Using Gaia for studying Milky Way star clusters

Transcription

1 Using Gaia for studying Milky Way star clusters Eugene Vasiliev Institute of Astronomy, Cambridge

2 Synopsis Overview of the Gaia mission and DR: scientific instruments, catalogue contents, measurement uncertainties, caveats and limitations. Using Gaia astrometry for studying internal kinematics of globular clusters: sample cleaning, membership determination, measurement of velocity dispersion and rotation in the presence of correlated systematic errors. Galactic population of globular clusters: distribution of clusters in 6d phase space, orbits, inference on the Milky Way potential.

3 Overview of Gaia mission Scanning the entire sky every couple of weeks Astrometry for sources down to 1 mag Broad-band photometry/low-res spectra G BP RP RVS Radial velocity down to 1 mag (end-of-mission) [Source: ESA]

4 Overview of Data Release astrometry Based on months of data collection Total number of sources: Sources with full astrometry (parallax ϖ, proper motions µ α, µ δ ): Colours (G BP, G RP ): Radial velocities: 7. 6 RV colours Effective temperature: 6 Stellar parameters (R, L ): 77 6 Extinction and reddening: 88 6 Variable sources:. 6 Number per.1 mag bin Teff vrad Variable Gaia-CRF ICRF3 prototype SSO Gaia DR1 Gaia DR 1 1 Mean [Brown+ ]

5 Photometry G-band uncertainty [mag] 1 13 M 3 (NGC 7), D= kpc 1 1 G 17 R-band uncertainty [mag] G BP G RP [Evans+ ]

6 Astrometry parallax uncertainty [mas] M 3 (NGC 7), D= kpc Standard uncertainty in parallax (¾$) [mas] 1 : : :1 : : :1 systematic error G G BP G RP Semi major axis of error ellipse in p:m: (¾pm;max) [mas yr 1 ] : G magnitude 1 : : :1 : : :1 proper motion uncertainty [mas/yr] systematic error 1 mas/yr =.7 km/s (D/1 Kpc) : G magnitude [Lindegren+ ]

7 Spectroscopy M 3 (NGC 7), D= kpc RV measurements only for stars with T eff [3 69] K and G RVS 1 radial velocity uncertainty [km/s] G G BP G RP systematic error [Katz+ ]

8 Correlations and systematic errors Y [deg] µ α X [deg] correlations between parallax and PM (ω Cen region) Y [deg] µ δ X [deg] Y [deg] µ α µ δ X [deg] mean parallax and PM (Large Magellanic Cloud region) Y [deg] X [deg] Y [deg] µα X [deg] Y [deg] µδ X [deg]

9 Limitations No special processing for binary stars (but a poor astrometric solution is marked clearly astrometric excess noise) Colour photometry has lower spatial resolution (a quality control flag is provided phot bp rp excess factor) Poor completeness at faint magnitudes in crowded regions Need to apply various filters to clean up the sample (but do it wisely, e.g., don t just throw away stars with negative parallaxes ϖ < [see Luri+ for a discussion]) Should not generally use 1/ϖ as a proxy for distance (unless ɛ ϖ /ϖ.) [Arenou+ ]

10 Example: NGC 139 (ω Cen) All stars 3 Sky map PM map Y [arcmin] µδ [mas/yr] N= Color - magnitude diagram 3 3 X [arcmin] Magnitude vs. distance from center 1 1 µα [mas/yr] Magnitude vs. PM error BP-RP [mag] R [arcmin] δµα [mas/yr]

11 Example: NGC 139 (ω Cen) Stars with full astrometry (ϖ, µ α, µ δ ) Y [arcmin] 3 Sky map µδ [mas/yr] PM map N= Color - magnitude diagram 3 3 X [arcmin] Magnitude vs. distance from center 1 1 µα [mas/yr] Magnitude vs. PM error BP-RP [mag] R [arcmin] δµα [mas/yr]

12 Example: NGC 139 (ω Cen) Parallax cut: ϖ ϖ < 3 δϖ, ϖ =. mas Y [arcmin] 3 Sky map µδ [mas/yr] PM map N= Color - magnitude diagram 3 3 X [arcmin] Magnitude vs. distance from center 1 1 µα [mas/yr] Magnitude vs. PM error BP-RP [mag] R [arcmin] δµα [mas/yr]

