11institutetext: Department of Physics, University of Rome “Tor Vergata”, Via
della Ricerca Scientifica 1, 00133, Rome, Italy
11email<EMAIL_ADDRESS>22institutetext: Department of Physics and Astronomy, University of Florence, Largo Enrico Fermi 5, 50125 Firenze 33institutetext: INAF – Turin Astrophysical Observatory, via Osservatorio 20, 10025, Pino Torinese, Italy 44institutetext: Max Planck Institute for Astronomy, Königstuhl 17, 69117 – Heidelberg, Germany 55institutetext: INAF – Osservatorio Astronomico di Roma, via Frascati 33, 00040, Monte Porzio Catone (RM), Italy 66institutetext: Aix-Marseille Université, CNRS, CNES, LAM, Marseille, France 77institutetext: INAF – Osservatorio Astronomico di Padova, Vicolo dell’Osservatorio 5, 35122, Padova, Italy 88institutetext: Dipartimento di Fisica, Universitá degli Studi di Torino, via Pietro Giuria 1, 10125, Torino, Italy 99institutetext: Centro de Astrobiología, Depto. de Astrofísica, ESAC campus, 28692, Villanueva de la Cañada (Madrid), Spain 1010institutetext: Center for Astrophysics | Harvard & Smithsonian, 60 Garden Street, Cambridge, MA 02138, USA 1111institutetext: Department of Earth and Planetary Sciences, Harvard University, 20 Oxford Street, Cambridge, MA 02138, USA 1212institutetext: INAF – Osservatorio Astronomico di Trieste, via Tiepolo 11, 34143, Trieste, Italy 1313institutetext: INAF – Osservatorio Astronomico di Capodimonte, Salita Moiariello 16, 80131, Naples, Italy 1414institutetext: INAF – Osservatorio Astrofisico di Catania, via S. Sofia 78, 95123, Catania, Italy 1515institutetext: INAF – Osservatorio Astronomico di Palermo, Piazza del Parlamento, 1, 90134, Palermo, Italy 1616institutetext: INAF – Osservatorio Astronomico di Cagliari, via della Scienza 5, 09047, Selargius (CA), Italy 1717institutetext: Dipartimento di Fisica e Astronomia “Galileo Galilei” – Università di Padova, Vicolo dell’Osservatorio 2, 35122, Padova, Italy 1818institutetext: INAF – Osservatorio Astronomico di Brera, Via E. Bianchi 46, 23807, Merate (LC), Italy 1919institutetext: Fundación Galileo Galilei - INAF, Rambla José Ana Fernandez Pérez 7, 38712 Breña Baja, TF, Spain 2020institutetext: European Southern Observatory, Karl-Schwarzschild-Straße 2, D-85748 Garching, Germany 2121institutetext: Institut d’Astrophysique de Paris, CNRS, UMR 7095 & Sorbonne Université, UPMC Paris 6, 98bis Bd Arago, 75014 Paris, France 2222institutetext: Observatoire de Haute-Provence, CNRS, Université d’Aix-Marseille, 04870 Saint-Michel-l’Observatoire, France 2323institutetext: Aix-Marseille Université, CNRS, CNES, LAM, Marseille, France 2424institutetext: Université Grenoble Alpes, CNRS, IPAG, 38000 Grenoble, France 2525institutetext: Department of Physics, Shahid Beheshti University, Tehran, Iran 2626institutetext: Laboratoire J.-L. Lagrange, OCA, Université de Nice-Sophia Antipolis, CNRS, Campus Valrose, 06108 Nice Cedex 2, France 2727institutetext: IRAP, Université de Toulouse, CNRS, UPS, CNES, F-31400 Toulouse, France 2828institutetext: Department of Physics and Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 2929institutetext: NASA Goddard Space Flight Center, 8800 Greenbelt Road, Greenbelt, MD 20771, USA 3030institutetext: Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 3131institutetext: Kavli Institute for Astrophysics and Space Research, Massachusetts Institute of Technology, Cambridge, MA 02139, USA 3232institutetext: Harvard-Smithsonian Center for Astrophysics, 60 Garden Street, Cambridge, MA 02138, USA 3333institutetext: Department of Physics and Astronomy, University of New Mexico, 210 Yale Blvd NE, Albuquerque, NM 87106, USA 3434institutetext: NASA Ames Research Center, Moffett Field, CA 94035, USA 3535institutetext: SETI Institute, Mountain View, CA 94043, USA 3636institutetext: Space Science & Astrobiology Division MS 245-3 3737institutetext: INAF-Telescopio Nazionale Galileo, Apartado 565, E-38700, Santa Cruz de La Palma, Spain 3838institutetext: Department of Astrophysical Sciences, Princeton University, Princeton, NJ 08544, USA 3939institutetext: Astrophysics Group, Keele University, Staffordshire, ST5 5BG, UK 4040institutetext: Department of Aeronautics and Astronautics, MIT, 77 Massachusetts Avenue, Cambridge, MA 02139, USA 4141institutetext: INAF – Osservatorio Astrofisico di Arcetri, Largo E. Fermi 5, 50125, Florence, Italy

# The GAPS Programme at TNG XL:
A puffy and warm Neptune-sized planet and an outer Neptune-mass candidate
orbiting the solar-type star TOI-1422

L. Naponiello 1122 L. Mancini 113344 M. Damasso 33 A. S. Bonomo 33 A. Sozzetti
33 D. Nardiello 6677 K. Biazzo 55 R. G. Stognone 88 J. Lillo-Box 99 A. F.
Lanza 1414 E. Poretti 18181919 J. J. Lissauer 35353636 L. Zeng 10101111 A.
Bieryla 3232 G. Hébrard 21212222 M. Basilicata 11 S. Benatti 1515 A. Bignamini
1212 F. Borsa 1818 R. Claudi 77 R. Cosentino 1919 E. Covino 1313 A. de
Gurtubai 1919 X. Delfosse 2424 S. Desidera 77 D. Dragomir 3333 J. D. Eastman
3232 Z. Essack 30303131 A. F. M. Fiorenzano 1919 P. Giacobbe 33 A. Harutyunyan
1919 N. Heidari 252526262323 C. Hellier 3939 J. M. Jenkins 3434 C. Knapic 1212
P.-C. König 20202121 D. W. Latham 3232 A. Magazzù 3737 A. Maggio 1515 J.
Maldonado 1515 G. Micela 1515 E. Molinari 1616 M. Molinaro 1212 E. H. Morgan
2828 C. Moutou 2727 V. Nascimbeni 77 E. Pace 22 I. Pagano 1414 M. Pedani 1919
G. Piotto 1717 M. Pinamonti 33 E. V. Quintana 2929 M. Rainer 1818 G. R. Ricker
3131 S. Seager 282830304040 J. D. Twicken 34343535 R. Vanderspek 2828 J. N.
Winn 3838

###### Abstract

Context. Neptunes represent one of the main types of exoplanets and have
chemical-physical characteristics halfway between rocky and gas giant planets.
Therefore, their characterization is important for understanding and
constraining both the formation mechanisms and the evolution patterns of
planets.

Aims. We investigate the exoplanet candidate TOI-1422 b, which was discovered
by the _TESS_ space telescope around the high proper-motion G2 V star TOI-1422
($V=10.6$ mag), 155 pc away, with the primary goal of confirming its planetary
nature and characterising its properties.

Methods. We monitored TOI-1422 with the HARPS-N spectrograph for 1.5 years to
precisely quantify its radial velocity variation. We analyze these radial
velocity measurements jointly with TESS photometry and check for blended
companions through high-spatial resolution images using the AstraLux
instrument.

Results. We estimate that the parent star has a radius and a mass of
$R_{\star}=1.019_{-0.013}^{+0.014}\,R_{\sun}$,
$M_{\star}=0.981_{-0.065}^{+0.062}\,M_{\sun}$, respectively. Our analysis
confirms the planetary nature of TOI-1422 b and also suggests the presence of
a Neptune-mass planet on a more distant orbit, the candidate TOI-1422 c, which
is not detected in _TESS_ light curves. The inner planet, TOI-1422 b, orbits
on a period $P_{\rm b}=12.9972\pm 0.0006$ days and has an equilibrium
temperature $T_{\rm eq,b}=867\pm 17$ K. With a radius of $R_{\rm
b}=3.96^{+0.13}_{-0.11}\,R_{\oplus}$, a mass of $M_{\rm
b}=9.0^{+2.3}_{-2.0}\,M_{\oplus}$ and, consequently, a density of $\rho_{\rm
b}=0.795^{+0.290}_{-0.235}$ g cm-3, it can be considered a warm Neptune-size
planet. Compared to other exoplanets of similar mass range, TOI-1422 b is
among the most inflated ones and we expect this planet to have an extensive
gaseous envelope that surrounds a core with a mass fraction around $10\%-25\%$
of the total mass of the planet. The outer non-transiting planet candidate,
TOI-1422 c, has an orbital period of $P_{\rm c}=29.29^{+0.21}_{-0.20}$ days, a
minimum mass, $M_{\rm c}\sin{i}$, of $11.1^{+2.6}_{-2.3}\,M_{\oplus}$, an
equilibrium temperature of $T_{\rm eq,c}=661\pm 13$ K and, therefore, if
confirmed, it could be considered as another warm Neptune.

###### Key Words.:

planetary systems – techniques: photometric – techniques: spectroscopic –
techniques: radial velocities stars: individual: TOI-1422 – method: data
analysis

##  1 Introduction

Exoplanetary science has expanded quickly from the simple detection of new
worlds to their in-depth characterization. The latter is especially feasible
for planets orbiting bright stars on a plane almost aligned to our line of
sight, so that their radius and mass can both be derived by transit photometry
and radial velocity measurements, respectively. The population of known
transiting planets has increased significantly in the last two decades mainly
thanks to dedicated ground-based surveys, which were then followed by surveys
from space that resulted to be much more efficient considering the total
number of discoveries.

So far, the _Kepler_ and the K2 space missions (Borucki et al., 2010; Howell
et al., 2014) have had a very important impact on the exoplanet field by
discovering thousands of confirmed and candidate planets, many of which are
not amenable to radial velocity (RV) follow up due to the faintness of their
host stars. The Transiting Exoplanet Survey Satellite (_TESS_) (Ricker et al.,
2014), currently at the end of its first extended mission with a second one
already proposed, was designed to target nearby and bright stars on a great
portion of the sky (around 85% sky coverage during the primary mission alone)
because such stars are easier to follow-up by means of RV and result in
refined measurements of their own exoplanet masses, atmospheres, sizes, and
therefore densities.

The possibility to use a large exoplanet sample, like that of _Kepler_ , which
is based on homogeneous data and has a minimal pollution from false positives
($<10\%$, Fressin et al. 2013), has allowed us to distinguish between several
distinct exoplanet regimes (Weiss & Marcy, 2014; Buchhave et al., 2014; Zeng
et al., 2019): the terrestrial-like planets ($R_{\rm p}<1.7\,R_{\oplus}$), the
gas dwarf planets with rocky cores and hydrogen–helium envelopes, the
H2O-dominated ices/fluids water worlds (both of the latter two classes have
$1.7\,R_{\oplus}<R_{\rm p}<3.9\,R_{\oplus}$) and the ice or gas giant planets
($R_{\rm p}>3.9\,R_{\oplus}$).

Planet occurrence around main-sequence stars has been early investigated
thanks to Doppler surveys (e.g., Cumming et al., 2008; Wright et al., 2012).
In particular, the Keck Eta-Earth survey (Howard et al., 2010) and the
CORALIE+HARPS survey (Mayor et al., 2011) first explored the domain of low-
mass ($3-30\,M_{\oplus}$) close-in ($P_{\rm orb}\sim 50$ days) planets. These
planets resulted an order of magnitude more common than giant planets.

