Dynamical Modelling of Stellar Systems in the Gaia Era

Home , FSR 1758, IC 4499, Milky Way (mythology), NGC 4372, NGC 5824, NGC 5897

Dynamical modelling of stellar systems in the Gaia era

Eugene Vasiliev

Institute of Astronomy, Cambridge Synopsis

Overview of dynamical modelling

Overview of the Gaia mission

Examples: Large Magellanic Cloud Globular clusters Measurement of the Milky Way gravitational potential Fred Hoyle vs. the Universe

What does “dynamical modelling” mean?

It does not refer to a simulation (e.g. N-body) of the evolution of a stellar system. Most often, it means “modelling a stellar system in a dynamical equilibrium” (used interchangeably with “steady state”). vs. the Universe

What does “dynamical modelling” mean?

Fred Hoyle What does “dynamical modelling” mean?

Fred Hoyle vs. the Universe 3D Steady-state assumption =⇒ Jeans theorem:

f (x, v)= f I(x, v;Φ)

observations: 3D – 6D integrals of motion (≤ 3D?), e.g., I = {E, L,... }

Why steady state?

Distribution function of stars f (x, v, t) satisﬁes [sometimes] the collisionless Boltzmann equation:

∂f (x, v, t) ∂f (x, v, t) ∂Φ(x, t) ∂f (x, v, t) + v − = 0. ∂t ∂x ∂x ∂v

Potential ⇔ mass distribution ∂f (x, v, t) , t , t , t + ∂t

observations: 3D – 6D integrals of motion (≤ 3D?), e.g., I = {E, L,... }

Why steady state?

Distribution function of stars f (x, v, t) satisﬁes [sometimes] the collisionless Boltzmann equation:

∂f (x, v ) ∂Φ(x ) ∂f (x, v ) v − = 0. ∂x ∂x ∂v

Steady-state assumption =⇒ Jeans theorem:

f (x, v)= f I(x, v;Φ) ∂f (x, v, t) , t , t , t + ∂t

Why steady state?

Distribution function of stars f (x, v, t) satisﬁes [sometimes] the collisionless Boltzmann equation:

∂f (x, v ) ∂Φ(x ) ∂f (x, v ) v − = 0. ∂x ∂x ∂v

3D Steady-state assumption =⇒ Jeans theorem:

f (x, v)= f I(x, v;Φ)

observations: 3D – 6D integrals of motion (≤ 3D?), e.g., I = {E, L,... } Flavours of dynamical modelling

Tracers individual stars integrated light Photometry star counts surface brightness proﬁle

Kinematics line-of-sight velocities (vLOS), LOS velocity distribution f (vLOS), proper motions (PM) v, σ, Gauss–Hermite moments

Methods Jeans eqns; distribution functions f (I); orbit superposition ZZZ Dynamical ∇2Φ(x) = 4π G ρ(x) = 4π G f (x, v) d 3v self-consistency?

Outcome total potential Φ; dark matter distribution; internal kinematics – All-purpose galaxy modeling architecture

I Extensive collection of gravitational potential models (analytic profiles, azimuthal- and spherical-harmonic expansions) constructed from smooth density profiles or N-body snapshots; I Conversion to/from action/angle variables; I Self-consistent multicomponent models with action-based DFs; I Schwarzschild orbit-superposition models; I Generation of initial conditions for N-body simulations; I Various math tools: 1d,2d,3d spline interpolation, penalized spline fitting and density estimation, multidimensional sampling; I Efficient and carefully designed C++ implementation, examples, Python and Fortran interfaces, plugins for Galpy, NEMO, AMUSE.

arXiv:1802.08239, 1802.08255, 1912.04288; http://agama.software Astrometry 101

Position on the sky α, δ Parallax $ = 1/distance Proper motion µα, µδ Line-of-sight velocity vlos Binary orbit parameters How Gaia astrometry works Overview of Gaia Data Release 2 Scanning the entire sky every couple of weeks I G I DR2 based on 22 months of observations I Astrometry for sources down to 21 mag (1.3 × 109) BP RP RVS I Broad-band blue/red photometry (1.4 × 109) I Radial velocity down to ∼ 13 mag (∼ 7 × 106 stars) ! No special treatment of binary stars

