1. Dynamical Models James Binney Oxford University

2. Outline Importance of models Importance of models Modelling methods Modelling methods Schoenrich’s model Schoenrich’s model

3. Importance of dynamical models Equilibrium models Equilibrium models –Track DM –Reduce 6d phase space to 3d integral space –Relate near to far, seen to unseen Secular evolution Secular evolution –Galaxies works in progress –Evolution driven by gas infall, clouds, spirals, the bar, satellites,.. –Chemical & dynamical evolution entangled: by modelling together, reconstruct past, understand present

4. Features peculiar to MW Large-scale structure hidden – e.g. rotation curve hard to plot Large-scale structure hidden – e.g. rotation curve hard to plot Proper motions, trig parallaxes + spectra for À 10 7 stars Proper motions, trig parallaxes + spectra for À 10 7 stars Most classes of stars only seen nearby Most classes of stars only seen nearby –Need many independent samples nearby, fewer far away (Brown Velazquez & Aguilar 05) –Problematic for particle-based models Solar nhd studied in exquisite detail Solar nhd studied in exquisite detail –Use dynamics to leverage to global understanding

5. Science requirements Complexity of system, richness of data, mandate hierarchical modelling Complexity of system, richness of data, mandate hierarchical modelling –Axisymmetric ! barred ! spiral ! metallicity distribution ! warp ! streams !.. Need DF so we can resample & calculate likelihoods Need DF so we can resample & calculate likelihoods Need to calculate secular evolution Need to calculate secular evolution

6. Modelling methods N-body (TQ) N-body (TQ) Schwarzschild & Torus models Schwarzschild & Torus models –MW a linear combination of orbits

7. Schwarzschild’s problems Orbits not naturally characterised Orbits not naturally characterised DF not returned DF not returned Poisson noise Poisson noise Eqs under-determined so no unique soln; should count # of solutions Magorrian (2006) Eqs under-determined so no unique soln; should count # of solutions Magorrian (2006) Sampling problem Sampling problem Messy: need to store M phase-space p ® for N orbits ! N*M matrix to invert Messy: need to store M phase-space p ® for N orbits ! N*M matrix to invert Solution: replace time-series orbits with orbital tori Solution: replace time-series orbits with orbital tori

8. Orbital tori (e.g. McMillan & Binney 08) Orbit characterized by actions J – essentially unique unlike initial conditions Orbit characterized by actions J – essentially unique unlike initial conditions Compact analytic formulae for x(J, µ ) and v(J, µ ) Compact analytic formulae for x(J, µ ) and v(J, µ ) Can interpolate in J to new orbits Can interpolate in J to new orbits So can find at what µ star is at given x & get right v So can find at what µ star is at given x & get right v –If orbit integrated in t, star will just comes close & we have to search for closest x 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) 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) Sampling density apparent because d 6 w=(2 ¼ ) 3 d 3 J Sampling density apparent because d 6 w=(2 ¼ ) 3 d 3 J

9. Tori (cont) The likelihood of arbitrary data given a model can be calculated by doing 1-d integral for each star 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 © Given f(J) have a stable scheme for determining self- consistent © The J are adiabatic invariants – useful when © slowly evolving (mass-loss, 2-body relax, disc accretion…) The J are adiabatic invariants – useful when © slowly evolving (mass-loss, 2-body relax, disc accretion…) Fokker-Planck eqn takes exceptionally simple form Fokker-Planck eqn takes exceptionally simple form We are equipped to do Hamiltonian perturbation theory We are equipped to do Hamiltonian perturbation theory

10. Schoenrich’s model (arXiv:0809.3006 & MN submitted) Inputs Inputs –Standard chemical evolution 80 250pc annuli, Kennicutt law, C, O, Ca,.. Fe followed, SNIa / exp(-t/1.5Gyr) for t>0.12Gyr 80 250pc annuli, Kennicutt law, C, O, Ca,.. Fe followed, SNIa / exp(-t/1.5Gyr) for t>0.12Gyr –Blurring Non-linear epicycles driven by ¾ / ¿ 0.33 Non-linear epicycles driven by ¾ / ¿ 0.33 –Churning Stars, gas swapped between annuli Stars, gas swapped between annuli –Gas flow in plane to ensure exp(-R/R d ) star-density – ½ (z) from f(W) and © (R 0,z) Fitted to Geneva-Copenhagen N(Z) and Hess diagram Fitted to Geneva-Copenhagen N(Z) and Hess diagram

11. Schoenrich’s results Select “GCS” sample Select “GCS” sample Snhd breaks up into Snhd breaks up into Thin disc Thin disc –Low ® –-0.65 < [Fe/H] < 0.15 Metal-poor thick disc Metal-poor thick disc –High ®, [Fe/H]<-0.8 Metal-rich thick disc Metal-rich thick disc –Overlaps thin disc in [Fe/H] but +0.3dex in [O/Fe] –Significant asymm drift –6Gyr < ¿ < 10.5Gyr Thin disc ¿ < 7Gyr Thin disc ¿ < 7Gyr Metal-poor thick disc ¿ > 10Gyr Metal-poor thick disc ¿ > 10Gyr

12. Schoenrich’s results (cont) Vertical changes in chemistry Vertical changes in chemistry

13. Schoenrich’s results (cont) Vertical profiles exponential Vertical profiles exponential –270pc, 820 pc (690+890) –Thick-d 14% at z=0 [Fe/H] and V strongly correlated (Haywood 08) [Fe/H] and V strongly correlated (Haywood 08)

14. Kinematic selection (Bensby+03) Seriously scrambles discs Seriously scrambles discs thin thick

15. Conclusions Dynamical modelling key to interpreting MW surveys Dynamical modelling key to interpreting MW surveys Hierarchical modelling essential Hierarchical modelling essential Secular evolution fundamental Secular evolution fundamental Must model chemical evolution in parallel Must model chemical evolution in parallel Torus modelling seems to meet spec Torus modelling seems to meet spec Simplest model of chemical evolution that includes secular heating with radial migration produces thin/thick divide Simplest model of chemical evolution that includes secular heating with radial migration produces thin/thick divide Consequently, no evidence for early merger Consequently, no evidence for early merger Thick disc in 2 parts Thick disc in 2 parts Kinematic selection badly blurs the picture Kinematic selection badly blurs the picture