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