11institutetext:  INAF - Osservatorio Astrofisico di Torino, via Osservatorio
20, 10025 Pino Torinese, Italy 22institutetext: Département d’astronomie de
l’Université de Genève, Chemin Pegasi 51, 1290 Versoix, Switzerland
33institutetext: Dipartimento di Fisica, Università degli Studi di Torino, via
Pietro Giuria 1, 10125 Torino, Italy 44institutetext: School of Physics &
Astronomy, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK
55institutetext: KICC & Astrophysics Group, Cavendish Laboratory, University
of Cambridge, J.J. Thomson Avenue, Cambridge CB3 0HE, UK 66institutetext:
Dipartimento di Fisica, Università degli Studi di Milano, Via Celoria 16,
I-20133 Milano, Italy 77institutetext: Dipartimento di Fisica e Astronomia
“Galileo Galilei”, Università degli Studi di Padova, Vicolo dell’Osservatorio
3, 35122 Padova, Italy 88institutetext: INAF - Osservatorio Astronomico di
Padova, vicolo dell’Osservatorio 5, 35122, Padova, Italy 99institutetext: DTU
Space, National Space Institute, Technical University of Denmark, Elektrovej
328, DK-2800 Kgs. Lyngby, Denmark 1010institutetext: Astrophysics Group,
University of Exeter, Exeter EX4 2QL, UK 1111institutetext: Université Côte
d’Azur, Observatoire de la Côte d’Azur, CNRS, Laboratoire Lagrange, Bd de
l’Observatoire, CS 34229, 06304 Nice cedex 4, France 1212institutetext: Center
for Astrophysics — Harvard & Smithsonian, 60 Garden Street, Cambridge, MA
02138, USA 1313institutetext: INAF - Osservatorio Astronomico di Cagliari, via
della Scienza 5, 09047 Selargius, Italy 1414institutetext: INAF - Osservatorio
Astronomico di Brera, Via E. Bianchi 46, 23807 Merate, Italy
1515institutetext: Fundación Galileo Galilei - INAF, Rambla José Ana Fernandez
Pérez 7, E-38712 Breña Baja, Tenerife, Spain 1616institutetext: INAF -
Osservatorio Astronomico di Palermo, Piazza del Parlamento, 1, 90134 Palermo,
Italy 1717institutetext: Instituto de Astrofísica de Canarias, C/Vía Láctea
s/n, E-38205 La Laguna (Tenerife), Canary Islands, Spain 1818institutetext:
Departamento de Astrofísica, Univ. de La Laguna, Av. del Astrofísico Francisco
Sánchez s/n, E-38205 La Laguna (Tenerife), Canary Islands, Spain
1919institutetext: SUPA, Institute for Astronomy, University of Edinburgh,
Blackford Hill, Edinburgh EH9 3HJ, UK 2020institutetext: Centre for Exoplanet
Science, University of Edinburgh, Edinburgh EH9 3FD, UK 2121institutetext:
Department of Physics and Kavli Institute for Astrophysics and Space Research,
Massachusetts Institute of Technology, Cambridge, MA 02139, USA

# Cold Jupiters and improved masses in 38 Kepler and K2 small planet systems
from 3661 HARPS-N radial velocities.

No excess of cold Jupiters in small planet systems.

A. S. Bonomo 11 X. Dumusque 22 A. Massa 33 A. Mortier 4455 R. Bongiolatti 66
L. Malavolta 7788 A. Sozzetti 11 L. A. Buchhave 99 M. Damasso 11 R. D.
Haywood, STFC Ernest Rutherford Fellow1010 A. Morbidelli 1111 D. W. Latham
1212 E. Molinari 1313 F. Pepe 22 E. Poretti 14141515 S. Udry 22 L. Affer 1616
W. Boschin 1515 17 17 18 18 D. Charbonneau 1212 R. Cosentino 1515 M.
Cretignier 22 A. Ghedina 1515 E. Lega 1111 M. López-Morales 1212 M. Margini 77
A. F. Martínez Fiorenzano 1515 M. Mayor 22 G. Micela 1616 M. Pedani 1515 M.
Pinamonti 11 K. Rice 19192020 D. Sasselov 1212 R. Tronsgaard 99 A. Vanderburg
2121<EMAIL_ADDRESS>
(Received 21 February 2023 / Accepted 6 April 2023)

The exoplanet population characterized by relatively short orbital periods
($P<100$ d) around solar-type stars is dominated by super-Earths and sub-
Neptunes. However, these planets are missing in our Solar System and the
reason behind this absence is still unknown. Two theoretical scenarios invoke
the role of Jupiter as the possible culprit: Jupiter may have acted as a
dynamical barrier to the inward migration of sub-Neptunes from beyond the
water iceline; alternatively, Jupiter may have considerably reduced the inward
flux of material (pebbles) required to form super-Earths inside that iceline.
Both scenarios predict an anti-correlation between the presence of small
planets and that of cold Jupiters in exoplanetary systems.
To test that prediction, we homogeneously analyzed the radial-velocity
measurements of 38 Kepler and K2 transiting small planet systems gathered over
nearly ten years with the HARPS-N spectrograph, as well as publicly available
radial velocities collected with other facilities. We used Bayesian
differential evolution Markov chain Monte Carlo techniques, which in some
cases were coupled with Gaussian process regression to model non-stationary
variations due to stellar magnetic activity phenomena. We detected five cold
Jupiters in three systems: two in Kepler-68, two in Kepler-454, and a very
eccentric one in K2-312. We also found linear trends caused by bound
companions in Kepler-93, Kepler-454, and K2-12, with slopes that are still
compatible with a planetary mass for outer bodies in the Kepler-454 and K2-12
systems.
By using binomial statistics and accounting for the survey completeness, we
derived an occurrence rate of $9.3^{+7.7}_{-2.9}\leavevmode\nobreak\ \%$ for
cold Jupiters with $0.3-13\leavevmode\nobreak\ \rm M_{\rm Jup}$ and $1-10$ AU,
which is lower but still compatible at $1.3\sigma$ with the value measured
from radial-velocity surveys for solar-type stars, regardless of the presence
or absence of small planets. The sample is not large enough to draw a firm
conclusion about the predicted anti-correlation between small planets and cold
Jupiters; nevertheless, we found no evidence of previous claims of an excess
of cold Jupiters in small planet systems.
As an important byproduct of our analyses, we homogeneously determined the
masses of 64 Kepler and K2 small planets, reaching a precision better than 5,
7.5, and 10$\sigma$ for 25, 13, and 8 planets, respectively. Finally, we
release the 3661 HARPS-N radial velocities used in this work to the scientific
community. These radial-velocity measurements mainly benefit from an improved
data reduction software that corrects for subtle prior systematic effects.

###### Key Words.:

Planetary systems – Planets and satellites: individual (Kepler-10, Kepler-19,
Kepler-20, Kepler-21, Kepler-22, Kepler-37, Kepler-68, Kepler-78, Kepler-93,
Kepler-102, Kepler-103, Kepler-107, Kepler-109, Kepler-323, Kepler-409,
Kepler-454, Kepler-538, Kepler-1655, Kepler-1876, K2-2/HIP 116454, K2-3,
K2-12, K2-36, K2-38, K2-79, K2-96/HD 3167, K2-106, K2-110, K2-111, K2-131,
K2-135/GJ 9827, K2-141, K2-167, K2-222, K2-262/Wolf 503, K2-263, K2-312/HD
80653, K2-418/EPIC-229004835) – Planets and satellites: detection – Planets
and satellites: formation – Planets and satellites: fundamental parameters –
Techniques: radial velocities.

##  1 Introduction

One of the most striking findings from the detection of almost 4000 transiting
planets so far is that the most common type of exoplanets in relatively close
orbits (orbital periods $P<100$ d) around solar-type stars are small planets
(SPs) with radii of $1<R_{\rm p}<4\leavevmode\nobreak\ \rm R_{\oplus}$. These
are hosted by about half of the solar-type stars in the Milky Way (e.g., Winn
& Fabrycky 2015 and references therein) and can be subdivided into two main
classes: i) high-density super-Earths with $1.0\lesssim R_{\rm p}\lesssim
1.7\leavevmode\nobreak\ \rm R_{\oplus}$ and a rocky composition; and ii)
lower-density sub-Neptunes with $1.7\lesssim R_{\rm p}\lesssim
4.0\leavevmode\nobreak\ \rm R_{\oplus}$, which are thought to be ice-rich
and/or have an atmospheric envelope of hydrogen and helium (e.g., Zeng et al.
2019). The rocky and ice-rich compositions, if not considerably altered by
post-formation processes such as core-powered mass loss (e.g., Ginzburg et al.
2018; Gupta & Schlichting 2019) or atmospheric photo-evaporation for the
hottest planets (e.g., Lopez & Fortney 2014; Owen & Wu 2017), would (in
principle) reflect different formation locations, namely, inside or beyond the
water iceline (at $\sim 1-3$ AU around a solar-type star), respectively.
Super-Earths and sub-Neptunes appear to be separated by the so-called radius-
valley at $R_{\rm p}\sim 1.7\leavevmode\nobreak\ \rm R_{\oplus}$ in the
distribution of planet radii from the _Kepler_ space mission (Fulton et al.,
2017; Van Eylen et al., 2018; Zeng et al., 2021), even though there may be a
certain mixing of the two populations that occurs (e.g., Lacedelli et al.
2021, 2022).

Despite being very abundant overall, super-Earths and sub-Neptunes are absent
in our Solar System – and the reason for that remains an open question. Some
theoretical efforts to explain the lack of SPs in the Solar System have been
undertaken, for instance, by Izidoro et al. (2015) and Lambrechts et al.
(2019). While relying on different frameworks, both works invoke the influence
of Jupiter as the possible culprit and generalize their outcomes to
exoplanetary systems, so as to place the Solar System in the exoplanet
context.

The scenario proposed by Izidoro et al. (2015) assumes that planet cores form
preferentially in the proximity of the different icelines (e.g., Drażkowska &
Alibert 2017). The innermost core is expected to grow faster due to the strong
dependence of the accretion timescale on orbital radius. If such a core
becomes a gas giant, it could then block the migration of the outer cores
towards the parent star. Occasionally, one core (or more) could jump over the
giant planet and move closer to the star, with a “jumping” probability
depending on the gas-disk profiles and the initial number and total mass of
the cores. In the simulations carried out by Izidoro et al. (2015), with five
cores and a total mass of $\sim 30\leavevmode\nobreak\ \rm M_{\oplus}$, the
jumping probability was found to be $\lesssim 10-20\%$ (see their Figure 3).

According to this scenario, the early formation of Jupiter in the Solar System
may have prevented the nuclei of Saturn, Uranus, and Neptune from migrating
towards the Sun and, hence, from becoming a compact system of short-period
sub-Neptunes such as those observed by the _Kepler_ , K2, and TESS space
telescopes.

The second framework, described in Lambrechts et al. (2019), is based on the
formation of super-Earths inside the water iceline through pebble accretion
(e.g., Ogihara et al. 2015). With extensive simulations, Lambrechts et al.
(2019) showed that the outcome in the formation of super-Earths is strongly
dependent on the amount of pebble flux drifting inwards from the outer regions
of the protoplanetary disk. Low pebble fluxes would generate Mars-mass
embryos, which may then grow to terrestrial planets at orbital distances
$a\gtrsim 0.4$ AU through mutual collisions after disk dissipation, as has
likely occurred in the Solar System (e.g., Raymond et al. 2014). Conversely,
higher fluxes of pebbles would produce more massive embryos in shorter time.
These could migrate towards their parent star if gas was still present in the
disk, thereby forming a compact system of close-in super-Earths (see their
Figure 3).

When a giant planet forms, it opens a gap in the disk, considerably reducing
or even halting the inward flux of pebbles from the regions outside its orbit.
The formation of Jupiter might thus explain why the Solar System contains no
short-period super-Earths, but terrestrial planets only: Jupiter may have
reduced the flux of material required to form bigger planets within the water
iceline.

Both scenarios above predict an anti-correlation between the presence of
short-period SPs and that of cold Jupiters111We define a cold Jupiter as a
planet with mass between 0.3 and 13 $\rm M_{Jup}$ and semi-major axis between
1 and 10 AU. Even though some studies have shown that giant planets might have
masses greater than the deuterium burning limit of 13 $\rm M_{Jup}$ (e.g.,
Chabrier et al. 2014), the above formation models generally refer to cold
Jupiters with $M_{\rm p}<13\leavevmode\nobreak\ \rm M_{Jup}$. (CJs) in
exoplanetary systems, which could be tested observationally. Previous works by
Zhu & Wu (2018) and Bryan et al. (2019) seem to contradict this anti-
correlation, reporting instead an excess of CJs in SP systems. Specifically,
by analyzing 65 transiting and non-transiting systems with low-mass ($1<M_{\rm
p}<10\leavevmode\nobreak\ \rm M_{\oplus}$) and short-period ($P<100$ d)
planets, Bryan et al. (2019) found an occurrence rate of $f_{\rm
CJ|SP}=36_{-6}^{+7}\leavevmode\nobreak\ \%$ for gaseous giant planets, with
masses of $M_{\rm p}=0.5-13\leavevmode\nobreak\ \rm M_{Jup}$ and semi-major
axes of $a\sim 1-10$ AU. This rate is higher than that found by Wittenmyer et
al. (2020), that is, $f_{\rm CJ}=20.2_{-3.4}^{+6.3}\%$222This value was
determined by summing up the occurrence rates of cold Jupiters from 300 to
10000 d in Table 3 of Wittenmyer et al. (2020)., for $M_{\rm
p}=0.3-13\leavevmode\nobreak\ \rm M_{Jup}$ and $a=1-10$ AU, from the AAT,
HARPS/ESO, HIRES/Keck, and CORALIE radial-velocity (RV) data of solar-type
stars with time spans longer than eight years, irrespective of the presence or
absence of SPs.

Other works, based on RV surveys only, have attempted to estimate the
frequency of CJs in low-mass planet systems and/or the frequency of low-mass
planets in CJ systems, sometimes with apparently conflicting results. For
instance, Barbato et al. (2018) reported no low-mass planets with $M_{\rm
p}=10-30\leavevmode\nobreak\ \rm M_{\oplus}$ and $P<150$ d in 20 CJ systems
around solar-type stars observed with HARPS. On the contrary, based on the
California Legacy Survey conducted with the HIRES/Keck and APF/Lick
spectrographs, Rosenthal et al. (2022) found that planets with $M_{\rm
p}=2-30\leavevmode\nobreak\ \rm M_{\oplus}$ and $P\lesssim 150$ d may occur
approximately twice as frequently around CJ-host stars. However, the latter
authors observed no significant differences in the occurrence of inner low-
mass planets with and without CJ siblings, when limiting their range in mass
to $M_{\rm p}=2-20\leavevmode\nobreak\ \rm M_{\oplus}$.

As also noted by Rosenthal et al. (2022), RV surveys are sensitive to more
massive inner planets than transit surveys, besides determining minimum masses
only. Moreover, the adopted ranges in semi-major axes by Rosenthal et al.
(2022) for both the inner low-mass planets and the outer CJs differ from those
in Bryan et al. (2019), and they include, for instance, warm Jupiters with
$a=0.23-1$ AU. This stands in the way of a straightforward comparison of their
results with those in Bryan et al. (2019).

In the present work, we aim to test the theoretical predictions of Izidoro et
al. (2015) and Lambrechts et al. (2019) by searching for CJs and determining
their occurrence rate in 38 transiting systems: 19 observed by _Kepler_ and 19
by K2; 14 of them are in common with the sample studied by Bryan et al.
(2019). For this purpose, we used 3661 high-precision HARPS-N radial
velocities, 3471 out of which were collected by the HARPS-N Guaranteed Time of
Observations (GTO) consortium, and the remaining 190 RVs by other groups,
mainly for the purpose of determining the masses and densities of _Kepler_ and
K2 transiting planets. Nonetheless, we monitored these systems over the years
specifically to look for outer giant planets. An important byproduct of our RV
analyses is the improvement in the precision and/or accuracy of planetary
masses and densities, thanks to the use of a significant number of yet
unpublished HARPS-N/GTO RVs as well as the first combination of HARPS-N/GTO
RVs with literature measurements.

##  2 Target selection and radial-velocity data

Among the _Kepler_ and K2 systems observed by the HARPS-N/GTO program, we
chose those: i) hosting small (low-mass) planets with a radius of $1<R_{\rm
p}<4\leavevmode\nobreak\ \rm R_{\oplus}$, mass of $1<M_{\rm
p}<20\leavevmode\nobreak\ \rm M_{\oplus}$, and orbital period of $P<100$ d;
and ii) having at least 15 HARPS-N RV measurements for a time span longer than
$\sim 1$ yr. This resulted in the selection of the vast majority of _Kepler_
and K2 systems monitored by the HARPS-N/GTO program. As in the work by Zhu &
Wu (2018), we adopted a wider range in $M_{\rm p}$ than Bryan et al. (2019),
given that several SPs are known to have $M_{\rm p}>10\leavevmode\nobreak\ \rm
M_{\oplus}$.

