Kuzmin and Stellar DynamicsKuzmin and Stellar Dynamics

IntroductionIntroduction

Dynamical modelsDynamical models

G.G. Kuzmin’s pioneering workG.G. Kuzmin’s pioneering work– Mass models, orbits, distribution functionsMass models, orbits, distribution functions

Structure of triaxial galaxiesStructure of triaxial galaxies

ConclusionsConclusions

Z=18

Z=0

Galaxy Formation and Galaxy Formation and EvolutionEvolution

Galaxies form by hierarchical accretion/mergingGalaxies form by hierarchical accretion/merging– Matter clumps through gravitationMatter clumps through gravitation– Primordial gas starts forming first starsPrimordial gas starts forming first stars– Stars produce heavier elements (‘metals’)Stars produce heavier elements (‘metals’)– Subsequent generations of stars contain more metalsSubsequent generations of stars contain more metals– Massive galaxies form from assembly of smaller unitsMassive galaxies form from assembly of smaller units

Galaxy encounters still occurGalaxy encounters still occur– Deformation, stripping, mergingDeformation, stripping, merging– Galaxies continue to evolveGalaxies continue to evolve

Central black hole also influences evolutionCentral black hole also influences evolution

Observational ApproachesObservational Approaches Study very distant galaxiesStudy very distant galaxies

– Observe evolution (far away = long ago)Observe evolution (far away = long ago)– Objects faint and small: little information Objects faint and small: little information

Study nearby galaxiesStudy nearby galaxies– Light not resolved in individual starsLight not resolved in individual stars– Objects large & bright: structure accessibleObjects large & bright: structure accessible– Infer evolution through archaeologyInfer evolution through archaeology– Fossil record is cleanest in early-type galaxiesFossil record is cleanest in early-type galaxies

Study resolved stellar populationsStudy resolved stellar populations– Ages, metallicities and motions of starsAges, metallicities and motions of stars– Archaeology of Milky Way and its neighborsArchaeology of Milky Way and its neighbors

Dynamical ModelsDynamical Models Aim: find phase-space distribution function fAim: find phase-space distribution function f

– Provides orbital structureProvides orbital structure– Mass-density distribution Mass-density distribution ρρ = ∫∫∫ = ∫∫∫ f df d33vv– Velocities v derive from gravitational potential VVelocities v derive from gravitational potential V– Self-consistentSelf-consistent model: 4 model: 4ππGGρρ= = 22VV

ApproachesApproaches– Assume f find Assume f find ρρ (but what to assume for (but what to assume for ff?)?)– Assume Assume ρρ find f (solve integral equation) find f (solve integral equation)

Use Jeans theorem f = f(I) to make progressUse Jeans theorem f = f(I) to make progress– Provides f(E,L) for spheres, f(E,LProvides f(E,L) for spheres, f(E,Lzz) for axisymmetry ) for axisymmetry

– f(E,If(E,I22,I,I33) for ) for separableseparable axisymmetric & triaxial models axisymmetric & triaxial models

SpheresSpheres Hamilton-Jacobi equation separates in (r,Hamilton-Jacobi equation separates in (r,θθ,,φφ))

– Four integrals of motion: E, LFour integrals of motion: E, Lxx, L, Lyy, L, Lzz

– All orbits regular: planar rosette’sAll orbits regular: planar rosette’s

Mass modelMass model– Defined by density profile Defined by density profile ρρ(r)(r)

Gravitational potential by two single integrationsGravitational potential by two single integrations

Selfconsistent modelsSelfconsistent models– Isotropic models f=f(E) via Abel inversion (Eddington 1916)Isotropic models f=f(E) via Abel inversion (Eddington 1916)– Circular orbit model: only orbits with zero radial actionCircular orbit model: only orbits with zero radial action– Many distribution functions: f=f(E), f=f(E+aL), f(E, L), Many distribution functions: f=f(E), f=f(E+aL), f(E, L),

corresponding to different velocity anisotropiescorresponding to different velocity anisotropies– Constrain f further by measuring kinematicsConstrain f further by measuring kinematics

