Modelling the Milky Way James Binney Oxford University TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA A A A A A A A AAA

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)

Stellar streams Belokurov et al (2007)

Tidal streams (Pal 5) Sloan digital sky survey (SDSS)

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 v c at R>R 0 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 = t 0 -t < t max ¾ / (age + const) ¯ Find – ¯ ' 0.35 –t max > 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 = -v R V = v Á -v c 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-R g ) 2 + const –Harmonic oscillator

Epicycle approx. R = R g + X cos( · t) From R 2 d Á /dt = L – Á =  g t - ° (X/R g ) sin( · t) – ° = 2  g / · ( ' √2 > 1) In T R = 2 ¼ / · star has not completed a revolution (  g T R < 2 ¼ ) precession)

Peculiar velocity Can show V = v Á -v c (r) = ( · X/ ° ) cos( · t) So peculiar V velocity has smaller amplitude than U = -v R velocity Consequently expect ¾ Á < ¾ R as observed Energy of harmonic oscillator E R = ½ [v R 2 + · 2 (R-R g ) 2 ] = ½(U 2 + ° 2 V 2 ) ( ) ¾ R / ¾ Á = ° ) 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 U 2 +V 2 = U 0 2 +V 0 2 –E R ! E R ’ such that –2(E R ’-E R ) = U U 2 + ° 2 (V V 2 ) = ( ° 2 -1)(V 0 2 -V 2 ) –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 T R (E,L) and T Á (E,L) 4 phase-space coords subject to 2 constraints ) orbit 2d Has 2 characteristic frequencies  R = 2 ¼ /T R,  Á = 2 ¼ /T Á

Scattering by spirals Consider steadily rotating spiral pattern In co-rotating frame (  p ) –E J = E -  p L is conserved – ) dE =  p dL But dE c =  c dL ! at R CR no change in E R = E-E c = ½ (U 2 + ° 2 V 2 ) but E R increases elsewhere given dL>0 at R 0 at R>R CR When m(  Á -  p ) = § · (Lindblad resonances) spiral resonantly excites E R (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

Orbits in axisymmetric © Orbits with same (E,L z ); but different I 3 ? If 9 I 3 then H=E, I 3 =const, z=0 (3 constraints on 4 coords R, p R, z, p z ) should make p R (R) only

Surface of section “Consequents” on a curve Similar to L 2 = const Conclude 3 integrals (E, L z, I 3 ) conserved We have only approximations to I 3 (R,v R,z,v z ) L 2 = constp z = 0

Age-metallicity distribution Traditional model of chemical evolution Disc divided into annuli Chemical evolution of each annulus isolated from others Consequently at R 0 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 ¾ Fe 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 feed slightly 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 feed slightly 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 feed slightly 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 R i some number dN of stars with guiding-centre radius R g = R i Salpeter IMF Stars added by Kennicutt (1998) law SFR / § gas 1.4 Gas density determined by requirement that § * (R) / exp(-R/R d ) at all times (no inside-out growth) Start with 8 £ 10 9 M ¯ of gas and add gas at rate –where b 1 =0.3Gyr, M 1 =4.5 £ 10 9 M ¯ and b 2 =14Gyr At each timestep a fraction of stars at R i transfer to R j (j=i § 1): –p ij / k ch M j /M max –Rule makes dM out =p i,i+1 M i equal to dM in =p i+1,i M i+1 as required by conservation of L z –Dimensional analysis suggests k ch independent of radius, number of rings, etc

Input physics (cont) When disc “observed” or stars die, the annulus they contribute to calculated by assuming Density of stars in phase space –F(L z ) calculated from N(R g ) – ¾ calculated by assuming that at R 0 1/2 / (age+const) 0.33 –Elsewhere / exp[(R 0 -R)/1.5R d ] (van der Kruit & Searle 82) Vertical structure assumes / (age+const) 0.33 and º (z) / exp[- © R (z)/ ¾ z 2 ] where © R (z)= © (z,R)- © (0,R) As stars migrate in R they take ¾ z with them

