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(abridged) The third data release of Gaia, has provided stellar parameters, metallicity [M/H], [{\alpha}/Fe], individual abundances, broadening parameter from its RVS spectra for about 5.6 million objects thanks to the GSP-Spec module. The catalogue publishes the radial velocity of 33 million sources. 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. 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. We find an excellent agreement between the Gaia and Gaia-ESO radial velocities given the uncertainties affecting each dataset. Less than 25 of ~2100 Gaia-ESO spectroscopic binaries are flagged as non-single stars by Gaia. The temperature scales are in good agreement. The calibrated GSP-Spec gravity should be preferred. We note that the quality (accuracy, precision) of the GSP-Spec parameters degrades quickly for objects fainter than G~11. 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 studied some properties of the open-cluster population: our combined sample traces very well the radial [Fe/H] and [Ca/Fe] gradients, the age-metallicity relations in different radial regions, and it places the clusters in the thin disc., Comment: Accepted for publication in Astronomy & Astrophysics

The Li abundance observed in pre-main sequence and main sequence late-type stars is strongly age-dependent, but also shows a complex pattern depending on several parameters, such as rotation, chromospheric activity and metallicity. The best way to calibrate these effects, with the aim of studying Li as an age indicator for FGK stars, is to calibrate coeval groups of stars, such as open clusters (OCs) and associations. We present a considerable target sample of 42 OCs and associations, ranging from 1 Myr to 5 Gyr, observed within the Gaia-ESO survey (GES), and using the latest data provided by GES iDR6 and the most recent release of Gaia that was then available, EDR3. As part of this study, we update and improve the membership analysis for all 20 OCs presented in our previous article. We perform detailed membership analyses for all target clusters to identify likely candidates, using all available parameters provided by GES and based on numerous criteria: from radial velocity distributions, to the astrometry and photometry provided by Gaia, to gravity indicators, [Fe/H] metallicity, and Li content. We obtain updated lists of cluster members for the whole target sample, as well as a selection of Li-rich giant contaminants obtained as an additional result of the membership process. Each selection of cluster candidates was thoroughly contrasted with numerous existing membership studies using data from Gaia to ensure the most robust results. These final cluster selections will be used in the third and last paper of this series, which reports the results of a comparative study characterising the observable Li dispersion in each cluster and analysing its dependence on several parameters, allowing us to calibrate a Li-age relation and obtain a series of empirical Li envelopes for key ages in our sample., Comment: 83 pages, 246 figures. To be published in A&A, accepted January 27, 2024

The Gaia-ESO Survey is an European Southern Observatory (ESO) public spectroscopic survey that targeted $10^5$ stars in the Milky Way covering the major populations of the disk, bulge and halo. The observations were made using FLAMES on the VLT obtaining both UVES high ($R\sim47,000$) and GIRAFFE medium ($R\sim20,000$) resolution spectra. The analysis of the Gaia-ESO spectra was the work of multiple analysis teams (nodes) within five working groups (WG). The homogenisation of the stellar parameters within WG11 (high resolution observations of FGK stars) and the homogenisation of the stellar parameters within WG10 (medium resolution observations of FGK stars) is described here. In both cases, the homogenisation was carried out using a bayesian Inference method developed specifically for the Gaia-ESO Survey by WG11. The WG10 homogenisation primarily used the cross-match of stars with WG11 as the reference set in both the stellar parameter and chemical abundance homogenisation. In this way the WG10 homogenised results have been placed directly onto the WG11 stellar parameter and chemical abundance scales. The reference set for the metal-poor end was sparse which limited the effectiveness of the homogenisation in that regime. For WG11, the total number of stars for which stellar parameters were derived was 6,231 with typical uncertainties for Teff, log g and [Fe/H] of 32~K, 0.05 and 0.05 respectively. One or more chemical abundances out of a possible 39 elements were derived for 6,188 of the stars. For WG10, the total number of stars for which stellar parameters were derived was 76,675 with typical uncertainties for Teff, log g and [Fe/H] of 64~K, 0.15 and 0.07 respectively. One or more chemical abundances out of a possible 30 elements were derived for 64,177 of the stars., Comment: 31 pages, 19 figures