Other studies for determining the occurrence rates of planets, based on the
_Kepler_ sample, agree in determining that for planets with less than 1-year
orbital period, their mean number per star is higher within the radius range
$1\,R_{\oplus}<R_{\rm p}<4\,R_{\oplus}$ compared to the range
$4\,R_{\oplus}<R_{\rm p}<16\,R_{\oplus}$ or similar (Howard et al., 2012;
Fressin et al., 2013; Petigura et al., 2013). The subsequently and gradually
refinement of parent-star properties (especially thanks to high-resolution
stellar spectra) revealed a clear bimodality of the radius distribution of
close-in ($P<100$ days), small-size ($R_{\rm p}<4.0\,R_{\oplus}$) planets
orbiting bright, main-sequence solar-type stars (Petigura et al., 2017; Fulton
& Petigura, 2018; Van Eylen et al., 2018)111For a possible explanation of
Fulton’s gap see Modirrousta-Galian et al. 2020. These two quite distinct
populations were identified as “super-Earths” ($R_{\rm p}<1.5\,R_{\oplus}$)
and “sub-Neptunes” ($R_{\rm p}=2-3\,R_{\oplus}$), which are also represented
in the intermediate region ($R_{\rm p}=1.5-2\,R_{\oplus}$) with fewer planets.
However, it is better to stress that, since we do not know for sure what they
are made of, the space of physical parameters ($R_{\rm p}$, $M_{\rm p}$), for
which the previous terms apply, are not strictly defined.

The advantage in studying transiting planets is the possibility, in many
cases, to measure both the planetary radius and mass and, therefore, determine
their density and bulk composition. Knowing the structural properties, one
should be able to distinguish among the various scenarios of exoplanet
formation and evolution. Unfortunately, theoretical models (e.g., Bitsch et
al., 2019; Turbet et al., 2020) tell us that the mass-radius relationships for
small planets present degeneracy due to the vastness of possible different
compositions and amounts of rock, ice and gas, especially in the transition
between rocky super-Earths and Neptune-like planets (e.g., Miller-Ricci et
al., 2009; Lozovsky et al., 2018). A detailed investigation of the mass-radius
relation for small planets can be useful for throwing light on several open
questions, such as the diversity of planet core masses and compositions, or
those regarding the place where they form (in situ or beyond the snowline) and
the existence of the radius gap. We refer the reader to the recent review by
Biazzo et al., 2022 for an exhaustive discussion on this topic.

It is, therefore, clear how RV follow-up observations and in general planetary
mass measurements play an important role in this understanding process and why
today there is a tremendous effort in this field by many teams (e.g.,
KESPRINT: Gandolfi et al., 2018; HARPS-N consortium: Cloutier et al., 2020;
NCORES: Armstrong et al., 2020; TESS-Keck Survey: Chontos et al., 2022; GAPS:
Carleo et al., 2021) in order to confirm TESS small-planet candidates.

Probing the chemical composition of the atmosphere of a large number of sub-
Neptune planets would also be helpful to unravel the skein. Various techniques
(like high-resolution spectroscopy, transmission and emission spectroscopy)
have been implemented and applied successfully using the HST (Hubble Space
Telescope) instruments or the high-resolution spectrographs mounted on large-
class ground-based telescopes (e.g., CRIRES: Snellen et al., 2010; HARPS:
Wyttenbach et al., 2015; LDSS3C: Diamond-Lowe et al., 2018; GIANO: Brogi et
al., 2018; CARMENES: Casasayas-Barris et al., 2019; HARPS-N: Pino et al.,
2020; ESPRESSO: Borsa et al., 2021). Unfortunately, these techniques for
probing the planetary atmospheres are currently effectively applicable only to
giant planets, as we know few sub-Neptune planets for which the transmission-
spectrum signal can be detected with a sufficient signal-to-noise ratio for
allowing to discriminate among different atmospheric models. The featureless
transmission spectra of GJ 436 b (Knutson et al., 2014) and GJ 1214 b
(Kreidberg et al., 2014) are emblematic.

The situation should improve soon thanks to the JWST (James Webb Space
Telescope) (Barstow et al., 2015), which is about to go into operation, and
with the next generation of space-based and large ground-based telescopes
(Ariel: Tinetti et al., 2021; ELT: Ramsay et al., 2021; TMT: Skidmore et al.
2015). In the meantime, it is important that we continue to work for
uncovering new exoplanets, especially those of small size ($R_{\rm
p}<5\,R_{\oplus}$) that orbit bright ($V<11$ mag) main-sequence dwarf stars.
This is currently possible thanks to the great number of planet candidates
(more than 5000) that _TESS_ is discovering at the present time. The recent
detection of water vapour in the atmosphere of the super-Neptune TOI-674 b
with the HST (Brande et al., 2022) is a successful example of this effort.

On the 6th of November 2019, the _TESS_ target star TIC 333473672 was
officially named TOI-1422 (_TESS_ Object of Interest; Guerrero et al. 2021),
following the Data Validation Report Summary (DRS) produced by the _TESS_
Science Processing Operations Center (SPOC) (Jenkins et al., 2016) pipeline at
NASA Ames Research Center through the Transiting Planet Search (TPS; Jenkins
2002; Jenkins et al. 2010, 2020) and Data Validation (DV; Twicken et al. 2018,
Li et al. 2019) modules. In particular, TOI-1422 01 was flagged as a potential
planet with an orbital period of $13.0020\,\pm\,0.0040$ days, a transit depth
of $1422\,\pm\,94$ ppm (parts per million) and a corresponding radius of
$3.85\,\pm\,0.90\,R_{\oplus}$, which is compatible with Neptune’s radius. The
candidate passed all SPOC DV diagnostic tests; furthermore, all TIC (version
8) objects other than the target star were excluded as sources of the transit
signal through the difference image centroid offsets (Twicken et al., 2018).

The long-term, multi-program Global Architecture of Planetary Systems (GAPS)
(Covino et al., 2013; Poretti et al., 2016) exploits Doppler measurements
taken with the HARPS-N (High Accuracy Radial velocity Planet Searcher for the
Northern hemisphere) (Cosentino et al., 2012) instrument at the Telescopio
Nazionale Galileo (TNG) in La Palma (Spain). This high-resolution spectrograph
(resolving power $R\approx 115\,000$) delivers the highest RV precision ($\sim
1$ m s-1) currently achievable in the northern hemisphere. One of the aims of
the GAPS programme is to confirm and obtain an accurate mass determination of
planets having an intermediate-mass between super-Earths and super-Neptunes;
for this reason, TOI-1422 was selected for RV follow-up observations, which
started in June 2020.

In the present work, we report the results of our measurements and analysis
that allowed us to confirm TOI-1422 as a new planetary system. The paper is
organized as follows: Sect. 2 contains the details of the instruments and the
photometric and RV measurements. The results of our analyses are presented in
Sect. 3 and discussed in Sect. 4, before finally addressing the conclusions in
Sect. 5.

##  2 Observations and data reduction

###  2.1 TESS photometry

Since late July 2018, _TESS_ observed more than $200\,000$ stars with its four
wide-field optical CCD cameras ($24\times 96$ degrees), each having a focal
ratio of $f/1.4$ and a broad-band filter range between 600 and 1000 nm. The
pre-selected target TIC 333473672 was observed in Sectors 16 and 17 between
2019-Sep-11 and 2019-Nov-02, and a first of a total of four transiting events
were recorded on 2019-Sep-19. The two-minute cadence photometry of TOI-1422
from _TESS_ spans a total of $\thickapprox 50$ days and, to analyze it, we
used the Presearch Data Conditioning Simple Aperture Photometry (PDC-SAP;
Stumpe et al. 2012, 2014, Smith et al. 2012) light curve, which is provided by
the _TESS_ SPOC pipeline and retrieved through the Python package lightkurve
(Lightkurve Collaboration et al., 2018) from the Mikulski Archive for Space
Telescopes (MAST). We jointly fitted the transit model and a Gaussian Process
(GP) using a simple (approximate) Matern kernel, which was implemented in the
Python modelling tool juliet222https://juliet.readthedocs.io (Espinoza et al.,
2019) via celerite (Foreman-Mackey et al., 2017), of the form:

$k(\tau_{i,j})=\sigma_{\rm
GP}^{2}\,M(\tau_{i,j},\,\rho)+(\sigma_{i}^{2}+\sigma_{w}^{2})\,\delta_{i,j}\,,$
(1)

where $\sigma_{i}$ is the errorbar of the $i$-th data point, $\sigma_{\rm GP}$
the amplitude of the GP in parts per million (ppm), $\sigma_{w}$ an added
jitter term (in ppm), $\delta_{i,j}$ the Kronecker’s delta, $k(\tau_{i,j})$
the element $i$,$j$ of the covariance matrix as a function of
$\tau_{i,j}=|t_{i}-t_{j}|$, with $t_{i}$ and $t_{j}$ being the $i$,$j$ GP
regressors (i.e. the observing times), while

$M(\tau_{i,j},\,\rho)=(1+1/\epsilon)\,e^{-[1-\epsilon]\sqrt{3}\tau_{i,j}/\rho}+(1-1/\epsilon)\,e^{-[1+\epsilon]\sqrt{3}\tau_{i,j}/\rho}$
(2)

is the kernel with its characteristic time-scale $\rho$. The parameter
$\epsilon$ controls the quality of the approximation since, in the limit
$\epsilon\to 0$, the Eq. (2) becomes the Matern-3/2 function. In juliet, the
possible polluting sources inside _TESS_ aperture333tpfplotter is a python
package developed by J. Lillo-Box and publicly available on
www.github.com/jlillo/tpfplotter. (Fig. 1), which might result in a smaller
transit depth compared to the real one, are taken into account with a dilution
factor ($D$) that, in this case, has been neglected because the PDC-SAP
photometry is already corrected for dilution from other objects contained
within the aperture using the Create Optimal Apertures (COA) module (Bryson et
al., 2010, 2020)444Since the release of the light curve products from Year 2,
the SPOC background estimation algorithm has been updated due to an over-
correction bias which was significant for dim and/or crowded targets. For this
particular TOI, we estimated this over-correction to be negligible for the
planetary radius estimation as it is significantly smaller than the transit
depth uncertainty.. For the purpose of efficiently sample the whole plausible
zone in the $(b,k)$ plane, where $b$ is the impact parameter and $k$ is the
planet-to-star radius ratio, we used the $(r_{1},r_{2})$ parametrization
described in Espinoza (2018). This is the same approach that we adopted for
the modelling of the transits in the joint analysis with the radial velocities
(see Sect. 3.3). Moreover, here we make use of the limb-darkening
parametrizations of Kipping 2013 for two-parameter limb-darkening laws
($q_{1},q_{2}\to u_{1},u_{2}$).

Figure 1: The TPF (target pixel file) from the _TESS_ observation Sector 16,
made with tpfplotter (Aller et al., 2020) and centred on TOI-1422, which is
marked with a white cross. The SPOC pipeline aperture is shown by shaded red
squares, and the Gaia satellite eDR3 catalogue (Brown et al., 2018; Prusti et
al., 2016) is also overlaid with symbol sizes proportional to the magnitude
difference with TOI-1422. The difference image centroid locates the source of
the transits within $1.89\pm 5$ arcsec of the target star’s location, as
reported by the TicOffset for the multisector DV report for this system.
Figure 2: Top panel: PDC-SAP light curve of TOI-1422 (blue points with error
bars) as collected by _TESS_ in Sectors 16 and 17 with a 2-minute cadence. The
black line represents the best-fit model obtained through GP detrending, as
detailed in Sect. 2.1. Bottom panel: residuals of the best-fitting model in
parts per million.

The PDC-SAP light curve of TOI-1422 and its detrending are plotted in Fig. 2.
We also analyzed the SPOC SAP photometry (Twicken et al., 2010; Morris et al.,
2020), which presents a small long-term variability that might be due to
systematics, but no other feature or modulation can be discerned within the
experimental uncertainties, aside from a possible single extra transit event
which is discussed at the end of Sect. 3.4, and a steep flux drop at the end
of both SAP and PDC-SAP light curves which is probably due to high levels of
background noise.

