Context. The third data release of Gaia, has provided stellar parameters, metallicity [M/H], [α/Fe], individual abundances, broadening parameter from its Radial Velocity Spectrograph (RVS) spectra for about 5.6 million objects thanks to the GSP-Spec module, implemented in the Gaia pipeline. The catalogue also publishes the radial velocity of 33 million sources. In recent years, many spectroscopic surveys with ground-based telescopes have been undertaken, including the public survey Gaia-ESO, designed to be complementary to Gaia, in particular towards faint stars. Aims. We took advantage of the intersections between Gaia RVS and Gaia-ESO to compare their stellar parameters, abundances and radial and rotational velocities. We aimed at verifying the overall agreement between the two datasets, considering the various calibrations and the quality-control flag system suggested for the Gaia GSP-Spec parameters. Methods. For the targets in common between Gaia RVS and Gaia-ESO, we performed several statistical checks on the distributions of their stellar parameters, abundances and velocities of targets in common. For the Gaia surface gravity and metallicity we considered both the uncalibrated and calibrated values. Results. Overall, there is a good agreement between the results of the two surveys. We find an excellent agreement between the Gaia and Gaia-ESO radial velocities given the uncertainties affecting each dataset. Less than 25 out of the ≈2100 Gaia-ESO spectroscopic binaries are flagged as non-single stars by Gaia. For the effective temperature and in the bright regime (G ≤ 11), we found a very good agreement, with an absolute residual difference of about 5 K (±90 K) for the giant stars and of about 17 K (±135 K) for the dwarf stars; in the faint regime (G ≥ 11), we found a worse agreement, with an absolute residual difference of about 107 K (±145 K) for the giant stars and of about 103 K (±258 K) for the dwarf stars. For the surface gravity, the comparison indicates that the calibrated gravity should be preferred to the uncalibrated one. For the metallicity, we observe in both the uncalibrated and calibrated cases a slight trend whereby Gaia overestimates it at low metallicity; for [M/H] and [α/Fe], a marginally better agreement is found using the calibrated Gaia results; finally for the individual abundances (Mg, Si, Ca, Ti, S, Cr, Ni, Ce) our comparison suggests to avoid results with flags indicating low quality (XUncer = 2 or higher). These remarks are in line with the ones formulated by GSP-Spec. We confirm that the Gaia vbroad parameter is loosely correlated with the Gaia-ESO v sin i for slow rotators. Finally, we note that the quality (accuracy, precision) of the GSP-Spec parameters degrades quickly for objects fainter than G ≈ 11 or GRVS ≈ 10. Conclusions. We find that the somewhat imprecise GSP-Spec abundances due to its medium-resolution spectroscopy over a short wavelength window and the faint G regime of the sample under study can be counterbalanced by working with averaged quantities. We extended our comparison to star clusters using averaged abundances, using not only the stars in common, but also the members of clusters in common between the two samples, still finding a very good agreement. Encouraged by this result, we studied some properties of the open-cluster population, using both Gaia-ESO and Gaia clusters: our combined sample traces very well the radial metallicity and [Fe/H] gradients, the age-metallicity relations in different radial regions, and allows us to place the clusters in the thin disc.
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