The GAPS programme at TNG XLIV. Projected rotational velocities of 273 exoplanet-host stars observed with HARPS-N

The leading spectrographs used for exoplanets' sceince offer online data reduction softwares (DRS) that yield as an ancillary result the full-width at half-maximum (FWHM) of the cross-correlation function (CCF) that is used to estimate the radial velocity of the host star. The FWHM also contains information on the stellar projected rotational velocity vsini We wanted to establish a simple relationship to derive the vsini directly from the FWHM computed by the HARPS-N DRS in the case of slow-rotating solar-like stars. This may also help to recover the stellar inclination i, which in turn affects the exoplanets' parameters. We selected stars with an inclination of the spin axis compatible with 90 deg by looking at exoplanetary transiting systems with known small sky-projected obliquity: for these stars, we can presume that vsini is equal to stellar equatorial velocity veq. We derived their rotational periods from photometric time-series and their radii from SED fitting. This allowed us to recover their veq, which we could compare to the FWHM values of the CCFs obtained both with G2 and K5 spectral type masks. We obtained an empirical relation for each mask, useful for slow rotators (FWHM<20 km/s). We applied them to 273 exoplanet-host stars observed with HARPS-N, obtaining homogeneous vsini measurements. We compared our results with the literature ones to confirm the reliability of our work, and we found a good agreement with the values found with more sophisticated methods for stars with log g>3.5. We also tried our relations on HARPS and SOPHIE data, and we conclude that they can be used also on FWHM derived by HARPS DRS with G2 and K5 mask, and they may be adapted to the SOPHIE data as long as the spectra are taken in the high-resolution mode. We were also able to recover or constrain i for 12 objects with no prior vsini estimation.


Introduction
Stable, high-resolution optical spectrographs are some of the leading instruments used for the search and characterization of the exoplanets: many of them are designed expressly for these studies (e.g.HARPS, HARPS-N, ESPRESSO), and as such they are equipped with dedicated data reduction softwares (DRS).One of the main deliverables of the DRS is the cross-correlation function (CCF) of the reduced spectra with a stellar mask chosen from the available library of spectral type templates (Baranne et al. 1996;Pepe et al. 2002).
The CCF allows computing the radial velocity of the host star with very high precision, and it also yields a number of additional parameters, such as the CCF's bisector span (which can be used as an activity indicator), the CCF's contrast, and the full-width at half-maximum (FWHM).The latter may be related to the stellar projected rotational velocity v eq sin i ⋆ if appropriately calibrated: in this paper, we present the work done to cali-⋆ Based on observations made with the Italian Telescopio Nazionale Galileo (TNG) operated by the Fundación Galileo Galilei (FGG) of the Istituto Nazionale di Astrofisica (INAF) at the Observatorio del Roque de los Muchachos (La Palma, Canary Islands, Spain).brate the FWHM of the CCF that is computed by the HARPS-N DRS (Cosentino et al. 2014) using the G2 and K5 stellar masks.HARPS-N is the high-resolution optical spectrograph installed at the Telescopio Nazionale Galileo (TNG) at the Roque de Los Muchachos Observatory (La Palma, Canary Islands, Spain).
The use of the CCF's FWHM to estimate the v eq sin i ⋆ is particularly important in the case of slowly rotating stars, for which the v eq sin i ⋆ computation via Fourier transform of the line profiles or fitting with a rotational profile is complicated by the combination of the rotational broadening with the effects of the resolution smearing (≈ 2.6 km s −1 in the case of HARPS-N, R=115,000), and the micro-(v micro ) and macro-(v macro ) turbulence broadening.Slowly rotating solar-like and M-type stars are also among the main targets in the exoplanet field, therefore it is particularly important to have a reliable method to estimate the v eq sin i ⋆ for these objects in order to better characterize the host stars.Using the FWHM given by the HARPS-N DRS allows everyone to recover the v eq sin i ⋆ values directly for the HARPS-N archival data.
Once it is obtained, the v eq sin i ⋆ value may be used along with estimates of the stellar rotational period P rot (for exam-ple from photometric time-series or spectroscopic time-series of activity indices) and the stellar radius R ⋆ (derived for example from Spectral Energy Distribution (SED) fitting, see Sec. 2) to recover the stellar inclination i ⋆ , which heavily affects exoplanets' parameters (Hirano et al. 2014): The stellar inclination is a fundamental step also in computing the spin-orbit angle of exoplanetary systems, that is an important observational probe of the origin and evolution of the systems (e.g., Queloz et al. 2000;Winn et al. 2005).
The approach of exploiting known stellar radii and rotational periods to infer the rotational velocity and to calibrate the width of the CCF vs. v eq sin i ⋆ is not completely new, as it was previously adopted by Nordström et al. (2004).However, in their case, the stellar inclination remains unknown and the additional uncertainty is treated statistically.Instead, in our work we took advantage of the known viewing geometry of stars which host a transiting planet with an orbit inclination close to 90 deg, and a good spin-orbit alignment as inferred by the measurement of the Rossiter-McLaughlin effect (Rossiter 1924;McLaughlin 1924).This allowed us to rely on a sample of objects for which the projected rotational velocity, linked to the CCF width, is similar to the equatorial velocity inferred from the rotational period and the stellar radius.Furthermore, the selection of a sample of transiting planets ensures the availability of high-quality photometric data (which were taken for the planet search itself) and in most cases of additional relevant literature studies from follow-up observations.
This paper is organized as follows: in Section 2 we describe the selection procedure for our calibrators.We used them in Section 3 to create our empirical relation, and then we applied it to a large set of exoplanet host stars in Section 4. We test the applicability of our relation to other spectrographs in Section 5, and finally we present our conclusions in Section 6.