###  2.2 High-spatial resolution imaging - AstraLux

We observed TOI-1422 with the AstraLux high-spatial resolution camera (Hormuth
et al., 2008), located at the 2.2 m telescope of the Calar Alto Observatory
(CAHA, Almería, Spain) using the lucky-imaging technique. This technique
obtains diffraction-limited images by acquiring thousands of short-exposure
frames and selecting the ones with the highest Strehl ratio (Strehl, 1902) to
finally combine them into a co-added high-spatial resolution image. We
observed this target on the night of 29th of September 2021 under good weather
conditions with a mean seeing of 1 arcsec and obtained 50 000 frames with 20
ms exposure time in the Sloan Digital Sky Survey $z$ filter (SDSSz), with a
field-of-view windowed to $6\times 6$ arcsec. The datacube was reduced by the
instrument pipeline (Hormuth et al., 2008) and we selected the best-quality
$10\%$ frames to produce the final high-resolution image. We obtained the
sensitivity limits of the co-added image by using our own developed astrasens
package555https://github.com/jlillo/astrasens with the procedure described in
Lillo-Box et al. (2012, 2014). The 5$\sigma$ sensitivity curve is shown in
Fig. 3. We can discard sources down to 0.2 arcsec with a magnitude contrast of
$\Delta Z<4$ mag, corresponding to a maximum contamination level of 2.5%. By
using this high-spatial resolution image, we also estimate the probability of
an undetected blended source. This probability (fully described in Lillo-Box
et al. 2014) is called the blended source confidence (BSC). We use a python
implementation of this approach (bsc, by J. Lillo-Box) which uses the
TRILEGAL666http://stev.oapd.inaf.it/cgi-bin/trilegal galactic model (v1.6;
Girardi et al., 2012) to retrieve a simulated source population of the region
around the corresponding target777This is done in python by using the
astrobase implementation by Bhatti et al. (2020).. This simulated population
is used to compute the density of stars around the target position (radius
$r=1^{\circ}$) and derive the probability of chance-alignment at a given
contrast magnitude and separation. When applied to the TOI-1422 location, we
use a maximum contrast magnitude of $\Delta m_{\rm b,max}=6.97$ mag in the
SDSSz passband, corresponding to the maximum contrast of a blended eclipsing
binary that could mimic the observed transit depth of planet b ($\sim$ 1000
ppm). Thanks to our high-resolution image, we estimate the probability of an
undetected blended source to be 0.28%. The probability of such an undetected
source to be an appropriate eclipsing binary is thus even lower and
consequently, we can assume that the transit signal is not due to a blended
eclipsing binary.

Figure 3: Blended Source Confidence (BSC) curve from the AstraLux SDSSz image
(solid black line). The colour on each angular separation and contrast bin
represent the probability of a source aligned at the location of the target,
based on TRILEGAL model. The dotted horizontal line shows the maximum contrast
of a blended binary that is capable of imitating the planet’s transit depth.
The green region represents the regime that is not explored by the high-
spatial resolution image. The BSC curve corresponds to the integration of
$P_{\rm aligned}$ over this region.

###  2.3 HARPS-N radial velocities

Between June 2020 and January 2022, we collected a total of 112 RV
measurements of TOI-1422 with HARPS-N (Table 3). The RVs were calculated using
the TERRA pipeline (Anglada-Escudé & Butler, 2012), version 1.8, through the
YABI workflow interface (Hunter et al., 2012), which is maintained by the
Italian center for Astronomical Archive (IA2). TERRA is an algorithm based on
the template matching technique, and for the RVs retrieval is preferred, in
this paper, over the standard Data Reduction Software (DRS) pipeline, which
returned a slightly lower overall RV precision888For a comparison of the
performances of TERRA vs DRS see Perger, M. et al. 2017 on this target. We
used the RVs calculated using all the spectral orders and from the full sample
of RVs we removed four points following Chauvenet’s criterion. TERRA RVs have
an average measurement error of $2.6$ m s-1, a root mean square error of $4.5$
m s-1 and a signal-to-noise ratio S/N $\approx$ 35 measured at a reference
wavelength of 5500 $\AA$. A long linear trend is evident in HARPS-N RV data,
as we discuss in Sect. 3.2.

TOI-1422 was also observed with the SOPHIE instrument, a stabilized échelle
spectrograph mounted at the 193-cm Telescope of Observatoire de Haute-Provence
in France (Perruchot et al. 2008, Bouchy, F. et al. 2013), however, for
signals of low semi-amplitudes such as those we discuss in this work, the RV
measurements of SOPHIE, due to higher uncertainties compared to HARPS-N, do
not increase the significance of the results presented in Sect. 5 and
therefore have not been utilized.

##  3 System characterization

###  3.1 Parent star

From the co-added spectrum built out of individual HARPS-N spectra extracted
with the standard DRS pipeline, we derived the following atmospheric
parameters of the planet-host star TOI-1422: effective temperature $T_{\rm
eff}$, surface gravity $\log g$, microturbulence velocity $\xi$, iron
abundance [Fe/H], and rotational velocity $\upsilon\sin{i_{\star}}$. For
$T_{\rm eff}$, $\log g$, $\xi$, and [Fe/H] we applied a method based on
equivalent widths of iron lines taken from Biazzo et al. (2015) and the
spectral analysis package MOOG (Sneden 1973; version 2017). The Castelli &
Kurucz (2003) grid of model atmospheres was adopted. $T_{\rm eff}$ and $\xi$
were derived by imposing that the abundance of Fe i is not dependent on the
line excitation potentials and the reduced equivalent widths (i.e.
$EW/\lambda$), respectively, while $\log g$ was obtained by imposing the Fe
i/Fe ii ionization equilibrium condition. The $\upsilon\sin{i_{\star}}$ was
measured with the same MOOG code, by applying the spectral synthesis of three
regions around 5400, 6200, and 6700 Å, and adopting the same grid of model
atmosphere after fixing the macroturbulence velocity to the value of $3.4$ km
s-1 from the relationship by Doyle et al. (2014). From these results, the star
can be classified as a G2 V dwarf with a low projected rotation velocity
$\upsilon\sin{i_{\star}}$ of $1.9\pm 0.8$ km s-1, implying a maximum rotation
period of $27^{+19}_{-8}$ d at $1\sigma$. In analogy, using an empirical
relation based on the FWHM999This relation was calibrated using a set of well-
aligned transiting exoplanet systems, for which we could infer
$\upsilon\sin{i_{\star}}$ as equal to their equatorial velocities. We estimate
the equatorial velocities from the stellar radii and rotational period and
correlate these values directly to the FWHMs. derived by the HARPS-N DRS, we
find $\upsilon\sin{i_{\star}}\sim 2.2$ km s-1.

The field of TOI-1422 was also observed in 2004, 2006 and 2007 during the WASP
transit-search survey (Pollacco et al., 2006). A total of 20 000 photometric
data points were obtained by observing the field every $\sim$ 15 min on clear
nights, over spans of $\sim$ 120 days in each year. We searched the data for
any rotational modulation using the methods from Maxted et al. (2011) and
found no significant periodicity between 1 and 100 days, with a 95%-confidence
upper limit on the amplitude of 2 mmag. The TESS light curve shows no
modulation as well (Sect. 2.1), confirming that the star is rather
magnetically quiet over a period of $\sim$ 100 days.

Moreover, the spectrum of TOI-1422 clearly shows a lithium line at
$\lambda=6707.8$ Å. We, therefore, estimated the lithium abundance $\log
A{\rm(Li)}_{\rm NLTE}$ by measuring the lithium $EW$ and considering our
$T_{\rm eff}$, $\log g$, $\xi$, and [Fe/H] previously derived together with
the NLTE corrections by Lind et al. 2009. The value of the lithium abundance
is listed in Table 1 and its position in a $\log A{\rm(Li)}$-$T_{\rm eff}$
diagram is compatible with the M67 open cluster advanced age ($\sim 4.5$ Gyr;
see Pasquini et al. 2009) in agreement with the star’s low-activity level. The
physical parameters of TOI-1422 are also displayed in Table 1 and were
determined with the EXOFASTv2 Bayesian code (Eastman, 2017; Eastman et al.,
2019), by fitting the stellar Spectral Energy Distribution (SED) and by
employing the MESA Isochrones and Stellar Tracks (Dotter, 2016) to constrain
more precisely the stellar mass. In addition, in the table we report the
stellar magnitudes used for the SED modelling, while the SED best fit is shown
in Fig. 4. Gaussian priors were imposed on the Gaia eDR3 parallax (Gaia
Collaboration et al., 2021) as well as on the $T_{\rm eff}$ and $\rm[Fe/H]$ as
derived above from the analysis of the HARPS-N spectra. An upper limit was set
on the $V$-band extinction, $A_{\rm V}$, from reddening maps (Schlegel et al.,
1998; Schlafly & Finkbeiner, 2011).

Figure 4: Spectral energy distribution computed for TOI-1422, where the black
curve is the most likely atmospheric stellar model and the blue dots
correspond to the model fluxes over each passband. The horizontal and vertical
red error bars represent, respectively, the effective width of the passbands
and the reported photometric measurement uncertainties (refer to the
magnitudes in Table 1). Table 1: TOI-1422 parameters.

Parameter | Unit | Value | Source
---|---|---|---
Cross-identifications |  |  |
TOI |  | TOI-1422 | TOI catalogue
TIC ID |  | 333473672 | TIC
Tycho |  | 3235-00524-1 | Tycho
2MASS ID |  | J23365789+3938218 | 2MASS
Gaia ID |  | 1920333449169516288 | Gaia eDR3
Astrometric properties |  |  |
R.A. | J2016 | 354.240817 | Gaia eDR3
Dec | J2016 | +39.639275 | Gaia eDR3
Parallax | mas | $6.4418\pm 0.0138$ | Gaia eDR3
$\mu_{\alpha}$ | mas yr-1 | $-67.564\pm 0.015$ | Gaia eDR3
$\mu_{\delta}$ | mas yr-1 | $-31.180\pm 0.011$ | Gaia eDR3
Distance | pc | $154.56^{+0.037}_{-0.027}$ | VizieR
Photometric properties |  |  |
$B_{\rm T}$ | mag | $11.31\pm 0.07$ | Tycho
$V_{\rm T}$ | mag | $10.62\pm 0.05$ | Tycho
$J$ | mag | $9.585\pm 0.022$ | 2MASS
$H$ | mag | $9.275\pm 0.030$ | 2MASS
$K_{\rm S}$ | mag | $9.190\pm 0.022$ | 2MASS
$i^{\prime}$ | mag | $10.311\pm 0.075$ | APASS
$W1$ | mag | $9.161\pm 0.023$ | AllWISE
$W2$ | mag | $9.201\pm 0.020$ | AllWISE
$W3$ | mag | $9.161\pm 0.033$ | AllWISE
$A_{V}$ | mag | $<0.077$ | This work
Stellar parameters |  |  |
$L_{\star}$ | $L_{\sun}$ | $1.116\pm 0.037$ | This work
$M_{\star}$ | $M_{\sun}$ | $0.981^{+0.062}_{-0.065}$ | This work
$R_{\star}$ | $R_{\sun}$ | $1.019^{+0.014}_{-0.013}$ | This work
$T_{\rm eff}$ | K | $5840\pm 62$ | This work
$\log g_{\star}$ | cgs | $4.41\pm 0.11$ | This work
$\xi$ | km s-1 | $0.89\pm 0.07$ | This work
$\rm{[Fe/H]}$ | dex | $-0.09\pm 0.07$ | This work
Spectral type(a) |  | G2 V | This work
$\rho_{\star}$ | g cm-3 | $1.3\pm 0.1$ | This work
$\upsilon\sin{i_{\star}}$ | km s-1 | $1.7\pm 0.4$ | This work
$\log A{\rm(Li)_{\rm NLTE}}$ |  | $1.97\pm 0.05$ | This work
$\log R^{\prime}_{\rm HK}$ | dex | $-4.95\pm 0.03$ | This work
Age | Gyr | $5.1^{+3.9}_{-3.1}$ | This work

TESS Primary Mission TOI catalogue (Guerrero et al., 2021); TIC (Stassun et
al., 2018, 2019); Tycho (Høg et al., 2000); 2MASS (Skrutskie et al., 2006);
Gaia eDR3 (Brown et al., 2021); AllWISE (Cutri et al., 2021); APASS (Henden et
al., 2015); VizieR Online Data catalogue (Bailer-Jones et al., 2021). 101010
$a$$a$footnotetext: Spectral type defined according to the stellar spectral
classification of Gray & Corbally 2009.