13 Example: NGC 139 (ω Cen) 3 Cut on astrometric quality: astrometric excess noise < 1 mas renormalized unit weight error < 1. Y [arcmin] Sky map µδ [mas/yr] PM map N= Color - magnitude diagram 3 3 X [arcmin] Magnitude vs. distance from center 1 1 µα [mas/yr] Magnitude vs. PM error BP-RP [mag] R [arcmin] δµα [mas/yr]

14 Example: NGC 139 (ω Cen) Cut on photometric quality: phot bp rp excess factor < (G BP G RP ) Y [arcmin] 3 Sky map µδ [mas/yr] PM map N= Color - magnitude diagram 3 3 X [arcmin] Magnitude vs. distance from center 1 1 µα [mas/yr] Magnitude vs. PM error BP-RP [mag] R [arcmin] δµα [mas/yr]

15 Example: NGC 139 (ω Cen) Cut on proper motions: (µ α µ α, ) + (µ δ µ δ, ) < µ, µ α, = 3. mas/yr, µ δ, = 6.7 mas/yr, µ = mas/yr N=86 Y [arcmin] 3 3 Sky map µδ [mas/yr] PM map 1 3 Color - magnitude diagram 3 3 X [arcmin] Magnitude vs. distance from center 1 1 µα [mas/yr] Magnitude vs. PM error BP-RP [mag] R [arcmin] δµα [mas/yr]

16 Membership determination by proper motions PM map A hard cutoff in PM space is not always possible and is conceptually unsatisfactory. A more mathematically well-grounded alternative: gaussian mixture modelling. µδ [mas/yr] 1 1 µα [mas/yr] f (µ i ) = q cl N (µ i µ cl, Σ cl;i ) + (1 q cl ) N (µ i µ fg, Σ fg;i ) N (µ µ, Σ) exp [ 1 (µ µ)t Σ 1 (µ µ) ] π det Σ where the mean PMs µ and dispersions Σ of the cluster and foreground distributions, and the fraction of cluster members q cl, are all inferred by maximizing the likelihood of the observed stellar PMs.,

17 Membership determination by proper motions Take into account the measurement errors ɛ µα, ɛ µδ, ρ µαµ δ for each star i: ( ) σ cl (r i ) + ɛ µ α ρ µαµ δ ɛ µα ɛ µδ Σ cl;i = ρ µαµ δ ɛ µα ɛ µδ σcl (r i) + ɛ µ δ ( ) S αα + ɛ µ α S αδ + ρ µαµ δ ɛ µα ɛ µδ Σ fg;i = S αδ + ρ µαµ δ ɛ µα ɛ µδ S δδ + ɛ µ δ And allow for a spatially-dependent density of cluster members: q cl (r i ). Maximize ln L Nstars ln f (µ i ) by adjusting free parameters: i=1 µ cl, µ fg, S αα, S δδ, S αδ, radius and normalization of q cl (r), normalization of σ cl (r). Posterior membership probability for each star: p cl;i = q cl (r i ) N (µ i µ cl, Σ cl;i ) q cl (r i ) N (µ i µ cl, Σ cl;i ) + [1 q cl (r i )] N (µ i µ fg, Σ fg;i )

18 Examples of membership determination [Fe/H]=-1.3, E(B-V)=.1, D=. kpc BP-RP [mag] [Fe/H]=-.7, E(B-V)=., D=.6 kpc Y [arcmin] Y [arcmin] 3 3 NGC 139 (ω Cen) [376 / 7698] 3 X [arcmin] NGC 63 [11 / 967] 3 µδ [mas/yr] µδ [mas/yr] µα=-3.±.39, µδ=-6.7±.39, σµ= µα [mas/yr] µα=-.17±.7, µδ=-.±.7, σµ= BP-RP [mag] X [arcmin] 6 µα [mas/yr]

19 Inferring the internal velocity dispersion The intrinsic PM distribution is broadened by observational errors, but if they are properly taken into account, the inferred PM dispersion σ cl should be independent of the selected subset of stars (bright or faint). G< G< 3 G<: data with errors G<: deconvolved G<: bootstrapped G<: data with errors G<: deconvolved G<: bootstrapped µδ [mas/yr] 6 # of stars µα [mas/yr] µα [mas/yr]