Chemical evolution models Traditional (e.g. Chiappini+ 01) –Disc divided into annuli –Each annulus evolves independently –No connection with kinematics With radial migration (Schoenrich & B 2009a,b) –Annuli exchange stars & gas –Stars: In each cohort ¾ (R,t)= ¾ 0 (R)(t+t 0 ) ¯ /t 1 ¯ so epicycle amplitude grows (blurring) In each 15 Myr timestep, stars in annulus i have probability of transferring to annulus j=i § 1 (churning) –Gas: Distributed onflow + inflow + churning –Inflow and onflow controlled by f A and f B ; churning controlled by k ch –f B set by gradient of [Fe/H] in gas disc; f A and k ch fitted to local metallicity distribution

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 f B 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 Z ISM (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 1 st Gyr distinct: bimodal chemistry Schoenrich + B 09 Vertical profile 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 )

The disc divided

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 ~10 8 stars, proper motions of ~10 7 stars, spectra of ~10 6 stars, trig parallaxes of ~10 5 stars By end of decade will have trig parallaxes for ~10 9 stars and spectra of 10 8 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)=v 0 +As System appears hot but is actually cold Problem made worse by errors in ¹ “measured” proper motions Inferred velocities

What we need A hierarchy of dynamical models of variable complexity that yield number density of stars (DF) in (m app,Colour,Z,g,s,l,b,v los,v ®,v ± ) –unwise to bin data in > 10d space of observables Only an equilibrium model allows us to infer ½ DM 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)

Non-rotating barred   L (x,y) = ½v 0 2 ln(R c 2 +x 2 +y 2 /q 2 ) Potential supports boxes and loops

SoS confirms 9 I 2 Closed orbits parent each family

Staeckel  In spherical  orbits bounded by r=const In 2d harmonic oscillator bounded by x,y=const Orbits in  L bounded by ellipsoidal coords (u,v)

Action-angle coords Imagine we had n integrals J i (x,p) for Hamiltonian system with n coords Dream about using the J i as momenta Let conjugate variable be  i Then Hamilton’s equations would be

Action-angle coords (cont) So orbit would be J i =const,  i =  0 +  i t Fact that  i increases without limit while x j, p j bounded suggests x j, p j periodic fns of  i ; can prove conjecture (Arnold book) So Implies motion quasiperiodic:

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 J i d  i = 2  J i So J i to within factor 2  i th 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 – s A  i dq i dp i = s A  i dQ i dP i –On surface of a torus dJ i = 0 so s dx i dv i = 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 – s P  i p i dq i = s P  i P i dQ i –In particular s P v i dx i = s P J i d µ i

Orbits in axisymmetric © Orbits with same (E,L z ); but different I 3 ? If 9 I 3 then H=E, I 3 =const, z=0 (3 constraints on 4 coords R, p R, z, p z ) should make p R (R) only These orbits have same J Á ´ L z but on left J R is smaller and J z is larger than on right in such a way that E = H(J R,J z,J Á ) is the same

Surface of section “Consequents” on a curve Similar to L 2 = const Conclude 3 integrals (E, L z, I 3 ) conserved We have only approximations to I 3 (R,v R,z,v z ) L 2 = constp z = 0 The surface of section is a cross section through the torus

Non-rotating barred   L (x,y) = ½v 0 2 ln(R c 2 +x 2 +y 2 /q 2 ) 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 –n 1   +n 2   +n 3   =0 for some integers n 1, n 2, n 3 –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 /s D d 3  –This is the key to determining an orbit’s observables Action space: –Phase-space volume occupied by orbits in d 3 J is V=(2  ) 3 d 3 J –So action space is accurate representation of phase space

Adiabatic invariance After time t each phase-space point w ´ (x,v) ! new point w t =(x t,v t ) Defines Hamiltonian map H t : w ! w t 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, T 0 ! T t, a torus of  t Since H t : T 0 ! T t, actions of T t = actions of T 0

