Supermassive Black Hole
Mass Measurement Techniques
Daniel Batcheldor
Florida Institute of Technology
Department of Physics and Space Sciences
[email protected]
The Role of SBHs?
AGN evolution arguments imply SBHs are
expected to reside at the centers of most galaxies.
The AGN fuel in galaxies leads us to expect
relationships between galaxies and SBHs.
“Energy driven” models predict M• ∝ σ 5
(e.g., Silk & Rees, 1998)
“Momentum driven” models predict M• ∝ σ 4
(e.g., King 2003)
See: “Dynamics and Evolution of Galactic Nuclei”, D. Merritt (July 2013)
The Role of SBHs?
Gültekin et al. (2011)
McConnell et al. (2011)
β = 4.2 ± 0.4
β = 5.1 ± 0.3
β = 4.0 ± 0.3
Tremaine et al. (2002)
β = 3.8 ± 0.3
β = 5.12
Graham et al. (2011)
β = 4.8 ± 0.5
Gebhardt et al. (2000)
For M• ∝ σ β
Ferrarese & Merritt (2000)
The nature of the observed relationship is debatable.
The Basic Technique
Emsellem et al. (2004)
Φ(r)tot = Φ(r)gal + Φ•
???
High resolution nuclear
gas or stellar dynamics
Invert luminosity profile
to density
(assuming some M/L)
Determines the technique
used.
24
Integral field spectroscopy
Graham et al. (2003)
CHAPTER 2
Figure 2.2 Observed intensity profiles of two elliptical galaxies at visual wavelengths, showing fits of the data to standard model profiles [210]. The
galaxy on the right is well fit by Sérsic’s law, equation (2.3), shown
as the dashed curve. The galaxy on the left has a well-resolved core;
the solid curve shows a fit of the data to a core-Sérsic model, equa-
The Basic Technique
Spatial Resolution Requirements
When the mass in stars is comparable to the mass of the SBH.
M∗ (r < rm ) = 2M•
i.e., at r = rm 1/3 of gravitational force comes from
the SBH, 2/3 from the galaxy.
2σ 2
For a singular isothermal sphere M∗ (< r) =
r
G
So, when rm = rh (the influence radius) we find
GM•
rh =
σ2
Note that the MW SBH “signature” is not seen until 0.2rh.
See: “Dynamics and Evolution of Galactic Nuclei”, D. Merritt (July 2013)
Selection Effects
???
Batcheldor (2010)
OBSERVATIONS OF GALACTIC NUCLEI AND SUPERMASSIVE BLACK HOLES
35
Gas Kinematics
galaxies in the figure have cores similar to M87’s, and it is likely that
he resolution required to detect SBHs in these galaxies via stellar
motions is likewise higher than in galaxies like the Milky Way – in
other words, that a resolution of ∼ 0.1rh is probably not sufficient.
Method:
• Measure central wavelength of
emission lines vs. spatial position.
• Fit inclined rotating Keplerian disk.
Good:
• High S/N for emission lines
(comparatively)
• Relatively easy modeling.
Bad:
Gas not always present.
•
Macchetto
et al.
Figure 2.6 The rotationM87,
curve of
the ionized gas
disk(1997)
near the center of the giant
non-gravitational motions.
• Potential
elliptical galaxy
M87
[331].
The
data
are
from
the Faint-Object
Camera
M = 3.2 ± 0.9 × 109 M
•
⊙
on the Hubble Space Telescope. The solid and dotted curves are from
models that assume two different orientations for the gas disk; all such
models are found to imply the presence of a SBH of a few 109 M! . The
best-fit value is claimed to be 3.2 ± 0.9 × 109 M! . The stellar-dynamical
data for this galaxy (Figure 2.5) show now indication of a central rise
and the value of M• derived from those data is accordingly much more
Gas Kinematics
8
NGC 4258, Moran et al. (1999)
CHAPTER 1
Masers
Sufficient emission from AGN for
population inversions in
molecular gas cloud
Stimulated emission at GHz
wavelengths + VLBI gives m.a.s.
resolution.
Model rotating Keplerian disk.
M• = 3.9 × 107 M⊙
Figure 1.1 The top panel shows the distribution on the plane of the sky of the water
masers in NGC 4258. The units are milliarcseconds (mas); one mas
corresponding to 0.035 pc at the distance of the galaxy. The bottom
panel shows the rotation curve traced by the maser clouds [392].
s−1 . Such velocities can only be maintained if SgrA* is roughly 4
million times more massive than our Sun. The current, best estimate
Favourable inclination needed.