SpheresSpheres Large literature on construction of spherical modelsLarge literature on construction of spherical models

Popular mass models includePopular mass models include– Hénon’s (1961) isochrone Hénon’s (1961) isochrone – The The -models (e.g., Dehnen 1993)-models (e.g., Dehnen 1993)

Already found by e.g., Franx in ~1988Already found by e.g., Franx in ~1988 Include the Jaffe (1982) and Hernquist (1990) modelsInclude the Jaffe (1982) and Hernquist (1990) models

Many of these were studied much earlier by Kuzmin Many of these were studied much earlier by Kuzmin and collaborators and collaborators – In particular Veltmann (and later Tenjes) In particular Veltmann (and later Tenjes) – Density profiles and distribution functionsDensity profiles and distribution functions– Results not well known in Western literature, but Results not well known in Western literature, but

summarized in IAU 153, 363-366 (1993)summarized in IAU 153, 363-366 (1993)

The Milky WayThe Milky Way Stellar motions near the SunStellar motions near the Sun

– If Galaxy oblate and f=f(E, LIf Galaxy oblate and f=f(E, Lzz) then ) then vvRR22= = vvzz

22 and and vvRRvvzz=0 =0

– Observed: Observed: vvRR22 vv

22 vvzz22 and and vvRRvvzz0 0

– Galactic potential must support a third integral of motion IGalactic potential must support a third integral of motion I33

Separable potentials known to have three exact Separable potentials known to have three exact integrals of motion, E, Iintegrals of motion, E, I22 and I and I33, quadratic in velocities, quadratic in velocities– Stäckel (1890), Eddington (1915), Clark (1936)Stäckel (1890), Eddington (1915), Clark (1936)

Chandrasekhar Chandrasekhar assumedassumed f=f(E+aI f=f(E+aI22+bI+bI33) to find ) to find – This is the ‘Ellipsoidal Hypothesis’This is the ‘Ellipsoidal Hypothesis’– Model self-consistent only if Model self-consistent only if spherical: limited applicability spherical: limited applicability

Little interest in opposite route: from Little interest in opposite route: from to f to f – G.B.van Albada (1953): oblate separable potentials not G.B.van Albada (1953): oblate separable potentials not

associated with sensible mass distributions (associated with sensible mass distributions ())

Kuzmin’s ContributionKuzmin’s Contribution Set of seminal papers based on his 1952 PhD thesis* Set of seminal papers based on his 1952 PhD thesis*

Considers mass models with potential Considers mass models with potential

in spheroidal coordinates (in spheroidal coordinates (, , , , ) ) and F(and F() a smooth function () a smooth function ( = = , , ))

These potentials haveThese potentials have– Three exact integrals of motion E, LThree exact integrals of motion E, Lzz and I and I33

– Useful associated densities, given by simple formulaUseful associated densities, given by simple formula– (R, z) (R, z) 0 if and only if 0 if and only if (0, z) (0, z) 0 (Kuzmin’s Theorem) 0 (Kuzmin’s Theorem)

*Translated by Tenjes in 1996, including additions from 1969*Translated by Tenjes in 1996, including additions from 1969

)()( FF

V

Kuzmin’s ContributionKuzmin’s Contribution Assumption: Assumption:

n=3n=3– Fair approximation to Milky Way potential (no dark halo)Fair approximation to Milky Way potential (no dark halo)– Flattened generalisation of Hénon’s isochrone (1961)Flattened generalisation of Hénon’s isochrone (1961)

n=4 n=4 – Exactly spheroidal model with Exactly spheroidal model with

In limit of extreme flatteningIn limit of extreme flattening– Models Models Kuzmin disk; surface density Kuzmin disk; surface density – Rediscovered by Toomre (1963)Rediscovered by Toomre (1963)

