11institutetext:  Instituto de Astrofísica de Canarias (IAC), 38205 La Laguna,
Tenerife, Spain
11email<EMAIL_ADDRESS>22institutetext:  Departamento de Astrofísica, Universidad
de La Laguna (ULL), 38206 La Laguna, Tenerife, Spain 33institutetext:
Institute of Astronomy, Faculty of Physics, Astronomy and Informatics,
Nicolaus Copernicus University, Grudzia̧dzka 5, 87-100 Toruń, Poland
44institutetext:  Instituto de Astrofísica de Andalucía (IAA-CSIC), Glorieta
de la Astronomía s/n, 18008 Granada, Spain 55institutetext:  DTU Space,
National Space Institute, Technical University of Denmark, Elektrovej 328,
DK-2800 Kgs. Lyngby, Denmark 66institutetext:  Thomas Jefferson High School,
6560 Braddock Rd, Alexandria, VA 22312 USA 77institutetext:  George Mason
University, Department of Physics and Astronomy, 4400 University Drive MS 3F3,
Fairfax, VA 22030 USA 88institutetext:  Centro de Astrobiología (CSIC-INTA),
ESAC Campus, Camino Bajo del Castillo s/n, Villanueva de la Cañada, E-28692
Madrid, Spain 99institutetext:  NASA Exoplanet Science Institute,
Caltech/IPAC, Mail Code 100-22, 1200 E. California Blvd., Pasadena, CA 91125,
USA 1010institutetext:  NASA Ames Research Center, Moffett Field, CA 94035,
USA 1111institutetext:  Department of Physics and Astronomy, Michigan State
University, East Lansing, MI 48824, USA 1212institutetext:  Max Planck
Institute for Astronomy, Königstuhl 17, D-69117 Heidelberg, Germany
1313institutetext:  Department of Earth and Planetary Sciences, University of
California Riverside, 900 University Ave, Riverside, CA 92521, USA
1414institutetext:  NSF Astronomy and Astrophysics Postdoctoral Fellow
1515institutetext:  Department of Astronomy & Astrophysics, 525 Davey
Laboratory, The Pennsylvania State University, University Park, PA, 16802, USA
1616institutetext:  Center for Exoplanets and Habitable Worlds, 525 Davey
Laboratory, The Pennsylvania State University, University Park, PA, 16802, USA
1717institutetext:  Department of Physics and Astronomy, University of New
Mexico, 210 Yale Blvd NE, Albuquerque, NM 87106, USA 1818institutetext:
Department of Earth, Atmospheric and Planetary Sciences, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA 1919institutetext:
Department of Aeronautics and Astronautics, MIT, 77 Massachusetts Avenue,
Cambridge, MA 02139, USA 2020institutetext:  NCCR/PlanetS, Centre for Space &
Habitability, University of Bern, Bern 3012, Switzerland

2121institutetext:  Department of Physics and Kavli Institute for Astrophysics
and Space Research, Massachusetts Institute of Technology, Cambridge, MA
02139, USA 2222institutetext:  Department of Astrophysical Sciences, Peyton
Hall, 4 Ivy Lane, Princeton, NJ 08544, USA 2323institutetext:  Department of
Physics and McDonnell Center for the Space Sciences, Washington University,
St. Louis, MO 63130, USA 2424institutetext:  NASA Goddard Space Flight Center,
8800 Greenbelt Road, Greenbelt, MD 20771, USA 2525institutetext:  SETI
Institute, 189 Bernardo Ave., Suite 200, Mountain View, CA 94043, USA
2626institutetext:  Pappalardo Fellow 2727institutetext:  Royal Astronomical
Society, Burlington House, Piccadilly, London W1J 0BQ 2828institutetext:
Center for Astrophysics $|$ Harvard & Smithsonian, 60 Garden Street,
Cambridge, MA 02138, USA 2929institutetext:  Planetary Discoveries,
Fredericksburg, VA 22405, USA

# Revisiting the warm sub-Saturn TOI-1710b

The impact of stellar activity on the mass measurement

J. Orell-Miquel Revisiting the warm sub-Saturn TOI-1710bRevisiting the warm
sub-Saturn TOI-1710bRevisiting the warm sub-Saturn TOI-1710bRevisiting the
warm sub-Saturn TOI-1710b I. Carleo Revisiting the warm sub-Saturn
TOI-1710bRevisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-
Saturn TOI-1710bRevisiting the warm sub-Saturn TOI-1710b F. Murgas Revisiting
the warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn
TOI-1710bRevisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-
Saturn TOI-1710b G. Nowak Revisiting the warm sub-Saturn TOI-1710bRevisiting
the warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn
TOI-1710bRevisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-
Saturn TOI-1710bRevisiting the warm sub-Saturn TOI-1710b E. Pallé Revisiting
the warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn
TOI-1710bRevisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-
Saturn TOI-1710b R. Luque Revisiting the warm sub-Saturn TOI-1710bRevisiting
the warm sub-Saturn TOI-1710b T. Masseron Revisiting the warm sub-Saturn
TOI-1710bRevisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-
Saturn TOI-1710bRevisiting the warm sub-Saturn TOI-1710b J. Sanz-Forcada
Revisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn
TOI-1710b D. Dragomir Revisiting the warm sub-Saturn TOI-1710bRevisiting the
warm sub-Saturn TOI-1710b P. A. Dalba Revisiting the warm sub-Saturn
TOI-1710bRevisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-
Saturn TOI-1710bRevisiting the warm sub-Saturn TOI-1710b R. Tronsgaard
Revisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn
TOI-1710b J. Wittrock Revisiting the warm sub-Saturn TOI-1710bRevisiting the
warm sub-Saturn TOI-1710b K. Kim Revisiting the warm sub-Saturn
TOI-1710bRevisiting the warm sub-Saturn TOI-1710b C. Stibbards Revisiting the
warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn TOI-1710b K. I.
Collins Revisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn
TOI-1710b P. Plavchan Revisiting the warm sub-Saturn TOI-1710bRevisiting the
warm sub-Saturn TOI-1710b S. B. Howell Revisiting the warm sub-Saturn
TOI-1710bRevisiting the warm sub-Saturn TOI-1710b E. Furlan Revisiting the
warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn TOI-1710b L. A.
Buchhave Revisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-
Saturn TOI-1710b C. L. Gnilka Revisiting the warm sub-Saturn
TOI-1710bRevisiting the warm sub-Saturn TOI-1710b A. F. Gupta Revisiting the
warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn TOI-1710bRevisiting
the warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn TOI-1710b Th.
Henning Revisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn
TOI-1710b K. V. Lester Revisiting the warm sub-Saturn TOI-1710bRevisiting the
warm sub-Saturn TOI-1710b J. E. Rodriguez Revisiting the warm sub-Saturn
TOI-1710bRevisiting the warm sub-Saturn TOI-1710b N. J. Scott Revisiting the
warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn TOI-1710b H. P. Osborn
Revisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn
TOI-1710bRevisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-
Saturn TOI-1710b S. Villanueva Jr Revisiting the warm sub-Saturn
TOI-1710bRevisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-
Saturn TOI-1710bRevisiting the warm sub-Saturn TOI-1710b S. Seager Revisiting
the warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn
TOI-1710bRevisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-
Saturn TOI-1710bRevisiting the warm sub-Saturn TOI-1710bRevisiting the warm
sub-Saturn TOI-1710b J. N. Winn Revisiting the warm sub-Saturn
TOI-1710bRevisiting the warm sub-Saturn TOI-1710b J. M. Jenkins Revisiting the
warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn TOI-1710b R.
Vanderspek Revisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-
Saturn TOI-1710b D. W. Latham Revisiting the warm sub-Saturn
TOI-1710bRevisiting the warm sub-Saturn TOI-1710b P. Rowden Revisiting the
warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn TOI-1710b D. Watanabe
Revisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn
TOI-1710b G. Torres Revisiting the warm sub-Saturn TOI-1710bRevisiting the
warm sub-Saturn TOI-1710b C. J. Burke Revisiting the warm sub-Saturn
TOI-1710bRevisiting the warm sub-Saturn TOI-1710b T. Daylan Revisiting the
warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn TOI-1710b T. Barclay
Revisiting the warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn
TOI-1710b J. D. Twicken Revisiting the warm sub-Saturn TOI-1710bRevisiting the
warm sub-Saturn TOI-1710bRevisiting the warm sub-Saturn TOI-1710bRevisiting
the warm sub-Saturn TOI-1710b G. R. Ricker Revisiting the warm sub-Saturn
TOI-1710bRevisiting the warm sub-Saturn TOI-1710b

(Received October 3, 2023; accepted January 12, 2024)

The Transiting Exoplanet Survey Satellite (TESS) provides a continuous suite
of new planet candidates that need confirmation and precise mass determination
from ground-based observatories. This is the case for the G-type star
TOI-1710, which is known to host a transiting sub-Saturn planet
($\mathrm{M_{p}}$ $=$ 28.3 $\pm$ 4.7 $\mathrm{M}_{\oplus}$) in a long-period
orbit ($P$ = 24.28 d). Here we combine archival SOPHIE and new and archival
HARPS-N radial velocity data with newly available TESS data to refine the
planetary parameters of the system and derive a new mass measurement for the
transiting planet, taking into account the impact of the stellar activity on
the mass measurement. We report for TOI-1710 b a radius of $\mathrm{R_{p}}$
$=$ 5.15 $\pm$ 0.12 $\mathrm{R}_{\oplus}$, a mass of $\mathrm{M_{p}}$ $=$ 18.4
$\pm$ 4.5 $\mathrm{M}_{\oplus}$, and a mean bulk density of $\rho_{\rm p}$ $=$
0.73 $\pm$ 0.18 $\mathrm{g\,cm^{-3}}$, which are consistent at 1.2$\sigma$,
1.5$\sigma$, and 0.7$\sigma$, respectively, with previous measurements.
Although there is not a significant difference in the final mass measurement,
we needed to add a Gaussian process component to successfully fit the radial
velocity dataset. This work illustrates that adding more measurements does not
necessarily imply a better mass determination in terms of precision, even
though they contribute to increasing our full understanding of the system.
Furthermore, TOI-1710 b joins an intriguing class of planets with radii in the
range 4–8 $\mathrm{R}_{\oplus}$ that have no counterparts in the Solar System.
A large gaseous envelope and a bright host star make TOI-1710 b a very
suitable candidate for follow-up atmospheric characterization.

###### Key Words.:

stars: individual: TOI-1710 – planetary systems: individual: TOI-1710b –
techniques: photometric – techniques: radial velocities

##  1 Introduction

The Transiting Exoplanet Survey Satellite (TESS; Ricker et al., 2015) is a
NASA-sponsored space telescope launched on April 18, 2018. The original TESS
mission was a two-year full-sky survey to search for transiting planets, but
the mission obtained a first extension of observations covering from July 2020
to September 2022. TESS is currently in its second extended mission, which
will last at least until September 2025. One of the TESS mission’s main goals
is to look for 50 small planets with $\mathrm{R_{P}}$¡4 $\mathrm{R}_{\oplus}$
suitable for atmospheric characterization. However, during that search TESS
has detected planets with a wide range of radii. In particular, TOI-216b
(Dawson et al., 2021), TOI-421c (Carleo et al., 2020), TOI-674b (Murgas et
al., 2021), LTT 9779b (Jenkins et al., 2020), and TOI-257b (Addison et al.,
2021) are some examples of confirmed planets alerted by TESS that have sizes
between Uranus (4 $\mathrm{R}_{\oplus}$) and Saturn (9.5
$\mathrm{R}_{\oplus}$); they are called sub-Saturns (as defined in Petigura et
al., 2017).

Sub-Saturns are a very interesting group of exoplanets to study because they
have no counterparts in our Solar System. Their large sizes can be explained
by a heavy metal core with a significant H/He envelope (Lopez & Fortney,
2014). Since sub-Saturns are a lighter version of Jovian planets, they could
potentially offer new benchmarks to study different envelope accretion
scenarios for gas-giant planets. Runaway accretion, suggested as a possible
mechanism for the formation of Jupiter-like planets (Pollack et al., 1996;
Hubickyj et al., 2005), does not manage to explain these low-density planets;
alternative processes have been proposed, such as accretion within a gas-
depleted disk (Lee & Chiang, 2015).

As is shown in Petigura et al. (2017) and Nowak et al. (2020), systems holding
sub-Saturn planets tend to present different characteristics and
architectures, depending on whether they are single-planet or multi-planet
systems. Lone sub-Saturns are often more massive and orbit their host stars on
more eccentric orbits than those in multi-planetary systems. These differences
suggest that the presence of another planet (or planets) might play an
important role in the formation of sub-Saturns. Dynamical instabilities
between planets could result in mergers or scattering to high-inclination
orbits, establishing the observed architectures with high-mass planets in low-
multiplicity systems (Petigura et al., 2016, 2017; Van Eylen et al., 2018).

TOI-1710 b (König et al., 2022) is a transiting sub-Saturn planet on a 24-day
orbit around a bright (V = 9.5 mag, J = 8.3 mag) G5-type star. TOI-1710 (BD+76
227) is at a distance of 81 pc and is located near the Camelopardalis
constellation. König et al. (2022) used SOPHIE and HARPS-N radial velocities
(RVs) to derive for TOI-1710 b a radius of $\mathrm{R_{p}}$ $=$ 5.34 $\pm$
0.11 $\mathrm{R}_{\oplus}$ and a mass of $\mathrm{M_{p}}$ $=$ 28.3 $\pm$ 4.7
$\mathrm{M}_{\oplus}$. They also explored the influence of the stellar
activity of the host star on their dataset, and found no significant impact.

Here we combined the previously published RVs from König et al. (2022) with
new HARPS-N observations to derive a refined mass measurement of TOI-1710 b
considering the impact of the stellar activity of its host star. Furthermore,
we included in our joint fit the new available TESS photometric data to refine
its orbital period, and other system parameters.

This paper is organized as follows. In Section 2 we describe the observations
used in this work. The stellar properties of TOI-1710 are reported in Section
3. In Section 4 we explain the methods used in the data analysis. The results
and conclusions are discussed and presented in Section 5.