] 10

! Poor completeness in crowded ﬁelds 1 ¡

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

8 m

10 [ )

Teff ICRF3 prototype x

a 2 m 7 vrad SSO ;

10 m Variable Gaia DR1 p

¾ 1 Gaia-CRF2 Gaia DR2 ( :

n 6

i 10 m : b 0 5

: p g n a 105 i e m s

1 0:2 i . l l

0 4 e 10 r r e o 0:1 r p r

r 3

10 f e o

b 0:05 s i m

2 x u

10 a systematic error N r o

j 0:02 1 a

10 m ¡ i 0:01 1 mas/yr = 4.7 km/s × (D/1 Kpc)

0 m 10 e S 0:005 5 10 15 20 25 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 Mean G [mag] G magnitude [Brown+ 2018] [Lindegren+ 2018] Example 1: Large Magellanic Cloud Example 1: Large Magellanic Cloud

[credit: ESA/Gaia/DPAC, 25/04/2018] Gaia DR2 stars in the Large Magellanic Cloud

12 12 12 All stars Filtered by parallax and 13 13 13 radial astrometric excess noise velocities 14 14 14

15 15 15

16 16 16 Foreground LMC

G [mag] 17 17 17

18 18 18

19 19 19

20 20 20 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 BP-RP [mag] BP-RP [mag] BP-RP [mag] 25 15 8

20 6 15 10 ] r 10 y 4 / s

a 5 5 m [ 2 Y 0 µ 5 0 0 10

15 5 2 20 15 10 5 0 5 10 15 20 10 5 0 5 10 4 2 0 2 4 6 µX [mas/yr] µX [mas/yr] µX [mas/yr] Gaia DR2 stars in the Large Magellanic Cloud

12 107 12 12 All stars Filtered by parallax and 13 6 13 13 radial 10 astrometric excess noise velocities 14 14 all 14 105 astrometry 15 15 15 104 red giants 16 16 16 3 Foreground LMC G [mag] 17 10 17 17 2 combined 18 number of stars 10 18 18 19 101 19 19 10 5 0 5 10 20 20 20 0.5 0.0 0.5 1.0 1.5 2.0 µ2.5Y, m3.0as/y0.5r 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 BP-RP [mag] BP-RP [mag] BP-RP [mag] 25 15 8 104 20 data with errors 3 6 15 10 deconvolved 10 ] r 10 bootstrapped y 2 4 /

s 10

a 5 5 m

[ 1 2 Y 0 10 µ

5 0 typical 0errors: number of stars 10 best 5% 0 median 10 worst 5% 10-1 15 0.5 0.0 0.5 1.0 1.55 2.0 2.5 3.0 2 20 15 10 5 0 5 10 15 20 10 5 0 5 10 4 2 0 2 4 6 µ , mas/yr µX [mas/yr] X µX [mas/yr] µX [mas/yr] Internal kinematics of the Large Magellanic Cloud

8 520 8 160 8 40 8 40 6 µX 500 6 µY 140 6 σX 6 σY 4 480 4 120 4 30 4 30 2 460 2 100 2 2 0 440 0 80 0 20 0 20

Y [deg] 2 420 2 60 2 2 4 400 4 40 4 10 4 10 observations 6 380 6 20 6 6 8 360 8 0 8 0 8 0 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 8 6 4 2 0 2 4 6 8 X [deg] 8 8 105 8 0.8 340 6 vLOS 6 Σ 6 r 0.6 320 4 4 4 0.4 300 4 2 2 10 2 0.2 280 0 0 0 0.0 260 Y [deg] 2 2 3 2 0.2 240 10 4 4 4 0.4 220 6 6 6 0.6 200 8 8 102 8 0.8 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] Internal kinematics of the Large Magellanic Cloud

8 520 8 160 8 40 8 40 6 µX 500 6 µY 140 6 σX 6 σY 4 480 4 120 4 30 4 30 2 460 2 100 2 2 0 440 0 80 0 20 0 20

