Using Gaia for Studying Milky Way Star Clusters

Home , IC 4499, NGC 4372, NGC 4833, NGC 5897, NGC 5986, NGC 6101

Using Gaia for studying Milky Way star clusters

Eugene Vasiliev

Institute of Astronomy, Cambridge Synopsis

I Overview of the Gaia mission and DR2: scientiﬁc instruments, catalogue contents, measurement uncertainties, caveats and limitations.

I 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.

I Galactic population of globular clusters: distribution of clusters in 6d phase space, orbits, inference on the Milky Way potential. Overview of Gaia mission G I Scanning the entire sky every couple of weeks BP RP RVS I Astrometry for sources down to 21 mag

I Broad-band photometry/low-res spectra

I Radial velocity down to ∼ 15 mag (end-of-mission)

[Source: ESA] Overview of Data Release 2 astrometry

I Based on 22 months of data collection

9 I Total number of sources: 1.69 × 10 RV I Sources with full astrometry (parallax $, 9 proper motions µα∗, µδ): 1.33 × 10 9 I Colours (GBP , GRP ): 1.38 × 10 6 I Radial velocities: 7.2 × 10 colours

6 108 I Eﬀective temperature: 160 × 10 Teff ICRF3 prototype 107 vrad SSO Variable Gaia DR1 Gaia-CRF2 Gaia DR2

6 n 6 i 10 b I Stellar parameters (R , L ): 77 × 10 g a 105 m

6 .

0 4

I Extinction and reddening: 88 × 10 r e p

r 103 e

6 b

Variable sources: 0.55 × 10 m I 2 u 10 N

101

100 5 10 15 20 25 Mean G [mag] [Brown+ 2018] Photometry G-band uncertainty [mag]

12 M 3 (NGC 5272), D=10 kpc 13

16 R-band uncertainty [mag] G 17

21 0.0 0.5 1.0 1.5

GBP GRP − [Evans+ 2018] Astrometry 10 parallax uncertainty [mas] 5 ] s a

m 2 [ ) $ ¾

( 1 x

12 a l l

a 0:5 r

M 3 (NGC 5272), D=10 kpc a p n i 0:2 y

13 t n i a t

r 0:1 e c n

u 0:05

14 d r a d n

a 0 02 t : S systematic error 15 0:01

0:005 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

16 ] 10 1 G magnitude ¡ r

y proper motion uncertainty [mas/yr] 5 s G a m [

17 ) x

a 2 m ; m p

¾ 1 ( 18 : m : 0:5 p n i e s

p 0:2 i l

19 l e r

o 0:1 r r e f

o 0:05

20 s i x

a systematic error r o

j 0:02 a m

21 ¡

i 0:01

0.0 0.5 1.0 1.5 m e 1 mas/yr = 4.7 km/s × (D/1 Kpc) S 0:005 GBP GRP 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 − G magnitude [Lindegren+ 2018] Spectroscopy

12 RV measurements only for stars with M 3 (NGC 5272), D=10 kpc Teﬀ ∈ [3500 ÷ 6900] K and GRVS ≤ 12 13

15 radial velocity uncertainty [km/s]

16 G 17

20 systematic error 21 0.0 0.5 1.0 1.5

GBP GRP − [Katz+ 2018] Correlations and systematic errors correlations between parallax and PM (ω Cen region) 1.0 1.0 1.0 1.0 µα µδ µα µδ ÷ ÷ ÷ 0.8 0.6 0.5 0.5 0.5 0.4

0.2

0.0 0.0 0.0 0.0

Y [deg] Y [deg] Y [deg] 0.2

0.4 0.5 0.5 0.5 0.6

0.8

1.0 1.0 1.0 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 1.0 0.5 0.0 0.5 1.0 X [deg] X [deg] X [deg] mean parallax and PM (Large Magellanic Cloud region)

0.10 0.05 0.00 0.05 0.10 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 8 8 8 µα µδ 6 6 6

4 ® 4 ® 4 ®

2 2 2

0 0 0 Y [deg] Y [deg] Y [deg] 2 2 2

4 4 4

6 6 6

8 8 8 8 6 4 2 0 2 4 6 8 8 6 4 2 0 2 4 6 8 8 6 4 2 0 2 4 6 8 X [deg] X [deg] X [deg] Limitations

I No special processing for binary stars (but a poor astrometric solution is marked clearly – astrometric excess noise)

I Colour photometry has lower spatial resolution (a quality control ﬂag is provided – phot bp rp excess factor)

I Poor completeness at faint magnitudes in crowded regions

I 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 $ < 0 [see Luri+ 2018 for a discussion])

I Should not generally use 1/$ as a proxy for distance (unless $/$ . 0.2)