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 ' 2  R Treat z-oscillations as adiabatically invariant as star oscillates in R

Chaos & diffusion In 2d regular orbits (2d) can form barrier on 3d surface E=const In 3d regular orbits (3d) cannot divide 5d surface E=const So in general expect common sea outside space of regular orbits But stars may move slowly because repeatedly trapped (Arnold diffusion) Likely key process in galactic dynamics No formalism to handle it

Schwarzschild Modelling (Schwarzschild 1979) Standard for modelling external galaxies (e.g., SAURON project) Define observables O j to be quantities such as § (x), §, § 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 w i 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 x J ( µ ), v J ( µ ) Actions J=(J 1,J 2,J 3 ) 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 x J ( µ ), v J ( 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 O Jk, the contribution of orbit J to observable k (more on this later)

How to weight orbits? Could simply adapt Schwarzschild: –create orbit library x J ( µ ) v J ( µ ) for some sampling of 3d action space –Seek non-negative weights w J such that K observational constraints satisfied by  J w J O Jk But it’s better to introduce fewer parameters Simple analytic functions f(J) prove able to provide good fits to data

Example: vertical profiles MN 401, 2318 (2010) Vertical profile simply fitted GCS isothermal prediction GCS model

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

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

Preliminary data (Burnett +10) RAVE internal d.r. Binney 10 model

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

V ¯ (arXiv ; 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 h V i ( ¾ 2 ) to ¾ = 0 (Dehnen & B 98) Underpinned by Stromberg’s eqn

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 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 u 1,u 2,… Then calculate log likelihood L =   ln(p i ) 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 d 3 J 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(M I ) be the luminosity function Then dP = F(M I )dM I f(J) d 3 J d 3 µ is probability of finding a star of abs mag M I in an element of phase space From M I, J, µ we can predict observables u = (l,b,I, , ¹ ®, ¹ ±,v los ) If true values form u’, probability of observing u in d 7 u given errors ¾ i is –P o d 7 u = d 7 u s dM I s d 3 J d 3 µ G(u-u’, ¾ )F(M I )f(J) –where G(u, ¾ ) =  i=1  e -u 2 /2 ¾ 2 /(√2 ¼ ¾ i ) is error Gaussian and u’(M I,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 – s s dM I F(M I ) s d 3 J d 3 µ 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 M I,crit = I lim -5log(s/10pc) and Á += s - 1 MI,crit dM I F(M I ) After repeating above for N points Á /= N Now constraint on f is s d 3 J Á (J)f(J) = const

Further detail For each star in the catalogue we have to evaluate –P o d 7 u = d 7 u s dM I s d 3 J d 3 µ G(u-u’, ¾ )F(M I )f(J) By Markov-Chain Monte-Carlo (MCMC) sampling we obtain N points that sample action space with density f(J) Then P o ' N -1  i s dM I F(M I ) s d 3 µ 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 M I 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

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 ¾ i on velocities of stars without spectra are infinite Likely pairing RAVE + 2MASS

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 J r axis ! spherical radially anisotropic model Shift stars onto J Á axis ! cold disc Spreading stars from J Á axis towards J µ, J r warms disc f(J) = s(J)f 0 (H) with s(J) “shift function” – s d 3 J ± (E-H(J)]s(J) = s d 3 J ± [E-H(J)] –or s dJ µ dJ Á /   s(J) = s dJ µ dJ Á /   Hernquist models ¯ = ½ ¯ = -½

Choice of df: Halo (stellar & dark) Assume ½ (r) and level of anisotropy ¯ =1- ¾ t /2 ¾ r Determine f(H,L)=L -2 ¯ f 1 (H), where L = J µ +|J Á | from Analytic solution for ½ < ¯ <3/2 For ¯ = ½ Express H(J) and use approximation in f 1

