Revisiting the warm sub-Saturn TOI-1710b

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-1710b 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-1710b 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-1710b a very suitable candidate for follow-up atmospheric characterization.


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 R P <4 R ⊕ 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 R ⊕ ) and Saturn (9.5 R ⊕ ); 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(Petigura et al. , 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 R p = 5.34 ± 0.11 R ⊕ and a mass of M p = 28.3 ± 4.7 M ⊕ .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

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. 1  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 (MAST 2 ).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. 2012Stumpe et al. , 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 A.1.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 catalogue 3 (Guerrero et al. 2021).A signal with a period of 24.28 days and a transit depth (∆F) of 3070.0 ppm, corresponding to a planet radius of about 5.4 R ⊕ , 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 ± 2.6") and excluded all other TIC objects.

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 groundbased 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.
2.4.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 ∼ 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 HK ) were measured using an online version of the DRS, the YABI tool,4 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 templatematching, chromatic index (CRX), differential line width (dLW), and Hα, sodium Na D1 and Na D2 indexes.The uncertainties of the RVs measured with serval are in the range 0.8-2.6 m s −1 , with a mean value of 1.4 m s −1 .The uncertainties of absolute RVs measured with DRS are in the range 0.8-3.2m s −1 , with a mean value of 1.5 m s −1 .Table B.1 gives the time stamps of the spectra in BJD TDB serval relative RVs along with their 1σ error bars, and spectral activity indicators measured with YABI and serval.König et al. (2022) also performed a ground-based followup 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.

High-resolution imaging with Gemini/'Alopeke
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.Addition-

nm
Fig. 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"×1.2"reconstructed image of TOI-1710 at 832 nm band, centred on the star.ally, '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. 5'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 1000×0.06sec 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.

Stellar companions
Given the fainter TOI-1710 companion labelled #2 in Figure A.1 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 G RP = 17.4 mag is the only target within one TESS pixel, but it is much fainter than TOI-1710 (G RP = 8.9 mag).Stars with differences in Gaia G RP -band larger than 8 mag are not detected in the TESS images (Figure A.1).Only Gaia DR2 1116612783096856960, which has ∆ G RP = 4.2 mag, is detected.We identified this star as TIC 445805957, which is the southern star labelled #2 in Figure A.1.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 G RP -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 TESS ) using its definition from Eq. 6 in Espinoza et al. (2019).We calculated a D 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 TESS is 0.999 (i.e.consistent with little to no flux contamination).

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 ′ HK measured with YABI, we estimated a rotation period of TOI-1710 for 20.9 ± 4.2 days and 20.6 ± 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 2.2-3.9Gyr.
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.
We searched for photometric periodicities in the total of 117 NSVS measurements using the generalized Lomb-Scargle (GLS) periodogram6 (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 rot reported by König et al. (2022) of ∼22 d.

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 ± 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 ± 0.05 km s −1 was also derived by requiring no trend of Fe line abundances against their equivalent widths.The output metallicity (+0.12 ± 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 eff = 5720 ± 28, log g = 4.49 ± 0.04, and [M/H] = +0.06± 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 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 ± 32 K and 5719 ± 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(Rodrigues et al. , 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 0.95 ± 0.02 R ⊙ and 0.99 ± 0.07 M ⊙ , 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 ± 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 ± 0.6 Gyr) derived from the empirical R ′ 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 ⋆ , R ⋆ , log(g), metallicity, age, and log(R ′ HK ) are still in agreement within the error bars.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 HK .All the activity indicators except CRX and Na D2 also present significant peaks at ∼ P rot /2.König et al. (2022) also presented the GLS periodograms for FWHM, Hα, and log(R' HK ) with significant peaks close to the stellar rotation period.Moreover, the HARPS-N CTR, FWHM, dLW, and Hα index have strong correlations with the HARPS-N RVs (see Fig. B.2). 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.