##  2 Observations

###  2.1 TESS photometry

Listed as TIC 445805961 in the TESS Input Catalog (TIC; Stassun et al., 2018)
and then later classified as the TESS object of interest TOI-1710, this star
was observed by TESS in 2 min cadence integrations in Sectors 19, 20, 26, 40,
53, 59, and 60. It is not scheduled to be observed again until Sector 73,
according to TESS-point Web Tool.111https://heasarc.gsfc.nasa.gov/wsgi-
scripts/TESS/TESS-point_Web_Tool/TESS-point_Web_Tool/wtv_v2.0.py/

The TESS raw data are reduced by the Science Processing Operations Center
(SPOC; Jenkins et al., 2016) at the NASA Ames Research Center and are publicly
available at the Mikulski Archive for Space Telescopes
(MAST222https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html). The
light curves were extracted using simple aperture photometry (SAP; Morris et
al., 2020) and corrected from systematics using the Presearch Data
Conditioning (PDC) pipeline (Smith et al., 2012; Stumpe et al., 2012, 2014).
For this work we used the PDC-corrected SAP photometry, which is the TESS
product also used by König et al. (2022). The TESS pixels considered to
compute the SAP and PDC-corrected SAP for Sectors 19, 20, 26, 40, 53, 59, and
60 are shown in Figure 8. A single transit was detected in the Sector 19 PDC-
SAP flux time series, and the correct period can be identified in a combined
search of the available sectors.

On February 19, 2020, the star was announced under the TESS Object of Interest
(TOI) number 1710 as a possible host for a transiting planet in the TOI
catalogue333https://tev.mit.edu/data/ (Guerrero et al., 2021). A signal with a
period of $24.28$ days and a transit depth ($\Delta F$) of 3070.0 ppm,
corresponding to a planet radius of about 5.4 $\mathrm{R}_{\oplus}$, was
detected in the Quick Look Pipeline (QLP; Huang et al., 2020a, b ). All SPOC
data validation (Twicken et al., 2018; Li et al., 2019) diagnostic test
results for this signal are fully consistent with a transiting planet
associated with the host star (within 0.7 $\pm$ 2.6$"$) and excluded all other
TIC objects.

###  2.2 Ground-based photometry

We observed a full transit of TOI-1710.01 on the night of February 29, 2020,
with the George Mason University Observatories 0.8 m telescope. We used the
TESS Transit Finder to schedule our transit observation (Jensen, 2013). We
utilized AstroImageJ (Collins et al., 2017) for data reduction, plate-solving,
aperture photometry, light curve extraction, and detrending. Due to
differences in quality and scattering between space- and ground-based
measurements, we analysed the two datasets individually, focusing our
photometric analyses on the TESS light curves, and we did not include these
observations in the joint fit modelling in Sect. 4.3. However, ground-based
observations are key to the independent confirmation and validation of TESS
candidates.

###  2.3 High-resolution spectroscopy with SOPHIE

König et al. (2022) performed a ground-based follow-up campaign with the
SOPHIE spectrograph. They obtained a total of 30 RV measurements acquired from
September 14, 2020, to May 7, 2021. The details of these observations are
explained in Sect. 2.2.2 of König et al. (2022).

###  2.4 High-resolution spectroscopy with HARPS-N

TOI-1710 was observed with the High Accuracy Radial velocity Planet Searcher
for the Northern hemisphere (HARPS-N; Cosentino et al., 2012) mounted on the
3.6m Telescopio Nazionale Galileo (TNG) at the Roque de los Muchachos
Observatory, La Palma. The star was monitored first from October 27, 2020, to
April 5, 2021, and then from September 12, 2021, to April 4, 2022. We obtained
a total of 54 high-resolution ($R\sim 115\,000$) spectra. The observations
were carried out as part of observing programs CAT20B_41 and CAT21A_24 (PI:
Pallé). The exposure times varied from 680 to 2400 seconds, depending on
weather conditions and scheduling constraints, leading to a S/N per pixel of
33–106 at 5500 Å. The spectra were extracted using the offline version of the
HARPS-N DRS pipeline (Cosentino et al., 2014), version 3.7. Doppler
measurements (absolute RVs) and spectral activity indicators (cross-
correlation function full width at half maximum, FWHM; cross-correlation
function contrast, CTR; cross-correlation function bisector, BIS; and $R_{\rm
HK}$) were measured using an online version of the DRS, the YABI
tool,444Available at http://ia2-harps.oats.inaf.it:8000. by cross-correlating
the extracted spectra with a G2 mask (Baranne et al., 1996). We also used the
serval code (Zechmeister et al., 2018) to measure relative RVs by template-
matching, chromatic index (CRX), differential line width (dLW), and H$\alpha$,
sodium Na D1 and Na D2 indexes. The uncertainties of the RVs measured with
serval are in the range 0.8–2.6 $\mathrm{m\,s^{-1}}$, with a mean value of 1.4
$\mathrm{m\,s^{-1}}$. The uncertainties of absolute RVs measured with DRS are
in the range 0.8–3.2 $\mathrm{m\,s^{-1}}$, with a mean value of 1.5
$\mathrm{m\,s^{-1}}$. Table 6 gives the time stamps of the spectra in BJDTDB
serval relative RVs along with their $1\sigma$ error bars, and spectral
activity indicators measured with YABI and serval.

König et al. (2022) also performed a ground-based follow-up campaign with
HARPS-N, and obtained 31 measurement acquired from October 3, 2020, to April
19, 2021. König et al. (2022) measurements from SOPHIE and HARPS-N are
contemporaneous with our first observing campaign. Moreover, there are no
significant differences between the two HARPS-N datasets: consistent exposure
times, similar S/N per pixel, and measured RV uncertainties, and the same
instrumental mode. For consistency, we reduced their HARPS-N spectra together
with our HARPS-N observations, with the YABI tool and the HARPS-N DRS
pipeline. The RV values (from the König et al. 2022 spectra) extracted using
the DRS pipeline are exactly the same as those presented in König et al.
(2022, Table C.2). We ran the serval pipeline over both datasets, considering
them a single dataset of 85 HARPS-N RVs. The RVs extracted by serval are used
in the analyses of this work.

###  2.5 High-resolution imaging with Gemini/‘Alopeke

Figure 1:  Gemini/‘Alopeke high-resolution image of TOI-1710 take on February
9, 2021. TOI-1710 is a single star to contrast limits of 5.5 to 7 mag within
1.6 AU out to 97 AU (d = 81 pc). The inset shows a 1.2”$\times$1.2”
reconstructed image of TOI-1710 at 832 nm band, centred on the star.

If an exoplanet host star has a spatially close companion, that companion
(bound or line of sight) can create a false-positive transit signal if it is,
for example, an eclipsing binary. Additionally, ‘third-light’ flux from the
close companion star can lead to an underestimated planetary radius if not
accounted for in the transit model (Ciardi et al., 2015) and cause non-
detections of small planets residing within the same exoplanetary system
(Lester et al., 2021). The discovery of close-in bound companion stars, which
exist in nearly one-half of FGK-type stars (Matson et al., 2018), provides
crucial information towards our understanding of exoplanetary formation,
dynamics, and evolution (Howell et al., 2021). Thus, to search for close-in
bound companions, unresolved in TESS or other ground-based follow-up
observations, we obtained high-resolution imaging speckle observations of
TOI-1710.

TOI-1710 was observed on February 9, 2021, using the ‘Alopeke speckle
instrument on the Gemini North 8 m
telescope.555https://www.gemini.edu/sciops/instruments/alopeke-zorro/ ‘Alopeke
provides simultaneous speckle imaging in two bands (562 nm and 832 nm) with
output data products including a reconstructed image with robust contrast
limits on companion detections (e.g. Howell et al., 2016). Seven sets of
1000$\times$0.06 sec exposures were collected and subjected to Fourier
analysis in our standard reduction pipeline (see Howell et al., 2011). Figure
1 shows our final contrast curves and the 832 nm reconstructed speckle image.
We find that TOI-1710 is a single star with no companion brighter than 5.5–7
magnitudes below that of the target star from the diffraction limit (20 mas)
out to 1.2”. At the distance of TOI-1710 (d = 81 pc) these angular limits
correspond to spatial limits of 1.6 to 97 AU.

##  3 Stellar system analysis

Table 1:  Stellar parameters of TOI-1710.

Parameter | Value | Reference
---|---|---
Name and identifiers
TIC | 445805961 | TESS
TOI | 1710 | TOI
BD | +76 227 | BD
TYC | 4525–1009–1 | TYC
2MASS | J06170789+7612387 | 2MASS
Gaia DR2 | 1116613161053977472 | Gaia
Coordinates and spectral type
$\alpha$ (J2000) | 06${}^{\mathrm{h}}\,$17m 07${}^{\mathrm{s}}\\!$.86 | Gaia
$\delta$ (J2000) |  $+$76º 12′ 38$"\\!$.81 | Gaia
Spectral type | G5 V | W03
Parallax and kinematics
$\pi$ [mas] | 12.2823 $\pm$ 0.0266 | Gaia
$d$ [pc] | 81.42 $\pm$ 0.18 | Gaia
$\mu_{\alpha}\cos{\delta}$ [mas year-1] | 59.837 $\pm$ 0.037 | Gaia
$\mu_{\delta}$ [mas year-1] | 55.610 $\pm$ 0.044 | Gaia
$V_{\rm r}$ [$\mathrm{km\,s^{-1}}$] |  $-$39.40 $\pm$ 0.35 | Gaia
|  $-$38.8134 $\pm$ 0.0007 | Sec. 2.4
Magnitudes
B [mag] | 10.20 $\pm$ 0.03 | TYC
V [mag] | 9.54 $\pm$ 0.02 | TYC
G [mag] | 9.3598 $\pm$ 0.0002 | Gaia
T [mag] | 8.9134 $\pm$ 0.006 | TESS
J [mag] | 8.319 $\pm$ 0.019 | 2MASS
H [mag] | 8.003 $\pm$ 0.034 | 2MASS
K [mag] | 7.959 $\pm$ 0.026 | 2MASS
Stellar parameters
Mass $M_{\star}$ [$M_{\odot}$] | 0.99 $\pm$ 0.07 | Sec. 3.3
| 0.984${}^{+0.050}_{-0.059}$ | K22
Radius $R_{\star}$ [$R_{\odot}$] | 0.95 $\pm$ 0.02 | Sec. 3.3
| 0.968${}^{+0.016}_{-0.014}$ | K22
Luminosity $L_{\star}$ [$L_{\odot}$] | 0.895 $\pm$ 0.003 | Gaia
Effective temperature $T_{\rm eff}$ [K] | 5730 $\pm$ 30 | Sec. 3.3
| 5665 $\pm$ 55 | K22
Surface gravity $\log$(g) | 4.54 $\pm$ 0.09 | Sec. 3.3
| 4.46 $\pm$ 0.10 | K22
Metallicity $\rm[Fe/H]$ | 0.12 $\pm$ 0.06 | Sec. 3.3
| 0.10 $\pm$ 0.07 | K22
Age [Gyr] | 2.8 $\pm$ 0.6 | Sec. 3.2
| 4.2${}^{+4.1}_{-2.7}$ | K22
log(R${}^{{}^{\prime}}_{HK}$) |  $-$4.786 $\pm$ 0.012 | Sec. 3.2
|  $-$4.78 $\pm$ 0.03 | K22

TESS: Stassun et al. (2018, 2019), TOI: Guerrero et al. (2021), BD: Argelander
(1903), TYC: Høg et al. (2000); Gaia: Gaia Collaboration et al. (2018); Gaia
Collaboration (2020); 2MASS: Cutri et al. (2003); W03: Wright et al. (2003);
K22: König et al. (2022).

###  3.1 Stellar companions

Given the fainter TOI-1710 companion labelled $\\#$2 in Figure 8 and the large
TESS pixel size of 21”, it is crucial to verify that no visually close-by
targets are present. In Section 2.5 we ruled out this scenario within 1.2”
around TOI-1710 (see Fig. 1). To go wider, we extended the exploration of
close companions until 60” around the star position using the Gaia DR2 (Gaia
Collaboration et al., 2018) database. The main astrometric parameters and
properties of the five found objects, relative to TOI-1710, are included in
Table 2.

Gaia DR2 1116613161052594560 at 7.2” with GRP = 17.4 mag is the only target
within one TESS pixel, but it is much fainter than TOI-1710 (GRP = 8.9 mag).
Stars with differences in Gaia GRP-band larger than 8 mag are not detected in
the TESS images (Figure 8). Only Gaia DR2 1116612783096856960, which has
$\Delta$ GRP = 4.2 mag, is detected. We identified this star as TIC 445805957,
which is the southern star labelled $\\#$2 in Figure 8. Since star $\\#$2 has
a parallax and proper motion similar to that of TOI-1710, we can assume that
they are probably gravitationally bounded. This is in agreement with Mason et
al. (2001) and Tian et al. (2020), who reported this pair of stars as a wide
binary system.

Because the Gaia GRP-band (630–1050 nm) and the TESS band (600–1000 nm) are
very much alike, we could estimate the dilution factor for the TESS photometry
(D${}_{\textit{TESS}}$) using its definition from Eq. 6 in Espinoza et al.
(2019). We calculated a D${}_{\textit{TESS}}$ = 0.988 considering all the
nearby stars, corresponding to 1.2% flux contamination. However, the PDC-
corrected SAP light curve, which is the data used in the photometry and joint
fit analyses (Sections 4.2 and 4.3), already takes into account the possible
flux contamination by the nearby companion, which is by far the brightest
close star. Excluding the flux from TIC 445805957 from the calculations,
D${}_{\textit{TESS}}$ is 0.999 (i.e. consistent with little to no flux
contamination).

Table 2:  Relative properties to TOI-1710 of the nearby stars. TOI-1710 wide
binary companion Gaia DR2 identifier is marked in bold.

Gaia | Separation |  $\Delta$ Parallax |  $\Delta$ Proper motion |  $\Delta$ GRP
---|---|---|---|---
DR2 | [ ” ] | [ mas ] | [ mas/year ] | [ mag ]
1116613161052594560 | 7.2 $\pm$ 0.1 | 12.0 $\pm$ 0.2 | 72.66 $\pm$ 0.14 | 8.545 $\pm$ 0.025
1116613225476795776 | 30.8 $\pm$ 0.1 | 12.30 $\pm$ 0.25 | 79.35 $\pm$ 0.14 | 9.153 $\pm$ 0.020
1116613195412077696 | 37.1 $\pm$ 0.1 | 10.9 $\pm$ 0.7 | 72.2 $\pm$ 0.5 | 10.292 $\pm$ 0.050
1116613225476796288 | 42.1 $\pm$ 0.1 | 12.1 $\pm$ 0.3 | 80.02 $\pm$ 0.17 | 9.592 $\pm$ 0.020
1116612783096856960 | 43.3 $\pm$ 0.1 | 0.0 $\pm$ 0.06 | 0.087 $\pm$ 0.003 | 4.2307 $\pm$ 0.0018

###  3.2 Stellar rotation

Based on the Noyes et al. (1984) and Mamajek & Hillenbrand (2008) activity-
rotation relations and using (B-V) of 0.658 and the
$\log\,R^{{}^{\prime}}_{HK}$ measured with YABI, we estimated a rotation
period of TOI-1710 for 20.9 $\pm$ 4.2 days and 20.6 $\pm$ 3.2 days,
respectively. Using the activity-age relation of Mamajek & Hillenbrand (2008),
we also found the age of TOI-1710 to be in the range of 2.2–3.9 Gyr.