[Arenou+ 2018] Example: NGC 5139 (ω Cen) 40 10 Sky map PM map All stars 30 5 20

10 ]

r 0 y / s

0 a m [

Y [arcmin] 5 10 µ

20 10 N=248494 30

40 15 100 101 102 103 40 30 20 10 0 10 20 30 40 20 15 10 5 0 5 X [arcmin] µα [mas/yr] 10 10 10 Color - magnitude diagram Magnitude vs. distance from center Magnitude vs. PM error

12 12 12

14 14 14

16 16 16 G [mag] G [mag] G [mag]

18 18 18

20 20 20

0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 25 30 35 40 0.05 0.1 0.2 0.5 1 2 4 BP-RP [mag] R [arcmin] δµα [mas/yr] Example: NGC 5139 (ω Cen) 40 10 Sky map PM map 30 Stars with full astrometry 5 20

($, µα, µδ)

10 ]

r 0 y / s

0 a m [

Y [arcmin] 5 10 µ

20 10 N=156227 30

40 15 100 101 102 103 40 30 20 10 0 10 20 30 40 20 15 10 5 0 5 X [arcmin] µα [mas/yr] 10 10 10 Color - magnitude diagram Magnitude vs. distance from center Magnitude vs. PM error

12 12 12

14 14 14

16 16 16 G [mag] G [mag] G [mag]

18 18 18

20 20 20

0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 25 30 35 40 0.05 0.1 0.2 0.5 1 2 4 BP-RP [mag] R [arcmin] δµα [mas/yr] Example: NGC 5139 (ω Cen) 40 10 Sky map PM map 30 5 Parallax cut: 20

$ − $0 < 3 δ$,

10 ]

r 0 y / s

$0 = 0.2 mas 0 a m [

Y [arcmin] 5 10 µ

20 10 N=146313 30

40 15 100 101 102 103 40 30 20 10 0 10 20 30 40 20 15 10 5 0 5 X [arcmin] µα [mas/yr] 10 10 10 Color - magnitude diagram Magnitude vs. distance from center Magnitude vs. PM error

12 12 12

14 14 14

16 16 16 G [mag] G [mag] G [mag]

18 18 18

20 20 20

0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 25 30 35 40 0.05 0.1 0.2 0.5 1 2 4 BP-RP [mag] R [arcmin] δµα [mas/yr] Example: NGC 5139 (ω Cen) 40 10 Sky map PM map 30 Cut on astrometric quality: 5 20 astrometric excess noise < 1 mas

10 ]

r 0 y / s renormalized unit weight error < 1.4 0 a m [

Y [arcmin] 5 10 µ

20 10 N=111336 30

40 15 100 101 102 103 40 30 20 10 0 10 20 30 40 20 15 10 5 0 5 X [arcmin] µα [mas/yr] 10 10 10 Color - magnitude diagram Magnitude vs. distance from center Magnitude vs. PM error

12 12 12

14 14 14

16 16 16 G [mag] G [mag] G [mag]

18 18 18

20 20 20

0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 25 30 35 40 0.05 0.1 0.2 0.5 1 2 4 BP-RP [mag] R [arcmin] δµα [mas/yr] Example: NGC 5139 (ω Cen) 40 10 Sky map PM map 30 Cut on photometric quality: 5 20 phot bp rp excess factor <

10 ]

r 0 y /

2 s

1.3 + 0.06 (GBP − GRP ) 0 a m [

Y [arcmin] 5 10 µ

20 10 N=47857 30

40 15 100 101 102 103 40 30 20 10 0 10 20 30 40 20 15 10 5 0 5 X [arcmin] µα [mas/yr] 10 10 10 Color - magnitude diagram Magnitude vs. distance from center Magnitude vs. PM error

12 12 12

14 14 14

16 16 16 G [mag] G [mag] G [mag]

18 18 18

20 20 20

0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 25 30 35 40 0.05 0.1 0.2 0.5 1 2 4 BP-RP [mag] R [arcmin] δµα [mas/yr] Example: NGC 5139 (ω Cen) 40 10 Sky map PM map 30 Cut on proper motions: 5 20 2 2 2 (µα − µα,0) + (µδ − µδ,0) < ∆µ ,

10 ]

r 0 y / s

µα,0 = −3.2 mas/yr, 0 a m [

µδ,0 = −6.7 mas/yr, Y [arcmin] 5 10 µ ∆µ = 2 mas/yr 20 10 N=22865 30

40 15 100 101 102 103 40 30 20 10 0 10 20 30 40 20 15 10 5 0 5 X [arcmin] µα [mas/yr] 10 10 10 Color - magnitude diagram Magnitude vs. distance from center Magnitude vs. PM error

12 12 12

14 14 14

16 16 16 G [mag] G [mag] G [mag]

