Modelling the Milky Way

James Binney

Oxford University

Outline • A quick tour of the MW

– Disc, bar/bulge, stellar & dark halos; • kinematics of solar nhd

– secular heating, age– Hercules stream, bar

• Epicycle motion, peculiar velocities• Anharmonic motion, frequencies• Scattering by spirals, Lindblad resonances and corotation, heating, churning• Vertical motion, third integral, SoS• Local age-Z distribution, Fe/H to Ra correlation• Thin & thick discs, nature and possible origins• Schoenrich & Binney model• Models

– why we need them– What we plan to do with them– Why we need a hierarchy of models

• Models – how: N-body or Schwarzschild?• Angle-action coordinates – the basics

– Time-averages theorem– Action space– Adiabatic invariance, applications

• Schwarzschild modelling in more detail ! torus modelling• Weighting orbits with analytic DFs, examples, solar motion• Choosing the DF – galaxies in action space• Fitting the DF to survey data

Disc & bulge

2MASS

COBE

2MASS

Near IR Photometry

• Galaxy brighter on left of GC

• Individual objects (eg HB stars) also brighter on left

If we could look down

Stellar halo(SDSS)

• Count millions of faint stars

• Mostly main-sequence stars

• So colour strongly correlated with distance– Red nearest

• Can probe halo in shells of increasing distance

• Not smooth! residuals Bell et al (2007)

Halo & disc globular clusters

Disk

Halo

• Disk clusters – At small radii

– more metal-rich

• Also a population of field stars traced by blue horizontal branch stars & RR Lyrae stars

• many from destroyed globular clusters

Dark Halo

• To determine vc at R>R0 need to know distance to tracer

• Hard to track DM around MW

NGC 3198

Milky Way

Stars near the Sun

• Stars born on nearly circular orbits

• Gradually move to less circular and more inclined orbits

• Stars then have random velocities

• Spiral structure and molecular clouds increase random velocities over time

Hipparcos data

Age & SF history of disc(Aumer & Binney 09)

• Fit ¾(B-V) and N(B-V) from Hipparcos

• SFR / exp(-t/T) for age = t0-t < tmax

• ¾ / (age + const)¯

• Find – ¯ ' 0.35– tmax> 11 Gyr– T > 10. Gyr (T ' 11.5Gyr)

• Solar nhd very old!• SFR rather steady

Velocity space from Hipparcos• Distribution of stars lumpy in velocity

space • Pointer to the Galactic bar and spiral

structure• U = -vR

• V = vÁ-vc

Hercules stream

Hercules stream

Stars trapped by the bar

Melnik & Rautiainen 10

Orbits: epicycle approximation

• In a spherical potential– ) const = L ´ r£v– So motion in a plane

• R motion oscillation in asymmetric ©eff

• For small amplitudes – © ' ½ ·2(R-Rg)2 + const

– Harmonic oscillator

ÄR = ¡d©e®dR

; ©e® = ©+L 2

2R2

Epicycle approx.

• R = Rg + X cos(·t)

• From R2 dÁ/dt = L– Á = gt - ° (X/Rg) sin(· t)

– ° = 2g/· ('√2 > 1)

• In TR = 2¼/· star has not completed a revolution (

gTR< 2¼ ) precession)

Peculiar velocity

• Can show V = vÁ-vc(r) = (· X/°) cos(· t)• So peculiar V velocity has smaller amplitude than U = -vR

velocity• Consequently expect ¾Á< ¾R as observed• Energy of harmonic oscillator• ER = ½ [vR

2 + ·2(R-Rg)2] = ½(U2 + °2V2) () ¾R/¾Á = °)• Suppose star encounters massive molecular cloud on

circular orbit– Then in cloud’s rest frame star will scatter elastically, so (U,V) !

