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
How was Milky Way formed?
Stars formed
Ji
DM
Jf
Chemical enrichment
Gas
Gas Accretion
Secular heating,
non circular orbits
2
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
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
The Besancon model (BGM).

p(r,t) p(m|t) p(Z|t) the main assumption

Padova Isochrones
4
The kinematic model: Gaussian distribution function
Age Velocity dispersion (AVR)
Asymmetric Drift
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
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
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 given L.
•Naturally handles asymmetric distribution.
•Note vertical and planar motion are decoupled, potential separable
in Φ(R,z)=Φ(R)+Φ(z).
The vertical dependence of kinematics
11
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
12
Model parameters explored
θ={Set of
parameter
s}
13
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
14
RAVE analysis
p(θ | l,b,vlos) ~ p(l,b,vlos | θ ) p(θ)

No use of proper motion or distances

No use of J-K, Teff, log g
15
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
16
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,
–
Rσthin negative.
–
Since local V☉ ~ 10 km/s, LSR might be on a non
circular orbit.
17
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
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.
Problems with Gaussian models

In Gaussian models Rσthin is strongly correlated with V☉.

Moreover, for RAVE data Rσthin is negative while for GCS data it
is positive.

So the Gaussian model gives inconsistent results when applied
to RAVE and GCS data sets.
20
RAVE-GAUSS
• Marginalised
posterior
distribution of
model
parameters.
• The anticorrelation of
Rσthin and V☉
sun can be seen
• V☉ and vc
affected by αz
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.
22
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.
23
Best fit parameters for the Shu model
αz free makes
vc go up.
232.8+7.53=240.3
3 km/s.
GCS and RAVE
also match,
σR similar for thin
and thick
Quantities in
magenta were
kept fixed during
fitting. Units are
24
in km/s and kpc
Rave velocity
distribution compared
to models.
• The Shu model fits the
RAVE data well.
25
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.
26
Systematics

Dependence on gradient dvc/dR

Rsun dependence

Isochrone magnitudes

p(r,t) the BGM model.
27
Systematics



Kinematic models are not fully self-consistent
The vertical dependence handled by a fudge
factor.
Dynamically self-consistent models based on
action integrals.
28
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).
29
Publicly available codes that might be of interest to you
GALAXIA- http://galaxia.sourceforge.net
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.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.
30
– API available in IDL, MATLAB, Python,C,C++, Fortran, Java
Comparison with Besancon
43.1
27.8
17.5
31