Transcript Black Hole Demographics

Laura Ferrarese
Rutgers University
Lecture 2: Stellar Dynamics
SIGRAV Graduate School in Contemporary
Relativity and Gravitational Physics
Lecture 2: Outline
1. As spectacular as unique: the Milky Way
2. Observational requirements (beyond the Milky Way) and Instrumentation ground and space based
3. Mathematical formalism
4. Application: M87
5. Biases and Systematics
6. Results and Concluding Remarks
The Milky Way
 The galactic center is only
8kpc away. Because of heavy
dust absorption, a clear view
of the stars in the vicinity of
the galactic center can only
be obtained in the nearinfrared.
 In the K-band (2.1 m), the
diffraction limited resolution
is 0.06 arcsec:

A star which travels at a
speed of 27 km/s, would
move relative to the
background stars by
one resolution element
in one month. Proper
motion studies are
therefore feasible.
The Milky Way
 Schödel et al. 2002 (Nature 419, 694): CONICA/NAOS adaptive optics
images at the
ESO/VLT.
 Movie showing the proper motion of the ‘S2’ star over a period of 10 years. The star is on a
bound, highly elliptical orbit around SgrA*, with an orbital period of 15.2 years, a pericenter
distance of 17 light hours, and an orbital semi-major axis of 5.5 light days.
 This single stellar orbit allowed to measure the enclosed mass: 3.7106 M! There are
virtually no assumptions made, except for those regarding the stellar mass distribution
Time™ an d a YU V4 20 cod ec dec ompre ss or a re nee de d to se e th is p ictu re.
(which needs to be known to determine how much of the Quick
enclosed
mass is in a central
point source.)
 The measurements imply a central mass density in
excess of 1017 M pc-3, excluding with high confidence
that the central dark mass consists of a cluster of
unusual stars or elementary particles. This leaves little
doubt of the presence of a supermassive black hole at
the centre of the galaxy in which we live.
The Milky Way
 Further support to the existence of a SBH in the MW come from the detection of an X-ray
flare, using the Chandra satellite (Baganoff et al. 2001, Nature, 413, 45). The flare is
interpreted as due to instability in the mass accretion rate. The timescale of the event is
consistent with the light crossing time for the inner accretion disk (~10 Schwartzchild radii)
September 1999
October 2000
Beyond the Milky Way
 The most nearby large galaxy, M31, is 100 times farther away than the galactic center.
 proper motion timescales are a factor 100 longer;
 stellar mass density is a factor 1003 larger, therefore individual stars are not resolved.
QuickTime™ and a Video decompressor are needed to see this picture.
 In (almost) every galaxy beyond the Milky Way, proper motion studies of individual stars
are not feasible, and we have to resort to extract the velocity field from the integrated
(along the line of sight) stellar light. This presents several problems:
 Stellar absorption lines are faint and long exposure times are required to collect the
necessary signal-to-noise. For instance, it would take the Hubble Space Telescope
more than 100 orbits (more than any TAC would allocate to the project) to obtain
spectra suitable for stellar dynamical studies in M87, the cD galaxy in the Virgo
cluster.
 The modeling is difficult: the orbital structure is unknown and difficult to derive from
the data because of line of sight projections
 There might be a fundamental mathematical limitation to the accuracy of the mass
measurements that can be derived from dynamical modeling.
Requirements
 Stellar kinematics (bottom figure): need
to cover the Ca or Mg absorption lines at
8500 and 5200 Å respectively, with
spectral resolution > 60 km/s (depending
on the mass of the BH).
 Gas kinematics (top figure): H+NII
region at ~6500 Å, with spectral
resolution ~100 km/s
QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture.
Instrumentation - HST
 Challenge:
 to detect the dynamical signature of a supermassive black hole, we need to observe
the region of space - the sphere of influence - within which the black hole
gravitational potential dominates over that of the surrounding stars. The farther in we
are within this region, the more markedly keplerian the rotation and velocity
dispersion curves will be. The ‘black hole radius of influence’ is:
rinfl = GMBH/2 ~ 50 (MBH/109 M)/(2/300 km s-1) pc ~ 0.7 at Virgo
where  is the stellar velocity dispersion and MBH is the black hole mass.
 Even at the distance of Virgo (15 Mpc), the BH sphere of influence is very small.

