
main overdensities are the mini-Neptunes (above the line) and the super-Earths
(below the line). The scarcity of planets in the upper left part of the
diagram is the so-called photoevaporation desert, hot-Neptune desert, or hot-
super-Earth desert (Lundkvist et al. 2016).

###### Acknowledgements.

Funding for the TESS mission is provided by NASA’s Science Mission
Directorate. We acknowledge the use of public TESS data from pipelines at the
TESS Science Office and at the TESS Science Processing Operations Center. This
research has made use of the Exoplanet Follow-up Observation Program website,
which is operated by the California Institute of Technology, under contract
with the National Aeronautics and Space Administration under the Exoplanet
Exploration Program. Resources supporting this work were provided by the NASA
High-End Computing (HEC) Program through the NASA Advanced Supercomputing
(NAS) Division at Ames Research Center for the production of the SPOC data
products. This paper includes data collected by the TESS mission that are
publicly available from the Mikulski Archive for Space Telescopes (MAST). We
are grateful to the ESO/La Silla staff for their support. This work has made
use of data from the European Space Agency (ESA) mission Gaia
(https://www.cosmos.esa.int/gaia), processed by the Gaia Data Processing and
Analysis Consortium (DPAC,
https://www.cosmos.esa.int/web/gaia/dpac/consortium). Funding for the DPAC has
been provided by national institutions, in particular the institutions
participating in the Gaia Multilateral Agreement. This work makes use of
observations from the LCOGT network. Part of the LCOGT telescope time was
granted by NOIRLab through the Mid-Scale Innovations Program (MSIP). MSIP is
funded by NSF. Based in part on observations obtained at the Southern
Astrophysical Research (SOAR) telescope, which is a joint project of the
Ministério da Ciência, Tecnologia e Inovações (MCTI/LNA) do Brasil, the US
National Science Foundation’s NOIRLab, the University of North Carolina at
Chapel Hill (UNC), and Michigan State University (MSU). This work has been
supported by a grant from Labex OSUG@2020 (Investissements d’avenir – ANR10
LABX56). Based on data collected under the ExTrA project at the ESO La Silla
Paranal Observatory. ExTrA is a project of Institut de Planétologie et
d’Astrophysique de Grenoble (IPAG/CNRS/UGA), funded by the European Research
Council under the ERC Grant Agreement n. 337591-ExTrA. OA, VB and MT’s work
has been carried out in the frame of the National Centre for Competence in
Research PlanetS supported by the Swiss National Science Foundation (SNSF). OA
and VB acknowledge funding from the European Research Council (ERC) under the
European Union’s Horizon 2020 research and innovation programme (project Spice
Dune; grant agreement No 947634). M.T. thanks the Gruber Foundation for its
support to this research. N. A.-D. acknowledges the support of FONDECYT
project 3180063.

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##  Appendix A Additional figures and tables

Figure 22: TESS target pixel file image of GJ 3090 in Sectors 2, 3, and 29
(created with tpfplotter, Aller et al. 2020). The electron counts are colour-
coded. The pixels highlighted in red are used for the simple aperture
photometry. The positions of the stars in the Gaia DR2 (Gaia Collaboration et
al. 2018) are indicated with red circles (and labelled with numbers according
to the distance to the main target in the aperture labelled ”1”), their size
is proportional to the Gaia DR2 magnitudes. Table 6: Parameters measured on
HARPS spectra of GJ 3090.