Gas Kinematics
Spectroastrometry
Method: Measure the photocenter of emission lines in
different velocity channels.
Greatly improves spatial resolution compared to measuring
A. Gnerucci et al.: Spectroastrometry of rotating gas disks. II. Application to Centaurus A
central wavelength
of emission lines vs. spatial position.
Fig. 4. Spectroastrometric curves of the
Paβ (black
points)
and [Fe II] (red A)
points)
lines (ISAAC et
spectra)
at the three slit position angles. Left panel:
NGC
5128
(Centaurus
, Gnerucci
al. (2011)
PA1. Central panel: PA2. Right panel: PA3. The dashed vertical lines on each panel represent the limits of the HV range.
7
M• = 9.6+2.5
−1.8 × 10 M⊙
Stellar Dynamics
Method: Based on orbital superposition technique
(Schwarzschild 1979).
Good: Always stars, always gravitational motion.
Bad: Complex models, low S/N.
Cappellari et al. (2004)
Stellar Dynamics
χ2 degeneracy between # of orbits modeled
& # of constraining data points.
84
78
VALLURI,
MERRITT,
& EMSELLEM
VALLURI,
MERRITT,
& EMSELLEM
Vol. 602
Vol. 602
[left] model, [right] M32,Valluri et al. (2004)
Fig. 5.—One-dimensional
cuts through
!2V plots
Fig. data,
3, all taken at Fig. 22.—One-dimensional cuts through Fig. 21. This figure shows that for
Fig. 20.—One-dimensional
cuts through
Fig. 19 the
at !
¼ 2. of
These
verticaltoarrow
location
of the
true
model
which are !
superior
in quality
most indicates
HST STISthe
nuclear
data,
place
only
veryparameter,
the largest orbit library, the minimum in the !2 valley is still reasonably
V ¼ 2. The
6
a
M% ¼ 2:625
suggesting that the HST STIS data for M32 may yield tight constraints
weak constraints
on M&
.. 10 M'. When the number of orbits used is small, there isnarrow,
definite, but spurious, !2 minimum. As No is increased, this minimum
on M. in this galaxy.
broadens into the perfectly flat plateau characteristic of underdetermined
problems.
The true
model parameters
on that plateau
but two
cannot be unshows two
minima,
although
displacedlieslightly
from the
ambiguously
recovered.
best fit. That model lies between the two minima seen in the
in the top
right-hand
panel of Figure 23. However, as N is
M• ∼ 1.5 − 5 × 106 M⊙
o
increased, we find that the two minima merge into a single,
top right-hand panel of Figure 23 and somewhere near the
2
Stellar Dynamics
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
30
Signal-to-Noise Ratio
Residuals Normalized Flux
Most stellar dynamical estimates come from single slit
observations that have extremely (prohibitively?) low S/N.
0.3
0
25
20
15
10
5
-0.3
8400
8500
8600o
0
8700
Wavelength (A)
-0.9 -0.6 -0.3
0
0.3 0.6 0.9
Radius (arc seconds)
30
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Signal-to-Noise Ratio
Residuals Normalized Flux
NGC 821, Richstone et al. (2004), log M• = 7.93+0.15
−0.23
0.3
0
-0.3
8400
8500
8600o
Wavelength (A)
8700
25
20
15
10
5
0
-0.9 -0.6 -0.3
0
0.3 0.6 0.9
Radius (arc seconds)
NGC 2778, Gebhardt et al. (2003), log M• = 7.15+0.19
−0.45
Stellar Dynamics
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
30
Signal-to-Noise Ratio
Residuals Normalized Flux
Most stellar dynamical estimates come from single slit
observations that have extremely (prohibitively?) low S/N.
0.3
0
25
20
15
10
5
-0.3
8400
8500
8600o
0
8700
-0.9 -0.6 -0.3
0
0.3 0.6 0.9
Radius (arc seconds)
Wavelength (A)
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
30
Signal-to-Noise Ratio
Residuals Normalized Flux
NGC 3585, Gültekin et al. (2009), log M• = 8.53+0.16
−0.08
0.3
0
25
20
15
10
5
-0.3
8400
8500
8600o
Wavelength (A)
8700
0
-0.9 -0.6 -0.3
0
0.3 0.6 0.9
Radius (arc seconds)
NGC 5576, Gültekin et al. (2009), log M• = 8.26+0.06
−0.11
Other Significant Issues
Comparing Methods
10
log M
1
.
