1. (Collaborators: James Binney, Tilmann Piffl, Jason Sanders)
Studying the dynamics of the Milky Way: Where are we, and where are we going?
Paul McMillan
(Collaborators: James Binney, Tilmann Piffl, Jason Sanders)

2. Two fundamental problems
We can only observe positions & velocities, not acceleration.
We’re inside the Milky Way, moving with it. So we don’t even observe positions or velocities.
So we need good forward models

3. What produces the potential?
We can approximate that the system is ‘collisionless’ (components are smooth)
Stellar disc – scale length 2-4kpc.
Gas disc – local surface density 5-10 M/pc2 
~8.3 kpc
Dark matter?

4. Peculiar velocity of the sun?
The Sun is not on a circular orbit.
From Hipparcos data:
Relatively easy to find vR & vz (-11 & 7 km/s)
Credit: NASA/JPL-Caltech/R. Hurt (SSC)
Not so easy to find vϕ – asymmetric drift means that <vϕ> ≠ vc - can try to extrapolate to zero dispersion ( = zero asymmetric drift)
Dehnen & Binney – vϕ = 5 km/s

5. Determining the potential from position-velocity data
We need to make some assumptions, otherwise any potential is possible
Usually* this assumption has to be statistical equilibrium.
Steady state = f(integrals of motion)
Which integrals…
*Some special cases e.g. stellar streams – not equilibrium but all from ~same place

6. Action-angle variables
Regular orbits have conserved action J, with conjugate angle θ.
θ = θ0 + Ωt (with 2π period)
Only known analytically for simple spherically symmetric potentials. Via 1D integral for Stäckel potentials

7. Action-angle variables
Regular orbits have conserved action J, with conjugate angle θ.
θ = θ0 + Ωt (with 2π period)
Only known analytically for simple spherically symmetric potentials. Via 1D integral for Stäckel potentials

8. We have made substantial progress in improving approximations
We can still use them
We have made substantial progress in improving approximations
Moderate accuracy – high speed
Adiabatic approximation (motion decouples in R & z, e.g. Binney & McMillan 2011)
Stackel ‘fudge’ (motion decouples in ellipsoidal coordinates, Binney 2012, Sanders & Binney 2014
coming soon: github.com/GalacticDynamics-Oxford)
Higher accuracy - slower
With a generating function (For a given action: Kaasalainen 1994, McM. & Binney 2008,
github.com/PaulMcMillan-Astro/Torus
For a given starting point, Sanders & Binney 2014
github.com/jlsanders/genfunc )
(see also galpy: Bovy, 2014, github.com/jobovy/galpy)

9. Suitable dfs for disc galaxies
f(J) is in equilibrium, but what form to use?
Density profile
(R ~ Jϕ/vc)
(Spitzer 1942)
First DF (Binney 2010) – Can fit rho z and local V, but…
(Shu 1969, see also Dehnen 1999)
(Binney 2010, Binney & McMillan 2011)

10. Fit to local velocity & density
Add many together, with varying σ(stellar age), can fit to local kinematics and density profile
(Binney 2010) vφ
local vR
local
But note that a shift in vϕ would give a better fit – can do this by assuming different vϕ,
ρ(z)
local
Velocity shift by 7 km/s (see also McMillan & Binney, Schonrich BD)
vφ(z)
local
c.f. McMillan & Binney 2010 (masers),
Schönrich, Dehnen & Binney 2010 (asymmetric drift),
But also Bovy et al 2012, 2015 (APOGEE)

11. Finding the Galactic potential
As with Schwartzchild modelling, if f(J) in Φ fits the data, that’s the ‘best’ potential.
Demand potential fits other constraints (e.g. Sgr A* proper motion), for given halo
Fit f(J) to (binned) kinematics of RAVE giants
Iterate until density profiles of stellar discs in df & potential are consistent.
Compare to vertical density profile from literature (Juric et al 2008, 0.7<r-i<0.8)
Piffl et al
(Piffl, Binney, McMillan, & RAVE 2014)

12. Local dark matter
We’re left with effectively two free parameters for the potential:
Local DM density & halo flattening.
For spherical halo:
ρDM, = M/pc3
= 0.48 GeV/cm3
Note that statistical error bars are tiny (~0.4%)
Piffl et al results

13. With systematic uncertainties and varying halo flattening
Where q is axis ratio of DM halo, and α = 0.89
More Piffl et al results
Largest component of the uncertainty is the systematic uncertainty in the distance scale (affects density profile & velocities)
Emphasises the importance of Gaia parallaxes for Galactic modelling

14. Other results – similar methodology
Bovy & Rix (2013) applied a similar methodology to 16,000 G-dwarfs observed by SEGUE
Report vertical force as a function of R
Gives:
Stellar scale length = 2.15 ± 0.14 kpc
M*(MW) = 46 ± 3 × 109 M (for R0=8kpc)
Vc,disc/Vc,total = 0.83 ± 0.04
Maximum disc, Vc At R=2.2 Rd

15. Conclusions
Action-angle coordinates are immensely valuable, and now they’re easy to use.
We’ve already used them to analyse Milky Way data, and determined the local DM density
Models based on action-angle coordinates will be vital for extracting information about Galactic structure from Gaia data.
Conclusions…