To appear in "The Central Kpc of Starbursts
and AGN", ed. J. H. Knapen, J. E. Beckman, I. Shlosman & T. J. Mahoney;
astro-ph/0107134
Abstract. Supermassive black holes appear to be uniquely
associated with galactic bulges.
The mean ratio of black hole mass to bulge mass was until recently
very uncertain, with ground-based, stellar kinematical data
giving a value for
<M /
Mbulge> roughly an order
of magnitude larger than other techniques.
The discrepancy was resolved with the discovery of the
M
-
relation,
which simultaneously established a tight corrrelation between black
hole mass and bulge velocity dispersion, and confirmed that the
stellar kinematical mass estimates were systematically too large due to
failure to resolve the black hole's sphere of influence.
There is now excellent agreement between the various techniques
for estimating
<M
/
Mbulge>,
including dynamical mass estimation in quiescent galaxies;
reverberation mapping in active galaxies and quasars;
and computation of the mean density of compact objects based on
integrated quasar light. All techniques now give
<M
/
Mbulge>
10-3 and
3 × 105
M
/
Mpc-3. Implications of the
M
-
relation for the formation
of black holes are discussed.
Table of Contents
The argument that active galaxies and quasars are powered
by accretion onto massive compact objects was first made
almost four decades ago
(Salpeter 1964;
Zeldovich 1964).
Since that time, the existence of supermassive black holes
has been confirmed in the nuclei of
nearby galaxies and in a handful of more distant galaxies
by direct dynamical measurement of their masses.
The best determinations are in our own Galaxy
(M
3 × 106
M
,
Genzel et al. 2000;
Ghez et al. 1998),
NGC 4258
(Miyoshi et al. 1995),
and M87
(Macchetto et al. 1997).
The data for each of these galaxies exhibit a clear rise
very near the center in the orbital velocity of stars or gas,
suggestive of motion around a compact object.
Data of this quality are unfortunately still rare,
and the majority of black hole detections have necessarily
been based on stellar-kinematical data which do not
exhibit a clear signature of the presence of a black hole. These data
(Magorrian et al. 1998;
Richstone et al. 1998)
imply a mean black hole mass
that is uncomfortably large compared with values predicted from
quasar light.
The inconsistency has been taken as evidence for low radiative
efficiencies during the optically bright phase of quasars (e.g.
Haehnelt, Natarajan &
Rees 1998)
or for continued growth of black holes after the quasar epoch (e.g.
Richstone et al. 1998).
It is now clear that this discrepancy was due almost entirely
to systematic errors in the stellar kinematical mass estimates.
The first convincing demonstration of this came from the
M -
relation,
a tight empirical correlation between black hole
mass and bulge velocity dispersion. The
M
-
relation was discovered
by ranking black hole detections in terms of their believability
and excluding the least secure cases. The remarkable and unexpected result
(Ferrarese & Merritt
2000)
was an almost perfect correlation between
M
and
for the best-determined black hole masses, compared
to a much weaker correlation for the less secure masses.
Ground-based, stellar kinematical estimates of
M
were found to scatter above the
M
-
relation by as much as
two orders of magnitude, suggesting that many of the published
masses were spurious and that most were substantial overestimates.
The ability of the
M -
relation to "separate the
wheat from the chaff" has led to a rapid advance in our understanding of
black hole demographics.
We review that progress in Section 2 and
Section 3; we argue that there is now
almost embarrassingly good agreement between the results from the various
techniques for estimating the mean mass density of black holes
in the universe.
Black hole masses determined dynamically in nearby quiescent
galaxies are now fully consistent with masses inferred
in active galaxies and quasars,
and with estimates of the density of dark relic objects
produced by accretion during the quasar epoch.
The need for non-standard accretion histories in order to reproduce
a large density of black holes in the current universe has disappeared.
Two recurrent themes in this review are the importance of
adequate data when estimating black hole masses;
and the much greater usefulness of accurate mass estimates
compared with simple detections.
When the first black hole detections were being published,
there was much discussion about whether the observations
(all ground-based at the time)
were of sufficient quality to resolve the black holes'
sphere of influence, rh =
GM /
2.
We now know that the ground-based data almost always failed
to do this, sometimes by a large factor, and that this
failure, coupled with shortcomings in the modelling,
led to systematic overestimates of
M
(Section 2).