18 18 18

20 20 20

0.0 0.5 1.0 1.5 2.0 0 5 10 15 20 25 30 35 40 0.05 0.1 0.2 0.5 1 2 4 BP-RP [mag] R [arcmin] δµα [mas/yr] Membership determination by proper motions 10 PM map

5 A hard cutoﬀ in PM space is not always possible ]

r 0 y

and is conceptually unsatisfactory. / s a m [

δ 5 A more mathematically well-grounded alternative: µ gaussian mixture modelling. 10

15 20 15 10 5 0 5 µα [mas/yr]

f (µi ) = qcl N (µi | µcl, Σcl;i ) + (1 − qcl) N (µi | µfg, Σfg;i )

exp − 1 (µ − µ)T Σ−1 (µ − µ) N (µ | µ, Σ) ≡ 2 √ , 2π det Σ where the mean PMs µ and dispersions Σ of the cluster and foreground distributions, and the fraction of cluster members qcl, are all inferred by maximizing the likelihood of the observed stellar PMs. Membership determination by proper motions

Take into account the measurement errors µα , µδ , ρµαµδ for each star i:

2 2 ! σcl(ri ) + µα ρµαµδ µα µδ Σcl;i = ρ σ2 (r ) + 2 µαµδ µα µδ cl i µδ

2 ! Sαα + µα Sαδ + ρµαµδ µα µδ Σfg;i = S + ρ S + 2 αδ µαµδ µα µδ δδ µδ

And allow for a spatially-dependent density of cluster members: qcl(ri ). N Pstars Maximize ln L ≡ ln f (µi ) by adjusting free parameters: i=1 µcl, µfg, Sαα, Sδδ, Sαδ, radius and normalization of qcl(r), normalization of σcl(r). Posterior membership probability for each star:

qcl(ri ) N (µi | µcl, Σcl;i ) pcl;i = qcl(ri ) N (µi | µcl, Σcl;i ) + [1 − qcl(ri )] N (µi | µfg, Σfg;i ) Examples of membership determination

10 40 [Fe/H]=-1.35, E(B-V)=0.12, D=5.2 kpc NGC 5139 (ω Cen) [23706 / 47698] µ =-3.24 0.039, µ =-6.72 0.039, σ =0.314 α ± δ ± µ 30 2 12 20 0

10 ] r 2 y / s

0 a

16 m [

4 G [mag] δ Y [arcmin] 10 µ

18 6 20

30 8 20

40 0.0 0.5 1.0 1.5 2.0 2.5 3.0 40 30 20 10 0 10 20 30 40 14 12 10 8 6 4 2 0 BP-RP [mag] X [arcmin] µα [mas/yr]

10 10 [Fe/H]=-0.70, E(B-V)=0.22, D=5.6 kpc NGC 6352 [2141 / 10967] µ =-2.17 0.057, µ =-4.40 0.057, σ =0.111 0 α ± δ ± µ

12 5 2

14 ] r

y 4 / s

0 a

16 m [

G [mag] δ Y [arcmin]

µ 6

18 5 8

20 10 10 0.0 0.5 1.0 1.5 2.0 2.5 3.0 10 5 0 5 10 6 4 2 0 2 BP-RP [mag] X [arcmin] µα [mas/yr] 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 σcl should be independent of the selected subset of stars (bright or faint).

4 G<20: data with errors 3 2 10 G<20: deconvolved G<20 G<18 G<20: bootstrapped G<18: data with errors 0 G<18: deconvolved G<18: bootstrapped 2

] 2 r 4 10 y / s a m [

δ 6 # of stars µ

1 10 10

14 10 8 6 4 2 0 2 4 6 8 10 8 6 4 2 0 2 4 6 8

µα [mas/yr] µα [mas/yr] Caveat: spatially correlated systematic errors

V (θij ) = hµi µj i, averaged over pairs of sources separated by angular distance θ. Large scale: quasars (∼ 0.5 × 106 sources across the entire sky). Small scale: stars in LMC (∼ 0.4 × 106 sources, subtract the mean PM).

0.0036 exp(−θ/0.25◦) (cov.fnc.1) V (θ) = 0.0008 exp(−θ/20◦) + 0.004 sinc(θ/0.5◦ + 0.25) (cov.fnc.2)

On small scales, the typical systematic error is pV (0) ∼ 0.06 mas/yr; ◦ on scales & 0.5 , it is ∼ 0.03 mas/yr. ]

2 4 0.10 ) r µα µα µδ µδ y Covariance function 1 / 0.004 − − 0.08 s 3 a Covariance function 2 ® ® m 0.06 ( [