Calibrators' selection and characterization
To calibrate our empirical relation as accurately as possible, we relied on a very strict selection of calibrators.We queried the NASA exoplanet archive1 to obtain a list of all known exoplanet host stars with a) declination > −25 deg (to ensure they were observable with the TNG), and b) an absolute value of the system sky-projected obliquity λ smaller than 30 deg, as derived from the Rossiter-McLaughlin effect and reported in the TEPCat catalog (Southworth 2011).The latter value is a compromise between the need to have systems that can be considered aligned in such a way that the stellar projected rotational velocity v eq sin i ⋆ can be considered approximately equal to the stellar equatorial velocity v eq , and the need to have a good number of useful calibrators (at least some tens of objects).
This selection resulted in a list of 66 targets.We then searched the TNG archive for public HARPS-N spectra of these stars, to combine with the proprietary data obtained within the Global Architecture of Planetary Systems (GAPS) program, which is an Italian project dedicated to the search and characterization of exoplanets (PI G. Micela; Covino et al. 2013).We thus found 44 stars with useful HARPS-N CCFs.
The stellar masks available in the DRS library are optimized for main sequence stars with stellar types G2, K5, and M2.With the new upgrades to the DRS, more masks are starting to be available for different spectral types, and they will have to be calibrated accordingly, but in this work we focused on the original masks that have been used so far, and that are still available in the DRS.Unfortunately, the M2 CCFs are useless for our purposes because the use of the M2 mask results in deformed CCF profiles with large bumps in the wings.In a previous work (Rainer et al. 2020), we created an improved M-type mask to overcome this problem, but we will not consider this mask here because it is not publicly available: our scope is to enable astronomers to use the public HARPS-N archival data.Thus, we focused on the G2, and K5 CCFs: while this optimized our work for solar-like stars, still some M-type stars may be reduced using the K5 mask in order to recover the v eq sin i ⋆ estimate from the CCF FWHM.
Our selection criteria ensure that sini ⋆ ≈ 1, which means that we can consider v eq sin i ⋆ ≈ v eq for all our calibrators.If we are able to estimate the equatorial velocity v eq , then we can build a relation between FWHM and v eq sin i ⋆ in a straightforward way.In order to compute v eq we needed estimates of the rotational periods P rot and the radii R ⋆ of our calibrators: We derived the rotation period P rot mainly from TESS (Ricker et al. 2015) and SuperWASP (Butters et al. 2010) photometry.In the case of TESS, we used the PDCSAP light curves (Stumpe et al. 2012) as downloaded from the MAST archive 2 , where systematic artifacts are likely removed by the PDCSAP pipeline.PDCSAP light curves were analysed using the Generalized Lomb-Scargle periodogram (GLS; (Zechmeister & Kürster 2009) and the detected periods are listed in Table 1 In the case of the SuperWASP photometric time-series, we first disregarded possible outliers, i.e. data points that deviated more than 3 standard deviations from the mean of the whole data series.Then, we computed a filtered version of the light curve by means of a sliding median boxcar filter with a boxcar extension equal to 2 hours.This filtered light curve was then subtracted from the original light curve, and all the points deviating more than 3 standard deviations of the residuals were discarded.Finally, we computed normal points by binning the data on time intervals having the duration of about 2 hours.The rotation period search was performed by using the GLS and the CLEAN (Roberts et al. 1987) periodogram analysis.All the periodicities detected by GLS, with a false alarm probability smaller than 0.1% (see Horne & Baliunas 1986), and recovered with the same value, within the uncertainty, also by CLEAN, were considered as the star's rotation period and listed in Table 1.To compute the error associated with the period, we followed the method used by Lamm et al. (2004).