The situation has improved somewhat with the Space Telescope,
but not dramatically:
we argue (Section 4) that the number of
galaxies with secure dynamical estimates of
M
will
increase only modestly over the next few
years in spite of ambitious ongoing programs with HST.
This is due partly to the fact that these observations were
planned at a time when the mean black hole mass was believed
to be much larger than it is now.
Progress in our understanding of black hole demographics is
more likely to come from techniques with higher effective resolution
than stellar or gas kinematics, notably reverberation mapping in AGN
(Peterson 1993).
While the ability of the
M -
relation to clarify the
data has been an enormous boon,
the existence of such a tight correlation must also be telling us something
fundamental about the way in which black holes form and about the
connection between black holes and bulges.
Unfortunately, the theoretical understanding of this connection has
lagged behind the phenomenology.
We summarize the proposed explanations for the origin of the
M
-
relation in
Section 5 and discuss their strengths and
weaknesses.
Epimetheus: Wie vieles ist denn dein?
Prometheus: Der Kreis, den meine Wirksamkeit erfüllt!
Epimetheus: What then do you possess?
Prometheus: My sphere of influence - nothing more and nothing less!
Goethe, Prometheus
2.1. A Discrepancy, and its Resolution
By 1999, a clear discrepancy was emerging between black hole masses derived from stellar kinematical studies and most other techniques. The former sample included many "standard bearers" like M31 (Richstone, Bower & Dressler 1990), NGC 3115 (Kormendy et al. 1996a) and NGC 4594 (Kormendy et al. 1996b). The size of the discrepancy was difficult to pin down since there were (and still are) essentially no galaxies for which black hole masses had been independently derived using more than one technique. However the masses derived from ground-based stellar kinematics were much larger, by roughly an order of magnitude on average, than those inferred from other techniques when galaxies with similar properties were compared, or when estimates of the cosmological density of black holes or the mean ratio of black hole mass to bulge mass were made. The discrepancy was clearest in two arenas:
Serious inconsistencies like these only appeared when comparisons were
made with black hole masses derived from the stellar kinematical data;
all other techniques gave roughly consistent values for
and
<M
/
Mbulge>.
Nevertheless, most authors accepted the correctness of the
stellar dynamical mass estimates and looked elsewhere to explain
the discrepancies.
Ho (1999)
suggested that the reverberation mapping masses had been
systematically underestimated.
Wandel (1999)
proposed that black holes in active galaxies were
smaller on average than those in quiescent galaxies, due either
to different accretion histories or to selection effects.
Richstone et al. (1998)
and Faber (1999)
suggested that the inconsistency
between their group's masses and the masses inferred from quasar light
could be explained if black holes had acquired 80% of
their mass after the quasar epoch
through some process that produced no observable radiation.
![]() |
Figure 1. Discovery of the
M |
What particularly caught our attention was the gulf between black hole
masses derived from high- and low-resolution data, and
(to a lesser extent) between gas- and stellar dynamical data;
the former (e.g.
Ferrarese, Ford &
Jaffe 1996;
Ferrarese & Ford
1999)
were typically taken at higher resolution than the latter.
Black hole masses derived from the highest resolution data,
in galaxies like the Milky Way
(Ghez et al. 1998;
Genzel et al. 2000)
and M87
(Macchetto et al. 1997),
were the
smallest when expressed as a fraction of the bulge mass, with
M /
Mbulge
10-3.
The largest fractional black hole masses - in galaxies like
NGC 3377
(Kormendy et al. 1998)
or NGC 4486b
(Kormendy et al. 1997) -
were mostly derived from stellar absorption-line spectra obtained from the
ground, at resolutions of ~ 1", corresponding to typical
linear scales of 10 - 100 pc. The mean value of
M
/
Mbulge for these galaxies was claimed
to be about 10-2
(Magorrian et al. 1998;
Richstone et al. 1998),
roughly an order of magnitude
greater than the value derived from the high-resolution data.
We began to suspect that some of the masses derived from the
lower-quality data might be serious over-estimates
- or, even worse, that some of the "detections" based on these
data were spurious.
To test this hypothesis, we tabulated all of the published black
hole masses that had been derived from stellar- or gas kinematical
data (excluding the reverberation mapping masses) and divided
them into two groups based on their expected accuracy.
This is not quite as easy as it sounds, since the "accuracy"
of a black hole mass estimate is not necessarily related
in any simple way to its published confidence range.