This explains why most secure SBH detections
have been made using the Hubble Space
Telescope:



Primary mirror is 2.4m in diameter
(small for ground-based standards) but
Resolution (in the optical): 0.1 arcsec
(5-10 times better than ground based
telescopes)
Point Spread Function is very stable
Quic kT ime™ and a T IFF (Unc ompres sed) dec ompres sor are needed to see this pic ture.
Instrumentation - Ground Based
 Compared to HST, ground based telescopes have the great advantage of a larger collective
area, but they generally lack the spatial resolution for systematic SBH searches.
 this will change when adaptive optics will be implemented in the optical.
 Integral Field Units (IFU) are much more suited than long-slit spectroscopy for kinematical
(both stellar and gas) studies on large scales. Available IFUs:
 WHT: Sauron
http://www.strw.leidenuniv.nl/sauron/
 CFHT: GrIF (Near IR)
http://www.cfht.hawaii.edu/Instruments/Spectroscopy/GriF/
 Gemini: GMOS
http://www.gemini.edu/sciops/instruments/gmos/gmosIFU.html
 VLT: VIRMOS
http://www.eso.org/instruments/vimos/
 AAT: SPIRAL
http://www.aao.gov.au/astro/spiral.html
QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture.

Sauron data for NGC 4365, a galaxy known to have a kinematically decoupled core
Mathematical Formulation
 Galaxies are composed of several hundred billion stars. In the most general case, each of
these stars:
 moves through the gravitational potential of the entire galaxy (produced by the effect
of every other star, and any “dark” mass distribution).
 interacts with nearby stars through stellar encounters.
 Observationally, except for a handful of cases, we do not observe individual stars. All we
can observe is the integrated light of stars along the line of sight:
 stellar surface brightness (projected on the plane of the sky).
 projected stellar rotation and velocity dispersion (integrated along the line of sight).
 higher moments of the absorption line profiles.
 From this, we want to arrive to a self-consistent description of the stellar system: derive
the mass density (luminous + dark matter) which reproduces the observables.
 This seems hopelessly complicated! Thankfully, there are a number of simplifications we
can make.
Collisionless Stellar-Dynamical Systems
 Real galaxies can be approximated as collisionless systems, i.e. the time scale over which
interactions between stars lead to a significant change in each star’s velocity is long
compared to the galaxy’s age.
 The Crossing Time is defined as the typical time scale for orbital motion; i.e. the time it
takes a typical star to cross the system:
tc  (Gh )1/2
where h is the average stellar density inside the half mass radius.
 The Relaxation Time is defined as the characteristic time scale over which, due to the
cumulative effect of stellar
 encounters, a typical star acquires a transverse velocity
comparable to its initial velocity:
tr 
N
tc
8ln N
where N is the total number of stars in the galaxy.

Collisionless Stellar-Dynamical Systems
 A typical galaxy has N = 1011 stars but is only t = 100 crossing times old, so the cumulative
effect of stellar encounters is insignificant:
tr  5 108 tc  5 106 t
 In this limit, each star moves in the smooth gravitational field (x,t) of the galaxy. This
greatly simplifies the problem, since instead of thinking about the motion of one point in a
phase space of 6N dimensions (3 spatial coordinates + 3 velocity coordinates, times N
stars), we can think about the motion of N points in a phase space of just 6 dimensions.

