Kinematic Modelling of the Milky Way using RAVE and GCS.

1. Sanjib Sharma Joss Bland-Hawthorn (University of Sydney, Australia) RAVE collaboration Kinematic Modelling of the Milky Way using RAVE and GCS. http://physics.usyd.edu.au/~sanjib/rave_sharma.pdf

2. How was Milky Way formed? Stars formed Jf Ji DM Chemical enrichment Gas Gas Accretion Secular heating, non circular orbits

3. Basic model of the Milky Way • fC(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 • fC(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 Age Velocity dispersion (AVR) Asymmetric Drift

6. The kinematic model: Gaussian distribution function Age Velocity dispersion (AVR) Asymmetric Drift

7. The kinematic model: Gaussian distribution function Age Velocity dispersion (AVR) Asymmetric Drift

8. The kinematic model: Gaussian distribution function Age Velocity dispersion (AVR) Asymmetric Drift

9. Cause of Asymmetric drift • 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>Rc. Sun

10. The kinematic model: The Shu distribution function • L is angular momentum, • ER=E-Ec(L)is the energy in excess of that required for circular motion with a givenL. • Naturally handles asymmetric distribution. • Note vertical and planar motion are decoupled, potential separable in Φ(R,z)=Φ(R)+Φ(z).

11. The vertical dependence of kinematics

12. Circular velocity profile • The circular velocity can have both radial and vertical dependence. • vc(R,z)=[v0+αR(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,vlos). 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☉, vc(R☉)? • Schonrich et al (2010) (11.1, 12.24, 7.25) km/s, 220+- 30 km/s W V Sun U Galactic Center

17. Proper motion of Sgr A* • vc , V☉andR☉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, • Rσthin negative. • Since local V☉ ~ 10 km/s, LSR might be on a non circular orbit.

18. Method of fitting analytical models to data • 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

19. 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.

20. Problems with Gaussian models • In Gaussian models Rσthinis strongly correlated with V☉. • Moreover, for RAVE dataRσthinis negative while for GCS data it is positive. • So the Gaussian model gives inconsistent results when applied to RAVE and GCS data sets.

21. RAVE-GAUSS • Marginalised posterior distribution of model parameters. • The anti-correlation of Rσthin and V☉ sun can be seen • V☉ and vcaffected 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 Rσthin – V☉degeneracy.

23. Best fit Gaussian vs Shu models • 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.

24. Best fit parameters for the Shu model αz free makes vc 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. How well does the RAVE best fit model fit the GCS data? • 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.

27. Systematics • Dependence on gradient dvc/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. Conclusions • 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 Rσthin) • Vc, V; ; • model dependent • Neglecting vertical dependence of kinematics can lead to undestimation of Vc. • 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.

30. Publicly available codes that might be of interest to you GALAXIA- http://galaxia.sourceforge.net • 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 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. EBF-http://ebfformat.sourceforge.netEfficient Binary data Format“Put fun back into numerical computing, Use EBF”

31. Comparison with Besancon 43.1 27.8 17.5