Kinematic Modelling of the Milky Way using RAVE and

GCS.

Sanjib Sharma Joss Bland-Hawthorn

(University of Sydney, Australia)

RAVE collaboration

http://physics.usyd.edu.au/~sanjib/rave_sharma.pdf

2

How was Milky Way formed?

DM

Gas

Ji Jf

Gas Accretion

Chemical enrichment

Stars formed

Secular heating, non circular orbits

3

Basic model of the Milky Way f

C(r,v,t,m,Z)

Thin disc, Thick Disc, Stellar Halo, Bar-Bulge.

f(r,v,t,m,Z) ∝ p(r,t) p(m|t) p(Z|t) p(v|r,t) position-age mass FeH velocity

4

Basic model of the Milky Way f

C(r,v,t,m,Z)

Thin disc, Thick Disc, Stellar Halo, Bar-Bulge.

f(r,v,t,m,Z) ∝ p(r,t) p(m|t) p(Z|t) p(v|r,t) position-age mass FeH velocity

The Besancon model (BGM). p(r,t) p(m|t) p(Z|t) the main assumption Padova Isochrones

5

The kinematic model: Gaussian distribution function

Asymmetric Drift

Age Velocity dispersion (AVR)

6

The kinematic model: Gaussian distribution function

Asymmetric Drift

Age Velocity dispersion (AVR)

7

The kinematic model: Gaussian distribution function

Asymmetric Drift

Age Velocity dispersion (AVR)

8

The kinematic model: Gaussian distribution function

Asymmetric Drift

Age Velocity dispersion (AVR)

•Vφ is shifted and has asymmetric

distribution

•Cause of asymmetric drift

– More stars with small L (exponential disc)

– Stars with small L are hotter (larger σ) so more likely on elliptical orbits.

– In elliptical orbit most time spent at apogee so at large R>R

c.

Cause of Asymmetric drift

Sun

•L is angular momentum,

•ER=E-E

c(L)

is the energy in excess of that required for circular

motion with a given L.•Naturally handles asymmetric distribution.•Note vertical and planar motion are decoupled, potential separable in Φ(R,z)=Φ(R)+Φ(z).

The kinematic model: The Shu distribution function

11

The vertical dependence of kinematics

12

Circular velocity profile The circular velocity can have both radial and vertical dependence.

vc(R,z)=[v

0+α

R(R-R

☉ )] [ 1/(1+αz(z/kpc)1.34) ]

Two popular models of Milky Way's 3D potential DB98 and LM10 shown below, have αz ~ 0.0374

In reality vertical motion is coupled to planar motion, so asymmetric drift also has a dependence on z. We incorporate it into αz.

We expect αz

> 0.0374

13

Model parameters explored

θ={Set of parameters}

14

Kinematic analysis using RAVE and GCS

GCS: A color magnitude limited sample of stars with (x,v). Very local 120 pc. (about 5000 stars)

RAVE a spectroscopic survey of about 500,000 stars with accurate, (l,b,v

los). 1- 2kpc

15

RAVE analysis

No use of proper motion or distances

No use of J-K, Teff

, log g

p(θ | l,b,vlos

) ~ p(l,b,vlos

| θ ) p(θ)

16

Questions? Is the Gaussian model correct? What are the correlations between different

parameters? Does GCS and RAVE give similar values? What are U☉,V☉,W☉, v

c(R☉) ?

Schonrich et al (2010) (11.1, 12.24, 7.25) km/s, 220+- 30 km/s

U

V

W

Galactic CenterSun

17

Proper motion of Sgr A* vc , V☉ and R☉ are difficult to measure but

Sgr A* at the center of galaxy, 8 yr baseline Ω☉=(vc+V ☉ )/R ☉ ~ 30.24 km/s/kpc (Reid & Brunthaler

2004). For R☉=8.0 kpc, (vc+V☉) ~ 242 km/s

Recently Bovy using APOGEE data find,

– vc=218 (+- 6) km/s, V ☉ = 26 (+- 3) km/s,

– Rσ

thin negative.

– Since local V ☉ ~ 10 km/s, LSR might be on a non circular orbit.

Bayesian data analysis. We use MCMC to explore the full posterior distribution of model parameters. – p(θ | l,b,vlos) ~ p(l,b,vlos | θ ) p(θ)

Data is color magnitude limited sample of stars with (l,b,vlos).