9
8
7
6
6
.
Batcheldor et al (2013)
7
8
log M
9
2
10
Other Significant Issues
M87
(a)
Offset SBHs
(b)
N
200 pc
Batcheldor et al. (2010).
Other Significant Issues
M87
(a)
Offset SBHs
(b)
N
200 pc
Projected offset ~ 7 pc
Batcheldor et al. (2010).
Other Significant Issues
Nuclear Star Clusters
No. 2, 2003
BLACK HOLE AT CENTER OF NGC 4041
NGC4041, Marconi et al. (2003)
877
OBSERVATIONS OF GALACTIC NUCLEI AND SUPERMASSIVE BLACK HOLES
25
NGC 205, Merritt (2009)
Fig. 10.—Left: Fit of the light profile obtained from the WFPC2/F814W data. Right: Fit of the NICS/K light profile with the central star cluster and
the 2.3 Surface brightness data of the Local Group dwarf galaxy NGC 205
Figure
extended component (the geometrical parameters of the star cluster were determined in the previous fit). In both cases the lower panel shows the fit residuals.
showing the resolved NSC [355]. The observations, shown as the open
The dashed line in the right-hand panel shows the K-band fit obtained by fixing the cluster normalization to a value that is 0.6 dex larger than the best-fit value.
circles, were made in the I-band, which centers around λ ≈ 8000 Å[522,
The open squares show the corresponding residuals.
279]. The data have been fit by a two-component model; the dashed
• NSCs require high spatial resolution to be resolved.
• Functional form of the density profile unclear.
• Can exceed the central SBH mass.
that the rotation curves are very similar beyond r > 0>5.
However, they differ at smaller radii because of the different
contribution of the star cluster to the total mass budget.
This difference is due to the fact that the mass-to-light ratio
has been assumed constant in each band.
4.2. Kinematics
and solid curves show the model before, and after, convolution with
the instrumental point-spread function. The lower panel shows the fit
residuals. This galaxy is roughly 800 kpc away, and one arc second
corresponds to about 3 pc. Surface brightness units (magnitudes per
square arc second) are explained in the caption to Figure 2.2.
In order to model the gas kinematical data (velocities and
widths measured along the slit), we select the simplest possiVarious generalizations of Sérsic’s profile have also been proposed
for fitting the excess nuclear light observed in fainter spheroids. But
ble approach and assume that the ionized gas is circularly
rotating in a thin disk located in the principal plane ofbecause
the NSCs are typically poorly resolved, many different functional
galaxy potential. The latter assumption is not needed ifforms
the are found to do an equally good job. Local Group galaxies
provide the only exceptions: at least two of these contain NSCs that
galaxy potential has a spherical symmetry. We assume that
are near enough to be resolved. One is the dwarf elliptical galaxy
the disk is not pressure supported, and we neglect all hydroNGC 205, a satellite companion to the “Andromeda nebula,” the giant
dynamical effects. Thus, the disk motion is completely deterspiral galaxy M31. Figure 2.3 shows the intensity profile measured
mined by the gravitational potential, which is made of two
components: one is stellar and is completely determined by
the mass distribution, derived in the previous section, and
The Best Estimates
OBSERVATIONS OF GALACTIC NUCLEI AND SUPERMASSIVE BLACK HOLES
33
Keplerian Signatures of SBHs
If the L and σ profiles vary slowly,
then the relationship between Φ
and the kinematics has the form:
G[M∗ (r) + M• ] = r(σ 2 + v̄φ2 )
∝
Close to the SBH v2 1/r, hence
“Keplerian”.
In “Dynamics and Evolution of Galactic Nuclei”,
D. Merritt.
Figure 2.5 Stellar kinematical data for galaxies with putative SBHs. The horizontal axis is the angular distance from the center of the galaxy in
seconds of arc. The vertical axis is the mean square line of sight stellar
velocity. In the case of M31, which has a lopsided nucleus, data are
plotted with respect to two possible centers: the point of peak velocity dispersion (circles) and the point of zero mean velocity (crosses).
The Best Estimates
SBHs that show a
Keplerian rise.
Summary
• SBHs play a fundamental role in galaxy evolution.
• The most common mass measurement techniques are
plagued by uncertainties.
• Only a handful of SBH measurements (using any
technique) are convincing.
• Other pitfalls (offsets, NSCs) potentially limit precision.
• New mass measurement techniques are encouraging
and necessary for progress in understanding the
interplay of SBHs and galaxies.