Choice of df: disc From § (R) determine f(H,J Á )=f 1 (J Á ) ± (J Á -L c (H)) for cold disc Replace ± -fn with fn of non-zero width, e.g. exp[- · (L c ) J r / ¾ r 2 (L c )] exp[- º (L c )J µ / ¾ z 2 (L c )] Where · in-plane epicycle frequency, º epicycle frequency ? plane Then 1/2 ¼ ¾ r, 1/2 ¼ ¾ z

Generation of tori

History 1989 – 1990 Colin McGill launched torus project Mikko Kaasalainen moved it on Walter Dehnen rewrote code 2006 – Paul McMillan revived it

Tori 3-torus = cube with opposite sides identified Integrable bound orbit lies on 3 torus J 1 (x,p)=const, J 2 (x,p)=const, J 3 (x,p)=const µ i give position on/within torus How to define J i ? Choose 3 non-equivalent loops ° i Then is indep of ° i and µ i is the conjugate variable Note on torus S=   s dp i dq i =  s dJ i d µ i =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 I i but numerical integration required for J i, µ i 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 = (2 ¼ ) -1 s dx p = ½X 2  So X = (2J/  ) 1/2 Also µ =  t So x = (2J/  ) 1/2 cos µ and p=-(2J  ) 1/2 sin µ is the equation of our 1-torus

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

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. 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

Box orbits (cont) 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’,  ) ! (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 b and parameters in f(u), g(v) to minimize h (H- h H i ) 2 i over torus

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

General © (x,y,z) No significant modifications required for general ©

Final solution r’= x(  ) r ;  ’= y(r) z(  ) S=xrp r ’+ yzp  ’ p r = x p r ’ + (dy/dr) z p  ’ p  = r (dx/d  ) p r ’+y (dz/d  ) p  ’ x determined as in simple scheme Choose to have p  ’=\surd(L 2 -L z 2 /cos 2  ’) as in spherical  Then equations for p r & p  (known fns of  ) yield o.d.e.s for Y(  )=y[r(  )] and z(  ) dY/d  =(p r /p  ’z) (dr/d  ) dz/d  =p  /(p  ’ y) Actually make indep variable  defined by  =  max sin 

What have we achieved? Analytic formulae x(J, µ ) and p(J, µ ) So can find at what µ star is at given x & get corresponding p 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, accretion..) Real-space characteristics of orbits naturally related to J so can design DF f(J) to give component of specified shape & kinematics

What have we achieved Numerically orbit given by parameters of toy \Phi 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 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 H 0 (J)= h H i J ± H ´ H-H 0 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 ) = H 0 (J) + ± H(J, µ ) ± H =  H n (J) e in. µ We define H 0 as h H i on our tori Consider d=2: resonance k 1  1 + k 2  2 = 0 Canonical transf with g.f. S = (k. µ )J à + µ 2 J » defines – à = (k. µ ) and » = µ 2 –J à = J 1 /k 1 and J » = J 2 - (k 2 /k 1 )J 1 On resonance  à = 0 so à a slow variable Averaging over » and adjusting phase of à H = H 0 (J à ) + 2  m  H mk cos(m à ) Problem reduced to d=1 so now integrable

Pendulum eq Simplify by –Taking leading term in series (e.g. m=2) –Replacing H mk by constant F –Taylor expanding H 0 using 0 =  à =  H/  J à when ± J à =0 Then H ' ½G ( ± J à ) 2 + F cos m à –Equation of pendulum

Failure of pendulum Problems with pendulum model – ± H doesn’t vanish outside island –Island symmetric in ± J Ã Pendulum Resonance

Solution Expand H mk to 2 nd order in ± J à H mk (J Ã0 + ± J à ) = H mk (J Ã0 +½ ®± J à + ¼ ¯ ( ± J à ) 2 ) H = ½(G+ ¯ cos m à )( ± J à ) 2 + ( ®±J à +F) cos m à ® > 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.