So we have to marginalise over – (t 'Age',m 'mass', Z '[Fe/H]')

– and also (d, vl, vb).

p(l,b,d,t|S) = ∬ p(l,b,d,t,m,Z) S(l,b,t,m,Z) dm dZ– p(l,b,d,t,m,Z) comes from the Besancon Galaxy model.

– S(l,b,t,m,Z) is the selection function unique to each survey

– Computation done numerically using the code GALAXIA

Method of fitting analytical models to data

Full exploration of parameter space using MCMC is computationally costly. In RAVE we have more than 200,000 stars which makes the task even more challenging.

• Instead of doing integrals– We set up the problem as a hierarchical Bayesian model.

– Metropolis-Within-Gibbs was found to be useful.

Method of fitting analytical models to data

20

Problems with Gaussian models

In Gaussian models Rσ

thin is strongly correlated with V☉.

Moreover, for RAVE data Rσ

thin is negative while for GCS data it is

positive.

So the Gaussian model gives inconsistent results when applied to RAVE and GCS data sets.

RAVE-GAUSS

• Marginalised posterior distribution of model parameters.

• The anti-correlation of R

σthin and V☉

sun

can be seen

• V ☉ and v

c

affected by αz

22

RAVE-SHU

In the Shu model the azimuthal motion is coupled to radial motion so it has 3 fewer parameters.

This helps to resolve the R

σthin

– V☉ degeneracy.

23

The Gaussian model fails to properly fit the wings of the distribution.

The reason that the Gaussian model gives inconsistent results for RAVE and GCS is because it is not a good description of the data.

Best fit Gaussian vs Shu models

24

Best fit parameters for the Shu model

αz

free makes v

c go up.

232.8+7.53=240.33 km/s.

GCS and RAVE also match,

σR

similar for thin and thick

Quantities in magenta were kept fixed during fitting. Units are in km/s and kpc

25

Rave velocity distribution compared

to models.

• The Shu model fits the RAVE data well.

26

• The Shu-RAVE slightly overestimates the right wing of the V distribution.

• The discrepancy is probably due to significant amount of substructure in the GCS data which can potentially bias the GCS best fit model.

How well does the RAVE best fit model fit the GCS

data?

27

Systematics Dependence on gradient dv

c/dR

Rsun

dependence

Isochrone magnitudes p(r,t) the BGM model.

28

Systematics Kinematic models are not fully self-consistent The vertical dependence handled by a fudge

factor. Dynamically self-consistent models based on

action integrals.

29

• Using only (l,b,vlos

) data from RAVE survey we obtain– good constraints on a number of kinematic parameters of the Milky

Way.

• Deficiency of a Gaussian model is clearly exposed by RAVE data– (high V☉ ,negative R

σthin)

• Vc,

V; ;

– model dependent

– Neglecting vertical dependence of kinematics can lead to undestimation of V

c.

• We find vcirc

~ 232 km/s and V

☉ ~7.54 km/s, which gives Ω☉=(vc+V

☉ )/R ☉ ~

30.04 km/s/kpc – in good agreement with Sgr A* proper motion of 30.24 km/s/kpc

(Reid & Brunthaler 2004).

• The best fit RAVE model also fits the GCS data well.– GCS V☉

is lower by 2 km/s.

• In future, fitting a dynamical model that takes into account both the spatial and kinematic components together should lead to more robust analysis.

Conclusions

30

Publicly available codes that might be of interest to you

Galaxia is a code for generating a synthetic model of the galaxy. The input model can be analytical or one obtained from N-body simulations. The code outputs a catalog of stars according to user specified color magnitude limits.

GALAXIA- http://galaxia.sourceforge.net

EBF-http://ebfformat.sourceforge.net Efficient Binary data Format“Put fun back into numerical computing, Use EBF”• A general purpose binary file format publicly available at• Automatic endian conversion, data type conversion• Store multiple data items in same file

– Time to locate an item, almost independent of number of items. Due to use of inbuilt hashtable

• Support for homogeneous multidimensional arrays and structures (also nested structures)

• Not tied to one programming languange.

– API available in IDL, MATLAB, Python,C,C++, Fortran, Java

31

Comparison with Besancon

43.127.817.5