We also included Kepler-22, even though it meets the second criterion only,
because the possible presence of CJs may provide valuable information on the
architecture of a system with a planet in the habitable zone (Borucki et al.,
2012). However, it was not counted in the computation of $f_{\rm CJ|SP}$ in
short-period SP systems, because Kepler-22b has an orbital period of $289$ d
$>100$ d.

The HARPS-N radial velocities used in this work were extracted with the
original HARPS-N Data Reduction Software version 3.7 from the stellar spectra
obtained before the early failure of the red side of the HARPS-N charge-
coupled device (CCD) in late September 2012 (e.g., Bonomo et al. 2014), and
with the updated DRS version 2.3.5 from the spectra gathered afterwards. This
latter version of the pipeline, adapted from the ESPRESSO spectrograph to
HARPS-N (Dumusque, 2021), computes a more stable wavelength solution through a
careful selection of the lines of the Thorium-Argon calibration lamp, by
avoiding saturated Thorium and Argon lines. It also corrects for: i) possible
RV long-term variations due to changing levels in the flux of the Thorium-
Argon calibration lamp with time; and ii) an offset in the DRS v3.7 data,
occurring at the beginning of June 2020 for the replacement of the Thorium-
Argon calibration lamp.

We observed the majority of the stars in our sample in $\rm OBJ\\_AB$
observing mode, that is, with fiber A on the target and fiber B on the sky to
monitor possible contamination by moonlight. For the brightest stars in our
sample, namely Kepler-21, Kepler-37, Kepler-68, Kepler-93, Kepler-409,
K2-96/HD 3167, K2-167, K2-222, K2-262/Wolf 503, and K2-312/HD 80563, we used
simultaneous calibration with fiber A on the target and fiber B on the
calibration Thorium-Argon or Fabry-Perot lamp, to achieve higher accuracy on
the relative RVs. We extracted the RVs by cross-correlating the spectra with a
stellar template close to the stellar spectral type (e.g., Pepe et al. 2002).
The only exception is the early-M-late-K-dwarf K2-3, for which we used the
TERRA software (Anglada-Escudé & Butler, 2012) to overcome the issue of
distorted cross-correlation functions (CCFs) for cooler stars (Rainer et al.,
2020), thereby achieving a reduced RV scatter (Damasso et al., 2018). For
Kepler-10, we performed an additional reduction using the Yarara-v2 tool
(Cretignier et al., 2021, 2022), because it proved to slightly enhance the
detectability of the planet-induced Doppler signals (Bonomo et al., in prep.).

Possible contamination of the HARPS-N spectra by moonlight was checked
following Malavolta et al. (2017) and corrected by computing the CCF after
subtracting the flux of fiber B from the flux of fiber A. This procedure led
to a reduced RV scatter in a few systems, the most evident cases being
Kepler-19, Kepler-107, and K2-110.

Five systems, namely K2-96/HD 3167, K2-106, K2-111, K2-131, and K2-135/GJ
9827, were also observed with HARPS-N by other groups. To obtain homogeneous
HARPS-N datasets across longer time spans and to take advantage of the
aforementioned improvements of the HARPS-N pipeline, we recomputed all the
HARPS-N RVs with the DRS-v2.3.5 from both the spectra acquired by the GTO and
the other publicly available spectra.

For each system, we also collected the published RVs gathered with
spectrographs other than HARPS-N, such as HIRES/Keck, HARPS/ESO, ESPRESSO/VLT,
PFS/MagellanII, and APF, and analyzed them along with the HARPS-N RVs (see
Sect. 3.2). This combination is needed to improve the constraints on the
presence or lack of CJs, determine more precise (and accurate) orbital and
physical parameters of the detected CJs, and achieve a better precision on the
masses and densities of the inner small (low-mass) planets. A few RV datasets
with a limited number of RVs and/or considerably lower precision than our
HARPS-N RVs were discarded, as they do not yield any improvement in the
orbital solution, while requiring additional free parameters (the radial-
velocity zero point and the uncorrelated jitter term; cf. Sect. 3.2). We did
not use the 71 available HIRES/Keck RVs (Weiss et al., 2016) for the analysis
of the Kepler-10 system, because they tend to reduce the RV semi-amplitudes of
the long-period SPs Kepler-10c and Kepler-10d (Sect. 4.3) compared to those
obtained with the 291 HARPS-N RV measurements only (Bonomo et al., in prep.),
even though the simultaneous modeling of both datasets is mainly driven by the
much more numerous HARPS-N RVs. In fact, Weiss et al. (2016) also showed that
the signal of Kepler-10c is practically undetected in the HIRES data.

We searched for possible outliers in the RV datasets for each system using the
Chauvenet’s criterion333The Chauvenet’s criterion states that a value from a
set of N measurements can be excluded if its deviation from the mean value is
so high that the normal distribution probability that one of the N
measurements with an equal or greater deviation may occur is less than 1/2N
(see, e.g., Bol’shev & Ubaidullaeva 1975). and removed them. For systems with
long-period RV signals, such as long-term slopes and/or Keplerians of CJs, we
applied the Chauvenet’s criterion after removing those signals. Through visual
inspection we checked that this criterion efficiently removes all the clear
outliers.

Table LABEL:table_system_parameters lists the 38 systems in our sample, the
multiplicity (single planet or multiple planets) of the transiting SPs, the
stellar parameters, the number of both total RVs and HARPS-N RVs used in this
work after the removal of outliers, and the total time span of the
observations. For each target, Table 9 reports the epochs, values, and formal
uncertainties of the HARPS-N RVs, the activity indicators of the CCF, namely,
the full width at half maximum (FWHM), the contrast and bisector span of the
CCF, as well as the spectroscopic activity indexes $S$ index and
$\log{R^{{}^{\prime}}_{\rm HK}}$. We warn that correlated variations of the
FWHM and contrast may have occurred at certain epochs due to changes in the
focus of HARPS-N, but do not affect the RVs because the product $\rm[FWHM\cdot
contrast]$ remains constant.

##  3 Data analysis

###  3.1 Updated atmospheric and physical stellar parameters

To maintain uniformity with previous studies of the systems published by the
HARPS-N/GTO consortium, we derived the atmospheric parameters, that is,
effective temperature $T_{\rm eff}$, metallicity [Fe/H], and surface gravity
$\log{g}$, for the host stars Kepler-22, Kepler-109, Kepler-323, Kepler-409,
Kepler-1876, K2-12, K2-38, K2-106, K2-131, and K2-167 from the HARPS-N
spectra. To this end, we employed two independent methods, ARES+MOOG (e.g.,
Mortier et al. 2014) and SPC (Buchhave et al., 2012, 2014), and adopted as
final parameters and uncertainties the average values and error bars provided
by the two methods (see Mortier et al. 2018 for more details).

To determine the stellar physical parameters, we used the EXOFASTv2 tool
(Eastman, 2017; Eastman et al., 2019), which adjusts the stellar radius, mass,
and age through a Bayesian differential evolution Markov chain Monte Carlo
(DE-MCMC) method (Ter Braak, 2006), by simultaneously fitting the stellar
spectral energy distribution (SED) and employing the MESA Isochrones and
Stellar Tracks (MIST) (Paxton et al., 2015). To sample the stellar SED, we
utilized the WISE W1, W2, W3, and W4 infrared magnitudes (Cutri et al., 2021),
the 2-MASS near-infrared J, H, and K magnitudes (Cutri et al., 2003), the
optical Tycho $B_{\rm T}$ and $V_{\rm T}$ magnitudes (Høg et al., 2000),
and/or the APASS Johnson B, V and Sloan $g^{{}^{\prime}}$, $r^{{}^{\prime}}$,
$i^{{}^{\prime}}$ magnitudes (Henden et al., 2016). We imposed Gaussian priors
on the $T_{\rm eff}$ and [Fe/H] atmospheric parameters, as derived from the
analyses of the HARPS-N spectra, as well as on the Gaia EDR3 parallax (Gaia
Collaboration et al., 2021). A uniform prior was instead used for the V-band
extinction, $A_{\rm V}$, with upper limits provided by reddening maps
(Schlegel et al., 1998; Schlafly & Finkbeiner, 2011).

For the remaining 27 systems, we adopted the previously published stellar
parameters, giving preference to those derived using the Gaia parallaxes
and/or asteroseismic analyses of the _Kepler_ light curves (e.g., Kepler-10,
Kepler-454, and Kepler-107). In one case (Kepler-20), we redetermined the
stellar parameters because the prior on the Gaia parallax yields slightly more
precise and accurate stellar radius and mass. The atmospheric and physical
parameters of all the host stars in our sample are listed in Table
LABEL:table_system_parameters.

###  3.2 Orbital fitting

We modeled the RV data of all the 38 systems in our sample with non-
interacting Keplerian orbits and a slope by maximizing a Gaussian likelihood
function (e.g., Ford 2006) through a DE-MCMC technique. The parameters of the
Keplerian model for each planet in a given system are the inferior conjunction
time, $T_{\rm c}$, which is equivalent to the transit midtime for transiting
planets; the orbital period, $P$; the widely adopted parameterization
$\sqrt{e}\cos(\omega)$ and $\sqrt{e}\sin(\omega)$ of eccentricity, $e,$ and
argument of periastron, $\omega$; and the RV semi-amplitude, $K$. We included
linear slopes to check for significant long-term trends and fit for the RV
zero point, $\gamma_{\rm i}$, and jitter term, $\sigma_{\rm jit,i}$, for the
i-th RV dataset gathered with the i-th spectrograph. The jitter terms,
$\sigma_{\rm jit}$, were summed in quadrature to the formal RV uncertainties,
$\sigma_{\rm RV}$, to account for additional white noise of unknown (stellar
or instrumental) origin (e.g., Gregory 2005).

Correlated noise in the RV time series caused by stellar magnetic activity was
modeled through Gaussian process (GP) regression (e.g., Haywood et al. 2014;
Haywood 2015; Grunblatt et al. 2015) within the same DE-MCMC tools, using a
covariance matrix described by either the quasi-periodic (QP) kernel in the
original form of Rasmussen & Williams (2006):

$k(t,t^{\prime})=h^{2}\cdot\exp{\left[-\frac{(t-t^{\prime})^{2}}{2\lambda_{1}^{2}}-\frac{2\sin^{2}{\left(\frac{\pi(t-t^{\prime})}{P_{\rm
rot}}\right)}}{\lambda_{2}^{2}}\right]}+\left[\sigma_{\rm
RV}^{2}(t)+\sigma_{\rm jit}^{2}\right]\cdot\delta_{\rm t,t^{\prime}},$ (1)

or the simpler squared-exponential (SE) kernel

$k(t,t^{\prime})=h^{2}\cdot\exp{\left[-\frac{(t-t^{\prime})^{2}}{2\lambda_{1}^{2}}\right]}+\left[\sigma_{\rm
RV}^{2}(t)+\sigma_{\rm jit}^{2}\right]\cdot\delta_{\rm t,t^{\prime}},$ (2)

where $h$ is the semi-amplitude of the correlated noise, $\lambda_{1}$ is the
correlation decay timescale, $P_{\rm rot}$ is the period of the quasi-periodic
variations, and $\lambda_{2}$ is the inverse complexity harmonic parameter.
The hyper-parameters $\lambda_{1}$, $P_{\rm rot}$, and $\lambda_{2}$ can be
associated respectively with the decay timescale of the active regions, the
stellar rotation period and the complexity of the activity signals (with
$\lambda_{2}\sim 3-5$ approaching simpler sinusoidal signals). We point out
that the parameter $\lambda_{2}$ in Eq. 1 is twice the equivalent parameter
$w$ in other implementations of the quasi-periodic kernel (e.g., Grunblatt et
al. 2015; Damasso et al. 2018).

The QP kernel was used to model correlated noise with a quasi-periodic
behavior, namely, in the presence of stellar rotation signals. The latter were
identified when a periodicity in the Generalized Lomb-Scargle (GLS)
periodogram of the RVs (Zechmeister & Kürster, 2009) with a false alarm
probability $FAP<10^{-3}$ was also found in the _Kepler_ /K2 light curves
and/or in the CCF or _S_ -index activity indicators. This concerns Kepler-21,
Kepler-78, Kepler-102, K2-3, K2-36, K2-131, K2-135/GJ 9827, K2-141, and
K2-312/HD 80653.

The SE kernel was employed for Kepler-93 and K2-2/HIP 116454 to account for RV
variations on timescales longer than the stellar rotation, which are likely
due to magnetic activity cycles and/or shorter Rieger-type cycles (Rieger et
al., 1984). The DE-MCMC analysis with the SE kernel proved to model such
variations efficiently, producing flat and Gaussian-distributed residuals. The
analysis of the same data with the QP kernel yielded very low acceptance
rates, which indicates that the GP-QP model is more complex than needed to
model long-term variations in both systems, leaving the QP hyper-parameters
$P_{\rm rot}$ and $\lambda_{2}$ practically unconstrained.

We imposed several priors on the model parameters as well as on the GP hyper-
parameters in case GP regression was used (i.e., in the presence of correlated
noise). Specifically, we used Gaussian priors on the transit time, $T_{\rm
c}$, and period, $P,$ of the inner transiting SPs, as provided by the transit
ephemerides derived in previous analyses of the _Kepler_ and K2 light curves
(see the second and third columns, and references in Table
LABEL:table_planet_parameters). With regard to orbital eccentricities, we
adopted: i) circular orbits for the closest SPs, whose orbital circularization
times are considerably shorter than the stellar age (e.g., Matsumura et al.
2008, 2010). Null eccentricities for these planets are also consistent with
the observation of their secondary eclipses at orbital phases $\Delta\phi\sim
0.5$ from transits (Singh et al., 2022); ii) half-Gaussian priors with zero
mean and $\sigma_{\rm e}=0.098$ for the transiting SPs in multiple systems,
following the finding of Van Eylen et al. (2019). This prior prevents the fit
from converging to spurious eccentricities, a well-known critical effect
occurring for low signal-to-noise Doppler signals (e.g., Zakamska et al. 2011;
Hara et al. 2019), given that the typical RV semi-amplitudes of the transiting
SPs in our sample are usually comparable to the RV scatter. Spurious high
eccentricities may also be unphysical, as they would lead to dynamical
instabilities (e.g., Giuppone et al. 2013); iii) uniform priors $U[0,1[$ for
the single-transiting SPs, except for Kepler-22b, Kepler-454b, Kepler-409b,
and Kepler-1876b, for which we used half-Gaussian priors with zero mean and
$\sigma_{\rm e}=0.45,0.35,0.42,0.37$, respectively, as derived from
asteroseismic and transit light-curve analyses (Van Eylen et al., 2019); and
iv) uniform priors for the long-period, non-transiting CJs. We adopted
uninformative priors on the RV semi-amplitudes, $K$, zero points, $\gamma$,
jitter terms, $\sigma_{\rm jit}$, and slopes, $\dot{\gamma}$.

Figure 1: Recovery rate (completeness) maps of cold Jupiters. _Top panel_ :
Completeness of Kepler-1876, one of the systems with the worst completeness
mainly due to the relatively short temporal baseline of the HARPS-N radial-
velocity observations. Black and white indicate 0% and 100% recovery rates,
respectively. _Middle panel_ : Completeness of Kepler-93, showing a very high
detection rate, given the long time span of observations. _Bottom panel_ :
Mean survey completeness obtained by averaging out the individual
completenesses of the 37 systems.

As for the GP hyper-parameters, we imposed uniform priors with bounds wide
enough to encompass the expected values of $\lambda_{1}$, $P_{\rm rot}$, and
$\lambda_{2}$, and only a lower bound of 0 $\rm m\,s^{-1}$ for $h$. Table 10
lists the adopted priors on the GP hyper-parameters for the systems that
required GP regression to model the non-stationary activity variations along
with the Keplerian signals. For three systems, namely Kepler-102, K2-141, and
K2-132, we further imposed $\lambda_{1}>P_{\rm rot}/2$, otherwise
$\lambda_{1}$ converged to very low values, on the order of a couple of days,
making the quasi-periodic term practically irrelevant and, hence,
unconstrained. Nonetheless, the RV semi-amplitudes obtained with and without
this prior (i.e., by also allowing $\lambda_{1}$ to converge towards very low
values) are fully consistent.

Additional signals attributed to non-transiting planets were searched for in
the RV residuals with GLS periodograms and they were included in the DE-MCMC
RV analysis if i) their $FAP<10^{-3}$; ii) their periodicity does not appear
in the GLS periodograms of any activity indicator; and iii) the difference in
the Bayesian information criterion ($BIC$) (Burnham & Anderson, 2004; Liddle,
2007) between the model with the additional planet and the model without it is
$\Delta BIC>10$ (Kass & Raftery, 1995). Similarly, possible RV long-term
slopes were considered significant if they were detected at more than
$3\sigma$ and the $\Delta BIC$ in favor of the model with the slope is greater
than 10.