Model n=nModel n=n00 is weighted sum of models with n>n is weighted sum of models with n>n00

– This built on his pioneering 1943 work on construction of This built on his pioneering 1943 work on construction of models by superposition of inhomogeneous spheroids models by superposition of inhomogeneous spheroids

2/2/1 )(')( nFd

d

220

)1( m

2/320

)1( R

2222 / qzRm

Kuzmin’s ContributionKuzmin’s Contribution Orbits in oblate separable modelsOrbits in oblate separable models

– All short-axis tubes (bounded by coordinate surfaces)All short-axis tubes (bounded by coordinate surfaces)– Similar to orbits in Milky Way found numerically by Similar to orbits in Milky Way found numerically by

Ollongren (1962) using Schmidt’s (1956) mass modelOllongren (1962) using Schmidt’s (1956) mass model

Distribution function f is function of single-valued Distribution function f is function of single-valued integrals of motion onlyintegrals of motion only– Rediscovered by Lynden-Bell (1962)Rediscovered by Lynden-Bell (1962)

f(E, Lf(E, Lzz) for model n=3 (with Kutuzov, 1960) ) for model n=3 (with Kutuzov, 1960) – (R, z) can be written explicitly as (R, z) can be written explicitly as (R, V) without any (R, V) without any

reference to spheroidal coordinatesreference to spheroidal coordinates

– Allows computing f(E, LAllows computing f(E, Lzz) via series expansion à la Fricke) via series expansion à la Fricke

– f(E, Lf(E, Lzz, I, I33) found by Dejonghe & de Zeeuw (1988) making ) found by Dejonghe & de Zeeuw (1988) making

full use of the elegant properties of the model full use of the elegant properties of the model

Kuzmin 1972Kuzmin 1972 Generalization of earlier work to triaxial shapesGeneralization of earlier work to triaxial shapes

– Very concise summary in Alma Ata conference 1972Very concise summary in Alma Ata conference 1972– English translation in IAU 127, 553-556 (1987)English translation in IAU 127, 553-556 (1987)

Potentials separable in ellipsoidal coordinates (Potentials separable in ellipsoidal coordinates (,,,,))– Three exact integrals of motion E, IThree exact integrals of motion E, I22 and I and I33

– (x, y, z) (x, y, z) 0 if and only if 0 if and only if (0, 0, z) (0, 0, z) 0 0– Elegant formula for densityElegant formula for density– Includes ellipsoidal model: with Includes ellipsoidal model: with – Four major orbit familiesFour major orbit families

Rediscovered in 1982-1985 (de Zeeuw)Rediscovered in 1982-1985 (de Zeeuw)– Via completely independent routeVia completely independent route

220

)1( m

222222 // qzpyxm

Separable Triaxial ModelsSeparable Triaxial Models Four orbit familiesFour orbit families

Same four orbit families found in Schwarschild’s Same four orbit families found in Schwarschild’s (1979) numerical model for stationary triaxial galaxy(1979) numerical model for stationary triaxial galaxy

2. Inner long-axis tube orbit

3. Outer long-axis tube orbit

4. Short-axis tube orbit

1. Box orbit

x y

z

Separable Triaxial ModelsSeparable Triaxial Models Mass modelsMass models

– Defined by short-axis density profile & central axis ratiosDefined by short-axis density profile & central axis ratios– Stationary triaxial shape, with central coreStationary triaxial shape, with central core– Gravitational potential by two single integrationsGravitational potential by two single integrations– Each model is weighted integral of constituent ellipsoidsEach model is weighted integral of constituent ellipsoids

Weight function follows via Stieltjes transformWeight function follows via Stieltjes transform Projection is same weighted integral of constituent elliptic disks: Projection is same weighted integral of constituent elliptic disks:

new method for finding potential of disksnew method for finding potential of disks