 Therefore, a complete description of the system is given by the one-particle distribution
function:
f (x,v;t)d3 x d3 v
f is equal to the number of stars in the phase space volume d3xd3v centered on (x,v) at time t.
 The distribution function obeys a continuity equation, i.e. the rate of change of the number
of stars within a phase space volume d3xd3v is equal to the amount of inflow minus the
amount of outflow: 
f
f
f
   ( f x)    ( f v)   v  f  (x)
0
t
t
v
 This is known as the Collisionless Boltzmann Equation (CBE)
Collisionless Stellar Systems
 The distribution function tells us all we ever want to know about the dynamics of the
system: once we know the distribution function, the CBE can be solved to give the total
gravitational potential, which is related to the total mass density (including dark matter and
a putative central SBH) by Poisson’s equation:
 2(x,t)  4G(x,t)
 But how can we derive the distribution function from the observables?
 Apart for a multiplicative factor (which we will assume of order unity for the sake of
argument), the stellar mass density is just the integral of the DF over all velocities:
Velocity dispersion, arising

 (x,t) 
f (x,v,t)d 3v
because not all the stars
near a given point x have the
same velocity
The mean stellar velocity is:


1
1
3
vi  f vi d v; viv j 
v
v



 f vv
i
d v    vi v j
3
j
2
ij
Streaming
motion
So, in principle, if we know the stellar mass density, the three components of the
streaming motion, and the three diagonal components of the velocity dispersion, then
we can derive the distribution function, the total gravitational potential (from the
CBE), and the total mass density (from Poisson’s equation).
Unfortunately, even assuming that none of the observables depends on time (i.e. the
system is in a steady state), we simply cannot measure all seven quantities.
Collisionless Stellar-Dynamical Systems
 To gain further insight onto the system, it is often convenient to take moments of the CBE,
which directly relate the gravitational potential to the mean streaming velocity, velocity
dispersion and stellar mass density. The zeroth and 1st order moments give the Jeans
equations:

Integrating the CBE over all velocities:
v (vvi )

0
t
xi

Multiplying the CBE by a velocity component vj and then integrating over all
velocities:
2
v j
v j
  (v ij )

v
 vvi
 v

t
xi
x j
xi
 If further
assumptions are made, the Jeans equations can sometimes be simplified to such
an extent that they can be solved analytically or numerically.
Spherical Systems
 Let’s consider the very simple case of a spherically symmetric, steady-state stellar system
for which vr = v = 0. In this case, all derivatives with respect to t,  and  vanish, and the
first moment (wrt vr) of the CBE becomes (in spherical coordinates):
(vv2r ) v

2
2
2
 2v r  (v  v )  v
r
r
r


Since d/dr=GM(r)/r2, this equation can be rearranged as:


  2  
2
2 

d
lnv
d
ln



r
 1  
GM(r)  rv2  r r2 

 
1
  2    2 
d
ln
r
d
lnr


r  
r 


 Therefore, M(r) is a function of five variables, namely the stellar mass density, the
streaming circular velocity, and the three components of the velocity dispersion.
real observations only allow us to measure three quantities, namely the
 Unfortunately,
surface brightness profile at each projected radius, the projected circular velocity, and the
line of sight velocity dispersion. It follows that we cannot build a unique model of an
external galaxy (even in the oversimplified case of a spherical system) based on the
observables only. More assumptions need to be made.
Two Simple Cases


  2  
2
2 
d lnv d ln r


 1  
GM(r)  rv2  r r2 

 
1
  2    2 
d
ln
r
d
lnr


r  
r 


(1)
 The easiest case is to assume that the stellar mass density follows the luminosity density,
the galaxy does not rotate and that the velocity ellipsoids are spherical throughout the
galaxy, i.e. the velocity field is isotropic. In this case:

d ln v d ln  2 
r

v2  v2r  v2  GM(r)   r r2 


d ln r d ln r 

(2)
 The second case is to assume that the mass density follows the luminosity density, i.e. (r)
= (r), with  = constant. In this case, (1) becomes:

d lnv d ln 2

2
r
4G   (r)dr  r 

 2;   1 2


r
d ln r d ln r

2
r
(3)
 The problem is: both cases can reproduce exactly the same observational constraints!
Application: M87
 M87 was the first galaxy for which the presence of a supermassive black hole was claimed
(Sargent et al. 1978, ApJ, 221, 731).
 Central galaxy in the Virgo Cluster.
 Very well known radio galaxy.
QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture.
QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture.
Application: M87
 Projected surface brightness profile (left) and velocity dispersion profile (right) for M87
(from Young et al. 1978, ApJ, 221, 721; Sargent et al. 1978, ApJ, 221, 731)
Steady-state, spherical, isotropic dynamical
models fits to the data:
a)
b)
c)
with black hole
with black hole & seeing convolved
no black hole
Application: M87
 Steady-state, spherical, isotropic dynamical models applied to M87 by Sargent et al. (1979,
ApJ, 221, 731). The data is interpreted as evidence of a central 5  109 M SBH.
Application: M87
 However, lets now consider a case in which we
impose the mass to light ratio throughout the
galaxy to be constant: M(R)/L(R)=constant.

