Kuzmin and Stellar Dynamics
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Introduction
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Dynamical models
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G.G. Kuzmin’s pioneering work
– Mass models, orbits, distribution functions
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Structure of triaxial galaxies
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Conclusions
Galaxy Formation and Evolution
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Galaxies form by hierarchical accretion/merging
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Matter clumps through gravitation
Primordial gas starts forming first stars
Stars produce heavier elements (‘metals’)
Subsequent generations of stars contain more metals
Massive galaxies form from assembly of smaller units
Galaxy encounters still occur
– Deformation, stripping, merging
– Galaxies continue to evolve
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Central black hole also influences evolution
Z=18
Z=0
Observational Approaches
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Study very distant galaxies
– Observe evolution (far away = long ago)
– Objects faint and small: little information
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Study nearby galaxies
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Light not resolved in individual stars
Objects large & bright: structure accessible
Infer evolution through archaeology
Fossil record is cleanest in early-type galaxies
Study resolved stellar populations
– Ages, metallicities and motions of stars
– Archaeology of Milky Way and its neighbors
Dynamical Models
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Aim: find phase-space distribution function f
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Provides orbital structure
Mass-density distribution ρ = ∫∫∫ f d3v
Velocities v derive from gravitational potential V
Self-consistent model: 4πGρ= 2V
Approaches
– Assume f find ρ (but what to assume for f?)
– Assume ρ find f (solve integral equation)
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Use Jeans theorem f = f(I) to make progress
– Provides f(E,L) for spheres, f(E,Lz) for axisymmetry
– f(E,I2,I3) for separable axisymmetric & triaxial models
Spheres
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Hamilton-Jacobi equation separates in (r,θ,φ)
– Four integrals of motion: E, Lx, Ly, Lz
– All orbits regular: planar rosette’s
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Mass model
– Defined by density profile ρ(r)
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Gravitational potential by two single integrations
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Selfconsistent models
– Isotropic models f=f(E) via Abel inversion (Eddington 1916)
– Circular orbit model: only orbits with zero radial action
– Many distribution functions: f=f(E), f=f(E+aL), f(E, L),
corresponding to different velocity anisotropies
– Constrain f further by measuring kinematics
Spheres
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Large literature on construction of spherical models
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Popular mass models include
– Hénon’s (1961) isochrone
– The -models (e.g., Dehnen 1993)
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Already found by e.g., Franx in ~1988
Include the Jaffe (1982) and Hernquist (1990) models
Many of these were studied much earlier by Kuzmin
and collaborators
– In particular Veltmann (and later Tenjes)
– Density profiles and distribution functions
– Results not well known in Western literature, but
summarized in IAU 153, 363-366 (1993)
The Milky Way
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Stellar motions near the Sun
– If Galaxy oblate and f=f(E, Lz) then vR2= vz2 and vRvz=0
– Observed: vR2 v2 vz2 and vRvz0
– Galactic potential must support a third integral of motion I3
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Separable potentials known to have three exact
integrals of motion, E, I2 and I3, quadratic in velocities
– Stäckel (1890), Eddington (1915), Clark (1936)
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Chandrasekhar assumed f=f(E+aI2+bI3) to find 
– This is the ‘Ellipsoidal Hypothesis’
– Model self-consistent only if  spherical: limited applicability
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Little interest in opposite route: from  to f
– G.B.van Albada (1953): oblate separable potentials not
associated with sensible mass distributions ()
Kuzmin’s Contribution
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Set of seminal papers based on his 1952 PhD thesis*
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Considers mass models with potential
V 
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F (  )  F ( )
 
in spheroidal coordinates (, , )
and F() a smooth function ( = , )
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These potentials have
– Three exact integrals of motion E, Lz and I3
– Useful associated densities, given by simple formula
– (R, z)  0 if and only if (0, z)  0 (Kuzmin’s Theorem)
*Translated by Tenjes in 1996, including additions from 1969
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Kuzmin’s Contribution
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Assumption:
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n=3
d
d
(   )
1/ 2
F ' ( )  
n / 2
– Fair approximation to Milky Way potential (no dark halo)
– Flattened generalisation of Hénon’s isochrone (1961)
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n=4
– Exactly spheroidal model
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 
0
(1  m )
2
2
with
In limit of extreme flattening
– Models  Kuzmin disk; surface density
– Rediscovered by Toomre (1963)
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m  R  z /q

