Gaia EDR3 view on Galactic globular clusters
MMNRAS , 000–000 (0000) Preprint 22 February 2021 Compiled using MNRAS L A TEX style file v3.0
Gaia EDR3 view on Galactic globular clusters
Eugene Vasiliev , (cid:63) , Holger Baumgardt Institute of Astronomy, Madingley road, Cambridge, CB3 0HA, UK Lebedev Physical Institute, Leninsky prospekt 53, Moscow, 119991, Russia School of Mathematics and Physics, The University of Queensland, St.Lucia, QLD 4072, Australia
22 February 2021
ABSTRACT
We use the data from
Gaia
Early Data Release 3 (EDR3) to study the kinematic properties of Milky Way globularclusters. We measure the mean parallaxes and proper motions (PM) for 170 clusters, determine the PM dispersionprofiles for more than 100 clusters, uncover rotation signatures in more than 20 objects, and find evidence for radialor tangential PM anisotropy in a dozen richest clusters. At the same time, we use the selection of cluster members toexplore the reliability and limitations of the
Gaia catalogue itself. We find that the formal uncertainties on parallaxand PM are underestimated by 10 −
20% in dense central regions even for stars that pass numerous quality filters.We explore the the spatial covariance function of systematic errors, and determine a lower limit on the uncertaintyof average parallaxes and PM at the level 0.01 mas and 0.025 mas yr − , respectively. Finally, a comparison of meanparallaxes of clusters with distances from various literature sources suggests that the parallaxes (after applying thezero-point correction suggested by Lindegren et al. 2020b) are overestimated by ∼ . − .
009 mas. Despite thesecaveats, the quality of
Gaia astrometry has been significantly improved in EDR3 and provides valuable insights intothe properties of star clusters.
The most recent data release (EDR3) from the
Gaia mis-sion (Gaia Collaboration 2020) does not provide new dataproducts, but instead improves upon the previous DR2 invarious aspects related to the photometric and astrometriccatalogues. In particular, the statistical uncertainties on par-allaxes (cid:36) and proper motions (PM) µ have been reduced by afactor of two on average, and the systematic uncertainties arereduced even further. Already after DR2, it became possibleto measure the mean parallaxes (Chen et al. 2018; Shao &Li 2019) and PM of almost all Milky Way globular clusters(Gaia Collaboration 2018; Baumgardt et al. 2019; Vasiliev2019b) and even to study the internal kinematics of manyof these systems: sky-plane rotation (Bianchini et al. 2018;Vasiliev 2019c; Sollima et al. 2019) and PM dispersion andanisotropy (Jindal et al. 2019). The improved data quality inEDR3 prompted us to reanalyze these properties, and at thesame time to explore the fidelity and limitations of EDR3 it-self. Since all stars in a given globular cluster have the sametrue parallax (with negligible spread) and share the samekinematic properties, we may use these datasets (amountingto tens of thousands stars in the richest clusters) to test thereliability of measurements in the Gaia catalogue and theirformal uncertainties.We apply a number of strict quality filters to select starsthat are believed to have reliable astrometric measurements,and use these “clean” subsets to determine the properties ofthe cluster and the foreground populations, and individualmembership probabilities for each star, in a mixture mod-elling procedure detailed in Section 2. We then use the se-lection of high-probability members to assess the statisticaland systematic uncertainties on (cid:36) and µ in Section 3. After calibrating the recipes for adjusting these uncertainties, weproceed to the analysis of mean PM and parallaxes of clus-ters in Section 4. Namely, we compare the parallaxes withliterature distances and find an empirical parallax offset of (cid:46) .
01 mas (on top of the zero-point correction applied toeach star according to Lindegren et al. 2020b) and an addi-tional scatter at the same level. We then explore the orbitalproperties of the entire cluster population, including someof the poorly studied objects with no prior measurements.Then in Section 5 we analyze the internal kinematics of starclusters: PM dispersions, anisotropy, and rotation signatures.Section 6 wraps up.
We follow the mixture modelling approach to simultaneouslydetermine the cluster membership probability for each starand to infer its properties, in particular, the mean parallax,proper motion, its dispersion, and other structural parame-ters. The procedure consists of several steps: • First we retrieve all sources from the Gaia archive thatare located within a given radius from the cluster centre (thisradius is adjusted individually for each cluster and is typ-ically at least a few times larger than its half-light radius;in some cases we increase it even further to obtain a suffi-cient number of field stars, which are a necessary ingredientin the mixture modelling). At this stage, we do not imposeany cuts on sources, other than the requirement to have 5-or 6-parameter astrometric solutions (5p and 6p for short). • Then we determine a ‘clean’ subset of sources thathave more reliable astrometry, following the recommen- © a r X i v : . [ a s t r o - ph . GA ] F e b Vasiliev & Baumgardt dations of Fabricius et al. (2020), but with tighterlimits on some parameters. This clean subset excludessources that have (a)
G <
13, or (b) RUWE > .
15, or (c) astrometric excess noise sig >
2, or (d) ipd gof harmonic amplitude > exp (cid:2) .
18 ( G − (cid:3) , or (e) ipd frac multi peak >
2, or (f) visibility periods used <
10, or (g) phot bp rp excess factor exceeding the colour-dependent mean trend described by equation 2 and Table 2of Riello et al. (2020) by more than 3 times the magnitude-dependent scatter given by equation 18 of that paper, or (h)flagged as a duplicated source. If the number of 5p sourcessatisfying these criteria exceeds 200, we use only these, oth-erwise take both 5p and 6p sources. The severity of thesequality cuts depends on the density of stars: in the central1 − (cid:36) and µ by a density-dependent factor discussedin the next section. • Then we run a simplified first pass of mixture mod-elling in the 3d astrometric space only. The distribution of allsources in the space { (cid:36), µ α , µ δ } is represented by two (if thetotal number of stars is below 200) or three Gaussian compo-nents, and we use the Extreme Deconvolution (XD) approach(Bovy et al. 2011) to determine the parameters (mean val-ues and covariance matrices) of these components, which, af-ter being convolved with observational uncertainties, best de-scribe the actual distribution of sources. One of these compo-nent (usually the narrowest, except NGC 104 and NGC 362,which sit on top of the Small Magellanic Cloud with a lowerPM dispersion than the cluster stars) is identified with thecluster, and the remaining one or two components – with field(usually foreground) stars. • Then we run a full mixture model, whichl differs fromthe XD model in several aspects. We use the distance fromthe cluster centre as an additional property of each star, andfit the cluster surface density profile by a simple Plummermodel with two free parameters updated in the course ofthe MCMC run. Moreover, the intrinsic parallax dispersionof cluster members is set to zero, and the intrinsic PM dis-persion is represented by a cubic spline in radius with 2 − µ t ( R ) = µ rot R/R ) / (cid:2) R/R ) (cid:3) with R equal to thescale length of the Plummer density profile and a free ampli-tude µ rot . The radial PM component of cluster members isassumed to be caused by perspective expansion (an assump-tion that is tested a posteriori in Section 5) and hence doesnot add free parameters. We use the Markov Chain MonteCarlo (MCMC) code emcee (Foreman-Mackey et al. 2013)to explore the parameter space, starting from the astrometricparameters determined by XD and reasonable initial valuesfor the remaining parameters. • After the MCMC runs converged, we determine themembership probability for each star (including those thatdid not pass the initial quality cutoffs), averaging the resultsof classification scheme from 100 realizations of model param-eters drawn from the MCMC chain. The colour-magnitude di-agram (CMD) of cluster members is visually inspected to ver- ify the outcome of the mixture model, which did not use thephotometric information. The final number of members foreach cluster ranges from a few up to more than 10 stars, withtypically between 10 and 50% of stars satisfying all qualityfilters and hence contributing to the determination of meanproperties of the cluster. • The mean parallax and PM of the cluster and their un-certainties are taken from the MCMC chain, but these val-ues do not take into account spatially correlated systematicerrors, which we include by running an additional postpro-cessing step, as described in Section 5. At this step, we alsore-evaluate the internal PM field of cluster members (rotationprofile and radial expansion/contraction rate) using a moreflexible (spline) parametrization and taking into account sys-tematic errors.This procedure differs from the one used in Vasiliev(2019b,c) in several aspects: (a) we typically use two ratherthan one Gaussian component for the foreground populationin the mixture model; (b) we fit the PM dispersion profilesimultaneously with other parameters, rather than determin-ing it a posteriori from the list of members; (c) we explorethe parameter space with MCMC, rather than picking upthe single maximum-likelihood solution, which implies prop-agation of uncertainties in all nuisance parameters (includingmembership probabilities) into the quantities of interest suchas mean PM and its dispersion. The analysis of the entirelist of clusters takes a few dozen CPU hours. Similar mix-ture modelling approaches have been used by Sollima (2020),who focused on the outer parts of clusters and additionallyused photometric information in membership classification,and Vitral (2021), who employed a fat-tailed Pearson dis-tribution for the field star PM, since it better describes theactual distribution than a single Gaussian (for this reason, weused two Gaussian components for the field population whenpossible). Mixture models have an advantage over more tradi-tional cleaning procedures such as iterative n -sigma clipping,allowing a statistically rigorous estimation of parameters ofthe distribution even in cases of low contrast between thecluster and the field populations. Gaia astrometry has been significantly improved in EDR3,both in terms of lower statistical uncertainties and bettercalibration. In addition, a larger number of quality criteriaare available to filter out stars with possibly unreliable as-trometry. Nevertheless, there are still some remaining issueswith the statistical and systematic errors, which we explorein this section. These are grouped into several categories: (1)consistency between averaged values among stars of differentmagnitudes and spatial regions; (2) underestimated statisti-cal errors; and (3) spatially correlated systematic errors. Inthese experiments, we use the samples of stars in several ofthe largest clusters classified as high-confidence members bythe mixture model (with ≥
90% probability, although formost stars it actually exceeds 99%).The very first question that is natural to ask is whetherthe average values of parallax and PM are consistent betweendifferent subsets of stars split by magnitude or spatial loca-tion. Figure 1 shows the results of this analysis for the twolargest clusters with ∼
50 000 members: NGC 104 (47 Tuc)
MNRAS , 000–000 (0000) aia EDR3 view on Galactic globular clusters
0' < R < 10' 10' < R < 15' 15' < R < 30'0.200.210.220.230.24 p a r a ll a x G
13 16 18 20 G
13 16 18 20 G
13 16 18 20
NGC 1040' < R < 14' 14' < R < 20' 20' < R < 40'0.170.180.190.200.21 p a r a ll a x G
13 16 18 20 G
13 16 18 20 G
13 16 18 20
NGC 5139 Y [ a r c m i n ] NGC 104 parallax Y [ a r c m i n ] NGC 5139 parallax
Figure 1.