|  |  | CCF | CCF | CCF |  |  |  |  |  |  |  |  |  |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
Time | RV | $\pm 1\leavevmode\nobreak\ \sigma$ | FWHM | contrast | bisector span |  H$\alpha$ | $\sigma_{\rm H\alpha}$ |  H$\beta$ | $\sigma_{\rm H\beta}$ |  H$\gamma$ | $\sigma_{\rm H\gamma}$ | NaD | $\sigma_{\rm NaD}$ |  SHK | $\sigma_{\rm S_{\rm HK}}$
$[{\rm BJD}]$ |  [m s-1] |  [m s-1] |  [km s-1] |  |  [m s-1] |  |  |  |  |  |  |  |  |  |
2458451.685318 | 17400.46 | 1.51 | 3.35795 | 15.88118 | -0.00033 | 0.07895 | 0.00016 | 0.06984 | 0.00038 | 0.1392 | 0.0011 | 0.01584 | 0.00012 | 2.910 | 0.070
2458452.659111 | 17399.92 | 1.42 | 3.35402 | 15.89489 | 0.00470 | 0.07954 | 0.00015 | 0.06970 | 0.00036 | 0.1392 | 0.0010 | 0.01613 | 0.00011 | 3.035 | 0.068
2458453.659986 | 17400.81 | 1.57 | 3.35787 | 15.93346 | 0.00657 | 0.08006 | 0.00017 | 0.07260 | 0.00040 | 0.1466 | 0.0012 | 0.01603 | 0.00013 | 2.981 | 0.073
2458454.611072 | 17412.42 | 2.18 | 3.37451 | 16.24279 | -0.00216 | 0.08077 | 0.00025 | 0.07143 | 0.00050 | 0.1449 | 0.0013 | 0.01645 | 0.00020 | 2.995 | 0.074
2458455.689244 | 17415.93 | 2.16 | 3.36805 | 16.06612 | -0.00762 | 0.07892 | 0.00024 | 0.06795 | 0.00051 | 0.1419 | 0.0014 | 0.01592 | 0.00019 | 3.012 | 0.087
2458501.556874 | 17409.55 | 2.29 | 3.37032 | 16.05940 | 0.00948 | 0.08461 | 0.00026 | 0.07695 | 0.00059 | 0.1520 | 0.0016 | 0.01693 | 0.00021 | 3.163 | 0.097
2458501.568610 | 17412.18 | 2.16 | 3.36481 | 16.03897 | -0.00344 | 0.08469 | 0.00024 | 0.07623 | 0.00056 | 0.1502 | 0.0016 | 0.01654 | 0.00019 | 3.069 | 0.096
2458502.554355 | 17404.14 | 1.67 | 3.35973 | 15.98065 | 0.00704 | 0.08332 | 0.00019 | 0.07644 | 0.00044 | 0.1557 | 0.0013 | 0.01670 | 0.00014 | 3.038 | 0.081
2458502.566287 | 17403.38 | 1.68 | 3.35793 | 15.93048 | -0.00297 | 0.08305 | 0.00019 | 0.07536 | 0.00045 | 0.1494 | 0.0013 | 0.01641 | 0.00014 | 3.019 | 0.086
2458503.528851 | 17403.06 | 1.64 | 3.35135 | 15.90108 | -0.00137 | 0.08414 | 0.00018 | 0.07779 | 0.00044 | 0.1598 | 0.0013 | 0.01711 | 0.00014 | 2.941 | 0.079
2458504.530673 | 17400.19 | 1.70 | 3.36297 | 16.01077 | 0.00884 | 0.07866 | 0.00019 | 0.06956 | 0.00041 | 0.1414 | 0.0012 | 0.01605 | 0.00014 | 2.700 | 0.070
2458505.528804 | 17401.71 | 1.67 | 3.36931 | 16.07097 | -0.00459 | 0.07771 | 0.00019 | 0.06757 | 0.00039 | 0.1402 | 0.0011 | 0.01591 | 0.00014 | 2.748 | 0.066
2458505.540007 | 17401.50 | 1.55 | 3.36802 | 16.11554 | 0.00304 | 0.07785 | 0.00017 | 0.06848 | 0.00037 | 0.1420 | 0.0010 | 0.01582 | 0.00013 | 2.762 | 0.062
2458506.533579 | 17409.04 | 1.49 | 3.36660 | 16.08706 | 0.00080 | 0.07702 | 0.00017 | 0.06586 | 0.00035 | 0.1371 | 0.0010 | 0.01553 | 0.00012 | 2.772 | 0.062
2458506.545199 | 17410.67 | 1.46 | 3.36419 | 16.07681 | -0.00320 | 0.07717 | 0.00016 | 0.06739 | 0.00035 | 0.1368 | 0.0010 | 0.01556 | 0.00012 | 2.776 | 0.064
2458507.541294 | 17415.10 | 1.10 | 3.36956 | 16.13506 | 0.00257 | 0.07554 | 0.00012 | 0.06448 | 0.00025 | 0.1311 | 0.0007 | 0.01536 | 0.00009 | 2.688 | 0.037