We searched for photometric time series from automated ground-based surveys to
detect long-term photometric modulations associated with the stellar rotation.
The Northern Sky Variability Survey (NSVS; Woźniak et al., 2004) is the only
public survey in which the search was successful. The search parameters were
the TOI-1710 coordinates from Table 1 and a radius of 1’ around it.

The query returned two objects labelled with different identifiers: ($i$)
$\\#$594837 (RA = 06:17:07.86, Dec = +76:12:38.23, and magnitude = 9.657 $\pm$
0.011 mag with a scatter magnitude of 0.024 mag) and ($ii$) $\\#$628193 (RA =
06:17:07.88, Dec = +76:12:38.41, and magnitude = 9.671 $\pm$ 0.011 mag with a
scatter magnitude of 0.021 mag). Objects $\\#$594837 and $\\#$628193 are only
separated from each other by 0.20” and they are 0.58” and 0.41” apart from
TOI-1710, respectively. Their magnitudes are coincident within the scatter
magnitudes and they are only 0.1 mag apart from TOI-1710 V-band magnitude.
Since we already confirmed in Section 3.1 that there is no other star of
similar magnitude within 60” around TOI-1710, we can safely assume that the
NSVS observations of $\\#$594837 and $\\#$628193 correspond to the same
object, TOI-1710.

Figure 2:  Dataset (top) and generalized Lomb-Scargle periodogram (bottom) of
Northern Sky Variability Survey photometric data for TOI-1710. The vertical
red dashed line marks the 24.28 d period. The 10%, 1%, and 0.1% FAP levels are
indicated by horizontal grey dotted, dash-dotted, and dashed lines,
respectively.

We searched for photometric periodicities in the total of 117 NSVS
measurements using the generalized Lomb-Scargle (GLS)
periodogram666https://github.com/mzechmeister/GLS (Zechmeister & Kürster,
2009). In addition, we computed the theoretical false alarm probability (FAP),
as described in Zechmeister & Kürster (2009). The photometry and its GLS
periodogram are shown in Figure 2. A baseline of 216 days should be enough to
detect any significant photometric modulation in the expected range for the
stellar rotational period. The only relevant signal is a broad double peak in
the range 20–24 d with a FAP close to the 1%, and consistent with the $P_{\rm
rot}$ reported by König et al. (2022) of $\sim$22 d.

###  3.3 Stellar parameters

The analysis of the stellar spectrum was carried out by using the BACCHUS code
(Masseron et al., 2016) relying on the MARCS model atmospheres (Gustafsson et
al., 2008) and using the co-added HARPS-N spectra. In brief, an effective
temperature of 5730 $\pm$ 10 K was derived by requiring no trend of the Fe I
lines abundances against their respective excitation potential. The surface
gravity was determined by requiring the ionization balance of the Fe I and Fe
II lines. A microturbulence velocity value of 0.89 $\pm$ 0.05
$\mathrm{km\,s^{-1}}$ was also derived by requiring no trend of Fe line
abundances against their equivalent widths. The output metallicity ($+0.12$
$\pm$ 0.06) is represented by the average abundance of the Fe I lines. An
alternative analysis of the HARPS-N spectrum was done with the SPC pipeline
(Buchhave et al., 2012). The results obtained with this pipeline are $T_{\rm
eff}$ = 5720 $\pm$ 28, $\log g$ = 4.49 $\pm$ 0.04, and $\rm[M/H]$ = $+$0.06
$\pm$ 0.04. These results are consistent with the parameters derived with the
BACCHUS pipeline. Nevertheless, the internal error in temperature that was
derived for example by the BACCHUS code (10 K) or the SPC code (28 K) are
unrealistically small as they do not include systematic errors. One of the
main known sources of uncertainties are NLTE or 3D effects that are not
considered in either of the 1D/LTE analyses of the HARPS-N spectrum. In order
to try to better estimate its global error, the effective temperature was also
derived with photometry. While the spectroscopic method we employed relies on
individual line strengths of the observed stellar spectrum, the photometric
method relies on the relative flux of the star and its observed colours. In
that sense, one can consider that the spectroscopic and the photometric method
provide fairly independent results, and thus better reflect the absolute
uncertainty on temperature. Nevertheless, the photometric method also has its
own uncertainties. To minimize the error, we chose the V-K colour index. This
index has been demonstrated to be the most robust and most reliable
temperature indicator because it offers a wider baseline of the spectral
energy distribution and because it is less dependent on other parameters, such
as metallicity and surface gravity (Bessell et al., 1998; Casagrande et al.,
2010). Another known source of uncertainty of the photometric method resides
in the absolute calibration that sets the zero-point of the $T_{\rm eff}$
scale; this calibration varies from one study to another. By using the V-K
colour-temperature relation of González Hernández & Bonifacio (2009) and that
of Casagrande et al. (2010) and assuming no reddening for such a close-by
star, we obtained a temperature of 5669 $\pm$ 32 K and 5719 $\pm$ 25 K,
respectively. While all photometric and spectroscopic temperatures are in good
agreement, we chose to compute the error in temperature by taking half of the
difference between the most extreme spectroscopic and photometric
temperatures, which are the BACCHUS and the González Hernández & Bonifacio
(2009) temperatures, leading to a total error of 30 K.

In a second step, we used the Bayesian tool PARAM (Rodrigues et al., 2014,
2017) to derive the stellar mass, radius, and age utilizing the spectroscopic
parameters and the updated Gaia luminosity along with our spectroscopic
temperature. However, these Bayesian tools underestimate the error budget as
they do not take into account the systematic errors between one set of
isochrones to another, due to the various underlying assumptions in the
respective stellar evolutionary codes. In order to take into account those
systematic errors, we combined the results of the two sets of isochrones
provided by PARAM (i.e. MESA and Parsec) and added the difference between the
two sets of results to the error budget provided by PARAM. We obtained a
stellar radius and mass of $\rm 0.95\pm 0.02\,R_{\odot}$ and $\rm 0.99\pm
0.07\,M_{\odot}$, respectively. We note, however, that although using two sets
of isochrones may mitigate underlying systematic errors, our formal error
budget for radius and luminosity may still be underestimated, as demonstrated
by Tayar et al. (2022). For solar-type stars such as TOI-1710, absolute errors
may instead be up to 4%, 2%, 5%, and 20% respectively for radius, luminosity,
mass, and age.

The derived age from the isochrones (3.2 $\pm$ 3.1 Gyr) do not provide
constraining information as age is very degenerate for stars on the main
sequence (see Fig. 3) . Therefore, we rely on the study of Mamajek &
Hillenbrand (2008) for the choice of the age (2.8 $\pm$ 0.6 Gyr) derived from
the empirical R${}^{{}^{\prime}}_{HK}$-age relation.

All stellar parameters adopted in our joint modelling of the system (presented
in Section 4) are summarized in Table 1. In Table 1 are also displayed the
values provided by König et al. (2022). The values are in good agreement,
although we derive a slightly higher temperature. The method employed to
derive the temperature in König et al. (2022) is very similar to ours, and the
small discrepancy is likely due to the differences in the model atmosphere,
radiative transfer codes, and the selection of Fe lines, as demonstrated in
Jofré et al. (2014). While such a difference in temperature consistently leads
to a slightly higher mass and smaller radius between the two studies,
$M_{\star}$, $R_{\star}$, log(g), metallicity, age, and
log(R${}^{{}^{\prime}}_{HK}$) are still in agreement within the error bars.

Figure 3: Location of the host star in the spectroscopic H-R diagram. The red
point and error bars indicate the spectroscopic parameters of the host star,
while the solid, short-dashed, and long-dashed lines show isochrones for 2.5,
3.0, and 4.0 Gyr, respectively.

##  4 Analysis

###  4.1 Frequency analysis and RV correlations

Figure 4: Generalized Lomb-Scargle periodograms of HARPS-N, SOPHIE, and
combined RV measurements (from top to bottom). Vertical red dotted line
indicates the period of TOI-1710 b at 24.28 d. Shaded orange region indicates
the 3$\sigma$ region of the $P_{\rm rot}$ from the joint fit (see Sect. 4.3).
The graph shows 10%, 1% and 0.1% FAP levels in grey dotted, dash-dotted, and
dashed, respectively.

Our RV dataset involves a total of 115 measurements from HARPS-N (85) and
SOPHIE (30) spectrographs with a time baseline of 217 days. We searched for
TOI-1710 b and other signals in the HARPS-N, SOPHIE, and combined RV datasets
computing their GLS periodograms, which are shown in Figure 4. The signal of
the planet is well detected in the RV datasets with FAP $\lesssim$ 1 %.
However, the biggest peak in the HARPS-N and the combined dataset periodograms
is at $\sim$20 days, which is consistent with the rotation period of the star
derived in Sect. 3.2 and by König et al. (2022). Thus, the RVs analysed in
this work have two clear signals that have to be modelled: the transiting
planet TOI-1710 b at 24.28 d, and the stellar rotation at 20–22 d. Due to the
proximity of the two signals, they should be treated with caution when
analysing the RVs.

We computed the GLS periodogram of the HARPS-N activity parameters measured
with YABI and serval (see Figure 12). In general, all the activity
periodograms present peaks in the region where the stellar rotation is
expected (20–24 d). These peaks are significant (FAP ¡ 1 %) for CTR, FWHM,
dLW, Na D1, and $R_{\rm HK}$. All the activity indicators except CRX and Na D2
also present significant peaks at $\sim P_{\rm rot}/2$. König et al. (2022)
also presented the GLS periodograms for FWHM, H$\alpha$, and log(R‘HK) with
significant peaks close to the stellar rotation period. Moreover, the HARPS-N
CTR, FWHM, dLW, and H$\alpha$ index have strong correlations with the HARPS-N
RVs (see Fig. 13). Similar correlations are obtained when we split the HARPS-N
dataset in two (before and after September 2021), indicating that the stellar
correlation did not come (only) from the second epoch and that it is present
over all the observations. The stellar rotation detected in the RV GLS
periodogram and the correlation of the RVs with the FWHM, for example, are not
negligible.

###  4.2 Photometric modelling

In a first step, previous to the RV and photometry simultaneous analysis, we
studied the photometric data alone. Although TOI-1710 was observed in Sectors
19, 20, 26, 40, 53, 59, and 60, the TOI-1710 b expected transit in Sector 60
fell in the mid-sector observing gap. Thus, we did not consider Sector 60 in
our analysis because it has no 1710 b transits, and to save computational
time. We analysed the TESS data from Sectors 19, 20, 26, 40, 53, and 59 with
the Python library juliet (Espinoza et al., 2019). Juliet performs the fitting
procedure using other public packages for modelling of transit light curves
(batman, Kreidberg, 2015) and GPs (celerite, Foreman-Mackey et al., 2017).
Instead of using the Markov chain Monte Carlo (MCMC) technique, juliet uses a
nested sampling algorithm (dynesty, Speagle, 2020; MultiNest, Feroz et al.,
2009; Buchner et al., 2014) to explore the whole the parameter space. We
considered the uninformative sample ($r_{1}$,$r_{2}$) parametrization
introduced in Espinoza (2018) to explore the impact parameter of the orbit
($b$) and the planet-to-star radius ratio ($p$ $=$
${R}_{\mathrm{p}}/{R}_{\star}$) values. In the fitting procedure we adopted a
quadratic limb-darkening law with the ($q_{1}$,$q_{2}$) parametrization
introduced by Kipping (2013). According to the flux contamination exploration
in Sect. 3.1, we can safely fix the dilution factor to 1. We added a celerite
GP exponential kernel to the transiting planet model to account for the extra
correlated noise in the photometric data.

The fitted parameters with their prior and posterior values, and the derived
parameters are shown in Table 5. The TESS data along with the transiting and
GP models are shown in Fig. 9. The TOI-1710 b phase-folded photometry from the
juliet anaysis is shown in Fig. 10. The results from the photometric data
analysis are consistent with the planetary characteristics from König et al.
(2022).

Furthermore, we analysed the ground-based TOI-1710 b transit modelling the
airmass-detrended light curve with ExoFASTv2 (Eastman et al., 2019) to
validate the detection. The independent analysis confirmed the results derived
from the TESS light curves, and we recovered a $8\sigma$ significance and on-
time and on-target transit with consistent depth, duration, and timing with
the TESS ephemerides. Ground-based photometry along with its transit fit is
shown in Figure 11.

###  4.3 Joint fit modelling

Figure 5:  Phase-folded RV (top) and TESS (bottom) data for TOI-1710 b. Upper panel: HARPS-N (blue) and SOPHIE (orange) RV data with the overplotted fit from the 1pGP model. Lower panel: Combination of TESS transits from Sectors 19, 20, 26, 26, 40, 53, and 59. The black line is the inferred 1pGP model.  Table 3:  Statistical criteria for the three tested models. Listed are the model names, BIC and AIC values, and number of free parameters. The best criteria model is marked in bold.  Model | BIC | AIC | $N_{\mathrm{free}}$
---|---|---|---
1p |  $-$61120.8 |  $-$61370.8 | 38
1pGP |  $-$61116.9 |  $-$61399.8 | 43
1pGP+FWHM |  $-$61340.0 |  $-$61643.4 | 46
Table 4: TOI-1710 parameters from the transit and RV joint fit, obtained with
the model 1pGP.