Mean parallax of stars in NGC 104 (47 Tuc, top rowand bottom left panel) and NGC 5139 ( ω Cen, centre row andbottom right panel) in different magnitude ranges and spatial re-gions. Three sub-panels in each plot correspond to different radialranges (centre, intermediate range, and outskirts), and each rangeis further split into four quadrants, which are shown by differ-ent colours and symbols. Each sub-panel is additionally split intothree magnitude intervals (bright, intermediate and faint stars).The number of stars contributing to each measurement ranges from ∼ −
200 for bright bins to ∼ − ∼ .
01 mas. Bottom row shows the 2d maps of mean parallax(uncertainty-weighted moving average over 200 neighbouring starsbrighter than G=18), which also demonstrates spatially correlatedresiduals at scales of 10 − (cid:48) . and NGC 5139 ( ω Cen). We split the stars into three magni-tude ranges, three radial intervals and four quadrants in eachinterval. The statistical uncertainties on the mean parallaxof stars in each bin are small enough to highlight that thedifference between these values exceeds these uncertainties,and is at the level ∼ . − .
01 mas, depending on thecluster. There are no universal trends with magnitude (thisstatement is also verified by analyzing a few less populousclusters), indicating that the parallax zero-point correctionsuggested by Lindegren et al. (2020b) adequately compen-sates these variations, but the remaining differences are likelycaused by spatial variations in the zero point, which are nothandled by the correction formula due to scarcity of referenceobjects (mainly quasars). The lower panels of that figure showthe 2d maps of mean parallax in both clusters, which indeedfluctuate on a scale of 10 − (cid:48) .It is known that the Gaia astrometry contains spatiallycorrelated systematic errors associated with the scanning law,which vary mostly on the scale ∼ . ◦ , but have some cor-relations over larger angular scales as well. Although theprominence of these ‘checkerboard patterns’ is significantlyreduced in EDR3 compared to DR2 (see, e.g., Figure 14 inLindegren et al. 2020a), they are not completely eliminated.In the first approximation, these spatially correlated errorsin astrometric quantities χ ≡ { (cid:36), µ } can be described bycovariance functions V χ ( θ ) ≡ χ i χ j , which depend on theangular separation θ between two points i , j (measured indegrees). Lindegren et al. (2020a) estimated these functionsin a number of ways, primarily based on the sample of ∼ quasars, which are found across the entire sky (except theregions close to the Galactic disc), but are relatively faint.They find V (cid:36) ( θ ) (cid:39) µ as on scales ∼ ◦ , and possibly afew times higher in the limit of small separation, althoughwith a large statistical uncertainty limited by the numberof close pairs of quasars. On the other hand, stars in theLarge Magellanic Cloud (LMC) are brighter and more dense,allowing them to extend the covariance function to smallerseparations, which turns out to be ∼ × smaller than forquasars. Ma´ız Apell´aniz et al. (2021) carried out an indepen-dent analysis of quasars and LMC stars, but restricted theirsample to quasars brighter than G = 19. They also obtainedlower values for V (cid:36) in the range 50 − µ as for separationssmaller than a few degrees, and their LMC sample yielded ayet lower V (cid:36) ∼ − µ as for θ (cid:46) ◦ . All these findingssuggest that the spatial correlations are less prominent forbrighter stars, but the dependence of the covariance functionon magnitude has not yet been quantified.We explored the spatial covariance of parallaxes and PMin the four richest clusters (NGC 104, 5139, 6397 and 6752),using only the 5p stars that pass all quality criteria, but con-sidering different magnitude ranges. Figure 2 shows that thebinned covariance functions have a similar behaviour for allclusters, and its limiting value at θ = 0 does depend on themagnitude cutoff, increasing towards fainter magnitudes. Thevalue for bright stars (13 < G <
18) is V (cid:36) (0) (cid:39) µ as , con-sistent with the findings of Ma´ız Apell´aniz et al. (2021), and afew times large if we consider all stars. Due to a finite spatialextent of cluster members, the estimated covariance functiondrops to zero or negative values at separations θ (cid:38) . − . ◦ comparable to the size of the cluster, which does not reflect itstrue behaviour on large scales (in other words, the systematicerror may have spatial correlations on larger scales, but they MNRAS , 000–000 (0000)
Vasiliev & Baumgardt θ [ deg ] V ( θ ) [ µ a s ] θ [ deg ] θ [ deg ] Figure 2.
Spatial covariance function for parallax as a function ofseparation between sources V (cid:36) ( θ ), estimated in four richest clus-ters in different ranges of magnitudes: from G = 13 up to G = 18(left), G = 19 (centre) or G = 21 (right panel), note that eachnext range includes the previous one. Pairs of stars are binnedinto 7 −
15 distance bins; the uncertainties in each bin are drivenby measurement errors, which increase with magnitude. In all pan-els, the values of V (cid:36) ( θ = 0) are significantly different from zero,but they increase with the inclusion of fainter stars. would be the same for all cluster members and hence simplyshift the mean (cid:36) , which is subtracted before computing V (cid:36) ).We thus augment our estimates of V (cid:36) ( θ ) with the ones fromLindegren et al. (2020a) and Ma´ız Apell´aniz et al. (2021) for θ (cid:38) . ◦ , while introducing an overall magnitude-dependentnormalization factor. Our approximation is V (cid:36) ( θ, G ) = (cid:8) / (1 + θ/ . ◦ ) + 70 exp (cid:2) − θ/ ◦ − ( θ/ ◦ ) (cid:3)(cid:9) × max( G − , µ as , (1)where G is the magnitude of the brighter star in the pair.We do not use the same functional form as Ma´ız Apell´anizet al. (2021) since the latter is unphysical (a valid covariancefunction must have a non-negative angular power spectrum,i.e., Legendre integral transform), but instead reproduce theoverall trends with a different function.To conduct a similar analysis for the PM covariance func-tion, we relied on the assumption (tested later in Section 5)that the radial component of PM is entirely caused by per-spective effects, thus after subtraction of these it should bezero on average. Unfortunately (in this context), unlike paral-lax, the intrinsic dispersion of PM is significantly larger thanmeasurement errors, thus reducing the statistical significanceof the estimated V µ ( θ ). Nevertheless, we obtain a reasonablyconsistent limiting value V µ (0) (cid:39)
400 [ µ as yr − ] for all clus-ters and magnitude ranges; combined with the larger-scaletrends from Lindegren et al. (2020a), we approximate V µ ( θ ) = 400 (1 + θ/ . ◦ ) + 300 exp( − θ/ ◦ ) [ µ as yr − ] . (2)These covariance functions yields a systematic uncertainty (cid:15) (cid:36) (cid:39) .
011 mas and (cid:15) µ (cid:39) .
026 mas yr − in the limit ofsmall separations, which should be viewed as the irreduciblesystematic floor on the precision of parallax and PM measure-ment for any compact stellar system. The overall uncertainty(statistical and systematic combined) for a given selectionof cluster members is derived using the method describedin Vasiliev (2019c), and may be smaller than these values ifa cluster spans more than a fraction of a degree on the sky. Table 1.
Coefficients for the multiplicative parallax uncertaintyscaling factor η = (1 + Σ / Σ ) ζ , and the additional systematicerror, for different subsets of stars in the range 13 < G < ζ Σ [stars arcmin − ] (cid:15) (cid:36), sys [mas]clean, 5p 0.04 10 0.01clean, 6p 0.04 5 0.01non-clean 0.15 20 0.04 Ma´ız Apell´aniz et al. (2021) discuss a similar method for com-bining statistical and systematic uncertainties, but in theirequations 5–7, the overall error is dominated by the largestvalues of V (cid:36) ( θ ), whereas our method essentially sums up thevalues of the inverse covariance matrix, and therefore putsmore emphasis on stars with smallest spatial covariances. Inpractice, though, the difference should be minor.Apart from confirming the systematic error, we also findthat the formal statistical uncertainties are somewhat under-estimated. This is most easily manifested in the distributionof parallaxes for cluster members: the intrinsic spread of (cid:36) is negligibly small even for the closest clusters ( (cid:46) − mas),and hence we should expect the uncertainty-normalized de-viations from the mean parallax for each star, ( (cid:36) i − (cid:36) ) /(cid:15) (cid:36),i ,to follow a standard normal distribution. Figure 3 shows thedistribution of these measurements for one of the richest clus-ters, NGC 5139 ( ω Cen), split into several groups by stellarmagnitude and distance from the cluster centre. It is clearthat even for the “clean” subset of stars that satisfy all qual-ity criteria, the distribution has more prominent tails than aGaussian, at least in the central regions with high density ofstars. We may quantify this effect by assuming that the actual(externally calibrated) measurement uncertainty ˜ (cid:15) (cid:36) is givenby the scaled formal uncertainty, summed in quadrature withan additive constant: (cid:15) (cid:36), ext = η (cid:15) (cid:36) + (cid:15) (cid:36), sys . (3)Figure 4 shows the error inflation factor η estimated inseveral magnitude bins from a dozen clusters, as a functionof source density. Despite some scatter, the overall trendsare consistent between all clusters, and indicate that theformal uncertainties should be scaled by η (cid:39) . − . η (cid:39) η is somewhat higher,but one can compensate for this by an additive systematicerror (cid:15) (cid:36), sys (cid:39) .
01 mas. These values are comparable withthose reported by Fabricius et al. (2020, their figures 19, 21),although they did not study the variation with source den-sity and did not apply all quality criteria that we used here,thus their η are somewhat higher on average. El-Badry et al.(2021) find a similarly mild ( (cid:46) .