Figure 5: GLS periodogram of HARPS-N radial velocities and of various activity
indexes specified in the labels, after the removal of a linear trend (Sect.
3.2). The main peak of the RV GLS and that of the TOI-1422 b period are
highlighted with a green and red dashed line, respectively. They do not
overlap with any of the peaks from the indexes, which in general do not
suggest any clear stellar rotation period. The period corresponding to the
highest peak in the RV GLS periodogram, and its FAP, are written on the top of
the first panel, while the horizontal dashed lines remark the $10\%$ and $1\%$
confidence levels (evaluated with the bootstrap method), respectively. The
three peaks surrounding the RVs main frequency can all be explained as aliases
of the 29-day signal due to the two highest frequencies of the window function
(190 and 390.3 days, as shown in Fig. 16 and Fig. 17).

###  3.2 RV and activity indexes periodogram analysis

We computed the Generalised Lomb Scargle (GLS) periodogram for the HARPS-N RVs
and different stellar activity indexes111111The Full-Width-at-Half-Maximum
(FWHM) and the Bisector inverse span (BIS) are calculated using the cross
correlation function (CCF) derived by the DRS pipeline. We also analysed the
chromospheric $\log{R^{\prime}_{\rm HK}}$ index, and additional activity
diagnostics derived from the spectroscopic lines He I, Na I, Ca I, H$\alpha$06
and H$\alpha$16 as defined in the code ACTIN
(https://github.com/gomesdasilva/ACTIN v.1.3.9, Gomes da Silva et al. 2018,
which has been used for the calculation. In particular, the two H-alpha
indices have 1.6 and $0.6\,\AA$ band-pass width, respectively. using the
Python package astropy v.4.3.1 (Price-Whelan et al., 2018). The periodogram of
the RVs shows the main peak around 29-days, other than a significant peak at
13 days (TOI-1422 b transiting period), after correcting for a linear trend of
$\sim 4$ m s-1 yr-1, observed in HARPS-N data. No index shows signs of 29-days
periodicity, but a linear trend is also present in the FWHM and $\log
R^{\prime}_{\rm HK}$ (see Fig. 15), with the former correlating the most with
the RVs, unveiling a moderate Spearman coefficient (Spearman, 1904) of $0.41$
($p$-value $0.01\%$). Therefore, in order to explain the nature of the main
peak in the RVs, we present the GLS periodogram of these coefficients
posterior to the removal of their linear trends in Fig. 5 (see Fig. 16 for a
closer look at the RVs panel), but again no trace of the 29-days signal is
found. We also perform a GP regression analysis, using a quasi-periodic model,
of the $\log R^{\prime}_{\rm HK}$ index corrected for the linear trend over
the time series, and find no evidence of any particular periodic modulation in
the posterior distribution of the periodic time-scale hyper-parameter. In
short, there is no evidence pointing to a specific periodic rotation of the
star TOI-1422, other than the tentative estimation from
$\upsilon\sin{i_{\star}}$.

A query from the Gaia eDR3 archive returns astrometric excess noise and
renormalized unit weight error (RUWE) values of 80 $\mu$as and 1.09,
respectively, for TOI-1422. The star is thus astrometrically quiet. The
analysis of Sect. 2.2 rules out the existence of obvious sub-arcsec stellar
companions, and no co-moving objects are present in Gaia eDR3 data in a 600
arcsec radius. The linear trend seen in the RV data along with a few activity
indexes can therefore be explained by a long star magnetic activity, rather
than by the presence of a companion121212In this case, at a projected
separation of 0.1 arcsec ($\sim 15$ au at the distance of TOI-1422), the lower
limit of the AstraLux imaging data, a maximum RV slope of the magnitude
measured in this work would be produced by a companion of $\sim 30\,M_{\rm
Jup}$ (i.e. either a very low-mass star or a massive sub-stellar companion)..

###  3.3 RV and photometry joint analysis

Figure 6: GLS periodogram of the transiting one-planet model RV residuals. The
main peak is highlighted in red and corresponds to a period of 29.2 days, with
a False Alarm Probability (FAP) of 0.45% (evaluated with the bootstrap
method), while the horizontal dashed lines show the 10% and 1% confidence
levels. Figure 7: The RV measurements of TOI-1422 versus time are shown on the
top panel, while their residuals over the model fit are in the bottom panel.
The circles with blue error bars are the RV data taken with HARPS-N. The large
and small error bars indicate $\sigma_{t}$ and $\sigma_{w}$ (the added jitter
term), respectively. In the top panel, the black line represents the two-
planet model fit.

A joint transit and RV analysis has been carried out with juliet, which
employs different Python tools:
batman131313https://lweb.cfa.harvard.edu/~lkreidberg/batman (Kreidberg, 2015)
for the modelling of transits, RadVel141414https://radvel.readthedocs.io
(Fulton et al., 2018) for the modelling of RVs and stochastic processes, which
are treated as GPs with the packages george151515https://george.readthedocs.io
(Ambikasaran et al., 2015) and celerite161616https://celerite.readthedocs.io.
The radial velocity model that we used in juliet is the following:

$M(t)=K(t)+\epsilon(t)+\overline{\mu}+A\,t+B\,,$ (3)

where $\epsilon(t)$ is a noise model for the HARPS-N instrument here assumed
to be white-gaussian noise, i.e. $\epsilon(t)\thickapprox
N(0,\sigma(t)^{2}+\sigma_{w}^{2})$, with $\sigma(t)^{2}$ being the formal
uncertainty of the RV point at time $t$, $\sigma_{w}^{2}$ being an added
jitter term and $N(\mu,\sigma^{2})$ denoting a normal distribution with mean
$\mu$ and variance $\sigma^{2}$. $K(t)$ is the Keplerian model of the RV star
perturbations due to the orbiting planet, $\overline{\mu}$ is the systemic
velocity linked to the instrument, and the coefficients $A$, $B$ (also
referred to as RV slope and RV intercept) represent an additional linear trend
used for modelling non-Keplerian signals with a period longer than the
observation span. For a total number of data points $N$, we assumed the model
likelihood to follow the likelihood of a $N$-dimensional multivariate
Gaussian:

$ln\,p(\@vec{y}|\@vec{\theta})=-\frac{1}{2}[N\,ln\,2\pi+ln\,|\Sigma|+\@vec{r}^{T}\Sigma^{-1}\@vec{r}]\,,$
(4)

where $\@vec{y}$ and $\@vec{\theta}$ are vectors containing respectively all
the RV data points and instrumental parameters, while $\@vec{r}$ is the
residual vector given by

$r\left(t_{i}\right)=y\left(t_{i}\right)-M\left(t\right)\,.$ (5)

The elements of the covariance matrix $\Sigma$ are:

$\Sigma(t_{i},t_{j})=k(\mu_{i},\mu_{j})+(\sigma_{w}^{2}+\sigma_{t}^{2})\delta_{t_{i},t_{j}}\,,$
(6)

with $k$ equal to any GP kernel model or zero for a pure white-noise one. In
order to estimate the Bayesian posteriors and evidence, $\mathcal{Z}$, of
different models, we used a Dynamic Nested Sampling package dynesty (Speagle,
2020), which adaptively allocates samples based on a posterior structure and,
at the same time, estimates evidence and sampling from multi-modal
distributions. In general, dynamic nested sampling algorithms sample a dynamic
number of live points from the prior “volume” and sequentially replace the
point with the lowest likelihood with a new one, while updating the Bayesian
evidence by the difference $\Delta\mathcal{Z}$. Usually, the stopping
criterion is a defined value of $\Delta\mathcal{Z}$, below which the algorithm
is said to have converged ($\Delta\mathcal{Z}\thickapprox 0.5$). However, here
we used the default criterion described in Sect. 3.4 of Speagle (2020).

Figure 8: The residuals for the mid-transit timings of TOI-1422 b versus a
linear ephemeris, with 1-$\sigma$ error bars, are plotted in black. The green
circles, red diamonds and blue stars represent TTV predictions respectively in
the case of null, average or maximum eccentricities, with the error bars
showing the uncertainty due to $T_{0,c}$ (see Table 2). The points have been
slightly shifted on the x-axis to allow for more visibility.

In order to reveal the transiting object suggested by the TESS light curve, we
first run the RV and photometry joint analysis with a simple one-planet model
using as transit-related priors the parameters in the DVR produced by the SPOC
pipeline both with a fixed null and uniformly-sampled eccentricity via the
parametrization $S_{1}=\sqrt{e}\sin{\omega}$, $S_{2}=\sqrt{e}\cos{\omega}$,
that is described in Eastman et al. (2013). All the priors are defined in
Table 4. In particular, we have set Gaussian priors on both the limb-darkening
coefficients (from Claret, A. 2017) and the star mean density $\rho_{\star}$
(from Sect. 3.1), which is implemented here instead of the scaled semi-major
axis, $a/R_{\star}$, because the latter can be recovered using Kepler’s third
law using only the period of the respective planet, which is a direct result
of any juliet run. In this way, from the single value of $\rho_{\star}$ we can
evenly derive $a/R_{\star}$ in the case of multiple planets.

The best one-planet RV model fit is found with $e=0$
($\Delta\ln{\mathcal{Z}}^{e_{b}=0}_{e_{b}\neq 0}=0.7$), but the scatter of the
residuals is higher than the average photon-noise uncertainties for this kind
of star. In fact, the same peak of 29-days, which was found in the RVs GLS
periodogram, is also distinctly found in the residuals of the transiting one-
planet model (see Fig. 6). Consequently, we proceed to test two-planet models,
whose priors are summed in Table 5. Since they have comparable statistical
significance ($\Delta\ln{\mathcal{Z}}^{e_{b,c}=0}_{e_{b,c}\neq 0}=0.4$) for
the rest of the paper we use the results of the eccentric model. The two-
planet eccentric model is plotted on top of the RVs in Fig. 7, along with its
residuals. TOI-1422 b RV semi-amplitude and orbital period result to be
$K_{\rm b}=2.47^{+0.50}_{-0.46}$ m/s and $P_{\rm b}=12.9972\pm 0.0006$ days,
respectively. While the second planet, candidate TOI-1422 c, has RV semi-
amplitude $K_{\rm c}=2.36^{+0.42}_{-0.40}$ m/s, orbital period $P_{\rm
c}=29.29^{+0.21}_{-0.20}$ days and $T_{0,{\rm c}}=2458776.6\pm 4.6$ BJD (see
the posteriors in Fig. 14 and Table 6). The eccentricities turn out to be
$e_{\rm b}=0.04^{+0.05}_{-0.03}$ and $e_{\rm c}=0.14^{+0.17}_{-0.10}$, but it
is important to note that when they are fixed to zero, the orbital parameters
of TOI-1422 b and TOI-1422 c remain, within 1-$\sigma$, compatible with those
of the eccentric model.

###  3.4 Results

TOI-1422 c’s orbital period explains both the main peak found in the residuals
of the one-planet model (Fig. 6) and in the RV GLS periodogram (Fig. 5); it is
also in 9:4 orbital resonance with the first planet. The difference between
the Bayesian evidence of the two-planet eccentric model and the one-planet
model ($\Delta\ln{\mathcal{Z}}^{2p}_{1p}=5.1$) is barely above the very strong
evidence threshold defined in Kass & Raftery (1995)
($\Delta\ln{\mathcal{Z}}>5$), so even if the existence of candidate planet c
remains unproven, we believe the two-planet model is currently the better one
to explain the 29-days signal observed in the RVs, due to the lack of evidence
for star activity.

Furthermore, the two-planet analysis has been replicated with different
numbers of data points in order to understand how and if new measurements were
impacting the significance of the second planet detection and, as shown in
Fig. 11, both the RV semi-amplitude and the period seem to stabilise after
$\approx 60$ measurements, which matches the beginning of the second
observation season, while the significance of the 29-days peak also grows
(Fig. 18). It is noteworthy to mention that the GLS periodogram of the
residuals of the two-planet model does not show peaks below 50% FAP and hence
does not suggest the presence of additional detectable signals.