20 Caveat: spatially correlated systematic errors V (θ ij ) = µ i µ j, averaged over pairs of sources separated by angular distance θ. Large scale: quasars (. 6 sources across the entire sky). Small scale: stars in LMC (. 6 sources, subtract the mean PM). [.36 exp( θ/. V (θ) =.8 exp( θ/ ) (cov.fnc.1) ) +. sinc(θ/. +.) (cov.fnc.) PM covariance function V(θ) [(mas/yr) ] On small scales, the typical systematic error is V ().6 mas/yr; on scales., it is.3 mas/yr. µα µ α µδ µ δ Covariance function 1 Covariance function Data for quasar pairs Data for pairs of LMC stars angular distance θ [deg] Y [deg] X [deg] X [deg]

21 How to properly account for correlated systematic errors Likelihood function for the entire dataset (µ {µ i } N i=1 ): L = N (µ 1 µ, Σ) V () + ɛ 1 V (θ 1 ) V (θ 13 ) V (θ 1N ) V (θ 1 ) V () + ɛ V (θ 3 ) V (θ N ) Σ = V (θ N1 ) V (θ N ) V (θ N3 ) V () + ɛ N (ɛ 1..ɛ N are statistical errors of each datapoint). This is easily generalized to d case with covariance matrices of statistical errors, and allowing for spatially-dependent internal dispersion and mean value of µ. The downside is that one needs to invert the N N covariance matrix Σ for the entire dataset (for the optimal error-weighted estimate). N Alternatively, an unweighted estimate of the uncertainty on µ is 1 N N i=1 j=1 Σ ij.

22 Systematic uncertainty on the mean PM RMS systematic error of µ [mas/yr] Generate many realizations of mock PM maps with the given spatial covariance, for clusters with different spatial extent, and estimate the uncertainty in the mean PM cov.fnc.1, unweighted cov.fnc.1, error-weighted cov.fnc., unweighted cov.fnc., error-weighted angular size of the cluster [deg] Y [deg] Y [deg] mock systematic error maps µ µ µ µ X [deg] X [deg]

23 Example: mock PM maps for NGC 7 (M 3) y [arcmin] distance = kpc; v los = 1 km/s; σ = 6 km/s; v rot = km/s; stars distance spread x [arcmin] perspective expansion x [arcmin] intrinsic rotation x [arcmin] y [arcmin] velocity dispersion x [arcmin] systematic errors x [arcmin] statistical errors x [arcmin]

24 Spatial correlation should not be ignored! Doing so underestimates the error bars on fit parameters, even when systematic errors are much smaller than statistical errors. ignoring correlations taking correlations into account.. 1 ±. µ α ±. µ δ ±. µ α ±. 3 µ δ µ, σµ [mas/yr].. µ, σµ [mas/yr] R [arcmin] 1 R [arcmin]

25 Internal kinematics of globular clusters from Gaia PM Clear signature of rotation in clusters: confirming the results of Bianchini+ based on Gaia data..6 NGC 666 (M ) N=178, D=3. kpc. 8. NGC 9 (M ). N=, D=7.6 kpc 8 µt, σµ [mas/yr] vt, σ [km/s] µt, σµ [mas/yr] vt, σ [km/s]. 1 R [arcmin]. 1 R [arcmin]

26 Internal kinematics of globular clusters from Gaia PM Good match between line-of-sight velocity dispersion and PM dispersion: confirming the analysis of Gaia PM dispersion profiles by Baumgardt+, and complementing the HST measurements in central parts [Bellini+ 1, Watkins+ 1]. µt, σµ [mas/yr]. NGC N=, D=. kpc vt, σ [km/s] µt, σµ [mas/yr]. NGC 789 (M ) N=139, D=. kpc vt, σ [km/s].. 1 R [arcmin] R [arcmin]

27 Internal kinematics of globular clusters from Gaia PM Mismatch between σ los and PM dispersion in central parts: likely due to crowding issues and aggressive sample cleanup..6 NGC (7 Tuc) N=, D=. kpc...3 NGC 67 N=, D=. kpc 8 6 µt, σµ [mas/yr].. vt, σ [km/s] µt, σµ [mas/yr]..1 vt, σ [km/s] R [arcmin] 1 R [arcmin]

28 Internal kinematics of globular clusters from Gaia PM Overall scale mismatch between σ los and σ µ : a prime method for measuring the distance from kinematic analysis. µt, σµ [mas/yr] NGC 611 (M ) N=766, D=. kpc 3 vt, σ [km/s] µt, σµ [mas/yr]. NGC 6838 (M 71) N=36, D=. kpc vt, σ [km/s] R [arcmin] R [arcmin] 1