(U’,V’) such that U2+V2 = U0 2+V02

– ER ! ER’ such that – 2(ER’-ER) = U02 - U2 + °2(V02- V2) = (°2-1)(V02-V2)– E goes up if V goes up and U down: when scattering converts

radial to tangential motion

R) (cont)• For larger amplitude,

anharmonic, radial oscillations

• Can evaluate TR(E,L) and TÁ(E,L)

• 4 phase-space coords subject to 2 constraints

• ) orbit 2d• Has 2 characteristic

frequencies R = 2¼/TR, Á = 2¼/TÁ

E (r; _r; _Á); L = r2 _Á

Scattering by spirals

• Consider steadily rotating spiral pattern

• In co-rotating frame (p) – EJ = E - pL is conserved – ) dE = p dL

• But dEc = cdL• ! at RCR no change in ER= E-

Ec = ½ (U2+°2V2) but ER increases elsewhere given dL>0 at R<RCR and dL>0 at R>RCR

• When m(Á - p) = §· (Lindblad resonances) spiral resonantly excites ER (heating)

• At ILR excited stars stay on resonance

• At CR stars shifted between equally circular orbits: “churning”

• Effect of spiral averages to 0 away from resonances

Lindblad diagram

Churning (Sellwood & Binney 2002)

Churning

• Even weak spiral structure shifts stars by a few kpc radially in half Hubble time

Vertical motion in Axisymmetric

• Strictly conserved: E &

• Can study motion in (R,z) plane

L z = R2 _Á

ÄR = ¡@

@R©e®

Äz= ¡@@z

©e®©e® ´ ©+

L 2z

2R2

Orbits in axisymmetric ©

• Orbits with same (E,Lz); but different I3?

• If 9 I3 then H=E, I3=const, z=0 (3 constraints on 4 coords R, pR, z, pz) should make pR(R) only

Surface of section• “Consequents” on a

curve• Similar to L2 = const• Conclude 3 integrals

(E, Lz, I3) conserved• We have only

approximations to I3(R,vR,z,vz)

L2 = constpz = 0

Age-metallicity distribution

• Traditional model of chemical evolution

• Disc divided into annuli

• Chemical evolution of each annulus isolated from others

• Consequently at R0 predict Z(age)

• Low numbers of low-Z stars ) prolonged addition of metal-poor gas to each annulus

Age-Z distribution

• Ages hard to determine

• Increasing ¾Fe with B-V signals growing spread in [Fe/H] with age

Haywood 08

Z gradient in R

ISM (Sellwood & Binney 02)

Haywood 08

Core-collapse & deflagration supernovae

• In massive stars (M(0)>8M¯) the core collapses to form a BH or neutron star

• E released by collapse ejects envelope• Ejecta rich in C, O, Mg (® nuclides)• In less massive stars core cools to form CO white dwarf• later mass-transfer from binary companion can reignite

nucleosynthesis explosively• E released by nucleosynthesis completely destroys WD• Whole mass of WD converted to Fe• ° rays from decay of unstable Ni nuclei power type Ia SN• Result:

– from stellar population get early release (10 – 100 Myr) of ®-rich material

– After ~1Gyr get additional Fe– Stars formed in first Gyr “® enhanced” (really “Fe-poor”)

Thin & thick discs

• Non-exponential º(z)

• At root a chemical split?

• Thick disc like bulge?

Origin of disc

• Thin disc formed gradually over >10 Gyr• Thick disc old (& formed in first Gyr)• Old theory: thick disc old thin disc disrupted by

large satellite (plus satellite stars)• Note:

– fattening a disc requires no external energy, just scattering of stars

• But actually simplest model of chemodynamical evolution yields thin/thick dichotomy (latency of type Ia SN) (Schoenrich & B 2009)

Schoenrich & Binney Model

radial spacing 0.25 kpc

time spacing 30 Myrs

Schoenrich & Binney Model

direct onflow ~ 75% of feedslightly preenriched

outflow/loss < 10% of processed gas

radial spacing 0.25 kpc

time spacing 30 Myrs

Schoenrich & Binney Model

Inflow ~ 25% of feed through disk

direct onflow ~ 75% of feedslightly preenriched

outflow/loss < 10% of processed gas

radial spacing 0.25 kpc

time spacing 30 Myrs

Schoenrich & Binney Model

Inflow ~ 25% of feed through disk

direct onflow ~ 75% of feedslightly preenriched

outflow/loss < 10% of processed gas

radial spacing 0.25 kpc

time spacing 30 Myrs

Churning-mass exchange between neighbouring rings-cold gas and stars-no heating of the disc