Parameter |  Prior(a) |  Value(b)
---|---|---
_Model Parameters_ |
Orbital period $P_{\mathrm{orb}}$ [days] | $\mathcal{U}[24.18,24.38]$ | $24.283382\pm 0.000019$
Transit epoch $T_{0}$ [BJD - 2 450 000] | $\mathcal{U}[9025.00,9037.00]$ | $9031.23007_{-0.00037}^{+0.00041}$
$\sqrt{e}\sin\omega_{\star}$ | $\mathcal{U}(-1,1)$ | $0.42\pm 0.13$
$\sqrt{e}\cos\omega_{\star}$ | $\mathcal{U}(-1,1)$ | $-0.01\pm 0.10$
Scaled planetary radius $R_{\mathrm{p}}/R_{\star}$ | $\mathcal{U}[0,0.5]$ | $0.048\pm 0.0004$
Impact parameter, $b$ | $\mathcal{U}[0,1]$ | $0.081_{-0.058}^{+0.093}$
Radial velocity semi-amplitude variation $K$ [m s-1] | $\mathcal{U}[0,50]$ | $4.20\pm 1.0$
Derived parameters |
Planet radius [$R_{\rm J}$] | $\cdots$ | $0.460\pm 0.010$
Planet radius [$R_{\oplus}$] | $\cdots$ | $5.15\pm 0.12$
Planet mass [$M_{\rm J}$] | $\cdots$ | $0.058_{-0.014}^{+0.015}$
Planet mass [$M_{\oplus}$] | $\cdots$ | $18.4_{-4.5}^{+4.8}$
Eccentricity $ecc$ | $\cdots$ | $0.185_{-0.091}^{+0.12}$
Scaled semi-major axis $a/R_{\star}$ | $\cdots$ | $30.6_{-3.9}^{+3.1}$
Semi-major axis $a$ [AU] | $\cdots$ | $0.164\pm 0.004$
$\omega_{\rm P}$ (deg) | $\cdots$ |  91 $\pm$ 14
Orbital inclination $i$ (deg) | $\cdots$ | $89.8\pm 0.3$
Transit duration $T_{14}$ [days] | $\cdots$ | $0.263_{-0.024}^{+0.037}$
Transit duration $T_{23}$ [days] | $\cdots$ | $0.238_{-0.021}^{+0.033}$
Equilibrium temperature(c) $T_{\rm eq}$ [K] | $\cdots$ | $730_{-30}^{+36}$
Planet instellation ${S}$ [$\mathrm{S}_{\oplus}$] | $\cdots$ | $33\pm 2$
_Other system parameters_ |
Jitter term $\sigma_{\rm HARPS-N}$ [m s-1] | $\mathcal{U}[0,60]$ | $1.40_{-0.33}^{+0.35}$
Jitter term $\sigma_{\rm SOPHIE}$ [m s-1] | $\mathcal{U}[0,60]$ | $4.4_{-1.2}^{+1.3}$
Stellar density $\rho_{\star}$ [$\rho_{\odot}$] | – | $0.65\pm 0.22$
Limb darkening $q_{1}$ | $\mathcal{N}[0.226,0.090]$ | $0.293\pm 0.062$
Limb darkening $q_{2}$ | $\mathcal{N}[0.52,0.21]$ | $0.34\pm 0.11$
_Stellar activity GP model Parameters_
$h_{\rm HARPS-N}$ [m s-1] | $\mathcal{U}[0,100]$ | $6.5_{-1.1}^{+1.4}$
$h_{\rm SOPHIE}$ [m s-1] | $\mathcal{U}[0,100]$ | $4.4_{-2.3}^{+2.4}$
$\lambda$ [days] | $\mathcal{U}[5,2000]$ | $62_{-12}^{+15}$
$\omega$ | $\mathcal{U}[0.01,1.50]$ | $0.375_{-0.088}^{+0.110}$
$\theta$ (Prot) [days] | $\mathcal{U}[2,40]$ | $21.51_{-0.24}^{+0.23}$

_Note_ – (a) $\mathcal{U}[a,b]$ refers to uniform priors between $a$ and $b$;
$\mathcal{N}[a,b]$ refers to Gaussian priors with median $a$ and standard
deviation $b$.
(b) Parameter estimates and corresponding uncertainties are defined as the
median and the 16th and 84th percentiles of the posterior distributions.
(c) Assuming zero Bond albedo.

We performed a joint fit using only TESS photometry and RV data using
PyORBIT777Available at https://github.com/LucaMalavolta/PyORBIT (Malavolta et
al. 2016, 2018) in order to obtain the planetary system parameters. For the
fit of transit light curves, this code makes use of the public package batman
(Kreidberg, 2015) and allows Gaussian processes (GPs) to be included through
the george package (Ambikasaran et al., 2014) in order to model the presence
of correlated noise or systematic effects in the data. The parameter space is
sampled by using the MCMC technique with the ensemble sampler emcee (Foreman-
Mackey et al., 2013). The initial conditions are obtained with the global
optimization code PyDE.888Available at https://github.com/hpparvi/PyDE We use
the GP quasi-periodic kernel, as defined by Grunblatt et al. (2015), imposing
a uniform prior on the rotational period $P_{\rm rot}$. To run the GPs, we
used 10 000 steps, 184 walkers, a burn-in cut of 1 000, and a thinning factor
of 100. The confidence intervals of the posterior distributions are computed
considering the 34.135th percentile from the median.

Regarding the photometric TESS light curves, we modelled the time of first
transit $T_{c}$, the orbital period $P$, the limb darkening (LD) with Kipping
(2013) parametrization, the impact parameter $b$, and the scaled planetary
radius $R_{P}/R_{\star}$. The eccentricity $ecc$ and argument of periastron
$\omega$ were modelled with the parametrization from Eastman et al. 2013
($\sqrt{e}\cos\omega$,$\sqrt{e}\sin\omega$). We used Gaussian priors both for
the stellar mass and radius (as obtained in Sect. 3.3) and for the LD
coefficients (calculated with PyLDTk;999Available at
https://github.com/hpparvi/ldtk Parviainen & Aigrain 2015; Husser et al.
2013). The stellar density and the impact parameter $b$ were left free. In
order to save computational time, we only used the TESS data around TOI-1710 b
transits.

The RV data are modelled taking into account the offset between different
instruments, and a jitter term for systematics and stellar noise. The joint
fit was performed for three models: i) one planet without GP (1p); ii) one
planet with GP (1pGP); iii) one planet with GP and FWHM included in the GP
(1pGP+FWHM). This last model, where the GP is fed with the FWHM values, was
tested because of the strong correlation between the FWHM and the HARPS-N RVs
(Fig. 13). We computed the Bayesian criteria in order to evaluate their
significance. In particular, we computed the Bayesian information criterion
(BIC) and the Akaike information criterion (AIC); the 1pGP+FWHM model is
strongly preferred over the other two models (see Table 3), indicating that
the activity plays an important role in the data. This is also proved by the
RV residuals, which are quite flat with no extra significant signals
(FAP$\gg$10%), when adopting the GP in the model (see Fig 14). The priors,
posteriors, and derived planetary parameters from the 1pGP+FWHM model are
presented in Table 4. The phase-folded TESS photometry and RVs are shown in
Fig. 5.

König et al. (2022) also detected the $P_{\rm rot}$ in their activity
indicators GLS periodograms, but the stellar rotation was not strong in their
RV periodogram. No correlation plots (e.g. Fig. 13) are shown. They tested
three RV models with different GP kernels (one of which is the quasi-periodic
kernel used in our analysis), and found values and uncertainties in good
agreement with those obtained when no GP was included. However, they modelled
the RV data with the juliet code (Espinoza et al., 2019), whose GPs can only
be fed by the time stamps, as in our 1pGP model. The 1p and 1pGP models have
no clear Bayesian criterion preference, but 1pGP+FWHM does (Table 3). Although
the $P_{\rm rot}$ and the planet period peaks are clear in the GLS periodogram
(Fig. 4), they are close. Thus, it is expected that, when modelling the two
models simultaneously, the semi-amplitude is smaller than the one obtained
when only the planet Keplerian is considered (as in König et al., 2022). The
semi-amplitudes obtained in this work and by König et al. (2022) differ by
1.6$\sigma$. Moreover, although our analyses included 54 more RVs, the fitted
semi-amplitude has the same uncertainties ($\pm$1.0 m s-1) as that of König et
al. (2022). This may be explained by the addition of the GP in the model, and
in fact the semi-amplitude uncertainty obtained with our 1p model is 0.8 m
s-1.

By adding two more TESS transits in the joint fit analysis, we improved the
photometric parameters. We reduced the $P$, $T_{0}$, and $R_{\rm p}/R_{\star}$
uncertainties from 4.3$\cdot$10-5, 4.2$\cdot$10-4, and 6$\cdot$10-4 in König
et al. (2022) to 1.9$\cdot$10-5, 3.7$\cdot$10-4, and 4$\cdot$10-4,
respectively. However, the uncertainty for $R_{\rm p}$ did not improve (this
work: 0.12 $\mathrm{R}_{\oplus}$; König et al. (2022): 0.11
$\mathrm{R}_{\oplus}$), meaning that the main source of uncertainty comes from
$R_{\star}$.

##  5 Discussion and conclusions

From our analyses, we derive for the transiting sub-Saturn planet TOI-1710 b a
radius of $R_{\rm p}$ $=$ 5.15 $\pm$ 0.12 $\mathrm{R}_{\oplus}$, a mass of
$M_{\rm p}$ $=$ 18.4 ${}^{+4.8}_{-4.5}$ $\mathrm{M}_{\oplus}$, and a mean bulk
density of $\rho_{\rm p}$ $=$ 0.73 $\pm$ 0.18 $\mathrm{g\,cm^{-3}}$, adopting
the stellar parameter values from Section 3.3. Our planetary parameters are
consistent with those previously derived by König et al. (2022), and we
confirmed the eccentric orbit of TOI-1710 b (ecc =
0.185${}_{-0.091}^{+0.12}$). The mass uncertainties are very similar between
the two works, but our mean mass measurement value presents a precision at the
4$\sigma$ level, and it is $\sim$1.5$\sigma$ away from that of König et al.
(2022, $M_{\rm p}$ $=$ 28.3 $\pm$ 4.7 $\mathrm{M}_{\oplus}$). Although this
difference is not significant, we needed to add a GP component accounting for
the stellar rotation to successfully fit the RV dataset from this work (see
Table 3 and Fig. 14).

After subtracting the 1pGP+FWHM model, the GLS periodograms of the HARPS-N,
SOPHIE, and combined RVs are well below the 10 % FAP (see Fig. 14). König et
al. (2022) reported an extra peak at 30 d with FAP $\simeq$ 2.2 % after
modelling the TOI-1710 b signal. However, here with more RV measurements and
after accounting for the stellar rotation, we do not find evidence of any
extra signal in the TOI-1710 planetary system.

###  5.1 Potential atmospheric characterization of TOI-1710 b

TOI-1710 b is a gaseous planet with a scale height ($H$) of $\sim$390 km. As
it is expected to posses a significant envelope, we evaluated its suitability
for atmospheric characterization via the transmission spectroscopy metric
(TSM) proposed by Kempton et al. (2018). TSM takes into account the planet
radius, mass, equilibrium temperature, stellar radius, and J magnitude. The
formula also includes a scale factor to account for 10 hours of observing time
with the James Webb Space Telescope (JWST; Gardner et al., 2006) Near Infrared
Imager and Slitless Spectrograph (NIRISS) instrument. According to Table 1
from Kempton et al. (2018), the scale factor for sub-Saturn planets is 1.15.
The computed TSM for TOI-1710 b is $\sim$150, which is greater than the
threshold TSM of 90 for planets ranging 1.5–10 $\mathrm{R}_{\oplus}$ to be
selected as high-quality targets for its atmospheric characterization.

There are only a few sub-Saturn planets around bright stars (J $\leq$ 9) with
higher TSM: GJ 436b (J = 6.9 mag, TSM = 481; Butler et al., 2004), HAT-P-11b
(J = 7.6 mag, TSM = 202; Bakos et al., 2010), GJ 3470b (J = 8.8 mag, TSM =
270; Bonfils et al., 2012), HD 89345b (J = 8.1 mag, TSM = 114; Van Eylen et
al., 2018), WASP-166b (J = 8.4 mag, TSM = 233; Hellier et al., 2019), HD
219666b (J = 8.6 mag, TSM = 142; Esposito et al., 2019), LTT 9779b (J = 8.4
mag, TSM = 185; Jenkins et al., 2020), and TOI-421c (J = 8.5 mag, TSM = 161;
Carleo et al., 2020). Among the many species that are expected in TOI-1710 b’s
atmosphere, one of the most interesting and accessible would be the He i
triplet, which has proved to be a powerful tool for studying extended
planetary atmospheres, tracking mass-loss and winds in the upper atmospheres,
and even detecting cometary-like atmospheric tails (Nortmann et al., 2018). He
i has already been detected in similar sub-Saturns, such as GJ 3470 b (Palle
et al., 2020; Ninan et al., 2020), HAT-P-11 b (Allart et al., 2018), and
HAT-P-26 b (Vissapragada et al., 2022). We used the rotational period of the
star to calculate the expected X-ray and extreme UV emission, using the
relations in Wright et al. (2011) and Sanz-Forcada et al. (2011). We estimate
an X-ray luminosity of $\log L_{\rm X}$ $=$ 28.07 and a calculated $F_{\rm
XUV}$ $\sim$1 W m-2 irradiance in the 5–504 $\AA$ spectral range at the planet
b orbit, similar to the flux received by HD 209458 b, where the He i triplet
was positively detected by Alonso-Floriano et al. (2019). This is further
supported by the high estimated transmission spectroscopy S/N calculated by
König et al. (2022), placing TOI-1710 b as the fourth-best candidate within
the sub-Saturn population. It is thus clear that TOI-1710 b is a very
appealing target for the atmospheric studies, and strong atmospheric features
(e.g. the Na i doublet and K i line at 7701 $\AA$ in the visible or the He i
triplet and molecules in the near-infrared) are expected in TOI-1710 b, if
they are not veiled by clouds or hazes.

Figure 6: Mass-radius (left) and period-radius (right) diagrams for all known
planets with $M_{\rm p}$ $\in$ [1,60] M⊕ and $R_{\rm p}$ $\in$ [1,10] R⊕
determined with a precision better than 30% (grey points). The coloured dashed
lines are the theoretical mass-radius models from Zeng et al. (2019). Sub-
Saturns located in multi-planet systems are marked with green squares, and
sub-Saturns located in single-planet systems are orange circles. TOI-1710 b is
marked with a red star.  Figure 7:  Mass-density (top), mass-eccentricity
(middle), and mass-metalicity (bottom) diagrams for the sub-Saturn planets
from Fig. 6. Sub-Saturns located in multi-planet systems are marked with green
squares and sub-Saturns located in single-planet systems are orange circles.
TOI-1710 b is marked with a red star.

###  5.2 TOI-1710 b within the sample of sub-Saturns

Figure 6 compares TOI-1710 b with known exoplanets, with masses and radii
measured with a precision better than 30%, from the NASA Exoplanet
Archive101010https://exoplanetarchive.ipac.caltech.edu/ along with theoretical
composition models of Zeng et al.
(2019).111111https://lweb.cfa.harvard.edu/~lzeng/planetmodels.html The
displayed models regard planets with Earth-like composition cores (32.5% Fe +
67.5% MgSiO2), 100% H2O composition, and 95% Earth-like rocky core surrounded
by 5% H2 gaseous envelopes. The models are truncated at a reference pressure
of 1 mbar and consider an isothermal atmosphere, for which we chose an
equilibrium temperature closest to that of TOI-1710 b. Its position in the
mass-radius diagram (Fig. 6, left panel), well above the lighter planetary
model available (Earth-like composition core with 5% H2 gaseous envelope), and
its bulk mean density ($\rho_{\rm p}$ $=$ 0.75 $\pm$ 0.18
$\mathrm{g\,cm^{-3}}$) suggest that TOI-1710 b may hold a large H–He envelope
that could be suitable for spectroscopical exploration, due to the brightness
of its host star.