Data for quasar pairs 2 ) 0.003

θ 0.04 (

V Data for pairs of LMC stars 1 n 0.02 o i t 0.002 c 0 0.00 n u f

Y [deg] 0.02

e 1 c 0.001 n 0.04 a i 2 r a 0.06 v o 3 c 0.000

0.08 M

P 4 0.10 0 1 2 3 4 5 6 4 3 2 1 0 1 2 3 44 3 2 1 0 1 2 3 4 angular distance θ [deg] X [deg] X [deg] How to properly account for correlated systematic errors N Likelihood function for the entire dataset (µ ≡ {µi }i=1):

L = N (µ | 1 µ, Σ)

 2  V (0) + 1 V (θ12) V (θ13) ··· V (θ1N )  V (θ ) V (0) + 2 V (θ ) ··· V (θ )  Σ =  21 2 23 2N   ......   . . . . .  2 V (θN1) V (θN2) V (θN3) ··· V (0) + N

(1..N are statistical errors of each datapoint). This is easily generalized to 2d case with 2 × 2 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). q 1 PN PN Alternatively, an unweighted estimate of the uncertainty on µ is N i=1 j=1 Σij . Systematic uncertainty on the mean PM mock systematic error maps 4 0.10 µ µ Generate many realizations of mock PM maps 3 − 0.08 ® 0.06 with the given spatial covariance, 2 for clusters with diﬀerent spatial extent, 0.04 1 0.02 and estimate the uncertainty in the mean PM. 0 0.00

Y [deg] 1 0.02 0.04 0.07 2

] 0.06

r cov.fnc.1, unweighted

y 3

/ 0.08

s 0.06 cov.fnc.1, error-weighted

a 4 0.10

m cov.fnc.2, unweighted 4 3 2 1 0 1 2 3 4 [

µ 0.05 X [deg]

cov.fnc.2, error-weighted f

o 1.0 0.10 µ

r µ

o 0.04 − 0.08 r

r ®

e 0.06 0.5 c i 0.03 0.04 t a 0.02 m e

t 0.02 0.0 0.00 s y

Y [deg] 0.02 s

S 0.01 0.04

M 0.5

R 0.06 0.00 0.08 -2 -1 0 1 2 10 10 10 10 10 1.0 0.10 angular size of the cluster [deg] 1.0 0.5 0.0 0.5 1.0 X [deg] Example: mock PM maps for NGC 5272 (M 3)

distance = 10 kpc; vlos = −150 km/s; σ = 6 km/s; vrot = 2 km/s; 4000 stars 0.015 0.015 0.045 20 distance spread 0.012 20 perspective expansion 0.012 20 intrinsic rotation

0.009 0.009 0.030

10 0.006 10 0.006 10 0.015 0.003 0.003

0 0.000 0 0.000 0 0.000

y [arcmin] 0.003 0.003 0.015 10 0.006 10 0.006 10

0.009 0.009 0.030

20 0.012 20 0.012 20 0.045 0.015 0.015 20 10 0 10 20 20 10 0 10 20 20 10 0 10 20 x [arcmin] x [arcmin] x [arcmin]

0.20 0.100 0.5

20 velocity dispersion 20 systematic errors 20 statistical errors 0.4 0.15 0.075 0.3 0.10 0.050 10 10 10 0.2 0.05 0.025 0.1

0 0.00 0 0.000 0 0.0

y [arcmin] 0.1 0.05 0.025 10 10 10 0.2 0.10 0.050 0.3 0.15 0.075 20 20 20 0.4

0.20 0.100 0.5 20 10 0 10 20 20 10 0 10 20 20 10 0 10 20 x [arcmin] x [arcmin] x [arcmin] 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

0. 155 0. 005 2. 707 0. 004 0. 168 0. 044 2. 681 0. 043 − ± − ± − ± − ± µα 0. 123 µδ 2. 628 µα 0. 123 µδ 2. 628 0.20 − − 0.20 − −

0.15 0.15 0.2 0.1 0.0 2.7 2.6 2.5 0.2 0.1 0.0 2.7 2.6 2.5 ] ] r r y y / 0.10 / 0.10 s s a a m m [ [

µ µ σ σ

, 0.05 , 0.05 µ µ

0.00 0.00

0.05 0.05

0 5 10 15 20 25 0 5 10 15 20 25 R [arcmin] R [arcmin] Internal kinematics of globular clusters from Gaia PM

Clear signature of rotation in ∼ 10 clusters: conﬁrming the results of Bianchini+ 2018 based on Gaia data.

0.25 0.6 NGC 6656 (M 22) NGC 5904 (M 5) 8 N=4171, D=3.2 kpc N=4476, D=7.6 kpc 8 0.20 0.5 ] ]

r r 6 ] ] y y

/ / 0.15 s s / / s 0.4 s

a 6 a m m k k m m 4 [ [ [ [

0.10 σ σ µ µ

0.3 σ σ , ,

t t , , v v t 4 t , ,

µ µ 2 r r

0.05 v v , ,

r 0.2 r µ µ 2 0.00 0 0.1

0.05 0.0 0 2 0 5 10 15 0 5 10 15 R [arcmin] R [arcmin] 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+ 2018, and complementing the HST measurements in central parts [Bellini+ 2014, Watkins+ 2015].