We determined the values and the $1\sigma$ uncertainties of the model and
derived parameters from the medians and the 15.87%-84.14% quantiles of their
posterior distributions. For distributions consistent with zero, such as those
of eccentricities or RV semi-amplitudes in cases of non-detection of the
Doppler signals, we provided only the $1\sigma$ upper limits defined as the
0%-68.27% quantiles. In Table LABEL:table_accelerations we report for each
system the HARPS-N systemic velocities and jitter terms, the GP hyper-
parameters in the presence of stellar activity signals, and the linear
accelerations. In Tables 13 and 9, we give the parameters of the non-
transiting CJs and low-mass planets, respectively. In Table
LABEL:table_planet_parameters, we give the parameters of the 64 Kepler and K2
transiting SPs in our sample (see Sect. 4).

###  3.3 Survey completeness

To determine the frequency of CJs in our sample of _Kepler_ and K2 systems, we
first need to evaluate the sensitivity of our survey to the presence of such
planets. Indeed, CJs might not have been detected in some systems because of
limited temporal baselines, poor temporal sampling, and/or relatively low
precision of the RV measurements. Therefore, our measure of the occurrence
rate of CJs has to take the completeness (or recovery rate) of our survey into
account.

The completeness can be estimated with experiments of injection and recovery
of planetary signals for each system, by considering the real times of the
observations and the RV uncertainty of each measurement at time $t$, that is,
$\sigma(t)=\sqrt{\sigma_{\rm RV}^{2}(t)+\sigma_{\rm jit}^{2}}$. Following
Bryan et al. (2019), we simulated signals of CJs in a logarithmic grid of
30x30 cells of planetary mass, $\Delta M_{\rm p}$, versus semi-major axis,
$\Delta a$, covering the ranges of $0.3-20\leavevmode\nobreak\ \rm M_{Jup}$ in
$M_{\rm p}$ and $0.5-20$ AU in $a$. For each cell of a given system, we
simulated 300 RV signals of CJs at the epochs of our RV observations by
randomly varying i) $M_{\rm p}$ and $a$ within the cell bounds; ii) $T_{\rm
c}$ within the orbital period corresponding to $a$ and the stellar mass
$M_{\star}$ (Table LABEL:table_system_parameters) from Kepler’s third law;
iii) $\cos{i}$ from 0 to 1, where $i$ is the orbital inclination; and iv) the
argument of periastron, $\omega,$ from 0 to $2\pi$, while drawing the orbital
eccentricity, $e,$ from a beta distribution (Kipping, 2013). We then shifted
every RV point at time $t$ according to a Gaussian distribution with a mean
equal to the RV value and a standard deviation $\sigma(t)$. For simplicity, we
did not simulate stellar magnetic activity signals, assuming that those
signals would be efficiently modeled with GP regression, as shown in Sect. 4.

To establish the recovery rate in every $\Delta M_{\rm p}$-$\Delta a$ cell, we
fit the injected signals with a slope, a quadratic trend and a Keplerian orbit
(with input parameters close to the simulated ones) and compared these models
with a constant model (i.e., no signal) through the $\Delta BIC$ criterion: if
$\Delta BIC>10$ in favor of the model with the planet-induced signal, we
recorded a detection of the simulated signal, otherwise its non-detection.
Figure 1 shows the completeness of two systems, Kepler-1876 (top panel), and
Kepler-93 (middle panel) as one of the worst and best cases in our sample,
respectively, and the average completeness of the 37 systems (bottom panel).

##  4 Results

###  4.1 Search for and characterization of cold Jupiters

We first present the results of the search for CJs in our survey.

####  4.1.1 Systems with no long-term trends

Altogether 31 out of 38 systems do not show any significant long-term trend as
caused by sufficiently massive outer companions within the completeness limits
(Table LABEL:table_accelerations). The RVs of two of these systems, K2-110
(Osborn et al., 2017) and Kepler-78 (Pepe et al., 2013; Howard et al., 2013),
are shown in Figures 2 and 3 as representative cases. Possible extra noise in
the K2-110 system was taken into account through the white noise term,
$\sigma_{\rm jit}$, only, because the host star is not magnetically active. On
the contrary, GP regression with a quasi-periodic kernel was needed to model
the strongly correlated variations of the active star Kepler-78 (Fig. 3; see
also Grunblatt et al. 2015).

Figure 2: Radial-velocity measurements of K2-110. _Top panel_ : Radial
velocities as a function of time, showing no long-term trends. The blue and
light blue points display the measurements obtained with the HARPS-N and HARPS
spectrographs, respectively, and the black line indicates the Keplerian best-
fit model. _Bottom panel_ : Same radial velocities as in the top panel, but
phase folded at the ephemeris of K2-110b ($P=13.86$ d).

Figure 3: Radial-velocity measurements of Kepler-78. _Top panel_ : Radial
velocities as a function of time, showing variations due to both stellar
activity and the ultra-short-period planet Kepler-78b, but no significant
long-term trends. The blue and red circles indicate the radial-velocity points
gathered with the HARPS-N and HIRES spectrographs, respectively. _Middle
panel_ : Zoom on the first 100 days, mainly showing the non-stationary stellar
activity signal, which was modeled with Gaussian process regression and a
quasi-periodic kernel (black solid line). _Bottom panel_ : Radial velocities
phase-folded at the ephemeris of Kepler-78b ($P=0.35$ d), after removing the
Gaussian process model for the stellar activity variations.

####  4.1.2 Systems with long-term linear trends

Five systems show long-term linear trends, namely Kepler-93 (Dressing et al.
2015; Fig. 6), Kepler-454 (Gettel et al. 2016; Fig. 8), K2-12 (Mayo et al.
2018; Fig. 6), K2-96/HD 3167 (Christiansen et al. 2017; Gandolfi et al. 2017;
Fig. 6), and K2-262/Wolf 503 (Polanski et al., 2021). The trends in Kepler-93,
Kepler-454 and K2-12 are caused by bound companions, while those in K2-96/HD
3167 and K2-262/Wolf 503 are due to stellar magnetic activity, because similar
trends are also seen in the activity indicators (see Fig. 6 for the case of
K2-96/HD 3167; cf. also Bourrier et al. 2022). Gaussian processes with a SE
kernel were used to model the RVs of Kepler-93 to account for long-term
variations overlapping with the linear trend and the signal of the planet
Kepler-93b (Fig. 6). Such variations are likely due to stellar activity, as
they seem to follow those of the _S_ -index activity indicator after 7000 $\rm
BJD_{TDB}-2450000$ with a Pearson correlation coefficient of $0.26\pm 0.03$
(Fig. 6, right panel).

From the temporal baseline, $\Delta T$, of the RV time series and the
amplitude of the trend, $\Delta K=\dot{\gamma}\cdot\Delta T$, we could
estimate the lower limits of orbital period, $P,$ and minimum mass, $M_{\rm
p}\sin{i}$, of the companions producing the trends, by assuming circular or
quasi-circular orbits, namely: $P\geq 4\cdot\Delta T$ and $M_{\rm
p}\sin{i}\geq(\Delta K/28.4\leavevmode\nobreak\ {\rm m\,s^{-1}})\cdot(M_{\rm
s}/{\rm M_{\odot}})^{2/3}\cdot(P/{\rm yr})^{1/3}$. Table 12 reports the lower
limits of the orbital parameters and $M_{\rm p}\sin{i}$ of the outer
companions in the Kepler-93, Kepler-454, and K2-12 systems. The companion
generating the linear trend in Kepler-93 is a brown dwarf or a low-mass star,
having $M_{\rm p}\sin{i}\geq 21\leavevmode\nobreak\ \rm M_{Jup}$, while the
masses of the companions producing the slopes in the Kepler-454 and K2-12
systems are currently compatible with a planetary companion, but further
monitoring is needed to unveil their nature.

Figure 4: Radial-velocity and CaII _S_ -index measurements of Kepler-93. _Left
panel:_ HIRES (red circles) and HARPS-N (blue circles) radial velocities. The
black line shows the best-fit model with a long-term linear slope caused by a
non-planetary companion, a Keplerian orbit for the transiting planet
Kepler-93b ($P=4.73$ d), and Gaussian process regression with a squared
exponential kernel to account for long-term stellar activity variations.
_Right panel_ : Residuals of the HARPS-N radial velocities after subtracting
the linear long-term trend and the signal of Kepler-93b (top), and
measurements of the stellar activity $S$ index (bottom). The increase in both
the RVs and the $S$ index from $\sim 7000$ to $\sim 8300$ $\rm
BJD_{TDB}-2450000$ indicates that the small-amplitude long-term variations
superposed to the linear slope seen in the left panel are likely due to
stellar magnetic activity.

Figure 5: Radial-velocity and activity index measurements of K2-12. _Left
panel:_ HARPS-N radial velocities showing the long-term linear slope due to a
companion of yet unknown nature. The Doppler signal of the transiting planet
K2-12b ($P=8.28$ d) is undetected, hence, only an upper limit to its mass
could be given. _Right panel_ : FWHM of the cross-correlation function (top)
and CaII _S_ -index (bottom) time series showing no significant linear trends.
This would indicate that the radial-velocity linear trend in the left panel is
caused by a physically bound companion.

Figure 6: Radial-velocity and CCF FWHM measurements of K2-96/HD 3167. _Left
panel_ : Radial velocities showing a long-term linear trend: red, green, blue,
and light blue points show the measurements obtained with the HIRES, APF,
HARPS-N, and HARPS spectrographs, respectively. _Right panel_ : Linear trend
observed in the FWHM of the HARPS-N cross-correlation functions. A similar
behavior is observed in the CaII $S$ index. This implies that the slope
observed in the RVs (left panel) is produced by stellar activity, and not by a
physical companion.

####  4.1.3 Systems with Keplerian signals of cold Jupiters

Resolved Keplerian orbits of CJs are observed only in three of the thirty-
eight systems, namely: Kepler-68 (Gilliland et al., 2013), Kepler-454 (Gettel
et al., 2016), and K2-312/HD 80653 (Frustagli et al., 2020).

The two CJs, Kepler-68d and Kepler-454c, with $P=633$ and 524 d and $M_{\rm
p}\sin{i}=0.75$ and 4.51 $\rm M_{Jup}$, were previously discovered by
Gilliland et al. (2013) and Gettel et al. (2016). Moreover, a long-term
quadratic trend in Kepler-68 (Mills et al., 2019) and a slope in Kepler-454
(Gettel et al., 2016) were also found in the RV data, revealing the presence
of additional outer companions in both systems, given that no trends were seen
in the activity indicators. Our analysis not only allowed us to refine the
parameters of the Kepler-68d and Kepler-454c giant planets, but also unveiled
two additional CJs, namely: Kepler-68e and Kepler-454d, with $P\sim 3450$ and
4070 d and minimum masses of 0.27 and 2.31 $\rm M_{Jup}$, respectively (see
also Margini et al. in prep. for Kepler-68). The RVs of both systems as a
function of time and the phase-folded RV signals of the CJs Kepler-68d,
Kepler-68e, Kepler-454c, and Kepler-454d are shown in Figs. 7 and 8.

Figure 7: Radial-velocity measurements of Kepler-68. _Top panel:_ HIRES (red
circles) and HARPS-N (blue circles) radial velocities and the best-fit model
(black line) with four Keplerians corresponding to the two transiting planets
Kepler-68b ($P=5.40$ d) and Kepler-68c ($P=9.60$ d), and the two cold giant
planets Kepler-68d ($P=633$ d $=1.7$ yr) and Kepler-68e ($P\sim 3450$ d $=9.4$
yr). _Middle and bottom panels:_ Phase-folded radial-velocity signals of the
cold giant planets Kepler-68d and Kepler-68e. Note: orbital phases equal to 0
and 1 correspond to inferior conjunction.

Figure 8: Radial-velocity measurements of Kepler-454. _Top left panel:_ HIRES
(red circles) and HARPS-N (blue circles) radial velocities and the best-fit
model (black line) with a long-term linear trend and three Keplerians
corresponding to the transiting planet Kepler-454b ($P=10.57$ d) and the two
cold giant planets Kepler-454c ($P=524$ d $=1.4$ yr) and Kepler-454d ($P\sim
4070$ d $=11.1$ yr). _Top right panel:_ Linear trend after removing the three
Keplerian signals, which is caused by an additional fourth companion of yet
unknown nature. _Bottom panels:_ Phase-folded radial-velocity signals of the
cold giant planets Kepler-454c (left) and Kepler-454d (right). Note: orbital
phases equal to 0 and 1 correspond to inferior conjunctions.  Figure 9:
HARPS-N radial velocities of K2-312 / HD 80653 and the best-fit model (black
line) with two Keplerians, which correspond to the ultra-short-period
transiting planet K2-312b ($P=0.72$ d) and the eccentric cold giant planet
K2-312c ($P\sim 921$ d $=2.5$ yr; $e=0.85$), and Gaussian process regression
with a quasi-periodic kernel.

The RV monitoring of the K2-312 system revealed that the linear trend observed
by Frustagli et al. (2020) is due to a very eccentric CJ, namely, K2-312c,
with $P=921$ d, $M_{\rm p}\sin{i}=5.41\leavevmode\nobreak\ \rm M_{Jup}$, and
$e=0.85$ (Fig. 9; Poretti et al. in prep.). The parameters of the five CJs
with resolved Keplerian orbits and their $1\sigma$ uncertainties are given in
Table 13.

###  4.2 Occurrence rate of cold Jupiters

By considering the three systems with resolved Keplerian orbits (Kepler-68,
Kepler-454, and K2-312) and possibly the K2-12 system, which shows a trend
that is currently compatible with a planetary companion at orbital distance
$a<10$ AU, we can derive the occurrence rate of CJs, $f_{\rm CJ|SP}$, in our
sample. From this sample, we have to remove Kepler-22 for the reason explained
in Sect. 2, which yields a total of 37 _Kepler_ and K2 systems. We used
binomial statistics, namely:

$b(d|N_{\star,\rm eff},f_{\rm CJ|SP})=\frac{N_{\star,\rm
eff}!}{d!(N_{\star,\rm eff}-d)!}f_{\rm CJ|SP}^{d}(1-f_{\rm
CJ|SP})^{N_{\star,\rm eff}-d},$ (3)

where $d$ is the number of systems with detected CJs, that is, $d=3$ or 4,
depending on whether K2-12 is included or not; and $N_{\star,\rm eff}$ is not
just the number of systems in our sample $N_{\star}=37$, but the “effective”
number of stars $N_{\star,\rm eff}=N_{\star}\cdot C$, where $C$ is the average
completeness obtained by computing the mean of the completeness maps of the 37
systems (Fig. 1, bottom panel).

To compare our results with those of Wittenmyer et al. (2020), we computed $C$
and $f_{\rm CJ|SP}$ for the range $\Delta M_{\rm p}=0.3-13\leavevmode\nobreak\
\rm M_{Jup}$ in planetary mass, and different intervals $\Delta a=1-2$, $2-4$,
$4-10$, and $1-10$ AU in semi-major axis (see Table 14). Similarly, we also
derived $C$ and $f_{\rm CJ|SP}$ for $\Delta M_{\rm p}=0.5-13$,
$0.5-20\leavevmode\nobreak\ \rm M_{Jup}$, and $\Delta a=1-10$ and $1-20$ AU,
for comparison with the $f_{\rm CJ|SP}$ found by Bryan et al. (2019). We
report our values and $1\sigma$ error bars of the occurrence rates of CJs as
well as those obtained by Wittenmyer et al. (2020) and Bryan et al. (2019) in
Table 8. In summary, we found $f_{\rm
CJ|SP}=9.3^{+7.7}_{-2.9}\leavevmode\nobreak\ \%$ for d=3 and $f_{\rm
CJ|SP}=12.3^{+8.1}_{-3.7}\leavevmode\nobreak\ \%$ for d=4, namely, by
considering the long-term trend in K2-12 as planetary in origin, for $\Delta
M_{\rm p}=0.3-13\leavevmode\nobreak\ \rm M_{Jup}$ and $\Delta a=1-10$ AU. The
former value is lower than that found by Wittenmyer et al. (2020), namely,
$f_{\rm CJ}=20.2^{+6.3}_{-3.4}\leavevmode\nobreak\ \%$, by a factor of 2;
however, given the large uncertainties, the two measurements are compatible at
1.3$\sigma$. Our $f_{\rm CJ|SP}$ is four times lower than that derived by
Bryan et al. (2019).

