1 / 23

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

ramona-wynn
Download Presentation

Kuzmin and Stellar Dynamics

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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

More Related