A phase-folded plot of both the transit and the radial velocities is shown in
Fig. 9 for the eccentric two-planet model. The radius for TOI-1422 b was
calculated with the transformations provided by Espinoza (2018) and, using the
stellar radius of Sect. 3.1, its revised value turns out to be $R_{\rm
b}=3.96^{+0.13}_{-0.11}\,R_{\oplus}$. Using the stellar radius from Table 1,
we derived the mass of both objects to be $M_{\rm
b}=9.0^{+2.3}_{-2.0}\,M_{\oplus}$ and $M_{\rm
c}\sin{i_{c}}=11.1^{+2.6}_{-2.3}\,M_{\oplus}$. Their final parameters are
reported in Table 2. An independent joint analysis of the HARPS-N radial
velocities and TESS photometry, after the transits have been normalised
through a local linear fitting, was also performed with a DE-MCMC method
(Eastman et al., 2013, 2019), following the same implementation as in Bonomo
et al. (2014, 2015). The obtained results are consistent, within 1-$\upsigma$,
with those reported in Table 2.

In order to evaluate possible transit time variations (TTVs) due to the
influence of candidate TOI-1422 c over TOI-1422 b, we plot the four mid-
transit times minus their expected values (based on the two-planet eccentric
model) in Fig. 8 along with different TTV predictions made with the code
described in Agol & Deck 2016. Unfortunately, there are not enough transits to
draw any conclusion as all the delays are compatible with zero within
1-$\sigma$, therefore further precise monitoring of TOI-1422 b transits is
here encouraged in order to confirm the existence of TOI-1422 c and, overall,
better characterise the planetary system.

Figure 9: Top panel: The TESS phase-folded light curve of TOI-1422 b transits,
compared to the best-fitting model. Bottom panels: HARPS-N RV data of TOI-1422
phase-folded to the period of planet b (middle) and candidate c (bottom),
along with their residuals over the model. The red circles represent the
average value of phased RV data points. Figure 10: Top panel: PDC-SAP light
curve with TOI-1422 b transits highlighted in red and the expected TOI-1422 c
transits, with their uncertainties, highlighted in blue. A single planetary-
transit event has also been marked with a green vertical line and is discussed
at the end of Sect. 3.4). Bottom panel: The single transit-like event as seen
in the PATHOS light curve and the corresponding fit. Table 2: Best fit median
values, with upper and lower 68% credibility bands as errors, of the fitted
and derived parameters for TOI-1422 b and TOI-1422 c, as extracted from the
posterior distribution of the two-planet eccentric model (Table 6 and Fig.
14).

| TOI-1422 b | TOI-1422 c
---|---|---
Transit and orbital parameters |  |
$K$ (m s-1) | $2.47^{+0.50}_{-0.46}$ | $2.36^{+0.42}_{-0.40}$
$P_{\rm orb}$ (days) | $12.9972\pm 0.0006$ | $29.29^{+0.21}_{-0.20}$
$T_{\rm 0}$ (BJD) | $2\,458\,745.9205^{+0.0012}_{-0.0011}$ | $2\,458\,776.6^{+4.6}_{-4.5}$
$T_{\rm 14}$ (hours) | $4.52\pm 0.16$ | –
$R_{\rm p}/R_{\star}$ | $0.0356^{+0.0007}_{-0.0005}$ | –
$b$ | $0.19^{+0.11}_{-0.10}$ | –
$i$ (deg) | $89.52^{+0.26}_{-0.28}$ | –
$a/R_{\star}$ | $22.72^{+0.31}_{-0.40}$ | $39.05^{+0.50}_{-0.73}$
$q_{1}$ | $0.28^{+0.11}_{-0.08}$ | –
$q_{2}$ | $0.30^{+0.05}_{-0.05}$ | –
$\sqrt{e}\sin\omega$ | $0.018^{+0.108}_{-0.095}$ | $0.120^{+0.221}_{-0.233}$
$\sqrt{e}\cos\omega$ | $-0.149^{+0.153}_{-0.128}$ | $-0.070^{+0.349}_{-0.304}$
Derived parameters |  |
$M_{\rm p}$ ($M_{\oplus}$) | $9.0^{+2.3}_{-2.0}$ | –
$M_{\rm p}\sin{i}$ ($M_{\oplus}$) | – | $11.1^{+2.6}_{-2.3}$
$R_{\rm p}$ ($R_{\oplus}$) | $3.96^{+0.13}_{-0.11}$ | –
$\rho_{\rm p}$ (g cm-3) | $0.795^{+0.290}_{-0.235}$ | –
$\log{g_{p}}$ (cgs) | $2.75^{+0.08}_{-0.14}$ | –
$a$ (AU) | $0.108\pm 0.003$ | $0.185\pm 0.006$
$T_{\rm eq}^{(\ddagger)}$ (K) | $867\pm 17$ | $661\pm 13$
$u_{1}$ | $0.32^{+0.12}_{-0.10}$ | –
$u_{2}$ | $0.21^{+0.10}_{-0.08}$ | –
$e$ | $0.04^{+0.05}_{-0.03}$ | $0.14^{+0.17}_{-0.10}$
$\omega$ (deg) | $153^{+20}_{-56}$ | $99^{+63}_{-64}$

171717(‡) This is the equilibrium temperature for a zero Bond albedo and
uniform heat redistribution to the night side.

Figure 11: RV semi-amplitude $K$ and orbital period $P$, along with their
1-$\sigma$ error bars, for candidate planet c as functions of the number of
data points used for the two-planet eccentric model analysis with juliet.

###  3.5 Other transit events

In the search for TOI-1422 c transits, we found a possible single transit-like
event around 2 458 756.35 BTJD days, as shown in Fig. 10, which can not be
related to either TOI-1422 b or TOI-1422 c. We have fitted this potential
transit using the light curve from the pipeline PATHOS (Nardiello et al.,
2019) and retrieved a possible radius $R_{\rm
d}=2.82_{-0.05}^{+0.38}\,R_{\oplus}$, which is compatible with the transit
depth observed in the PDC-SAP and SAP light curves as well. The duration of
the transit suggests an orbital period longer than TOI-1422 c’s but this is
very uncertain, while the lack of other transits in the TESS light curve
suggests an orbital period between 17-22, or longer than, 35 days, thus
incompatible with TOI-1422 c’s. PATHOS is a PSF-based approach to TESS data
which allows the minimization of dilution effects in crowded environments, and
here it is utilized to extract high-precision photometry of TOI-1422 to
independently confirm the presence of this transit even after the application
of a different neighbour-subtraction technique. Neither the single transit nor
TOI-1422 b transits show correlation with the X,Y pixels and the sky
background signal (Fig. 19), and the single transit depth also does not change
with different photometric apertures (Fig. 20). Nevertheless, the 3-planet
model for the joint transit-RV analysis is not statistically significant and
the lack of other transits makes the suggestion of another candidate
impossible to justify.

However, no transit compatible with the expected $T_{0,{\rm c}}$ and $P_{\rm
c}$ evaluated with the RV and photometry joint analysis, was found in the SPOC
(both SAP and PDC-SAP) light curves, even though a small part of the supposed
transiting window has been missed by _TESS_. When we take into account both
the time-span of _TESS_ light curve and TOI-1422 c expected (non-grazing)
transit duration, the probability that such transits would have been missed
can be estimated to be around 1% and 7%, with respectively 1$\sigma$ and
3$\sigma$ uncertainty on $T_{0,c}$. Other than misaligned orbits, another
possible explanation for the lack of TOI-1422 c transits is that despite its
mass, which is greater than planet b’s, its size could be much smaller (like
the high-density sub-Neptune, BD+20594b of Espinoza et al. 2016), as any
object with a radius approximately below 2.8$R_{\oplus}$ might be disguised in
the light curve noise (as proven by the, so far undetected and uncertain,
single transit-like event). Ultimately, it remains unknown if candidate planet
c is transiting or not, so further high-precision long photometric follow-up
observations will be important to clear up this possibility along with the
nature of the single transit event. The new _TESS_ observations of this
target, during Sector 57, are definitely welcome as they might shed some light
on both matters.

##  4 Discussion

###  4.1 Orbital resonance

As we have seen, candidate c is within 1-$\sigma$, in 9:4 orbital resonance
with planet b. This is likely coincidental since the resonance is 5th-order
and thus very weak unless one of the planets is quite eccentric181818We note
that, even with the $e$’s suggested by the eccentric fits, which are unusually
high compared to most multi-transiting planetary systems according to Xie et
al. 2016, the 9:4 would not be as strong as a first-order resonance. or the
mutual inclination is high. The exact 9:4 (or 2.25) resonance is within
uncertainty perhaps only because the uncertainty of TOI-1422 c’s orbital
period is large compared to the tight period uncertainties of transiting
planets. As a matter of fact, period ratios a bit above 2 have been found
within many exoplanetary systems (Winn & Fabrycky, 2015), but it is also
possible that the 9:4 resonance is actually the result of a resonant chain of
3 planets in 1st-order 3:2 resonances among each other, with the middle one
yet to be seen. If that is the case, since the period ratios of Kepler planets
near first order resonances are usually slightly wide of resonance, the likely
orbital period for this unknown exoplanet would be slightly more than 19.5
days, and thus compatible with the observed single transit discussed in Sect.
3.4. Given this orbital period and assuming that an RV semi-amplitude roughly
up to 2 m s-1 might be hidden in the residuals of the two-planet model, this
middle object should not have a mass higher than $\approx 8\,M_{\oplus}$, or a
density higher than $\approx 2$ g cm-3.

###  4.2 Mass-radius diagram and internal structure of planet b

TOI-1422 b is among one of the puffier planets with a density of $\sim 0.8$ g
cm-3, which is close to Saturn’s and, therefore, lower than most exoplanets in
this mass range. It lies towards the upper-left corner of the mass-radius
diagram (Fig. 12), making it very similar to Kepler-36 c (Vissapragada et al.,
2020) and especially to Kepler-11 e (Lissauer et al., 2013), which even shares
the same kind of host star but is on a longer orbit. On one hand, it has a
similar radius compared to Neptune and Uranus in our solar system, whereas, on
the other hand, its mass is only about $50\%-60\%$ that of our ice giants.
Thus, an extensive gaseous envelope, surrounding a massive core, is expected
to be found in TOI-1422 b. More precisely, the mass fraction of this envelope
is expected to be around $10\%-25\%$ of the total mass of the planet (using
the equations of state from Becker et al. 2014), suggesting that the
atmosphere has not been blown away by the stellar wind. The nature of this
extensive envelope as well as its core requires further investigation. For
this purpose, we assess the expected S/N of the JWST/NIRISS
measurements191919From a 10-hour observing program assuming a cloud-free,
solar-metallicity, $H_{2}$-dominated atmosphere. of TOI-1422 b transits
compared to planets of similar sizes, by evaluating the transmission
spectroscopy metric (TSM) defined in Kempton et al. 2018:

$\mbox{TSM}=\mbox{(Scale factor)}\times\frac{R_{\oplus}^{3}\,T_{\rm
eq}}{M_{\oplus}\,R_{\star}^{2}}\times 10^{-0.2\,J}\,,$ (7)

where the scale factor is a dimensionless normalization constant, equal to
1.28 for planets with $2.75<R_{\oplus}<4.0$, and $J$ is the apparent magnitude
of the host star in the $J$ band (a filter that is near the middle of the
NIRISS bandpass). As a result (Fig. 13), TOI-1422 b ranks
fourth202020Following TOI-561 c (Lacedelli et al., 2020, 2022), HD 136352 c
(Kane et al., 2020; Delrez et al., 2021) and HD 63935 b (Scarsdale et al.,
2021) among Neptunes ($2.75<R_{\oplus}<4.0$) orbiting G-F dwarfs
($T_{eff}>5400$K), but being the one with the lowest density, it is definitely
an interesting candidate for atmospheric characterization by the JWST.

Figure 12: Planetary masses and radii of the known transiting exoplanets
(values taken from the Transiting Extrasolar Planet catalogueue, TEPCat, which
is available at http://www.astro.keele.ac.uk/jkt/tepcat/ catalogue)
(Southworth, 2010, 2011) with equilibrium temperature $T_{\rm eq}$ between 600
and 1000 K and host star radius between 0.6 and 1.5 $R_{\sun}$. Different
lines correspond to different mass fractions of relatively cold hydrogen
envelopes. The ice giants of the Solar System are displayed in black-filled
circles. TOI-1422 b is on the low-density envelope of planets with precise
mass/radius estimations ($\sigma M_{p}/M_{p}\leq 30\%$; $\sigma
R_{p}/R_{p}\leq 10\%$), one of the reasons that make it potentially valuable
for transit spectroscopy. Figure 13: Transmission spectroscopy observations
TSM values with JWST over the equilibrium temperature for planets with a
measured mass in the radius range $2.75<R_{\oplus}<4.0$, including TOI-1422 b
(green star). Black filled dots and empty squares identify the sample of
planets around stars with $T_{eff}>5400\,K$ and $T_{eff}<5400\,K$,
respectively.