###  4.3 Non-transiting low-mass planets

By employing GLS periodograms of the RV residuals and Bayesian model
comparison through the $BIC$ criterion (Sect. 2), we confirmed the presence of
additional RV signals that can be attributed to the non-transiting planets
Kepler-10d ($P=151.0$ d, $M_{\rm p}\sin{i}=12.8\leavevmode\nobreak\ \rm
M_{\oplus}$; Bonomo et al. in prep.), K2-96b/HD 3167b ($P=8.4$ d, $M_{\rm
p}\sin{i}=4.3\leavevmode\nobreak\ \rm M_{\oplus}$; Christiansen et al. 2017),
K2-96e/HD 3167e ($P=96.6$ d, $M_{\rm p}\sin{i}=8.4\leavevmode\nobreak\ \rm
M_{\oplus}$; Bourrier et al. 2022), and K2-111c ($P=15.7$ d, $M_{\rm
p}\sin{i}=11.1\leavevmode\nobreak\ \rm M_{\oplus}$; Mortier et al. 2020);
their orbital parameters and minimum masses are given in Table 9. Concerning
K2-96e/HD 3167e, we found a slightly different $P$ than Bourrier et al.
(2022), namely, $P_{\rm e}=102.09\pm 0.52$ d, which is likely due to a
different treatment of the activity signal: we fit a slope to all the RVs (see
Fig. 6), while Bourrier et al. (2022) included in their MCMC analysis two
activity-decorrelation terms for the HARPS and HARPS-N spectrographs. In any
case, as noted by Bourrier et al. (2022), the observing spectral window allows
for different solutions of $P_{\rm e}$ with multiple peaks in the posterior
distribution (see their Fig. 12).

We report a new planet candidate, Kepler-1876c ($P=15.8$ d, $M_{\rm
p}\sin{i}=11.0\leavevmode\nobreak\ \rm M_{\oplus}$), in the single transiting
system Kepler-1876 (Fig. 10). The GLS periodogram of the HARPS-N RVs of
Kepler-1876 shows a significant periodicity at $P=15.8$ d with FAP of
$5.6\cdot 10^{-5}$, which does not appear in any of the activity indicators.
Nonetheless, the $\Delta BIC$ in favor of the two-planet model is currently
$\Delta BIC=5.3<10$; thus, more RVs are needed to confirm the planetary nature
of this signal, also by checking that its phase and amplitude do not change
with time.

Figure 10: Phase-folded HARPS-N radial velocities (blue dots) of Kepler-1876
at the period $P=15.8$ d of the possible planet candidate Kepler-1876c. The
black solid line shows the best-fit model.

We cannot confirm the signals of the non-transiting planet Kepler-20g
(Buchhave et al., 2016) or the non-transiting candidate K2-2c (Vanderburg et
al., 2015) with respective periods of $P=35$ and 45 d and RV semi-amplitudes
of $K=4.1$ and $\sim 2\leavevmode\nobreak\ \rm m\,s^{-1}$. The former signal
is no longer present after the improvement of the DRS software to extract the
HARPS-N RVs (Sect. 2). The latter was likely due to a combination of the
spectral window of the original RVs in Vanderburg et al. (2015) and an
activity signal with periodicity of $\sim 270$ d, which is also seen in the
GLS periodograms of the _S_ -index and FWHM activity indicators, and was
modeled with a GP-SE approach (see Table LABEL:table_accelerations). The fact
that the 45 d signal could have originated from stellar activity variations
was already discussed and considered by Vanderburg et al. (2015).

As specified in Sect. 3.2, we used non-interacting Keplerians to model RV
planetary signals and GLS periodograms to search for non-transiting planets in
the RV residuals. This implies that additional planets revealed by transit
timing variations (TTVs), with undetectable signal in the RVs only (according
to the criteria given in Sect. 3.2), such as Kepler-19c and d (Malavolta et
al., 2017), were not modeled in this study for uniformity.

###  4.4 Improved physical and orbital parameters of 64 Kepler and K2 small
planets

As mentioned in Sect. 1, the DE-MCMC analyses of new HARPS-N RVs, combined (in
some cases for the first time) with the literature RVs obtained with other
instruments, allowed us to update the orbital and physical parameters of the
64 transiting planets. In particular, from the stellar parameters given in
Table LABEL:table_system_parameters, the transit parameters ($P$, $R_{\rm p}$
and $i$) from the literature444For the systems with updated $R_{\star}$ values
in Table LABEL:table_system_parameters, we recomputed $R_{\rm p}$ from the
$R_{\rm p}/R_{\star}$ transit parameter and $R_{\star}$. In cases where the
impact parameter $b$ only is provided in literature assuming circular orbits,
we derived the orbital inclination as $i=\arccos{(b\cdot R_{\star}/a)}$., and
our updated RV semi-amplitudes $K$, we re-derived the masses, densities, and
surface gravities for all the planets as in Table
LABEL:table_planet_parameters. We also report in the same table the planet
equilibrium temperature, $T_{\rm eq}$, assuming a null Bond albedo and full
redistribution of heat from the dayside to the nightside (e.g., López-Morales
& Seager 2007), and the incident flux, $F_{\rm p}$. Both $T_{\rm eq}$ and
$F_{\rm p}$ were updated in case the stellar $T_{\rm eff}$ needed to be re-
determined from our HARPS-N spectra (Table LABEL:table_system_parameters).

Table LABEL:table_planet_parameters also emphasizes the fundamental
contribution of HARPS-N in determining precise masses and densities to infer
the composition of SPs. Specifically, the masses of 25 planets are determined
with a precision higher than $5\sigma$ ($M_{\rm p}/\sigma_{M_{\rm p}}>5$), 13
of which with $M_{\rm p}/\sigma_{M_{\rm p}}>7.5$, and 8 with $M_{\rm
p}/\sigma_{M_{\rm p}}>10$. Only upper limits on $M_{\rm p}$ are given for 20
planets whose induced Doppler signal was not detected. We note that a
discussion of the planetary compositions from the measure of planetary masses
and densities is beyond the scope of this work555In particular, we refer the
reader to companion papers, which discuss in detail the updated compositions
of Kepler-10b and c (Bonomo et al., in prep.), Kepler-68b and c (Margini et
al., in prep.) and K2-418b/EPIC-229004835b (Tronsgaard et al., in prep.).

##  5 Discussion and conclusions

The value of the occurrence rate of CJs from the sample of 37 Kepler and K2
systems monitored by the HARPS-N/GTO consortium over the long term, namely,
$f_{\rm CJ|SP}=9.3^{+7.7}_{-2.9}\%$, is lower than the frequency of $f_{\rm
CJ}=20.2^{+6.3}_{-3.4}\%$ derived by Wittenmyer et al. (2020) for solar-type
stars from RV surveys, regardless of the possible presence of inner SPs (Table
8). This might hint at the theoretical anti-correlation between the presence
of SPs and CJs predicted by Izidoro et al. (2015) and Lambrechts et al.
(2019). However, the large uncertainty on our $f_{\rm CJ|SP}$ associated with
the inevitably limited target sample does not allow us to draw a firm
conclusion. Moreover, the sample considered by Wittenmyer et al. (2020) likely
contains a certain fraction of stars hosting SPs, which went undetected
through the Doppler method and, thus, a comparison of cold-Jupiter occurrence
rates for stars with and without SPs is not straightforward. Nonetheless,
assuming that the aforementioned anti-correlation does hold, the possible
“contamination” of the sample of Wittenmyer et al. (2020) by SP systems would
tend to make $f_{\rm CJ}$ lower than it really is (in other words, $f_{\rm
CJ}$ might be higher than $\sim 20\%$ for solar-type stars hosting no SPs).

Our results do not support the claim by Bryan et al. (2019) regarding an
excess of CJs in SP systems (see Table 8): according to their $f_{\rm CJ|SP}$,
we should have discovered CJs in $12\pm 2$ of our systems, while we only found
them in 3 systems. The reason for such a discrepancy lies at least in part in
an incorrect interpretation of the planetary origin of some linear trends by
Bryan et al. (2019) (see their Fig. 3): for instance, the trend in the GJ 273
system is due to secular acceleration, the slope in HD 3167 to stellar
activity (Fig. 6), and those in Kepler-93b and Kepler-407b are compatible with
being generated by brown dwarfs or low-mass stellar companions. In addition,
$f_{\rm CJ|SP}$ was computed by Bryan et al. (2019) in a different way,
namely, by assuming a double power law $f_{\rm CJ|SP}=A\cdot M_{\rm
p}^{\alpha}\cdot a^{\beta}$, deriving the coefficients $A$, $\alpha,$ and
$\beta$ with a likelihood approach, and integrating $f_{\rm CJ|SP}$ over the
parameter space. Nonetheless, the exclusion of approximately half of the long-
term trends found by Bryan et al. (2019), since they were not due to CJs,
would considerably reduce their $f_{\rm CJ|SP}$ and make it more compatible
with our estimate.

The present work will be extended to a larger sample, including the TESS SPs
monitored with HARPS-N since 2019 (e.g., Cloutier et al. 2021; Lacedelli et
al. 2021; Naponiello et al. 2022), as well as the K2 and TESS SP systems
observed from the Southern hemisphere with other spectrographs such as
ESPRESSO and HARPS. Performing the same analyses of RV data in Sect. 3.2 on a
sample at least three times as large will allow us to

  * (i)

derive a more precise $f_{\rm CJ|SP}$ in SP systems.

  * (ii)

compute $f_{\rm CJ|SP}$ as a function of the multiplicity of SPs. Indeed, the
predicted anti-correlation between CJs and SPs is expected to be more
pronounced for systems with a higher level of multiplicity of SPs, because
single cores have a higher probability of ”jumping” inside the orbit of the
cold gas giant during their inward migration (see Fig. 3 in Izidoro et al.
2015). Detecting non-transiting planets with RVs (Sect. 4.3), especially in
single transiting systems, is therefore crucial for more detailed studies of
the possible dependence of $f_{\rm CJ|SP}$ on the multiplicity of SPs.

  * (iii)

compute $f_{\rm CJ|SP}$ as a function of the cold-Jupiter multiplicity: if
multiple CJs were formed, it would be (in principle) even more difficult for
the farther icy cores to migrate inward because the multiple CJs would act as
a stronger dynamical barrier than a single CJ. The Jupiter-Saturn pair in the
Solar System might have prevented the Uranus and Neptune cores from migrating
towards the Sun. Nevertheless, the CJ pairs we discovered in the Kepler-68 and
Kepler-454 systems did not hinder the formation of the inner SPs Kepler-68b/c
and Kepler-454b.

  * (iv)

determine $f_{\rm CJ|SP}$ as a function of the composition of SPs, which would
be mainly ice-rich (sub-Neptunes) in the scenario of Izidoro et al. (2015)
(cf. also Zeng et al. 2019) or rocky (super-Earths) in the scenario of
Lambrechts et al. (2019); this is because they are expected to form beyond or
inside the water iceline, respectively. Unveiling a possible anti-correlation
between CJs and predominantly ice-rich or rocky SPs may thus yield important
clues on the mechanisms of formation of short-period SPs. An anti-correlation
between rocky super-Earths and CJs would not be expected instead, if the
former mainly originate inside rings of silicate-rich planetesimals at
approximately 1 AU and then migrate inward (Batygin & Morbidelli, 2023).
Actually, a positive correlation could exist if the mass initially in the ring
of silicate-rich planetesimals, leading to rocky super-Earths, is correlated
with the mass of the ice-rich planetesimals, leading to the formation of the
cores of outer giant planets. In this regard, the improvement in the mass
determination of several of the 64 Kepler and K2 transiting SPs from our RV
analyses (Table LABEL:table_planet_parameters) is useful to distinguish
between rocky and non-rocky compositions.

For these purposes, we recommend continuing to follow up on SP systems, even
after achieving the desired precision on planetary masses for the
investigation of SP compositions. This can be done at a very low cost, given
that just a few RV measurements over the years would, in principle, be
sufficient to search for outer giant planets, provided that the spectrograph
is stable. For the brightest targets, additional, complementary information on
the presence of CJs will also be provided by the Gaia mission (e.g., Holl et
al. 2022), starting with the publication of Data Release 4, slated for the end
of 2025.

The 3661 HARPS-N RVs and activity indicators used for this work are released
to the scientific community via the CDS666http://vizier.u-strasbg.fr/cgi-
bin/VizieR and DACE777https://dace.unige.ch/dashboard/ databases to serve as
the basis for further studies.

###### Acknowledgements.

This work is based on observations made with the Italian Telescopio Nazionale
Galileo (TNG) operated on the island of La Palma by the Fundación Galileo
Galilei of the INAF at the Spanish Observatorio del Roque de los Muchachos of
the Instituto de Astrofisica de Canarias (GTO programme). The HARPS-N project
was funded by the Prodex Program of the Swiss Space Office (SSO), the Harvard-
University Origin of Life Initiative (HUOLI), the Scottish Universities
Physics Alliance (SUPA), the University of Geneva, the Smithsonian
Astrophysical Observatory (SAO), the Italian National Astrophysical Institute
(INAF), the University of St. Andrews, Queen’s University Belfast and the
University of Edinburgh. This paper is based on small-size planetary systems
discovered by the _Kepler_ mission. Funding for the Kepler mission was
provided by the NASA Science Mission directorate. 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 project has received
funding from the European Research Council (ERC) under the European Union’s
Horizon 2020 research and innovation program (grant agreement SCORE No
851555). This work has been carried out within the framework of the National
Centre of Competence in Research PlanetS supported by the Swiss National
Science Foundation under grants 51NF40_182901 and 51NF40_205606. The authors
acknowledge the financial support of the SNSF. We acknowledge financial
contribution from the agreement ASI-INAF n.2018-16-HH.0. F.P.E. would like to
acknowledge the Swiss National Science Foundation (SNSF) for supporting
research with HARPS-N through the SNSF grants nr. 140649, 152721, 166227 and
184618. R.D.H. is funded by the UK Science and Technology Facilities Council
(STFC)’s Ernest Rutherford Fellowship (grant number ST/V004735/1).