We also checked the spectroscopic activity indicators timeseries: we investigated the R ′ HK activity index using GLS.In general, we did not find any conclusive results given that for most stars only a small number of observations sparsely obtained over a few years were available.In a few cases, the periodogram analysis provided P rot detection, which was always consistent with the photometrically determined period.For the sake of sample homogeneity, we thus considered only the photometric periods.
The stellar radii R ⋆ were obtained by fitting the Spectral Energy Distribution (SED) via the MESA Isochrones and Stellar Tracks (MIST, Dotter 2016;Choi et al. 2016) through the EXOFASTv2 suite (Eastman et al. 2019).Specifically, we fitted References. 1 SuperWASP; 2 TESS; 3 this work; 4 Nikolov et al. (2014); 5 Ment et al. (2018); 6 Stassun et al. (2017); 7 Bonomo et al. (2017); 8 Mancini  et al. (2018); 9 Sada & Ramón-Fox (2016); 10 Buchhave et al. (2010); 11  Bakos et al. (2011); 12 Esposito et al. (2017); 13 Fischer et al. (2007); 14  Mann et al. (2020); 15 https://exofop.ipac.caltech.edu; 16Santerne et al. (2016); 17   Crouzet et al. (2012) the available archival magnitudes of each star in the sample imposing Gaussian priors on the effective temperature T eff and metallicity [Fe/H] based on the respective literature values listed in Table 1 and on the parallax π based on the Gaia EDR3 astrometric measurement (Gaia Collaboration et al. 2016, 2021).Since the SED primarily constrains R ⋆ and T eff , the stellar parameters are simultaneously constrained by the SED and the MIST isochrones, and a penalty for straying from the MIST evolutionary tracks ensures that the resulting star realization is physical in nature (see Eastman et al. 2019, for more details on the method).In Fig. 1 we show our results compared with the R ⋆ and T eff of the exoplanet-host stars present in the NASA archive, while in Fig. 2 we show the correlation and residuals between our values and those from the literature.
Thus we obtained our semi-final calibrators' list, which is shown in Table 1: 27 stars with known P rot and R ⋆ .In the end, all our calibrators have λ < 21.2 degrees, strengthening our assumption of v eq ≈ v eq sin i ⋆ .We also checked the Gaia DR3 archive to ensure that we are working with single stars: K2-29 has a fainter companion separated by ≈4.4 arcsec with ∆V=1.8, and TrES-4 has a fainter companion separated by ≈1.6 arcsec with ∆V=4.9.We considered that in both cases the combination of the faintness and the distance of the companions allowed us to keep the stars in our calibrators' list.
Using the stellar parameters T eff and log g from the literature, we estimated the micro-(v micro ) and macro-(v macro ) turbulence velocities for each object.In particular, v micro was obtained with Adibekyan et al. (2012) relationships valid for stars with 4500 < T eff < 6500 K, 3.0 < log g < 5.0, and -1.4 < [Fe/H] < 0.5 dex.Regarding v macro , it was computed with the calibration obtained by Doyle et al. (2014) using asteroseismic rotational velocities for the stars with T eff > 5700 K, while for the stars with T eff < 5700 K we used the empirical relationship by Brewer et al. (2016).Both relations are valid for dwarf stars (see also Biazzo et al. 2022).To estimate the errors on our v micro and v macro , we considered the root-mean-square error (rms) given in the papers, which is larger than the errors derived from the parameters.The rms are 0.18 km s −1 for v micro , 0.73 km s −1 for v macro from Doyle et al. ( 2014) (T eff > 5700 K), and 0.5 km s −1 for v macro from Brewer et al. (2016) (T eff < 5700 K).
We note that HAT-P-2 has P rot =2.82±0.05days from TESS photometry, but a completely different value from SuperWASP (97±10 days).Applying Eq. 2, the TESS value yields v eq =30.12 km s −1 , and the SuperWASP value v eq =0.88 km s −1 .The TESS value is nearer to the v eq sin i ⋆ = 20.12±0.9km s −1 result obtained from the Fourier transform of the CCF and with the 20.8±0.03 km s −1 value from the literature (Bonomo et al. 2017), but there is still a large discrepancy.In any case, this fast rotation excludes this star from being a useful calibrator (see Section 3): the final calibrators' list thus contains the stars in Table 1 with the exception of HAT-P-2.

