The final Gaia-ESO Survey (GES) cluster-membership catalogue (Jackson et al. 2021, MNRAS 509, 1664) assigns membership probabilities to 43,211 spectroscopic targets in 63 Galactic open clusters and 7 globular clusters by maximum-likelihood modelling of their 3D kinematics (Gaia EDR3 proper motions plus GES radial velocities), using a mixture of Gaussian velocity distributions with independent, generally unequal per-component dispersions in fixed sky-aligned coordinates, with binary-induced radial-velocity broadening and an empirically fitted field-star background. The paper's own discussion acknowledges that this simplified kinematic model fixes the velocity ellipsoid to the sky-aligned coordinate axes and neglects free rotation of the cluster axes in velocity space, bulk rotation and perspective expansion; that the inferred radial-velocity dispersions and membership likelihoods depend on assumed unresolved-binary population parameters; that the pre-selected globular-cluster fields lack the field stars needed to characterise the background; that no satisfactory kinematic solution was found for Loden 165; and that GES-Gaia cross-match imperfections cause incompleteness. No published follow-up has quantified or removed these systematics for this catalogue, while Wright et al. (2024) have meanwhile shown that anisotropic velocity dispersions are in fact pervasive in the young GES clusters. The problem of extending the catalogue's reliability to all 70 clusters with quantified residual systematic uncertainties remains open.

Background

Open clusters (gravitationally bound groups of a few hundred to a few thousand stars born together) and globular clusters (dense, ancient, 10^5-10^6-star systems) are standard laboratories for stellar evolution, the Galactic age-distance-chemistry structure, and dynamical theory; almost every such study begins by deciding which observed stars are actually cluster members rather than unrelated foreground/background field stars. The Gaia-ESO Survey (GES) obtained high-resolution spectroscopy of about 100,000 stars with FLAMES on the VLT, including tens of thousands of targets in star clusters, delivering among other products radial velocities (RVs, the line-of-sight velocity component measured from Doppler shifts). Jackson et al. (2021) combined the final GES internal data release with astrometry from Gaia EDR3 (the European Space Agency mission's early third data release, providing positions, parallaxes and proper motions, i.e. angular sky motions that convert to tangential velocities given a distance) to compute, for 43,211 targets towards 70 clusters, the probability P3DP_{\rm 3D} that each target is a cluster member.

The method is maximum-likelihood fitting of a two-component mixture in 3D velocity space: cluster stars are modelled as a Gaussian with independent, generally unequal intrinsic dispersions DRVD_{\rm RV}, DRAD_{\rm RA}, DDecD_{\rm Dec} in a fixed right-ascension/declination-aligned coordinate frame (a diagonal, generally triaxial velocity ellipsoid locked to the sky axes), and field stars as a separately fitted background distribution. Binary stars unresolved in the GES single-epoch spectra broaden the observed RV distribution; following Cottaar et al. (2012), this broadening is modelled by simulating RV offsets for a population of binaries with fixed solar-neighbourhood parameters from Raghavan et al. (2010): binary fraction fB=0.46f_{\rm B}=0.46, a lognormal period distribution (mean log⁡P=5.03\log P = 5.03 in days, dispersion 2.28 dex) and a flat mass-ratio distribution. The resulting catalogue of 13,985 high-probability members (P3D>0.9P_{\rm 3D}>0.9) is essentially purely kinematic, and so is well suited as an unbiased membership basis for photometric, chemical and rotation-activity studies; the discrimination between members and non-members is quantified per cluster by the area under the ROC curve (the probability that a random member scores above a random non-member), which exceeds 0.975 for all open clusters.

The same discussion sections, however, identify the catalogue's unquantified systematics. First, the fitted velocity ellipsoid is locked to the sky-aligned coordinate axes: real clusters need not have their velocity ellipsoid aligned with those axes, they can rotate bodily, and they can show apparent expansion caused purely by perspective — for a finite-size cluster, the perspective expansion (van Leeuwen 2009; Kuhn et al. 2019) makes the near-side and far-side stars of a moving or expanding group differ in measured proper motion and RV even with no internal dynamics — and the paper asserts, without quantification, that these effects are not crucial for membership. Second, the inferred DRVD_{\rm RV} and the membership likelihoods inherit any error in the assumed binary population (binary fraction, period and mass-ratio distributions); the predecessor analysis (Jackson et al. 2020) explored this sensitivity only for one prototype cluster, NGC 2516. Third, for the globular clusters the GES fibres were preferentially allocated to previously known likely members, leaving too few measured field stars (about 5 per cent of targets) to characterise the background velocity distribution, so non-members cannot be clearly identified there. Fourth, for the sparsely studied cluster Loden 165 the likelihood surface had no identifiable peak at all, and a dispersion-capped forced fit produced an obviously false solution, so no membership probabilities could be published. Fifth, GES targets are matched to Gaia sources through catalogue cross-identification that the paper itself describes as imperfect: it adopts the Gaia IDs given in the GESiDR6 catalogue (cited as 'Hourihane et al. in preparation'), 'a better, but still not a perfect, cross-match', and 167 targets newly assigned membership in the 2021 paper carry different Gaia source IDs than in the DR2-era predecessor analysis; the residual difficulty is a source of incompleteness rather than contamination.

Problem Statement

For the Gaia-ESO Survey cluster sample (63 open clusters and 7 globular clusters, observed with the FLAMES-fed GIRAFFE multi-fibre spectrograph (mostly in its HR15N setup) and/or UVES (the Ultraviolet and Visual Echelle Spectrograph on the same FLAMES facility) and cross-matched with Gaia EDR3), quantify and, where possible, remove the systematic uncertainties in the 3D-kinematic membership probabilities and intrinsic velocity dispersions published by Jackson et al. (2021), so that the catalogue's reliability extends to all 70 clusters with a defensible systematic error budget in which every effect is quantified and any effect that cannot be removed is at least bounded. Concretely: (1) determine how much the neglected velocity-space structure - free rotation of the cluster axes in velocity space, bulk rotation and perspective expansion in spatially extended clusters - biases the published membership probabilities P3DP_{\rm 3D} and dispersions DRVD_{\rm RV}, DRAD_{\rm RA}, DDecD_{\rm Dec}; (2) propagate the sensitivity of the probabilities and dispersions to the assumed unresolved-binary population (binary fraction and the period and mass-ratio distributions) across the whole catalogue, beyond the existing single-cluster NGC 2516 test; (3) characterise the field-star background velocity distribution in the pre-selected globular-cluster fields well enough to assign reliable membership probabilities (and identify non-members) for outlier targets there; (4) establish the applicability limits of the method - in particular, decide whether Loden 165 is a genuine cluster for which deeper or wider data would recover a kinematic solution, or a case beyond the method's reach, and more generally what target selection, depth and spatial coverage the procedure requires; and (5) quantify and, where possible, remove the GES-Gaia cross-match incompleteness through a documented, improved source identification. These five components serve one objective - a membership catalogue with quantified residual systematics covering every cluster in the survey, with Loden 165, for which no kinematic solution could be published, addressed through the applicability determination of strand (4). For each cluster with a published kinematic solution (all except Loden 165), the quantities to be established are the size of each systematic effect relative to the quoted formal uncertainties and the identity of the published probabilities or dispersions that change materially.

References

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