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Table 1: Kepler and K2 systems in our sample. System | Multiple/Single | $M_{\rm s}$ | $R_{\rm s}$ | $T_{\rm eff}$ | $\rm[Fe/H]$ | Age | $N_{\rm RV}$ | $N_{\rm RV}$ | $N_{\rm Dat}$ | Duration | System and/or literature
---|---|---|---|---|---|---|---|---|---|---|---
| Transiting System |  [$\mathrm{M}_{\odot}$] |  [$\mathrm{R}_{\odot}$] | [K] | [dex] | [Gyr] | tot | HN |  | [day] | data reference
Kepler-10 | m | $0.910\pm 0.021$ | $1.065\pm 0.009$ | $5708\pm 28$ | $-0.15\pm 0.04$ | $10.6_{-1.3}^{+1.5}$ | 291 | 291 |  21 | 4021 | 1
Kepler-19 | s | $0.936\pm 0.040$ | $0.859\pm 0.018$ | $5541\pm 60$ | $-0.13\pm 0.06$ | $1.9\pm 1.7$ | 104 | 104 |  21 | 2912 | 2, 3
Kepler-20 | m | $0.929\pm 0.053$ | $0.9164_{-0.0077}^{+0.0087}$ | $5495\pm 50$ | $0.07\pm 0.08$ | $5.6_{-3.5}^{+4.5}$ | 161 | 131 | 2 | 3669 |  4, this work2
Kepler-21 | s | $1.408_{-0.030}^{+0.021}$ | $1.902_{-0.012}^{+0.018}$ | $6305\pm 50$ | $-0.03\pm 0.10$ | $2.84\pm 0.35$ | 98 | 98 | 1 | 1972 | 5, 6
Kepler-22 | s | $0.857_{-0.043}^{+0.051}$ | $0.869\pm 0.011$ | $5596\pm 61$ | $-0.255\pm 0.065$ | $7.0_{-4.2}^{+4.0}$ | 70 | 55 |  31 | 3913 |  7, this work2
Kepler-37 | m | $0.790_{-0.030}^{+0.033}$ | $0.7890_{-0.0056}^{+0.0064}$ | $5357\pm 68$ | $-0.36\pm 0.05$ | $7.6_{-3.1}^{+3.4}$ | 145 | 114 | 2 | 3424 | 7, 8
Kepler-68 | m | $1.057_{-0.020}^{+0.022}$ | $1.2564\pm 0.0084$ | $5847\pm 75$ | $0.11\pm 0.06$ | $6.84_{-1.04}^{+0.90}$ | 225 | 143 | 2 | 4521 | 9, 10
Kepler-78 | s | $0.779_{-0.046}^{+0.032}$ | $0.7475_{-0.0078}^{+0.0077}$ | $5058\pm 50$ | $-0.18\pm 0.08$ | N.A. | 201 | 117 | 2 | 2312 | 11, 12, 13
Kepler-93 | s | $0.911\pm 0.033$ | $0.919\pm 0.011$ | $5669\pm 75$ | $-0.18\pm 0.10$ | $6.6\pm 0.9$ | 153 | 121 | 2 | 4435 | 14, 15
Kepler-102 | m | $0.803\pm 0.021$ | $0.724\pm 0.018$ | $4909\pm 98$ | $0.11\pm 0.04$ | $1.1_{-0.5}^{+3.6}$ | 146 | 74 |  31 | 3746 | 16
Kepler-103 | m | $1.212_{-0.033}^{+0.024}$ | $1.492_{-0.022}^{+0.024}$ | $6009\pm 64$ | $0.16\pm 0.04$ | N.A. | 60 | 60 | 1 | 1594 | 17
Kepler-107 | m | $1.238\pm 0.029$ | $1.447\pm 0.014$ | $5854\pm 61$ | $0.321\pm 0.065$ | $4.29_{-0.56}^{+0.70}$ | 121 | 121 | 1 | 2180 | 18
Kepler-109 | m | $1.094_{-0.078}^{+0.086}$ | $1.387\pm 0.021$ | $5950\pm 62$ | $-0.020\pm 0.065$ | $6.2_{-2.2}^{+2.7}$ | 66 | 51 | 2 | 3663 |  7, this work2
Kepler-323 | m | $1.015_{-0.070}^{+0.072}$ | $1.120_{-0.018}^{+0.020}$ | $6004\pm 71$ | $-0.14\pm 0.07$ | $5.4_{-2.8}^{+3.4}$ | 48 | 48 | 1 | 2237 |  this work2
Kepler-409 | s | $0.913_{-0.048}^{+0.050}$ | $0.897_{-0.011}^{+0.012}$ | $5421\pm 64$ | $0.05\pm 0.07$ | $5.8_{-3.6}^{+4.5}$ | 65 | 40 | 2 | 3362 |  7, this work2
Kepler-454 | s | $1.03_{-0.03}^{+0.04}$ | $1.066\pm 0.012$ | $5687\pm 49$ | $0.32\pm 0.08$ | $5.25_{-1.39}^{+1.41}$ | 147 | 111 | 2 | 4404 | 19
Kepler-538 | s | $0.892_{-0.035}^{+0.051}$ | $0.8717_{-0.0061}^{+0.0064}$ | $5534\pm 61$ | $-0.09\pm 0.065$ | $5.3_{-3.0}^{+2.4}$ | 111 | 85 | 2 | 3336 | 20
Kepler-1655 | s | $1.03\pm 0.04$ | $1.03\pm 0.02$ | $6148\pm 71$ | $-0.24\pm 0.05$ | $2.56\pm 1.06$ | 97 | 97 | 1 | 1566 | 21
Kepler-1876 | s | $1.187_{-0.088}^{+0.075}$ | $1.477_{-0.022}^{+0.021}$ | $6104\pm 57$ | $0.01\pm 0.06$ | $4.4_{-1.4}^{+2.1}$ | 70 | 70 | 1 | 464 |  this work2
K2-2 / HIP 116454 | s | $0.755\pm 0.027$ | $0.716\pm 0.024$ | $5089\pm 50$ | $-0.16\pm 0.08$ | N.A. | 108 | 108 | 2 | 2367 | 22
K2-3 | m | $0.62\pm 0.06$ | $0.60\pm 0.06$ | $3835\pm 70$ | $-0.01\pm 0.09$ | $>1$ | 327 | 195 | 3 | 898 | 23
K2-12 | s | $0.965_{-0.048}^{+0.064}$ | $1.117\pm 0.014$ | $5672\pm 63$ | $0.00\pm 0.07$ | $8.9_{-3.3}^{+3.0}$ | 50 | 50 | 1 | 2535 |  this work2
K2-36 | m | $0.79\pm 0.01$ | $0.718_{-0.006}^{+0.008}$ | $4916\pm 37$ | $-0.09_{-0.04}^{+0.06}$ | N.A. | 86 | 86 | 1 | 1865 | 24
K2-38 | m | $1.054_{-0.063}^{+0.070}$ | $1.141\pm 0.012$ | $5705\pm 66$ | $0.235\pm 0.070$ | $6.1_{-3.1}^{+3.4}$ | 97 | 40 | 4 | 1576 |  25, this work2
K2-79 | s | $1.066_{-0.070}^{+0.057}$ | $1.265_{-0.027}^{+0.041}$ | $5897\pm 118$ | $0.035\pm 0.060$ | $6.5\pm 1.3$ | 77 | 77 | 1 | 1517 | 26
K2-96 / HD 3167 | m | $0.837_{-0.043}^{+0.053}$ | $0.880_{-0.013}^{+0.012}$ | $5261\pm 60$ | $0.04\pm 0.05$ | $7.8\pm 4.3$ | 471 | 213 | 5 | 1940 | 13, 27, 28, 29
K2-106 | s | $0.950_{-0.048}^{+0.060}$ | $0.988\pm 0.011$ | $5532\pm 62$ | $0.11\pm 0.07$ | $7.5_{-4.0}^{+3.8}$ | 111 | 44 | 4 | 1231 |  30, 31, this work2
K2-110 | s | $0.738\pm 0.018$ | $0.713\pm 0.020$ | $5010\pm 50$ | $-0.34\pm 0.03$ | $8\pm 3$ | 32 | 15 | 2 | 1470 | 32
K2-111 | s | $0.84\pm 0.02$ | $1.25\pm 0.02$ | $5775\pm 70$ | $-0.46\pm 0.05$ | $13.5_{-0.9}^{+0.4}$ | 155 | 114 | 1 | 1590 | 33, 34
K2-131 | s | $0.803\pm 0.034$ | $0.7595_{-0.0089}^{+0.0087}$ | $5120\pm 71$ | $-0.04\pm 0.07$ | $5.3_{-3.7}^{+4.9}$ | 114 | 83 | 2 | 463 |  35, this work2
K2-135 / GJ 9827 | m | $0.606_{-0.014}^{+0.015}$ | $0.602_{-0.004}^{+0.005}$ | $4305\pm 49$ | $-0.26\pm 0.09$ | $10_{-5}^{+3}$ | 127 | 50 | 4 | 3922 | 36, 37, 38
K2-141 | m | $0.708\pm 0.028$ | $0.681\pm 0.018$ | $4570\pm 100$ | $0.00\pm 0.06$ | $6.3_{-4.7}^{+6.6}$ | 74 | 49 | 2 | 1501 | 39, 40
K2-167 | s | $1.010_{-0.071}^{+0.081}$ | $1.494_{-0.034}^{+0.036}$ | $6011\pm 60$ | $-0.40\pm 0.06$ | $7.5_{-2.1}^{+2.4}$ | 82 | 82 | 1 | 2208 |  this work2
K2-222 | s | $0.989_{-0.065}^{+0.070}$ | $1.115\pm 0.029$ | $5942\pm 119$ | $-0.315\pm 0.060$ | $7.1_{-1.7}^{+1.5}$ | 70 | 70 | 1 | 1599 | 26
K2-262 / Wolf 503 | s | $0.688_{-0.016}^{+0.023}$ | $0.689_{-0.020}^{+0.021}$ | $4716\pm 60$ | $-0.47\pm 0.08$ | $11\pm 2$ | 84 | 25 | 4 | 1116 | 41, 42
K2-263 | s | $0.88\pm 0.03$ | $0.85\pm 0.02$ | $5368\pm 44$ | $-0.08\pm 0.05$ | $7\pm 4$ | 95 | 95 | 1 | 1545 | 43
K2-312 / HD 80653 | s | $1.18\pm 0.04$ | $1.22\pm 0.01$ | $5959\pm 61$ | $0.255\pm 0.065$ | $2.67\pm 1.20$ | 208 | 208 | 1 | 1068 | 44
K2-418 / EPIC-229004835 | s | $0.97\pm 0.04$ | $0.999\pm 0.008$ | $5868\pm 60$ | $-0.12\pm 0.05$ | $4.9_{-1.7}^{+1.9}$ | 126 | 126 | 1 | 845 | 45
Table 1: continued. 888From left to right, the columns report the name of the
system, the multiplicity of transiting planets, the stellar parameters (mass,
radius, effective temperature, metallicity and isochronological age), the
number of radial velocities from all surveys ($N_{\rm RV}$ tot), the number of
HARPS-N radial velocities ($N_{\rm RV}$ HN), the number of radial-velocity
datasets ($N_{\rm Dat}$), the total duration of the radial-velocity time
series, and the literature references for both the stellar parameters and the
radial-velocity measurements. Table also available at the CDS.

11footnotemark: 1

Two datasets were considered for the HARPS-N data, because the replacement of
the red side of the HARPS-N CCD in late September 2012 resulted in a different
zero point for the RVs gathered after that epoch. 22footnotemark: 2 The new
system parameters $M_{\rm s}$, $R_{\rm s}$, and age were derived with the
public EXOFASTv2 tool by fitting the stellar SED and using the MIST
evolutionary tracks. Gaussian priors were imposed on the $T_{\rm eff}$ and
$\rm[Fe/H]$, as derived from the analysis of the HARPS-N spectra, and on
stellar parallax from Gaia EDR3 (see text for more details).

(1) Dumusque et al. 2014; (2) Ballard et al. 2011; (3) Malavolta et al. 2017;
(4) Buchhave et al. 2016; (5) Howell et al. 2012; (6) López-Morales et al.
2016; (7) Bryan et al. 2019; (8) Rajpaul et al. 2021; (9) Mills et al. 2019;
(10) Margini et al. in prep.; (11) Pepe et al. 2013; (12) Howard et al. 2013;
(13) Dai et al. 2019; (14) Dressing et al. 2015; (15) Marcy et al. 2014; (16)
Brinkman et al. 2023; (17) Dubber et al. 2019; (18) Bonomo et al. 2019; (19)
Gettel et al. 2016; (20) Mayo et al. 2019; (21) Haywood et al. 2018; (22)
Vanderburg et al. 2015; (23) Damasso et al. 2018; (24) Damasso et al. 2019;
(25) Toledo-Padrón et al. 2020; (26) Nava et al. 2022; (27) Christiansen et
al. 2017; (28) Gandolfi et al. 2017; (29) Bourrier et al. 2022; (30) Sinukoff
et al. 2017; (31) Guenther et al. 2017; (32) Osborn et al. 2017; (33) Fridlund
et al. 2017; (34) Mortier et al. 2020; (35) Dai et al. 2017; (36) Teske et al.
2018; (37) Prieto-Arranz et al. 2018; (38) Rice et al. 2019; (39) Malavolta et
al. 2018; (40) Barragán et al. 2018; (41) Peterson et al. 2018; (42) Polanski
et al. 2021; (43) Mortier et al. 2018; (44) Frustagli et al. 2020; (45)
Tronsgaard et al. in prep..

Table 2: HARPS-N measurements of radial velocity and activity indicators.

System | Time | RV | $\sigma_{\rm RV}$ | FWHM | $\sigma_{\rm FWHM}$ | C | $\sigma_{\rm C}$ | BIS | $\sigma_{\rm BIS}$ | $S_{\rm MW}$ | $\sigma_{\rm S_{\rm MW}}$ | $\log{R^{{}^{\prime}}_{\rm HK}}$ | $\sigma_{\log{R^{{}^{\prime}}_{\rm HK}}}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  [$\rm BJD_{TDB}-2.45\mbox{\sc{e}06}$] |  [ m s-1] |  [ m s-1] |  [ m s-1] |  [ m s-1] | [%] | [%] |  [ m s-1] |  [ m s-1] |  |  | dex | dex
Kepler-10 | 6072.682383 | -98740.32 | 1.76 | 6762.45 | 9.56 | 45.65 | 0.05 | -23.88 | 2.49 | 0.163 | 0.004 | -4.99 | 0.02
Kepler-10 | 6072.704768 | -98740.14 | 1.86 | 6753.98 | 9.55 | 45.70 | 0.05 | -27.56 | 2.64 | 0.170 | 0.005 | -4.96 | 0.02
… | … | … | … | … | … | … | … | … | … | … | … | … | …
999From left to right, che columns report the name of the system, the epoch of
the observation, the radial velocity and its uncertainty, the activity
indicators full width at half maximum (FWHM), contrast (C), and bisector span
(BIS) of the cross-correlation function, and the CaII H&K Mount Wilson $S$
index ($S_{\rm MW}$) and $\log{R^{{}^{\prime}}_{\rm HK}}$ along with their
uncertainties. Data are available at both the CDS and DACE databases. A
portion is shown here for guidance regarding its form and content.

Table 3: Priors imposed on the hyper-parameters of the Gaussian process
regression for the stars showing significant magnetic activity variations.

System | Adopted GP kernel | $h$ | $\lambda_{1}$ | $\lambda_{2}$ | $P_{\rm rot}$
---|---|---|---|---|---
| SE or QP | [ m s-1] | [day] | - | [day]
Kepler-21 | QP | $U[0,+\infty[$ | $U[0,300]$ | $U[0.1,5]$ | $U[10,15]$
Kepler-78 | QP | $U[0,+\infty[$ | $U[0,300]$ | $U[0.1,5]$ | $U[0,20]$
Kepler-93 | SE | $U[0,+\infty[$ | $U[0,500]$ | - | -
Kepler-102 | QP | $U[0,+\infty[$ | $\lambda_{1}>P_{\rm rot}/2$ | $U[0.1,5]$ | $U[15,40]$
K2-2 / HIP116454 | SE | $U[0,+\infty[$ | $U[0,50]$ | - | -
K2-3 | QP | $U[0,+\infty[$ | $U[20,60]$ | $U[0.1,5]$ | $U[35,43]$
K2-36 | QP | $U[0,+\infty[$ | $U[0,300]$ | $U[0.1,5]$ | $U[15,20]$
K2-131 | QP | $U[0,+\infty[$ | $U[0,300]$ | $U[0.1,5]$ | $U[5,15]$
K2-135 / GJ 9827 | QP | $U[0,+\infty[$ | $U[0,300]$ | $U[0.1,5]$ | $U[25,35]$
K2-141 | QP | $U[0,+\infty[$ | $\lambda_{1}>P_{\rm rot}/2$ | $U[0.1,5]$ | $U[10,20]$
K2-312 / HD 80653 | QP | $U[0,+\infty[$ | $\lambda_{1}>P_{\rm rot}/2$ | $U[0.1,5]$ | $U[15,25]$
101010From left to right, the columns report the name of the system, the
adopted kernel for the covariance function – either squared exponential (SE)
or quasi-periodic (QP) kernel – and the priors on the hyper-parameters of the
covariance function, namely the radial-velocity semi-amplitude, $h$, the
exponential decay time, $\lambda_{1}$, the inverse harmonic complexity term,
$\lambda_{2}$, and the rotation period, $P_{\rm rot}$ (the SE kernel has only
the two hyper-parameters $h$ and $\lambda_{1}$). $U$ stands for uniform
(uninformative) prior.