Y [deg] 2 420 2 60 2 2 4 400 4 40 4 10 4 10 observations 6 380 6 20 6 6 8 360 8 0 8 0 8 0 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 8 6 4 2 0 2 4 6 8 X [deg] 8 520 8 116005 8 400.8 8 40 340 6 vµLXOS 500 6 ΣµY 140 6 σrX 0.6 6 σY 320 4 480 4 120 4 300.4 4 30 300 4 2 460 2 10010 2 0.2 2 280 0 440 0 80 0 200.0 0 20 260 Y [deg] Y [deg] 2 420 2 603 2 0.2 2 240 10 4 400 4 40 4 100.4 4 10 220 6 380 6 20 6 0.6 6 200 8 360 8 1002 8 00.8 8 0 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 8 6 4 2 0 2 4 6 8 X [deg] X [deg] X [deg] 8 20 8 20 8 20 8 20 ∆µ ∆µ ∆σ ∆σ 6 X 15 6 Y 15 6 X 15 6 Y 15 4 10 4 10 4 10 4 10 2 5 2 5 2 5 2 5 0 0 0 0 0 0 0 0

Y [deg] 2 5 2 5 2 5 2 5 residuals4 model 10 4 10 4 10 4 10 6 15 6 15 6 15 6 15 8 20 8 20 8 20 8 20 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 8 6 4 2 0 2 4 6 8 X [deg] X [deg] X [deg] X [deg] Internal dynamics of the Large Magellanic Cloud [1805.08157] z Method: axisymmetric Jeans Anisotropic model (a generalization of Cappellari 2002). Assume that the velocity ellipsoid is aligned with cylindrical coordinates and 2 2 has an axis ratio σR /σz = b = const. Total gravitational potential: R R [Kpc] Φ(R, z) = Φ + Φ . 0 1 2 3 4 5 6 stars DM halo 100 Density of tracers (stars): 1 2 80 irc ρ(R, z) = 4πG ∇ Φstars. v c vφ ]

s 60 Compute σR,z from Jeans eqns: / m k 2 [ ∂Φ ∂(ρ σ ) 1 σ R , 40 0 = ρ + v ∂z ∂z b σR, φ 20 2 2 2 ∂Φ ∂(ρ σ ) ρ σ − v σz 0 = ρ + R + R φ ∂R ∂R R 0 0 1 2 3 4 5 6 7 8 R [deg] 1 Semi major axis of error ellipse in p:m: (¾pm;max) [mas yr¡ ]

0 ¡ 0 0 0 Example 2: globular clusters : 0 : : : 0 0 0 1 0 0 0 0 : : : 0 1 2 5 1 2 5 5 1 2 5 6 Challenges: 7 – crowding (can’t study central regions) 8 – contamination from galactic stars 9 – systematic error is of order σ 1 0 1 CMD of a typical cluster at 10 kpc 1 1 2

12 G M 3 (NGC 5272), D=10 kpc m 1 3 a

13 g n 1 i t 4 14 u d e 1 15 5 1 6

16 error systematic 1 G 17 7 1 18 8 1 19 9 2 20 0 2 21 1 0.0 0.5 1.0 1.5 G G 1 5 25 km/s BP − RP Determination of cluster membership CMD sky plane proper motion space Determination of cluster membership Probabilistic membership determination

A hard cutoﬀ in PM space is not always possible and is conceptually unsatisfactory. A more mathematically well-grounded alternative: gaussian mixture modelling.

f (µi ) = q N (µi | µcl, Σcl;i ) + (1 − q) 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 q, are all inferred by N Pstars maximizing the likelihood of the observed stellar PMs ln L ≡ ln f (µi ). i=1 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 ) Internal kinematics of globular clusters [1811.05345]

Clear signature of rotation in ∼ 10 clusters: see also Bianchini+ 2018, Sollima+ 2019, Jindal+ 2019, all based on Gaia data

0.25 0.6 NGC 6656 (M 22) NGC 5904 (M 5) 8 N=4166, D=3.2 kpc N=4475, D=7.6 kpc 8 0.20 0.5 ] ]

r r 6 ] ] y y / /

s 0.15 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

r 2 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 2 4 6 8 10 12 14 16 R [arcmin] R [arcmin] Internal kinematics of globular clusters [1811.05345]

Good match between line-of-sight velocity dispersion and PM dispersion: see also Baumgardt+ 2018, Jindal+ 2019 (Gaia PM) and 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 [1811.05345]

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 0 5 10 15 20 R [arcmin] R [arcmin] The globular cluster FSR 1758

“Sequoia in the forest” recently discovered by Barb´a+2019 in the DECaPS legacy survey of the Galactic bulge [Schlaﬂy+2018]. Despite being located in a highly crowded region, it stands out very clearly in Gaia PM.

