Dynamical ModelsDynamical Models

James BinneyJames Binney

Oxford UniversityOxford University

OutlineOutline

• Importance of modelsImportance of models

• Modelling methodsModelling methods

• Schoenrich’s modelSchoenrich’s model

Importance of dynamical Importance of dynamical modelsmodels• Equilibrium modelsEquilibrium models

– Track DMTrack DM– Reduce 6d phase space to 3d integral spaceReduce 6d phase space to 3d integral space– Relate near to far, seen to unseenRelate near to far, seen to unseen

• Secular evolution Secular evolution – Galaxies works in progressGalaxies works in progress– Evolution driven by gas infall, clouds, spirals, Evolution driven by gas infall, clouds, spirals,

the bar, satellites,..the bar, satellites,..– Chemical & dynamical evolution entangled: by Chemical & dynamical evolution entangled: by

modelling together, reconstruct past, modelling together, reconstruct past, understand presentunderstand present

Features peculiar to MWFeatures peculiar to MW

• Large-scale structure hidden – e.g. rotation Large-scale structure hidden – e.g. rotation curve hard to plotcurve hard to plot

• Proper motions, trig parallaxes + spectra Proper motions, trig parallaxes + spectra for for ÀÀ 10 1077 stars stars

• Most classes of stars only seen nearbyMost classes of stars only seen nearby– Need many independent samples nearby, Need many independent samples nearby,

fewer far away (Brown Velazquez & Aguilar 05)fewer far away (Brown Velazquez & Aguilar 05)– Problematic for particle-based modelsProblematic for particle-based models

• Solar nhd studied in exquisite detailSolar nhd studied in exquisite detail– Use dynamics to leverage to global Use dynamics to leverage to global

understandingunderstanding

Science requirementsScience requirements

• Complexity of system, richness of Complexity of system, richness of data, mandate hierarchical modellingdata, mandate hierarchical modelling– Axisymmetric Axisymmetric !! barred barred !! spiral spiral !!

metallicity distribution metallicity distribution !! warp warp !! streams streams !! ....

• Need DF so we can resample & Need DF so we can resample & calculate likelihoodscalculate likelihoods

• Need to calculate secular evolutionNeed to calculate secular evolution

Modelling methodsModelling methods

• N-body (TQ)N-body (TQ)

• Schwarzschild & Torus modelsSchwarzschild & Torus models– MW a linear combination of orbitsMW a linear combination of orbits

Schwarzschild’s problemsSchwarzschild’s problems

• Orbits not naturally characterisedOrbits not naturally characterised• DF not returned DF not returned • Poisson noisePoisson noise• Eqs under-determined so no unique soln; Eqs under-determined so no unique soln;

should count # of solutions Magorrian should count # of solutions Magorrian (2006)(2006)

• Sampling problemSampling problem• Messy: need to store M phase-space pMessy: need to store M phase-space p®® for for

N orbits N orbits !! N*M matrix to invert N*M matrix to invert• Solution: replace time-series orbits with Solution: replace time-series orbits with

orbital toriorbital tori

Orbital tori Orbital tori (e.g. McMillan & Binney 08) (e.g. McMillan & Binney 08)

• Orbit characterized by actions J – essentially Orbit characterized by actions J – essentially unique unlike initial conditionsunique unlike initial conditions

• Compact analytic formulae for x(J,Compact analytic formulae for x(J,µµ) and v(J,) and v(J,µµ))• Can interpolate in J to new orbitsCan interpolate in J to new orbits• So can find at what So can find at what µµ star is at given x & get right star is at given x & get right

vv– If orbit integrated in t, star will just comes close & we If orbit integrated in t, star will just comes close & we

have to search for closest xhave to search for closest x

• Real-space characteristics of orbits naturally Real-space characteristics of orbits naturally related to J so can design DF f(J) to give related to J so can design DF f(J) to give component of specified shape & kinematics component of specified shape & kinematics (GDII (GDII sec 4.6)sec 4.6)

• Sampling density apparent because dSampling density apparent because d66w=(2w=(2¼¼))33dd33JJ

Tori (cont)Tori (cont)