6 0.5 NGC 6397 0.25 NGC 7089 (M 2) N=10000, D=2.4 kpc 5 N=1296, D=10.5 kpc 0.4 0.20 10 ] ] r 4 r ] ] y y / / s s

/ 0.15 / s 0.3 s a a m m k k m 3 m [ [ [ [

0.10 5 σ σ µ µ

σ σ 0.2 , ,

t t , , v v t 2 t , ,

µ µ 0.05 r r

v v , , r r

µ 0.1 µ 1 0.00 0

0.0 0 0.05

0 5 10 15 20 0 2 4 6 8 10 12 R [arcmin] R [arcmin] Internal kinematics of globular clusters from Gaia PM

Mismatch between σlos and PM dispersion in central parts: likely due to crowding issues and aggressive sample cleanup.

0.6 NGC 104 (47 Tuc) NGC 6752 0.4 8 N=10000, D=4.4 kpc N=10000, D=4.2 kpc 10 ] ]

r 0.4 r 0.3 6 ] ] y y / / s s / / s s a a m m k k m 5 m [ [ [ [ 0.2 4 0.2 σ σ µ µ

σ σ , ,

t t , , v v t t , , µ µ r r

v v , , 0.1 2 r 0.0 0 r µ µ

0.0 0 0.2 5

0 5 10 15 20 25 30 0 5 10 15 20 R [arcmin] R [arcmin] Internal kinematics of globular clusters from Gaia PM

Overall scale mismatch between σlos and σµ: a prime method for measuring the distance from kinematic analysis.

0.6 4 NGC 6121 (M 4) 0.20 NGC 6838 (M 71) N=7538, D=2.0 kpc 5 N=1983, D=4.0 kpc 0.5 3 0.15

] 4 ] r r ] ] y 0.4 y / / s s / / s s

a a 2 3 m 0.10 m k k m m [ [

[ 0.3 [

σ σ µ µ

σ σ , ,

t t , , v v t 2 t 0.05 1 0.2 , , µ µ r r

v v , , r r µ 0.1 1 µ 0.00 0

0.0 0 0.05 1 0 5 10 15 20 0 1 2 3 4 5 6 7 8 R [arcmin] R [arcmin] Distribution of globular clusters in galactic coordinates

∼ 150 globular clusters in the Milky Way Distribution of globular clusters in distance and # of stars

47 Tuc ω Cen

NGC 6397 NGC 6752 104

M 4 NGC 3201

M 10

M 5 HST (Sohn+ 2018) M 22 M 55 M 13 M 3 M 12 Gaia (Helmi+ 2018) NGC 6362 M 71 NGC 288 NGC 6366 NGC NGC 6352 4372 M 15 Gaia (Vasiliev 2018) NGC 4833 NGC 5927 M 53 M 92 NGC 6723 NGC 2808 NGC 362

M 68 NGC 6541 M 30 M 2 M 107 M 62 NGC 6101 3 M 19 10 Pal 7 NGC 5897 NGC 5466 M 54 NGC NGC 6388 5986 M 56 NGC 1851 M 14 M 70 NGC 5053 NGC 6760 M 79 NGC 6356 NGC 6544 NGC 6539 NGC 1261 M 80 M 9 NGC 6139 NGC 2298 NGC 6749 NGC 5286 M 28 NGC 6584 IC 4499 Terzan 3 NGC 5824 NGC 6496 NGC 6553 NGC 6934 M 72 NGC NGC 6535 6712 NGC 6287 M 69 NGC 6441 Terzan 8 Pal 10 NGC 4147 NGC 6517 NGC 6652NGC 5946 M 75 NGC 6342 Pal 5 E 3 NGC 6293 NGC 5634 NGC 63806569 Pal 8Pal 11 NGC 6624 NGC 6235 NGC 6304 NGC 6440

Number of stars in Gaia NGC 6325 NGC 6284 NGC 6426 BH 184 Arp 2 NGC 7006 NGC 6256 NGC 2419 NGC 7492 NGC 6144 Rup 106 NGC 6229 Pismis 26 NGC 6316 BH 176 Terzan 12 2 Pal 12 NGC 5694 10 Pal 9 NGC 6642 NGC NGC 6355 6638 Terzan 7

Pal 2 Terzan 5 Terzan 1 FSR 1716 Pyxis Pal 15 Pal 14

Terzan 9 NGC Pal 6401 1 Pal 6 NGC NGC 6558 6522 BH 229 ESO280 Pal 13 Liller 1 Djorg 1 Terzan 10 ESO452 Terzan Terzan 4 2 NGC 6453