In Figure 6 (and in Figure 7 as well), we marked all other known sub-Saturns
detected in multi-planet systems as green squares, while sub-Saturns in
single-planet systems (e.g. TOI-1710 b) are shown as orange circles. Figure 6,
left panel, shows that lone sub-Saturns tend to be more massive, which is a
trend already reported in previous studies (e.g. Petigura et al., 2017, Van
Eylen et al., 2018, and Nowak et al., 2020). However, in the intermediate
region at 20–30 $\mathrm{M}_{\oplus}$, where TOI-1710 b is located, the two
populations are mixed, and there is not a clear boundary line between them.
Its closer lone sub-Saturn companion in the mass-radius diagram is TOI-674 b
(Murgas et al., 2021), which has an orbital period of less than 2 days ($P$ =
1.97 d).

The right panel in Fig. 6 displays the period-radius diagram, where the
TOI-1710 b position seems to be an outlier of the single-planet sub-Saturn
population. From this figure we can clearly see that sub-Saturns in multi-
planet systems tend to orbit host stars with longer orbital periods than those
in single-planet systems. This is unlikely to be an observational or
instrumental bias because the lighter planets are the ones farther from their
stars. However, because single sub-Saturns have short periods, their possible
companions may be in outer orbits. Thus, the hypothetical companions are less
likely to transit and should be massive enough to be detectable via radial
velocity or astrometry methods. TOI-1710 b has a period of 24 d, and is well
within the parameter space of the multi-planet systems; it is the sub-Saturn
planet in a single-planet system with the longest period in the diagram.
However, there are other sub-Saturn planets with longer periods than TOI-1710
b whose properties have not been determined with precision, for example
Kepler-413A b (Kostov et al., 2014) with $P$ = 66 d, and $M_{\rm p}$ $=$ 67
$\pm$ 21 $\mathrm{M}_{\oplus}$. From both panels of Fig. 6, we conclude that
single sub-Saturns are more massive and have shorter periods than accompanied
sub-Saturns, which are lighter and have longer periods.

Petigura et al. (2017) and Nowak et al. (2020) also explored the possible
relations between metallicity, density, and eccentricity of sub-Saturns with
planetary mass. In Figure 7 we reproduced these diagrams; TOI-1710 b is
indicated with a red star. Other sub-Saturns are marked following the criteria
in Fig. 6. Those works found a marginal correlation (Spearman correlation
coefficient $r$ = 0.57) between the stellar metallicity and the mass of the
sub-Saturns, where massive planets orbit metal-rich stars. A slighly enriched
metallicity of $\rm[Fe/H]$ = 0.12 $\pm$ 0.06 dex places TOI-1710 at the centre
of the metallicity distribution (see Fig. 7, bottom panel). According to the
mass-metallicity correlation, the planet mass should also be in the centre of
the planetary masses, which TOI-1710 b is. TOI-1710 b is displayed in central
positions in all the panels of Fig. 7, and in the mass-radius diagram (Fig. 6,
left panel). Thus, although we have not found evidence of extra planetary
signals, further follow-up of the TOI-1710 system will allow us to keep
refining the characteristics of the transiting planet, and confirm if TOI-1710
b is alone orbiting its host star or if it is the inner planet of a multi-
planet system.

###### Acknowledgements.

This work is partly financed by the Spanish Ministry of Economics and
Competitiveness through grants PGC2018-098153-B-C31. We acknowledge the
contributions of Deven Combs, Sudhish Chmaladinne, Kevin Eastridge and Michael
Bowen in the collection and anaylsis of the TOI 1710.01 ground-based light
curve from George Mason Observatory. G.N. thanks for the research funding from
the Ministry of Education and Science programme the ”Excellence Initiative -
Research University” conducted at the Centre of Excellence in Astrophysics and
Astrochemistry of the Nicolaus Copernicus University in Toruń, Poland. T.M.
acknowledges financial support from the Spanish Ministry of Science and
Innovation (MICINN) through the Spanish State Research Agency, under the
Severo Ochoa Program 2020-2023 (CEX2019-000920-S) T.H. acknowledges support
from the European Research Council under the Horizon 2020 Framework Program
via the ERC Advanced Grant Origins 83 24 28. P. D. is supported by a National
Science Foundation (NSF) Astronomy and Astrophysics Postdoctoral Fellowship
under award AST-1903811. D. D. acknowledges support from the TESS Guest
Investigator Program grant 80NSSC19K1727 and NASA Exoplanet Research Program
grant 18-2XRP18-2-0136. J.O.M. agraeix el recolzament, suport i ànims que
sempre ha rebut per part de padrina Conxa, padrina Mercè, Jeroni, Mercè i més
familiars i amics. J.O.M acknowledges the special support from Maite, Guillem,
Alejandro, Benet, Joan, and Montse. This research has made use of resources
from AstroPiso collaboration. J.O.M. acknowledges the contributions of Jorge
Terol Calvo, i molt especialment a tu, Yess. 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 (Istituto Nazionale di
Astrofisica) at the Spanish Observatorio del Roque de los Muchachos of the
Instituto de Astrofisica de Canarias This work has been carried out within the
framework of the NCCR PlanetS supported by the Swiss National Science
Foundation. This paper includes data collected by the TESS mission, which are
publicly available from the Mikulski Archive for Space Telescopes (MAST).
Funding for the TESS mission is provided by NASA’s Science Mission
directorate. We acknowledge the use of public TESS data from pipelines at the
TESS Science Office and at the TESS Science Processing Operations Center.
Resources supporting this work were provided by the NASA High-End Computing
(HEC) Program through the NASA Advanced Supercomputing (NAS) Division at Ames
Research Center for the production of the SPOC data products. This research
has made use of the Exoplanet Follow-up Observation Program website, which is
operated by the California Institute of Technology, under contract with the
National Aeronautics and Space Administration under the Exoplanet Exploration
Program. 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. Some of the
observations in the paper made use of the High-Resolution Imaging instrument
‘Alopeke under Gemini LLP Proposal Number: GN/S-2021A-LP-105.‘Alopeke was
funded by the NASA Exoplanet Exploration Program and built at the NASA Ames
Research Center by Steve B. Howell, Nic Scott, Elliott P. Horch, and Emmett
Quigley. ‘Alopeke was mounted on the Gemini North telescope of the
international Gemini Observatory, a program of NSF’s OIR Lab, which is managed
by the Association of Universities for Research in Astronomy (AURA) under a
cooperative agreement with the National Science Foundation. on behalf of the
Gemini partnership: the National Science Foundation (United States), National
Research Council (Canada), Agencia Nacional de Investigación y Desarrollo
(Chile), Ministerio de Ciencia, Tecnología e Innovación (Argentina),
Ministério da Ciência, Tecnologia, Inovações e Comunicações (Brazil), and
Korea Astronomy and Space Science Institute (Republic of Korea). This work
made use of tpfplotter by J. Lillo-Box (publicly available in
www.github.com/jlillo/tpfplotter), which also made use of the python packages
astropy, lightkurve, matplotlib and numpy. This work made use of corner.py by
Daniel Foreman-Mackey (Foreman-Mackey, 2016).

## References

  * Addison et al. (2021) Addison, B. C., Wright, D. J., Nicholson, B. A., et al. 2021, MNRAS, 502, 3704
  * Allart et al. (2018) Allart, R., Bourrier, V., Lovis, C., et al. 2018, Science, 362, 1384
  * Alonso-Floriano et al. (2019) Alonso-Floriano, F. J., Snellen, I. A. G., Czesla, S., et al. 2019, A&A, 629, A110
  * Ambikasaran et al. (2014) Ambikasaran, S., Foreman-Mackey, D., Greengard, L., Hogg, D. W., & O’Neil, M. 2014
  * Argelander (1903) Argelander, F. W. A. 1903, Eds Marcus and Weber’s Verlag, 0
  * Bakos et al. (2010) Bakos, G. Á., Torres, G., Pál, A., et al. 2010, ApJ, 710, 1724
  * Baranne et al. (1996) Baranne, A., Queloz, D., Mayor, M., et al. 1996, A&AS, 119, 373
  * Bessell et al. (1998) Bessell, M. S., Castelli, F., & Plez, B. 1998, A&A, 333, 231
  * Bonfils et al. (2012) Bonfils, X., Gillon, M., Udry, S., et al. 2012, A&A, 546, A27
  * Buchhave et al. (2012) Buchhave, L. A., Latham, D. W., Johansen, A., et al. 2012, Nature, 486, 375
  * Buchner et al. (2014) Buchner, J., Georgakakis, A., Nandra, K., et al. 2014, A&A, 564, A125
  * Butler et al. (2004) Butler, R. P., Vogt, S. S., Marcy, G. W., et al. 2004, ApJ, 617, 580
  * Carleo et al. (2020) Carleo, I., Gandolfi, D., Barragán, O., et al. 2020, AJ, 160, 114
  * Casagrande et al. (2010) Casagrande, L., Ramírez, I., Meléndez, J., Bessell, M., & Asplund, M. 2010, A&A, 512, A54
  * Ciardi et al. (2015) Ciardi, D. R., Beichman, C. A., Horch, E. P., & Howell, S. B. 2015, ApJ, 805, 16
  * Collins et al. (2017) Collins, K. A., Kielkopf, J. F., Stassun, K. G., & Hessman, F. V. 2017, AJ, 153, 77
  * Cosentino et al. (2012) Cosentino, R., Lovis, C., Pepe, F., et al. 2012, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 8446, Ground-based and Airborne Instrumentation for Astronomy IV, ed. I. S. McLean, S. K. Ramsay, & H. Takami, 84461V
  * Cosentino et al. (2014) Cosentino, R., Lovis, C., Pepe, F., et al. 2014, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9147, Ground-based and Airborne Instrumentation for Astronomy V, 91478C
  * Cutri et al. (2003) Cutri, R. M., Skrutskie, M. F., van Dyk, S., et al. 2003, VizieR Online Data Catalog, II/246
  * Dawson et al. (2021) Dawson, R. I., Huang, C. X., Brahm, R., et al. 2021, AJ, 161, 161
  * Eastman et al. (2013) Eastman, J., Gaudi, B. S., & Agol, E. 2013, PASP, 125, 83
  * Eastman et al. (2019) Eastman, J. D., Rodriguez, J. E., Agol, E., et al. 2019, arXiv e-prints, arXiv:1907.09480
  * Espinoza (2018) Espinoza, N. 2018, Research Notes of the American Astronomical Society, 2, 209
  * Espinoza et al. (2019) Espinoza, N., Kossakowski, D., & Brahm, R. 2019, MNRAS, 490, 2262
  * Esposito et al. (2019) Esposito, M., Armstrong, D. J., Gandolfi, D., et al. 2019, A&A, 623, A165
  * Feroz et al. (2009) Feroz, F., Hobson, M. P., & Bridges, M. 2009, MNRAS, 398, 1601
  * Foreman-Mackey (2016) Foreman-Mackey, D. 2016, The Journal of Open Source Software, 1, 24
  * Foreman-Mackey et al. (2017) Foreman-Mackey, D., Agol, E., Angus, R., & Ambikasaran, S. 2017, AJ, 154, 220
  * Foreman-Mackey et al. (2013) Foreman-Mackey, D., Hogg, D. W., Lang, D., & Goodman, J. 2013, PASP, 125, 306
  * Gaia Collaboration (2020) Gaia Collaboration. 2020, VizieR Online Data Catalog, I/350
  * Gaia Collaboration et al. (2018) Gaia Collaboration, Brown, A. G. A., Vallenari, A., et al. 2018, A&A, 616, A1
  * Gardner et al. (2006) Gardner, J. P., Mather, J. C., Clampin, M., et al. 2006, Space Sci. Rev., 123, 485
  * González Hernández & Bonifacio (2009) González Hernández, J. I. & Bonifacio, P. 2009, A&A, 497, 497
  * Grunblatt et al. (2015) Grunblatt, S. K., Howard, A. W., & Haywood, R. D. 2015, ApJ, 808, 127
  * Guerrero et al. (2021) Guerrero, N. M., Seager, S., Huang, C. X., et al. 2021, ApJS, 254, 39
  * Gustafsson et al. (2008) Gustafsson, B., Edvardsson, B., Eriksson, K., et al. 2008, A&A, 486, 951
  * Hellier et al. (2019) Hellier, C., Anderson, D. R., Triaud, A. H. M. J., et al. 2019, MNRAS, 488, 3067
  * Høg et al. (2000) Høg, E., Fabricius, C., Makarov, V. V., et al. 2000, A&A, 355, L27
  * Howell et al. (2016) Howell, S. B., Everett, M. E., Horch, E. P., et al. 2016, ApJ, 829, L2
  * Howell et al. (2011) Howell, S. B., Everett, M. E., Sherry, W., Horch, E., & Ciardi, D. R. 2011, AJ, 142, 19
  * Howell et al. (2021) Howell, S. B., Matson, R. A., Ciardi, D. R., et al. 2021, AJ, 161, 164
  * Huang et al. (2020a) Huang, C. X., Vanderburg, A., Pál, A., et al. 2020a, Research Notes of the American Astronomical Society, 4, 204
  * Huang et al. (2020b) Huang, C. X., Vanderburg, A., Pál, A., et al. 2020b, Research Notes of the American Astronomical Society, 4, 206
  * Hubickyj et al. (2005) Hubickyj, O., Bodenheimer, P., & Lissauer, J. J. 2005, Icarus, 179, 415
  * Husser et al. (2013) Husser, T.-O., Wende-von Berg, S., Dreizler, S., et al. 2013, A&A, 553, A6
  * Jenkins et al. (2016) Jenkins, J. M., Twicken, J. D., McCauliff, S., et al. 2016, in Society of Photo-Optical Instrumentation Engineers (SPIE) Conference Series, Vol. 9913, Software and Cyberinfrastructure for Astronomy IV, ed. G. Chiozzi & J. C. Guzman, 99133E
  * Jenkins et al. (2020) Jenkins, J. S., Díaz, M. R., Kurtovic, N. T., et al. 2020, Nature Astronomy, 4, 1148
  * Jensen (2013) Jensen, E. 2013, Tapir: A web interface for transit/eclipse observability
  * Jofré et al. (2014) Jofré, P., Heiter, U., Soubiran, C., et al. 2014, A&A, 564, A133
  * Kempton et al. (2018) Kempton, E. M. R., Bean, J. L., Louie, D. R., et al. 2018, PASP, 130, 114401
  * Kipping (2013) Kipping, D. M. 2013, MNRAS, 435, 2152
  * König et al. (2022) König, P. C., Damasso, M., Hébrard, G., et al. 2022, A&A, 666, A183
  * Kostov et al. (2014) Kostov, V. B., McCullough, P. R., Carter, J. A., et al. 2014, ApJ, 784, 14
  * Kreidberg (2015) Kreidberg, L. 2015, Publications of the Astronomical Society of the Pacific, 127, 1161
  * Lee & Chiang (2015) Lee, E. J. & Chiang, E. 2015, ApJ, 811, 41
  * Lester et al. (2021) Lester, K. V., Matson, R. A., Howell, S. B., et al. 2021, arXiv e-prints, arXiv:2106.13354
  * Li et al. (2019) Li, J., Tenenbaum, P., Twicken, J. D., et al. 2019, PASP, 131, 024506
  * Lopez & Fortney (2014) Lopez, E. D. & Fortney, J. J. 2014, ApJ, 792, 1
  * Malavolta et al. (2018) Malavolta, L., Mayo, A. W., Louden, T., et al. 2018, AJ, 155, 107
  * Malavolta et al. (2016) Malavolta, L., Nascimbeni, V., Piotto, G., et al. 2016, A&A, 588, A118
  * Mamajek & Hillenbrand (2008) Mamajek, E. E. & Hillenbrand, L. A. 2008, ApJ, 687, 1264
  * Mason et al. (2001) Mason, B. D., Wycoff, G. L., Hartkopf, W. I., Douglass, G. G., & Worley, C. E. 2001, AJ, 122, 3466
  * Masseron et al. (2016) Masseron, T., Merle, T., & Hawkins, K. 2016, BACCHUS: Brussels Automatic Code for Characterizing High accUracy Spectra
  * Matson et al. (2018) Matson, R. A., Howell, S. B., Horch, E. P., & Everett, M. E. 2018, AJ, 156, 31
  * Morris et al. (2020) Morris, R. L., Twicken, J. D., Smith, J. e. C., et al. 2020, Kepler Data Processing Handbook: Photometric Analysis, Kepler Science Document KSCI-19081-003, id. 6. Edited by Jon M. Jenkins.
  * Murgas et al. (2021) Murgas, F., Astudillo-Defru, N., Bonfils, X., et al. 2021, A&A, 653, A60
  * Ninan et al. (2020) Ninan, J. P., Stefansson, G., Mahadevan, S., et al. 2020, ApJ, 894, 97
  * Nortmann et al. (2018) Nortmann, L., Pallé, E., Salz, M., et al. 2018, Science, 362, 1388
  * Nowak et al. (2020) Nowak, G., Palle, E., Gandolfi, D., et al. 2020, MNRAS, 497, 4423
  * Noyes et al. (1984) Noyes, R. W., Hartmann, L. W., Baliunas, S. L., Duncan, D. K., & Vaughan, A. H. 1984, ApJ, 279, 763
  * Palle et al. (2020) Palle, E., Nortmann, L., Casasayas-Barris, N., et al. 2020, A&A, 638, A61
  * Parviainen & Aigrain (2015) Parviainen, H. & Aigrain, S. 2015, MNRAS, 453, 3821
  * Petigura et al. (2016) Petigura, E. A., Howard, A. W., Lopez, E. D., et al. 2016, ApJ, 818, 36
  * Petigura et al. (2017) Petigura, E. A., Sinukoff, E., Lopez, E. D., et al. 2017, AJ, 153, 142
  * Pollack et al. (1996) Pollack, J. B., Hubickyj, O., Bodenheimer, P., et al. 1996, Icarus, 124, 62
  * Ricker et al. (2015) Ricker, G. R., Winn, J. N., Vanderspek, R., et al. 2015, Journal of Astronomical Telescopes, Instruments, and Systems, 1, 014003
  * Rodrigues et al. (2017) Rodrigues, T. S., Bossini, D., Miglio, A., et al. 2017, MNRAS, 467, 1433
  * Rodrigues et al. (2014) Rodrigues, T. S., Girardi, L., Miglio, A., et al. 2014, MNRAS, 445, 2758
  * Sanz-Forcada et al. (2011) Sanz-Forcada, J., Micela, G., Ribas, I., et al. 2011, A&A, 532, A6+
  * Smith et al. (2012) Smith, J. C., Stumpe, M. C., Van Cleve, J. E., et al. 2012, PASP, 124, 1000
  * Speagle (2020) Speagle, J. S. 2020, MNRAS, 493, 3132
  * Stassun et al. (2019) Stassun, K. G., Oelkers, R. J., Paegert, M., et al. 2019, AJ, 158, 138
  * Stassun et al. (2018) Stassun, K. G., Oelkers, R. J., Pepper, J., et al. 2018, AJ, 156, 102
  * Stumpe et al. (2014) Stumpe, M. C., Smith, J. C., Catanzarite, J. H., et al. 2014, PASP, 126, 100
  * Stumpe et al. (2012) Stumpe, M. C., Smith, J. C., Van Cleve, J. E., et al. 2012, PASP, 124, 985
  * Tayar et al. (2022) Tayar, J., Claytor, Z. R., Huber, D., & van Saders, J. 2022, ApJ, 927, 31
  * Tian et al. (2020) Tian, H.-J., El-Badry, K., Rix, H.-W., & Gould, A. 2020, ApJS, 246, 4
  * Twicken et al. (2018) Twicken, J. D., Catanzarite, J. H., Clarke, B. D., et al. 2018, PASP, 130, 064502
  * Van Eylen et al. (2018) Van Eylen, V., Dai, F., Mathur, S., et al. 2018, MNRAS, 478, 4866
  * Vissapragada et al. (2022) Vissapragada, S., Knutson, H. A., Greklek-McKeon, M., et al. 2022, AJ, 164, 234
  * Woźniak et al. (2004) Woźniak, P. R., Vestrand, W. T., Akerlof, C. W., et al. 2004, AJ, 127, 2436
  * Wright et al. (2003) Wright, C. O., Egan, M. P., Kraemer, K. E., & Price, S. D. 2003, AJ, 125, 359
  * Wright et al. (2011) Wright, N. J., Drake, J. J., Mamajek, E. E., & Henry, G. W. 2011, ApJ, 743, 48
  * Zechmeister & Kürster (2009) Zechmeister, M. & Kürster, M. 2009, A&A, 496, 577
  * Zechmeister et al. (2018) Zechmeister, M., Reiners, A., Amado, P. J., et al. 2018, A&A, 609, A12
  * Zeng et al. (2019) Zeng, L., Jacobsen, S. B., Sasselov, D. D., et al. 2019, Proceedings of the National Academy of Science, 116, 9723