Notice that M(R) here is not the stellar
mass, but includes a central
supermassive black hole (and anything
else).

In this case, the system is not isotropic
and equation (2) can be used to determine
the degree of anisotropy.
 Right: The computed profile for the radial
velocity dispersion and anisotropy parameter 
in M87. This model assumes a constant
M(R)/L(R) (i.e. no central SBH!) and gives just as
good a fit to the observables as the isotropic
model by Sargent et al. (1979). From Binney &
Mamon (1982, MNRAS, 200, 361)
Application: M87
 Spherical, isotropic and anisotropic models for M87, using more recent data (from van der
Marel 1994, MNRAS, 270, 271).

M/L vs Anisotropy
 In other words, anisotropy in the velocity dispersion, and a varying mass to light ratio, can
mimic the same observables. Here is a more intuitive way to see how this ambiguity comes
about. Consider the Jeans equation once more:


  2  
2
2 
d
lnv
d
ln



r
 1  
GM(r)  rv2  r r2 

 
1
  2    2 
d
ln
r
d
lnr


r  
r 


(1)
 In a real galaxy, -dln/dlnr is of order or less than +1, while -dlnr2/dlnr must always be
smaller than +1.

If the galaxy is isotropic, then the last two terms drop out, and we are left with a
positive solution for M(r).

Let’s take the extreme case of a highly anisotropic galaxy for which v=v=0. In this
case, the last two terms offset the first two, with the effect of reducing M(r)!
Breaking the Degeneracy
 Tangentially anisotropic DFs are generally more flat-topped and those from radially
anisotropic models are more peaked than an isotropic profile (figure from van der Marel &
Franx 1993, ApJ, 407, 525).
Anisotropy in Giant Ellipticals
Relation between v/ and ellipticity for an isotropic system
From Illingworth 1981
More General Galaxy Models
 Let’s look at the systems we just discussed in terms of the distribution function.
 Let’s define an integral of motion as any function of velocity and coordinates whose time
derivative vanishes for all orbits:
d
J(x(t),v(t))  0
dt
It can be shown easily that any integral of motion, or any function which depends on (x,v,t)
only through an integral of motion, is a solution of the time dependent CBE. Again, let’s
limit ourselves to time-independent systems.

More General Galaxy Models
 A spherical, isotropic system, admits only one integral of motion: the total energy. In other
words, the distribution function depends only on E: f = f(E)
In this case, there is a one-to-one correspondence between mass density and
distribution function: given (r) we can analytically find f(E) which self-consistently
generates (r) (and vice versa).
 If the distribution function depends on more than one integral of motion, the velocity
dispersion tensor cannot be isotropic. The simplest case is one where f=f(E,Lz), with Lz the
z-component of the angular momentum. These “2-Integral” models can yield reasonable
approximations to some stellar systems.
As in the spherical, isotropic model. there is only one distribution function that can
generate a given (r) and the net streaming motion if the  direction
 However, in most systems, two integrals are not sufficient to fully characterize the stellar
orbits. In fact, numerical calculations show that most orbits are not completely described
by just two integrals, i.e. they admit a third integral. There is no general expression for this
integral of motion, although in nearly spherical systems it is approximated by the
magnitude of the total angular momentum.
In full ‘3-integral models’, there is not a unique distribution function which generates
a given (r) and net streaming motion. In other words, a change in the potential can
always be compensated with an appropriate change in the distribution functions to
give exactly the same observables
Biases and Systematics
 This indeterminacy might preclude us from pinning down the gravitational potential