NGC 6528 Pal 4 FSR 1735 IC 1257 Eridanus NGC 6540 BH 261

Whiting 1 1 10 ESO456 Crater

AM 1

Pal 3 AM 4 Terzan 6 2 3 4 5 7 10 20 30 50 100 Heliocentric distance, Kpc Previous measurements of mean PM of globular clusters 10 10 10 10 HST Dinescu Chemel Kharchenko

5 5 5 5 ] r y /

s 0 0 0 0 a m [ r e t

i 5 5 5 5 l , α µ 10 10 10 10

15 15 15 15 15 10 5 0 5 10 15 10 5 0 5 10 15 10 5 0 5 10 15 10 5 0 5 10 µα, this [mas/yr] µα, this [mas/yr] µα, this [mas/yr] µα, this [mas/yr] 5 5 5 5

0 0 0 0 ] r y /

s 5 5 5 5 a m [ r e t

i 10 10 10 10 l , δ µ 15 15 15 15

20 20 20 20 20 15 10 5 0 5 20 15 10 5 0 5 20 15 10 5 0 5 20 15 10 5 0 5 µδ, this [mas/yr] µδ, this [mas/yr] µδ, this [mas/yr] µδ, this [mas/yr] 1.0 6 6 15 ] r 4 4 10 y /

s 0.5 a NGC 6838 M 71 2 2 5 m [ s i

h Terzan 5

Terzan 5 11 t

, 0.0 0 0 0 δ Terzan 8 µ NGC 6101 − Pyxis 2 2 5 r NGC 5466 e t

i 0.5 l ,

δ 4 4 10 µ

1.0 6 6 15 1.0 0.5 0.0 0.5 1.0 6 4 2 0 2 4 6 6 4 2 0 2 4 6 15 10 5 0 5 10 15 µ µ [mas/yr] µ µ [mas/yr] µ µ [mas/yr] µ µ [mas/yr] α, liter − α, this α, liter − α, this α, liter − α, this α, liter − α, this Distribution of globular clusters in position/velocity space

[Fe/H]

2 1 0 200

Crater Crater Crater

AM 1 AM 1 AM 1

Pal 4 Pal 4 Pal 4

100 Pal 3 Eridanus Eridanus Pal 3 Eridanus Pal 3 NGC 2419 NGC 2419 NGC 2419

Pal 14 Pal 14 Pal 14

Pyxis Pal 15 Pyxis Pyxis NGC 7006Pal 15 NGC 7006 NGC 7006Pal 15 Pal 2 Pal 2 Whiting 1 Pal 2 Whiting 1 Whiting 1

NGC 5694 NGC 6229 NGCNGC 6229 5694 NGC 5824 NGC 5694 Pal 13 Pal 13 Pal 13 NGC 6229 NGC 7492 NGC 7492 NGC 5824 NGC 5824 NGC 5634 NGC 7492 NGC 4147 NGCNGC 4147 5634 NGC 4147 Arp 2 Arp 2 Arp 2 M 79 NGC 1261M 79 NGC 5634 20 Rup 106 NGC 1261M 79 Terzan 8 NGCM 5053 53 Pal 5 M 53 Pal 5 M 53 M 54 NGC 1851 TerzanM 854 Pal 5 NGC 5053 NGC 1261 M 54 Rup 106 Terzan 8 IC 1257NGC 1851 NGC 5053 Pal 1 Pal 1 NGC 5466 IC 4499 Pal 1 NGC 1851 NGCIC 12572298 Rup 106 Pal 12 IC 1257 IC 4499 NGC 5466 NGC 2298 NGC 5466 Pal 12 NGC 2298 IC 4499 Pal 12 Terzan 7 NGC 6426 Terzan 7 BH 176 M 75 M 75 M 75 M 72 Terzan 7 NGC 6426 ESO280 NGC 6426