2) error inflation factor fromanalysis of parallaxes of wide binaries (after applying somequality cuts similar to ours). We also examined the perfor-mance of the composite quality filter (“astrometric fidelity“)suggested by Rybizki et al. (2021), but found that it failsto remove many astrometrically unreliable sources in densecentral regions, producing much broader distributions of nor-malized parallax deviations with η ∼ . − η is slightly higher for 6p sources that pass all other qualityfilters. We may approximately describe its dependence onsource density Σ as η = (1 + Σ / Σ ) ζ ; the values of Σ , ζ and (cid:15) sys for various subsets are reported in Table 1.It is natural to expect that the PM uncertainties could MNRAS , 000–000 (0000) aia EDR3 view on Galactic globular clusters -4 -3 -2 -1 η =1.203N=20830 Distribution of uncertainty-normalized deviations from the mean parallax for stars in the clean subset in NGC 5139 ( ω Cen).Histograms are split by distance range (from top to bottom row) and magnitude (from left to right, with the last column showing 6psources, which are typically fainter than G = 20). If the formal uncertainties were correctly estimated, the histograms should have followeda standard normal distribution (shown by gray dotted lines), however in practice, the distribution is broader by a factor η (cid:39) . − . 10 1003 30 3000.951.001.051.101.151.201.251.30 e rr o r i n f l a t i o n f a c t o r η Figure 4. Parallax error inflation factor η as a function of source density and magnitude, for a dozen clusters with ≥ (cid:36) i − (cid:36) ) /(cid:15) (cid:36),i , similarto the one shown in Figure 3 for one particular cluster. The scaling factor η that the uncertainties should be multiplied by to match thestandard normal distribution is generally larger in higher-density central regions; the gray dotted lines show the trend η = (1 + Σ / Σ ) ζ with Σ = 10 stars arcmin − and ζ = 0 . 04. The first and the last panel exceed this trend line on average, and a better fit could beobtained by adding in quadrature a systematic error (cid:15) sys = 0 . 01 mas (first panel – bright stars) or lowering the value of Σ (last panel –6p sources). MNRAS000 01 mas (first panel – bright stars) or lowering the value of Σ (last panel –6p sources). MNRAS000 , 000–000 (0000) Vasiliev & Baumgardt Radius [arcmin] † µ σ µ NGC 3201 (nominal errors)13.0 − − − Radius [arcmin] † µ σ µ NGC 3201 ( η -scaled errors)13.0 − − − Radius [arcmin] † µ σ µ NGC 5272 ( η -scaled errors)13.0 − − − Figure 5. PM dispersion profiles for NGC 3201 (left and centre panels) and NGC 5272 (M 3, right panel) as a function of distance,split into several magnitude bins. Left panel: when using nominal measurement uncertainties (the average (cid:15) µ for each bin is printed inthe legend), the deconvolved intrinsic dispersion is higher for fainter stars, indicating that their uncertainties are underestimated. In theremaining two panels, we use the same prescription for the density-dependent error inflation factor η as for the parallax, which producesprofiles that are in reasonable agreement between different magnitude bins. Shaded areas show the 68% confidence intervals on σ µ ( R )from the MCMC runs of the full mixture model pipeline, in which this scaling factor was also applied and faint stars have been excluded. be underestimated by a similar factor, but this is more dif-ficult to test empirically, since the intrinsic PM dispersion isnon-negligible for most clusters with a sufficiently high num-ber of stars, and is indeed among the free parameters in themixture model. One possibility is to check whether the error-deconvolved PM dispersion is the same when computed fromdifferent magnitude ranges. Figure 5 demonstrates that whenusing formal uncertainties from the Gaia catalogue withoutany correcting factors, the internal PM dispersion appears tobe higher for fainter stars, indicating that their uncertaintiesare likely underestimated. On the other hand, when adoptingthe same prescription for the PM uncertainty scaling factor η as for the parallax, the inferred dispersion is usually con-sistent between different magnitude ranges. Based on theseexperiments, we adopted this scaling prescription for the en-tire mixture model fitting procedure. We also checked thatthe results are largely insensitive ( (cid:46) . 01 mas yr − difference)to 5% variations in the adopted value of η (cid:39) . 15 in thehighest-density regions (and corresponding changes at lowerdensities).Nevertheless, to avoid possible biases caused by cluster-to-cluster variations of the scaling factor, we conservativelyused only those stars from the clean subset that have suf-ficiently small uncertainties (usually these are the brighterones). Namely, we select only stars with (cid:15) µ < κσ µ / ( η − η ),where κ = 0 . η (Σ) ≥ η = 0 . 9. The idea is that in high-density central regions, η is not only higher, but also more uncertain, and thus wereduce the maximum acceptable statistical error of stars inthis region to limit the bias from incorrect assumptions aboutthis scaling factor. Obviously, this filter already requires theknowledge of the intrinsic dispersion σ µ , so we apply it itera-tively, using the profiles σ µ ( R ) obtained in previous runs. Allstars in the clean subset that do not pass this filter are stillused in the astrometric fit, but are convolved with a fixed(previous) σ µ ( R ) profile instead of the one inferred duringthe fit, so do not bias its properties. If the number of starswith small enough uncertainties is too low ( (cid:46) HST -derived PM was appliedby Watkins et al. (2015), who removed all stars with uncer-tainties larger that 0 . σ µ (this is a more stringent criterionthan we use, but their PM uncertainties are typically at thelevel 0.05 mas yr − , which is lower than most Gaia stars). Having established the necessary adjustments and correctionfactors for the mixture modelling procedure, we now discussits outcomes pertaining to the global properties of the clusters– mean parallaxes and PM.Figure 6 shows the comparison of Gaia parallaxes with thedistances compiled by Baumgardt (2021) from various litera-ture sources. The agreement is generally good, with the ma-jority of points lying within the shaded region representingsystematic uncertainties on parallax. There are some anoma-lies, too: for instance, a number of clusters at low Galacticlatitudes have systematically lower parallaxes than impliedby the literature distances, but they are in highly-reddenendregions and it is plausible that the CMD-derived distancesare biased. In particular, for Pal 6, Pal 7 (IC 1276), Pal 10and Terzan 12 it appears that using the parallax distanceand slightly adjusting the extinction coefficient produces abetter fit to the CMD than the literature values. Parallaxesof NGC 4147, Terzan 1 and Terzan 5 are also significantlysmaller than 1 /D , and despite rather large uncertainties, thedifference is statistically unlikely, possibly indicative of un-detected problematic sources in Gaia .On the other hand, there is a general tendency of parallaxto be slightly larger than 1 /D , illustrated by a large frac-tion of clusters with 5 < D < 20 lying above the (cid:36) D = 1line. This may indicate that the parallax zero-point correc-tion suggested by Lindegren et al. (2020b) is overshootingby ∼ . − . 01 mas on average, although it is also pos-sible that there are some general systematic biases in theliterature distances. Finally, for some bright, low-extinctionclusters such as NGC 6205 (M 13), NGC 6341 (M 92) andNGC 7078 (M 2), the disagreement between CMD-derived MNRAS , 000–000 (0000) aia EDR3 view on Galactic globular clusters 102 3 5 7 20 30distance [Kpc]0.5120.60.81.21.5 p a r a ll a x × d i s t a n c e T u c N G C P a l N G C E S O - N G C R u p B H M N G C ω C e n N G C N G C I C N G C B H F S R M M M N G C M M M N G C N G C N G C M N G C N G C N G C T e r z a n N G C B H F S R N G C P a l N G C T e r z a n N G C N G C N G C N G C P a l T e r z a n M M P a l M P a l N G C P a l M T e r z a n P a l M M N G C M M N G C nu m b e r o f s t a r s Figure 6. Comparison of mean parallaxes (cid:36) of globular clusters derived in this work with distances D from literature, determinedby other methods (dynamical, photometric, etc.). The vertical axis shows the Gaia parallax multiplied by the distance: values above 1correspond to the parallax distance being smaller than the literature value, and vice versa. The vertical error bars take into account thestatistical uncertainties both on the mean parallax and the distance, but the horizontal error bars for the distance are not displayed; allclusters with (cid:15) (cid:36) D < . E ( B − V ) ≥ 1, for which many photometricdistance determination methods may be less reliable. The agreement between Gaia parallaxes and other distance measurements is fairlygood, but at large D the parallaxes seem to be higher on average than 1 /D , and the scatter is larger than the statistical uncertainties.The gray-shaded region shows Monte Carlo samples from a statistical relation (4) reproducing this offset and scatter, with parametersshown in Figure 7. and parallax distances cannot be explained other than by thesystematic variation of parallax zero-point across the sky. Tomake a quantitative statistical estimate, we assume that therelation between the Gaia EDR3 parallax (cid:36) (after zero-pointcorrection) and the literature distance D follows a normaldistribution (cid:36) i − /D i ∼ N (cid:0) ∆ (cid:36), (cid:15) (cid:36),i + (cid:15) (cid:36), sys + [ (cid:15) D,i + (cid:15) D, sys ] /D (cid:1) , (4)where (cid:15) (cid:36),i is the formal statistical parallax uncertainty for i -th cluster, (cid:15) (cid:36), sys is the additional systematic parallax un-certainty (same for all clusters), (cid:15) D,i is the relative distanceuncertainty for i -th cluster from the combination of literaturemeasurements, and (cid:15) D, sys is the additional systematic relativeuncertainty in distance, all added in quadrature. The param-eters of this relation, estimated by a Monte Carlo simulation,are shown in Figure 7 (we restricted the sample to clusterswith low extinction, E ( B − V ) < . 0, and tried a number ofother quality cuts, though they do not have a strong effect onthe results). The offset in parallax is ∆ (cid:36) ∼ . − . 009 masdepending on the subset (that is, zero-point corrected paral-laxes are slightly too high), and the additional systematicerrors in distance and parallax are, unsurprisingly, anticorre-lated, but while the former is consistent with zero, the lat- ter is (cid:15) (cid:36) ∼ . ± . 