##  5 Conclusions

In this paper, we have confirmed the planetary nature of the TESS transiting
planet TOI-1422 b, which turns out to be a low-density and warm Neptune-size
planet orbiting around an astrometrically, and overall magnetically, quiet G2
V star. Therefore, TOI-1422 b is the latest addition to the low-populated
range of exoplanets with the size of Neptune but Saturn-like density. In order
to well constrain the mass of TOI-1422 b, a long RV monitoring with more than
a hundred observations was necessary with the HARPS-N instrument at the TNG in
La Palma, which resulted in fully characterized orbital and physical
parameters of this new planetary system. On top of that, our RV measurements
also suggest the presence in the system of a, possibly non-transiting, heavier
candidate planet TOI-1422 c in a weak 9:4 orbital resonance with its inner
brother, which will require further study to validate.

###### Acknowledgements.

We acknowledge the use of public TESS data from pipelines at the TESS Science
Office and at the TESS Science Processing Operations Center. 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. The research has
made use of the SIMBAD database, operated at CDS, Strasbourg, France, NASA’s
Astrophysics Data System and the NASA Exoplanet Archive, which is operated by
the California Institute of Technology, under contract with the National
Aeronautics and Space Administration under the Exoplanet Exploration Program.
This research was also partly supported by the Department of Energy under
awards DE-NA0003904 and DE-FOA0002633 (to S.B.J., principal investigator, and
collaborator Li Zeng) with Harvard University and by the Sandia Z Fundamental
Science Program. This research represents the authors’ views and not those of
the Department of Energy. The work is based on observations made with the
Italian Telescopio Nazionale Galileo (TNG) operated on the island of La Palma
by the Fundacion Galileo Galilei of the INAF (Istituto Nazionale di
Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the
Instituto de Astrofisica de Canarias. This work has also 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. L. M. acknowledges support
from the “Fondi di Ricerca Scientifica d’Ateneo 2021” of the University of
Rome “Tor Vergata”. J.L-B. acknowledges financial support received from “la
Caixa” Foundation (ID 100010434) and from the European Unions Horizon 2020
research and innovation programme under the Marie Slodowska-Curie grant
agreement No 847648, with fellowship code LCF/BQ/PI20/11760023. This research
has also been partly funded by the Spanish State Research Agency (AEI)
Projects No. PID2019-107061GB-C61 and No. MDM-2017-0737 Unidad de Excelencia
“María de Maeztu”- Centro de Astrobiología (INTA-CSIC). We acknowledge
financial contribution from the agreement ASI-INAF n.2018-16-HH.0. D.D.
acknowledges support from the _TESS_ Guest Investigator Program grants
80NSSC21K0108 and 80NSSC22K0185. L.N. acknowledges the support of the ARIEL
ASI-INAF agreement 2021-5-HH.0.

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##  Appendix A HARPS-N RV datapoints

Table 3: HARPS-N RV data points and activity indexes (used in Fig. 5) obtained
with the TERRA reduction pipeline between June 8, 2020 and January 21, 2022.
The four lines in bold highlight the RV data points that have been removed
because they do not fit Chauvenet’s criterion.