Our criterion was simply the quality of the data:
"accurate" black hole masses were those derived
from HST data, at resolutions of ~ 0.1", as well as
M for
the Milky Way black hole (which is by far the
nearest) and the black hole in NGC 4258 (for which VLBI gives
a resolution of ~ 0.1 pc).
The velocity data for these galaxies (our "Sample A") was always
found to exhibit a clear rise in the inner few data points,
suggesting that the black hole's sphere of influence
rh
GM
/
2 had been well
resolved. The remaining black hole masses ("Sample B") were all those
derived from lower-resolution data, typically ground-based
stellar kinematics, including most of the masses in the
Magorrian et al. (1998)
study. Sample A contained 12 galaxies, Sample B 31.
Our first attempt to compare "Sample A" and "Sample B" masses
was disappointing (Figure 1a). In the
M -
Lbulge plane,
the Sample A masses do fall slightly below those from Sample B,
but the intrinsic scatter in Lbulge is apparently so
large that there is no clear difference in the relations defined by the
two samples.
But when we plotted
M versus
the velocity dispersion
of the bulge stars, something magical happened
(Figure 1b): now the
Sample A galaxies clearly defined the lower edge of the
relation, while the Sample B galaxies scattered above,
some by as much as two orders of magnitude in
M
!
Furthermore the correlation defined by the Sample A galaxies
alone was very tight.
What particularly impressed us about the
M -
plot was the fact that
the Sample A galaxies, which are diverse in their properties,
showed such a tight correlation; while the Sample B galaxies, which are much
more homogeneous, exhibited a large scatter.
For instance, Sample A contains two spiral galaxies, two lenticulars,
and both dwarf and giant ellipticals; while the Sample B
galaxies are almost exclusively giant ellipticals.
Furthermore the black hole masses in Sample A were derived using
a variety of techniques, including absorption-line stellar kinematics
(M32, NGC 4342),
dynamics of gas disks (M87, NGC 4261),
and velocities of discrete objects (MW, NGC 4258);
while in Sample B all of the black hole masses were derived from
stellar spectra obtained from the ground.
This was circumstantial, but to us compelling, evidence
that the Sample A masses were defining the true relation and
that the Sample B masses were systematically in error.
Fitting a regression line to
logM vs.
log
for
the Sample A galaxies alone, we found
![]() | (1) |
with = 4.80 ± 0.5
(Ferrarese & Merritt
2000).
We defined the quantity
c to be the rms
velocity of stars in an aperture
of radius re/8 centered on the nucleus, with
re the half-light radius of the bulge.
This radius is large enough that the stellar
velocities are expected to be affected at only the few percent
level by the gravitational force from the black hole, but small enough that
c can easily be
measured from the ground.
A striking feature of the
M -
relation is its negligible
scatter. The reduced
2 of
Sample A about the best-fit line of Eq. 1,
taking into account measurement errors in both variables,
is only 0.74, essentially a perfect fit.
Such a tight correlation seemed almost too good to be true
(and may in fact be a fluke resulting from the small sample size)
but we felt we could not rule it out given the existence of
other, similarly tight correlations in astronomy, e.g.
the near-zero thickness of the elliptical galaxy fundamental plane.
In fact the scatter in the
M -
relation is so small that it is
reasonable to use the relation to predict black hole masses,
even in galaxies for which determinations of
M
based on detailed modelling have previously been published.
One can then ask, galaxy by galaxy,
whether the observations on which the published estimate of
M
was
based were of sufficiently high quality to
resolve the black hole's sphere of influence.
Table 1 and Figure 2 show
the results.
Table 1 is a ranked list of the most secure black
hole detections to date.
The galaxies are listed in order of increasing
FWHM / 2rh, i.e. the ratio of the size of the
resolution
element to twice the radius of influence of the black hole.
In the case of HST observations, for which the PSF is undersampled, FWHM
is the diameter of the FOS aperture or the width of the STIS slit.
For ground-based observations, FWHM refers to the seeing disk.
Figure 2 plots the same quantities for
essentially all galaxies with published estimates of
M
based
on stellar or gas kinematics.
Not surprisingly, only the black holes in the Milky Way and in NGC 4258 have been observed at a resolution greatly exceeding rh. The Sample A galaxies of Ferrarese & Merritt (2000) also satisfy FWHM / 2rh < 1, although sometimes marginally. By contrast, almost none of the ground-based data resolved rh, sometimes failing by more than a factor of 10.