001 mas. This value is very similar tothe estimates of spatially correlated systematic uncertaintyin parallax discussed above, and hence likely represents thetrue limit of the parallax precision in Gaia EDR3.Only in special circumstances it may be possible to es-timate the zero-point offset more precisely at the locationof a given cluster: Chen et al. (2018) combined the sparsequasars with the much more numerous stars of the SmallMagellanic Cloud (SMC) to derive corrected parallaxes forNGC 104 (47 Tuc) and NGC 362 from Gaia DR2. Their val-ues (cid:36) = 0 . ± . (cid:36) = 0 . ± . 007 agree wellwith our measurements (0.225 and 0.110 respectively). Re-cently Soltis et al. (2020) derived the parallax for NGC 5139( ω Cen) from Gaia EDR3 to be (cid:36) = 0 . ± . (cid:15) (cid:36) (cid:39) . 01 mas, correspond-ing to a 5% distance uncertainty, significantly larger thanthey optimistically assumed, and reducing the precision oftheir calibration of cosmological distance indicators. We stillneed to wait until further Gaia data releases to bring downthe systematic uncertainty to competitive levels. MNRAS , 000–000 (0000) Vasiliev & Baumgardt † D = . +0 . − . . . . . † † = . +0 . − . . . . † D . . . ∆ . . . . † . . . ∆ ∆ = . +0 . − . Figure 7. Parameters of the statistical relation (4) between Gaia parallaxes and literature distances, sampled from a Monte Carlochain. (cid:15) D is the relative distance error added to the formal sta-tistical uncertainty of literature distance values, and is consistentwith zero. (cid:15) (cid:36) is the systematic uncertainty on Gaia parallaxes thatneeds to be added to random uncertainties to make them statis-tically consistent with inverse distances, while ∆ (cid:36) is the parallaxoffset (positive values indicate that Gaia parallaxes are on averagehigher than 1 /D ). 10 1002 5 20 50distance [Kpc]1101000.5252050 v e l o c i t y un c e r t a i n t y A M E r i d a nu s N G C K o P a l P a l C r a t e r K o A M M un o z B H P a l R y u P a l P a l D j o r g U K S G r a n T e r z a n P a l R y u L a e v e n s S e g u e nu m b e r o f s t a r s Figure 8. Uncertainty in the transverse velocity v ⊥ ≡ µ D as afunction of heliocentric distance D has two components: distanceuncertainty µ (cid:15) D is shown by circles (individual estimates whereavailable, or assuming a rather optimistic 5% relative error other-wise), and PM uncertainty (cid:15) µ D is shown by crosses (taking intoaccount systematic errors (cid:15) µ (cid:39) . √ − , which usuallydominate over statistical errors). For most clusters, the first factoris more important. Colours show the number of cluster memberswith reliable astrometry for each object. Comparing our parallax values with the ones derived byShao & Li (2019) from Gaia DR2, we find a general agree-ment within error bars, after correcting for the mean parallaxoffset in DR2 ∆ (cid:36) (cid:39) − . 03 (i.e., DR2 parallaxes are smalleron average). Only a few clusters showed a statistically signif-icant disagreement, e.g., NGC 5272 (M 3) ( (cid:36) our − (cid:36) S (cid:39) . . − . Gaia DR2 data: for 80% clusters, the total difference inboth PM components is within 0.1 mas yr − , comparableto the systematic uncertainty of DR2 (0 . 066 mas yr − percomponent). Clusters with the largest PM difference usuallyhave very few stars or are located in dense regions: AM 4( ∼ . − ), FSR 1735 ( ∼ . ∼ 100 globular clusters from Gaia EDR3, which are in a very good agreement with our re-sults (except for two clusters with deviations ∼ . − ,NGC 5024 and NGC 5634, for which we visually inspectedthe distribution of stars in the PM space and confirmed thecorrectness of our measurements).The newly derived mean PM are dominated by systematicuncertainties at the level ∼ . 025 mas yr − for most clusters(unless the number of members is below ∼ 100 or the contrastbetween the cluster and the field stars in the PM space islow, in which case statistical uncertainties may be higher).However, we are ultimately interested not in the value of thePM, but of the transverse physical velocity v ⊥ ≡ µ D , and theuncertainty in distance usually is the dominant limiting factorin the precision of the velocity, as illustrated in Figure 8.Overall, the velocity uncertainty is of order ∼ − 20 km s − for most clusters, except the most distant ones.Out of 157 objects in the Harris (1996, 2010) catalogue,we could not determine the PM for only a few clusters whichare located in highly extincted regions and are not visibleto Gaia : 2MASS–GC01, 2MASS–GC02, GLIMPSE01 andGLIMPSE02. We have also added a number of recently dis-covered globular cluster candidates to our list: FSR 1716(Minniti et al. 2017), FSR 1758 (Barb´a et al. 2019), VVV–CL001 (Minniti et al. 2011), VVV–CL002 (Moni Bidin et al.2011), BH 140 (Cantat-Gaudin et al. 2018), Gran 1 (Gran etal. 2019), Pfleiderer 2 (Ortolani et al. 2009), ESO 93–8 (Bicaet al. 1999), Mercer 5 (Mercer et al. 2005), Segue 3 (Belokurovet al. 2010), Ryu 059, Ryu 879 (Ryu & Lee 2018), Kim 3(Kim et al. 2016), Crater / Laevens 1 (Belokurov et al. 2014;Laevens et al. 2014), Laevens 3 (Laevens et al. 2015), Mu˜noz1 (Mu˜noz et al. 2012), BLISS 1 (Mau et al. 2019), bringingthe total count to 170. Many of these additional objects arepoorly studied and lack line-of-sight velocity measurements,and the nature of them is not well established. We shall seebelow that a few of these are located at low Galactic latitudesand move within the disc plane, so may well be open ratherthan globular clusters.Given the full 6d phase-space coordinates of clusters, onemay examine their orbital properties or the distribution inthe space of integrals of motion: this has become a popularexercise especially after the advent of precise PM from Gaia MNRAS , 000–000 (0000) aia EDR3 view on Galactic globular clusters i n c li n a t i o n ( L z / L ) e cc e n t r i c i t y N G C M N G C M F S R M N G C B H N G C N G C N G C N G C N G C N G C N G C M N G C N G C M N G C M N G C M M N G C N G C M P a l N G C N G C N G C B H N G C N G C N G C N G C M M M N G C M M N G C N G C M N G C M M N G C M M N G C N G C N G C F S R N G C M M N G C B H N G C N G C N G C M N G C P a l N G C M N G C P a l N G C N G C T e r z a n N G C T e r z a n N G C N G C T e r z a n N G C N G C T e r z a n N G C P a l N G C M N G C M M N G C N G C I C B H N G C T e r z a n N G C N G C N G C N G C N G C N G C D j o r g N G C N G C B H P a l N G C T e r z a n G r a n T e r z a n D j o r g T e r z a n N G C N G C P a l N G C E S O - F S R R u p I C T e r z a n P a l E S O N G C B li ss E U K S A r p P a l P a l R y u P f l e i d e r e r P a l R y u T e r z a n N G C S e g u e VVV C L M e r c e r P a l K i m P a l P yx i s K o VVV C L M un o z E S O P a l P a l L ill e r E r i d a nu s P a l W h i t i ng A M C r a t e r K o L a e v e n s A M N G C T o n T e r z a n N G C N G C M M N G C N G C M N G C T u c ω C e n i n c li n a t i o n ( L z / L ) e cc e n t r i c i t y M M F S R N G C N G C N G C M N G C N G C B H M M N G C N G C N G C M N G C M M M N G C N G C M N G C M N G C N G C N G C N G C N G C M M P a l M N G C N G C N G C M N G C N G C M N G C N G C N G C N G C M M N G C B H M M N G C M N G C M N G C N G C N G C M M N G C N G C M F S R N G C M N G C N G C P a l N G C B H N G C N G C N G C N G C N G C N G C N G C P a l N G C N G C N G C N G C T e r z a n N G C N G C T e r z a n T e r z a n T e r z a n N G C N G C T e r z a n B H M I C N G C P a l N G C M N G C N G C T e r z a n To n B H P a l N G C N G C N G C N G C T e r z a n D j o r g N G C N G C N G C D j o r g N G C T e r z a n T e r z a n N G C N G C G r a n P f l e i d e r e r N G C N G C N G C E S O - P a l F S R R u p T e r z a n P a l T e r z a n L ill e r E S O A r p U K S N G C E P a l P a l P yx i s I C E S O P a l VVV C L P a l R y u P a l P a l B li ss M e r c e r VVV C L R y u Pa l C r a t e r P a l E r i d a nu s S e g u e K i m L a e v e n s W h i t i n g A M M un o z K o K o A M N G C M T u c ω C e n Figure 9. Orbital parameters of clusters in the McMillan (2017) best-fit potential (top) and in Bovy (2015) MWPotential2014 (bottom).Horizontal axis shows the semimajor axis, vertical – eccentricity, and colour – inclination (prograde in red, retrograde in blue, and polarorbits in green). Each cluster is shown by a cloud of points sampled from the measurement uncertainties, which in most cases are dominatedby distance uncertainties. We note that the changes in orbit parameters in different Galactic potentials sometimes exceed the measurementuncertainties, especially in the outer part of the Galaxy. DR2 (e.g., Binney & Wong 2017; Myeong et al. 2018; Mas-sari et al. 2019; Piatti 2019; Forbes 2020; Kruijssen et al.2020; P´erez-Villegas et al. 2020; Bajkova et al. 2020). Theentire population of clusters can be used to probe the gravi-tational potential of the Milky Way, using Jeans equations ordistribution function-based dynamical models (e.g., Watkinset al. 2019; Posti & Helmi 2019; Vasiliev 2019b; Eadie & Ju-ric 2019). However, the outer parts of the Galaxy, where thisexercise is most useful, are subject to the non-equilibrium distortions caused by the recent passage of the Large Magel-lanic Cloud (LMC), which perturbs the velocities by as muchas few tens km s − and can bias the inference on the grav-itational potential (Erkal et al. 2020a,b; Cunningham et al.2020; Petersen & Pe˜narrubia 2020). Although the effect ofthe LMC can be accounted for in dynamical models, as il-lustrated by Deason et al. (2021) in the context of a simplescale-free model for halo stars, we leave a proper treatmentof globular cluster dynamics for a future study. MNRAS000 DR2 (e.g., Binney & Wong 2017; Myeong et al. 2018; Mas-sari et al. 2019; Piatti 2019; Forbes 2020; Kruijssen et al.2020; P´erez-Villegas et al. 2020; Bajkova et al. 2020). Theentire population of clusters can be used to probe the gravi-tational potential of the Milky Way, using Jeans equations ordistribution function-based dynamical models (e.g., Watkinset al. 2019; Posti & Helmi 2019; Vasiliev 2019b; Eadie & Ju-ric 2019). However, the outer parts of the Galaxy, where thisexercise is most useful, are subject to the non-equilibrium distortions caused by the recent passage of the Large Magel-lanic Cloud (LMC), which perturbs the velocities by as muchas few tens km s − and can bias the inference on the grav-itational potential (Erkal et al. 2020a,b; Cunningham et al.2020; Petersen & Pe˜narrubia 2020). Although the effect ofthe LMC can be accounted for in dynamical models, as il-lustrated by Deason et al. (2021) in the context of a simplescale-free model for halo stars, we leave a proper treatmentof globular cluster dynamics for a future study. MNRAS000 , 000–000 (0000) Vasiliev & Baumgardt Figure 9 shows the orbital properties of the entire popu-lation of globular clusters: semimajor axis, eccentricity andinclination, computed in two variants of static Milky Waypotentials: McMillan (2017) or Bovy (2015) (these proper-ties remain qualitatively similar if we use other reasonablechoices for the potential). Each cluster is shown by a cloudof points representing the measurement uncertainties of its6d phase-space coordinates, which, as discussed above, aremostly dominated by distance uncertainties (also propagatedinto the transverse velocity). For the 9 clusters lacking line-of-sight velocity measurements, the missing velocity was drawnfrom a global distribution function, resulting in a plausiblemean value with a large uncertainty (cid:38) 100 km s − . An alter-native depiction is the rhomboid action-space diagram (e.g.,Figure 5 in Vasiliev 2019b), which shows the same informa-tion in a different projection of the 3d space of integrals of mo-tion. Note however that peri- and apocentres (equivalently,eccentricity and semimajor axis) are computed by numericalorbit integration, whereas actions are computed in the St¨ackelapproximation; in practice, this distinction is unimportant.Objects located in the same region of the plot and havingsimilar colours are close in the 3d integral space and may bephysically related, such as the population of globular clus-ters associated with the Sgr stream. Note that the proximityin the integral space does not imply that the objects lineup on the same path on the sky plane, therefore the asso-ciation with the stream should be examined in the space ofobservables – celestial coordinates, distances, PM and line-of-sight velocities. This connection was explored in a numberof papers, e.g., Law & Majewski (2010b), Bellazzini et al.(2020), Arakelyan et al. (2020); however, these studies all re-lied on the old model of the Sgr stream from Law & Majewski(2010a), which was conceived before the more recent obser-vations of its trailing arm, and does not match its features(primarily the distance to the apocentre). When using themodel of the Sgr stream from Vasiliev et al. (2021), whichadequately matches all currently available observational con-straints, we find that only the following clusters can be un-ambiguously associated with the Sgr stream: four clusters inthe Sgr remnant – NGC 6715 (M 54, which sits at its centre),Terzan 7, Terzan 8, Arp 2, three clusters in the trailing arm –Pal 12, Whiting 1 and NGC 2419, and one faint cluster Ko 1,which previously lacked PM measurements (and still has noline-of-sight velocity data), but now coincides with the streamin position, distance and both PM components. Interestingly,it sits in the region where the second wrap of the trailing armintersects with the second wrap of the leading arm, so is con-sistent with both interpretations. The line-of-sight velocityis expected to be rather different: 100 to 150 km s − for thetrailing arm or − 100 to 0 km s − for the leading arm. Ko 1was previously conjectured to belong to Sgr stream by Paustet al. (2014) together with its sibling Ko 2; however, the lat-ter does not match the stream neither in distance nor in PM.A few distant outer halo clusters – Pal 3, Pal 4 and Crater– also might be associated with the Sgr debris scattered be-yond the apocentre of the trailing arm, but they do not matchthe model track in at least one dimension (though the modelmight be unreliable beyond 100 kpc, not being constrainedby observational data).Other conspicuous features in Figure 9 include the popu-lations of high-eccentricity clusters with semimajor axes inthe range 6 − 20 kpc, which are believed to be associated with an ancient merger of a satellite galaxy on a nearly ra-dial orbit (Myeong et al. 2018). Finally, a significant fractionof clusters in the inner part of the Galaxy (with semima-jor axes below 6 − | b | < ◦ ), the PM component perpendicular to theGalactic plane ( µ corr b ) is significantly smaller than the parallelcomponent ( µ corr l ). We may conclude that the orbits of theseclusters necessarily stay close to the disc plane, even if theline-of-sight velocity is not known (these cases are markedby italic). This subset of disc-like clusters includes BH 140,BH 176, BLISS 1 , ESO 93–8, Ko 2 , Mercer 5, Pfleiderer 2 ,Segue 3. Some of these objects may well be old open clustersrather than globular clusters. The orbit of Ryu 059 also liesclose to the disc plane but is retrograde and highly eccentric(although its line-of-sight velocity is not known, the reflex-corrected sky-plane velocity already exceed 400 km s − ), withthe estimated apocentre radii exceeding 100 kpc. It is quiteplausible that the distance to this cluster is overestimated– a similar story happened with Djorg 1, for which a 30%downward distance revision by Ortolani et al. (2019) made itsorbit much less eccentric and more realistic. Finally, Kim 3 ,Laevens 3, Mu˜noz 1, Ryu 879 have high-inclination orbits.As is clear from Figure 9, the orbital parameters often havesignificant uncertainties, but these are strongly correlated,therefore quoting the confidence intervals on r peri / apo or ec-centricity makes little sense without a full covariance matrix,or better, the full posterior distribution. Rather than provid-ing these quantities in a tabular form, we list the observables(distance, PM and line-of-sight velocity) and their uncertain-ties (which are nearly uncorrelated) in Table A1, and providea Python script for computing orbits in any given potential,sampling from the possible range of initial conditions for eachcluster; the integrations are carried out with the Agama li-brary for galactic dynamics (Vasiliev 2019a). Figure A1 in the Appendix shows the derived PM dispersionand rotation profiles for ∼ 100 clusters that have a sufficientlylarge number of stars and are not too distant, comparing ourresults with other studies.We explored the sky-plane rotation signatures in all clus-ters, using the method detailed in the Appendix A6 ofVasiliev (2019c), which takes into account spatially corre-lated systematic errors. Namely, we fitted a general linearmodel with N + 3 free parameters to the PM field of starswith high membership probability, where the free parametersare the two components of the mean PM, the slope ξ of theradial PM component ( µ R = ξ R ), and N amplitudes of aB-spline representation of the tangential PM component µ t ,with N = 2 − https://github.com/GalacticDynamics-Oxford/GaiaTools MNRAS , 000–000 (0000) aia EDR3 view on Galactic globular clusters ξ expected ξ m e a s u r e d M M M M N G C N G C N G C M T u c ω C e n ( ξ meas − ξ exp ) /† ξ c o un t Figure 10. Slope of the radial PM component ξ ≡ µ R /R (inunits of mas yr − per degree). Left panel shows the measured val-ues for 10 clusters with uncertainties (cid:15) ξ smaller than 0.05, whichagree fairly well with values expected from perspective effects.Right panel shows the distribution of deviation of measured val-ues from expectations, normalized by measurement uncertainties (cid:15) ξ , for ∼ 150 clusters with at least 100 stars. The red curve showsthe standard normal distribution, which adequately describes theactual histogram of deviations (though the latter has a slight ex-cess of objects around zero, for which the uncertainties (cid:15) ξ mightbe overestimated). membership, so we used it as a post-processing step afterthe main MCMC run, and accounted for uncertain member-ship by considering 16 realizations of the subset of clustermembers, selecting stars in proportion to their membershipprobability. The PM dispersion profile was kept fixed, as theprevious analysis indicated that it is little affected by thespatial correlations (unlike the mean PM field).The radial PM component is expected to be caused entirelydue to perspective contraction or expansion: for a cluster ata distance D moving with line-of-sight velocity v los , the ex-pected µ R at an angular distance R (measured in degrees)is ξ R , with ξ expected = − v los /D × ( π/ ◦ / . ξ for 10 clusters with small measurement uncertainties (cid:15) ξ ,demonstrating a very good agreement. The cluster with thelargest amplitude of the perspective expansion is NGC 3201,for which the value of ξ is measured with (cid:46) 6% relative error;for NGC 5139 ( ω Cen) the relative error is ∼ (cid:15) ξ might be overesti-mated). We stress that these uncertainties take into accountspatially correlated systematic errors: if we use only statisti-cal errors, (cid:15) ξ would be considerably smaller for rich clusters,and deviations between measured and expected ξ would ex-ceed (3 − (cid:15) ξ for quite a few objects. This exercise validatesour approach for treating correlated systematic errors andgives credence to the similar analysis of rotation signaturesin the PM.The uncertainty in the PM rotation profile is typically atthe level 0 . − . 015 mas yr − for sufficiently rich clusters, be-ing dominated by systematic errors. We detect unambiguousrotation (at more than 3 σ level) in 17 clusters, with further6 showing indicative signatures exceeding 0 . 03 mas yr − at ∼ σ level; they are listed in Table 2. In most cases, our find-ings agree with previous studies based on Gaia DR2 (also Table 2. Rotation signatures in star clusters detected in thiswork (last column) and in some previous studies based on Gaia data: Gaia Collaboration (2018), Bianchini et al. (2018), Vasiliev(2019c), Sollima et al. (2019). “+” indicates a firm detection, “?”– a tentative detection, and “ − ” stands for no clear signature.Cluster G18 B18 V19 S19 this workNGC 104 + + + + +NGC 1904 − ? +NGC 3201 ? − ? +NGC 4372 + ? ? +NGC 5139 + + + + +NGC 5272 + + ? ? +NGC 5904 + + + + +NGC 5986 − − ? ?NGC 6139 +NGC 6218 − − ? +NGC 6266 ? + + +NGC 6273 + + + +NGC 6333 ?NGC 6341 − − ? +NGC 6388 − − − ?NGC 6402 ? − ? +NGC 6539 ? − ? ?NGC 6656 + + + + +NGC 6715 − − ?NGC 6752 + + ? ? +NGC 6809 + + ? ? ?NGC 7078 + + + + +NGC 7089 + + + + shown in the table). The new additions are NGC 6139, inwhich we find a rather prominent rotation signature despitea lack of it in DR2; NGC 6333 (M 9) and NGC 6388, inwhich the signal in EDR3 is stronger than in DR2 but stillnot unambiguous; and NGC 6715 (M 54), in which Alfaro-Cuello et al. (2020) detected rotation in line-of-sight velocities(although this cluster is a rather special case, sitting in thecentre of the Sgr dwarf galaxy, and our kinematic analysisdoes not separate it from the stars of the galaxy itself). Onthe other hand, we do not detect significant rotation in sev-eral clusters examined and found rotating by Sollima et al.