Hubble Space Telescope 3”x3”Lauer et al. AJ 1998

3 pc

Statistical mechanics of stellar systems near galaxy

centers

Hubble Space Telescope 3”x3”Lauer et al. AJ 1998

3 pc

1000” = 3600 pc

1” = 3.6 pc

0.1 pc

radio source Sagittarius A*

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

trelax = 5 Gyr1M ¯

m¾

100 km s¡ 1

³r

1 pc

´2

• 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

2 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”

trelax = 5 Gyr1M ¯

m¾

100 km s¡ 1

³r

1 pc

´2

• thus: the relaxed stellar system is also a near-Keplerian stellar system

0.1 pc

2 pc

r¤ ´ GM ²¾2 = 4:3 pc M ²

107M ¯

³100 km s¡ 1

¾

´2108 M¯

4£106 M¯

Ghez et al. (2005) - UCLA Eisenhauer et al. (2005) - MPE

The black hole in the Galactic center

0.01 pc10 X Sun-Pluto distance

• center of attraction is located at the radio source Sagittarius A* which is presumably the black hole

• 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)

Sgr A*

only alternatives to a black hole are:

• boson star

• cluster of 1010 planet-mass black holes

NGC 4258

NGC 4258• 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

Miyoshi et al. (1995)

Herrnstein et al. (1999)

Humphreys et al. (2007)

0.15 pc

xx

Gültekin et al. (2009)

• by now there are ~40-50 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)

bla

ck h

ole

mass

(so

lar

mass

es)

Gültekin et al. (2009)

• 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)

bla

ck h

ole

mass

(so

lar

mass

es)

• 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 densityn(r) / exp

¡¡ ©

kT

¢/ exp

¡GM mkT r

¢

• 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)

Boltzmann constant

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)

trelax(r) » torb(r)M

mN 1=2(1)

mass of a starnumber of stars inside radius r

mass of the black hole

orbital period

1”current resolution limit

1”current resolution limit

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 semi-major axes

eccentricities

Toy model #1:

• simplify this drastically by assuming equal masses, semi-major axes, eccentricities and neglecting all harmonics > l=2

Resulting Hamiltonian is

H = ¡ 12C

P Ni;j =1 cos2 µi j

Kocsis & Tremaine (2009)

Thermodynamic equilibrium from resonant relaxation – toy model #1

• assumes equal masses, semi-major axes, eccentricities and neglects all harmonics > l=2

• this is the Maier-Saupe model for the isotropic-nematic phase transition in liquid crystals

• above temperature

T=Tcrit=0.0743 CN/k

the only equilibrium is isotropic. Below Tcrit there is a phase transition to a disk

H = ¡ 12C

P Ni;j =1 cos2 µi j

Boltzmann constant Tcrit

unstable

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=h Eki, C=h 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

• 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”)

data from Bartko et al. (2009); animation from Kocsis (2009)

Toy model #2 (Laplace-Lagrange model)

arb

itra

ry s

cale

blue = high-mass (visible) stars

red = low-mass (invisible) stars

red noise blue noise

warp of disk / thickness of disk

thickness near inner edge / thickness near outer edge

observed value (63±/10±)

orbit normal

outer radius

inner 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?