Galaxy | Type | Distance |
M![]() | ![]() | FWHM/2rh | Reference |
Galaxies for which rh has been resolved | ||||||
MW | SbI-II | 0.008 | 0.0295±0.0035 | 100±20 | 7.3-4 | Genzel et al. 2000 |
N4258 | SAB(s)bc | 7.2 | 0.390±0.034 | 138±18 | 1.1-3 | Miyoshi et al. 1995 |
N4486 | E0pec | 16.7 | 35.7±10.2 | 345±45 | 0.03 | Macchetto et al. 1997 |
N3115 | S0- | 9.8 | 9.2±3.0 | 278±36 | 0.04 | Emsellem et al. 1999 |
N221 | cE2 | 0.8 | 0.039±0.009 | 76±10 | 0.06 | Joseph et al. 2000 |
N5128 | S0pec | 4.2 | 2.4 3.6-1.7 | 145±25 | 0.10 | Marconi et al. 2001 |
N4374 | E1 | 18.7 | 17 +12-6.7 | 286±37 | 0.10 | Bower et al. 1998 |
N4697 | E6 | 11.9 | 1.7 +0.2-0.3 | 163±21 | 0.10 | "Nuker" group, unpubl.3 |
N4649 | E2 | 17.3 | 20.6 +5.2-10.2 | 331±43 | 0.10 | "Nuker" group, unpubl.3 |
N4261 | E2 | 33.0 | 5.4 +1.2-1.2 | 290±38 | 0.18 | Ferrarese et al. 1996 |
M81 | SA(s)ab | 3.9 | 0.68 0.07-0.13 | 174±17 | 0.19 | STIS IDT, unpubl.3 |
N4564 | E | 14.9 | 0.57 +0.13-0.17 | 153±20 | 0.33 | "Nuker" group, unpubl.3 |
I1459 | E3 | 30.3 | 4.6±2.8 | 312±41 | 0.35 | Verdoes Kleijn et al. 2000 |
N5845 | E* | 28.5 | 2.9 +1.7-2.7 | 275±36 | 0.40 | "Nuker" group, unpubl.3 |
N3379 | E1 | 10.8 | 1.35±0.73 | 201±26 | 0.44 | Gebhardt et al. 2000a |
N3245 | SB(s)b | 20.9 | 2.1±0.5 | 211±19 | 0.48 | Barth et al. 2001 |
N4342 | S0- | 16.7 | 3.3 +1.9-1.1 | 261±34 | 0.56 | Cretton & van den Bosch 1999 |
N7052 | E | 66.1 | 3.7 +2.6-1.5 | 261±34 | 0.66 | van der Marel & van den Bosch 1998 |
N4473 | E5 | 16.1 | 0.8 +1.0-0.4 | 188±25 | 0.77 | "Nuker" group, unpubl.3 |
N6251 | E | 104 | 5.9±2.0 | 297±39 | 0.84 | Ferrarese & Ford 1999 |
N2787 | SB(r)0+ | 7.5 | 0.41 0.04-0.05 | 210±23 | 0.87 | Sarzi et al. 2001 |
N3608 | E2 | 23.6 | 1.1 +1.4-0.3 | 206±27 | 0.98 | "Nuker" group, unpubl.3 |
Galaxies for which rh has not been resolved | ||||||
N3384 | SB(s)0- | 11.9 | 0.14 +0.05-0.04 | 151±20 | 1.0 | "Nuker" group, unpubl.3 |
N4742 | E4 | 15.5 | 0.14 0.04-0.05 | 93±10 | 1.0 | STIS IDT, unpubl.3 |
N1023 | S0 | 10.7 | 0.44±0.06 | 201±14 | 1.1 | STIS IDT, unpubl.3 |
N4291 | E | 26.9 | 1.9 +1.3-1.1 | 269±35 | 1.1 | "Nuker" group, unpubl.3 |
N7457 | SA(rs)0- | 13.5 | 0.036 +0.009-0.011 | 73±10 | 1.1 | "Nuker" group, unpubl.3 |
N821 | E6 | 24.7 | 0.39 +0.17-0.15 | 196±26 | 1.3 | "Nuker" group, unpubl.3 |
N3377 | E5+ | 11.6 | 1.10 +1.4-0.5 | 131±17 | 1.3 | "Nuker" group, unpubl.3 |
N2778 | E | 23.3 | 0.13 +0.16-0.08 | 171±22 | 2.8 | "Nuker" group, unpubl.3 |
Galaxies in which dynamical studies are inconclusive | ||||||
N224 | Double nucleus, system not in dynamical equilibrium. | Bacon et al. 2001 | ||||
N598 | Data imply upper limit only,
![]() ![]() | Merritt, Ferrarese & Joseph 2001 | ||||
N1068 | Velocity curve is sub-Keplerian. | Greenhill et al. 1996 | ||||
N3079 | Masers do not trace a clear rotation curve. | Trotter et al. 1998 | ||||
N4459 | Data do not allow unconstrained fits. | Sarzi et al. 2001 | ||||
N4486B | Double nucleus, system not in dynamical equilibrium. | STIS IDT, unpubl.2 | ||||
N4945 | Asymmetric velocity curve; velocity is sub-Keplerian. | Greenhill et al. 1997 | ||||
1 Type is revised Hubble type. Black hole
masses are in 108 solar masses, velocity dispersions are in km
s-1, and distances are in Mpc.