(2019): NGC 2808, NGC 6205 (M 13), NGC 6397, NGC 6541,NGC 6553, NGC 6626 (M 62): that study considered bothPM and line-of-sight velocities, and all these clusters haveinclination angles exceeding 60 ◦ , i.e., the rotation signal ismostly seen in the line-of-sight velocity field (though we dosee weak signatures in the PM field of the last two objects).We also excluded Terzan 5 due to a small number of starssatisfying our quality cuts.Turning to the analysis of PM dispersion profiles, we com-pare them with the profiles derived from Gaia DR2 byVasiliev (2019c) and Baumgardt et al. (2019), finding gener-ally a good agreement, with the present study having some-what smaller uncertainties due to improvements in Gaia as-trometry. In addition, the inferred PM dispersion profilescan be compared with the HST -derived PM dispersions inthe central parts of 22 clusters studied in Watkins et al.(2015) and 9 clusters studied in Cohen et al. (2021). Unfortu-nately, due to the strict quality cutoffs adopted in the presentstudy, there is very little (if any) spatial overlap between Gaia and HST measurements from Watkins et al. (2015), but ingeneral, the PM dispersion profiles agree remarkably well.Some discrepancy is seen in NGC 5139 ( ω Cen), where our MNRAS , 000–000 (0000) Vasiliev & Baumgardt σ µ , t / σ µ , R − NGC 104 σ µ , t / σ µ , R − NGC 2808 σ µ , t / σ µ , R − NGC 3201 σ µ , t / σ µ , R − NGC 4372 σ µ , t / σ µ , R − NGC 5139 σ µ , t / σ µ , R − NGC 5272 σ µ , t / σ µ , R − NGC 5904 σ µ , t / σ µ , R − NGC 6121 σ µ , t / σ µ , R − NGC 6205 σ µ , t / σ µ , R − NGC 6218 σ µ , t / σ µ , R − NGC 6254 σ µ , t / σ µ , R − NGC 6397 σ µ , t / σ µ , R − NGC 6656 σ µ , t / σ µ , R − NGC 6752 σ µ , t / σ µ , R − NGC 6809 σ µ , t / σ µ , R − NGC 7078 Figure 11. PM anisotropy profiles for the richest clusters. Shown is the ratio σ µ,t /σ µ,R − ∼ 500 stars per bin (except the two largest clusters);only the stars from the clean subset with small uncertainties are used. The profiles are rather diverse: half of the clusters (NGC 104,NGC 2808, NGC 3201, NGC 4372, NGC 5272, NGC 6205, NGC 6397, NGC 6752 and especially NGC 7078) are radially anisotropic, some(NGC 6121, NGC 6809) show weak tangential bias, NGC 5139 transitions from radial to tangential anisotropy, and remaining objects areconsistent with isotropy. PM dispersion profile is lower (although there are almost nostars in our clean subset to anchor it in the crowded centralparts), NGC 6341 (M 92), where Gaia σ µ is slightly higher,NGC 6535, where both profiles have large uncertainties but Gaia is lower, and NGC 6681 (M 70), where the Gaia σ µ pro-file is significantly lower. The last case is the most puzzlingdiscrepancy, which appears to be robust against variations inthe assumed error inflation factor η or various quality filters.The 9 clusters with HST measurements from Cohen et al.(2021) have a larger spatial extent and agree very well with Gaia in all cases except NGC 6355 and NGC 6401, where Gaia σ µ is slightly higher, and NGC 6380, where Gaia dis-persion is ∼ 15% lower.The PM dispersion profiles can also be compared with line-of-sight velocity dispersion profiles from various spectroscopicstudies; here we use three such datasets – Kamann et al.(2018) used MUSE IFU in central regions of some clusters,while Ferraro et al. (2018) and Baumgardt et al. (2019) pro-vide wide-field coverage. The conversion from σ µ to σ LOS involves distance, and is one of the methods for its deter-mination. Using the average distances from the literature,we generally get good agreement between PM and line-of-sight dispersions, but in some cases the discrepancy is sig-nificant and may indicate some problems in either dataset,or alternatively, calls for a revision of the distance. For well-populated clusters, the discrepancies are usually less than10%. The most notable outliers are NGC 1904, NGC 5272(M 3), NGC 6388, where Gaia PM is ∼ − 15% higher forthe adopted distances; NGC 6304, NGC 6553, NGC 6626,NGC 6779 (M 56), where Gaia PM is ∼ 15% lower; andNGC 6681 (M 70), which is problematic as described above.Although we used isotropic PM dispersion in the mixturemodel by default, in the richest clusters we were able to ex-plore the anisotropy by fitting the radial and tangential PMdispersions separately. Figure 11 shows the radial profiles ofPM dispersion anisotropy, σ t /σ R − 1, for 16 clusters with suf-ficient number of stars and satisfying the consistency checksthat the PM dispersion and anisotropy agree between stars MNRAS , 000–000 (0000) aia EDR3 view on Galactic globular clusters in different magnitude ranges. There is a considerable diver-sity in the anisotropy profiles: half of these clusters have pre-dominantly radially anisotropic PM dispersion, a few othershave tangential anisotropy, and NGC 5139 ( ω Cen) transi-tions from being radially anisotropic in the inner part to tan-gentially anisotropic in the outskirts. The difference between σ R and σ t is typically at the level 10 − never becalled “radial velocity”!).PM anisotropy was explored by Watkins et al. (2015) inthe central parts of 22 clusters observed by HST , who founddeviations from isotropy at a few per cent level. However, asdiscussed above, there is almost no spatial overlap between HST and the clean Gaia subset. Our findings can be moredirectly compared to Jindal et al. (2019), who explored PManisotropy in 10 clusters using Gaia DR2. Our results agreefor all clusters except NGC 6656 (M 22), which they found tobe radially anisotropic but we do not see a strong evidence forthis, and NGC 6397, which appeared to be isotropic in theiranalysis but weakly radial in our study. The radial anisotropyin NGC 3201 has also been detected by Bianchini et al. (2019)using Gaia DR2, consistent with our measurement. Gaia EDR3 is a quantitative rather than qualitative improve-ment upon the revolutionary DR2 catalogue, yet its precisionis materially better. In the first part of this study, we scru-tinized the fidelity and accuracy of Gaia astrometry with avariety of methods, using the data from a dozen richest glob-ular clusters, after applying a number of stringent filters toremove possibly unreliable sources. • Formal statistical uncertainties on parallax and PM ad-equately describe the actual errors in regions with low stellardensity, but appear to be underestimated by 10 − 20% inhigher-density regions; Table 1 provides the suggested cor-rection factors. • The parallax zero-point correction proposed by Linde-gren et al. (2020b) might be too large (overcorrecting) by ∼ . − . 009 mas. • Spatially correlated systematic errors in parallax andPM are considerably lower than in DR2: for bright stars,the residual systematic error in (cid:36) is at the level 0 . 01 mas (afourfold improvement), while for stars fainter than G = 18it may be twice higher. The systematic uncertainty in PM is ∼ . 025 mas yr − , or 2 . × better than in DR2.In the second part, we re-examined the kinematics of al-most all Milky Way globular clusters, derived the mean par-allaxes and PM, galactic orbits, and analyzed the internal ro-tation, dispersion and anisotropy profiles of sufficiently richclusters. While the improvements in precision for mean PMwith respect to studies based on DR2 (Gaia Collaboration2018, Baumgardt et al. 2019, Vasiliev 2019b) is substantial,the precision of the phase-space coordinates is typically lim- ited by distance uncertainties, so has not improved as much. Gaia parallaxes will eventually deliver the most precise dis-tance estimates for nearby clusters, but at present, the sys-tematic uncertainty at the level 0 . 01 mas limits their useful-ness in practice. On the other hand, the internal kinematics(PM dispersions and rotation signatures) have also improvedconsiderably, and agree well with various independent esti-mates. We find evidence of rotation in more than 20 clusters,and measure the PM dispersion profiles in more than a hun-dred systems, down to the level 0 . 05 mas yr − , i.e. at least 2 × better than in DR2. These data can be used to improve dy-namical models of clusters and provide independent distanceestimates, which are examined in Baumgardt (2021). DATA AVAILABILITY We provide the catalogues of all Gaia sources in the fieldof each cluster with their membership probabilities andthe tables of radial profiles of PM dispersion and rota-tion amplitudes, available at https://zenodo.org/record/4549398 , as well as the summary table of mean paral-laxes and PM and scripts for computing cluster orbitsin a given potential, available at https://github.com/GalacticDynamics-Oxford/GaiaTools . REFERENCES Alfaro-Cuello M., Kacharov N., Neumayer N., et al., 2020, ApJ,892, 20Arakelyan N., Pilipenko S., Sharina M., 2020, Ast. 