1. The challenge of Galaxy modelling James Binney Oxford University TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A AA A A

2. Outline The why & how of modelling the Galaxy The why & how of using actions Simple DFs for the disc(s) Determining the solar motion

3. Impetus Drive to understand galaxy formation central to contemporary astrophysics Our Galaxy is fiducial: galaxies like it dominate current cosmic SFR We are in the middle of an era of giant Galaxy surveys (Hipparcos, UCAC2, DENIS, 2MASS, SDSS, RAVE, HERMES, APOGEE, Gaia) Measure (x,v), [Fe/H], [ ® /H] for > 10 7 stars ( ® = C, O, Ne, Mg, S, Ca) The data come from many surveys and each has a complex set of observational biases Dynamics & chemistry intertwined Gaia an optical instrument – extinction a crucial output Models an essential tool for combining heterogeneous & biased data

4. Key questions How violent is our history? –Mergers vs accretion –Relation to WHIM (>50% of baryons) Can we identify relics? –By groupings in chemistry & orbit space What did the Local Group look like at z? –Vital for interpretation of high-z obs When did bar form? –Has it been renewed? –Impact on DM? –Role in secular heating? What is the structure of the ISM? –Extinction, exchange of matter with WHIM

5. Hierarchical modelling MW complex system, data heterogeneous & rich Multi-stage modelling essential Require models that can be –Evaluated in different © s –Viewed at varying resolution solar nhd, far MW with & without chemistry –Capable of refinement –Provide understanding perturbation theory

6. DF f(J) DF allows resampling – “zooming” Actions allow –Mapping of orbits between © s –Straightforward solution for self-consistent © –Use of perturbation theory

7. Constructing an integrable H Given time-independent H = ½p 2 + © (x) integrate orbits Suppose © / ln(x 2 +q 1 2 y 2 +q 2 2 z 2 ) Orbits come in families Time series x(t) etc are quasiperiodic

8. Numerical orbits Binney & Spergel 82

9. Angles & actions Quasiperiodic orbits ) exist integrals J 1, J 2, J 3 that can be complemented by coordinates µ 1, µ 2, µ 3 with trivial eqns of motion J i = constat and µ i =  i t + const Orbits 3-tori labelled by J with µ defining position on torus Torus null in sense s torus dx ¢ dv = 0 Question is: how to find (x, v)(J, µ ) for given © ?

10. Analytic models (de Zeeuw MNRAS 1985) Most general: –Staeckel © defined in terms of confocal ellipsoidal coordinates © separable in x,y,z and © (r) limiting cases Staeckel © yields analytic I i but numerical integration required for J i, µ i Everything analytic for 3d harmonic oscillator and isochrone

11. Torus programme Map toy torus from harmonic oscillator or isochrone into target phase space Use canonical mapping, so image is also null Adjust mapping so H = const on image

12. e.g. Box orbits (Kaasalainen & Binney 1994) Orbits » bounded by confocal ellipsoidal coords (u,v) x’=  sinh(u) cos(v); y’=  cosh(u) sin(v) When (u,v) cover rectangle, (x’,y’) cover realistic box orbit

13. Box orbits (2) Drive (u,v) with equations of motion when x=f(u), y=g(v) execute s.h.m. p u (x,p x )=df/du p x ; p v =dg/dv p y x=(2J x /  x ) 1/2 sin(  x ), p x = etc So (J,  ) ! (x,p x,..) ! (u,p u,..) ! (x’,p x ’,..) Requires orbit to be bounded by ellipsoidal coord curves – insufficiently general

14. Box orbits (3) So make transformation (J’,  ) ! (J,  ) by S( ,J’) = .J’+ 2  S n (J’) sin(n.  ) J =  S/  =J’+ 2  nS n (J’) cos(n.  ) The overall transformation – (J’,  ) ! (J,  ) ! (x,p x,..) ! (u,p u,..) ! (x’,p x ’,..) is now general (x,y) are not quite bounded by a rectangle, so (x’,y’) are not quite bounded by ellipsoidal coordinates Determine ¢, S n and parameters in f(u), g(v) to minimize h (H- h H i ) 2 i over torus

15. Kaasalainen & B (1994) Log © Staeckel ©

16. Orbits in © (R,z) Ignorable Á ! motion in (R,z) with H = ½p 2 + ½L z 2 /R 2 + © Orbits nearly bounded by (u,v) so can proceed as above Or do

17. General © (x,y,z) No significant modifications required for general © (including rotating frame of reference; Kaasalainen 1995)

18. What have we achieved? Analytic formulae x(J, µ ) and v(J, µ ) So can find at what µ star is at given x & get corresponding v If orbit integrated in t, star will just come close, & we have to search for closest x Orbit characterized by actions J –essentially unique unlike initial conditions Sampling density apparent because d 6 w=(2 ¼ ) 3 d 3 J The J are adiabatic invariants – useful when © slowly evolving –mass-loss, 2-body relax, disc accretion…

19. What have we achieved (2) Real-space characteristics of orbits naturally related to J so can design DF f(J) to give component of specified shape & kinematics (GDII sec 4.6) Numerically orbit given by parameters of toy plus point transformations plus <~100 S n (cf 1000s of (x,p) t if orbit integrated in t) S n are continuous fns of J, so we can interpolate between orbits The likelihood of arbitrary data given a model can be calculated by doing 1-d integral for each star Given f(J) have a stable scheme for determining self-consistent © Fokker-Planck eqn exceptionally simple in a-a coordinates We are equipped to do Hamiltonian perturbation theory

20. Choice of DF Represent DF with analytic f(J) –To give physical insight –To keep DF smooth low entropy null hypothesis

21. DF for disk (arXiv0910.1512) Vertical profile simply fitted

22. DF for disk But need many distinct sub-populations Imagine each cohort is quasi-isothermal Consistent with data for young stars In-plane quasi-isothermal

23. Obtain full disc by integrating over age with ¾ / (t+t 0 ) ¯ ( ¯ = 0.38) and adding quasi- isothermal thick-disc Thick-d parameters fitted to Ivezic + 08, Bond + 09

24. DF for thisc Fit thin-d parameters to GCS stars

25. V ¯ (arXiv0910.1512; Schoenrich + 10) Shapes of U and V distributions related by dynamics If U right, persistent need to shift oberved V distribution to right by ~6 km/s Problem would be resolved by increasing V ¯ Standard value obtained by extrapolating h V i ( ¾ 2 ) to ¾ =0 (DehnenB 98) Underpinned by Stromberg’s eqn

26. V ¯ (cont) Actually h V i (B-V) and ¾ (B-V) and B-V related to metallicity as well as age On account of the radial decrease in Fe/H, in Schoenrich & Binney (09) model, Stromberg’s square bracket varies by 2 with colour SB09 Aumer & B 09 Schoenrich + 10

27. V ¯ (cont) Safer to fit theoretical U, V, W distribs to data Conclude V ¯ =12 § 2 km/s

28. Conclusions Urgent requirement for hierarchy of models that incorporate dynamics & chemistry Important to know DF Torus dynamics seems to fit the bill Basic reference models have analytic DFs Fit to current data imperfect but data rather than models may be at fault Next steps –fit in space of observables (M V, , ¹, l, b) taking proper account of inhomogeneous errors –Upgrade SB09 dynamics Use tori Securer basis for scattering probabilities