![]() ![]() ![]() ![]() | ||||||
2 For the reasons outlined in the text, the masses from Magorrian et al. (1998) are omitted from this tabulation. This includes NGC 4594, which was included in Kormendy & Gebhardt (2001). | ||||||
3 Preliminary masses tabulated in Kormendy & Gebhardt (2001). Data and modelling for these mass estimates are not yet available. |
The latter point is important, since precisely these
data were used to define the canonical relation between
black hole mass and bulge luminosity
(Magorrian et al. 1998;
Richstone et al. 1998;
Faber 1999)
that has served as the basis for so many subsequent studies (e.g.
Haehnelt, Natarajan &
Rees 1998;
Catteneo, Haehnelt &
Rees 1999;
Salucci et al. 1999;
Kauffmann & Haehnelt
2000;
Merrifield, Forbes &
Terlevich 2000).
Figure 3 plots the likely "error" in the
ground-based mass estimates (defined as the ratio of the quoted mass,
Mfit, to the mass implied by Eq. 1)
as a function of the effective resolution
FWHM / 2rh. The error
is found to correlate strongly with the quality of the data.
For the best-resolved of the Magorrian et al. candidates,
FWHM/2rh
1,
the average error in
M
appears to be a factor of ~ 3, rising roughly linearly with
FWHM/rh to values of
~ 102 for the most poorly-resolved candidates.
An important quantity is the mean ratio of black hole
mass to bulge mass,
<M /
Mbulge>.
Figure 4 compares the distribution of
Mfit/Mbulge, the mass ratio computed by
Magorrian et al. (1998),
to the distribution obtained when Mfit is replaced
by M
as
computed from the
M
-
relation.
The mean value of
(M
/
Mbulge) drops from 1.7 × 10-2 to
2.5 × 10-3, roughly an order of magnitude.
The mean value of
log10(M
/ Mbulge) shifts downward by -0.7
corresponding to a factor ~ 5 in
M
/
Mbulge.
The density of black holes in the local universe implied by the
lower value of
<M
/
Mbulge> is
~ 5 ×
105
M
Mpc-3
(Merritt & Ferrarese
2001a),
consistent with the value
required to explain quasar luminosities assuming
a standard accretion efficiency of 10%
(Chokshi & Turner
1992;
Salucci et al. 1999;
Barger et al. 2001).
![]() |
Figure 4. Frequency function of
black-hole-to-bulge mass ratios (adapted from
Merritt & Ferrarese
2001a).
The dashed curve is the "Magorrian relation"
(Magorrian et al. 1998)
based on black hole masses derived from ground-based kinematics
and two-integral modelling.
The solid curve is the frequency function obtained when
black hole masses are instead computed from the
M |
2.2. Pitfalls of Stellar Dynamical Mass Estimation
Why were most of the stellar dynamical mass estimates
so poor; why were they almost always over-estimates;
and what lessons do past mistakes have for the future?
The answer to the first question is simple in retrospect.
Figure 5 shows how the signal of the black hole - a
sudden rise in the rms stellar velocities at a distance of
~ rh
GM
/
2 from the black
hole - is degraded by seeing. For
FWHM/2rh
2, the signal is so
small as to be almost unrecoverable except with data of exceedingly high
S/N.
Most of the ground-based observations fall into this regime
(Figure 2).