$\mathrm{BJD_{\textsf{UTC}}}$ | RV |  ±$1\upsigma_{\textsf{RV}}$ | FWHM |  BIS(†) |  Exp.(∗) | $\mathrm{S/N}$ | $I_{\rm Ca\leavevmode\nobreak\ {\sc ii}}$ | $I_{\rm H\upalpha 06}$ | $I_{\rm He\leavevmode\nobreak\ {\sc i}}$ | $I_{\rm Na\leavevmode\nobreak\ {\sc i}}$ | $I_{\rm Ca\leavevmode\nobreak\ {\sc i}}$ | $I_{\rm H\upalpha 16}$
---|---|---|---|---|---|---|---|---|---|---|---|---
$-2457000\,[days]$ | $\mathrm{\leavevmode\nobreak\ \leavevmode\nobreak\ [m\,s^{-1}]}$ | – | – | $\mathrm{[sec]}$ | – | – | – | – | – | – |  –
2008.69349316 | -7.95 | 1.95 | 7174.86 | -7.56 | 900 | 42.8 | 0.549907 | 0.084427 | 0.097028 | 0.180378 | 0.527652 | 0.683496
2009.71160774 | -3.78 | 2.25 | 7181.28 | -6.09 | 900 | 34.3 | 0.555074 | 0.078803 | 0.097526 | 0.179596 | 0.522912 | 0.63963
2026.66187880 | -4.51 | 2.31 | 7164.91 | 0.894 | 900 | 44.2 | 0.554022 | 0.082134 | 0.100171 | 0.187731 | 0.505988 | 0.687073
2027.72062839 | 4.586 | 2.2 | 7153.54 | -4.7 | 900 | 36.4 | 0.558749 | 0.083288 | 0.107005 | 0.187897 | 0.530667 | 0.687811
2028.69301741 | 3.085 | 2.06 | 7178.27 | -6.91 | 900 | 39.3 | 0.587418 | 0.078941 | 0.098429 | 0.181754 | 0.53219 | 0.657997
2037.70925340 | -1.04 | 2.78 | 7190.11 | 1.317 | 900 | 27.1 | 0.588273 | 0.087717 | 0.10143 | 0.181427 | 0.533928 | 0.670313
2038.71004788 | -6.86 | 2.35 | 7174.7 | -12.8 | 900 | 33.8 | 0.589738 | 0.08641 | 0.102117 | 0.18419 | 0.531931 | 0.671326
2039.71645615 | -2.51 | 2.17 | 7176.79 | -5.84 | 900 | 40.5 | 0.590585 | 0.078021 | 0.099401 | 0.182083 | 0.530063 | 0.658114
2040.70404346 | -0.98 | 2.65 | 7182.67 | -18.1 | 900 | 31.6 | 0.593109 | 0.085039 | 0.099033 | 0.182093 | 0.528559 | 0.650582
2050.69789104 | -1.68 | 1.94 | 7172.47 | -1.01 | 900 | 40.2 | 0.594075 | 0.087503 | 0.099635 | 0.183098 | 0.534144 | 0.666092
2051.69861363 | -4.63 | 1.99 | 7177.72 | -11.5 | 900 | 38.6 | 0.591485 | 0.076251 | 0.099171 | 0.182165 | 0.525447 | 0.654697
2054.72404480 | -1.22 | 3.15 | 7162.98 | -4.59 | 900 | 30.1 | 0.564703 | 0.094168 | 0.100699 | 0.182831 | 0.523618 | 0.663779
2068.61577511 | -3.21 | 6.73 | 7165.8 | -6.58 | 900 | 12.7 | 0.584758 | 0.089195 | 0.098718 | 0.179674 | 0.52911 | 0.641145
2069.66437348 | 1.928 | 1.46 | 7167.86 | -1.31 | 900 | 52.5 | 0.562065 | 0.080459 | 0.103277 | 0.183022 | 0.530759 | 0.690607
2070.70250746 | -3.5 | 1.57 | 7172.32 | -3.47 | 900 | 47.8 | 0.561804 | 0.078625 | 0.09964 | 0.18108 | 0.529827 | 0.668034
2071.70780293 | -0.88 | 2.12 | 7195.44 | -4.27 | 900 | 39.4 | 0.589073 | 0.083154 | 0.099452 | 0.180534 | 0.52584 | 0.65915
2072.70309705 | -2.2 | 1.86 | 7185.56 | -14.3 | 900 | 42.3 | 0.557986 | 0.082279 | 0.098576 | 0.180502 | 0.525855 | 0.654666
2075.63970105 | -3.37 | 1.9 | 7169.71 | 0.109 | 900 | 45.6 | 0.594915 | 0.083487 | 0.100564 | 0.181682 | 0.527055 | 0.685465
2076.71092255 | -6.68 | 2.31 | 7174.73 | -9.8 | 900 | 33.1 | 0.554061 | 0.07931 | 0.103664 | 0.184892 | 0.530864 | 0.665231
2078.68129258 | -6.1 | 1.94 | 7175.62 | -16.9 | 900 | 44.2 | 0.588419 | 0.082137 | 0.100369 | 0.183009 | 0.534477 | 0.67444
2079.69077171 | -4.53 | 1.95 | 7176.88 | -8.31 | 900 | 45.3 | 0.560252 | 0.079819 | 0.100953 | 0.183346 | 0.530948 | 0.673179
2091.68437135 | 8.017 | 1.77 | 7185.69 | -8.84 | 900 | 41.1 | 0.549246 | 0.082682 | 0.099392 | 0.181514 | 0.526944 | 0.682402
2092.62561704 | -1.48 | 2.43 | 7172.71 | -7.71 | 900 | 34.1 | 0.581979 | 0.086567 | 0.100783 | 0.183421 | 0.530712 | 0.666342
2093.65087003 | 7.487 | 7.04 | 7164.43 | -16.6 | 900 | 13.4 | 0.564012 | 0.107488 | 0.101191 | 0.188208 | 0.50388 | 0.653611
2094.60798244 | -1.1 | 1.7 | 7168.72 | -6.22 | 900 | 42.4 | 0.5533 | 0.084541 | 0.100049 | 0.184004 | 0.527844 | 0.70821
2095.61169299 | -0.2 | 2.3 | 7152.3 | 3.952 | 900 | 36.4 | 0.585249 | 0.077758 | 0.105133 | 0.187703 | 0.525175 | 0.697218
2096.62962752 | -0.29 | 1.86 | 7166.3 | -11.5 | 900 | 46.4 | 0.541437 | 0.080363 | 0.09865 | 0.182509 | 0.530522 | 0.677679
2097.66412402 | -2.95 | 2.64 | 7180.27 | -4.25 | 900 | 33.7 | 0.55499 | 0.084606 | 0.102655 | 0.182256 | 0.527347 | 0.650539
2099.63975353 | -8.99 | 2.53 | 7178.63 | -1 | 900 | 32.3 | 0.511441 | 0.083238 | 0.09935 | 0.183486 | 0.53274 | 0.664088
2106.65854394 | -2.12 | 2.29 | 7175.27 | -17.8 | 900 | 34.3 | 0.542303 | 0.089511 | 0.101622 | 0.184597 | 0.529016 | 0.670419
2110.71566068 | -6.18 | 3.41 | 7186.25 | 5.57 | 900 | 25.3 | 0.512902 | 0.082166 | 0.101099 | 0.184164 | 0.530842 | 0.67291
2111.55269319 | -6.16 | 1.92 | 7162.99 | -2.12 | 900 | 41 | 0.525014 | 0.080415 | 0.100104 | 0.185605 | 0.529595 | 0.68975
2112.58796961 | -7.39 | 1.88 | 7169.58 | -11.6 | 900 | 40 | 0.534984 | 0.083924 | 0.099656 | 0.182888 | 0.53337 | 0.680635
2119.69014174 | 11.98 | 2.85 | 7190.32 | -9.13 | 900 | 25.8 | 0.545065 | 0.087481 | 0.098395 | 0.182445 | 0.534123 | 0.626241
2120.67679558 | -2.05 | 2.87 | 7208.68 | -6.74 | 900 | 27.8 | 0.528515 | 0.086578 | 0.102032 | 0.186926 | 0.532501 | 0.627348
2125.54442015 | -5.53 | 3.13 | 7169.97 | -3.52 | 900 | 28.1 | 0.555249 | 0.085601 | 0.101368 | 0.186582 | 0.523009 | 0.665638
2126.56835804 | -6.72 | 2.67 | 7174.3 | -16.4 | 1200 | 32.1 | 0.567202 | 0.083452 | 0.099046 | 0.183561 | 0.527745 | 0.651399
2127.63415799 | 0.931 | 2.18 | 7180.78 | -4.92 | 900 | 35.3 | 0.539365 | 0.085257 | 0.098857 | 0.18364 | 0.524582 | 0.655533
2130.59628332 | -1.27 | 3.36 | 7187.06 | -7.56 | 900 | 23.7 | 0.576932 | 0.083983 | 0.098713 | 0.183139 | 0.531644 | 0.651165
2134.62011272 | 0.717 | 3.58 | 7200.6 | -10.9 | 900 | 22.7 | 0.580358 | 0.091434 | 0.100845 | 0.184286 | 0.525551 | 0.661751
2137.55086797 | -6.08 | 1.75 | 7184.04 | -1.32 | 900 | 39.9 | 0.587977 | 0.087623 | 0.097656 | 0.18263 | 0.528299 | 0.65772
2153.56077137 | -4.39 | 7.54 | 7144.15 | -24 | 900 | 14.5 | 0.561764 | 0.077853 | 0.102889 | 0.189554 | 0.527182 | 0.670189
2156.53992496 | -0.87 | 2.45 | 7194.74 | -1.33 | 900 | 34.9 | 0.596947 | 0.089431 | 0.103586 | 0.187334 | 0.531839 | 0.639791
2157.58172574 | 15.06 | 6.49 | 7189.21 | 8.093 | 900 | 15.9 | 0.554652 | 0.093102 | 0.098102 | 0.18062 | 0.525648 | 0.651384
2169.31333487 | 0.713 | 5.22 | 7202.4 | -9.03 | 900 | 19.2 | 0.569926 | 0.086208 | 0.102926 | 0.186679 | 0.532988 | 0.706365
2170.34375539 | -3.22 | 2.75 | 7181.57 | -5.96 | 900 | 26.3 | 0.573886 | 0.092071 | 0.106226 | 0.189905 | 0.524569 | 0.677237
2171.31709456 | 5.22 | 2.1 | 7199.72 | -2.35 | 900 | 40 | 0.567555 | 0.080705 | 0.097921 | 0.182549 | 0.526484 | 0.643617
2172.31291994 | -1.16 | 1.78 | 7183.03 | -13.1 | 900 | 51.7 | 0.602331 | 0.082135 | 0.100467 | 0.184348 | 0.521872 | 0.655749
2189.39389821 | -5.12 | 2.18 | 7187.35 | 0.742 | 900 | 34.9 | 0.553612 | 0.087636 | 0.106345 | 0.186232 | 0.524683 | 0.636639
2190.35888875 | -5.56 | 2.1 | 7181.28 | -1.38 | 900 | 37.8 | 0.553642 | 0.08599 | 0.10436 | 0.185399 | 0.52397 | 0.649955
2192.33613311 | -1.58 | 1.8 | 7190.22 | -4.64 | 900 | 41.2 | 0.594328 | 0.086471 | 0.099245 | 0.182607 | 0.521139 | 0.661821
2212.37354384 | 7.092 | 3.59 | 7203.08 | -9.36 | 900 | 23.9 | 0.593524 | 0.09262 | 0.102766 | 0.183774 | 0.521448 | 0.630614
2213.42063681 | 16.78 | 4.42 | 7211.7 | 3.168 | 900 | 19 | 0.596428 | 0.099179 | 0.101745 | 0.181668 | 0.520206 | 0.654342
2216.40383735 | -8.94 | 2.19 | 7203.82 | -10.3 | 900 | 33.9 | 0.594565 | 0.089359 | 0.100295 | 0.182115 | 0.519823 | 0.629985
2235.36719701 | 6.639 | 2.74 | 7177.77 | 0.334 | 900 | 25.5 | 0.595733 | 0.091527 | 0.101648 | 0.182908 | 0.520753 | 0.626604
2236.31768047 | 5.56 | 2.2 | 7188.86 | -8.92 | 900 | 38.7 | 0.567033 | 0.087864 | 0.100285 | 0.182334 | 0.518575 | 0.610068
2237.32364626 | 4.933 | 3.21 | 7181.9 | -17.9 | 1200 | 28.2 | 0.559051 | 0.086238 | 0.104913 | 0.186654 | 0.525845 | 0.664309
2239.31266431 | 21.6 | 2.85 | 7219.49 | 1.592 | 900 | 31.5 | 0.591982 | 0.088633 | 0.099598 | 0.180553 | 0.524177 | 0.630278
2240.31312271 | 7.368 | 2.31 | 7208.34 | -2.51 | 900 | 37.6 | 0.594946 | 0.081703 | 0.10278 | 0.183971 | 0.522467 | 0.662409
2244.31706817 | 6.052 | 2.84 | 7195.35 | 0.352 | 900 | 32.5 | 0.594812 | 0.084492 | 0.100398 | 0.18777 | 0.524697 | 0.655945
2245.31980936 | 0.237 | 2.24 | 7209.07 | -5.47 | 900 | 34.9 | 0.589965 | 0.087524 | 0.103149 | 0.183372 | 0.519622 | 0.663637
2246.32103509 | 27.1 | 7.78 | 7248.46 | 34.51 | 1200 | 12.7 | 0.594927 | 0.090211 | 0.100742 | 0.184404 | 0.521228 | 0.74003
2412.68868670 | -3.7 | 2.54 | 7163.71 | -3.37 | 900 | 32.8 | 0.550281 | 0.090724 | 0.100293 | 0.183955 | 0.523314 | 0.6709
2413.66378329 | -1.5 | 2.16 | 7179.72 | -6.74 | 900 | 34.4 | 0.558588 | 0.088191 | 0.099293 | 0.1792 | 0.526702 | 0.652927
2414.68886021 | 2.937 | 2.15 | 7166.9 | 4.331 | 900 | 37.1 | 0.555402 | 0.084723 | 0.098655 | 0.183102 | 0.530929 | 0.676626
2416.63114629 | 2.289 | 1.95 | 7191.8 | -11.7 | 900 | 48 | 0.584258 | 0.083699 | 0.099042 | 0.180173 | 0.526418 | 0.68478
2417.65137234 | 3.912 | 1.73 | 7191.95 | -8.14 | 900 | 40.8 | 0.583166 | 0.087257 | 0.098779 | 0.18156 | 0.52857 | 0.664774
2418.65486069 | 4.286 | 3.33 | 7195.12 | -13.2 | 900 | 21.6 | 0.587824 | 0.10659 | 0.10043 | 0.180994 | 0.498526 | 0.662161
2427.73181821 | 1.618 | 2.29 | 7191.43 | -0.36 | 900 | 34.2 | 0.595541 | 0.089463 | 0.100339 | 0.183087 | 0.530979 | 0.649579
2428.66018094 | 0.03 | 3.07 | 7187.2 | -11.9 | 900 | 28.1 | 0.587265 | 0.089537 | 0.098082 | 0.181868 | 0.526651 | 0.668995
2430.66988978 | -0.96 | 2.26 | 7186.32 | -3.5 | 900 | 36 | 0.558113 | 0.085708 | 0.102817 | 0.184679 | 0.529491 | 0.668573
2431.71360583 | -5.64 | 2.64 | 7190.54 | -22.7 | 900 | 32.1 | 0.550955 | 0.07889 | 0.100986 | 0.181203 | 0.529026 | 0.662741
2443.66087057 | 8.667 | 3.5 | 7188.34 | -4.35 | 900 | 25.2 | 0.590039 | 0.092999 | 0.096614 | 0.179042 | 0.526005 | 0.696157
2444.57982389 | 9.299 | 3.32 | 7180.37 | -2.39 | 900 | 26.8 | 0.589859 | 0.091057 | 0.098188 | 0.179557 | 0.522807 | 0.666047
2445.59632825 | 10.81 | 1.86 | 7178.02 | -8.87 | 900 | 36.9 | 0.557645 | 0.085087 | 0.10075 | 0.181234 | 0.52886 | 0.678235
2446.60873425 | 5.874 | 1.87 | 7182.03 | 2.719 | 900 | 45 | 0.552972 | 0.085681 | 0.101935 | 0.183638 | 0.525628 | 0.682197
2447.61736597 | 1.855 | 2.1 | 7181.21 | -2 | 900 | 39.9 | 0.585974 | 0.086557 | 0.101268 | 0.182021 | 0.529423 | 0.6903
2448.59166621 | 2.524 | 1.79 | 7183.26 | -6.53 | 900 | 46.5 | 0.550409 | 0.081722 | 0.099201 | 0.181717 | 0.528627 | 0.676518
2449.61666267 | -2.34 | 1.85 | 7181.85 | -9.02 | 900 | 46.7 | 0.55726 | 0.081361 | 0.100242 | 0.183494 | 0.527072 | 0.693437
2453.58138876 | -5.57 | 2.17 | 7179.71 | -10.5 | 900 | 38.1 | 0.576 | 0.087524 | 0.101564 | 0.182982 | 0.529699 | 0.666481
2454.66732062 | -2.11 | 5 | 7157.53 | -13.6 | 900 | 16.4 | 0.534792 | 0.105417 | 0.106316 | 0.188685 | 0.531533 | 0.672621
2455.68136136 | 1.242 | 2.34 | 7201.87 | -15 | 900 | 37.6 | 0.559782 | 0.085698 | 0.097922 | 0.182038 | 0.527513 | 0.642527
2456.70358427 | 1.194 | 1.71 | 7193.3 | -2.59 | 900 | 39.1 | 0.539261 | 0.083335 | 0.100678 | 0.18192 | 0.526882 | 0.654302
2457.65444875 | 0.232 | 1.53 | 7186.93 | -4.04 | 900 | 45.9 | 0.544456 | 0.082158 | 0.103781 | 0.185289 | 0.52979 | 0.678245
2458.62416725 | 1.797 | 2.1 | 7191.62 | -2.73 | 900 | 42.5 | 0.571849 | 0.083688 | 0.10082 | 0.181891 | 0.525159 | 0.670374
2459.58868774 | 0 | 2.42 | 7166.31 | -9.47 | 900 | 32 | 0.534143 | 0.084143 | 0.101018 | 0.183983 | 0.531303 | 0.671843
2460.68008744 | 2.859 | 2 | 7199.43 | -10 | 900 | 39 | 0.567005 | 0.085291 | 0.09952 | 0.182178 | 0.53004 | 0.641539
2461.61247483 | -3.43 | 2.01 | 7177.71 | -14.6 | 900 | 39.8 | 0.531283 | 0.081882 | 0.101482 | 0.18371 | 0.532312 | 0.662684
2462.59699290 | -1.49 | 2.41 | 7193.46 | -10.2 | 900 | 34 | 0.53251 | 0.091757 | 0.1007 | 0.182059 | 0.53047 | 0.678952
2464.59087617 | 5.604 | 5.45 | 7198.52 | -24.4 | 900 | 17.8 | 0.536789 | 0.095036 | 0.099161 | 0.183787 | 0.528055 | 0.686225
2465.55285533 | 1.953 | 1.95 | 7189.61 | -10.6 | 900 | 37.5 | 0.525385 | 0.081987 | 0.102391 | 0.184078 | 0.528833 | 0.659363
2472.64437339 | 9.088 | 3.09 | 7206.96 | -15.6 | 900 | 28.8 | 0.582423 | 0.088182 | 0.102267 | 0.18471 | 0.526636 | 0.671635
2473.58211801 | -2.46 | 3.71 | 7182.04 | -22.5 | 900 | 24 | 0.548713 | 0.090743 | 0.100974 | 0.184532 | 0.522132 | 0.670676
2475.57083577 | 3.505 | 2.35 | 7184.54 | -16.3 | 1800 | 32.2 | 0.580897 | 0.092011 | 0.101234 | 0.182594 | 0.528772 | 0.656172
2476.56282582 | 7.827 | 1.91 | 7191.68 | -6.25 | 900 | 44 | 0.556952 | 0.083328 | 0.09885 | 0.181388 | 0.523729 | 0.660628
2477.54247641 | 5.777 | 1.95 | 7190.45 | -13.2 | 900 | 40.1 | 0.584375 | 0.084401 | 0.105974 | 0.188686 | 0.528026 | 0.684554
2478.53938306 | 3.887 | 2.42 | 7199.21 | 1.209 | 900 | 33.4 | 0.542519 | 0.091376 | 0.099644 | 0.180595 | 0.522094 | 0.662569
2479.53060547 | 0.796 | 2.4 | 7200.54 | -6.9 | 900 | 35.5 | 0.578876 | 0.086324 | 0.099936 | 0.181075 | 0.527574 | 0.654195
2481.49382164 | 3.548 | 2.82 | 7186.71 | -6.58 | 900 | 31.2 | 0.53899 | 0.091801 | 0.101227 | 0.183502 | 0.527366 | 0.676158
2513.44982261 | 1.106 | 2.07 | 7176.01 | -3.44 | 900 | 41.6 | 0.561363 | 0.088328 | 0.100405 | 0.183709 | 0.523996 | 0.678603
2513.50218232 | -4.57 | 2.69 | 7172.76 | -4.86 | 900 | 33.2 | 0.566953 | 0.08974 | 0.10596 | 0.188059 | 0.524511 | 0.666513
2515.41917706 | -2.34 | 1.93 | 7184.92 | -5.97 | 900 | 41.7 | 0.564153 | 0.089897 | 0.101247 | 0.182411 | 0.526736 | 0.673459
2515.44453511 | -3.02 | 1.76 | 7178.72 | -6.73 | 900 | 49.2 | 0.567076 | 0.088259 | 0.098628 | 0.181831 | 0.520853 | 0.679953
2516.53077561 | -4.93 | 2.92 | 7190.8 | 1.972 | 900 | 31.9 | 0.602149 | 0.092247 | 0.099896 | 0.182444 | 0.527717 | 0.666103
2565.36649887 | 0.845 | 1.64 | 7187.23 | -3.3 | 900 | 54.5 | 0.596593 | 0.088658 | 0.09841 | 0.180042 | 0.522949 | 0.675468
2566.34542939 | -5.47 | 2.15 | 7179.09 | -6.14 | 900 | 45.4 | 0.601739 | 0.091592 | 0.099575 | 0.18055 | 0.525212 | 0.678527
2575.40497038 | 3.93 | 2.12 | 7181.99 | -8.36 | 900 | 40.1 | 0.566264 | 0.087701 | 0.098051 | 0.182169 | 0.523518 | 0.671318
2579.38524573 | 5.059 | 2.18 | 7191.98 | -5.62 | 900 | 38.3 | 0.562049 | 0.089366 | 0.104764 | 0.183719 | 0.525329 | 0.674917
2580.41266409 | 10.09 | 5.97 | 7187.22 | 4.876 | 900 | 19.9 | 0.572011 | 0.107006 | 0.107628 | 0.185394 | 0.52275 | 0.687077
2584.33980519 | 0.821 | 2.95 | 7183.34 | 0.433 | 900 | 29.2 | 0.56581 | 0.092026 | 0.104575 | 0.185035 | 0.514433 | 0.687819
2588.34483394 | 1.991 | 5.05 | 7181.58 | 9.001 | 900 | 19.2 | 0.560173 | 0.094073 | 0.110324 | 0.186892 | 0.524911 | 0.680451
2601.31675212 | 5.918 | 2.03 | 7196.44 | -6.76 | 900 | 45.2 | 0.565656 | 0.087291 | 0.10094 | 0.17769 | 0.523886 |  0.667702