Table 4: HARPS-N systemic radial velocities and jitter terms, Gaussian process hyper-parameters, and linear accelerations from the DE-MCMC radial-velocity modeling. System | $\gamma_{\rm HN-1}$ | $\gamma_{\rm HN-2}$ | $\sigma_{\rm jit,HN-1}$ | $\sigma_{\rm jit,HN-2}$ | $h$ | $\lambda_{1}$ | $\lambda_{2}$ | $P_{\rm rot}$ | $\dot{\gamma}$ | $\Delta BIC>10$ | Comment
---|---|---|---|---|---|---|---|---|---|---|---
|  [ m s-1] |  [ m s-1] |  [ m s-1] |  [ m s-1] |  [ m s-1] | [day] | - | [day] |  [$\rm m\,s^{-1}\,day^{-1}$] |  |
Kepler-10 | $-98737.12\pm 0.47$ |  $0.09\pm 0.17$1 | $2.52_{-0.41}^{+0.44}$ | $2.13\pm 0.15$ | - | - | - | - | $-2.7\leavevmode\nobreak\ \mbox{\sc{e}-04}\pm 1.6\leavevmode\nobreak\ \mbox{\sc{e}-04}$ | - |
Kepler-19 | $-10614.72\pm 1.31$ | $-10762.08_{-0.74}^{+0.76}$ | $4.78_{-0.96}^{+1.20}$ | $3.51_{-0.42}^{+0.47}$ | - | - | - | - | $1.7\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 0.9\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
Kepler-20 | - | $-21070.23\pm 0.95$ | - | $4.06_{-0.41}^{+0.45}$ | - | - | - | - | $0.7\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 1.2\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
Kepler-21 | - | $-19322.87_{-1.41}^{+1.38}$ | - | $2.61\pm 0.44$ | $4.28_{-0.89}^{+1.45}$ | $46_{-21}^{+90}$ | $0.88_{-0.25}^{+0.47}$ | $13.42_{-0.47}^{+0.24}$ | $-5.6_{-1.7}^{+1.9}\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
Kepler-22 | $-4076.15_{-1.81}^{+1.91}$ | $-4221.01\pm 0.99$ | $5.09_{-1.45}^{+1.97}$ | $2.47_{-0.58}^{+0.62}$ | - | - | - | - | $1.8\leavevmode\nobreak\ \mbox{\sc{e}-04}\pm 8.0\leavevmode\nobreak\ \mbox{\sc{e}-04}$ | - |
Kepler-37 | - | $-30820.12\pm 0.56$ | - | $2.59_{-0.19}^{+0.21}$ | - | - | - | - | $6.9\leavevmode\nobreak\ \mbox{\sc{e}-04}\pm 7.4\leavevmode\nobreak\ \mbox{\sc{e}-04}$ | - |
Kepler-68 | - | $-20903.57_{-0.36}^{+0.45}$ | - | $1.81\pm 0.17$ | - | - | - | - | $-1.2_{-0.48}^{+0.55}\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - | 2 CJs (ro)
Kepler-78 | - | $-3530.11\pm 3.21$ | - | $1.65\pm 0.33$ | $8.90_{-1.49}^{+2.13}$ | $26.7_{-5.9}^{+6.8}$ | $0.475_{-0.076}^{+0.086}$ | $12.92\pm 0.12$ | $-1.6\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 3.2\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
Kepler-93 | - | $27245.19_{-1.12}^{+1.23}$ | - | $1.25\pm 0.16$ | $3.23_{-0.73}^{+1.07}$ | $130_{-34}^{+46}$ | - | - | $3.92\leavevmode\nobreak\ \mbox{\sc{e}-02}\pm 0.11\leavevmode\nobreak\ \mbox{\sc{e}-02}$ | y | BD/LMS
Kepler-102 | $-28152.22\pm 1.50$ | $-28163.26\pm 1.06$ | $<2.20$ | $<0.72$ | $3.26_{-0.50}^{+0.54}$ | $21.3_{-4.6}^{+8.9}$ | $0.73_{-0.16}^{+0.20}$ | $27.95_{-1.67}^{+3.64}$ | $-7.3\leavevmode\nobreak\ \mbox{\sc{e}-04}\pm 7.0\leavevmode\nobreak\ \mbox{\sc{e}-04}$ | - |
Kepler-103 | - | $-28609.58\pm 0.71$ | - | $2.26_{-1.15}^{+0.98}$ | - | - | - | - | $-2.3\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 1.8\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
Kepler-107 | - | $5536.53\pm 0.87$ | - | $<2.34$ | - | - | - | - | $-1.2\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 1.4\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
Kepler-109 | - | $-37129.88\pm 1.53$ | - | $2.85_{-1.19}^{+1.06}$ | - | - | - | - | $-2.9\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 2.0\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
Kepler-323 | - | $13349.22_{-1.64}^{+1.61}$ | - | $<2.86$ | - | - | - | - | $3.2\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 1.4\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
Kepler-409 | - | $-28541.49\pm 0.57$ | - | $2.39_{-0.29}^{+0.36}$ | - | - | - | - | $0.0\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 0.4\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
Kepler-454 | - | $-71463.30_{-2.2}^{+1.2}$ | - | $2.26_{-0.33}^{+0.35}$ | - | - | - | - | $9.15_{-1.3}^{+2.6}\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | y | CJ/BD/LMS + 2 CJs (ro)
Kepler-538 | - | $-37467.97_{-0.85}^{+0.80}$ | - | $2.34_{-0.29}^{+0.32}$ | - | - | - | - | $-0.2\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 1.1\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
Kepler-1655 | - | $-40769.17\pm 0.64$ | - | $<0.91$ | - | - | - | - | $1.8\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 1.0\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
Kepler-1876 | - | $7681.39\pm 0.45$ | - | $1.87_{-0.36}^{+0.39}$ | - | - | - | - | $7.8\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 3.2\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
K2-2 / HIP 116454 | - |  $-3299.73\pm 0.61$2 | - | $0.97\pm 0.45$ | $2.72_{-0.42}^{+0.47}$ | $4.15_{-1.05}^{+2.44}$ | - | - | $0.2\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 1.4\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
K2-3 | - |  $-0.06\pm 0.55$3 | - | $0.98_{-0.31}^{+0.24}$ | $2.60_{-0.28}^{+0.32}$ | $41.6_{-8.8}^{+9.8}$ | $0.42_{-0.12}^{+0.21}$ | $40.71_{-1.36}^{+0.87}$ | $-0.4\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 1.7\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
K2-12 | - | $17000.18\pm 0.99$ | - | $4.36_{-0.96}^{+1.02}$ | - | - | - | - | $1.884\leavevmode\nobreak\ \mbox{\sc{e}-02}\pm 0.096\leavevmode\nobreak\ \mbox{\sc{e}-02}$ | y | CJ/BD/LMS
K2-36 | - | $13625.38\pm 4.25$ | - | $2.64\pm 1.06$ | $15.5_{-2.6}^{+3.8}$ | $99_{-29}^{+41}$ | $0.67_{-0.12}^{+0.13}$ | $17.13_{-0.10}^{+0.07}$ | $3.7\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 7.3\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
K2-38 | - | $-36040.15\pm 0.56$ | - | $2.07\pm 0.67$ | - | - | - | - | $-8.2\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 3.1\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
K2-79 | - | $-10314.99\pm 0.76$ | - | $<1.74$ | - | - | - | - | $1.8\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 1.3\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
K2-96 / HD 3167 | - | $19371.36\pm 0.24$ | - | $1.43_{-0.09}^{+0.10}$ | - | - | - | - | $1.32_{-0.30}^{+0.28}\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | y | Activity
K2-106 | - | $-15735.71\pm 1.12$ | - | $<1.17$ | - | - | - | - | $-1.4\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 2.1\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
K2-110 | - | $-21798.17\pm 0.93$ | - | $<1.91$ | - | - | - | - | $0.7\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 1.5\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
K2-111 | - | $-16291.29\pm 0.26$ | - | $<0.93$ | - | - | - | - | $-1.8\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 0.5\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | n |
K2-131 | - | $6855.45\pm 6.17$ | - | $<1.59$ | $21.38_{-3.35}^{+4.68}$ | $31.9_{-7.3}^{+9.8}$ | $0.435\pm 0.074$ | $9.194\pm 0.065$ | $3.8\leavevmode\nobreak\ \mbox{\sc{e}-02}\pm 3.3\leavevmode\nobreak\ \mbox{\sc{e}-02}$ | - |
K2-135 / GJ 9827 | - | $31940.16_{-1.33}^{+1.38}$ | - | $<0.95$ | $3.83_{-0.65}^{+0.82}$ | $54_{-29}^{+48}$ | $0.61_{-0.14}^{+0.17}$ | $28.90_{-0.70}^{+1.09}$ | $-0.6\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 1.3\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
K2-141 | - | $-3408.80\pm 4.92$ | - | $<1.17$ | $12.75_{-3.13}^{+6.11}$ | $16.4_{-6.5}^{+16.2}$ | $0.94_{-0.23}^{+0.36}$ | $15.17_{-2.36}^{+3.63}$ | $-1.0_{-7.4}^{+7.1}\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
K2-167 | - | $-17550.44\pm 0.50$ | - | $2.36_{-0.24}^{+0.20}$ | - | - | - | - | $1.2\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 0.6\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
K2-222 | - | $1681.65\pm 0.46$ | - | $1.79_{-0.31}^{+0.34}$ | - | - | - | - | $5.8\leavevmode\nobreak\ \mbox{\sc{e}-04}\pm 7.1\leavevmode\nobreak\ \mbox{\sc{e}-04}$ | - |
K2-262 / Wolf 503 | - | $-46781.43\pm 0.46$ | - | $1.97_{-0.34}^{+0.42}$ | - | - | - | - | $6.6\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 1.0\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | y | Activity
K2-263 | - | $29837.72\pm 0.29$ | - | $1.50\pm 0.37$ | - | - | - | - | $4.1\leavevmode\nobreak\ \mbox{\sc{e}-04}\pm 4.9\leavevmode\nobreak\ \mbox{\sc{e}-04}$ | - |
K2-312 / HD 80653 | - | $8323.37_{-0.71}^{+0.76}$ | - | $0.64\pm 0.30$ | $2.86_{-0.25}^{+0.28}$ | $18.8_{-3.5}^{+4.1}$ | $0.46_{-0.06}^{+0.07}$ | $19.55_{-0.50}^{+0.57}$ | $0.0\pm 1.4$ | - | 1 CJ (ro)
K2-418 / EPIC-229004835 | - | $-31487.88\pm 0.30$ | - | $2.48\pm 0.24$ | - | - | - | - | $-2.5\leavevmode\nobreak\ \mbox{\sc{e}-03}\pm 1.1\leavevmode\nobreak\ \mbox{\sc{e}-03}$ | - |
111111From left to right, the columns report the name of the system, the
systemic radial velocity ($\gamma$), the jitter terms ($\sigma_{\rm jit}$),
the Gaussian process hyper-parameters – namely radial-velocity semi-amplitude
($h$), exponential decay time ($\lambda_{1}$), inverse harmonic complexity
term ($\lambda_{2}$), and rotation period ($P_{\rm rot}$) – , the linear
acceleration ($\dot{\gamma}$) and the $\Delta BIC$ in favor of the model with
acceleration. Even though, for simplicity, $\gamma$ and $\sigma_{\rm jit}$ are
reported for the HARPS-N spectrograph only, the DE-MCMC radial-velocity
modeling was carried out by including the data gathered with other
spectrographs, when available. Accelerations $\dot{\gamma}$ evaluated as
significant, from both a confidence level higher than $3\sigma$ and an odds
ratio greater than 10 from the $\Delta BIC$, are highlighted in boldface. HN-1
and HN-2 refer to the HARPS-N data collected before and after September 2012,
respectively. The last column reports comments about the origin of the
detected trends, such as possible cold Jupiter (CJ), brown dwarf (BD), or low-
mass star (LMS) companions or stellar magnetic activity, and about the
presence of cold Jupiters in resolved orbits (ro), which were modeled with
Keplerians and are thus not responsible for the trends.

11footnotemark: 1

The radial-velocity zero point is close to zero because the radial velocities
were extracted with the Yarara-v2 tool (Cretignier et al. 2022) from the
HARPS-N spectra reduced with the DRS-v2.3.5 (see Bonomo et al. in prep.).
22footnotemark: 2The last observing season was fitted as an independent
dataset for the presence of an offset in the RVs of very likely activity
origin. The systemic velocity and jitter of this season were found to be
$\gamma=-3292.51_{-1.23}^{+1.26}$ m s-1and $\sigma_{\rm
jit}=1.70_{-1.13}^{+1.43}$ m s-1. 33footnotemark: 3The radial-velocity zero
point is close to zero because the radial velocities were extracted with the
TERRA software (see text for details). The systemic velocity is
$30149.2_{-0.7}^{+0.6}$ m s-1(Damasso et al. 2018).

Table 5: Orbital parameters and minimum masses of the companions causing the
long-term trends observed in the Kepler-93, Kepler-454, and K2-12 systems.

System | $\rm P$ | $\rm a$ | $\rm K$ | $M_{\rm p}\sin{i}$
---|---|---|---|---
| [yr] | [AU] | [ m s-1] | [$\rm M_{Jup}$]
Kepler-93 | $>48.6$ | $>13.0$ | $>174$ | $>21$
Kepler-454 | $>48.2$ | $>13.4$ | $>40.3$ | $>5.3$
K2-12 | $>27.8$ | $>9.1$ | $>47.8$ | $>5.0$
121212From left to right, the columns report the name of the system and the
lower limits to orbital period, semi-major axis, RV semi-amplitude and minimum
mass of the companions.

Table 6: Parameters of the long-period non-transiting giant planets with
resolved orbit.

Name | $T_{\rm c}$ | $P$ | $a$ | $e$ | $\omega$ | $K$ | $M_{\rm p}\sin{i}$
---|---|---|---|---|---|---|---
| [$\rm BJD_{TDB}-2.45\mbox{\sc{e}06}$] | [day] | [AU] |  | [deg] | [ m s-1] | $[\rm M_{Jup}]$
Kepler-68d | $5892.0\pm 5.4$ | $632.62\pm 1.03$ | $1.469\pm 0.010$ | $0.102\pm 0.016$ | $256.4\pm 10.4$ | $17.20\pm 0.29$ | $0.749\pm 0.017$
Kepler-68e | $7791_{-126}^{+134}$ | $3455_{-169}^{+348}$ | $4.60_{-0.16}^{+0.32}$ | $0.33\pm 0.11$ | $196_{-20}^{+17}$ | $3.73\pm 0.39$ | $0.272\pm 0.032$
Kepler-454c | $7151.87\pm 0.69$ | $524.19\pm 0.20$ | $1.287\pm 0.017$ | $<0.0053$ | - | $111.58\pm 0.55$ | $4.51\pm 0.12$
Kepler-454d | $8433\pm 26$ | $4073_{-186}^{+399}$ | $5.10_{-0.19}^{+0.34}$ | $0.089_{-0.027}^{+0.036}$ | $117_{-21}^{+25}$ | $28.4_{-1.3}^{+2.4}$ | $2.31_{-0.16}^{+0.27}$
K2-312c / HD 80653c | $8968.80\pm 0.41$ | $921.2\pm 10.8$ | $1.961\pm 0.027$ | $0.853\pm 0.011$ | $41.6\pm 2.2$ | $194_{-13}^{+17}$ | $5.41_{-0.44}^{+0.52}$

131313From left to right, the columns report the name of the system, the
inferior conjunction time, the orbital period, the semi-major axis, the
eccentricity, the argument of periastron, the radial-velocity semi-amplitude,
and the planet minimum mass.

Table 7: Completeness (recovery rate) and fraction of stars hosting both
short-period small planets and cold Jupiters for different intervals of
orbital separation.

Orbital separation [AU] | Completeness [$\%$] | $N_{\star,\rm eff}$ | $N_{\star,\rm CJ}$ | $f_{\rm CJ|SP}[\%]$
---|---|---|---|---
$1-2$ | 96.9 | 35.9 | 2 | $5.6^{+6.5}_{-1.8}$
$2-4$ | 92.2 | 34.1 | 2 | $5.9^{+6.8}_{-1.9}$
$4-10$ | 77.4 | 28.6 | 0 | $<3.80$
$1-10$ | 87.9 | 32.4 | 3 | $9.3^{+7.7}_{-2.9}$
141414From left to right, the columns report the interval of semi-major axis,
the survey completeness, the effective number of stars ($N_{\star,\rm eff}\leq
37$), the number of systems with detected cold Jupiters ($N_{\star,\rm CJ}$)
in the sample, and the fraction of stars hosting both short-period small
planets and cold Jupiters ($f_{\rm CJ|SP}$).

Table 8: Fraction of stars with cold Jupiters in our sample compared with
those from previous works.

Planetary Mass [$\rm M_{Jup}$] | Orbital separation [AU] | $f_{\rm CJ|SP}[\%]$ | $f_{\rm CJ|SP}[\%]$ | $f_{\rm CJ}[\%]$ |  $f_{\rm CJ|SP}[\%]$ 1
---|---|---|---|---|---
|  | from Keplerians | from Keplerians and trends | (Wittenmyer et al. 2020) | (Bryan et al. 2019)
$0.3-13$ | $1-2$ | $5.6^{+6.5}_{-1.8}$ | - | $8.0^{+3.7}_{-2.2}$ | -
$0.3-13$ | $2-4$ | $5.9^{+6.8}_{-1.9}$ | - | $5.3^{+2.8}_{-1.5}$ | -
$0.3-13$ | $4-10$ | $<3.8$ | - | $6.9^{+4.2}_{-2.1}$ | -
$0.3-13$ | $1-10$ | $9.3^{+7.7}_{-2.9}$ | $12.3^{+8.1}_{-3.7}$ | $20.2^{+6.3}_{-3.4}$ | -
$0.5-13$ | $1-10$ | $8.8^{+7.4}_{-2.8}$ | $11.8^{+7.7}_{-3.5}$ | - | $36^{+7}_{-6}$
$0.5-13$ | $1-20$ | $8.3^{+7.0}_{-2.6}$ | $11.1^{+7.4}_{-3.3}$ | - | $41^{+8}_{-7}$
$0.5-20$ | $1-10$ | $8.7^{+7.3}_{-2.7}$ | $11.3^{+7.7}_{-3.4}$ | - | $38\pm 7$
$0.5-20$ | $1-20$ | $9.6^{+7.9}_{-3.0}$ | $12.7^{+8.3}_{-3.9}$ | - | $39\pm 7$

151515From left to right, the columns report the intervals of planetary mass
and semi-major axis; the fraction of stars with cold Jupiters ($f_{\rm
CJ|SP}$) in our sample by considering Keplerian signals only (third column) as
well as including the long-term trend of K2-12 (fourth column), because it is
still compatible with a planetary companion; the occurrence rates of cold
Jupiters derived by Wittenmyer et al. (2020) and Bryan et al.
(2019).11footnotemark: 1

$f_{\rm CJ|SP}$ for $1<M_{\rm p}<10\leavevmode\nobreak\ \rm M_{\oplus}$
instead of the wider range $1<M_{\rm p}<20\leavevmode\nobreak\ \rm M_{\oplus}$
used in both this work and Zhu & Wu (2018).

Table 9: Parameters of the non-transiting small planets.