2458515.529434 | 17407.78 | 2.52 | 3.36153 | 16.04329 | 0.00875 | 0.08690 | 0.00028 | 0.08175 | 0.00067 | 0.1654 | 0.0021 | 0.01738 | 0.00024 | 3.373 | 0.160
2458516.534667 | 17401.38 | 1.73 | 3.36335 | 16.07780 | -0.00141 | 0.08495 | 0.00019 | 0.07876 | 0.00046 | 0.1573 | 0.0014 | 0.01664 | 0.00015 | 3.286 | 0.104
2458517.533062 | 17408.08 | 1.93 | 3.36611 | 16.22144 | -0.00005 | 0.08836 | 0.00022 | 0.08685 | 0.00051 | 0.1720 | 0.0015 | 0.01780 | 0.00017 | 3.476 | 0.106
2458518.538737 | 17411.35 | 3.34 | 3.35353 | 16.03431 | -0.01362 | 0.08586 | 0.00036 | 0.07910 | 0.00090 | 0.1682 | 0.0033 | 0.01774 | 0.00034 | 3.187 | 0.323
2458520.545345 | 17418.12 | 2.36 | 3.36035 | 15.98638 | 0.00927 | 0.08614 | 0.00026 | 0.08452 | 0.00065 | 0.1803 | 0.0020 | 0.01776 | 0.00022 | 3.326 | 0.149
2458521.528513 | 17410.08 | 2.48 | 3.35432 | 15.88085 | 0.00351 | 0.09670 | 0.00029 | 0.10741 | 0.00075 | 0.2247 | 0.0023 | 0.02109 | 0.00024 | 4.316 | 0.153
2458522.526715 | 17402.21 | 2.04 | 3.36694 | 15.89182 | 0.00245 | 0.07871 | 0.00022 | 0.07066 | 0.00052 | 0.1476 | 0.0017 | 0.01618 | 0.00018 | 2.833 | 0.141
2458528.520417 | 17414.55 | 2.89 | 3.38350 | 15.99426 | -0.00201 | 0.08100 | 0.00031 | 0.07312 | 0.00071 | 0.1458 | 0.0022 | 0.01694 | 0.00028 | 2.916 | 0.185
2458530.523479 | 17411.69 | 2.09 | 3.36487 | 16.02049 | 0.00570 | 0.08518 | 0.00023 | 0.08002 | 0.00056 | 0.1662 | 0.0018 | 0.01740 | 0.00019 | 3.212 | 0.142
2458531.518947 | 17408.19 | 1.73 | 3.36275 | 16.02831 | -0.00712 | 0.08503 | 0.00019 | 0.07717 | 0.00046 | 0.1559 | 0.0015 | 0.01726 | 0.00015 | 3.081 | 0.116
2458537.519705 | 17410.79 | 2.21 | 3.35849 | 16.02056 | -0.00164 | 0.08591 | 0.00024 | 0.07931 | 0.00060 | 0.1595 | 0.0021 | 0.01691 | 0.00020 | 3.381 | 0.178
2458538.517565 | 17407.23 | 2.41 | 3.36078 | 16.03019 | -0.00210 | 0.08303 | 0.00026 | 0.07459 | 0.00063 | 0.1424 | 0.0022 | 0.01642 | 0.00022 | 3.021 | 0.189
2458539.504916 | 17395.49 | 1.75 | 3.36987 | 15.91665 | 0.00852 | 0.08257 | 0.00020 | 0.07377 | 0.00045 | 0.1451 | 0.0015 | 0.01648 | 0.00015 | 2.767 | 0.103
2458541.502377 | 17395.66 | 2.52 | 3.37507 | 15.88192 | -0.00129 | 0.07896 | 0.00028 | 0.06899 | 0.00058 | 0.1346 | 0.0018 | 0.01641 | 0.00023 | 2.467 | 0.126
2458741.680613 | 17412.50 | 1.52 | 3.39381 | 16.14142 | 0.00439 | 0.08017 | 0.00018 | 0.07138 | 0.00036 | 0.1441 | 0.0010 | 0.01652 | 0.00012 | 3.198 | 0.038
2458744.791074 | 17404.74 | 1.92 | 3.38396 | 16.23178 | -0.01427 | 0.08456 | 0.00023 | 0.07694 | 0.00047 | 0.1556 | 0.0014 | 0.01757 | 0.00015 | 3.634 | 0.065
2458746.611509 | 17421.12 | 3.99 | 3.38644 | 16.22588 | 0.01386 | 0.08533 | 0.00045 | 0.07616 | 0.00098 | 0.1570 | 0.0031 | 0.01671 | 0.00039 | 3.284 | 0.197
2458748.654565 | 17427.93 | 2.07 | 3.38101 | 16.26426 | 0.00572 | 0.08422 | 0.00024 | 0.07616 | 0.00052 | 0.1461 | 0.0015 | 0.01644 | 0.00017 | 3.229 | 0.076
2458754.630341 | 17390.93 | 2.21 | 3.38283 | 16.14074 | 0.00072 | 0.08424 | 0.00026 | 0.07554 | 0.00055 | 0.1524 | 0.0016 | 0.01680 | 0.00019 | 3.098 | 0.081
2458756.690323 | 17404.88 | 2.39 | 3.39395 | 16.08237 | 0.01304 | 0.08697 | 0.00029 | 0.08087 | 0.00060 | 0.1660 | 0.0018 | 0.01728 | 0.00021 | 3.247 | 0.087