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 midsector 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 p /R ⋆ ) 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 A

Joint fit modelling
We performed a joint fit using only TESS photometry and RV data using PyORBIT 7 (Malavolta et al. 2016(Malavolta et al. , 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 7 Available at https://github.com/LucaMalavolta/PyORBIT  We use the GP quasi-periodic kernel, as defined by Grunblatt et al. (2015), imposing a uniform prior   (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.
on the rotational period P 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 ⋆ .The eccentricity ecc and argument of periastron ω were modelled with the parametrization from Eastman et al. 2013 ( √ e cos ω, √ e sin ω).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; 9 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 9 Available at https://github.com/hpparvi/ldtkFWHM and the HARPS-N RVs (Fig. B.2).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≫10%), when adopting the GP in the model (see Fig B .3).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 rot in their activity indicators GLS periodograms, but the stellar rotation was not strong in their RV periodogram.No correlation plots (e.g.Fig. B.2) 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 rot and the planet period peaks are clear in the GLS peri-odogram (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 1.6σ.Moreover, although our analyses included 54 more RVs, the fitted semi-amplitude has the same uncertainties (±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 .

Discussion and conclusions
From our analyses, we derive for the transiting sub-Saturn planet TOI-1710 b a radius of R p = 5.15 ± 0.12 R ⊕ , a mass of M p = 18.4 +4.8 −4.5 M ⊕ , and a mean bulk density of ρ p = 0.73 ± 0.18 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 +0.12 −0.091 ).The mass uncertainties are very similar between the two works, but our mean mass measurement value presents a precision at the 4σ level, and it is ∼1.5σ away from that of König et al. ( 2022, M p = 28.3 ± 4.7 M ⊕ ).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. B.3).
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. B.3). König et al. (2022) reported an extra peak at 30 d with FAP ≃ 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.

Potential atmospheric characterization of TOI-1710 b
TOI-1710 b is a gaseous planet with a scale height (H) of ∼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 ∼150, which is greater than the threshold TSM of 90 for planets ranging 1.5-10 R ⊕ to be selected as high-quality targets for its atmospheric characterization.
There are only a few sub-Saturn planets around bright stars (J ≤ 9) with higher TSM: GJ 436b (J = 6.9 mag, TSM = 481; Butler et al. 2004) 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 cometarylike 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 X = 28.07 and a calculated F XUV ∼1 W m −2 irradiance in the 5-504 Å 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 Å 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.

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 Archive10 along with theoretical composition models of Zeng et al. (2019). 11The displayed models regard planets with Earth-like composition cores (32.5% Fe + 67.5% MgSiO 2 ), 100% H 2 O composition, and 95% Earth-like rocky core surrounded by 5% H 2 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% H 2 gaseous envelope), and its bulk mean density (ρ p = 0.75 ± 0.18 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 M ⊕ , 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 p = 67 ± 21 M ⊕ .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 [Fe/H] = 0.12 ± 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.The shaded orange region indicates the 3σ region of the P 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.

4. 1 .Fig. 3 .Fig. 4 .
Fig. 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.
.1.The TESS data along with the transiting and GP models are shown in Fig. A.2.The TOI-1710 b phase-folded photometry from the juliet anaysis is shown in Fig. A.3.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 8σ significance and ontime 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 A.4.
a) U[a, b] refers to uniform priors between a and b; N[a, b] refers to Gaussian priors with median a and standard deviation b.

Fig. 6 .
Fig.6.Mass-radius (left) and period-radius (right) diagrams for all known planets with M p ∈ [1,60] M ⊕ and R p ∈ [1,10] R ⊕ determined with a precision better than 30% (grey points).The coloured dashed lines are the theoretical mass-radius models fromZeng 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.

Fig. 7 .
Fig. 7. Mass-density (top), mass-eccentricity (middle), and massmetalicity (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.

Fig
Fig. A.1.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 (×) and labelled #1.

4 .
Fig. A.4.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.

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

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.
et al. 2013).The initial conditions are obtained with the global optimization code PyDE.

Table 4 .
TOI-1710 parameters from the transit and RV joint fit, obtained with the model 1pGP.