##  Appendix A Additional light curve figures and tables

Figure 8: TESS target pixel file image of TOI-1710 (TIC 445805961) observed in
Sectors 19, 20, 26, 40, 53, 59, and 60 (made with tpfplotter). The pixels
highlighted in red show the aperture used by TESS to obtain the photometry.
The electron counts are colour-coded. The positions and sizes of the red
circles respectively represent the positions and TESS magnitudes of nearby
stars. TOI-1710 is marked with a cross ($\times$) and labelled $\\#$1\.  Table
5:  Prior and posterior distributions from the juliet photometric fit. Prior
labels $\mathcal{U}$, $\mathcal{N}$, $\mathcal{F}$, and $\mathcal{J}$
represent uniform, normal, fixed, and Jeffrey’s distribution, respectively.

Parameter | Prior | Posterior
---|---|---
$P$ [d] | $\mathcal{N}(24.28,0.01)$ | 24.283385 (18)
$t_{0}$ (a) | $\mathcal{N}(1836.96,0.01)$ | 1836.96299 (48)
ecc (deg) | $\mathcal{F}(0)$ | –
$\omega$ (deg) | $\mathcal{F}(90)$ | –
$r_{1}$ | $\mathcal{U}(0,1)$ | 0.409${}^{+0.059}_{-0.052}$
$r_{2}$ | $\mathcal{U}(0,1)$ | 0.04990$\pm$0.00037
$\rho_{\star}$ [kg m-3] | $\mathcal{N}(1620,100)$ | 1617${}^{+36}_{-64}$
$\mu_{\textit{TESS}}$ (ppm) | $\mathcal{N}(0.0,0.1)$ | 135$\pm$6
$\sigma_{\textit{TESS}}$ (ppm) | $\mathcal{J}(10^{-6},10^{6})$ | 225$\pm$20
$q_{1,\textit{TESS}}$ | $\mathcal{U}(0,1)$ | 0.23${}^{+0.10}_{-0.07}$
$q_{2,\textit{TESS}}$ | $\mathcal{U}(0,1)$ | 0.52${}^{+0.21}_{-0.17}$
GPσ (ppm) | $\mathcal{J}(10^{-6},10^{6})$ | 224$\pm$16
GPρ [d] | $\mathcal{J}(10^{-3},10^{3})$ | 0.93${}^{+0.08}_{-0.07}$

121212 (a) Central time of transit ($t_{0}$) units are BJD $-$ 2 457 000.

Figure 9: TOI-1710 2 min cadence light curve from TESS Sectors 19, 20, 26, 40,
53, and 59 (blue points with error bars) along with the transit plus GP model
(black line) from juliet. Upward-pointing red triangles mark the TOI-1710 b
transits. Figure 10: TOI-1710 TESS photometry (blue dots with error bars)
phase-folded to the period $P$ and central time of transit $t_{0}$ (shown
above each panel, $t_{0}$ units are BJD $-$ 2 457 000) derived from the juliet
fit. The black line is the best transit model for TOI-1710 b. The orange
points show the binned photometry for ease of visualization. The GP model was
removed from the data. Figure 11: Light curve from the George Mason University
Observatory’s 0.8 m telescope along with the best-fit transit model (red
line). The blue points show the original photometric data with a cadence of 20
s; the black points show 30 min binned photometry.