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Kinematic profiles of Milky Way globular clusters derived in this work and in other studies.Blue and red solid lines show the radial profiles of internal PM dispersion σ µ and mean rotation µ rot ; shaded bands depict 68% confidenceintervals taking into account systematic errors. Violet diamonds show the PM dispersion profiles from HST (Watkins et al. 2015; Cohenet al. 2021); orange upward triangles, yellow downward triangles and greenish-gray stars – line-of-sight velocity dispersions from Kamannet al. (2018), Ferraro et al. (2018) and Baumgardt & Hilker (2018), correspondingly. N refers to the number of cluster members withgood astrometry, and the number in brackets – to the number of stars with small enough uncertainties to be used in the measurementof PM dispersion; the light-shaded part of the PM dispersion profile shows the ranges of radii containing less than 5 stars at both ends. (Continued on next page) MNRAS000 Kinematic profiles of Milky Way globular clusters derived in this work and in other studies.Blue and red solid lines show the radial profiles of internal PM dispersion σ µ and mean rotation µ rot ; shaded bands depict 68% confidenceintervals taking into account systematic errors. Violet diamonds show the PM dispersion profiles from HST (Watkins et al. 2015; Cohenet al. 2021); orange upward triangles, yellow downward triangles and greenish-gray stars – line-of-sight velocity dispersions from Kamannet al. (2018), Ferraro et al. (2018) and Baumgardt & Hilker (2018), correspondingly. N refers to the number of cluster members withgood astrometry, and the number in brackets – to the number of stars with small enough uncertainties to be used in the measurementof PM dispersion; the light-shaded part of the PM dispersion profile shows the ranges of radii containing less than 5 stars at both ends. (Continued on next page) MNRAS000 , 000–000 (0000) Vasiliev & Baumgardt NGC 4833 N=2940 (544), D=6.6 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 5024 (M 53) N=3398 (120), D=18.6 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 5139 ( ω Cen) N=53123 (37323), D=5.5 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 5272 (M 3) N=8992 (775), D=10.2 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 5286 N=847 (250), D=11.1 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 5466 N=2210 (50), D=16.3 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 5634 N=339 (12), D=25.8 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 5824 N=514 (28), D=31.8 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 5897 N=2233 (34), D=12.8 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 5904 (M 5) N=9163 (809), D=7.5 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 5927 N=1303 (455), D=8.3 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 5946 N=179 (54), D=9.5 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 5986 N=1118 (138), D=10.8 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6093 (M 80) N=1245 (122), D=10.5 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6101 N=2420 (177), D=14.7 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Figure A1 – continued MNRAS , 000–000 (0000) aia EDR3 view on Galactic globular clusters NGC 6121 (M 4) N=5210 (5207), D=1.9 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6139 N=335 (150), D=10.6 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6144 N=1586 (67), D=8.4 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6171 (M 107) N=1668 (258), D=5.6 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6205 (M 13) N=9637 (2442), D=7.5 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6218 (M 12) N=6213 (1201), D=5.1 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6229 N=256 (12), D=30.4 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6235 N=270 (22), D=12.0 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6254 (M 10) N=7270 (1788), D=5.1 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6256 N=102 (63), D=7.1 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6266 (M 62) N=1100 (619), D=6.7 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6273 (M 19) N=1435 (391), D=8.5 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6284 N=152 (33), D=15.0 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6287 N=201 (77), D=8.4 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6293 N=194 (84), D=9.3 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Figure A1 – continued MNRAS000 N=194 (84), D=9.3 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Figure A1 – continued MNRAS000 , 000–000 (0000) Vasiliev & Baumgardt NGC 6304 N=105 (102), D=6.1 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6316 N=62 (46), D=11.9 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6325 N=106 (47), D=7.4 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6333 (M 9) N=804 (226), D=8.2 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6341 (M 92) N=4365 (610), D=8.6 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6342 N=193 (50), D=8.6 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6352 N=2336 (228), D=5.5 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6355 N=63 (52), D=8.8 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6356 N=441 (81), D=15.7 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6362 N=5680 (503), D=7.6 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6366 N=1663 (754), D=3.5 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6380 (Ton 1) N=103 (42), D=9.6 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6388 N=535 (304), D=11.4 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6397 N=12318 (10935), D=2.5 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6401 N=40 (32), D=7.7 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Figure A1 – continued MNRAS , 000–000 (0000) aia EDR3 view on Galactic globular clusters NGC 6402 (M 14) N=646 (459), D=9.6 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6440 N=81 (81), D=8.1 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6441 N=118 (114), D=13.0 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6453 N=23 (23), D=10.4 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6517 N=147 (74), D=10.7 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6522 N=63 (55), D=7.6 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6528 N=24 (11), D=7.3 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6535 N=400 (33), D=6.5 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6539 N=391 (296), D=7.9 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6541 N=2556 (296), D=7.3 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6544 N=322 (319), D=2.6 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6553 N=207 (207), D=5.3 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6558 N=57 (16), D=7.0 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6569 N=102 (91), D=10.5 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6584 N=856 (50), D=13.7 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Figure A1 – continued MNRAS000 N=856 (50), D=13.7 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Figure A1 – continued MNRAS000 , 000–000 (0000) Vasiliev & Baumgardt NGC 6624 N=120 (68), D=8.1 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6626 (M 28) N=321 (229), D=5.2 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6637 (M 69) N=387 (68), D=8.8 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6638 N=151 (43), D=9.8 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6642 N=60 (43), D=8.3 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6652 N=319 (30), D=9.6 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6656 (M 22) N=4337 (3575), D=3.3 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6681 (M 70) N=856 (50), D=9.4 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6712 N=291 (118), D=7.4 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6715 (M 54) N=1122 (116), D=26.7 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6717 (Pal 9) N=285 (17), D=7.5 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6723 N=2331 (151), D=8.3 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6749 N=256 (72), D=7.7 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6752 N=16348 (10504), D=4.2 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6760 N=324 (196), D=8.6 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Figure A1 – continued MNRAS , 000–000 (0000) aia EDR3 view on Galactic globular clusters NGC 6779 (M 56) N=1566 (179), D=10.4 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6809 (M 55) N=7482 (1594), D=5.4 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6838 (M 71) N=3022 (837), D=4.0 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6864 (M 75) N=367 (47), D=19.5 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6934 N=745 (20), D=16.1 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 6981 (M 72) N=764 (28), D=16.8 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 7078 (M 15) N=4697 (588), D=10.7 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 7089 (M 2) N=3092 (352), D=11.6 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] NGC 7099 (M 30) N=2989 (220), D=8.5 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] BH 140 N=1197 (287), D=5.0 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] BH 184 (Lynga 7) N=147 (64), D=8.0 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] BH 229 (HP 1) N=61 (25), D=6.9 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Djorg 1 N=56 (15), D=9.3 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] FSR 1758 N=296 (154), D=11.5 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Pal 7 (IC 1276) N=578 (416), D=4.0 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Figure A1 – continued MNRAS000 N=578 (416), D=4.0 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Figure A1 – continued MNRAS000 , 000–000 (0000) Vasiliev & Baumgardt IC 4499 N=589 (38), D=19.1 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Pal 10 N=107 (40), D=5.9 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Pal 6 N=64 (26), D=7.5 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Terzan 1 (HP 2) N=96 (23), D=5.7 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Terzan 2 (HP 3) N=56 (12), D=7.0 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Terzan 3 N=331 (36), D=8.1 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Terzan 5 (Terzan 11) N=133 (18), D=6.1 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Terzan 9 N=51 (11), D=5.2 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Ton 2 (Pismis 26) N=80 (14), D=6.4 kpc µ r o t , σ µ [ m a s / y r ] σ l o s [ k m / s ] Figure A1 – continued MNRAS , 000–000 (0000) aia EDR3 view on Galactic globular clusters Table A1. Catalogue of mean parallaxes and PM of Milky Way globular clusters, derived from Gaia EDR3 astrometry, taking intoaccount spatially correlated systematic errors (hence the uncertainty does not decrease below 0 . 01 mas and 0 . 02 mas yr − ). Last columnshows the number of member stars that pass all quality filters and are used to determine the cluster properties.Name α [deg] δ [deg] µ α [mas yr − ] µ δ [mas yr − ] corr µ (cid:36) [mas] N memb NGC 104 (47 Tuc) 6 . − . 081 5 . ± . − . ± . 022 0 . 00 0 . ± . 009 39931NGC 288 13 . − . 583 4 . ± . − . ± . 025 0 . 01 0 . ± . 011 4689NGC 362 15 . − . 849 6 . ± . − . ± . 024 0 . 00 0 . ± . 011 3878Whiting 1 30 . − . − . ± . − . ± . 057 0 . 02 0 . ± . 046 40NGC 1261 48 . − . 216 1 . ± . − . ± . 025 0 . 00 0 . ± . 011 1217Pal 1 53 . 333 79 . − . ± . 034 0 . ± . 038 0 . 07 0 . ± . 022 92E 1 (AM 1) 58 . − . 615 0 . ± . − . ± . − . − . ± . 060 58Eridanus 66 . − . 187 0 . ± . − . ± . − . 09 0 . ± . 033 44Pal 2 71 . 525 31 . 381 1 . ± . − . ± . 031 0 . 04 0 . ± . 021 180NGC 1851 78 . − . 047 2 . ± . − . ± . 024 0 . 00 0 . ± . 011 2497NGC 1904 (M 79) 81 . − . 