2458757.604314 | 17407.92 | 2.49 | 3.38763 | 16.11348 | 0.00621 | 0.08342 | 0.00029 | 0.07718 | 0.00061 | 0.1597 | 0.0019 | 0.01697 | 0.00022 | 3.295 | 0.100
2458760.714200 | 17416.04 | 2.28 | 3.38513 | 16.06580 | 0.00450 | 0.08022 | 0.00026 | 0.07023 | 0.00055 | 0.1425 | 0.0016 | 0.01612 | 0.00019 | 2.972 | 0.090
2458760.725612 | 17410.24 | 2.10 | 3.38636 | 16.11989 | 0.00107 | 0.08019 | 0.00024 | 0.06915 | 0.00050 | 0.1381 | 0.0015 | 0.01615 | 0.00018 | 2.912 | 0.082
2458761.865381 | 17408.40 | 1.91 | 3.36370 | 15.93534 | -0.00729 | 0.08335 | 0.00022 | 0.07495 | 0.00052 | 0.1470 | 0.0017 | 0.01664 | 0.00016 | 3.084 | 0.117
2458761.876793 | 17407.12 | 2.16 | 3.37341 | 15.93166 | -0.00303 | 0.08387 | 0.00024 | 0.07517 | 0.00058 | 0.1466 | 0.0019 | 0.01724 | 0.00018 | 3.192 | 0.134
2458763.557347 | 17424.26 | 1.61 | 3.38776 | 16.18228 | 0.00645 | 0.08525 | 0.00019 | 0.07743 | 0.00041 | 0.1530 | 0.0012 | 0.01674 | 0.00013 | 3.265 | 0.054
2458763.568863 | 17424.55 | 1.54 | 3.38635 | 16.16532 | 0.00706 | 0.08374 | 0.00019 | 0.07494 | 0.00038 | 0.1493 | 0.0011 | 0.01656 | 0.00012 | 3.208 | 0.049
2458764.859828 | 17417.30 | 1.47 | 3.35755 | 15.94340 | 0.00174 | 0.08793 | 0.00017 | 0.08263 | 0.00042 | 0.1666 | 0.0014 | 0.01736 | 0.00011 | 3.335 | 0.079
2458764.871344 | 17419.59 | 1.51 | 3.35725 | 15.92168 | -0.00758 | 0.08945 | 0.00018 | 0.08653 | 0.00045 | 0.1739 | 0.0015 | 0.01809 | 0.00012 | 3.400 | 0.083
2458766.552229 | 17421.07 | 1.77 | 3.39656 | 16.29566 | -0.00427 | 0.08907 | 0.00022 | 0.08453 | 0.00046 | 0.1610 | 0.0013 | 0.01795 | 0.00015 | 3.310 | 0.054
2458766.563548 | 17417.80 | 1.67 | 3.40127 | 16.27750 | -0.00614 | 0.08924 | 0.00021 | 0.08587 | 0.00044 | 0.1655 | 0.0012 | 0.01800 | 0.00014 | 3.419 | 0.050
2458773.576203 | 17406.84 | 4.44 | 3.40203 | 16.14521 | 0.00887 | 0.08427 | 0.00052 | 0.07264 | 0.00103 | 0.1497 | 0.0032 | 0.01696 | 0.00045 | 2.869 | 0.190
2458773.692496 | 17404.76 | 2.58 | 3.38599 | 16.13724 | 0.00387 | 0.08447 | 0.00030 | 0.07690 | 0.00063 | 0.1823 | 0.0021 | 0.01717 | 0.00023 | 3.116 | 0.109
2458774.661052 | 17419.05 | 1.42 | 3.38376 | 16.29187 | -0.00735 | 0.08296 | 0.00018 | 0.07487 | 0.00035 | 0.1492 | 0.0010 | 0.01644 | 0.00011 | 3.504 | 0.037
2458774.753132 | 17420.22 | 1.74 | 3.37372 | 16.05479 | -0.00494 | 0.08526 | 0.00021 | 0.08016 | 0.00046 | 0.1614 | 0.0014 | 0.01723 | 0.00014 | 3.747 | 0.070
2458776.768016 | 17423.26 | 4.61 | 3.38822 | 15.96856 | -0.02925 | 0.08095 | 0.00051 | 0.06867 | 0.00112 | 0.1496 | 0.0039 | 0.01664 | 0.00046 | 2.744 | 0.254
2458776.835004 | 17422.78 | 4.87 | 3.37391 | 15.89282 | 0.01259 | 0.08212 | 0.00053 | 0.07226 | 0.00121 | 0.1523 | 0.0043 | 0.01630 | 0.00050 | 2.740 | 0.302
2458777.586320 | 17425.70 | 3.31 | 3.38196 | 16.17172 | -0.00043 | 0.08125 | 0.00037 | 0.07014 | 0.00079 | 0.1439 | 0.0025 | 0.01674 | 0.00031 | 3.038 | 0.153
2458777.723363 | 17422.05 | 2.52 | 3.38064 | 16.13028 | 0.00371 | 0.08470 | 0.00030 | 0.07866 | 0.00063 | 0.1585 | 0.0020 | 0.01741 | 0.00022 | 3.271 | 0.115

