Kuzmin and Stellar Dynamics

1
Kuzmin and Stellar Dynamics

• Introduction
• Dynamical models
• G.G. Kuzmins pioneering work
• Mass models, orbits, distribution functions
• Structure of triaxial galaxies
• Conclusions

2
Galaxy Formation and Evolution

• Galaxies form by hierarchical accretion/merging
• 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
• Central black hole also influences evolution

3
Observational Approaches

• Study very distant galaxies
• Observe evolution (far away long ago)
• Objects faint and small little information
• Study nearby galaxies
• 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

4
Dynamical Models

• Aim find phase-space distribution function f
• Provides orbital structure
• Mass-density distribution ? ??? f d3v
• Velocities v derive from gravitational potential
  V
• Self-consistent model 4pG? ?2V
• Approaches
• Assume f find ? (but what to assume for f?)
• Assume ? find f (solve integral equation)
• 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

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Spheres

• Hamilton-Jacobi equation separates in (r,?,f)
• Four integrals of motion E, Lx, Ly, Lz
• All orbits regular planar rosettes
• Mass model
• Defined by density profile ?(r)
• Gravitational potential by two single
  integrations
• Selfconsistent models
• Isotropic models ff(E) via Abel inversion
  (Eddington 1916)
• Circular orbit model only orbits with zero
  radial action
• Many distribution functions ff(E), ff(EaL),
  f(E, L), corresponding to different velocity
  anisotropies
• Constrain f further by measuring kinematics

6
Spheres

• Large literature on construction of spherical
  models
• Popular mass models include
• Hénons (1961) isochrone
• The ?-models (e.g., Dehnen 1993)
• 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)

7
The Milky Way

• Stellar motions near the Sun
• If Galaxy oblate and ff(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
• Separable potentials known to have three exact
  integrals of motion, E, I2 and I3, quadratic in
  velocities
• Stäckel (1890), Eddington (1915), Clark (1936)
• Chandrasekhar assumed ff(EaI2bI3) to find ?
• This is the Ellipsoidal Hypothesis
• Model self-consistent only if ? spherical
  limited applicability
• Little interest in opposite route from ? to f
• G.B.van Albada (1953) oblate separable
  potentials not associated with sensible mass
  distributions (?)

8
Kuzmins Contribution

• Set of seminal papers based on his 1952 PhD
  thesis
• Considers mass models with potential
  
• in spheroidal coordinates (?, ?, ?)
  and F(?) a smooth function (?
  ?, ?)
• 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 (Kuzmins
  Theorem)
• Translated by Tenjes in 1996,
  including additions from 1969

?
?
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Kuzmins Contribution

• Assumption
• n3
• Fair approximation to Milky Way potential (no
  dark halo)
• Flattened generalisation of Hénons isochrone
  (1961)
• n4
• Exactly spheroidal model with
• In limit of extreme flattening
• Models ? Kuzmin disk surface density
• Rediscovered by Toomre (1963)
• Model nn0 is weighted sum of models with ngtn0
• This built on his pioneering 1943 work on
  construction of models by superposition of
  inhomogeneous spheroids

10
Kuzmins Contribution

• 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 Schmidts (1956) mass
  model
• Distribution function f is function of
  single-valued integrals of motion only
• Rediscovered by Lynden-Bell (1962)
• f(E, Lz) for model n3 (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

11
Kuzmin 1972

• Generalization of earlier work to triaxial shapes
• Very concise summary in Alma Ata conference 1972
• English translation in IAU 127, 553-556 (1987)
• Potentials separable in ellipsoidal coordinates
  (?,?,?)
• 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 with
• Four major orbit families
• Rediscovered in 1982-1985 (de Zeeuw)
• Via completely independent route

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Separable Triaxial Models

• Four orbit families
• Same four orbit families found in Schwarschilds
  (1979) numerical model for stationary triaxial
  galaxy

1. Box orbit
2. Inner long- axis tube orbit
3. Outer long- axis tube orbit
4. Short-axis tube orbit
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Separable Triaxial Models

• Mass models
• 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
• 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 (pn or n/2) generates
  similar family
• de Zeeuw
  Pfenniger (1988) Evans de Zeeuw (1992)

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Separable Triaxial Models

• Jeans equations obtain ?vi2? directly to ? 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
• 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 ?(x, y, z)
• Abel models f S fi(EaiI2biI3) Dejonghe van
  de Ven et al. 2008
• Through Kuzmins work and subsequent follow-up
  the theory of stationary triaxial dynamical
  models is now as comprehensive as that for
  spheres

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Early-type Galaxies

• Structure
• Mildly triaxial shape
• Central cusp in density profile
• Super-massive central black hole
• 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
• Dynamical models
• Construct by numerical orbit superposition
• Use separable models for testing and insight
• Use kinematic data to constrain f

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Stellar Orbits in Galaxies
T1
T10
T50
T200
Image of orbit on sky

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

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Schwarzschilds Approach
Observed galaxy image
Images of model orbits

• Many different orbits possible in a given galaxy
• Find combination of orbits that are occupied by
  stars in the galaxy ? dynamical model (i.e. f)

Schwarzschild 1979 Vandervoort 1984
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Numerical Orbit Superposition

• No restriction on form of potential
• Arbitrary geometry
• Multiple components (BH, stars, dark halo)
• No restriction on distribution function
• No need to know analytic integrals of motion
• Full range of velocity anisotropy
• 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
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The E3 Galaxy NGC 4365

• Kinematically Decoupled Core
• Long-axis rotator, core rotates
  around short axis (Surma Bender
  1995)
• SAURON kinematics
• Rotation axes of main body and
  core misaligned by 82o
• Consistent with triaxial shape, both
  long-axis short-axis tubes
  occupied
• Customary interpretation
• Core is distinct, and remnant of
  last major accretion 12 Gyr
  ago

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Triaxial Dynamical Model

• Parameters
• Two axis ratios, two viewing angles, M/L,
  MBH
• Best-fit model
• Fairly oblate (0.70.951)
• 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
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Dynamics of Slow Rotators

• 11 slow rotators in representative SAURON sample
• Range of triaxiality 0.2 ? T ? 0.7 ? no prolate
  objects
• Mildly radially anisotropic
• Most have KDC
• 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
• KDC not distinct from main body
• In harmony with smooth Mgb and Fe gradients

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

• Kuzmin was a very gifted dynamicist
• 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
• Pereks (1962) review did help advertize the
  results, but even so, much of his work was
  independently rediscovered
• Kuzmins work has substantially increased our
  understanding of galaxy dynamics
• And increased the luminosity of Tartu Observatory

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