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Kuzmin and Stellar Dynamics

Kuzmin and Stellar Dynamics. Introduction Dynamical models G.G. Kuzmin’s pioneering work Mass models, orbits, distribution functions Structure of triaxial galaxies Conclusions. Z=18. Z=0. Galaxy Formation and Evolution. Galaxies form by hierarchical accretion/merging

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Kuzmin and Stellar Dynamics

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  1. Kuzmin and Stellar Dynamics • Introduction • Dynamical models • G.G. Kuzmin’s pioneering work • Mass models, orbits, distribution functions • Structure of triaxial galaxies • Conclusions

  2. Z=18 Z=0 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 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) • 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 separableaxisymmetric & triaxial models

  5. Spheres • Hamilton-Jacobi equation separates in (r,θ,φ) • Four integrals of motion: E, Lx, Ly, Lz • All orbits regular: planar rosette’s • Mass model • Defined by density profile ρ(r) • Gravitational potential by two single integrations • 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

  6. Spheres • Large literature on construction of spherical models • Popular mass models include • Hénon’s (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 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 • 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 f=f(E+aI2+bI3) 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.vanAlbada (1953): oblate separable potentials not associated with sensible mass distributions ()

  8. Kuzmin’s 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 (Kuzmin’s Theorem) *Translated by Tenjes in 1996, including additions from 1969  

  9. Kuzmin’s Contribution • Assumption: • n=3 • Fair approximation to Milky Way potential (no dark halo) • Flattened generalisation of Hénon’sisochrone (1961) • n=4 • Exactly spheroidal model with • In limit of extreme flattening • Models  Kuzmin disk; surface density • Rediscovered by Toomre (1963) • 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

  10. Kuzmin’s 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 Schmidt’s (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 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

  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

  12. z x y Separable Triaxial Models • Four orbit families • Same four orbit families found in Schwarschild’s (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

  13. 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 (p=n or n/2) generates similar family de Zeeuw & Pfenniger (1988); Evans & de Zeeuw (1992)

  14. Separable Triaxial Models • 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 • 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 = Σfi(E+aiI2+biI3) Dejonghe; van de Ven et al. 2008 • Through Kuzmin’s work and subsequent follow-up the theory of stationary triaxial dynamical models is now as comprehensive as that for spheres

  15. 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

  16. Stellar Orbits in Galaxies • Galaxies are made of stars • Stars move on orbits (with integrals of motion) • Galaxies are collections of orbits T=1 T=10 T=50 T=200 Image of orbit on sky

  17. Schwarzschild’s Approach • 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) Observed galaxy image Images of model orbits Schwarzschild 1979; Vandervoort 1984

  18. 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

  19. 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

  20. Triaxial Dynamical Model • Parameters • Two axis ratios, two viewing angles, M/L, MBH • 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

  21. 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.

  22. 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 • Perek’s (1962) review did help advertize the results, but even so, much of his work was independently rediscovered • Kuzmin’s work has substantially increased our understanding of galaxy dynamics • And increased the luminosity of Tartu Observatory

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