29 Distribution of globular clusters in galactic coordinates 1 globular clusters in the Milky Way

30 Distribution of globular clusters in distance and # of stars 7 Tuc ω Cen NGC 6397 NGC 67 M NGC 31 Number of stars in Gaia 3 1 M NGC 6 NGC 6366 M 71 M M 1 Terzan 1 M Pal 7 NGC NGC M 8 NGC 6 NGC 63 Pal 6 NGC 63 Pal Terzan M M 13 NGC 88 M 3 M 1 NGC 833 NGC 97 M 9 NGC 673 NGC 88 M 3 NGC 36 M 7 M 6 Terzan Terzan 1 BH 61 ESO6 NGC NGC Pal 9 Terzan 9 Terzan 6 NGC 636 NGC 61 M 3 NGC 676 NGC 639 M 9 NGC 679 Terzan 3 M 19 M 68 M 6 NGC NGC M 7 M 1 NGC Pal 61 1 NGC 1 M 8 NGC 6139 NGC 98 NGC 86 NGC 897 M 79 NGC 66 NGC E 3 NGC 63 NGC 693 NGC 66 NGC NGC Pal 8 Pal 11 NGC 6 63 NGC BH 63 NGC 66 NGC 61 Pismis 6 NGC 63 NGC NGC 68 6 BH 9 Liller 1 ESO Terzan Terzan NGC 68 M 69 NGC 66 NGC 687 NGC NGC FSR 17 M NGC 696 FSR 173 NGC 61 NGC 63 NGC 68 Djorg Heliocentric distance, Kpc NGC 61 NGC 66 NGC 636 NGC 1 NGC 693 M 7 NGC 68 NGC 3 IC 99 NGC 17 M 7 BH 176 Pal 1 NGC 66 Rup 6 ESO8 Pal Terzan 7 M NGC 63 IC 17 Terzan 8 Pal Whiting 1 NGC 8 Arp NGC 79 NGC 69 Pal 13 AM NGC 69 Pyxis NGC 76 Pal 1 HST (Sohn+ ) Gaia (Helmi+ ) Gaia (Vasiliev ) Pal 1 NGC 19 Eridanus Pal 3 Pal AM 1 Crater

31 Previous measurements of mean PM of globular clusters µα, liter [mas/yr] HST 1 1 µα, this [mas/yr] Dinescu 1 1 µα, this [mas/yr] Chemel 1 1 µα, this [mas/yr] Kharchenko 1 1 µα, this [mas/yr] µδ, liter [mas/yr] Terzan 11 µδ, liter µδ, this [mas/yr] 1 µδ, this [mas/yr] Terzan Terzan 8 NGC 6838 M 71 NGC 61 Pyxis NGC µα, liter µα, this [mas/yr] 1 µδ, this [mas/yr] µα, liter µα, this [mas/yr] 1 µδ, this [mas/yr] µα, liter µα, this [mas/yr] 1 µδ, this [mas/yr] µα, liter µα, this [mas/yr]