##  Appendix B Additional radial velocity figures and tables

Table 6: Radial velocities and spectral activity indicators measured from TNG/HARPS-N spectra with serval and DRS. Time stamps from König et al. (2022) spectra are marked in bold. Time series of spectra continue into Table 8.  BJDTBD |  RV serval | BIS | CCF_FWHM | CCF_CTR | $\mathrm{\log{R^{`}_{HK}}}$ | CRX | dlW
---|---|---|---|---|---|---|---
$-$2457000 |  [$\mathrm{m\,s^{-1}}$] |  [$\mathrm{m\,s^{-1}}$] |  [$\mathrm{km\,s^{-1}}$] | (%) | — |  ($\mathrm{m\,s^{-1}\,Np^{-1}}$) |  [$\mathrm{m^{2}\,s^{-2}}$]
Val. | Val. | $\sigma$ | Val. | $\sigma$ | Val. | Val. | Val. | $\sigma$ | Val. | $\sigma$ | Val. | $\sigma$
2125.664739 | 10.7 | 1.84 | -12.99 | 2.559 | 7.277 | 48.411 | -4.7656 | 0.0105 | 4.747 | 14.819 | 15.794 | 3.639
2126.618253 | 5.06 | 1.67 | -11.391 | 2.08 | 7.266 | 48.511 | -4.7681 | 0.0084 | 2.654 | 13.866 | 7.393 | 2.634
2127.719329 | 2.24 | 1.28 | -10.508 | 1.711 | 7.267 | 48.597 | -4.7574 | 0.0122 | 6.085 | 10.514 | -8.641 | 2.77
2130.73596 | -5.98 | 2.54 | -22.531 | 4.878 | 7.26 | 48.304 | -4.8279 | 0.0149 | 5.319 | 20.869 | 18.585 | 6.192
2133.711696 | -7.68 | 1.35 | -15.256 | 2.305 | 7.254 | 48.632 | -4.7963 | 0.0088 | 8.868 | 10.824 | -11.958 | 3.119
2134.756264 | -8.96 | 1.86 | -18.148 | 2.451 | 7.257 | 48.622 | -4.7663 | 0.0117 | 3.603 | 15.075 | -10.429 | 3.526
2149.682376 | 5.25 | 1.5 | -11.46 | 2.096 | 7.24 | 48.723 | -4.785 | 0.0071 | -7.997 | 12.821 | -4.596 | 2.747
2150.688883 | 9.76 | 1.2 | -11.969 | 1.788 | 7.244 | 48.698 | -4.7903 | 0.0061 | -16.119 | 9.969 | -0.516 | 1.78
2151.723635 | 10.56 | 1.58 | -11.029 | 2.38 | 7.246 | 48.671 | -4.7727 | 0.0097 | 0.421 | 12.7 | 1.356 | 3.282
2156.742015 | -6.31 | 1.12 | -19.71 | 1.724 | 7.266 | 48.596 | -4.7887 | 0.0073 | -7.344 | 9.163 | -5.44 | 2.048
2158.686506 | -5.64 | 1.54 | -12.062 | 2.384 | 7.221 | 48.765 | -4.7668 | 0.0326 | -7.901 | 12.921 | -12.779 | 3.158
2160.659656 | -7.27 | 1.24 | -17.6 | 1.771 | 7.226 | 48.806 | -4.7938 | 0.0056 | 9.02 | 10.47 | -13.702 | 2.157
2172.605447 | 9.48 | 1.65 | -11.241 | 2.006 | 7.267 | 48.546 | -4.794 | 0.0054 | 1.762 | 13.263 | 0.551 | 3.3
2188.596847 | 3.8 | 1.3 | -14.871 | 1.96 | 7.272 | 48.48 | -4.7884 | 0.0151 | 10.722 | 10.837 | 11.812 | 2.93
2189.64318 | -2.69 | 1.02 | -12.606 | 1.601 | 7.273 | 48.569 | -4.7596 | 0.0173 | 5.66 | 8.758 | 0.28 | 1.761
2190.610672 | -1.96 | 1.41 | -15.521 | 1.949 | 7.262 | 48.616 | -4.7659 | 0.0222 | 3.008 | 12.432 | -5.422 | 2.553
2191.782237 | 1.06 | 1.07 | -17.256 | 1.394 | 7.264 | 48.64 | -4.771 | 0.0185 | 6.011 | 8.591 | -11.408 | 1.747
2204.493933 | -7.09 | 1.35 | -20.915 | 2.192 | 7.229 | 48.741 | -4.7981 | 0.0049 | 5.84 | 11.869 | -2.912 | 2.467
2204.673882 | -5.85 | 1.12 | -20.256 | 1.545 | 7.223 | 48.788 | -4.7929 | 0.0047 | 9.047 | 9.829 | -11.423 | 1.735
2205.499679 | -1.43 | 1.02 | -14.768 | 1.408 | 7.224 | 48.78 | -4.7944 | 0.0066 | 13.736 | 8.672 | -8.102 | 2.041
2205.685433 | -2.5 | 1.17 | -10.185 | 1.919 | 7.226 | 48.75 | -4.7814 | 0.0096 | 6.871 | 9.888 | -8.062 | 2.571
2206.4971 | -0.27 | 1.08 | -19.432 | 1.585 | 7.223 | 48.743 | -4.8021 | 0.0079 | -9.944 | 9.4 | -4.913 | 2.244
2206.679276 | -4.27 | 2.53 | -8.329 | 4.476 | 7.236 | 48.531 | -4.8129 | 0.0078 | -29.964 | 20.502 | 3.099 | 4.807
2212.534392 | 1.0 | 1.29 | -16.177 | 2.127 | 7.271 | 48.518 | -4.824 | 0.0309 | -11.427 | 10.351 | 3.176 | 3.029
2215.622998 | 15.87 | 1.01 | -20.647 | 1.54 | 7.273 | 48.514 | -4.7824 | 0.0079 | -3.803 | 8.19 | 6.178 | 2.435
2248.433766 | -1.07 | 1.01 | -27.121 | 1.326 | 7.217 | 48.844 | -4.82 | 0.0142 | -14.392 | 8.621 | -18.587 | 1.964
2248.444309 | -0.28 | 1.06 | -23.395 | 1.285 | 7.219 | 48.846 | -4.8332 | 0.0133 | 4.076 | 9.451 | -19.282 | 1.514
2265.415755 | 2.88 | 1.93 | -1.549 | 2.569 | 7.231 | 48.685 | -4.826 | 0.0123 | 21.158 | 15.537 | -1.462 | 3.367
2265.42869 | 2.53 | 1.63 | -21.879 | 2.93 | 7.229 | 48.657 | -4.8417 | 0.033 | 6.113 | 13.459 | -2.513 | 3.663
2265.560543 | 2.91 | 2.07 | -6.222 | 3.436 | 7.234 | 48.634 | -4.8317 | 0.0337 | -1.784 | 16.663 | -1.651 | 4.335
2265.57402 | 3.21 | 2.01 | -7.527 | 3.0 | 7.236 | 48.693 | -4.8179 | 0.0259 | -5.787 | 15.995 | -3.218 | 4.034
2268.426272 | -0.98 | 0.77 | -16.31 | 1.206 | 7.227 | 48.803 | -4.8096 | 0.0119 | -10.589 | 6.668 | -12.481 | 1.832
2268.438999 | -0.13 | 0.83 | -14.652 | 1.18 | 7.231 | 48.795 | -4.8162 | 0.0118 | -11.144 | 7.11 | -13.971 | 1.68
2268.505815 | 0.17 | 1.14 | -12.733 | 1.455 | 7.227 | 48.799 | -4.8228 | 0.0172 | 2.831 | 9.839 | -14.944 | 1.93
2268.519837 | -0.18 | 1.06 | -14.503 | 1.857 | 7.232 | 48.759 | -4.8353 | 0.0102 | 0.231 | 8.992 | -9.382 | 2.118
2275.338169 | 4.42 | 0.93 | -9.129 | 1.361 | 7.282 | 48.475 | -4.8152 | 0.0121 | -16.989 | 7.845 | 42.678 | 2.134
2276.48642 | -1.04 | 1.25 | -11.031 | 1.735 | 7.269 | 48.524 | -4.7992 | 0.0107 | -23.013 | 10.447 | 5.285 | 2.225
2277.409061 | -0.89 | 1.15 | -11.649 | 1.486 | 7.265 | 48.585 | -4.8085 | 0.0118 | -26.776 | 9.186 | -2.286 | 2.006
2278.43132 | -3.63 | 1.33 | -14.517 | 1.629 | 7.268 | 48.589 | -4.7943 | 0.0117 | -19.579 | 11.132 | -3.436 | 1.95
2287.3862 | 0.35 | 1.44 | -18.127 | 2.659 | 7.248 | 48.629 | -4.8129 | 0.0103 | -6.68 | 11.78 | -5.117 | 3.686
2289.40115 | -3.2 | 1.31 | -16.69 | 1.773 | 7.255 | 48.632 | -4.8181 | 0.0101 | -15.986 | 10.631 | -11.952 | 2.457
2290.341493 | 1.41 | 1.08 | -20.809 | 1.347 | 7.259 | 48.672 | -4.8113 | 0.0132 | -13.846 | 8.778 | 19.952 | 1.925
2294.435475 | 11.65 | 1.34 | -19.321 | 1.759 | 7.282 | 48.512 | -4.7985 | 0.0179 | -21.659 | 10.892 | 3.553 | 2.2
2297.388529 | 4.62 | 1.11 | -12.736 | 1.367 | 7.277 | 48.519 | -4.7866 | 0.0161 | -21.198 | 8.954 | 6.262 | 2.071
2298.383859 | 7.51 | 1.46 | -13.666 | 1.986 | 7.272 | 48.512 | -4.847 | 0.0223 | -29.481 | 11.714 | 5.659 | 2.732
2299.452091 | 1.09 | 1.98 | -16.992 | 3.415 | 7.275 | 48.391 | -4.8706 | 0.0307 | -11.599 | 16.296 | 17.651 | 4.4
2303.371341 | -0.39 | 1.37 | -16.247 | 2.125 | 7.261 | 48.583 | -4.8367 | 0.0441 | -6.505 | 10.725 | -7.953 | 3.33
2304.382026 | -7.29 | 1.8 | -25.52 | 2.417 | 7.266 | 48.606 | -4.815 | 0.0165 | -16.113 | 14.563 | -6.493 | 3.38
2307.349721 | -2.59 | 1.15 | -18.116 | 1.685 | 7.256 | 48.595 | -4.8167 | 0.0182 | -8.408 | 9.098 | 24.798 | 2.865
2309.358376 | -0.15 | 1.3 | -16.503 | 1.663 | 7.219 | 48.808 | -4.8179 | 0.0091 | 2.781 | 10.239 | -12.948 | 2.071
2309.371202 | -4.7 | 1.29 | -14.803 | 1.615 | 7.221 | 48.788 | -4.8248 | 0.0104 | -3.381 | 10.303 | -16.168 | 2.449
2310.348335 | 2.24 | 2.7 | -17.647 | 4.171 | 7.218 | 48.549 | -4.8028 | 0.01 | -21.734 | 20.574 | 38.414 | 5.807
2310.388924 | -0.26 | 1.01 | -18.428 | 1.663 | 7.226 | 48.777 | -4.8568 | 0.0167 | -18.036 | 7.356 | -11.074 | 2.285
2322.354105 | -4.91 | 2.04 | -6.607 | 2.953 | 7.252 | 48.389 | -4.8559 | 0.0308 | 10.087 | 17.174 | 44.35 | 3.227
2323.352913 | -1.43 | 1.26 | -20.516 | 1.744 | 7.258 | 48.563 | -4.7567 | 0.0136 | 2.532 | 10.796 | 27.362 | 2.544
2324.363167 | -1.08 | 1.52 | -14.854 | 2.11 | 7.247 | 48.657 | -4.7616 | 0.0102 | 7.458 | 12.643 | -20.307 | 3.142
2469.724254 | -9.49 | 1.48 | -27.555 | 2.283 | 7.219 | 48.807 | -4.7641 | 0.0077 | 21.531 | 12.308 | -17.313 | 2.829
2469.735454 | -11.03 | 1.32 | -16.553 | 2.147 | 7.22 | 48.824 | -4.8477 | 0.0423 | -3.082 | 11.466 | -19.027 | 3.0
2469.745825 | -9.52 | 1.4 | -17.314 | 2.067 | 7.221 | 48.815 | -4.8086 | 0.0134 | 26.204 | 11.819 | -14.525 | 2.95
2470.700788 | -5.58 | 2.52 | -24.563 | 3.877 | 7.22 | 48.715 | -4.808 | 0.0151 | 15.515 | 20.619 | -17.841 | 4.693
2470.722284 | -7.18 | 2.48 | -12.336 | 4.005 | 7.224 | 48.735 | -4.7924 | 0.0085 | -22.477 | 20.357 | -18.624 | 5.05
2470.741717 | -10.08 | 2.04 | -14.927 | 3.423 | 7.213 | 48.738 | -4.7746 | 0.0102 | -17.234 | 16.714 | -9.654 | 3.832
2471.706179 | -4.7 | 1.49 | -24.021 | 2.049 | 7.215 | 48.854 | -4.7648 | 0.0101 | -1.31 | 12.23 | -21.572 | 2.816
2471.71689 | -4.41 | 1.34 | -19.861 | 2.018 | 7.218 | 48.829 | -4.7567 | 0.0072 | 5.007 | 11.085 | -17.867 | 2.701
2548.691205 | -2.68 | 1.2 | -20.396 | 2.187 | 7.201 | 48.946 | -4.7777 | 0.0104 | 8.304 | 10.706 | -32.646 | 2.794
2548.715527 | -2.12 | 1.05 | -23.792 | 1.561 | 7.209 | 48.93 | -4.7743 | 0.0061 | -14.167 | 9.375 | -28.177 | 2.083
2549.68441 | -2.72 | 1.39 | -19.188 | 1.844 | 7.21 | 48.891 | -4.7593 | 0.0109 | 6.16 | 12.284 | -23.636 | 2.341
2549.694997 | -2.85 | 1.28 | -17.059 | 1.745 | 7.209 | 48.896 | -4.7545 | 0.0065 | 12.199 | 11.302 | -20.913 | 2.058
2571.606297 | -10.24 | 1.31 | -21.194 | 1.797 | 7.205 | 48.898 | -4.7508 | 0.0055 | 8.551 | 12.454 | -25.188 | 2.151
2571.617012 | -8.73 | 1.46 | -20.827 | 1.819 | 7.202 | 48.9 | -4.7545 | 0.0081 | 9.234 | 13.847 | -25.299 | 2.076
Table 7: Continuation of radial velocities and spectral activity indicators from Table 6.  BJDTBD | DRS RV | $\mathrm{H\alpha}$ | $\mathrm{NaD_{1}}$ | $\mathrm{NaD_{2}}$ | SNR
---|---|---|---|---|---
$-$2457000 |  [$\mathrm{m\,s^{-1}}$] | — | — | — | (@550nm)
Val. | Val. | $\sigma$ | Val. | $\sigma$ | Val. | $\sigma$ | Val. | $\sigma$ | Val.
2125.664739 | -38807.06 | 1.79 | 0.4358 | 0.0014 | 0.3151 | 0.0017 | 0.4121 | 0.0021 | 50.4
2126.618253 | -38813.01 | 1.46 | 0.4375 | 0.0012 | 0.3169 | 0.0015 | 0.4211 | 0.0018 | 61.3
2127.719329 | -38815.83 | 1.2 | 0.4381 | 0.001 | 0.3201 | 0.0012 | 0.4195 | 0.0015 | 71.9
2130.73596 | -38826.8 | 3.42 | 0.4356 | 0.0026 | 0.3195 | 0.0032 | 0.4395 | 0.0039 | 30.2
2133.711696 | -38826.62 | 1.62 | 0.4303 | 0.0012 | 0.3105 | 0.0016 | 0.4196 | 0.002 | 54.8
2134.756264 | -38828.23 | 1.72 | 0.4326 | 0.0013 | 0.3084 | 0.0017 | 0.4236 | 0.0021 | 52.3
2149.682376 | -38812.1 | 1.48 | 0.4368 | 0.0014 | 0.3134 | 0.0015 | 0.429 | 0.0019 | 61.7
2150.688883 | -38804.86 | 1.26 | 0.4361 | 0.0011 | 0.3172 | 0.0013 | 0.4168 | 0.0016 | 71.4
2151.723635 | -38808.6 | 1.68 | 0.4358 | 0.0012 | 0.3072 | 0.0016 | 0.4167 | 0.002 | 54.7
2156.742015 | -38823.64 | 1.21 | 0.4337 | 0.001 | 0.3135 | 0.0012 | 0.4215 | 0.0015 | 71.8
2158.686506 | -38823.02 | 1.68 | 0.429 | 0.0014 | 0.3199 | 0.0016 | 0.4174 | 0.002 | 55.3
2160.659656 | -38825.07 | 1.25 | 0.4284 | 0.0011 | 0.3132 | 0.0012 | 0.4202 | 0.0016 | 71.8
2172.605447 | -38807.53 | 1.4 | 0.4345 | 0.0011 | 0.3162 | 0.0014 | 0.4183 | 0.0017 | 62.4
2188.596847 | -38814.75 | 1.38 | 0.4329 | 0.0012 | 0.3177 | 0.0014 | 0.4216 | 0.0018 | 63.9
2189.64318 | -38821.64 | 1.12 | 0.4312 | 0.0011 | 0.3129 | 0.0012 | 0.4161 | 0.0015 | 76.8