[Barb´a+2019] The globular cluster FSR 1758

[Barb´a+2019] Dynamical modelling of FSR 1758 [1904.03185]

Fit a mixture model to the spatial and PM distribution of the cluster and ﬁeld stars, using a generalized King model for the cluster density and PM dispersion proﬁles:

R [pc] R [pc] 0 5 10 15 20 25 30 35 40 0 5 10 15 20 25 30 35 40 1000 10 all stars 0.20

field stars ] 100 8 n i astrometry m

c cluster members 0.15 r a / s r ]

a 10 6 r t ]

astrometric members y s s / / [ s

a m

k y m

0.10 [ t

[ i

n 1 4 e d

e c a f

r 0.05

u 0.1 2 S

0.01 0.00 0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 R [arcmin] R [arcmin] Photometric modelling of FSR 1758

Compare the CMD of member stars with that of several analogous globular clusters, determine the total mass by scaling the magnitude distribution:

4 4 NGC 6205 (M 13) 3 NGC 6656 (M 22) 3 NGC 6254 (M 10) FSR 1758 2 2 MIST PARSEC 1 1

G 0 G 0

1 1

NGC 6205 (M 13): M=4.61e5 M ¯ 2 2 NGC 6656 (M 22): M=4.05e5 M ¯ NGC 6254 (M 10): M=1.88e5 M ¯ 3 3 NGC 6752: M=2.29e5 M ¯ FSR 1758: M=2.5e5 M ¯ 4 4 0.5 0.0 0.5 1.0 1.5 2.0 10-5 10-4 10-3 10-2 G G N(

The brightest stars happened to be recorded by Gaia RVS;

Simpson 2019 used them to determine vLOS and the orbit of the cluster, which turned out to be strongly retrograde.

σLOS from 3 stars (!) in Gaia RVS

0.6 227.3 1.0 km/s ± 0.5

0.4 15.9% 84.1% 2.3%

0.3 median: 1.1 km/s 97.7% probability 0.2 84.1%: 3.2 km/s

0.1 97.7%: 8.9 km/s

0.0 222 224 226 228 230 232 0 2 4 6 8 10 vLOS [km/s] σLOS [km/s] Line-of-sight velocity dispersion of FSR 1758

The brightest stars happened to be recorded by Gaia RVS;

Simpson 2019 used them to determine vLOS and the orbit of the cluster, which turned out to be strongly retrograde.

σ with additional 9 stars from σ from 3 stars (!) in Gaia RVS LOS LOS ground-based spectroscopy [Villanova+2019] 0.6 0.6 227.3 1.0 km/s 227.0 1.2 km/s ± ± 0.5 0.5

0.4 0.4 15.9% 15.9% 84.1% 84.1% 2.3% 2.3%

0.3 median: 1.1 km/s 0.3 median: 3.8 km/s 97.7% 97.7% probability probability 0.2 0.2 84.1%: 3.2 km/s 84.1%: 5.1 km/s

0.1 0.1 97.7%: 8.9 km/s 97.7%: 7.0 km/s

0.0 0.0 222 224 226 228 230 232 0 2 4 6 8 10 220 225 230 235 0 2 4 6 8 10 vLOS [km/s] σLOS [km/s] vLOS [km/s] σLOS [km/s] Example 3: the entire Milky Way globular cluster system

1807.09775, see also Gaia Collaboration (Helmi+) 2018, Baumgardt+ 2018, de Boer+ 2019

∼ 150 globular clusters in the Milky Way 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| Distribution of globular clusters in position/velocity space

eccentricity 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

400 150 σr 350

300 100 σ ] θ s

250 / m k

[ σφ

200 σ

, 50 v

150 Total velocity, km/s vφ 100 0 v 50 θ vr 0 50 1 10 100 1 10 100 Galactocentric distance, kpc r [kpc] I Signiﬁcant overall rotation, especially within the central 10 kpc. I Nearly isotropic dispersion at r < 10 kpc, more radially anisotropic in outer parts. I A population of ∼ 10 clusters on eccentric orbits [Myeong+ 2018]. I Correlated orbits (e.g., Sgr stream: M 54, Terzan 7, Terzan 8, Arp 2, Pal 12, Whiting 1). Distribution of globular clusters in action space

polar (Jφ =0)