32 NGC 6 NGC 669 Terzan 1 Terzan M 7 NGC 638 Terzan 1 M 6 NGC 63 NGC 63 Terzan 9 NGC 63 Distribution of globular clusters in position/velocity space [Fe/H] 1 Crater Crater Crater NGC 19 Pal AM 1 Pal 3 Eridanus Pal Eridanus NGC 19 AM 1 Pal 3 AM 1 Pal Eridanus NGC 19 Pal 3 Pal 1 Pal 1 Pal 1 r [kpc] 1. NGC 3 7 Tuc E 3 NGC 37 NGC 6 M 1 NGC 696 M 8 M 1 Pal 11 Pal 8 NGC 61 Terzan 3 NGC 63 M 79 BH 176 NGC 88 M 13 NGC 636 NGC 97 NGC 617 NGC 61 Pal 9 Pal 1 Pal Pal 1 M 71 M 8 M 9 NGC 69 IC 17 ESO8 M NGC 96 M 1 NGC 63 NGC 63 NGC 68 NGC 6397 NGC 67 NGC 679 NGC 986 BH M 7 NGC 638 NGC 6139 NGC 61 Pal 7 NGC 63 M NGC 63 NGC 6388 NGC 639 NGC 61 NGC 673 NGC 6638 Terzan NGC 63 Pal NGC 6 M 6 NGC 63 NGC 66 NGC 63 Terzan 9 NGC 636 ω Cen M 9 NGC 6366 NGC 687 M 69 NGC 68 NGC 79 M 3 ESO FSR 173 Arp NGC 1 NGC 98 NGC 31 NGC 676 Terzan NGC 63 NGC 63 Terzan 1 Pal NGC 66 NGC 8 M 7 M 3 BH 61 Liller 1 NGC 17 NGC 1 M 68 NGC 833 NGC 897 NGC 66 Terzan Terzan 8 Terzan 7 NGC 63 M 3 NGC 88 NGC 36 NGC 66 M 19 NGC 68 M M Terzan 6 Pal 1 Whiting 1 NGC 671 NGC 61 BH 9 NGC 76 ESO6 NGC 6 IC 99 NGC 86 NGC 69 NGC v r [km/s] M 6 NGC 66 Pal 6 M Pismis 6 Pyxis Rup 6 NGC 693 M Pal 13 M 7 M 7 M NGC 68 NGC 693 NGC 1 IC 17 BH 176 NGC 88 M NGC 679 NGC 17 NGC 69 NGC 63 M 79 NGC 97 NGC 669 NGC 63 Pal NGC 69 NGC 1 NGC 66 NGC 693 NGC 676 ESO6 Pal 7 Liller 1 NGC 88 NGC 36 NGC 833 NGC 63 NGC 61 NGC 63 NGC 61 NGC 693 NGC 66 Terzan 6 Terzan NGC 76 Pal 11 Pal 9 Pal 1 M 7 M 1 M M 3 NGC 897 Pal Pal 1 M 3 Pal 8 M 1 NGC 6388 NGC 63 NGC 63 Pal 7 Tuc M 71 NGC 617 ESO M 3 NGC 98 ESO8 NGC 86 NGC 986 Terzan M 7 NGC 67 NGC 66 NGC 37 ω Cen M 8 M 1 NGC 61 M 9 NGC 79 M 9 NGC 6 NGC 6638 M 69 M 13 NGC 6366 M NGC 696 NGC 66 NGC 6 Terzan 9 Rup 6 M 6 NGC 3 NGC 68 NGC 636 NGC 6 BH NGC 66 NGC 68 NGC 96 M 68 NGC 636 Terzan 1 M 8 NGC 673 Terzan Pal 13 E 3 NGC 6397 NGC 63 BH 61 NGC 66 NGC 63 NGC 6 M NGC 671 Terzan 3 NGC 6139 M 19 Whiting 1 NGC 8 M 7 NGC 68 IC 99 NGC 31 M Terzan FSR 173 BH v θ M NGC 639 NGC 61 Pismis 6 Pal 6 Pal 1 NGC 61 NGC 63 NGC 68 Pyxis NGC 63 Arp Terzan 8 M Terzan 7 NGC 687 NGC 61 NGC 31 NGC 61 M 19 NGC 61 NGC 66 NGC 673 NGC 63 Pal 13 IC 99 ω Cen NGC 63 NGC 6388 NGC 693 NGC 69 NGC 1 NGC 98 IC 17 NGC 88 NGC 86 M 3 Terzan 6 NGC 76 NGC 687 M 13 NGC 638 NGC 6 Terzan Liller 1 BH 9 Pyxis NGC 69 NGC 17 NGC 1 M FSR 173 ESO Pal 1 Pal NGC 79 M 79 M 7 M 6 M 9 NGC 36 NGC 68 M NGC v φ ESO8 NGC 63 NGC 88 NGC 833 NGC 96 NGC 986 NGC 63 M 8 NGC 671 Pal 6 M 7 NGC 66 NGC 6638 NGC 617 NGC 66 Terzan 8 Terzan 7 M 8 NGC 66 M 1 M 7 Terzan 1 Terzan Arp M 7 NGC 63 NGC 66 M M NGC 61 Terzan NGC 6139 M 69 NGC 68 Whiting 1 Rup 6 NGC 8 M 3 NGC 3 NGC 66 NGC 693 M 3 NGC 37 NGC 897 NGC 6 NGC 636 NGC 68 M NGC 6397 NGC 636 M NGC 679 M 6 NGC 6366 M 1 BH NGC 6 NGC 639 Pal 9 Terzan M 1 Pal 8 Pal 11 NGC 676 ESO6 NGC 68 Pal 7 Tuc Pal M NGC 67 Terzan 1 NGC 63 NGC 63 M 71 NGC 669 Pal 1 BH 61 E 3 NGC 97 NGC 63 Terzan 3 NGC 61 Pismis 6 NGC 63 BH 176 Pal 7 M 9 M 68 Pal 1 NGC 696