Creating the empirical relation
In order to create our empirical relation, we used the following inputs: the FWHM of the CCFs of the HARPS-N spectra, as computed by the HARPS-N DRS and stored in the keyword HIERARCH TNG DRS CCF FWHM of the CCF FITS files; the stellar radii R ⋆ from Table 1; the rotational periods P rot from Table 1; v micro and v macro from Table 1.
Using the archival CCFs, we are limited by the standard CCF half-window of the HARPS-N DRS (20 km s −1 ): while it may be manually changed, the majority of the archival data will have this value.We also note that a more precise v eq sin i ⋆ could be recovered for faster rotating stars using rotational fitting or the Fourier transform method, instead of any empirical relation.We thus limit the applicability range of our relation to FWHM up to 20 km s −1 , which is a slightly larger value than the maximum FWHM that can be reliably computed with an half-window of 20 km s −1 , i.e. ≈16-18 km s −1 .
To check this applicability range, we built a range of synthetic CCF profiles by convolving a Gaussian function with the same FWHM of the HARPS-N resolution (≈2.6 km s −1 ) with different rotational profiles (v eq sin i ⋆ ranging from 0.2 to 50 km s −1 with a step of 0.2 km s −1 ).The rotational profiles were built using the following equation from Gray (2008): where a is the depth of the profile, x 0 the centre (i.e., the RV value), x l the v eq sin i ⋆ of the star, u the linear limb darkening (LD) coefficient, which we kept fixed as u=0.6.
We fitted the resulting profiles with a Gaussian (see Fig. 3) and compared the Gaussian FWHM with the input v eq sin i ⋆ to check their correlation.We chose a Gaussian fit to be consistent with HARPS-N DRS, that recovers both the radial velocity and the FWHM with a Gaussian fit of the CCF.
Using a single fit for the whole range resulted in some discrepancy at the borders, in particular for low FWHM values (FWHM < 6.5 km s −1 ), i.e. the range we are more interested in (see Fig. 4).As such we decided to try and improve the fit at lower values and limit our FWHM fitting range to 0-20 km s −1 : in this case, while higher-order polynomials behave well enough down to FWHM=5km s −1 , the linear fit residuals lie below 5% down to FWHM=3.5kms −1 (see Fig. 5).Considering that we  4. Correlation between the Gaussian fit's FWHM and the input v eq sin i ⋆ of the synthetic line profiles in the whole 0-50 km s −1 v eq sin i ⋆ (0-70 km s −1 FWHM) range.Upper panel: correlation between FWHM and v eq (black line) and relative linear fit (blue dotted line), quadratic fit (orange dotted line), cubic fit (green dashed line) and 4 th degree polynomial fit (red dashed line).Lower panel: residuals of the fits.The horizontal grey lines outline the 5% difference between the fit and the data have a small sample of calibrators (which hinders our ability to constrain an high degree polynomial), and that the linear fit recovers the v eq sin i ⋆ values with a 5% error at worst, we can then reasonably assume that using a linear fit on the calibrators with FWHM < 20 km s −1 would give us useful results.This is a simple test, that does not take into account all the other non-constant cause of broadening: for example the effects of v micro and v macro , that highly depend on the stellar type, are not considered.A more detailed test would involve studying the CCFs obtained on a range of synthetic spectra with different v eq sin i ⋆ and stellar parameters: unfortunately the HARPS-N DRS works only on real raw HARPS-N data, so we cannot perform this analysis.However, we were still able to test our final results in this sense, because while our calibrators' sample is quite small, the total number of stars for which we computed v eq sin i ⋆ , and that have literature values of v eq sin i ⋆ to compare to, is large enough to allow us to look for trends or misbehaviour related to the stellar parameters (see Sec. 4).
Taking into account all the previous considerations, such as the default half-window value of the CCFs, the aim to optimize the FWHM-v eq sin i ⋆ relation for the lower FWHM values, and above all the small sample of calibrators of which only one object (HAT-P-2) has FWHM > 20km s −1 , we then excluded HAT-P-2 from the final calibrators' list and consider our work reliably applicable only for FWHM < 20 km s −1 .We created our relation first by using the CCFs computed with the G2 mask, and then we repeated the work described hereafter also for the K5 CCFs.We built four data sets: the original FWHM computed by the DRS (FWHM DRS ), the FWHM DRS minus the v micro broadening: the FWHM DRS minus the v macro broadening: and the FWHM DRS minus both v micro and v macro broadening: We considered also removing the instrumental broadening, but since this is a constant effect in HARPS-N spectra it will simply be included in the empirical relation.
We fitted a linear relation to each one of our four data sets (Fig. 6): a) FWHM DRS vs. v eq , b) FWHM mic vs. v eq , c) FWHM mac vs. v eq , and d) FWHM mic+mac vs. v eq .The three leftmost points (TrES-4, Kepler-25, and HAT-P-8 from lower to higher FWHM respectively) may appear as outliers, but we decided to keep them for several reasons: we have very few calibrators with FWHM > 10 km s −1 , we have no solid reason to mistrust the P rot and R ⋆ values used in our work, and the v eq sin i ⋆ computed with the resulting calibrations for hundreds of exoplanet-host stars agree well with the literature values (see Sec. 4).
We used as final relation the most simple and straightforward one, that links linearly the FWHM DRS as it is and the v eq sin i ⋆ (Fig. 6, upper left panel), as this is the relation that may be more widely useful, because it does not depend on the knowledge of v micro and v macro .The resulting calibrations using the G2 and K5 masks respectively are thus: G2 mask : v eq sin i ⋆ = 1.09446 × FWHM DRS − 5.45380 K5 mask : v eq sin i ⋆ = 1.26952 × FWHM DRS − 6.06771 (7) For completeness' sake, we give here also the calibrations obtained for FWHM mic (Eq.8), FWHM mac (Eq.9), and FWHM mic+mac (Eq.10): G2 mask : v eq sin i ⋆ = 1.0438 × FWHM mic+mac − 4.13 K5 mask : v eq sin i ⋆ = 1.21346 × FWHM mic+mac − 4.57564 (10) To estimate the errors on our v eq sin i ⋆ measurements, we applied the error propagation theory.Considering that all our equations are linear fits structured as v eq sin i ⋆ = aFWHM + b, we could derive the error on v eq sin i ⋆ using the following equation: where σ a and σ b are the uncertainties in the fit parameters, while ρ(a, b) is the correlation coefficient: The values of σ a , σ b , and ρ(a, b) for all the Eqs.7,8,9,10 are listed in Table 2.We have no estimate on the error of FWHM DRS , because unfortunately this information is not stored in the header of the FITS files, but we tried to recover it by checking the standard deviation of the FWHM DRS values when more than one CCF was available.We found a standard deviation of the order of 4%, which is much lower than the other contributions to the error budget.Thus we deemed Eq. 11 sufficient to estimate the errors in v eq sin i ⋆ derived from Eq. 7. Concerning Eqs. 8, 9, 10 instead, we need to consider also the error on v micro and v macro , that will propagate and give us σ FWHM mic , σ FWHM mac , and σ FWHM mic+mac .The total error then will be: As stated before, we used the rms as errors on v micro and v macro (0.18 km s −1 for v micro , either 0.5 of 0.73 km s −1 for v micro depending of the star's temperature -the former for T eff < 5700 K, the latter for T eff > 5700 K).These values are larger than what we would obtain propagating the errors on the stellar parameters.
We compared the results obtained with the different calibration on our calibrators set (see Table 3), and the v eq sin i ⋆ agree to the order of 0.2-0.3km s −1 with the exception of WASP-14, where Eqs.7 and 8 give very different results from Eqs. 9 and 10: WASP-14 is the hottest star in our calibrators' set, with the largest v micro and v macro values, and the problems may arise from an over-estimating of these values due to the stellar T eff being at the edge of the applicability range of the relationships used to compute them.

