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Statistical mechanics of stellar systems near galaxy centers 3 pc Hubble Space Telescope 3”x3” Lauer et al. AJ 1998 3 pc 1” = 3.6 pc 1000” = 3600 pc Hubble Space Telescope 3”x3” Lauer et al. AJ 1998 radio source Sagittarius A* 0.1 pc infrared astronomy group, MPE Relaxation in the centers of galaxies • in contrast to laboratory gases, galaxies are collisionless, e.g., time for stars near the Sun to relax to a Maxwell-Boltzmann distribution is 1014 years = 104 times the age of the Galaxy • however, at 0.1 pc from the center of our Galaxy the density of stars is 108 times higher than around the Sun • in a spherical stellar system with 1-dimensional velocity dispersion ¾, the relaxation time at radius r is t r elax = 5 Gyr 1Mm ¯ ³ ¾ 100 r ¡ 1 km s 1 ´2 pc 2 pc • in Andromeda, stars are old but relaxed region is barely resolved • in the Milky Way, the relaxed region is well-resolved but there is a lot of recent star formation 0.1 pc Relaxation in the centers of galaxies • most nearby galaxies contain massive black holes at their centers • black holes dominate the dynamics inside the “sphere of influence” r¤ ´ GM ² ¾2 = 4:3 pc 10M7 M² ¯ t r elax = 5 Gyr 1Mm ¯ 100 ³ 100 ¡ 1 km s ´2 ¾ ³ ¾ r ¡ 1 km s 1 ´2 pc 2 pc • thus: the relaxed stellar system is also a near-Keplerian stellar system 4£106 M¯ 0.1 pc 108 M¯ The black hole in the Galactic center 10 X Sun-Pluto distance 0.01 pc Ghez et al. (2005) - UCLA Eisenhauer et al. (2005) - MPE • center of attraction is located at the radio source Sagittarius A* which is presumably the black hole Sgr A* • closest approach to black hole is only a few times the size of the solar system • orbits are closed ellipses so central mass must be not much bigger than the solar system (i.e. < 1010 km) • M = (3.95§0.06)£ 106 M¯ if distance R0 = 8000 pc = 8 kpc • R0 = 8.33§0.35 kpc (Gillessen et al. 2009) only alternatives to a black hole are: • boson star • cluster of 1010 planetmass black holes NGC 4258 Miyoshi et al. (1995) Herrnstein et al. (1999) NGC 4258 Humphreys et al. (2007) 0.15 pc • high-velocity maser emission from disk shows Keplerian rotation at r ~ 0.2 pc • M=(3.9±0.1)×107 solar masses • masers at systemic velocity show angular speed 31.5±1 milliarcseconds/yr and acceleration 9.3±0.3 km/s/yr • proper motion and acceleration yield independent distance estimates of 7.1±0.2 and 7.2±0.2 million parsecs – the most accurate distance we have to any object outside our own Galaxy xx • by now there are ~40black hole mass (solar masses) 50 detections of a massive dark object in nearby galaxies, 106-109 M¯ • mass determinations from • stellar dynamics • gas dynamics • maser disks • roughly, M / mass or luminosity in stars, M ~ 0.002 Mstars • promise of better data: • new maser disks • laser guide star adaptive optics stellar luminosity of host galaxy (solar units) Gültekin et al. (2009) black hole mass (solar masses) • tighter correlation is with velocity dispersion ¾ of host galaxy; roughly M / ¾4 with scatter of 0.3 in log10M for elliptical galaxies velocity dispersion of host galaxy (km/s) Gültekin et al. (2009) • at distances from the Galactic center < a few pc the relaxation time due to gravitational encounters is less than the age of the Galaxy • thermodynamic equilibrium in potential ©=-GM/r yields density ¡ n(r ) / exp ¡ © kT ¢ / exp ¡ GM m ¢ kT r Boltzmann constant • this doesn’t apply because stars at small radii are eaten by the black hole • correct solution including absorbing boundary condition is (Bahcall & Wolf 1976) n(r ) / r ¡ 7=4 • this is not easy to test because (i) wide range of masses, distribution not well known; (ii) recent star formation; (iii) possible dark remnants stellar mass temperature (1063K) Preto et al. (2004) ½ / r-7/4 Resonant relaxation in dense stellar systems • on timescales longer than the orbital period each stellar orbit can be thought of as an eccentric wire • orbits are specified by semi-major axis, , orbit normal, and eccentricity vector • each wire exerts steady force on all other wires, leading to secular evolution of the angular momentum and eccentricity vector • energy (semi-major axis) of each orbit is conserved, but angular momentum and eccentricity vectors evolve to a relaxed state (thermodynamic equilibrium) • sometimes called “scalar” resonant relaxation because the scalar eccentricity relaxes Rauch & Tremaine (1996) Resonant relaxation in dense stellar systems • in the Galactic center the stellar system is roughly spherical. At observable radii its mass is a significant fraction of the black hole’s (~5%) • the eccentricity vector precesses much faster than the angular momentum vector • orbit can be thought of as an axisymmetric disk or annulus • semi-major axis and eccentricity of each orbit is conserved, but angular momentum vector or orbit normal is not (“vector” resonant relaxation) t r el ax (r ) » t or b (r ) orbital period mass of a star M mN 1=2 mass of the black hole number of stars inside radius r (1) current resolution limit 1” current resolution limit 1” Thermodynamic equilibrium from vector resonant relaxation • mutual torques can lead to relaxation of orbit normals or angular momenta • energy (semi-major axis a) and scalar angular momentum (or eccentricity e) of each orbit is conserved, but vector angular momentum or orbit normal is not • interaction energy between stars i and j is mimjf(ai,aj,ei,ej,cos µij) where µij is the angle between the orbit normals masses Toy model #1: eccentricities