33 Velocity dispersion and rotation profiles Main kinematical features of the entire population of globular clusters: Significant overall rotation, especially within the central kpc (more prominent for metal-rich clusters). Nearly isotropic dispersion at r < kpc, more radially anisotropic in outer parts; a population of clusters on eccentric orbits [Myeong+ ]. Correlated orbits (e.g., Sgr stream: M, Terzan 7, Terzan 8, Arp, Pal 1, Whiting 1). v, σ [km/s] 1 σ r σ θ σ φ v φ v r v θ 1 r [kpc]

34 M 3 Pal 1 M Distribution of globular clusters in action space inner Galaxy circular (Jr = ) NGC 61 NGC 63 polar (Jφ = ) BH 9 NGC 673 ESO M 8 NGC 66 M 69 NGC 6 M 7 NGC 63 1 R circ [Kpc] NGC 639 circular (Jr = ) Terzan 3 Terzan 7 Terzan 8 M Arp NGC 3 M 3 Pal outer Galaxy NGC 66 NGC 63 NGC 6139 NGC 63 Pal 9 Whiting 1 NGC 636 M 1 NGC 897 NGC 8 NGC 63 NGC 61 M 19 NGC 63 M 6 retrograde in-plane (Jz = ) NGC 6388 NGC 63 ω Cen Terzan 6 Terzan NGC 6 NGC 638 NGC 693 NGC 687 FSR 173 NGC 6 M 7 NGC 6638 NGC 66 Pal 6 NGC 671 NGC 986 NGC 66 NGC 68 NGC 68 NGC 96 Terzan 9 M M 1 NGC 617 M 9 NGC 63 M 8 NGC 61 Pismis 6 NGC 61 Terzan Terzan 1 Terzan BH 61 M NGC 6397 ESO6 NGC 6366 Pal 8 NGC 679 in-plane (Jz = ) NGC 669 Pal 1 M 3 7 Tuc BH NGC 67 Terzan 1 NGC 63 NGC 676 retrograde Pal 7 NGC 6 NGC 63 NGC 97 NGC 63 prograde J z polar prograde J φ NGC 31 NGC 61 ω Cen IC 99 NGC 88 Pal 13 NGC 98 NGC 86 IC 17 M 13 NGC 76 NGC 66 NGC 1 NGC 69 Pyxis M NGC 1 NGC 79 M 6 NGC 36 NGC 17 M 7 NGC 69 NGC 68 M 9 NGC 63 M 7 ESO8 NGC 68 Rup 6 M 68 NGC 636 M 1 M NGC 6397 NGC 696 NGC 66 Pal 11 NGC 37 NGC 67 E 3 Pal 7 Pal Pal 1 NGC 97 M 71 BH 176 M radial (Jφ = ) radial J r Pal M 79 NGC 693 NGC 833 NGC 88

35 Dynamical modelling of the entire globular cluster population Assume an equilibrium distribution function (in action space): f (J) = [ ( ) η ] Γ/η [ M J (π J ) h(j) ( g(j) J ) η ] B/η ( ) κj φ 1 + tanh, J r + J z + J φ g(j) g r J r + g z J z + (3 g r g z ) J φ, h(j) h r J r + h z J z + (3 h r h z ) J φ, and a potential bulge, disk and a flexible halo profile: ( ) γ ( ) α ] (γ β)/α r r ρ(r) = ρ h [1 +. r h Maximize the likelihood of drawing the observed positions and velocities of clusters (taking into account their uncertainties) by varying the parameters of potential and DF. r h

36 Results: distribution of clusters flattening and density profile velocity dispersion and anisotropy z/r data model β σ r σ θ σ φ v φ r 3 ρ 1 ρ r 3 ρ r v, σ [km/s].1 1 r [kpc] 1 r [kpc]

37 Results: constraints on the Milky Way potential Results are broadly consistent with other studies based on globular clusters [Binney&Wong 17, Sohn+, Watkins+, Posti&Helmi, Eadie&Juric ]; the potential from McMillan(17) is acceptable, the one from Bovy(1) has too low rotation curve. Clusters should be combined with other dynamical tracers (dsph, halo stars) for a more robust inference on the potential. vcirc [km/s] 3 1 circular velocity curve McMillan (17) Bovy (1) total disc halo 1 R [kpc]

38 Summary Gaia is an immense source of valuable data, and is complementary to other surveys It can be used to measure the internal kinematics of globular clusters (velocity dispersion, rotation) Motion of globular clusters in the Galaxy is an important probe of the Milky Way potential