NGC 6723 1 5 M 69 2 10 20 50 100

BH 229 NGC 6624 Rcirc [Kpc]

) ci 0 NGC 6342 rc inner Galaxy = M 80 ul J r NGC 6144 NGC 6522 a ( r outer Galaxy ESO452 ar (J ul r = rc NGC 6325 NGC 6539 0 ci ) NGC 6652 M 107

NGC 6362 Terzan 3

FSR 1735 NGC 6453 Pal 9 NGC 6139 M 12 M 19 NGC 6355 M 62 prograde NGC 6528 NGC 6569

NGC 6541 NGC 6401 Terzan 10 BH 261 M 55 NGC 6752

NGC 6352 M 70 NGC 6558 M NGC10 6397 NGC 6293 Ton 2 Terzan 4 NGC 6304 NGC 5927 NGC 6440 M 14 NGC 5053

NGC 6388 Terzan 12 NGC 6638 Pal 8 NGC 6540 NGC 6553

NGC 6287 retrograde NGC 6366 Pal 7 BH 184 NGC 6535 NGC 6316 Terzan 7 M 28 Terzan 8 NGC 5986 Pal 5 ω NGC 6760 C NGC 6642 e n NGC 6441 Arp 2 M 53 Terzan 2 ESO456 M 54 Pal 6 NGC 6749 NGC 6256 NGC 6712 NGC 5946 i NGC 6517 Terzan 5 n ) Whiting 1 - Terzan 9 p Terzan 6 0 la = n J z e Terzan 1 ( (J NGC 6380 ne NGC 5824 z = Djorg 1 la 0 NGC 6544 -p Pal 12 ) in NGC 5897 NGC 6235 NGC 4833 M 3

M 30 Pyxis

IC 4499 47 Tuc

NGC 6752E 3 M 4 NGC 6397 0 NGC 6356 Pal 11

NGC 288 M 13 Pal 1

NGC 7492 M 68 FSR 1716 NGC 5927 radial (J =0) M 71 φ M 15 BH 176 NGC 5466

NGC 6287 Pal 7 J → Pal 10 z NGC 6101 NGC 6496 NGC 4372 NGC 6284 polar NGC 3201 M 22

FSR 1758 NGC 5634 ω M 5

C M 9 e Pal 13 prograde n ESO280 M 92 NGC 6426

NGC 2298

Rup 106 Jφ NGC 362 180◦ ← inclination → 0◦ NGC 6584 M 75

NGCM 126156 IC 1257 NGC 5286 M 2 Djorg 1 retrograde NGC 4147

NGC 7006 NGC 4833 NGC 5694 NGC 6229

eccentricity M 72 NGC 6934 NGC 1851 NGC 2808 M 79 radialJ

r ← Pal 2 1 Distribution of globular clusters in action space

polar (Jφ =0)

NGC 6723 1 5 M 69 2 10 20 50 100

BH 229 NGC 6624 Rcirc [Kpc]

) ci 0 NGC 6342 rc inner Galaxy = M 80 ul J r NGC 6144 NGC 6522 a ( r outer Galaxy ESO452 ar (J ul r = rc NGC 6325 NGC 6539 0 ci ) NGC 6652 M 107

NGC 6362 Terzan 3

FSR 1735 NGC 6453 Pal 9 NGC 6139 Sagittarius M 12 M 19 NGC 6355 M 62 prograde NGC 6528 NGC 6569

NGC 6541 NGC 6401 Terzan 10 BH 261 M 55 NGC 6752 [Law&Majewski 2010, . . . ]