Projected rotational velocity of exoplanet-host stars
We decided to apply our relation to all the HARPS-N observed exoplanet-host stars found in the TNG archive.First, we queried again the NASA exoplanet archive to obtain a complete list of all known exoplanet-host stars with declination > −25 deg, without any other constraints.We obtained a preliminary list of 3750 exoplanets (2753 host stars).
We queried the TNG archive3 with a self-written python code using the pyvo module4 in an asynchronous TAP query, retrieving up to 10 public CCF FITS files for each target.We found data for 313 stars, but some of them are useless for different reasons, e.g.fast rotating stars, too low signal-to-noise ratio (SNR), M-type stars reduced with the M2 mask.
We point out here that the CCFs of M-type stars may be used if they are computed with the K5 mask: this results in a noisier, but more physically significant CCF.We were also able to recover the M-type stars reduced with the M2 mask that were observed within the GAPS program: in this case, we could reduce again the spectra with the K5 mask using the YABI platform (Hunter et al. 2012) hosted at the IA2 Data Center 5 .
In the end, we had to discard some non-GAPS stars having only M2-mask public CCFs, and others stars whose CCFs had too low SNR, or the wrong input radial velocity.We estimated the v eq sin i ⋆ for all the 273 remaining targets with FWHM DRS < 20 km s −1 .Our v eq sin i ⋆ values are reported in Table ??, the errors are computed using Eq.11.
Some of the objects in our sample have both G2 and K5 CCFs in the TNG archive, and so we were able to directly compare the results of the two calibrations, in order to quantify the effect of a spectral type mismatch on the resulting v eq sin i ⋆ (see Fig. 7).These objects have a relatively small range of v eq sin i ⋆ , but still the results agree with less than 0.5 km s −1 difference for v eq sin i ⋆ < 4 km s −1 , and with less than 1 km s −1 for v eq sin i ⋆ > 4 km s −1 .Still, to ensure the best possible result, care should be taken to reduce every star with the more appropriate mask.Usually this is already done, because the better the star-mask match, the smaller is the error of the radial velocity computed by the DRS, but sometimes the stellar type is unknown prior to the observations and a mismatch may occur.Possible mismatches between hotter stars (early F-type or above) and the G2 mask are not considered here because hotter stars are usually also fast rotators and they would naturally fall outside the applicability Fig. 6.Linear correlations (black solid lines) between the four data sets derived from the FWHM DRS computed by the HARPS-N DRS with the G2 mask (x-axis) and the stellar equatorial velocity v eq (y-axis) for our set of calibrators.The Spearman's correlation coefficient r and p-value are shown in the plots.Upper left: Linear correlation between FWHM DRS and v eq and relative residuals.Upper right: Linear correlation between FWHM mic and v eq and relative residuals.Lower left: Linear correlation between FWHM mac and v eq and relative residuals.Lower right: Linear correlation between FWHM mic+mac and v eq and relative residuals.).Because we relied on the public data present in the TNG archive, there are a few mismatches between stellar type and mask in our sample, but in all these cases we have v eq sin i ⋆ < 4 km s −1 , so the mismatches should not affect heavily the results.