NGC 6101 ESO280 BH 176 ESO280 M 72 NGC 288 M 72 M 3 NGC 288 NGC 6934 NGCM 288 3 NGC 2808 NGC 6934 BH 176 M 3 NGC 6934 NGC 2808 M 15 NGC 6101 NGC 2808 M 2 M 2 M 15 NGC 6101 M 68 M 68 M 2 M 68 M 92 M 15 M 92 NGC 5286 10 NGC 362 NGCM 5286 92 M 56 E 3 NGC 362 M 56 NGC 5286 M 56 E 3 NGC 3201 NGC 362 E 3 NGC 3201 NGC 3201 M 13 NGC 6284 M 13 M 13 NGC 6356 47 TucPal 11 Pal 11NGC 5897 Pal 11 47 Tuc NGC 483347 Tuc M 30 M 30 M 30 NGC 4372 NGC 4372 M 71 NGC 4833 M 71 NGC 4372 NGC 4833 NGC 5897 r [kpc] NGC 5897 M 71 ω Cen ω Cen NGC 6356NGC 6284 Cen NGC 6284 NGC 6356 Pal 10 Pal 10 M 4 Pal 10 NGC 6584 ω NGC 6584 NGC 6397 NGC 6397 M 5 M 5 M 4 NGCM 6584 5 M 4 NGC 6752 NGC 6366NGC 6544 NGC 6397 NGC 5946 NGCPal 8 6362 M 22 NGC 5946 NGC 5946 M 22 Pal 8 NGC 6760 Pal 8 M 12 M 10 NGC 6362 NGC 6362 NGC 6749M 12 NGC 6752 NGC 6544 NGC 6760NGC 6752M 22 5 NGC 6544 NGC 6760 FSR 1735 NGC 6749 NGC 5927 NGC 6366 NGC 5927 NGC 6235 NGC 6366 M 10 M 1210 NGC 5986 FSR 1735 NGC 6749 NGC 5986 NGC 5986 NGC 5927 NGC 6517BH 184 BH 184 BH 184 M 55 FSR 1735 Pal 7 NGC 6535 Terzan 12 NGC 6352 M 80 M 14 NGC 6535 M 55 M 14 M 14 NGC 6535 M 55 Pal 7 NGC 6380 NGC 6496 NGC 6380 NGC 6712 NGC 6496 NGC 6235 NGC 6496 Pal 7 NGC 6712 M 107NGC 6235 NGC 6517 M 80 M 80NGC 6517 NGC 6388

NGC 6540 NGC 6712 NGC 6569 NGC 6388 NGC 6540 NGC 6256 NGC 6441

NGC 6441 NGC NGC6352 6453 NGC 6441 M 107 NGC 6388 NGC 6139 M 107 NGC 6139 NGC 6453 NGC 6453 NGC 6139 Terzan 12 M 28 NGC 6539 Terzan 12 NGC 6539 NGC 6352 NGC 6539 NGC 6652 NGC 6144

NGC 6304 NGC 6316 NGC 6380 M 28 NGC 6401 NGC 6144 M 28 NGC 6144NGC 6723 NGC 6553 Pal 9 NGC 6540 NGC 6569 NGC 6569 NGC 6553 NGC 6256 NGC 6723 NGC 6256 Pal 6 NGC 6723 Terzan 3 Terzan 3 Terzan 10 Terzan 3 Terzan 10 Pal 9 NGC 6652 Terzan 10 Pal 9 NGC 6638 NGC 6287 NGC 6638 NGC 6541 Pal 6 NGC 6541 NGC 6401 Pal 6 NGC 6652 NGC 6401 ESO452 NGC 6316 NGC 6316 NGC 6304 ESO452NGC 6304 NGC 6541 NGC 6293 M 70 NGC 6293 M 70

NGC 6553 M 70 ESO452M 62 2 NGC 6287 ESO456 BH 261 BH 261 NGC 6287 BH 261

NGC 6293 M 62 NGC 6638 M 62 ESO456 ESO456 M 9 Terzan 1 NGC 6642 NGCTerzan 1 6642 M 9 M 69 M 9 NGC 6642 NGC 6342 M 69 Terzan 5 M 19 NGC 6342Terzan 5 M 19 M 19 M 69 NGC 6355 NGC 6355 NGC 6342 NGC 6440 Pismis 26 Pismis 26 Terzan 9 Pismis 26 Terzan 9 NGC 6440 NGC 6624 NGC 6440 Terzan 1

Terzan 4 Terzan 6 Terzan 6 Terzan 6 NGC 6325 Terzan 5 NGC 6624 NGC 6558 NGC 6355 NGC 6624 NGC 6325 NGC 6325

Terzan 9 NGC 6558

Terzan 4 Terzan 4 1 NGC 6558

Terzan 2 Terzan 2 NGC 6522 Terzan 2 Liller 1 Liller 1 Liller 1 NGC 6522 NGC 6528 NGC 6528 NGC 6528 NGC 6522

BH 229 0.5 BH 229 BH 229 0 50 100 150 200 250 300 0 50 100 150 200 250 300 300 200 100 0 100 200 300 v [km/s] v vφ | r| | θ| Velocity dispersion and rotation proﬁles

Main kinematical features of the

entire population of globular clusters: 150 σr I Signiﬁcant overall rotation,

especially within the central 10 kpc 100 σ ] θ (more prominent for metal-rich clusters). s / m k