2
2
2
2
0
(1  R )
2
3/2
Model n=n0 is weighted sum of models with n>n0
– This built on his pioneering 1943 work on construction of
models by superposition of inhomogeneous spheroids
Kuzmin’s Contribution
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Orbits in oblate separable models
– All short-axis tubes (bounded by coordinate surfaces)
– Similar to orbits in Milky Way found numerically by
Ollongren (1962) using Schmidt’s (1956) mass model
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Distribution function f is function of single-valued
integrals of motion only
– Rediscovered by Lynden-Bell (1962)
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f(E, Lz) for model n=3 (with Kutuzov, 1960)
– (R, z) can be written explicitly as (R, V) without any
reference to spheroidal coordinates
– Allows computing f(E, Lz) via series expansion à la Fricke
– f(E, Lz, I3) found by Dejonghe & de Zeeuw (1988) making
full use of the elegant properties of the model
Kuzmin 1972
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Generalization of earlier work to triaxial shapes
– Very concise summary in Alma Ata conference 1972
– English translation in IAU 127, 553-556 (1987)
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Potentials separable in ellipsoidal coordinates (,,)
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Three exact integrals of motion E, I2 and I3
(x, y, z)  0 if and only if (0, 0, z)  0
Elegant formula for density
Includes ellipsoidal model:   (1 m ) with m  x
Four major orbit families
2
0
2
2
Rediscovered in 1982-1985 (de Zeeuw)
– Via completely independent route
2
 y / p  z /q
2
2
2
2
Separable Triaxial Models
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Four orbit families
z
x
y
1. Box orbit
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2. Inner longaxis tube orbit
3. Outer longaxis tube orbit
4. Short-axis
tube orbit
Same four orbit families found in Schwarschild’s
(1979) numerical model for stationary triaxial galaxy
Separable Triaxial Models
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Mass models
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Defined by short-axis density profile & central axis ratios
Stationary triaxial shape, with central core
Gravitational potential by two single integrations
Each model is weighted integral of constituent ellipsoids
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Weight function follows via Stieltjes transform
Projection is same weighted integral of constituent elliptic disks:
new method for finding potential of disks
These properties shared by larger set of models
– Each ellipsoid
 
0
(1  m )
2
p
(p=n or n/2) generates similar family
de Zeeuw & Pfenniger (1988); Evans & de Zeeuw (1992)
Separable Triaxial Models
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Jeans equations: obtain vi2 directly to ρ and V
– Three partial differential equations for three unknowns
– Equations written down by Lynden-Bell (1960), and solved
by van de Ven et al. (2003). No guarantee that f  0
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Analytic selfconsistent models
– Thin-tube orbit models (only tubes with zero radial action)
– Existence of more than one major orbit family: f(E, I2, I3)
not uniquely defined by ρ(x, y, z)
– Abel models f = Σ fi(E+aiI2+biI3) Dejonghe; van de Ven et al. 2008
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Through Kuzmin’s work and subsequent follow-up
the theory of stationary triaxial dynamical models is
now as comprehensive as that for spheres
Early-type Galaxies
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Structure
– Mildly triaxial shape
– Central cusp in density profile
– Super-massive central black hole
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Implications for orbital structure
– No global extra integrals I2 and I3
– Three tube orbit families
– Box orbits replaced by mix of boxlets (higher-order
resonant orbits) and chaotic orbits: slow evolution
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Dynamical models
– Construct by numerical orbit superposition
– Use separable models for testing and insight
– Use kinematic data to constrain f
Stellar Orbits in Galaxies
T=1
T=10
T=50
T=200
Image of orbit on sky
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Galaxies are made of stars
 Stars move on orbits (with integrals of motion)
 Galaxies are collections of orbits
Schwarzschild’s Approach
Images of model orbits
Observed
galaxy image
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Many different orbits possible in a given galaxy
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Find combination of orbits that are occupied by
stars in the galaxy  dynamical model (i.e. f)
Schwarzschild 1979; Vandervoort 1984
Numerical Orbit Superposition
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No restriction on form of potential
– Arbitrary geometry
– Multiple components (BH, stars, dark halo)
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No restriction on distribution function
– No need to know analytic integrals of motion
– Full range of velocity anisotropy
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Include all kinematic observables
– Fit on sky plane
– Codes exist to do this for spherical, axisymmetric and
non-tumbling triaxial geometry
Leiden group: Cretton, Cappellari, van den Bosch; Gebhardt & Richstone; Valluri
The E3 Galaxy NGC 4365
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Kinematically Decoupled Core
– Long-axis rotator, core rotates
around short axis (Surma & Bender 1995)
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SAURON kinematics:
– Rotation axes of main body and
core misaligned by 82o
– Consistent with triaxial shape, both
long-axis & short-axis tubes occupied
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Customary interpretation:
– Core is distinct, and remnant of
last major accretion ~12 Gyr ago
Triaxial Dynamical Model
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Parameters
– Two axis ratios, two viewing
angles, M/L, MBH
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Best-fit model
– Fairly oblate (0.7:0.95:1)
– Short axis tubes dominate,
but ~50% counter rotate,
except in core; cf NGC4550
– Net rotation caused by
long-axis tubes, except in core
– KDC not a physical subunit,
but appears so because of
embedded counter-rotating structure
van den Bosch et al. 2008
Dynamics of Slow Rotators
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11 slow rotators in representative SAURON sample
– Range of triaxiality: 0.2  T  0.7  no prolate objects
– Mildly radially anisotropic
– Most have ‘KDC’
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Dynamical structure
– Short axis tubes dominate
– Smooth variation with radius
– ~similar to dry merger simulations
Jesseit et al. 2005; Hoffman et al. 2010
– No sudden transition at RKDC
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KDC not distinct from main body
– In harmony with smooth Mgb and Fe gradients
van den Bosch et al. 2011, in prep.
Conclusions
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Kuzmin was a very gifted dynamicist
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Much of this work was unknown in West
– Few read Russian; translations came later, but even today
most papers are not in, e.g., ADS
– Kuzmin sent short English synopses to key dynamicists,
but these were not widely distributed
– Perek’s (1962) review did help advertize the results, but
even so, much of his work was independently rediscovered
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Kuzmin’s work has substantially increased our
understanding of galaxy dynamics
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And increased the luminosity of Tartu Observatory