Blurring -stars on increasingly eccentric orbits(heating of the disc) broadening of the disc and increasing scale height

Input physics• At each timestep add to annulus radius Ri some number dN of stars

with guiding-centre radius Rg= Ri

• Salpeter IMF• Stars added by Kennicutt (1998) law SFR / §gas

1.4

• Gas density determined by requirement that §*(R)/ exp(-R/Rd) at all times (no inside-out growth)

• Start with 8£109M¯ of gas and add gas at rate

– where b1=0.3Gyr, M1=4.5£109 M¯ and b2=14Gyr• At each timestep a fraction of stars at Ri transfer to Rj (j=i§1):

– pij / kchMj/Mmax

– Rule makes dMout=pi,i+1Mi equal to dMin=pi+1,iMi+1 as required by conservation of Lz

– Dimensional analysis suggests kch independent of radius, number of rings, etc

_Mgas =M1

b1e¡ t=b1 +

M2

b2e¡ t=b2

Input physics (cont)

• When disc “observed” or stars die, the annulus they contribute to calculated by assuming

• Density of stars in phase space

– F(Lz) calculated from N(Rg)– ¾ calculated by assuming that at R0 <vR

2>1/2/ (age+const)0.33

– Elsewhere <vR2> / exp[(R0-R)/1.5Rd] (van der Kruit & Searle 82)

• Vertical structure assumes <¾z>/ (age+const)0.33 and º(z) / exp[-©R(z)/¾z

2] where ©R(z)=©(z,R)-©(0,R)• As stars migrate in R they take ¾z with them

f (ER ;L z) / F (L z)exp(¡ ER =¾2)

¾2(L z)

Infall, inflow, metal gradient

• Spatial dependence of infall unknown• Unknown rate of flow through disc• These constrained by Z(R) in ISM• Hence determine relevant free

parameter fB

inflow

model

ISM

A

B

Gas per unit area

Gas through ring

In solar nhd

• Stars in solar nhd come from many radii

• Heterogeneous collection – spread in Fe/H does not reflect history of local ISM

• Kinematic / chemical correlations

• Non-uniformity larger at higher ages

ISM

Solar nhd stars

ZISM(age)

Solar-nhd metallicity distribution

• 3 parameters quantify radial infall profile, radial drift & churning strength

• 1 parameter fixed by [Me/H] gradient in ISM

• 2 parameters by [Me/H] distrib of GCS stars– Red – model– Blue – no churning– Pink – no blurring– Mixing has biggest impact at high

Fe/H

Data Holmberg+07

Origin of thick disc• The surprise: double

exponential structure in z emerges naturally

• SNIa make stars formed in 1st Gyr distinct: bimodal chemistry

Schoenrich + B 09

Schoenrich + B 09

Vertical profile

ChemistryChemistry

Chemistry

chemistry

Model Data

Comments • MW a very complicated system• Dynamics, stellar evolution, cosmology all play big roles in fashioning the

MW• Relevant observational data come from many sources and are by no means

free from error• To progress:

– Identify the most important physical processes– Model these with good physics when possible, or parameterise them when

physics is lacking– Use model to predict observables and thus optimise parameters

• Should we infer from the S&B09 model that the thick disc is unrelated to a merger?

– No! The model merely falsifies claims that the thin/thick disc dichotomy implies an early merger

• What it does do:– It shows that churning - which is required by physics & not optional – is important

for the structure of the solar neighbourhood– It highlights the importance of knowing the pattern of gas infall and the history of

spiral structure• What we need to do now

– Use quality N-body and N-body + hydro simulations to quantify churning & heating by spiral structure

– Understand how gas is accreted by disc (Marinacci + 2010)

Models: why we need them

• Near-IR point-source catalogues– 2MASS, DENIS, UKIDS, VHS, ….