Name | $T_{\rm c}$ | $P$ | $a$ | $e$ | $K$ | $M_{\rm p}\sin{i}$
---|---|---|---|---|---|---
| [$\rm BJD_{TDB}-2.45\mbox{\sc{e}06}$] | [day] | [AU] |  | [ m s-1] | $[\rm M_{\oplus}]$
Kepler-10d | $7165.4_{-5.3}^{+4.7}$ | $151.04\pm 0.45$ | $0.5379\pm 0.0044$ | $<0.26$ | $1.68\pm 0.28$ | $12.68\pm 2.24$
Kepler-1876c1 | $6988.65_{-1.8}^{+1.0}$ | $15.76_{-0.16}^{+0.10}$ | $0.1302\pm 0.0033$ | $<0.10$ | $2.52\pm 0.59$ | $11\pm 2.7$
K2-96d / HD 3167d | $7745.37\pm 0.18$ | $8.4112\pm 0.0052$ | $0.0763\pm 0.0015$ | $<0.12$ | $1.55\pm 0.15$ | $4.33\pm 0.45$
K2-96e / HD 3167e | $7738.43\pm 2.67$ |  $96.63\pm 0.29$2 | $0.3885\pm 0.0079$ | $<0.15$ | $1.33\pm 0.16$ | $8.41\pm 1.02$
K2-111c | $8119.62\pm 0.33$ | $15.6805\pm 0.0064$ | $0.11569\pm 0.00094$ | $<0.071$ | $3.19\pm 0.29$ | $11.08\pm 1.03$

161616From left to right, the columns report the name of the planet, the
inferior conjunction time, the orbital period, the semi-major axis, the
eccentricity, the radial-velocity semi-amplitude, and the planet minimum mass.
The planet candidate Kepler-1876c is also included.11footnotemark: 1

Planet candidate. 22footnotemark: 2Other orbital periods are possible due to
aliasing effects (see also Bourrier et al. 2022).

##  Appendix A Orbital and physical parameters of the 64 transiting Kepler and
K2 planets.

Table 10: Orbital and physical parameters of the 64 transiting Kepler and K2 planets. Name | $T_{\rm c}$ | $P$ | $R_{\rm p}$ | $i$ | Ref. | $e$ | $K$ | $M_{\rm p}$ | $\rho_{\rm p}$ | $\log{g_{\rm p}}$ | $a$ | $T_{\rm eq}$ | $F_{\rm p}$
---|---|---|---|---|---|---|---|---|---|---|---|---|---
|  [$\rm BJD_{TDB}-2.45\mbox{\sc{e}06}$] | [day] |  [$\rm R_{\oplus}$] | [deg] | transits |  |  [ m s-1] |  [$\rm M_{\oplus}$] |  [$\rm g\;cm^{-3}$] | [cgs] | [AU] | [K] |  [$\rm F_{\oplus}$]
Kepler-10b | $5034.08687(18)$ | $0.83749070(20)$ | $1.470_{-0.020}^{+0.030}$ | $84.8_{-3.9}^{+3.2}$ | 1 | 0(fixed) | $2.34\pm 0.21$ | $3.26\pm 0.30$ | $5.57_{-0.59}^{+0.61}$ | $3.166_{-0.045}^{+0.042}$ | $0.01685\pm 0.00013$ | $2188\pm 16$ | $3820\pm 120$
Kepler-10c | $5062.26648(81)$ | $45.294301(48)$ | $2.355\pm 0.022$ | $89.623\pm 0.011$ | 1, 2 | $0.130\pm 0.050$ | $2.19\pm 0.24$ | $11.4\pm 1.3$ | $4.78_{-0.54}^{+0.56}$ | $3.302_{-0.051}^{+0.047}$ | $0.2410\pm 0.0019$ | $578\pm 4$ | $18.68\pm 0.57$
Kepler-19b | $4959.7074(14)$ | $9.2869900(00)$ | $2.209\pm 0.048$ | $89.940_{-0.44}^{+0.060}$ | 3, 4 | $<0.41$ | $2.13_{-0.81}^{+0.90}$ | $6.1_{-2.7}^{+2.8}$ | $3.1\pm 1.4$ | $3.09_{-0.25}^{+0.16}$ | $0.0846\pm 0.0012$ | $851\pm 14$ | $87.6\pm 6.0$
Kepler-20b | $4967.50225(33)$ | $3.6961049(16)$ | $1.773_{-0.030}^{+0.053}$ | $87.36_{-1.6}^{+0.22}$ | 5, 6 | $<0.083$ | $4.23\pm 0.54$ | $9.7\pm 1.3$ | $9.4_{-1.4}^{+1.5}$ | $3.474_{-0.066}^{+0.060}$ | $0.04565\pm 0.00089$ | $1187\pm 16$ | $331\pm 19$
Kepler-20c | $4971.60886(16)$ | $10.8540774(21)$ | $2.894_{-0.033}^{+0.036}$ | $89.815_{-0.63}^{+0.036}$ | 5, 6 | $<0.076$ | $3.38\pm 0.61$ | $11.1\pm 2.1$ | $2.51_{-0.47}^{+0.48}$ | $3.113_{-0.090}^{+0.075}$ | $0.0936\pm 0.0018$ | $828\pm 11$ | $78.7\pm 4.6$
Kepler-20d | $4997.7296(11)$ | $77.611455(96)$ | $2.606_{-0.039}^{+0.053}$ | $89.708_{-0.053}^{+0.17}$ | 5, 6 | $<0.082$ | $2.12\pm 0.57$ | $13.4_{-3.6}^{+3.7}$ | $4.1_{-1.1}^{+1.2}$ | $3.28_{-0.14}^{+0.11}$ | $0.3474\pm 0.0067$ | $430\pm 6$ | $5.71\pm 0.33$
Kepler-20e | $4968.93956(34)$ | $6.0984882(99)$ | $0.821\pm 0.022$ | $87.63_{-0.13}^{+1.1}$ | 5, 6 | $<0.092$ | $<0.28$ | $<0.76$ | $<7.5$ | $<3.0$ | $0.0637\pm 0.0012$ | $1004\pm 14$ | $169.8\pm 9.8$
Kepler-20f | $4968.19883(57)$ | $19.578328(48)$ | $0.952_{-0.087}^{+0.047}$ | $88.788_{-0.072}^{+0.43}$ | 5, 6 | $<0.094$ | $<0.35$ | $<1.4$ | $<8.4$ | $<3.2$ | $0.1387\pm 0.0027$ | $681\pm 9$ | $35.9\pm 2.1$
Kepler-21b | $5093.83716(85)$ | $2.7858212(32)$ | $1.639_{-0.015}^{+0.019}$ | $83.20_{-0.26}^{+0.28}$ | 7 | 0(fixed) | $2.70\pm 0.46$ | $7.5\pm 1.3$ | $9.3\pm 1.6$ | $3.435_{-0.081}^{+0.069}$ | $0.04340_{-0.00032}^{+0.00021}$ | $2015\pm 16$ | $2749\pm 90$
Kepler-22b | $4966.7001(68)$ | $289.863876(13)$ | $2.10\pm 0.12$ | $89.764_{-0.042}^{+0.025}$ |  8, 9, t.w.1 | $<0.72$ | $<1.6$ | $<9.1$ | $<5.2$ | $<3.3$ | $0.812_{-0.013}^{+0.011}$ | $279\pm 4$ | $1.013\pm 0.060$
Kepler-37b | $5017.0473(37)$ | $13.367020(60)$ | $0.3098_{-0.0076}^{+0.0059}$ | $88.63_{-0.53}^{+0.30}$ |  6, 10, t.w.1 | $<0.098$ | $<0.25$ | $<0.79$ | $<140$ | $<3.9$ | $0.1019\pm 0.0014$ | $718\pm 10$ | $44.5\pm 2.8$
Kepler-37c | $5024.83997(87)$ | $21.301848(18)$ | $0.755_{-0.055}^{+0.033}$ | $89.07_{-0.33}^{+0.19}$ |  6, 10, t.w.1 | $<0.099$ | $<0.34$ | $<1.3$ | $<15$ | $<3.3$ | $0.1390\pm 0.0020$ | $615\pm 9$ | $23.9\pm 1.5$
Kepler-37d | $5008.24982(13)$ | $39.7922622(65)$ | $2.030_{-0.039}^{+0.030}$ | $89.335_{-0.047}^{+0.043}$ |  6, 10, t.w.1 | $<0.10$ | $<0.44$ | $<2.0$ | $<1.3$ | $<2.7$ | $0.2109\pm 0.0030$ | $499\pm 7$ | $10.39\pm 0.65$
Kepler-68b | $5006.858780(76)$ | $5.39875259(52)$ | $2.357\pm 0.023$ | $87.23_{-0.17}^{+0.22}$ | 6, 11 | $<0.090$ | $2.83\pm 0.23$ | $8.03\pm 0.67$ | $3.37_{-0.29}^{+0.30}$ | $3.151_{-0.038}^{+0.036}$ | $0.06135\pm 0.00043$ | $1275\pm 17$ | $441\pm 25$
Kepler-68c | $4969.3821(11)$ | $9.605027(13)$ | $0.979\pm 0.019$ | $87.071_{-0.094}^{+0.087}$ | 6, 11 | $<0.099$ | $<0.37$ | $<1.3$ | $<7.5$ | $<3.1$ | $0.09008\pm 0.00063$ | $1052\pm 14$ | $204\pm 12$
Kepler-78b | $4953.95984(15)$ | $0.355007450(80)$ | $1.201\pm 0.028$ | $75.2_{-2.1}^{+2.6}$ |  12, t.w.1 | 0(fixed) | $1.75\pm 0.28$ | $1.68\pm 0.27$ | $5.33_{-0.93}^{+0.97}$ | $3.058_{-0.080}^{+0.069}$ | $0.00901_{-0.00019}^{+0.00012}$ | $2223\pm 32$ | $4070\pm 240$
Kepler-93b | $4944.29227(13)$ | $4.72673978(97)$ | $1.478\pm 0.019$ | $89.183\pm 0.044$ | 13 | 0(fixed) | $1.89\pm 0.21$ | $4.66\pm 0.53$ | $7.93_{-0.94}^{+0.96}$ | $3.320_{-0.053}^{+0.048}$ | $0.05343\pm 0.00065$ | $1133\pm 17$ | $275\pm 18$
Kepler-102b | $4968.8696(11)$ | $5.286965(12)$ | $0.460\pm 0.026$ | $89.78\pm 0.22$ |  6, 14, t.w.1 | $<0.100$ | $<0.47$ | $<1.1$ | $<62$ | $<3.7$ | $0.05521\pm 0.00049$ | $857\pm 20$ | $90.0\pm 9.0$
Kepler-102c | $4972.9746(24)$ | $7.071392(22)$ | $0.567\pm 0.028$ | $89.82\pm 0.15$ |  6, 14, t.w.1 | $<0.094$ | $<0.66$ | $<1.7$ | $<52$ | $<3.7$ | $0.06702\pm 0.00059$ | $777\pm 18$ | $61.0\pm 6.1$
Kepler-102d | $4967.091280(00)$ | $10.3117670(41)$ | $1.154\pm 0.058$ | $89.49\pm 0.11$ |  6, 14, t.w.1 | $<0.092$ | $1.02\pm 0.44$ | $3.0\pm 1.3$ | $10.7_{-4.6}^{+5.1}$ | $3.34_{-0.24}^{+0.16}$ | $0.08618\pm 0.00076$ | $686\pm 16$ | $36.9\pm 3.7$
Kepler-102e | $4967.75370(12)$ | $16.1456994(22)$ | $2.17\pm 0.11$ | $89.488\pm 0.051$ |  6, 14, t.w.1 | $<0.089$ | $1.37\pm 0.53$ | $4.7\pm 1.8$ | $2.50_{-0.98}^{+1.1}$ | $2.99_{-0.21}^{+0.15}$ | $0.1162\pm 0.0010$ | $590\pm 14$ | $20.3\pm 2.0$
Kepler-102f | $4978.0276(16)$ | $27.453592(60)$ | $0.861\pm 0.022$ | $89.320\pm 0.037$ |  6, 14, t.w.1 | $<0.10$ | $<1.1$ | $<4.3$ | $<37$ | $<3.8$ | $0.1656\pm 0.0015$ | $495\pm 11$ | $10.0\pm 1.0$
Kepler-103b | $5677.65243(28)$ | $15.9653287(92)$ | $3.486_{-0.054}^{+0.057}$ | $87.914_{-0.072}^{+0.073}$ | 15 | $<0.093$ | $2.21\pm 0.88$ | $9.8\pm 3.9$ | $1.28_{-0.50}^{+0.52}$ | $2.90_{-0.22}^{+0.15}$ | $0.13223_{-0.0013}^{+0.00085}$ | $973\pm 13$ | $149.9\pm 8.6$
Kepler-103c | $5667.15973(44)$ | $179.60978(20)$ | $5.45\pm 0.18$ | $87.704_{-0.055}^{+0.12}$ | 15 | $<0.095$ | $3.8\pm 1.7$ | $38_{-16}^{+17}$ | $1.29_{-0.55}^{+0.58}$ | $3.10_{-0.24}^{+0.16}$ | $0.6639_{-0.0063}^{+0.0043}$ | $434.0\pm 6.0$ | $5.95\pm 0.34$
Kepler-107b | $5701.08414(37)$ | $3.1800218(29)$ | $1.536\pm 0.025$ | $89.05\pm 0.67$ | 16 | $<0.10$ | $1.44\pm 0.68$ | $3.8_{-1.7}^{+1.8}$ | $5.8_{-2.6}^{+2.7}$ | $3.20_{-0.26}^{+0.17}$ | $0.04544\pm 0.00036$ | $1592\pm 19$ | $1073\pm 53$
Kepler-107c | $5697.01829(79)$ | $4.9014520(00)$ | $1.597\pm 0.026$ | $89.49_{-0.44}^{+0.34}$ | 16 | $<0.080$ | $3.29\pm 0.66$ | $10.0\pm 2.0$ | $13.5_{-2.8}^{+2.9}$ | $3.586_{-0.099}^{+0.081}$ | $0.06064\pm 0.00048$ | $1378\pm 16$ | $602\pm 30$
Kepler-107d | $5702.9547(60)$ | $7.95839(12)$ | $0.860\pm 0.060$ | $87.55_{-0.48}^{+0.64}$ | 16 | $<0.11$ | $<2.1$ | $<7.7$ | $<67$ | $<4.0$ | $0.08377\pm 0.00065$ | $1173\pm 14$ | $315\pm 16$
Kepler-107e | $5694.48550(46)$ | $14.749143(19)$ | $2.903\pm 0.035$ | $89.67\pm 0.22$ | 16 | $<0.10$ | $3.22\pm 0.74$ | $14.1\pm 3.3$ | $3.18_{-0.74}^{+0.75}$ | $3.216_{-0.11}^{+0.091}$ | $0.12638\pm 0.00099$ | $955\pm 11$ | $138.7\pm 6.8$
Kepler-109b | $4955.97791(60)$ | $6.4816307(48)$ | $2.49\pm 0.11$ | $87.06\pm 0.11$ |  6, 14, t.w.1 | $<0.11$ | $<1.6$ | $<4.9$ | $<1.7$ | $<2.9$ | $0.0701\pm 0.0019$ | $1276\pm 23$ | $442\pm 34$
Kepler-109c | $4970.5722(10)$ | $21.222650(27)$ | $2.65\pm 0.12$ | $89.63\pm 0.19$ |  6, 14, t.w.1 | $<0.098$ | $<1.8$ | $<8.2$ | $<2.4$ | $<3.1$ | $0.1546\pm 0.0042$ | $859\pm 16$ | $91.0\pm 7.1$
Kepler-323b | $4953.9478(14)$ | $1.6783280(15)$ | $1.381\pm 0.035$ | $88.3_{-3.9}^{+1.5}$ |  9, 17, t.w.1 | $<0.095$ | $<2.1$ | $<3.9$ | $<8.2$ | $<3.3$ | $0.02778\pm 0.00068$ | $1838\pm 35$ | $1900\pm 150$
Kepler-323c | $4956.9823(20)$ | $3.5538229(22)$ | $1.570\pm 0.040$ | $88.7\pm 1.3$ |  9, 17, t.w.1 | $<0.095$ | $2.8\pm 1.4$ | $6.8_{-3.2}^{+3.4}$ | $9.6_{-4.6}^{+5.0}$ | $3.43_{-0.28}^{+0.18}$ | $0.0458\pm 0.0011$ | $1431\pm 27$ | $700\pm 56$
Kepler-409b | $5012.0829(34)$ | $68.9583216(39)$ | $1.199\pm 0.043$ | $86.30\pm 0.13$ |  9, 14, t.w.1 | $<0.69$ | $<1.6$ | $<6.0$ | $<19$ | $<3.6$ | $0.3192\pm 0.0060$ | $438\pm 7$ | $6.15\pm 0.42$
Kepler-454b | $5008.06758(77)$ | $10.5737534(78)$ | $2.37\pm 0.13$ | $87.90\pm 0.20$ | 18 | $<0.32$ | $1.64\pm 0.41$ | $5.4\pm 1.4$ | $2.23_{-0.64}^{+0.75}$ | $2.98_{-0.14}^{+0.11}$ | $0.09528_{-0.00091}^{+0.0013}$ | $916\pm 10$ | $117.7\pm 5.6$
Kepler-538b | $5044.6789(11)$ | $81.73778(13)$ | $2.215_{-0.034}^{+0.040}$ | $89.730_{-0.060}^{+0.14}$ | 19 | $<0.21$ | $2.09\pm 0.47$ | $12.9\pm 2.9$ | $6.5\pm 1.5$ | $3.411_{-0.11}^{+0.089}$ | $0.3554_{-0.0047}^{+0.0068}$ | $417\pm 5$ | $5.07\pm 0.29$
Kepler-1655b | $5013.89795(69)$ | $11.8728787(85)$ | $2.213\pm 0.082$ | $87.62\pm 0.55$ | 20 | $<0.19$ | $1.51\pm 0.48$ | $5.4\pm 1.7$ | $2.72_{-0.90}^{+0.97}$ | $3.03_{-0.17}^{+0.13}$ | $0.1029\pm 0.0014$ | $938\pm 15$ | $129.1\pm 8.8$
Kepler-1876b | $4971.0510(50)$ | $6.992050(38)$ | $0.853_{-0.026}^{+0.039}$ | $89.13_{-2.0}^{+0.75}$ |  21, t.w.1 | $<0.098$ | $<0.71$ | $<2.4$ | $<19$ | $<3.5$ | $0.0758\pm 0.0019$ | $1299\pm 22$ | $475\pm 34$
K2-2b / HIP 116454b | $6907.887(29)$ |  $9.0949(26)$ 2 | $2.53\pm 0.18$ | $88.43\pm 0.40$ | 22 | $<0.089$ | $3.73\pm 0.42$ | $10.1_{-1.1}^{+1.2}$ | $3.41_{-0.72}^{+0.94}$ | $3.187_{-0.080}^{+0.079}$ | $0.07765\pm 0.00093$ | $745\pm 15$ | $51.4\pm 4.3$
K2-3b | $6813.41843(39)$ | $10.0546260(00)$ | $2.25\pm 0.23$ | $89.588_{-0.100}^{+0.12}$ |  23, t.w.1 | $<0.094$ | $2.66\pm 0.37$ | $6.47_{-0.99}^{+1.0}$ | $3.12_{-0.90}^{+1.3}$ | $3.10\pm 0.11$ | $0.0778\pm 0.0026$ | $513\pm 29$ | $11.6_{-2.4}^{+2.8}$
K2-3c | $6812.28013(95)$ | $24.646582(39)$ | $1.69\pm 0.17$ | $89.905_{-0.088}^{+0.066}$ |  23, t.w.1 | $<0.095$ | $1.01\pm 0.35$ | $3.3\pm 1.2$ | $3.7_{-1.5}^{+2.1}$ | $3.05_{-0.20}^{+0.16}$ | $0.1414\pm 0.0047$ | $381\pm 21$ | $3.51_{-0.73}^{+0.85}$
K2-3d | $6826.22347(53)$ | $44.556456(97)$ | $1.62\pm 0.18$ | $89.788_{-0.029}^{+0.033}$ |  23, t.w.1 | $<0.097$ | $<0.39$ | $<1.6$ | $<2.1$ | $<2.8$ | $0.2097\pm 0.0070$ | $312\pm 17$ | $1.60_{-0.33}^{+0.38}$
K2-12b | $6815.3701(29)$ | $8.28246(52)$ | $2.44_{-0.14}^{+0.30}$ | $87.6_{-5.4}^{+1.8}$ |  24, t.w.1 | $<0.50$ | $<2.1$ | $<5.2$ | $<1.6$ | $<2.9$ | $0.0792_{-0.0013}^{+0.0016}$ | $1026\pm 15$ | $185\pm 12$
K2-36b | $6810.8916(13)$ | $1.422614(38)$ | $1.430\pm 0.080$ | $84.45_{-0.48}^{+0.78}$ | 25 | $<0.093$ | $2.85\pm 0.92$ | $4.3\pm 1.4$ | $8.0_{-2.7}^{+3.1}$ | $3.31_{-0.17}^{+0.13}$ | $0.02288\pm 0.00010$ | $1328\pm 12$ | $519\pm 20$
K2-36c | $6812.84001(71)$ | $5.340888(86)$ | $3.20\pm 0.30$ | $86.917_{-0.056}^{+0.066}$ | 25 | $<0.089$ | $3.4\pm 1.2$ | $7.9\pm 2.8$ | $1.30_{-0.51}^{+0.71}$ | $2.88_{-0.20}^{+0.16}$ | $0.05528\pm 0.00023$ | $854\pm 8$ | $89.0\pm 3.4$
K2-38b | $6896.8786(54)$ | $4.01593(50)$ | $1.655\pm 0.096$ | $88.36_{-0.15}^{+0.17}$ |  26, 27, t.w.1 | $<0.11$ | $3.02\pm 0.43$ | $7.7_{-1.1}^{+1.2}$ | $9.3_{-1.9}^{+2.4}$ | $3.440_{-0.085}^{+0.080}$ | $0.0503\pm 0.0011$ | $1310\pm 22$ | $490\pm 35$
K2-38c | $6900.4752(33)$ | $10.56103(90)$ | $2.49\pm 0.22$ | $87.68_{-0.28}^{+0.31}$ |  26, 27, t.w.1 | $<0.086$ | $2.11\pm 0.37$ | $7.4_{-1.3}^{+1.4}$ | $2.63_{-0.72}^{+1.0}$ | $3.07\pm 0.11$ | $0.0959\pm 0.0022$ | $949\pm 16$ | $135.3\pm 9.5$
K2-79b | $7103.22750(84)$ | $10.99470(47)$ | $4.09_{-0.12}^{+0.17}$ | $88.44\pm 0.44$ | 28 | $<0.23$ | $2.63\pm 0.69$ | $9.2\pm 2.4$ | $0.73_{-0.20}^{+0.21}$ | $2.73_{-0.14}^{+0.11}$ | $0.0988_{-0.0023}^{+0.0017}$ | $1022\pm 20$ | $181\pm 15$
K2-96b / HD 3167b | $7394.37454(43)$ | $0.959641(11)$ | $1.670_{-0.100}^{+0.17}$ | $83.4_{-7.7}^{+4.6}$ |  29, t.w.1 | 0(fixed) | $3.56\pm 0.15$ | $4.97_{-0.23}^{+0.24}$ | $5.6_{-1.3}^{+1.2}$ | $3.227_{-0.072}^{+0.057}$ | $0.01796_{-0.00031}^{+0.00037}$ | $1774\pm 29$ | $1650\pm 110$
K2-96c / HD 3167c | $7394.9788(12)$ | $29.8454(12)$ | $3.00_{-0.21}^{+0.45}$ | $89.30_{-1.0}^{+0.50}$ |  29, t.w.1 | $<0.060$ | $2.58\pm 0.15$ | $11.13_{-0.74}^{+0.78}$ | $2.11_{-0.69}^{+0.51}$ | $3.060_{-0.11}^{+0.064}$ | $0.1776_{-0.0031}^{+0.0037}$ | $564\pm 9$ | $16.9\pm 1.1$
K2-106b | $7394.01140(00)$ | $0.571292(12)$ | $1.725\pm 0.039$ | $86.4_{-4.1}^{+2.5}$ |  30, t.w.1 | 0(fixed) | $6.50\pm 0.52$ | $8.21_{-0.74}^{+0.76}$ | $8.77_{-0.94}^{+1.0}$ | $3.432_{-0.045}^{+0.044}$ | $0.01326_{-0.00023}^{+0.00028}$ | $2300\pm 36$ | $4670\pm 300$
K2-106c | $7405.7316(44)$ | $13.33970(96)$ | $2.836_{-0.073}^{+0.086}$ | $89.35_{-0.46}^{+0.43}$ |  30, t.w.1 | $<0.13$ | $2.48\pm 0.66$ | $8.9\pm 2.4$ | $2.12_{-0.57}^{+0.58}$ | $3.03_{-0.13}^{+0.10}$ | $0.1083_{-0.0018}^{+0.0023}$ | $805\pm 12$ | $70.0\pm 4.5$
K2-110b | $7275.32992(61)$ | $13.86375(26)$ | $2.592_{-0.097}^{+0.098}$ | $89.35_{-0.24}^{+0.41}$ | 31 | $<0.13$ | $5.25\pm 0.89$ | $15.9\pm 2.7$ | $5.02_{-0.98}^{+1.1}$ | $3.366_{-0.087}^{+0.077}$ | $0.10207\pm 0.00083$ | $638\pm 11$ | $27.7\pm 2.0$
K2-111b | $7100.0768(19)$ | $5.35180(40)$ | $1.820_{-0.090}^{+0.11}$ | $86.43_{-0.21}^{+0.37}$ | 32 | $<0.100$ | $2.30\pm 0.30$ | $5.58_{-0.73}^{+0.74}$ | $5.0_{-1.0}^{+1.1}$ | $3.212_{-0.079}^{+0.073}$ | $0.05650\pm 0.00045$ | $1309\pm 19$ | $490\pm 30$
K2-131b | $7582.9360(11)$ | $0.3693038(91)$ | $1.690_{-0.058}^{+0.085}$ | $85.0_{-10.}^{+9.0}$ |  33, t.w.1 | 0(fixed) | $8.0\pm 1.3$ | $7.9\pm 1.3$ | $8.8_{-1.7}^{+1.9}$ | $3.426_{-0.084}^{+0.075}$ | $0.00936\pm 0.00014$ | $2223\pm 37$ | $4070\pm 280$
K2-135b / GJ 9827b | $7738.82586(26)$ | $1.2089819(71)$ | $1.577_{-0.031}^{+0.027}$ | $86.07_{-0.34}^{+0.41}$ | 34 | $<0.063$ | $4.31\pm 0.39$ | $5.14\pm 0.47$ | $7.19_{-0.76}^{+0.81}$ | $3.306_{-0.045}^{+0.042}$ | $0.01880\pm 0.00016$ | $1175\pm 14$ | $318\pm 16$
K2-135c / GJ 9827c | $7742.19930(73)$ | $3.648096(63)$ | $1.241_{-0.026}^{+0.024}$ | $88.19_{-0.18}^{+0.21}$ | 34 | $<0.094$ | $<0.75$ | $<1.3$ | $<3.7$ | $<2.9$ | $0.03925\pm 0.00033$ | $813\pm 10$ | $73.0\pm 3.7$
K2-135d / GJ 9827d | $7740.96115(45)$ | $6.201470(63)$ | $2.022_{-0.043}^{+0.046}$ | $87.443\pm 0.045$ | 34 | $<0.13$ | $1.73\pm 0.43$ | $3.53_{-0.88}^{+0.87}$ | $2.34_{-0.59}^{+0.62}$ | $2.926_{-0.13}^{+0.099}$ | $0.05590\pm 0.00046$ | $681\pm 8$ | $36.0\pm 1.9$
K2-141b | $7744.07160(22)$ | $0.2803244(15)$ | $1.510\pm 0.050$ | $86.3_{-3.6}^{+2.7}$ | 35 | 0(fixed) | $6.10\pm 0.39$ | $4.97_{-0.34}^{+0.35}$ | $7.93_{-0.91}^{+1.0}$ | $3.330_{-0.042}^{+0.041}$ | $0.00747\pm 0.00010$ | $2103\pm 56$ | $3260_{-330}^{+370}$
K2-141c | $7751.15460(00)$ | $7.74850(22)$ | $7.0_{-2.8}^{+4.6}$ | $87.2_{-2.0}^{+1.6}$ | 35 | $<0.092$ | $<3.2$ | $<8.0$ | $<0.081$ | $<1.9$ | $0.06830\pm 0.00091$ | $695\pm 18$ | $39.1\pm 4.3$
K2-167b | $6979.9368(25)$ | $9.97748(00)$ | $2.30_{-0.14}^{+0.21}$ | $88.6_{-2.0}^{+1.0}$ |  24, t.w.1 | $<0.47$ | $1.97_{-0.55}^{+1.1}$ | $6.5_{-1.5}^{+1.6}$ | $2.87_{-0.36}^{+0.24}$ | $3.073_{-0.071}^{+0.050}$ | $0.0910\pm 0.0025$ | $1174\pm 24$ | $317\pm 27$
K2-222b | $7399.0595(16)$ | $15.38857(88)$ | $2.350_{-0.070}^{+0.080}$ | $89.12_{-0.41}^{+0.55}$ | 28 | $<0.16$ | $2.29\pm 0.43$ | $8.7\pm 1.7$ | $3.68_{-0.77}^{+0.86}$ | $3.188_{-0.097}^{+0.085}$ | $0.1206\pm 0.0029$ | $871\pm 23$ | $96.0_{-9.7}^{+11}$
K2-262b / Wolf 503b | $8191.36145(11)$ | $6.001270(21)$ | $2.043\pm 0.069$ | $89.87\pm 0.13$ | 36 | $0.409\pm 0.085$ | $3.11\pm 0.39$ | $6.27_{-0.84}^{+0.85}$ | $4.03_{-0.64}^{+0.72}$ | $3.168_{-0.069}^{+0.063}$ | $0.05712_{-0.00045}^{+0.00063}$ | $789\pm 16$ | $64.7\pm 5.5$
K2-263b | $8111.1274(12)$ | $50.818947(94)$ | $2.41\pm 0.12$ | $89.240_{-0.070}^{+0.050}$ | 37 | $<0.15$ | $2.83\pm 0.39$ | $14.9\pm 2.1$ | $5.8_{-1.1}^{+1.3}$ | $3.399_{-0.078}^{+0.073}$ | $0.2573\pm 0.0030$ | $470\pm 7$ | $8.17\pm 0.52$
K2-312b / HD 80653b | $8134.42440(70)$ | $0.719573(21)$ | $1.613\pm 0.071$ | $82.1\pm 2.4$ | 38 | 0(fixed) | $3.62\pm 0.21$ | $5.72_{-0.35}^{+0.36}$ | $7.47_{-1.00}^{+1.2}$ | $3.333_{-0.046}^{+0.048}$ | $0.01661\pm 0.00019$ | $2463\pm 30$ | $6130\pm 310$
K2-418b / EPIC-22…35b | $7920.44584(80)$ | $16.141132(19)$ | $2.332_{-0.094}^{+0.080}$ | $88.08_{-0.24}^{+0.26}$ | 39 | $<0.23$ | $2.76\pm 0.39$ | $10.4_{-1.5}^{+1.6}$ | $4.49_{-0.81}^{+0.92}$ | $3.272_{-0.077}^{+0.070}$ | $0.1237\pm 0.0017$ | $804\pm 10$ | $69.7\pm 3.7$
Table 10: continued. 171717From left to right, the columns report the planet
name, the transit mid-time, the orbital period, the planetary radius, the
orbital inclination, the references for the transit parameters, the orbital
eccentricity, the radial-velocity semi-amplitude, the planet mass, density and
surface gravity, the semi-major axis, the equilibrium temperature by
considering a null Bond albedo and full heat redistribution from the day to
the night side, and the stellar incident flux. Table also available at the
CDS.