uniquely when using 3-integral models (Valluri, Merritt & Emsellem, 2002, astroph/0210379).
Simulated dataset designed to reproduce HST/STIS data for M32 - Results obtained for Nc =
number of constraints = 571 (masses, v , , h3 , h4) and varying number of orbits No
N0/Nc=2.5
N0/Nc=15.6
N0/Nc=10.1
N0/Nc=5.0
N0/Nc=10.1
N0/Nc=15.6
N0/Nc=5.0
N0/Nc=2.5
Valluri, Merritt & Emsellem (2002)
A Complete Census of SBH Masses
from Stellar Dynamical Studies
MBH
+
-
(108 solar masses)
Reference
30.3
26.0
1.1
1.1
Cappellari et al. 2002, 578, 787
cE2
0.8
0.025
0.005
0.005
Verolme et al. 2002, MNRAS, 335, 517
N3115
S0
9.8
9.2
3.0
3.0
Emsellem et al. 1999, MNRAS, 303, 495
N3379
E1
10.8
1.35
0.73
0.73
Gebhardt et al. 2000, AJ, 119, 1157
N821
E6
24.7
0.37
0.24
0.08
Gebhardt et al. 2003, ApJ, 583, 92
N1023
S0
10.7
0.44
0.06
0.06
Gebhardt et al. 2003, ApJ, 583, 92
N2778
E
23.3
0.14
0.08
0.09
Gebhardt et al. 2003, ApJ, 583, 92
N3377
E5
11.6
1.00
0.9
0.1
Gebhardt et al. 2003, ApJ, 583, 92
N3384
SB(s)0-
11.9
0.16
0.01
0.02
Gebhardt et al. 2003, ApJ, 583, 92
N3608
E2
23.6
1.9
1.0
0.6
Gebhardt et al. 2003, ApJ, 583, 92
N4291
E
26.9
3.1
0.8
2.3
Gebhardt et al. 2003, ApJ, 583, 92
N4473
E5
16.1
1.1
0.5
0.8
Gebhardt et al. 2003, ApJ, 583, 92
N4564
E
14.9
0.56
0.03
0.08
Gebhardt et al. 2003, ApJ, 583, 92
N4649
E2
17.3
20.0
4.0
6.0
Gebhardt et al. 2003, ApJ, 583, 92
N4697
E6
11.9
1.7
0.2
0.3
Gebhardt et al. 2003, ApJ, 583, 92
N5845
E*
28.5
2.4
0.4
1.4
Gebhardt et al. 2003, ApJ, 583, 92
N7457
SA(rs)0-
13.5
0.035
0.011
0.014
Gebhardt et al. 2003, ApJ, 583, 92
Galaxy
Type
I1459
E3
N221
Distance
(Mpc)
Points to Bring Home
1) Dynamical measurements which do not resolve the SBH sphere of influence can severely bias
the mass measurement. This has in fact happened in the vast majority of ground based studies
(e.g. Kormendy et al. 1992, 1997,1998; Lauer et al. 1992; Fisher et al. 1995; Gebhardt et al. 1997;
Magorrian et al. 1998)
Points to Bring Home
2. The more complex the dynamical models, the larger the number of observational
constraints needed. The combination of HST data (in the innermost regions) and ground
based/2D data (in the outer regions) for M32 constitutes the best suite of data for any
galaxy so far (Verolme et al. 2002, MNRAS, 335, 517)
STIS data
V
SAURON data

h3

V
h4

h3
h4
Points to Bring Home
 Model parameters and internal structure are much more strongly constrained with the use
of the 2D data (Verolme et al. 2002, MNRAS, 335, 517)
3 level
Four slits + STIS
SAURON + STIS
Suggested Readings
 The SBH in the Milky Way:
Schodel, et al. 2002, Nature, 419, 694
Ghez et al. 2003, astro-ph/0303151
 Stellar Dynamics:
Binney & Tremaine, ‘Galactic Dynamics’, Princeton University
Press, Chapter 4
 M87, a practical case:
Sargent et al. 1979, ApJ, 221, 731
Binney & Mamon 1982, MNRAS, 200, 361
van der Marel 1994, MNRAS, 270, 271