• Spectroscopic surveys– SDSS, RAVE, SEGUE, HERMES, APOGEE, …

• Astrometry– Hipparcos, UCAC-3, Pan-Starrs, Gaia, Jasmine, …

• Already have photometry of ~108 stars, proper motions of ~107 stars, spectra of ~106 stars, trig parallaxes of ~105 stars

• By end of decade will have trig parallaxes for ~109 stars and spectra of 108 stars

Why: the goals

• Structure of the discs– Scale lengths– Velocity field– Spiral structure– Chemical distribution functions

• Morphology of the bar• Gross & fine structure of stellar halo• The grav potential © ) DM distribution• The distribution functions f®(x,v) of N stellar populations• The history of the Galaxy

– Merger events– Gas accretion (& ejection)– Secular heating / radial migration ) diffusion coeffs ) history of

spiral structure

How: carry models into space of observables

• Distance errors never negligible & with proper motions dangerous

• Imagine observing a shear flow v(s)=v0+As• System appears hot but is actually cold• Problem made worse by errors in ¹

“measured” proper motionsInferred velocities

What we need

• A hierarchy of dynamical models of variable complexity that yield number density of stars (DF) in (mapp,Colour,Z,g,s,l,b,vlos,v®,v±)– unwise to bin data in > 10d space of observables

• Only an equilibrium model allows us to infer ½DM from DF:– any © consistent with given distrib in (x,v) if we don’t

assume steady state:• If © too deep, system will collapse• If © too shallow, system will explode

• Must be able to refine to non-steady model (bar, spirals)

• Need apparatus to optimise fit of model to data

How? N-bodies?

• Flexible and easy to relate to cosmology

• But – Doesn’t deliver DF– Limited resolution– Difficult to control– Difficult to characterise

• 6N phase-space coordinates non-unique

How? Schwarzschild/Torus models?

• Choose ©• Build a library of orbits in ©• For each orbit choose probability that it’s

occupied by a star of (m, ¿, Fe/H, ®/Fe)

• Predict observables

• Adjust probabilities to optimise fit to survey

• If fit poor, adjust © and try again

Dynamics refresher• Hamilton’s equations of motion given H(q,p))

• DF, probability density in phase space [ (q,p) space], satisfies continuity equation

• Implies f is constant of motion so f(constants) (Jeans’ theorem)

_q=@H@p

_p= ¡@H@q

@f@t

+@f _q@q

+@f _p@p

= 0

)@f@t

+ _q@f@q

+ _p@f@p

= 0

)dfdt

= 0

Action-angle coords

• Imagine we had n integrals Ji(x,p) for Hamiltonian system with n coords

• Dream about using the Ji as momenta

• Let conjugate variable be i

• Then Hamilton’s equations would be0 = _J i = ¡

@H@µi

) H ( J )

_µi =@H@J i

= const = i( J )

Action-angle coords (cont)

• So orbit would be Ji=const, i=0+i t

• Fact that i increases without limit while xj, pj bounded suggests xj, pj periodic fns of i; can prove conjecture (Arnold book)

• So

• Implies motion quasiperiodic:

x(µ; J ) =X

nX n( J )ein¢µ

x(t) =X

nX n( J )ein¢ t

Quasiperiodicity (Binney & Spergel 1981)

• Check by numerically integrating orbits and Fourier transforming coordinates

• Spectral lines should all be of form n. for just 3 fundamental frequencies i

• Quasiperiodicity implies that n-d orbit is an n-torus

• J labels torus, position on torus• We’ve scaled i so incrementing by 2 takes you once

around torus• Consider s p.dq around torus on path on which only i

increases• s p.dq =s Jidi = 2 Ji

• So Ji to within factor 2 ith cross section of torus

Poincare invariants

• If A is a two-dimensional surface in phase space and (q,p) and (Q,P) are any two sets of canonical phase-space coordinates. Then by definition– sAi dqidpi = sAi dQidPi

– On surface of a torus dJi = 0 so s dxidvi = 0 (tori are null)

• It follows that Hamilton’s equations have the same form in all canonical coordinates

• It follows that along a closed path P– sPi pidqi = sPi PidQi

– In particular sP vidxi =sP Jidµi

Orbits in axisymmetric ©

• Orbits with same (E,Lz); but different I3?