These properties shared by larger set of modelsThese properties shared by larger set of models– Each ellipsoid (p=n or n/2) generates similar familyEach ellipsoid (p=n or n/2) generates similar family

de Zeeuw & Pfenniger (1988); Evans & de Zeeuw (1992)de Zeeuw & Pfenniger (1988); Evans & de Zeeuw (1992)

pm )1( 20

Separable Triaxial ModelsSeparable Triaxial Models Jeans equations: obtain Jeans equations: obtain vvii

22 directly to directly to ρρ and V and V– Three partial differential equations for three unknownsThree partial differential equations for three unknowns– Equations written down by Lynden-Bell (1960), and solved Equations written down by Lynden-Bell (1960), and solved

by van de Ven et al. (2003). No guarantee that f by van de Ven et al. (2003). No guarantee that f 0 0

Analytic selfconsistent models Analytic selfconsistent models – Thin-tube orbit models (only tubes with zero radial action) Thin-tube orbit models (only tubes with zero radial action)

– Existence of more than one major orbit family: f(E, IExistence of more than one major orbit family: f(E, I22, I, I33) )

notnot uniquely defined by uniquely defined by ρρ(x, y, z)(x, y, z)

– Abel models f = Abel models f = ΣΣ f fii(E+a(E+aiiII22+b+biiII33) ) Dejonghe; van de Ven et al. 2008Dejonghe; van de Ven et al. 2008

Through Kuzmin’s work and subsequent follow-up Through Kuzmin’s work and subsequent follow-up the theory of stationary triaxial dynamical models is the theory of stationary triaxial dynamical models is now as comprehensive as that for spheresnow as comprehensive as that for spheres

Early-type GalaxiesEarly-type Galaxies StructureStructure

– Mildly triaxial shapeMildly triaxial shape– Central cusp in density profileCentral cusp in density profile– Super-massive central black holeSuper-massive central black hole

Implications for orbital structureImplications for orbital structure– No No globalglobal extra integrals I extra integrals I22 and I and I33

– Three tube orbit families Three tube orbit families – Box orbits replaced by mix of Box orbits replaced by mix of boxletsboxlets (higher-order (higher-order

resonant orbits) and resonant orbits) and chaoticchaotic orbits: slow evolution orbits: slow evolution

Dynamical modelsDynamical models– Construct by numerical orbit superpositionConstruct by numerical orbit superposition– Use separable models for testing and insightUse separable models for testing and insight– Use kinematic data to constrain fUse kinematic data to constrain f

Stellar Orbits in GalaxiesStellar Orbits in Galaxies

Galaxies are made of starsGalaxies are made of stars Stars move on orbits (with integrals of motion)Stars move on orbits (with integrals of motion) Galaxies are collections of orbitsGalaxies are collections of orbits

Image of orbit on skyImage of orbit on sky

T=1 T=10

T=50 T=200

Schwarzschild’s ApproachSchwarzschild’s Approach

Images of model orbitsImages of model orbits Observed Observed galaxy imagegalaxy image

Many different orbits possible in a given galaxyMany different orbits possible in a given galaxy

Find combination of orbits that are occupied by Find combination of orbits that are occupied by stars in the galaxy stars in the galaxy dynamical model (i.e. dynamical model (i.e. ff))

Schwarzschild 1979; Vandervoort 1984Schwarzschild 1979; Vandervoort 1984

Numerical Orbit Numerical Orbit SuperpositionSuperposition

No restriction on form of potentialNo restriction on form of potential– Arbitrary geometryArbitrary geometry– Multiple components (BH, stars, dark halo)Multiple components (BH, stars, dark halo)

No restriction on distribution functionNo restriction on distribution function– No need to know analytic integrals of motionNo need to know analytic integrals of motion– Full range of velocity anisotropyFull range of velocity anisotropy