101010The table lists the RV, CCF full width at half maximum (FWHM), contrast,
and bisector span, H$\alpha$, H$\beta$, H$\gamma$, NaD, and SHK indexes.

Table 7: Modelling of the SED.

Parameter |  | Prior | Posterior median
---|---|---|---
|  |  | and 68.3% CI
Effective temperature, $T_{\mathrm{eff}}$ | [K] |  $N$(3556, 70) |  3659${}^{+50}_{-63}$
Surface gravity, $\log g$ | [cgs] |  $U$(-0.5, 6.0) |  5.33${}^{+0.44}_{-0.56}$
Metallicity, $[\rm{Fe/H}]$ | [dex] |  $N$(-0.06, 0.12) | $-0.09\pm 0.11$
Distance | [pc] |  $N$(22.444, 0.013) | $22.444\pm 0.013$
$E_{\mathrm{(B-V)}}$ | [mag] |  $U$(0, 3) |  0.028${}^{+0.042}_{-0.021}$
Jitter Gaia | [mag] |  $U$(0, 1) |  0.107${}^{+0.24}_{-0.065}$
Jitter 2MASS | [mag] |  $U$(0, 1) |  0.032${}^{+0.061}_{-0.023}$
Jitter WISE | [mag] |  $U$(0, 1) |  0.120${}^{+0.092}_{-0.045}$
Radius, $R_{\star}$ |  [$R_{\odot}$] |  $U$(0, 100) |  0.531${}^{+0.016}_{-0.012}$
Luminosity |  [L⊙] |  |  0.0455${}^{+0.0018}_{-0.0016}$
111111$N$($\mu$,$\sigma$): Normal distribution prior with mean $\mu$, and
standard deviation $\sigma$. $U$(l,u): Uniform distribution prior in the range
[l, u].

Figure 23: Posterior of the trial frequency for the second planet search
(orange, representing the same colour as in Fig. 7), with the spectral window
function (grey, provided by dace.unige.ch) centred at the highest peak.
1111footnotetext:  Table 8: Two-Keplerian fit to the HARPS data.