Comparison with the literature
Out of the stars listed in Table ??, 206 had also v eq sin i ⋆ values from the literature, so we could compare our results with them (see Fig. 8).As a sanity check, we used this larger sample to test our relations: we calibrated the G2 and K5 FWHM DRS values using the the whole set of literature v eq sin i ⋆ values.The resulting relations are: Table 3.Comparison between v eq sin i ⋆ obtained with the different Eqs. 7, 8, 8, 10 for our calibrators, along with the standard deviation of the results.The spectral types are taken from SIMBAD.
Name v eq sin i ⋆ v eq sin i ⋆mic v eq sin i ⋆mac v eq sin i ⋆mic+mac Std.dev.Mask used Sp.type As it is shown in Fig. 9, there is almost no difference between the relation obtained using the whole literature data set and the original one obtained from the selected calibrators (Table 1) for the G2 mask, while the situation is different when us-ing the K5 mask (see black solid line and red dashed line in Fig. 10).In this case, the spread is larger (and the Spearman's r coefficient lower), and so is the difference between the original calibration and the new one.We also lack reliable data points with FWHM DRS > 12 km s −1 , and the literature v eq sin i ⋆ values are very spread out.The latter fact could be caused by the type of stars that are usually reduced using the K5 mask, i.e. mid and late K-type and early M-type stars: these objects may be very active and this could affect both the shape of the CCF (and thus the FWHM DRS ) and the v eq sin i ⋆ estimation performed in literature.To better investigate this behaviour, and to check the possible limitations of our relations' applicability range, we looked at the sample considering also the stellar parameters of the stars, i.e.T eff , log g, and [Fe/H].We recovered the parameters from SIMBAD6 (Wenger et al. 2000) using an automated python query.We show the results in Fig. 11  and the literature v eq sin i ⋆ values (y-axis), with the Spearman's correlation coefficient r and p-value shown in the plot.The black line shows the linear fit of the data, the red dashed line shows the relation obtained from our selected calibrators (Eq.7), the blue dotted line shows the linear fit after removing the stars with log g < 3.5.Lower panels: residuals of the linear fitting, of the relation from selected calibrators, and of the linear fitting after removing the stars with log g < 3.5, respectively.
looking at the results from the G2 relation, we can see that stars with log g < 3.5 tend to cluster below the one-to-one correlation when comparing the results from the K5 relation to the literature v eq sin i ⋆ values.If we perform a linear fit between our v eq sin i ⋆ and the literature v eq sin i ⋆ only for stars with log g > 3.5 (blue dotted line in Fig. 10), then the resulting relation agrees much better with that obtained from the selected calibrators: While we advise using Eq.7 to compute v eq sin i ⋆ because we better trust our selected calibrators, we list in Table 2 also the parameters' errors and correlation factors needed to compute the errors when using Eq.14 (G2 mask only), and Eq. 15 (K5 mask).We can assume that, at least in the case of the K5 sample, our relations are applicable only for stars with log g > 3.5, i.e. mostly main sequence stars, but also some subgiants and red giants stars may fall in the applicability range.Unfortunately, we do not have a wide enough range of log g values in our G2 sample to test the same behaviour (see Fig. 11, middle panel), but considering that the G2 mask used in the HARPS-N DRS is optimized for the Sun, we can infer that also the G2 relation is best suited for main-sequence stars.
Comparing our results with the literature v eq sin i ⋆ , we found no star where our v eq sin i ⋆ differs more the 3σ from the literature value, and only 4 where the difference is larger than 2σ (WASP-1, WASP-127, TYC 1422-614-1, and TYC 3667-1280-1).Taking into account the very different methods used in literature to compute v eq sin i ⋆ this is a good indicator of the robustness and reliability of our FWHM DRS -v eq sin i ⋆ relation.