semi-major axes • simplify this drastically by assuming equal masses, semi-major axes, eccentricities and neglecting all harmonics > l=2 Resulting Hamiltonian is H = ¡ 1 2C P N i ;j = 1 cos2 µi j Kocsis & Tremaine (2009) Thermodynamic equilibrium from resonant relaxation – toy model #1 H = ¡ 1 2C P • assumes equal masses, semi-major axes, eccentricities and neglects all harmonics > l=2 N i ;j = 1 cos2 µi j unstable • this is the Maier-Saupe model for the isotropic-nematic phase transition in liquid crystals • above temperature T=Tcrit=0.0743 CN/k Boltzmann constant the only equilibrium is isotropic. Below Tcrit there is a phase transition to a disk Tcrit Thermodynamic equilibrium from resonant relaxation – toy model #2 • consider N stars with masses mj, semi-major axes aj, inclinations Ij, nodes j • assume small inclinations and eccentricities ej,Ij ¿ |aj-ak|/aj • this system is described by the Laplace-Lagrange Hamiltonian • this is a quadratic Hamiltonian and thus integrable • conserved quantities are the Hamiltonian and the angular momentum deficit, • solution is a sum of independent normal modes • eigenvalues of the matrix A yield frequencies of the normal modes !k Thermodynamic equilibrium from resonant relaxation – toy model #2 • assume that higher-order terms and short-period terms allow energy to be exchanged between normal modes • this leads to microcanonical ensemble: probability distribution of an ensemble of systems with given H and C is • as usual, for NÀ1 the microcanonical ensemble is equivalent to the canonical ensemble, Laskar (2008) • 1000 integrations of solar system for 5 Gyr using simplified but accurate equations of motion • all four inner planets exhibit diffusion in eccentricity and inclination • thermodynamic equilibrium is not achieved because • solar system is not old enough • planets collide when orbits cross 1 curve per 0.25 Gyr Thermodynamic equilibrium from resonant relaxation – toy model #2 • model is specified by masses mi, semi-major axes ai, and integrals H and C (initial conditions) • since H and C are both quadratic in the coordinates and momenta, the only non-trivial parameter is H/C • in microcanonical ensemble the mean energy and angular-momentum deficit in mode k are where s and t are chosen to match the initial conditions H=kh Eki, C=kh Cki • t = inverse temperature = ( S/ H)C ; can be negative • s = 0 ! same energy in every mode (equipartition) ; s < 0 ! red noise; s > 0 ! blue noise Thermodynamic equilibrium from resonant relaxation The right way to do it: • model is specified by masses mi, semi-major axes ai, eccentricities ei, and initial orientations of orbit normals • interaction energy between stars i and j is mimjf(ai,aj,ei,ej,cos µij) where µij is the angle between the orbit normals. Evaluate f numerically as an expansion in Pl(cos µ) (can be done once and for all at the start) • evaluate interaction energy numerically and use Markov chain Monte Carlo to find equilibrium state Kocsis & Tremaine (2009) The disk(s) in the Galactic center • ~ 100 massive young stars found in the central parsec age » 6£ 106 yr; formation is a puzzle: • formation in situ from a disk? • disruption of an infalling cluster? • implied star-formation rate is so high that it must be episodic • line-of-sight velocities measured by Doppler shift and angular velocities measured by astrometry five of six phase-space coordinates • many of velocity vectors lie close to a plane, implying that many of the stars are in a disk (Levin & Beloborodov 2003) Bartko et al. (2009) 0.1 pc The disk(s) in the Galactic center data from Bartko et al. (2009); animation from Kocsis (2009) • strong evidence for a warped disk (best-fit normals in inner and outer image differ by 60±) – the “clockwise disk” • disk thickness ~10±; rms eccentricity 0.3 • disk is less well-formed at larger radii • weaker evidence for a second disk between 3” and 7” (the “counterclockwise disk”) arbitrary scale Toy model #2 (Laplace-Lagrange model) observed value (63±/10±) warp of disk / thickness of disk red noise blue = high-mass (visible) stars red = low-mass (invisible) stars thickness near inner edge / thickness near outer edge blue noise orbit normal inner radius outer radius Kocsis (2009) The disk(s) in the Galactic center facts: • if star formation is episodic, presence of two disks is improbable • presence of two disks of the same age is even more improbable • resonant relaxation time is short enough that the disks should be in approximate thermodynamic equilibrium data from Bartko et al. (2009); animation from Kocsis (2009) speculation: • the properties of the disk(s) are consistent with a system in thermodynamic equilibrium from resonant relaxation: flat at small radii, isotropic at large radii, with large fluctuations near the transition Summary • vector resonant relaxation in the Galactic center is much faster than two-body relaxation or scalar resonant relaxation • much of the region within ~ 1 pc of the Galactic center is likely to be in thermodynamic equilibrium under resonant relaxation (details depend on age of stars) • vector resonant relaxation conserves semi-major axis and eccentricity but not the direction of orbit normal • preliminary models suggest that some or all of the curious features of the stellar distribution in the Galactic center arise naturally in thermal equilibrium states • for the future: – scalar resonant relaxation in disks can lead to eccentric disks (work with Jihad Touma) – vector resonant relaxation in globular clusters?