• If 9 I3 then H=E, I3=const, z=0 (3 constraints on 4 coords R, pR, z, pz) should make pR(R) only

• These orbits have same JÁ´ Lz but on left JR is smaller and Jz is larger than on right in such a way that E = H(JR,Jz,JÁ) is the same

Surface of section• “Consequents” on a

curve• Similar to L2 = const• Conclude 3 integrals

(E, Lz, I3) conserved• We have only

approximations to I3(R,vR,z,vz)

L2 = constpz = 0

The surface of section is a cross section through the torus

Non-rotating barred

• L(x,y) = ½v02ln(Rc

2+x2+y2/q2)

• Potential supports boxes and loops

• Each picture is a projection into 2d of a 2-torus

Resonant and non-resonant orbits

• A resonant orbit is one on which – n1+n2+n3=0 for some integers n1, n2, n3

– a star visits only a subspace of its torus (room)

• The set of points in a 3d space for which coordinates satisfy resonance condition “a set of measure zero” so non-resonant orbits the norm

• Time-averages theorem:– On a non-resonant orbit, average time spent in D /sDd3– This is the key to determining an orbit’s observables

• Action space:– Phase-space volume occupied by orbits in d3J is V=(2)3d3J– So action space is accurate representation of phase space

Adiabatic invariance• After time t each phase-space point w´(x,v) !

new point wt=(xt,vt)• Defines Hamiltonian map Ht: w! wt

• This map in canonical. i.e. conserves Poincare invariant, so along anf closed path P

• During slow change in all stars on given orbit have same experience so torus of 0, T0! Tt, a torus of t

• Since Ht: T0! Tt, actions of Tt = actions of T0

X

i

Z

Ppidqi =

X

i

Z

PJ idµi

Examples (1)

• Distant tidal encounter deforms • Orbits deformed

• But after encounter potential as before

• So each star has same actions in same , i.e. is on original orbit

• L is an action, so L unchanged even though perturbation not axisymmetric

Examples (2)

• In solar nhd z ' 2R

• Treat z-oscillations as adiabatically invariant as star oscillates in R

Schwarzschild Modelling(Schwarzschild 1979)

• Standard for modelling external galaxies (e.g., SAURON project)

• Define observables Oj to be quantities such as §(x), §<vlos>, §<vlos

2> that are linear in DF• Use Runge-Kutta or similar to obtain orbit as time

sequence x(t), v(t)• During integration determine contribution of orbit to

observables• Choose weight wi for each orbit to optimise fit to

observations• Observables linear functions of the weights ) “linear-

programming” or “quadratic programming” problem

Torus modelling• Find orbit as 3d object xJ(µ), vJ(µ)• Actions J=(J1,J2,J3) play role of initial conditions in

Schwarzschild modelling– Actions essentially unique – facilitates comparison of models– Actions adiabatic invariants – facilitates study of secular

evolution• In Oxford we have code that takes J as input and returns

xJ(µ), vJ(µ) as analytic functions – this code essentially replaces Runge-Kutta etc integrator (Kaasalainen & Binney 1994 & refs)

• Given x, one can easily find v of star when it reaches x [hard/impossible when you have only x(t)]

• Hence calculate OJk, the contribution of orbit J to observable k (more on this later)

How to weight orbits?

• Could simply adapt Schwarzschild: – create orbit library xJ(µ) vJ(µ) for some sampling of 3d

action space

– Seek non-negative weights wJ such that K observational constraints satisfied by J wJ OJk

• But it’s better to introduce fewer parameters• Simple analytic functions f(J) prove able to

provide good fits to data

Example: vertical profilesMN 401, 2318 (2010)

• Vertical profile simply fitted

GCS

isothermal

prediction GCSmodel

f z(J z) =( z J z + V2

° )¡ °

2¼R1

0 dJ z ( z J z + V2° )¡ °

DF for disk

• We need many distinct sub-populations

• Imagine each cohort is quasi-isothermal

• Consistent with data for young stars

• In-plane quasi-isothermal

f ¾z (J z) ´e¡ z J z =¾2

z

2¼R1

0 dJ z e¡ z J z =¾2z

f ¾r (J r ;L z) = c§¼¾2

r ·

¯¯¯¯R c

[1+ tanh(L z=L0)]e¡ · J r =¾2r

Disc DF

• Obtain full disc by integrating over age with ¾/ (t+t0)¯ (¯ = 0.38) and adding quasi-isothermal thick-disc