Stellar inclination
We focused on the results we obtained for stars with no v eq sin i ⋆ literature value, to see if we were able to recover an estimate of the stellar inclination i ⋆ .We did not perform this work on the other targets because our results do not differ much from those already in the literature, and so we do not expect any substantial changes or improvements on i ⋆ .
We used Eq. 1 to compute i ⋆ , which means that we could work only with objects with known P rot and R ⋆ .In some cases, the exoplanetary orbit inclination was known: we could then compare it to i ⋆ , to check the spin-orbit alignment of the system.Because of the sometimes large errors on the various parame-ters, many i ⋆ results were compatible with the whole range of possible inclinations.
We show in Table 5 only the results that set some constrains on the stellar possible inclination.While in most cases our results are compatible with aligned, edge-on planetary systems, we still found one system that shows a difference between i ⋆ and i p around the 2σ level (K2-173), and hints that the HD 13931 system may be aligned, but not edge-on.

Extension to other spectrographs
The relations found in our work between FWHM DRS and v eq sin i ⋆ are optimized for a specific combination of instrument, software and stellar masks.While there are other spectrographs with dedicated DRS, and a few of them also deliver the spectra's CCFs as an output, the different resolution, instrumental effects, wavelength ranges, numerical codes used to compute the CCF, and stellar masks could heavily influence the FWHM DRSv eq sin i ⋆ relation.A possible exception could be the HARPS spectrograph (Mayor et al. 2003), of which HARPS-N is a twin, not only concerning the hardware, but also the software, as HARPS and HARPS-N have almost the same DRS.
To test this assumption, we checked the public archives of two spectrographs with a similar spectral range as HARPS-N: HARPS (which has also the same resolution, telescope aperture and DRS as HARPS-N) and SOPHIE 7 .Both spectrographs have been used for many years in the exoplanets' search and characterization field, guaranteeing the availability of a large amount of public data of exoplanet-host stars.The main characteristics of HARPS-N, HARPS, and SOPHIE are listed in Table 6.SOPHIE has an high-resolution (HR) and an high-efficiency (HE) mode, but for a more direct comparison with HARPS-N we focused on the HR mode spectra to start.Both HARPS and SOPHIE have dedicated DRS that deliver the spectra's CCFs and their FWHMs using stellar masks similar (or, in the case of HARPS, identical) to the HARPS-N ones.We note here that also SOPHIE DRS is adapted from the HARPS DRS, so the three instruments have the same or a very similar DRS.
We searched the dedicated HARPS 8 and SOPHIE 9 archives for objects listed in Table ?? to download their HARPS and SO-PHIE CCFs.We selected only the CCFs obtained with either the G2 or K5 mask in the high-resolution mode, up to a maximum of 50 per object, so that, when possible, we could recover a statistically robust median FWHM DRS for each object.We then computed the v eq sin i ⋆ from the median FWHM DRS using Eq. 7, and we compared the results with our HARPS-N v eq sin i ⋆ .Figure 13 shows the comparison between the HARPS-N and HARPS results, and Fig. 14 shows the comparison between the HARPS-N and SOPHIE results.
It is plainly visible that the twin status of the HARPS and HARPS-N spectrographs would allow us to use the HARPS-N calibration directly with the HARPS data.It is interesting to note that, because we used HARPS spectra observed both before and after 2015, this is true for HARPS data taken both before and after the change of fibers (Lo Curto et al. 2015), even if this change should have slightly affected the FWHM DRS .
The situation regarding the SOPHIE data is slightly different: applying the HARPS-N relation to the SOPHIE data results in v eq sin i ⋆ values consistently overestimated, in particular at the lower end of the range.This is not surprising, as the lower 7 http://www.obs-hp.fr/guide/sophie/sophie-eng.shtml 8 http://archive.eso.org/scienceportal/home 9 http://atlas.obs-hp.fr/sophie/resolution of SOPHIE as compared to HARPS-N will result in larger FWHM DRS values due to the greater instrumental broadening.Still, the effect is not simply a rigid shift, but it appears as a parabolic trend.We manipulated the SOPHIE FWHM DRS values in order to correct them for the different instrumental resolution, using the following equation: (16) where c is the speed of light in km s −1 , and R SOPHIE and R HARPS−N are the resolution of SOPHIE and HARPS-N (see Table 6).The v eq sin i ⋆ values computed with FWHM new are in much better agreement with those derived from HARPS-N data, as shown in Fig. 15.While the spread between HARPS-N and SOPHIE v eq sin i ⋆ values is a bit larger than that between HARPS-N and HARPS ones, still it seems that our relation could be used also with the SOPHIE data, once they are corrected for the difference in resolution.
To better test this assumption, we also selected the SOPHIE CCFs computed from the spectra observed in the HE mode and then we compared the v eq sin i ⋆ computed from both the FWHM DRS and FWHM new .The latter were derived using Eq.16 with the HE resolution.The results are shown in Fig. 16: while correcting for the resolution does improve the agreement between HARPS-N and SOPHIE HE v eq sin i ⋆ values, the results are still discrepant.It seems then that a simple correction for the different resolution is not enough to adapt our relation to a different spectrograph, at least when the resolution difference is large enough.This assumes that there aren't any other factors in play, as, for example, a difference in the code to compute HR and HE CCFs in the SOPHIE DRS.
Unfortunately, we cannot test our method further on any other instrument, because very few spectrographs are equipped with dedicated DRS that yield also the CCFs in addition to the reduced spectra.ESPRESSO has the same capabilities (and a DRS derived from the HARPS one), but there are not enough public data from this instrument for a meaningful comparison.We were unable to compare the HARPS-N results with those obtained with instruments with a very different spectral coverage (such as the visible and near-infrared spectrograph CARMENES or the near-infrared spectrograph GIANO-B), because their DRSs do not compute any CCF.Still, in case any other future DRS will yield also the CCFs, it will be of fundamental importance to calibrate or check and adapt this relation for each combination of instrument, wavelength range, spectral resolution, mathematical recipe (to compute both CCF and FWHM), and stellar mask.While the work is quite straightforward in the case of instruments such as HARPS-N (that offers a single, fixed choice of wavelength coverage and Article number, page 12 of 15 M. Rainer et al.: The GAPS programme at TNG XLV.Projected rotational velocities resolution), it may become slightly more complex when applied to instruments such as ESPRESSO (with 3 different resolving powers) or UVES (where a wide range of choices in both wavelength coverage and spectral resolution is available).
In any case, the strategy detailed in this paper in order to calibrate a FWHM DRS -v eq sin i ⋆ relation may be applied to any other relevant cases including self-made codes, allowing to better exploit the information carried in the CCFs.