2190.610672 | -38820.25 | 1.37 | 0.4303 | 0.0013 | 0.3171 | 0.0015 | 0.415 | 0.0018 | 63.8
2191.782237 | -38819.01 | 0.98 | 0.4332 | 0.0007 | 0.3182 | 0.001 | 0.4151 | 0.0012 | 87.7
2204.493933 | -38824.26 | 1.55 | 0.4318 | 0.0015 | 0.3187 | 0.0016 | 0.4142 | 0.002 | 59.3
2204.673882 | -38821.82 | 1.09 | 0.4338 | 0.0011 | 0.3107 | 0.0012 | 0.4277 | 0.0014 | 80.9
2205.499679 | -38817.96 | 0.99 | 0.4315 | 0.0009 | 0.3181 | 0.001 | 0.4193 | 0.0013 | 88.0
2205.685433 | -38819.52 | 1.35 | 0.433 | 0.0012 | 0.3167 | 0.0014 | 0.4157 | 0.0017 | 66.7
2206.4971 | -38818.84 | 1.12 | 0.4339 | 0.0011 | 0.3187 | 0.0012 | 0.4172 | 0.0015 | 79.2
2206.679276 | -38823.34 | 3.16 | 0.4351 | 0.0022 | 0.3229 | 0.0029 | 0.4216 | 0.0035 | 33.0
2212.534392 | -38817.57 | 1.49 | 0.4333 | 0.0011 | 0.3178 | 0.0015 | 0.4185 | 0.0018 | 58.8
2215.622998 | -38804.38 | 1.08 | 0.4338 | 0.0008 | 0.3178 | 0.0011 | 0.4176 | 0.0013 | 78.7
2248.433766 | -38817.36 | 0.94 | 0.4353 | 0.0009 | 0.3119 | 0.001 | 0.4201 | 0.0012 | 93.9
2248.444309 | -38817.29 | 0.91 | 0.4365 | 0.001 | 0.3103 | 0.001 | 0.4197 | 0.0012 | 96.6
2265.415755 | -38812.63 | 1.81 | 0.4353 | 0.0014 | 0.3143 | 0.0017 | 0.419 | 0.0022 | 51.8
2265.42869 | -38812.7 | 2.06 | 0.4364 | 0.0016 | 0.315 | 0.0019 | 0.4193 | 0.0024 | 46.7
2265.560543 | -38814.58 | 2.42 | 0.4368 | 0.0016 | 0.3094 | 0.0022 | 0.4159 | 0.0027 | 40.6
2265.57402 | -38812.17 | 2.11 | 0.435 | 0.0014 | 0.3144 | 0.0019 | 0.4202 | 0.0024 | 45.3
2268.426272 | -38816.12 | 0.85 | 0.4335 | 0.0008 | 0.3069 | 0.0009 | 0.4173 | 0.0011 | 103.8
2268.438999 | -38816.24 | 0.83 | 0.4351 | 0.0008 | 0.3076 | 0.0009 | 0.4187 | 0.0011 | 106.2
2268.505815 | -38816.0 | 1.03 | 0.4345 | 0.0009 | 0.3132 | 0.001 | 0.4181 | 0.0013 | 87.5
2268.519837 | -38815.97 | 1.31 | 0.4337 | 0.0011 | 0.314 | 0.0013 | 0.4197 | 0.0016 | 70.1
2275.338169 | -38812.94 | 0.95 | 0.4331 | 0.0009 | 0.3135 | 0.001 | 0.4218 | 0.0013 | 90.9
2276.48642 | -38820.71 | 1.22 | 0.4378 | 0.0012 | 0.3111 | 0.0013 | 0.42 | 0.0016 | 73.3
2277.409061 | -38820.58 | 1.04 | 0.4397 | 0.001 | 0.3129 | 0.0011 | 0.4224 | 0.0014 | 83.7
2278.43132 | -38822.2 | 1.14 | 0.4363 | 0.0011 | 0.3144 | 0.0012 | 0.4228 | 0.0015 | 77.4
2287.3862 | -38816.83 | 1.86 | 0.4379 | 0.0015 | 0.3131 | 0.0018 | 0.4257 | 0.0023 | 49.6
2289.40115 | -38820.9 | 1.24 | 0.4403 | 0.001 | 0.3121 | 0.0013 | 0.4202 | 0.0016 | 70.6
2290.341493 | -38816.6 | 0.94 | 0.4396 | 0.0008 | 0.3131 | 0.001 | 0.4215 | 0.0012 | 91.0
2294.435475 | -38806.25 | 1.23 | 0.4341 | 0.0011 | 0.3132 | 0.0012 | 0.42 | 0.0016 | 72.5
2297.388529 | -38813.86 | 0.96 | 0.4364 | 0.0009 | 0.3136 | 0.001 | 0.4224 | 0.0012 | 90.9
2298.383859 | -38810.32 | 1.39 | 0.4384 | 0.0012 | 0.3126 | 0.0014 | 0.4199 | 0.0018 | 65.1
2299.452091 | -38816.21 | 2.39 | 0.437 | 0.0018 | 0.311 | 0.0022 | 0.4233 | 0.0028 | 41.2
2303.371341 | -38819.51 | 1.49 | 0.4382 | 0.001 | 0.3153 | 0.0014 | 0.4228 | 0.0018 | 59.3
2304.382026 | -38824.48 | 1.69 | 0.4379 | 0.0013 | 0.3115 | 0.0016 | 0.4191 | 0.0021 | 54.1
2307.349721 | -38821.72 | 1.18 | 0.433 | 0.0009 | 0.3125 | 0.0011 | 0.4312 | 0.0014 | 73.5
2309.358376 | -38817.17 | 1.17 | 0.4319 | 0.0008 | 0.3092 | 0.0011 | 0.4224 | 0.0014 | 75.4
2309.371202 | -38820.39 | 1.14 | 0.433 | 0.0008 | 0.3082 | 0.0011 | 0.4223 | 0.0014 | 78.3
2310.348335 | -38814.24 | 2.94 | 0.4361 | 0.0017 | 0.3121 | 0.0025 | 0.4206 | 0.0031 | 34.2
2310.388924 | -38815.96 | 1.17 | 0.4368 | 0.0007 | 0.3121 | 0.0011 | 0.4248 | 0.0014 | 73.3
2322.354105 | -38822.48 | 2.07 | 0.4274 | 0.0016 | 0.3151 | 0.002 | 0.4295 | 0.0025 | 45.8
2323.352913 | -38818.13 | 1.22 | 0.432 | 0.0011 | 0.3104 | 0.0013 | 0.4193 | 0.0016 | 72.2
2324.363167 | -38821.12 | 1.48 | 0.431 | 0.0012 | 0.308 | 0.0015 | 0.4138 | 0.0018 | 61.0
2469.724254 | -38826.37 | 1.61 | 0.4368 | 0.0014 | 0.3161 | 0.0016 | 0.4203 | 0.002 | 57.9
2469.735454 | -38827.19 | 1.51 | 0.4364 | 0.0014 | 0.313 | 0.0015 | 0.4183 | 0.0019 | 61.0
2469.745825 | -38825.9 | 1.46 | 0.4328 | 0.0014 | 0.317 | 0.0015 | 0.4153 | 0.0019 | 63.0
2470.700788 | -38823.39 | 2.73 | 0.4331 | 0.002 | 0.3171 | 0.0025 | 0.4179 | 0.0031 | 37.1
2470.722284 | -38821.79 | 2.82 | 0.434 | 0.0021 | 0.316 | 0.0026 | 0.4202 | 0.0032 | 36.1
2470.741717 | -38826.41 | 2.41 | 0.4352 | 0.0018 | 0.3164 | 0.0023 | 0.4137 | 0.0028 | 41.0
2471.706179 | -38822.72 | 1.44 | 0.4295 | 0.0011 | 0.3179 | 0.0014 | 0.4189 | 0.0017 | 63.4
2471.71689 | -38821.18 | 1.42 | 0.4323 | 0.0012 | 0.3188 | 0.0014 | 0.4177 | 0.0017 | 64.6
2548.691205 | -38819.07 | 1.54 | 0.4306 | 0.0013 | 0.3183 | 0.0015 | 0.4301 | 0.0018 | 62.6
2548.715527 | -38818.63 | 1.1 | 0.4282 | 0.0009 | 0.3166 | 0.0011 | 0.4326 | 0.0013 | 84.5
2549.68441 | -38819.35 | 1.3 | 0.429 | 0.0011 | 0.3174 | 0.0012 | 0.4241 | 0.0015 | 72.9
2549.694997 | -38821.02 | 1.23 | 0.4304 | 0.001 | 0.3183 | 0.0012 | 0.4217 | 0.0015 | 76.9
2571.606297 | -38825.54 | 1.27 | 0.4273 | 0.0012 | 0.3173 | 0.0012 | 0.4239 | 0.0016 | 75.2
2571.617012 | -38825.13 | 1.28 | 0.4276 | 0.0012 | 0.3176 | 0.0013 | 0.4259 | 0.0016 | 74.5
Table 8: More radial velocities and spectral activity indicators measured from TNG/HARPS-N spectra with serval and DRS, continuation from Table 6.  BJDTBD |  RV serval | BIS | CCF_FWHM | CCF_CTR | $\mathrm{\log{R^{`}_{HK}}}$ | CRX | dlW
---|---|---|---|---|---|---|---
$-$2457000 |  [$\mathrm{m\,s^{-1}}$] |  [$\mathrm{m\,s^{-1}}$] |  [$\mathrm{km\,s^{-1}}$] | (%) | — |  ($\mathrm{m\,s^{-1}\,Np^{-1}}$) |  [$\mathrm{m^{2}\,s^{-2}}$]
Val. | Val. | $\sigma$ | Val. | $\sigma$ | Val. | Val. | Val. | $\sigma$ | Val. | $\sigma$ | Val. | $\sigma$
2572.618783 | -9.92 | 1.1 | -16.587 | 1.654 | 7.203 | 48.926 | -4.7687 | 0.0063 | 0.712 | 10.598 | -26.164 | 2.053
2573.609105 | -8.63 | 1.05 | -16.946 | 1.609 | 7.205 | 48.935 | -4.7727 | 0.0073 | -2.969 | 10.216 | -30.159 | 2.145
2573.620272 | -10.03 | 1.42 | -19.354 | 1.91 | 7.196 | 48.96 | -4.7861 | 0.0167 | -4.553 | 13.573 | -35.489 | 2.397
2574.616426 | -9.14 | 1.21 | -24.856 | 1.892 | 7.206 | 48.922 | -4.7958 | 0.009 | 2.954 | 11.396 | -32.579 | 2.343
2574.626909 | -5.63 | 1.22 | -20.424 | 1.761 | 7.203 | 48.981 | -4.7911 | 0.0057 | 6.312 | 11.505 | -36.546 | 2.263
2593.541588 | -12.46 | 1.45 | -21.182 | 2.575 | 7.195 | 48.881 | -4.7512 | 0.0078 | 23.799 | 13.114 | -28.16 | 3.367
2594.558159 | -13.36 | 2.09 | -25.055 | 3.025 | 7.201 | 48.767 | -4.7486 | 0.0054 | 12.099 | 18.98 | -16.529 | 3.449
2608.642185 | -2.63 | 2.25 | -16.671 | 4.002 | 7.202 | 48.897 | -4.7303 | 0.0093 | 10.776 | 20.233 | -28.475 | 4.352
2609.541599 | -6.15 | 1.3 | -12.365 | 2.229 | 7.189 | 48.99 | -4.7497 | 0.0215 | 8.681 | 11.756 | -36.94 | 2.724
2609.60981 | -2.56 | 1.92 | -14.308 | 2.344 | 7.203 | 48.998 | -4.7685 | 0.0109 | 5.883 | 17.278 | -34.373 | 3.209
2610.443516 | -4.76 | 1.31 | -16.717 | 1.539 | 7.202 | 49.008 | -4.8084 | 0.0151 | 32.462 | 12.065 | -34.381 | 1.831
2610.564053 | -9.59 | 1.17 | -16.357 | 1.646 | 7.202 | 49.018 | -4.8002 | 0.0084 | 29.505 | 10.587 | -36.771 | 2.288
2664.36496 | -8.65 | 1.36 | -15.026 | 1.651 | 7.209 | 48.987 | -4.8028 | 0.021 | 15.094 | 12.691 | -31.371 | 2.305
2676.357574 | -13.89 | 1.5 | -21.413 | 2.078 | 7.191 | 49.086 | -4.7978 | 0.0091 | 11.69 | 14.028 | -45.617 | 2.586
2677.355196 | -13.72 | 1.74 | -19.361 | 3.014 | 7.195 | 49.0 | -4.8078 | 0.0122 | -13.236 | 16.024 | -5.524 | 3.676
Table 9: Continuation of radial velocities and spectral activity indicators from Table 8.  BJDTBD | DRS RV | $\mathrm{H\alpha}$ | $\mathrm{NaD_{1}}$ | $\mathrm{NaD_{2}}$ | SNR
---|---|---|---|---|---
$-$2457000 |  [$\mathrm{m\,s^{-1}}$] | — | — | — | (@550nm)
Val. | Val. | $\sigma$ | Val. | $\sigma$ | Val. | $\sigma$ | Val. | $\sigma$ | Val.
2572.618783 | -38827.33 | 1.17 | 0.4286 | 0.0011 | 0.3194 | 0.0012 | 0.424 | 0.0015 | 81.0
2573.609105 | -38826.06 | 1.13 | 0.4317 | 0.0011 | 0.3146 | 0.0011 | 0.4231 | 0.0014 | 83.2
2573.620272 | -38826.21 | 1.35 | 0.4295 | 0.0013 | 0.3154 | 0.0014 | 0.4259 | 0.0017 | 71.2
2574.616426 | -38827.24 | 1.33 | 0.4349 | 0.0011 | 0.3223 | 0.0012 | 0.426 | 0.0015 | 75.0
2574.626909 | -38824.05 | 1.24 | 0.4309 | 0.001 | 0.319 | 0.0011 | 0.4264 | 0.0014 | 79.8
2593.541588 | -38829.8 | 1.82 | 0.426 | 0.0016 | 0.3166 | 0.0017 | 0.4228 | 0.0022 | 55.2
2594.558159 | -38831.89 | 2.13 | 0.4286 | 0.0018 | 0.3154 | 0.002 | 0.4269 | 0.0025 | 48.1
2608.642185 | -38823.24 | 2.82 | 0.4302 | 0.0022 | 0.3101 | 0.0025 | 0.4237 | 0.0032 | 38.9
2609.541599 | -38821.59 | 1.57 | 0.4266 | 0.0012 | 0.3101 | 0.0014 | 0.4285 | 0.0018 | 61.8
2609.60981 | -38818.43 | 1.65 | 0.4257 | 0.0012 | 0.3115 | 0.0015 | 0.4355 | 0.0019 | 59.2
2610.443516 | -38822.51 | 1.09 | 0.429 | 0.0009 | 0.3127 | 0.001 | 0.4246 | 0.0013 | 86.9
2610.564053 | -38826.56 | 1.16 | 0.4263 | 0.001 | 0.3145 | 0.0011 | 0.4255 | 0.0014 | 82.0
2664.36496 | -38826.09 | 1.16 | 0.4215 | 0.001 | 0.3137 | 0.0011 | 0.4346 | 0.0014 | 81.0
2676.357574 | -38830.48 | 1.46 | 0.4189 | 0.0014 | 0.3138 | 0.0015 | 0.4225 | 0.0019 | 65.9
2677.355196 | -38829.55 | 2.12 | 0.4224 | 0.0018 | 0.3151 | 0.002 | 0.4357 | 0.0025 | 48.6
Figure 12: GLS periodograms for the activity indicators computed from the
HARPS-N spectra (from top to bottom): cross-correlation function bisector
(BIS), cross-correlation function contrast (CTR), cross-correlation function
full width at half maximum (FWHM), chromatic index (CRX), differential line
width (dLW), and the H$\alpha$, Na D1, Na D2, and RHK indexes. The 10%, 1%,
and 0.1% FAP levels are indicated by the grey dotted, dash-dotted, and dashed
lines, respectively. The vertical red dotted line indicates the period of
TOI-1710 b at 24.28 d as a reference. The shaded orange region indicates the
3$\sigma$ region of the $P_{\rm rot}$ from the joint fit.  Figure 13:
Correlation between HARPS-N RVs and the HARPS-N activity indicators (orange
points with error bars). In each case a linear regression was performed
(overplotted blue line) and the Pearson correlation $r$ was computed, which is
indicated.

Figure 14: Generalized Lomb-Scargle periodograms of HARPS-N, SOPHIE, and
combined RV residuals (from top to bottom) after subtracting the 1pl model
(left), 1pGP model (centre), and 1pGP+FWHM model (right). The vertical red
dotted line indicates the period of TOI-1710 b at 24.28 d. The shaded orange
region indicates the 3$\sigma$ region of the $P_{\rm rot}$ from the joint fit
(see Sect. 4.3). The 10%, 1%, and 0.1% FAP levels are indicated by the grey
dotted, dash-dotted, and dashed lines, respectively.