• Thick disc parameters fitted to Ivezic + 08, Bond + 09

f thn(J r ; J z;L z) =

R¿m

0 d¿ e¿=t0 f¾r (J r ;L z)f¾z (J z)t0(e¿m =t0 ¡ 1)

¾z(L z;¿) = ¾z0

µ¿ + ¿1

¿m + ¿1

¶¯

eq(R 0¡ R c )=R d

DF for disc• Fit thin-disc parameters to GCS stars

thinthick

V¯

(arXiv0910.1512; Schoenrich + 10)

• Shapes of U and V distributions related by dynamics

• If U right, persistent need to shift observed V distribution to right by ~6 km/s

• Problem would be resolved by increasing V¯

• Standard value obtained by extrapolating hVi(¾2) to ¾ = 0 (Dehnen & B 98)

• Underpinned by Stromberg’s eqn

V¯ (cont)• Actually hVi(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

Schoenrich + 10

Stromberg [.]

How to fit model?• Procedure in these examples poor: fitted in physical rather than

observable space• With analytic DFs model yields probability density in space of

observables u1,u2,…

• Then calculate log likelihood L = ln(pi) and extremise it with respect to the parameters in f

• Unfortunately some tori invisible, some poorly sampled; introduce selection function Á(J) equal to fraction of angle space visible in survey.

• Constrain DF only where Á(J)>² and subject to s d3J Á(J)f(J) = const

In more detail..• Stars not standard objects but drawn from a population within which

apparent magnitude (e.g. I) and colour (e.g. I-K) vary significantly• Basic set of observations includes I and I-K, which carry some

distance information• Let F(MI) be the luminosity function• Then dP = F(MI)dMI f(J) d3J d3µ is probability of finding a star of abs

mag MI in an element of phase space• From MI, J, µ we can predict observables u = (l,b,I,,¹®,¹±,vlos)• If true values form u’, probability of observing u in d7u given errors

¾i is– Pod7u = d7us dMIs d3J d3µ G(u-u’,¾)F(MI)f(J) – where G(u,¾) = i=1

e-u2/2¾2/(√2¼ ¾i) is error Gaussian and u’(MI,J,µ) are the true observables

• We can show that f coincides with the true DF when log likelihood L is maximised with respect to f subject to the constraint – ss dMIF(MI)s d3J d3µ f(J) = constant– Where S indicates over survey volume only

Selection function

• We calculate selection function of survey Á(J) as follows

• Pick random point µ ) (s,l,b)• If (l,b) outside survey area Á +=0• If (l,b) in survey area • MI,crit= Ilim-5log(s/10pc) • and Á +=s-1

MI,crit dMI F(MI)• After repeating above for N points Á /= N• Now constraint on f is s d3JÁ(J)f(J) = const

Further detail• For each star in the catalogue we have to evaluate

– Pod7u = d7us dMIs d3J d3µ G(u-u’,¾)F(MI)f(J) • By Markov-Chain Monte-Carlo (MCMC) sampling we obtain N

points that sample action space with density f(J)• Then Po ' N-1is dMIF(MI)s d3µ G(u-u’i,¾)• The measurement errors in (l,b) negligible so approximate their

Gaussians by Dirac ±-fns• Then

– So for each star we have to integrate over possible distances (parallaxes) and possible absolute magnitudes

– For given distance, absolute magnitude is strongly constrained by apparent magnitude, so integral over MI not expensive

• Calculation numerically challenging– Key is to identify the tori that cross a given line of sight– Cost is dominated by these los integrals – in practice do not resample

action space for every change in parameters of f– Calculation extremely parallelisable

Po =1N

X

i

Zd$

¯¯¯¯

@(µ)@($ ;l;b)

¯¯¯¯

ZdM I F (M I )G(u ¡ u0

i ;¾)

Modelling RAVE, APOGEE,….