• The likelihood of arbitrary data given a model can The likelihood of arbitrary data given a model can be calculated by doing 1-d integral for each star be calculated by doing 1-d integral for each star

• Given f(J) have a stable scheme for determining Given f(J) have a stable scheme for determining self-consistent self-consistent ©©

• The J are adiabatic invariants – useful when The J are adiabatic invariants – useful when ©© slowly evolving (mass-loss, 2-body relax, disc slowly evolving (mass-loss, 2-body relax, disc accretion…)accretion…)

• Fokker-Planck eqn takes exceptionally simple Fokker-Planck eqn takes exceptionally simple formform

• We are equipped to do Hamiltonian perturbation We are equipped to do Hamiltonian perturbation theorytheory

Schoenrich’s model Schoenrich’s model (arXiv:0809.3006 & MN submitted)(arXiv:0809.3006 & MN submitted)

• InputsInputs– Standard chemical evolution Standard chemical evolution

• 80 250pc annuli, Kennicutt law, C, O, Ca,.. Fe 80 250pc annuli, Kennicutt law, C, O, Ca,.. Fe followed, SNIa followed, SNIa // exp(-t/1.5Gyr) for t>0.12Gyr exp(-t/1.5Gyr) for t>0.12Gyr

– BlurringBlurring• Non-linear epicycles driven by Non-linear epicycles driven by ¾¾ // ¿¿ 0.330.33

– ChurningChurning• Stars, gas swapped between annuliStars, gas swapped between annuli

– Gas flow in plane to ensure exp(-R/RGas flow in plane to ensure exp(-R/Rdd) star-density) star-density– ½½(z) from f(W) and (z) from f(W) and ©©(R(R00,z),z)

• Fitted to Geneva-Copenhagen N(Z) and Hess Fitted to Geneva-Copenhagen N(Z) and Hess diagramdiagram

Schoenrich’s Schoenrich’s resultsresults• Select “GCS” sampleSelect “GCS” sample• Snhd breaks up intoSnhd breaks up into• Thin discThin disc

– Low Low ®®– -0.65 < [Fe/H] < 0.15-0.65 < [Fe/H] < 0.15

• Metal-poor thick discMetal-poor thick disc– High High ®®, [Fe/H]<-0.8, [Fe/H]<-0.8

• Metal-rich thick discMetal-rich thick disc– Overlaps thin disc in [Fe/H] but Overlaps thin disc in [Fe/H] but

+0.3dex in [O/Fe]+0.3dex in [O/Fe]– Significant asymm driftSignificant asymm drift– 6Gyr < 6Gyr < ¿ ¿ < 10.5Gyr< 10.5Gyr

• Thin disc Thin disc ¿ ¿ < 7Gyr< 7Gyr• Metal-poor thick disc Metal-poor thick disc ¿ ¿ > 10Gyr> 10Gyr

Schoenrich’s Schoenrich’s results (cont)results (cont)

• Vertical profiles Vertical profiles exponentialexponential– 270pc, 820 pc 270pc, 820 pc

(690+890)(690+890)– Thick-d 14% at z=0Thick-d 14% at z=0

• [Fe/H] and V strongly [Fe/H] and V strongly correlated (Haywood correlated (Haywood 08)08)

Kinematic Kinematic selection selection (Bensby+03)(Bensby+03)• Seriously scrambles Seriously scrambles

discsdiscs

thin

thick

ConclusionsConclusions

• Dynamical modelling key to interpreting MW Dynamical modelling key to interpreting MW surveyssurveys

• Hierarchical modelling essentialHierarchical modelling essential• Secular evolution fundamentalSecular evolution fundamental• Must model chemical evolution in parallelMust model chemical evolution in parallel• Torus modelling seems to meet specTorus modelling seems to meet spec• Simplest model of chemical evolution that includes Simplest model of chemical evolution that includes

secular heating with radial migration produces secular heating with radial migration produces thin/thick dividethin/thick divide

• Consequently, no evidence for early mergerConsequently, no evidence for early merger• Thick disc in 2 partsThick disc in 2 parts• Kinematic selection badly blurs the pictureKinematic selection badly blurs the picture