Conclusions
Using a well-defined set of calibrators, we were able to obtain two straightforward relations to obtain an estimation of the stellar v eq sin i ⋆ directly from the FWHM DRS computed by the HARPS-N DRS using the G2 and K5 masks (see Eq. 7).These calibrations may be applied when the FWHM DRS value is less than 20 km s −1 .For larger values, other methods to compute the v eq sin i ⋆ are more accurate (i.e., Fourier transform or rotational profile fitting).Other relations were computed to be used when it is possible to estimate v micro and/or v macro , and thus remove their contribution to the FWHM DRS .
We applied our basic relations to all the exoplanet-host stars found in the HARPS-N public archive and in the GAPS private data with CCF computed with G2 or K5 mask and FWHM DRS < 20 km s −1 : we obtained a catalog of homogeneous v eq sin i ⋆ measurements for 273 exoplanet-host stars.Of these stars, 206 have literature values of v eq sin i ⋆ : comparing our results with those we found a very good agreement, with no object differing more than 3σ.Considering the stellar parameters when comparing our results with the literature, we constrain our relation to stars with log g > 3.5.
We can reliably affirm that our simple FWHM DRS -v eq sin i ⋆ relations give solid results, comparable with those obtained with more sophisticated methods such as, for example, spectral synthesis.While our errors may overall be larger than those obtained in the literature, our results would still be useful in characterizing exoplanetary properties, and they may be used as a starting point for a more detailed analysis of the exoplanetary systems.In fact, we were able to determine or constrain the stellar inclination for 12 exoplanet-host stars with no previous v eq sin i ⋆ measurements, finding hints of spin-orbit misalignment in the K2-173 system.
We also tested our relations on the FWHM DRS computed by the HARPS and SOPHIE DRS, and we conclude that Eq. 7 may be used as it is also with HARPS data taken in the high accuracy mode (R = 115, 000).It would be possible to use our relation on the SOPHIE HR data once they are corrected for the different resolution, while using the SOPHIE HE data would require some additional fine-tuning.Still, the strategy detailed in this paper (selection of the calibration, creation of the FWHM DRSv eq sin i ⋆ relation, test of the applicability range) may be used to calibrate other FWHM DRS -v eq sin i ⋆ , with different combination of instrument resolutions, wavelength ranges, mathematical codes (to compute both CCF and FWHM), and stellar masks.

Fig. 1 .Fig. 2 .
Fig. 1.Comparison in the T eff -R ⋆ parameter space between the sample of stars analysed in this work (red circles) and the currently known exoplanet-host stars (grey dots) as retrieved from the NASA Exoplanet Catalog.

Table 2 .
Fit parameters a and b, uncertainties σ a and σ b , and correlation factor ρ(a, b) for all the relevant equations obtained in this paper.relation (FWHM DRS < 20 km s −1

Fig. 7 .Fig. 8 .
Fig. 7. Upper panel: comparison between v eq sin i ⋆ obtained with the G2 relation (x-axis) and the K5 relation (y-axis).The red line shows the one-to-one correlation.Lower panel: residuals.

Fig. 9 .Fig. 10 .
Fig.9.Upper panel: correlation between the G2 FWHM DRS (x-axis) and the literature v eq sin i ⋆ values (y-axis), with the Spearman's correlation coefficient r and p-value shown in the plot.The black line shows the linear fit of the data, the blue dashed line shows the relation obtained from our selected calibrators (Eq.7).Middle panel: residuals of the linear fitting.Lower panel: residuals of the relation from selected calibrators.

Fig. 11 .
Fig. 11.Comparison between our v eq sin i ⋆ (x-axis) and the literature values (y-axis) when using the G2 relation, color-coded according to the stellar parameters T eff (upper panel), log g (middle panel), and [Fe/H] (lower panel).

Fig. 12 .
Fig. 12.Comparison between our v eq sin i ⋆ (x-axis) and the literature values (y-axis) when using the K5 relation, color-coded according to the stellar parameters T eff (upper panel), log g (middle panel), and [Fe/H] (lower panel).

Fig. 13 .Fig. 14 .
Fig.13.Upper panel: comparison between v eq sin i ⋆ computed from the HARPS-N FWHM DRS (x-axis) and those computed from the HARPS FWHM DRS (y-axis).The blue dots are the values computed with the G2 mask relation, the red triangles are those computed with the K5 mask relation.The black line shows the one-to-one correlation.Lower panel: residuals.

Fig. 15 .
Fig. 15.Upper panel: comparison between v eq sin i ⋆ computed from the HARPS-N FWHM DRS (x-axis) and those computed from the corrected SOPHIE high-resolution FWHM new (y-axis).The blue dots are the values computed with the G2 mask relation, the red triangles are those computed with the K5 mask relation.The black line shows the one-toone correlation.Lower panel: residuals.

Fig. 16 .
Fig.16.Upper panel: comparison between v eq sin i ⋆ computed from the HARPS-N FWHM DRS (x-axis) and those computed with SOPHIE in high-efficiency mode (y-axis).Orange triangles and light-blue dots are the results from SOPHIE high-efficiency FWHM DRS with the K5 and G2 relation respectively, while the red triangles and blue dots are the results from the corrected SOPHIE high-efficiency FWHM new (y-axis).The black line shows the one-to-one correlation.Lower panel: residuals for the corrected SOPHIE high-efficiency FWHM new only.

Table 1 .
Calibrators' list.HAT-P-2 is present here, but not used as a calibrator because of its large FWHM value (> 20 km s −1 ).

Table 6 .
Main characteristics of the HARPS-N, HARPS, and SOPHIE spectrographs.