• To extract information about © need to input both stellar number density and kinematics (e.g. vertical Jeans eq

• Usually kinematics measured for a subsample• hope that kinematics of subsample same as that of

photometrically complete sample• Then pretend that velocities have been measured for

complete sample but errors ¾i on velocities of stars without spectra are infinite

• Likely pairing RAVE + 2MASS

dº¾2

dz= ¡ º

d©dz

Further developments

• I’ve assumed that we have parallaxes • Currently more likely that we have photometric distances

s obtained from J, J-K etc• Burnett & Binney (2010) describe a Bayesian formalism

for extracting photometric s (and Fe/H, log(g), etc) from J, J-K etc

• For now we use these distances (for RAVE stars) in place of in algorithms I’ve described

• One can also extend the algorithms to include isochrones so distances are determined in parallel with f(J)

• In principle this extension is preferable to our current procedure, but it is even more challenging computationally, so we are not implementing it yet

Choice of df• In an “ergodic” model f(H)• So in action space f const on constant-

E surface• Nearly planar triangular surfaces• If move stars over H=const surfaces,

little change to spherically averaged density profile ½(r)

• Shift stars from ergodic df towards Jr axis ! spherical radially anisotropic model

• Shift stars onto JÁ axis ! cold disc• Spreading stars from JÁ axis towards

Jµ, Jr warms disc • f(J) = s(J)f0(H) with s(J) “shift function”

– s d3J ±(E-H(J)]s(J) =s d3J ±[E-H(J)]– or s dJµ dJÁ/s(J) =s dJµ dJÁ/

Hernquist models

¯= ½

¯ = -½

Choice of df: Halo (stellar & dark)

• Assume ½(r) and level of anisotropy ¯=1-¾t/2¾r

• Determine f(H,L)=L-2¯f1(H), where L = Jµ+|JÁ| from

• Analytic solution for ½ < ¯ <3/2 • For ¯ = ½

• Express H(J) and use approximation in f1

const £ r2¯ º(r) =Z ª (r )

0dE

f +1(E)(ª ¡ E)¯ ¡ 1=2

f 1(E) =1

2¼2

d(rº)dª

Choice of df: disc

• From §(R) determine f(H,JÁ)=f1(JÁ)±(JÁ-Lc(H)) for cold disc

• Replace ±-fn with fn of non-zero width, e.g. exp[-·(Lc) Jr/¾r

2(Lc)] exp[-º(Lc)Jµ/¾z

2(Lc)]

• Where · in-plane epicycle frequency, º epicycle frequency ? plane

• Then <vr2>1/2¼ ¾r, <vz

2>1/2¼¾z

Generation of tori

History

• 1989 – 1990 Colin McGill launched torus project• 1991 - 1994 Mikko Kaasalainen moved it on• 1994 - 1996 Walter Dehnen rewrote code• 2006 – Paul McMillan revived it

Tori

• 3-torus = cube with opposite sides identified• Integrable bound orbit lies on 3 torus J1(x,p)=const,

J2(x,p)=const, J3(x,p)=const• µi give position on/within torus• How to define Ji? Choose 3 non-equivalent loops °i • Then is indep of °i and µi is the

conjugate variable• Note on torus S=s dpidqi=s dJidµi=0; tori are null• Fact: any null surface in H = const is an orbit

Analytic models(de Zeeuw MNRAS 1985)

• Most general: – Staeckel © defined in terms of confocal ellipsoidal

coordinates

• © separable in x,y,z and ©(r) are limiting cases of Staeckel ©

• Staeckel © yields analytic Ii but numerical integration required for Ji,µi

• everything analytic for 3d harmonic oscillator and isochrone

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

Harmonic oscillator

• x = Xcos( t) ) p = -Xsin( t)• So

• So J = (2¼)-1 s dx p = ½X2• So X = (2J/)1/2

• Also µ = t• So x = (2J/)1/2 cosµ and p=-(2J)1/2 sinµ is the

equation of our 1-torus

Idxp=X 2 2

Z T

0dt sin2( t)

=X 2 2T=2= ¼X 2

Generating functions

• Let (q,p) and (Q,P) be two different sets of coordinates for phase space

• Let S(q,P) be any differentiable function on phase space

• If (q,p) are canonical, then so are (Q,P) if

• S is the generating function of the canonical transformation (q,p)$ (Q,P)

• Every canonical transformation has a generating function

• S = qP generates the identity transformation Q=q, p=P

p=@S@q

and Q =@S@P

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)

• As (u,v) covers rectangle, (x’,y’) covers realistic box orbit

Box orbits (cont)

• Drive (u,v) with equations of motion when x=f(u), y=g(v) execute s.h.m.