In fact the situation is even worse than
Figure 5 suggests, since for
FWHM
rh, the rise in
(R) will
be measured by only a single data point.
This is the case for many of the galaxies that are listed
as "resolved" in Table 1 (e.g. NGC 3379,
Gebhardt et al. 2000a).
![]() |
Figure 5. Degradation due to seeing of the
velocity dispersion spike
produced by a black hole in a hot (nonrotating) stellar system.
Heavy line is the profile unaffected by seeing; R is the
projected distance from the black hole and rh =
GM |
A short digression is in order at this point. Data taken from the ground often show an impressive central spike in the velocity dispersion profile; examples are NGC 4594 (Kormendy et al. 1996b) and NGC 4486b (Kormendy et al. 1997). However such features are due in part to blending of light from two sides of the nucleus where the rotational velocity has opposite signs and would be almost as impressive even if the black hole were not present. This point was first emphasized by Tonry (1984) in the context of his ground-based M32 observations. As he showed, the velocity dispersion spike in M32 as observed at ~ 1" resolution is consistent with rotational broadening and does not require any increase in the intrinsic velocity dispersion near the center.
Why should poor data lead preferentially to overestimates of
M,
rather than random errors?
There are two reasons. First, as pointed out by
van der Marel (1997),
much of the model-fitting prior to 1999 was carried out
using isotropic spherical models or their axisymmetric analogs,
the so-called "two-integral" (2I) models.
Such models predict a velocity dispersion profile
that gently falls as one moves inward,
for two reasons: non-isothermal cores, i.e.
~
r-
with
{0, 2}, generically have
central minima in the rms velocity (e.g.
Dehnen 1993);
and, when flattened, the 2I axisymmetric models
become dominated by nearly circular orbits (in order to maintain
isotropy in the meridional plane) further reducing the predicted
velocities near the center.
Figure 6 illustrates these effects for a set of
axisymmetric 2I models with
= 1.5.
Real galaxies almost always exhibit a monotonic rise in
vrms.
Adding a central point mass can correct this deficiency of the models,
but only an unphysically large value of
Mfit will affect the
stellar motions at large enough radii,
r
0.1
re, to do the trick.
This is probably the explanation for the factor ~ 3 mean error in
M
derived from the best ground-based data (Figure 3).
![]() |
Figure 6. Velocity dispersion profiles of the "two-integral" (2I) models that were used as templates for estimating black hole masses in many of the stellar kinematical studies (e.g. Magorrian et al. 1998). Model flattening is indicated as c/a; there are no central black holes. Ticks mark the point Rmax of maximum velocity; Rmax moves outward as the flattening is increased. |
The much larger values of
Mfit /
M
associated with the more distant galaxies in
Figure 3 are probably attributable to
a different factor.
When the data contain no useful information about the black hole mass,
only values of
Mfit that are much larger than the true
mass will significantly affect the
2 of the model fits.
The only black holes that can be "seen" in such data are
excessively massive ones.
Can these problems be overcome by abandoning 2I models in favor
of more general, three-integral (3I) models?
The answer, surprisingly, is "no": making the modelling algorithm
more flexible (without also increasing the amount or quality of the
data) has the effect of weakening the constraints on
M.
The reason is illustrated in Figure 7.
The rms velocities in 2I models are uniquely
determined by the assumed potential, i.e. by
Mfit and
M/L, the mass-to-light ratio assumed for the stars.
This means that the models are highly over-constrained by the
data - there are far more observational constraints (velocities)
than adjustable parameters (Mfit, M/L),
hence one expects to find a unique set of values for
Mfit and M/L that come closest to
reproducing the data.
This is the usual case in problems of statistical estimation
and it implies a well-behaved set of
2 contours with
a unique minimum.
When the same data are modeled using the more general distribution
of orbits available in a 3I model,
the problem becomes under-constrained:
now one has the freedom to adjust the phase-space distribution
function in order to compensate for changes in
Mfit and M/L,
so as to leave the goodness of fit precisely unchanged.
The result is a plateau in
2
(Figure 7), the width
of which depends in a complicated way on the ratio of
observational constraints to number of orbits or phase-space cells
in the modelling algorithm
(Merritt 1994).
Thus, 3I modelling of the ground-based data would
only show that the range of possible values of
Mfit
includes, but is not limited to, the values found using the 2I models;
it would not generate more precise estimates of
M
unless
the data quality were also increased.
