Transcript Thermodynamic equilibrium from resonant relaxation

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?