• pu(x,px)=df/du px ; pv=dg/dv py

• x=(2Jx/x)1/2 sin(x), px= etc

• So (J,) ! (x,px,..) ! (u,pu,..) ! (x’,px’,..)

• Requires orbit to be bounded by ellipsoidal coord curves – insufficiently general

Box orbits (cont)

• So make transformation (J’,) ! (J,) by• S(,J’) = .J’+ 2 Sn(J’) sin(n.)• J =S/=J’+ 2 nSn(J’) cos(n.)• The overall transformation

(J’,) ! (x’,px’,..) is now general• (x,y) are not quite bounded by a rectangle,

so (x’,y’) are not quite bounded by ellipsoidal coordinates

• Determine ¢, Sb and parameters in f(u), g(v) to minimize h(H-hHi)2i over torus

Orbits in ©(R,z)

• Ignorable Á ! motion in (R,z) with H = p2/2 + Lz

2/2R2 + ©

• Orbits nearly bounded by (u,v) so can proceed as above

• Or do

(J 0r ;µ

0r ; ::)

S=J µ0+¢¢¢

!(J r ;µr ; ::)

Isochr

!(pr ;r; ::)

General ©(x,y,z)

• No significant modifications required for general ©

What have we achieved

• Numerically orbit given by parameters of toy \Phi plus point transformations plus ~100 Sn (cf 1000s of (x,p)t if orbit integrated in t)

• Sn are continuous fns of J, so we can interpolate between orbits

• We are equipped to do Hamiltonian perturbation theory

Resonances

• Orbit family determined a priori by gross structure of mapping

• Can foliate phase space with tori at will

• Then define integrable H0(J)=hHiJ• ±H ´ H-H0 may cause qualitative change

when i are rationally related

• Orbit said to be “trapped” by resonance

Orbits in flattened isochrone

q = 0.7

q = 0.4

trapped

Secular perturbation theory(Kaasalainen 1994)

• H(J,µ) = H0(J) + ±H(J,µ)• ±H = Hn(J) ein.µ

• We define H0 as hHi on our tori• Consider d=2: resonance k11+ k22= 0• Canonical transf with g.f. S = (k.µ)JÃ+µ2J» defines

– Ã = (k.µ) and » = µ2 – JÃ= J1/k1 and J»= J2 - (k2/k1)J1

• On resonance Ã= 0 so à a slow variable• Averaging over » and adjusting phase of Õ H = H0(JÃ) + 2mHmkcos(mÃ)• Problem reduced to d=1 so now integrable

Pendulum eq

• Simplify by – Taking leading term in series (e.g. m=2)

– Replacing Hmk by constant F

– Taylor expanding H0 using 0 = Ã= H/JÃ when ±JÃ=0

• Then H' ½G (±JÃ)2 + F cos mÖ Equation of pendulum

Failure of pendulum

• Problems with pendulum model– ±H doesn’t vanish outside island– Island symmetric in ±JÃ

Pendulum

Resonance

Solution• Expand Hmk to 2nd order in ±JÃ

• Hmk(JÃ0+±JÃ) = Hmk(JÃ0+½®±Jà + ¼¯(±JÃ)2)

• H = ½(G+¯cos mÃ)(±JÃ)2 + (®±JÃ+F) cos mÃ

® > 0 ¯ > 0

Result

• Fit ®, ¯ to ±H from 3 tori through island

Direct integration

Torus + p theory

Conclusions

• Dynamical modelling of galaxies key for studies of black holes, DM & galaxy formation

• Current approaches rely on time integration• Seriously limited by Poisson noise, poor

characterization of orbits and sampling problem• All these difficulties eliminated if time series

replaced by tori• With tori can also use perturbation theory to study

fine structure and develop deeper understanding.