524 2 . ± . − . ± . 026 0 . 00 0 . ± . 011 1556NGC 2298 102 . − . 005 3 . ± . − . ± . 026 0 . 00 0 . ± . 011 1047NGC 2419 114 . 535 38 . 882 0 . ± . − . ± . 026 0 . 02 0 . ± . 017 333Ko 2 119 . 571 26 . − . ± . − . ± . − . 10 0 . ± . 160 7Pyxis 136 . − . 221 1 . ± . 032 0 . ± . 035 0 . 03 0 . ± . 022 69NGC 2808 138 . − . 863 0 . ± . 024 0 . ± . 024 0 . 00 0 . ± . 010 4740E 3 (ESO 37-1) 140 . − . − . ± . 027 7 . ± . 027 0 . 00 0 . ± . 013 252Pal 3 151 . 383 0 . 072 0 . ± . − . ± . − . 41 0 . ± . 050 61NGC 3201 154 . − . 412 8 . ± . − . ± . 022 0 . 00 0 . ± . 010 11398ESO 93-8 169 . − . − . ± . 034 1 . ± . 035 0 . 01 0 . ± . 018 32Pal 4 172 . 320 28 . − . ± . − . ± . − . − . ± . 033 96Crater (Laevens 1) 174 . − . − . ± . − . ± . − . 23 0 . ± . 143 15Bliss 1 177 . − . − . ± . 042 0 . ± . − . 05 0 . ± . 040 10Ko 1 179 . 827 12 . − . ± . − . ± . − . − . ± . 133 6NGC 4147 182 . 526 18 . − . ± . − . ± . − . 03 0 . ± . 013 347NGC 4372 186 . − . − . ± . 024 3 . ± . 024 0 . 00 0 . ± . 010 2552Rup 106 189 . − . − . ± . 027 0 . ± . 026 0 . 02 0 . ± . 013 301NGC 4590 (M 68) 189 . − . − . ± . 024 1 . ± . 024 0 . 00 0 . ± . 011 3283BH 140 193 . − . − . ± . 024 1 . ± . 024 0 . 00 0 . ± . 011 1197NGC 4833 194 . − . − . ± . − . ± . 025 0 . 01 0 . ± . 011 2940NGC 5024 (M 53) 198 . 230 18 . − . ± . − . ± . − . 01 0 . ± . 011 3398NGC 5053 199 . 113 17 . − . ± . − . ± . − . 02 0 . ± . 011 1452Kim 3 200 . − . − . ± . 176 3 . ± . − . 20 0 . ± . 158 9NGC 5139 ( ω Cen) 201 . − . − . ± . − . ± . 022 0 . 00 0 . ± . 009 53123NGC 5272 (M 3) 205 . 548 28 . − . ± . − . ± . 023 0 . 00 0 . ± . 010 8992NGC 5286 206 . − . 374 0 . ± . − . ± . 026 0 . 00 0 . ± . 011 847AM 4 209 . − . − . ± . − . ± . − . 42 0 . ± . 296 8NGC 5466 211 . 364 28 . − . ± . − . ± . 024 0 . 01 0 . ± . 011 2210NGC 5634 217 . − . − . ± . − . ± . − . 01 0 . ± . 012 339NGC 5694 219 . − . − . ± . − . ± . − . 06 0 . ± . 017 149IC 4499 225 . − . 214 0 . ± . − . ± . 025 0 . 00 0 . ± . 011 589Munoz 1 225 . 450 66 . − . ± . − . ± . − . 10 0 . ± . 138 5NGC 5824 225 . − . − . ± . − . ± . − . 01 0 . ± . 012 514Pal 5 229 . − . − . ± . − . ± . 028 0 . 00 0 . ± . 015 232NGC 5897 229 . − . − . ± . − . ± . − . 01 0 . ± . 011 2233NGC 5904 (M 5) 229 . 638 2 . 081 4 . ± . − . ± . − . 01 0 . ± . 010 9163NGC 5927 232 . − . − . ± . − . ± . 025 0 . 00 0 . ± . 011 1303NGC 5946 233 . − . − . ± . − . ± . − . 01 0 . ± . 012 179BH 176 234 . − . − . ± . − . ± . − . 03 0 . ± . 017 89NGC 5986 236 . − . − . ± . − . ± . 025 0 . 00 0 . ± . 011 1118FSR 1716 242 . − . − . ± . − . ± . − . 02 0 . ± . 023 40Pal 14 (Arp 1) 242 . 752 14 . − . ± . − . ± . 038 0 . 13 0 . ± . 032 80BH 184 (Lynga 7) 242 . − . − . ± . − . ± . 027 0 . 00 0 . ± . 012 147NGC 6093 (M 80) 244 . − . − . ± . − . ± . − . 01 0 . ± . 011 1245Ryu 059 (RLGC 1) 244 . − . 593 1 . ± . 054 0 . ± . 047 0 . − . ± . 046 78NGC 6121 (M 4) 245 . − . − . ± . − . ± . 023 0 . 00 0 . ± . 010 5210NGC 6101 246 . − . 202 1 . ± . − . ± . 024 0 . 00 0 . ± . 011 2420NGC 6144 246 . − . − . ± . − . ± . − . 01 0 . ± . 011 1586NGC 6139 246 . − . − . ± . − . ± . 026 0 . 00 0 . ± . 011 335Terzan 3 247 . − . − . ± . − . ± . 026 0 . 00 0 . ± . 012 331NGC 6171 (M 107) 248 . − . − . ± . − . ± . 025 0 . 00 0 . ± . 011 1668ESO 452-11 (1636-283) 249 . − . − . ± . − . ± . − . 05 0 . ± . 015 143MNRAS000 015 143MNRAS000 , 000–000 (0000) Vasiliev & Baumgardt Name α [deg] δ [deg] µ α [mas yr − ] µ δ [mas yr − ] corr µ (cid:36) [mas] N memb NGC 6205 (M 13) 250 . 422 36 . − . ± . − . ± . 023 0 . 00 0 . ± . 010 9637NGC 6229 251 . 745 47 . − . ± . − . ± . 026 0 . 01 0 . ± . 012 256NGC 6218 (M 12) 251 . − . − . ± . − . ± . 024 0 . 00 0 . ± . 010 6213FSR 1735 253 . − . − . ± . − . ± . − . 05 0 . ± . 110 27NGC 6235 253 . − . − . ± . − . ± . − . 01 0 . ± . 012 270NGC 6254 (M 10) 254 . − . − . ± . − . ± . 024 0 . 00 0 . ± . 010 7270NGC 6256 254 . − . − . ± . − . ± . − . 01 0 . ± . 013 102Pal 15 254 . − . − . ± . − . ± . 034 0 . 03 0 . ± . 025 126NGC 6266 (M 62) 255 . − . − . ± . − . ± . 026 0 . 00 0 . ± . 011 1100NGC 6273 (M 19) 255 . − . − . ± . 025 1 . ± . 025 0 . 00 0 . ± . 011 1435NGC 6284 256 . − . − . ± . − . ± . 027 0 . 01 0 . ± . 012 152NGC 6287 256 . − . − . ± . − . ± . 028 0 . 00 0 . ± . 013 201NGC 6293 257 . − . 582 0 . ± . − . ± . 028 0 . 00 0 . ± . 012 194NGC 6304 258 . − . − . ± . − . ± . 029 0 . 00 0 . ± . 011 105NGC 6316 259 . − . − . ± . − . ± . 031 0 . 01 0 . ± . 013 62NGC 6341 (M 92) 259 . 281 43 . − . ± . − . ± . 023 0 . 00 0 . ± . 010 4365NGC 6325 259 . − . − . ± . − . ± . 029 0 . 02 0 . ± . 013 106NGC 6333 (M 9) 259 . − . − . ± . − . ± . 026 0 . 00 0 . ± . 011 804NGC 6342 260 . − . − . ± . − . ± . 026 0 . 00 0 . ± . 012 193NGC 6356 260 . − . − . ± . − . ± . 026 0 . 00 0 . ± . 012 441NGC 6355 260 . − . − . ± . − . ± . 029 0 . 02 0 . ± . 012 63NGC 6352 261 . − . − . ± . − . ± . 025 0 . 01 0 . ± . 011 2336IC 1257 261 . − . − . ± . − . ± . 033 0 . 08 0 . ± . 027 64Terzan 2 (HP 3) 261 . − . − . ± . − . ± . 039 0 . 09 0 . ± . 027 56NGC 6366 261 . − . − . ± . − . ± . 024 0 . 01 0 . ± . 011 1663Terzan 4 (HP 4) 262 . − . − . ± . − . ± . 065 0 . 22 0 . ± . 063 43BH 229 (HP 1) 262 . − . 982 2 . ± . − . ± . 034 0 . 04 0 . ± . 014 61FSR 1758 262 . − . − . ± . 026 2 . ± . 025 0 . 01 0 . ± . 011 296NGC 6362 262 . − . − . ± . − . ± . 024 0 . 00 0 . ± . 011 5680Liller 1 263 . − . − . ± . − . ± . 099 0 . − . ± . 101 54NGC 6380 (Ton 1) 263 . − . − . ± . − . ± . 030 0 . 01 0 . ± . 015 103Terzan 1 (HP 2) 263 . − . − . ± . − . ± . 058 0 . 11 0 . ± . 034 96Ton 2 (Pismis 26) 264 . − . − . ± . − . ± . 028 0 . 04 0 . ± . 015 80NGC 6388 264 . − . − . ± . − . ± . 026 0 . 00 0 . ± . 011 535NGC 6402 (M 14) 264 . − . − . ± . − . ± . 026 0 . 00 0 . ± . 011 646NGC 6401 264 . − . − . ± . 035 1 . ± . 034 0 . 02 0 . ± . 013 40NGC 6397 265 . − . 674 3 . ± . − . ± . 022 0 . 00 0 . ± . 010 12318VVV CL002 265 . − . − . ± . 145 2 . ± . 087 0 . 17 0 . ± . 101 14Pal 6 265 . − . − . ± . − . ± . 033 0 . 05 0 . ± . 017 64NGC 6426 266 . 228 3 . − . ± . − . ± . 026 0 . 01 0 . ± . 012 130Djorg 1 266 . − . − . ± . − . ± . − . 01 0 . ± . 029 56Terzan 5 (Terzan 11) 267 . − . − . ± . − . ± . 075 0 . 25 0 . ± . 038 133NGC 6440 267 . − . − . ± . − . ± . 036 0 . 02 0 . ± . 013 81NGC 6441 267 . − . − . ± . − . ± . 028 0 . 00 0 . ± . 011 118Terzan 6 (HP 5) 267 . − . − . ± . − . ± . 039 0 . 12 0 . ± . 030 62NGC 6453 267 . − . 599 0 . ± . − . ± . 036 0 . 01 0 . ± . 013 23UKS 1 268 . − . − . ± . − . ± . − . 11 0 . ± . 083 136VVV CL001 268 . − . − . ± . − . ± . 117 0 . 22 0 . ± . 115 27Gran 1 269 . − . − . ± . − . ± . 037 0 . 08 0 . ± . 025 19Pfleiderer 2 269 . − . − . ± . − . ± . 031 0 . 10 0 . ± . 021 108NGC 6496 269 . − . − . ± . − . ± . 025 0 . 01 0 . ± . 013 460Terzan 9 270 . − . − . ± . − . ± . 047 0 . 19 0 . ± . 027 51Djorg 2 (ESO 456-38) 270 . − . 826 0 . ± . − . ± . 033 0 . 04 0 . ± . 018 16NGC 6517 270 . − . − . ± . − . ± . 028 0 . 03 0 . ± . 012 147Terzan 10 270 . − . − . ± . − . ± . 048 0 . 01 0 . ± . 042 52NGC 6522 270 . − . 034 2 . ± . − . ± . 035 0 . 02 0 . ± . 013 63NGC 6535 270 . − . − . ± . − . ± . 026 0 . 01 0 . ± . 012 400NGC 6528 271 . − . − . ± . − . ± . − . 01 0 . ± . 017 24NGC 6539 271 . − . − . ± . − . ± . 026 0 . 02 0 . ± . 011 391NGC 6540 (Djorg 3) 271 . − . − . ± . − . ± . 031 0 . 03 0 . ± . 016 17NGC 6544 271 . − . − . ± . − . ± . 030 0 . 00 0 . ± . 011 322NGC 6541 272 . − . 715 0 . ± . − . ± . 025 0 . 01 0 . ± . 011 2556ESO 280-06 272 . − . − . ± . − . ± . 033 0 . 10 0 . ± . 025 45NGC 6553 272 . − . 909 0 . ± . − . ± . 029 0 . 00 0 . ± . 011 207NGC 6558 272 . − . − . ± . − . ± . 033 0 . 02 0 . ± . 018 57Pal 7 (IC 1276) 272 . − . − . ± . − . ± . 026 0 . 00 0 . ± . 011 578MNRAS , 000–000 (0000) aia EDR3 view on Galactic globular clusters Name α [deg] δ [deg] µ α [mas yr − ] µ δ [mas yr − ] corr µ (cid:36) [mas] N memb Terzan 12 273 . − . − . ± . − . ± . 044 0 . 24 0 . ± . 032 70NGC 6569 273 . − . − . ± . − . ± . 028 0 . 01 0 . ± . 012 102BH 261 (ESO 456-78) 273 . − . 635 3 . ± . − . ± . 038 0 . 10 0 . ± . 022 21NGC 6584 274 . − . − . ± . − . ± . 025 0 . 00 0 . ± . 011 856Mercer 5 275 . − . − . ± . − . ± . 113 0 . 37 0 . ± . 091 40NGC 6624 275 . − . 361 0 . ± . − . ± . 029 0 . 00 0 . ± . 012 120NGC 6626 (M 28) 276 . − . − . ± . − . ± . − . 01 0 . ± . 011 321NGC 6638 277 . − . − . ± . − . ± . 029 0 . 02 0 . ± . 012 151NGC 6637 (M 69) 277 . − . − . ± . − . ± . 027 0 . 04 0 . ± . 012 387NGC 6642 277 . − . − . ± . − . ± . 030 0 . 01 0 . ± . 013 60NGC 6652 278 . − . − . ± . − . ± . 026 0 . 01 0 . ± . 012 319NGC 6656 (M 22) 279 . − . 905 9 . ± . − . ± . 024 0 . 01 0 . ± . 010 4337Pal 8 280 . − . − . ± . − . ± . 027 0 . 01 0 . ± . 012 134NGC 6681 (M 70) 280 . − . 292 1 . ± . − . ± . 026 0 . 01 0 . ± . 011 856Ryu 879 (RLGC 2) 281 . − . − . ± . − . ± . 078 0 . 39 0 . ± . 086 28NGC 6712 283 . − . 706 3 . ± . − . ± . 026 0 . 01 0 . ± . 011 291NGC 6715 (M 54) 283 . − . − . ± . − . ± . 025 0 . 01 0 . ± . 011 1122NGC 6717 (Pal 9) 283 . − . − . ± . − . ± . 026 0 . 03 0 . ± . 012 285NGC 6723 284 . − . 632 1 . ± . − . ± . 026 0 . 01 0 . ± . 011 2331NGC 6749 286 . 314 1 . − . ± . − . ± . 028 0 . 01 0 . ± . 012 256NGC 6752 287 . − . − . ± . − . ± . 022 0 . 00 0 . ± . 010 16348NGC 6760 287 . 800 1 . − . ± . − . ± . 027 0 . 01 0 . ± . 011 324NGC 6779 (M 56) 289 . 148 30 . − . ± . 025 1 . ± . 025 0 . 00 0 . ± . 011 1566Terzan 7 289 . − . − . ± . − . ± . 029 0 . 04 0 . ± . 015 116Pal 10 289 . 509 18 . − . ± . − . ± . 031 0 . 02 0 . ± . 015 107Arp 2 292 . − . − . ± . − . ± . 030 0 . 05 0 . ± . 021 156NGC 6809 (M 55) 294 . − . − . ± . − . ± . 024 0 . 00 0 . ± . 010 7482Terzan 8 295 . − . − . ± . − . ± . − . 03 0 . ± . 013 354Pal 11 296 . − . − . ± . − . ± . 028 0 . 04 0 . ± . 017 171NGC 6838 (M 71) 298 . 444 18 . − . ± . − . ± . 025 0 . 00 0 . ± . 011 3022NGC 6864 (M 75) 301 . − . − . ± . − . ± . 025 0 . 01 0 . ± . 012 367NGC 6934 308 . 547 7 . − . ± . − . ± . 026 0 . 01 0 . ± . 012 745NGC 6981 (M 72) 313 . − . − . ± . − . ± . 025 0 . 00 0 . ± . 012 764NGC 7006 315 . 372 16 . − . ± . − . ± . 026 0 . 00 0 . ± . 015 174Laevens 3 316 . 729 14 . 984 0 . ± . − . ± . 078 0 . 04 0 . ± . 088 21Segue 3 320 . 379 19 . − . ± . − . ± . − . − . ± . 121 21NGC 7078 (M 15) 322 . 493 12 . − . ± . − . ± . 024 0 . 00 0 . ± . 010 4697NGC 7089 (M 2) 323 . − . 823 3 . ± . − . ± . 024 0 . 01 0 . ± . 011 3092NGC 7099 (M 30) 325 . − . − . ± . − . ± . 025 0 . 01 0 . ± . 011 2989Pal 12 326 . − . − . ± . − . ± . 028 0 . 08 0 . ± . 018 185Pal 13 346 . 685 12 . 772 1 . ± . 047 0 . ± . 046 0 . − . ± . 034 72NGC 7492 347 . − . 611 0 . ± . − . ± . 027 0 . 02 0 . ± . 014 190MNRAS000