Parameter | Units | Prior | Posterior
---|---|---|---
$\gamma$ |  [ km s-1] |  $U$(17.395, 17.425) | $17.4098\pm 0.0046$
Jitter |  [m s-1] |  $J$(0.01, 5) |  $0.102^{-0.081}_{+0.46}$
_GJ 3090 b_ |  |  |
P | [d] |  $N$(2.8531042, $3.1\times 10^{-6}$) | $2.8531042\pm 3.2\times 10^{-6}$
T${}_{\rm c}-2,450,000$ | [BJD] |  $N$(8370.41862, 0.00030) | $8370.41862\pm 0.00030$
$\sqrt{e}\cos{\omega}$ |  |  $U$(-1, 1) | $-0.25^{-0.16}_{+0.25}$
$\sqrt{e}\sin{\omega}$ |  |  $U$(-1, 1) | $-0.10\pm 0.36$
e |  |  | $0.18^{-0.12}_{+0.14}$
$\omega$ | [°] |  | $207^{-78}_{+51}$
K |  [m s-1] |  $U$(0, 10) |  $2.39\pm 0.56$
_GJ 3090 c_ |  |  |
P | [d] |  $U$(12.5, 12.9) | $12.734^{-0.033}_{+0.026}$
T${}_{\rm c}-2,450,000$ | [BJD] |  $U$(8364, 8377) | $8370.52^{-0.97}_{+1.4}$
$\sqrt{e}\cos{\omega}$ |  |  $U$(-1, 1) | $0.40^{-0.30}_{+0.15}$
$\sqrt{e}\sin{\omega}$ |  |  $U$(-1, 1) | $-0.01\pm 0.23$
e |  |  | $0.22\pm 0.13$
$\omega$ | [°] |  | $-1^{-43}_{+36}$
K |  [m s-1] |  $U$(0, 20) |  $6.18\pm 0.76$
_QPC_ |  |  |
$h_{1}$ |  [m s-1] |  $J$(0.01, 50) | $9.6^{-2.3}_{+3.1}$
$h_{2}$ |  [m s-1] |  $J$(0.01, 50) | $0.49^{-0.46}_{+4.0}$
$P$ | [d] |  $U$(1, 30) | $17.733^{-0.039}_{+0.21}$
$\lambda$ | [d] |  $U$(1, 2000) |  $610^{-350}_{+450}$
121212$N(\mu,\sigma)$: Normal distribution with mean $\mu$, and standard
deviation $\sigma$. $U(a,b)$: A uniform distribution defined between a lower
$a$ and upper $b$ limit. $J(a,b)$: Jeffreys (or log-uniform) distribution
defined between a lower $a$ and upper $b$ limit. $\gamma$, P, Tc, e, $\omega$,
and K denote the systemic velocity, orbital period, time of conjunction,
eccentricity, argument of pericentre, and RV semi-amplitude, respectively.
$h_{1}$, $h_{2}$, $P$, and $\lambda$ are the hyperparameters of the QPC kernel
defined in Perger et al. (2021).

Figure 24: Keplerian modelling of transit photometry and RVs of GJ 3090. Top
panel: Transit photometry observations (top, blue error bars), median model
(black line), and 68% CI (grey band, barely visible) computed from 1000 random
posterior samples. For the ExTrA observations, the label numbers (1, 2, and 3)
correspond to each of the ExTrA telescopes (offset for clarity). Data (bottom,
blue dots) detrended with the MAP model (black line). For the Spitzer and
ExTrA observations, 5-minute bins are shown as blue open circles. Middle
panel: HARPS RVs (top, blue error bars, as computed from the spectra), median
model (black line), 68% CI (grey band), two-Keplerian MAP model (green line),
and the mean of the predictive distribution of the kernel for the MAP model
(orange line). Residuals to the median model (bottom). Bottom panel: HARPS RVs
(bottom, blue error bars) corrected for the MAP model of activity (the orange
line in the middle panel) and the Keplerian orbit of planet c, folded at the
period of planet b. The black line is the corresponding Keplerian model of GJ
3090 b with the parameters of the MAP. The same is shown for planet c in the
right panel. Figure 25: Two-parameter joint posterior distributions for the
most relevant model parameters of the photodynamical modelling with the
stability constrain. The 39.3, 86.5, and 98.9% two-variable joint confidence
regions are denoted by three different grey levels. In the case of a Gaussian
posterior, these regions project on to the one-dimensional 1, 2, and 3
$\sigma$ intervals. The histogram of the marginal distribution for each
parameter is shown at the top of each column, except for the parameter on the
last line, which is shown at the end of the line. Units are the same as in
Table 3. Figure 26: Posterior TTVs of GJ 3090 b from the photodynamical
modelling (blue) compared with the with individual transit time determinations
(black error bars). Top panel: One thousand random draws from the posterior
distribution are shown. To compute the TTVs, we used a different linear
ephemeris for each posterior sample and the period and time of conjunction
from Sect. 4.2 for the individual transit time determinations. Lower panel:
The 68.3%, and 95.4% CIs are plotted in different intensities of blue.