Include all kinematic observablesInclude all kinematic observables– Fit on sky planeFit on sky plane– Codes exist to do this for spherical, axisymmetric and Codes exist to do this for spherical, axisymmetric and

non-tumbling triaxial geometrynon-tumbling triaxial geometry

Leiden group: Cretton, Cappellari, van den Bosch; Gebhardt & Richstone; ValluriLeiden group: Cretton, Cappellari, van den Bosch; Gebhardt & Richstone; Valluri

The E3 Galaxy NGC 4365The E3 Galaxy NGC 4365 Kinematically Decoupled CoreKinematically Decoupled Core

– Long-axis rotator, core rotates Long-axis rotator, core rotates around short axis around short axis (Surma & Bender 1995) (Surma & Bender 1995)

SAURON kinematics:SAURON kinematics:– Rotation axes of main body and Rotation axes of main body and

core misaligned by 82core misaligned by 82o o

– Consistent with triaxial shape, both Consistent with triaxial shape, both long-axis & short-axis tubes occupiedlong-axis & short-axis tubes occupied

Customary interpretation:Customary interpretation:– Core is distinct, and remnant of Core is distinct, and remnant of

last major accretion ~12 Gyr agolast major accretion ~12 Gyr ago

Triaxial Dynamical ModelTriaxial Dynamical Model ParametersParameters

– Two axis ratios, two viewing Two axis ratios, two viewing angles, M/L, Mangles, M/L, MBHBH

Best-fit modelBest-fit model– Fairly oblate (0.7:0.95:1) Fairly oblate (0.7:0.95:1) – Short axis tubes dominate, Short axis tubes dominate,

but ~50% but ~50% counter rotatecounter rotate, , except in core; cf NGC4550except in core; cf NGC4550

– Net rotation caused by Net rotation caused by long-axis tubes, except in core long-axis tubes, except in core

– KDC KDC notnot a physical subunit, a physical subunit, but appears so because of but appears so because of embedded counter-rotating structureembedded counter-rotating structure

van den Bosch et al. 2008van den Bosch et al. 2008

Dynamics of Slow RotatorsDynamics of Slow Rotators 11 slow rotators in representative SAURON sample11 slow rotators in representative SAURON sample

– Range of triaxiality: 0.2 Range of triaxiality: 0.2 T T 0.7 0.7 no prolate objects no prolate objects– Mildly radially anisotropicMildly radially anisotropic– Most have ‘KDC’Most have ‘KDC’

Dynamical structure Dynamical structure – Short axis tubes dominateShort axis tubes dominate– Smooth variation with radiusSmooth variation with radius– ~similar to dry merger simulations ~similar to dry merger simulations

Jesseit et al. 2005; Hoffman et al. 2010Jesseit et al. 2005; Hoffman et al. 2010

– No sudden transition at RNo sudden transition at RKDC KDC

KDC not distinct from main bodyKDC not distinct from main body– In harmony with smooth Mgb and Fe gradients In harmony with smooth Mgb and Fe gradients

van den Bosch et al. 2011, in prep.van den Bosch et al. 2011, in prep.

ConclusionsConclusions Kuzmin was a very gifted dynamicistKuzmin was a very gifted dynamicist

Much of this work was unknown in WestMuch of this work was unknown in West– Few read Russian; translations came later, but even today Few read Russian; translations came later, but even today

most papers are not in, e.g., ADSmost papers are not in, e.g., ADS– Kuzmin sent short English synopses to key dynamicists, Kuzmin sent short English synopses to key dynamicists,

but these were not widely distributedbut these were not widely distributed– Perek’s (1962) review did help advertize the results, but Perek’s (1962) review did help advertize the results, but

even so, much of his work was independently rediscoveredeven so, much of his work was independently rediscovered

Kuzmin’s work has substantially increased our Kuzmin’s work has substantially increased our understanding of galaxy dynamicsunderstanding of galaxy dynamics

And increased the luminosity of Tartu ObservatoryAnd increased the luminosity of Tartu Observatory