11footnotemark: 1

This work: the planet radius was newly determined from the $R_{\rm p}/R_{\rm
s}$ transit parameter in the literature and the stellar radius $R_{\rm s}$ as
reported in Table LABEL:table_system_parameters. 22footnotemark: 2The orbital
period of K2-2b comes from the RVs by imposing a Gaussian prior on $T_{\rm c}$
only, and slightly differs from the value reported in Vanderburg et al.
(2015), which is affected by systematics in the photometric data of the MOST
satellite (A. Vanderburg, private communication).

(1) Dumusque et al. 2014; (2) Bonomo et al. in prep.; (3) Ballard et al. 2011;
(4) Malavolta et al. 2017; (5) Buchhave et al. 2016; (6) Gajdoš et al. 2019
(7) López-Morales et al. 2016; (8) Borucki et al. 2012; (9) Holczer et al.
2016; (10) Barclay et al. 2013; (11) Margini et al. in prep.; (12) Howard et
al. 2013; (13) Ballard et al. 2014; (14) Marcy et al. 2014; (15) Dubber et al.
2019; (16) Bonomo et al. 2019; (17) Rowe et al. 2015; (18) Gettel et al. 2016;
(19) Mayo et al. 2019; (20) Haywood et al. 2018; (21) Coughlin et al. 2016;
(22) Vanderburg et al. 2015; (23) Kosiarek et al. 2019; (24) Mayo et al. 2018;
(25) Damasso et al. 2019; (26) Sinukoff et al. 2016; (27) Toledo-Padrón et al.
2020; (28) Nava et al. 2022; (29) Christiansen et al. 2017; (30) Guenther et
al. 2017; (31) Osborn et al. 2017; (32) Mortier et al. 2020; (33) Dai et al.
2017; (34) Rice et al. 2019; (35) Malavolta et al. 2018; (36) Polanski et al.
2021; (37) Mortier et al. 2018; (38) Frustagli et al. 2020; (39) Tronsgaard et
al. in prep.;