212121(∗) Duration of each individual exposure. (†) Bisector spans; error bars
are twice those of RVs.

##  Appendix B Priors and posteriors

Table 4: Prior volume for the parameters of the one-planet model fit of Sect.
2.3 processed with juliet. $\mathcal{U}(a,\>b)$ indicates a uniform
distribution between $a$ and $b$; $\mathcal{L}(a,\>b)$ a log-normal
distribution, $\mathcal{N}(a,\>b)$ a normal distribution and
$\mathcal{T}(a,\>b)$ a truncated normal distribution (where lower possible
value equals zero) with mean $a$ and standard deviation $b$.

Parameter | Prior distribution
---|---
Keplerian Parameters: |
$\rho_{\star}$ | $\mathrm{[kg/m^{3}]}$ | $\mathcal{N}(1300,\>100)$
$T_{0,b}$ | $\mathrm{[BJD]}$ | $\mathcal{N}(2458745.921,\>0.003)$
$P_{b}$ | $\mathrm{[days]}$ | $\mathcal{N}(12.998,\>0.002)$
$e_{b}^{*}$ |  | $0$
$\omega_{b}^{*}$ |  | $90$
Transit Parameters: |
$R_{\rm p}/R_{\rm\star}$ |  | $\mathcal{U}(0.0,\>1.0)$
$D$ |  | $1.0$
$q_{1}$ |  | $\mathcal{N}(0.31,\>0.30)$
$q_{2}$ |  | $\mathcal{N}(0.25,\>0.10)$
Light curve GP Hyperparameters: |
$\sigma_{\textsf{TESS}}$ | $\mathrm{[ppt]}$ | $\mathcal{L}(10^{-3},\>10)$
$\rho_{\textsf{TESS}}$ | $\mathrm{[days]}$ | $\mathcal{L}(10^{-1},\>10)$
RV parameters: |
$K_{b}$ | $\mathrm{[m/s]}$ | $\mathcal{U}(0.0,\>10.0)$
$\sigma_{\textsf{HARPS-N}}$ | $\mathrm{[m\,s^{-1}]}$ | $\mathcal{U}(0,\>10)$
$A$ | $\mathrm{[m\,s^{-1}\,days^{-1}]}$ | $\mathcal{U}(-1,\>1)$
$B$ | $\mathrm{[m\,s^{-1}]}$ | $\mathcal{U}(-20,\>20)$
222222(∗) In the case of non-null eccentricity, the priors were set as
follows: $(\sqrt{e}\,\sin\omega,\sqrt{e}\,\cos\omega)$ in
$\mathcal{U}(-1.0,\>1.0)$.

Table 5: Prior volume for the parameters of the two-planet model fit of Sect.
2.3 processed with juliet.

Parameter | Prior distribution
---|---
Keplerian Parameters: |
$\rho_{\star}$ | $\mathrm{[kg/m^{3}]}$ | $\mathcal{N}(1300,\>100)$
$T_{0,b}$ | $\mathrm{[BJD]}$ | $\mathcal{N}(2458745.921,\>0.003)$
$P_{b}$ | $\mathrm{[days]}$ | $\mathcal{N}(12.998,\>0.002)$
$T_{0,c}$ | $\mathrm{[BJD]}$ | $\mathcal{N}(2458740,\>2458790)$
$P_{c}$ | $\mathrm{[days]}$ | $\mathcal{U}(1,\>100)$
$(e_{b},e_{c})^{*}$ |  | $0$
$(\omega_{b},\omega_{c})^{*}$ |  | $90$
Transit Parameters: |
$R_{\rm p}/R_{\rm\star}$ |  | $\mathcal{U}(0.0,\>1.0)$
$D$ |  | $1.0$
$q_{1}$ |  | $\mathcal{N}(0.31,\>0.30)$
$q_{2}$ |  | $\mathcal{N}(0.25,\>0.10)$
Light curve GP Hyperparameters: |
$\sigma_{\textsf{TESS}}$ | $\mathrm{[ppt]}$ | $\mathcal{L}(10^{-3},\>10)$
$\rho_{\textsf{TESS}}$ | $\mathrm{[days]}$ | $\mathcal{L}(10^{-1},\>10)$
RV parameters:  |
$K_{b}$ | $\mathrm{[m/s]}$ | $\mathcal{U}(0.0,\>10.0)$
$K_{c}$ | $\mathrm{[m/s]}$ | $\mathcal{U}(0,\>10)$
$\sigma_{\textsf{HARPS-N}}$ | $\mathrm{[m\,s^{-1}]}$ | $\mathcal{U}(0,\>10)$
$A$ | $\mathrm{[m\,s^{-1}\,days^{-1}]}$ | $\mathcal{U}(-1,\>1)$
$B$ | $\mathrm{[m\,s^{-1}]}$ | $\mathcal{U}(-20,\>20)$
232323(∗) In the case of non-null eccentricity, the priors were set as
follows: $(\sqrt{e}\,\sin\omega,\sqrt{e}\,\cos\omega)$ in
$\mathcal{U}(-1.0,\>1.0)$.

Table 6: Posteriors result for the parameters of the two-planet eccentric model fit of Sect. 2.3 processed with juliet. Parameter | Value ($\pm 1\sigma$)
---|---
Keplerian Parameters: |
$\rho_{\star}$ | $\mathrm{[kg/m^{3}]}$ | $1312_{-68}^{+55}$
$a_{b}/R_{\star}$ |  | $22.72_{-0.40}^{+0.31}$
$a_{c}/R_{\star}$ |  | $39.05_{-0.73}^{+0.50}$
$T_{0,b}$ | $\mathrm{[BJD]}$ | $2458745.9205_{-0.0011}^{+0.0012}$
$P_{b}$ | $\mathrm{[days]}$ | $12.9972\pm 0.0006$
$T_{0,c}$ | $\mathrm{[BJD]}$ | $2458776.6_{-4.6}^{+4.5}$
$P_{c}$ | $\mathrm{[days]}$ | $29.29_{-0.20}^{+0.21}$
Transit Parameters: |
$R_{\rm p_{b}}/R_{\rm\star}$ |  | $0.0356\pm_{-0.0005}^{+0.0007}$
$q_{1}$ |  | $0.28^{+0.11}_{-0.08}$
$q_{2}$ |  | $0.30^{+0.05}_{-0.05}$
$b_{b}$ |  | $0.19_{-0.10}^{+0.11}$
$i_{b}$ | [deg] | $89.52^{+0.26}_{-0.28}$
Light curve GP Hyperparameters: |
$\sigma_{\textsf{TESS}}$ | $\mathrm{[ppt]}$ | $0.19_{-0.02}^{+0.03}$
$\rho_{\textsf{TESS}}$ | $\mathrm{[days]}$ | $0.76_{-0.15}^{+0.19}$
RV parameters:  |
$K_{b}$ | $\mathrm{[m/s]}$ | $2.47_{-0.46}^{+0.50}$
$K_{c}$ | $\mathrm{[m/s]}$ | $2.36_{-0.40}^{+0.42}$
$\sigma_{\textsf{HARPS-N}}$ | $\mathrm{[m\,s^{-1}]}$ | $2.93_{-0.32}^{+0.35}$
$A$ | $\mathrm{[m\,s^{-1}\,days^{-1}]}$ | $0.0110\pm 0.0015$
$B$ | $\mathrm{[m\,s^{-1}]}$ | $-9.1\pm 1.3$

##  Appendix C Corner plots

Figure 14: Corner plot for the posterior distribution of the joint transit and
RV analysis of Sect. 3.3 in the case of 2 planets, elaborated with juliet.

##  Appendix D Additional plots

Figure 15: The FWHM and $\log R^{\prime}_{\rm HK}$ are plotted over time
respectively in the upper and lower panel, along with their linear trends
(orange line) and average value (dashed gray line). Figure 16: Close-up look
of the RV GLS periodogram, executed with the publicly available tool Exo-
Striker (Trifonov 2019; https://github.com/3fon3fonov/exostriker) after the
removal of a linear trend. The two vertical blue lines, around the 29-days
signal (indicated by a vertical yellow line), show the main peak aliases due
to the two highest frequencies of the window function respectively in the
upper and bottom panel. The three horizontal dotted lines represent the
$10\%$, $1\%$ and $0.1\%$ FAP levels. Figure 17: Window function of the
HARPS-N RV measurements, as evaluated with Exo-Striker. The two highest peaks,
excluding the 1-day peak and frequencies close to zero, are indicated by the
respective labels. Figure 18: Unnormalized GLS power for different number of
HARPS-N observations. The power of the 29-day signal increases with more
observations. The red and green dashed vertical lines indicate respectively
TOI-1422 b and TOI-1422 c orbital periods, while the horizontal dashed lines
signal respectively the 10% and 1% confidence levels (evaluated with the
bootstrap method). Figure 19: TOI-1422 b transits, as seen with PATHOS, folded
on the first row of the left column and the single transit event on the right
one, with X/Y and the sky background on the following rows, showing no
correlation with the transits. Figure 20: Single transit depth from PATHOS in
different apertures, with the three rows showing the transit depth at aperture
radius 2, 3 and 4 pixels respectively.