NGC 6352 M 70 NGC 6558 M NGC10 6397 NGC 6293 Ton 2 Terzan 4 NGC 6304 NGC 5927 NGC 6440 M 14 NGC 5053

NGC 6388 Terzan 12 NGC 6638 Pal 8 NGC 6540 NGC 6553

M 30 Pyxis

IC 4499 47 Tuc

NGC 6752E 3 M 4 NGC 6397 NGC 6356 Pal 11

NGC 288 M 13 Pal 1

NGC 7492 M 68 FSR 1716 NGC 5927 radial (J =0) M 71 φ M 15 BH 176 NGC 5466

NGC 6287 J Pal 7 Pal 10 z NGC 6101 NGC 6496 NGC 4372 NGC 6284 polar NGC 3201 M 22

FSR 1758 NGC 5634 ω M 5

C M 9 e Pal 13 prograde n ESO280 M 92 NGC 6426

NGC 2298

Rup 106 Jφ NGC 362 NGC 6584 radially M 75

NGCM 126156 IC 1257 NGC 5286 M 2 Djorg 1 retrograde NGC 4147 anisotropic

NGC 7006 NGC 4833 NGC 5694 Sequoia M 72NGC 6229 population NGC 6934 NGC 1851 NGC 2808 M 79 radial [Myeong+ 2019] [Myeong+ 2018, Jr Pal 2 Helmi+ 2018] Distribution of globular clusters in action space

Jackson Pollock, “Convergence” Kliment Redko, “Uprising” Dynamical modelling of the entire globular cluster population

Assume a steady-state equilibrium model for the Milky Way globular cluster distribution function (DF): f J | α ; actions: J x, v | Φ free parameters potential: Φ(x | β ) Maximize the likelihood of drawing the observed positions and velocities of all clusters (marginalizing over measurement uncertainties) by varying the free parameters α, β of the DF and the potential.

Nclusters Z X max ln L ≡ max ln f J(w | Φ) N (w | wi ,δ wi ) dw, w ≡ {x, v} α,β α,β i=1 Explore the parameter space via MCMC and derive conﬁdence intervals for α, β. Constraining the Milky Way potential [1807.09775]

Williams17 Sakamoto03 w Leo I circular velocity curve McMillan11 Deason12b Nesti&Saucci13NFW Eadie15 300 Irrgang13model II Eadie16 McMillan (2017) Irrgang13model III Eadie17 Bovy (2015) McMillan17 Posti19 Battaglia05 Eadie&Juric19 250 total Xue08 Vasiliev19 disc Gnedin10 Li19 Watkins10 Li19 w RC halo ] Kafle12 Lin95 200 1 10 Deason12 Law05 ] M s

1 Huang16 Koposov10 / 1 Ablimit&Zhao17 Gibbons14 m 0

k 150 Sohn18 Kupper15 [ 1 [ c Watkins19 Malhan&Ibata19 r i )

c Kochanek96 w leoI Penarrubia14 r v WE99 100 < ( M 50

0 1 10 100 100 R [kpc] [compilation from Wang+ 2019]

101 102 r[kpc]

I Results are broadly consistent with other studies based on globular clusters [Binney&Wong 2017, Sohn+ 2018, Watkins+ 2019, Posti&Helmi 2019, Eadie&Juric 2019]; I Mass model from McMillan(2017) is acceptable, the one from Bovy(2015) is too light. Summary

Gaia is a treasure cove for dynamical modelling!

Credit: Amanda Smith, IoA Summer school in applied galactic dynamics July 6 – August 16, 2020 Center for Computational Astrophysics / Flatiron Institute, New York City Connecting dynamics theory and galactic observations, the Applied Galactic Dynamics School will consist of one week of lectures and tutorials, explorations of available data sets, and project discussions, and 5 weeks for individual projects. During the projects there will be regular group meetings and plenty of informal gatherings to foster cross- project connections, and to make optimal use of the expertise present at the School.

Support for students includes roundtrip airfare, a meal-plan and accommodation for all six weeks of the school. Students are expected to attend all six weeks. Organizers, mentors and lecturers: Emily Cunningham, Elena D’Onghia, Victor Debattista, Keith Hawkins, Amina Helmi, Jason Hunt, Kathryn Johnston, Melissa Ness, Sarah Pearson, Adrian Price-Whelan, Robyn Sanderson, David Spergel, Tjitske Starkenburg, Monica Valluri, Eugene Vasiliev, Martin Weinberg, Larry Widrow. Application deadline: February 15; website: galacticdynamics.nyc