[ σφ

I Nearly isotropic dispersion at r < 10 kpc, σ

, 50 more radially anisotropic in outer parts; v

a population of ∼ 10 clusters on vφ 0 eccentric orbits [Myeong+ 2018]. vθ Correlated orbits (e.g., Sgr stream: I vr 50 M 54, Terzan 7, Terzan 8, Arp 2, Pal 12, Whiting 1). 1 10 100 r [kpc] Distribution of globular clusters in action space

polar (Jφ = 0)

inner Galaxy BH 229 M 69 outer Galaxy 1 2 5 10 20 50 100 NGC 6723

NGC 6624 Rcirc [Kpc] NGC 6342 NGC 5053 NGC 6144 c ) 0 ir Pal 5 NGC 6522 c = u r NGC 6539 la J M 80 Terzan 7 Terzan 8 ( r NGC 6325 r (J M 53 la Arp 2 ESO452 r u = c M 54 ir M 107 0 NGC 6235 ) NGC 6235 c Terzan 3 Whiting 1

NGC 6362

NGC 6652 NGC 5824

Pal 9 NGC 6453 NGC 5897 M 12 NGC 6569 NGC 6139

NGC 6401

M 62 47 Tuc M 3

FSR 1735 M 30 NGCM 19 6355 Pal 12

NGC 6752 NGC 6752 Pyxis

IC 4499

M 9

NGC 6528 M 55 E 3

NGC 6541 BHM 261 10 NGC 6558 NGC 6397 NGC 6397

M 70 Pal 11 NGC 6356

NGC 6293 prograde Pismis 26 NGC 6304 M 14

M 13

Pal 1 NGC 6638

Terzan 4

NGC 6440 NGC 288 NGC 7492 NGC 6388 NGC 5927 NGC 5927

NGC 6352 NGC 6287 M 68

M 71 M 15 Pal 8 NGC 5466 BH 176 NGC 6540 NGC 6553

BH 184 Terzan 12 retrograde Pal 7 NGC 6101 Pal 7 NGC 6496 NGC 6284 Pal 10 NGC 4372 NGC 6366

NGC 5634 NGC 6316 M 5

NGC 3201 M 28 M 22 NGC 6535 NGC 6642NGC 5986

Pal 13 NGC 6426

ESO280 M 92

ω ω

C NGC 6760 C e e n Pal 6 NGC 6441 n

Rup 106 Terzan 2 NGC 6712

NGC 6256 NGC 5946

NGC 2298 NGC 362 NGC 6584 NGC 6749 Jz i ESO456 n NGC 6517 ) - 0 p Terzan 9 polar M 75 la = NGC 1261 z n M 2 J prograde Pal 15 e ( NGC 4147 Terzan 5

( e NGC 6229 J Terzan 6 n M 56 z = NGC 6544 la J φ NGC 5286 p NGC 5694 0 NGC 6380 - M 72 ) n retrograde IC 1257 NGC 7006 Terzan 1 i

NGC 1851

M 79

NGC 6934 NGC 4833

NGC 2808

M 4 Pal 2 radial Jr radial (Jφ = 0) Dynamical modelling of the entire globular cluster population

Assume an equilibrium distribution function (in action space):

ηΓ/η η−B/η M J0 g(J) κJφ f (J) = 3 1 + 1 + 1 + tanh , (2π J0) h(J) J0 Jr + Jz + |Jφ|

g(J) ≡ gr Jr + gz Jz + (3 − gr − gz ) |Jφ|, h(J) ≡ hr Jr + hz Jz + (3 − hr − hz )|Jφ|,

and a potential – bulge, disk and a ﬂexible halo proﬁle:

r −γ r α(γ−β)/α ρ(r) = ρh 1 + . rh rh 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. Results: distribution of clusters

ﬂattening and density proﬁle velocity dispersion and anisotropy

1.0 1.0 0.8 0.8 0.6 0.6 R

β 0.4 /

z 0.4 0.2 0.2 0.0 0.0 0.2

data σr σθ σφ vφ 150 model

ρ r− ] s

∝ / 100 m ρ 1 k 3 [ r σ , v

ρ 50 ∝ r − 5

0.1 0 1 10 100 1 10 100 r [kpc] r [kpc] Results: constraints on the Milky Way potential

circular velocity curve Results are broadly consistent with 300 McMillan (2017) other studies based on globular clusters Bovy (2015) 250 total [Binney&Wong 2017, Sohn+ 2018, Watkins+ disc halo 200

2018, Posti&Helmi 2018, Eadie&Juric 2018]; ] s / m

k 150 the potential from McMillan(2017) is [ c r i c v acceptable, the one from Bovy(2015) 100 has too low rotation curve. 50 Clusters should be combined with 0 other dynamical tracers (dSph, 1 10 100 halo stars) for a more robust R [kpc] inference on the potential. Summary

I Gaia is an immense source of valuable data, and is complementary to other surveys

I It can be used to measure the internal kinematics of globular clusters (velocity dispersion, rotation)

I Motion of globular clusters in the Galaxy is an important probe of the Milky Way potential