The greater difficulty of interpreting results from 3I modelling
has not been widely appreciated; few authors make
a distinction between "indeterminacy" in
M
(the width of the
constant-
2 plateau in
Figure 7)
and "uncertainty" (the additional range in
M
allowed by measurement errors), or look carefully
at how their confidence range depends on the number of orbits used.
We illustrate these difficulties by examining two recently published
studies based on high quality, stellar kinematical data.
1. NGC 3379
(Gebhardt et al. 2000a):
The prima-facie evidence
for a central mass concentration in this galaxy consists of a single
data point, the innermost velocity dispersion as measured by HST/FOS;
the rotation curve exhibits no central rise,
in fact it drops monotonically toward the center.
Goodness-of-fit contours generated from 3I models show the expected
plateau (Fig. 7 of Gebhardt et al.), extending from ~ 106
M to
~ 3 × 108
M
.
In fact a model with
M
= 0
fits the data just as well: the
authors state that "the difference between the no-black hole and
black hole models is so subtle" as to be almost indiscernable
(cf. their Fig. 11). Gebhardt et al. nevertheless argue for
M
> 0
based on the poorly-determined
wings of stellar velocity distribution measured within
the central FOS resolution element.
In view of the fact that this velocity distribution
exhibits a puzzling unexplained asymmetry (their Fig. 4),
the stellar dynamical case for a black hole
in this galaxy should probably be considered marginal.
2. NGC 4342
(Cretton and van den
Bosch 1999):
The evidence
for a central mass concentration is again limited to a single data
point, the central FOS velocity dispersion.
Cretton & van den Bosch find that a black-hole-free model provides
"fits to the actual data [that] look almost indistinguishable
from that of Model B" (a model with
Mfit = 3.6 × 108
M).
Their
2 contours
(their Fig. 7) nevertheless seem to show a preferred
black hole mass; however they note that
2 is dominated by the
data at radii
R
5", far
outside of the radius of influence of the black hole.
The probable culprit here is the modest number of orbits (1400,
compared with ~ 250 constraints) in their 3I solutions.
Outer data points are always the most difficult to fit when
modelling via a finite orbit library since only a fraction of the orbits
extend to large radii; this is clear in their fits (cf. their Fig. 8)
which become progessively worse at large radii.
We emphasize that both of these modelling studies were based on
high-quality data, with FWHM / 2rh
0.4 (NGC 3379)
and 0.6 (NGC 4342) (Table 1).
Nevertheless, the extreme freedom associated with 3I models permits a
wide range of black hole masses to be fit to the
velocity data in both galaxies.
As Figure 2 shows, most of the galaxies in the
ongoing HST/STIS survey
of galactic nuclei will be observed at even lower effective resolutions;
hence we predict that the black hole masses in many of these galaxies
will turn out to be consistent with zero and that the range of allowed
masses will usually be large.
(To be fair, we note that these observations were planned at a time when
<M
/
Mbulge> was believed to be much larger than it is now.)
We therefore urge caution when interpreting results like
Kormendy & Gebhardt's
(2001)
recent compilation
of black hole masses derived from unpublished 3I modelling.
The techniques that allow us to detect supermassive black holes in quiescent galaxies are rarely applicable to the hosts of bright AGNs. In the Seyfert 1 galaxies and in the handful of QSOs that are close enough that the black hole's sphere of influence has some chance of being resolved, the presence of the bright non-thermal nucleus (e.g. Malkan, Gorjian & Tam 1998) severely dilutes the very features which are necessary for dynamical studies. The only bright AGN in which a supermassive black hole has been detected by spatially-resolved kinematics is the nearby (Herrnstein et al. 1999; Newman et al. 2001) Seyfert 2 galaxy NGC 4258, which is blessed with the presence of an orderly water maser disk (Watson & Wallin 1994; Greenhill et al. 1995; Miyoshi et al. 1995). The radius of influence of the black hole at its center, ~ 0 ".15, can barely be resolved by HST but can be fully sampled by the VLBA at 22.2 GHz. Unfortunately, water masers are rare and of the handful that are known, only in NGC 4258 are the maser clouds distributed in a simple geometrical configuration that exhibits clear Keplerian motion around the central source (Braatz et al. 1996; Greenhill et al. 1996, 1997; Greenhill, Moran & Herrnstein 1997; Trotter et al. 1998). Black hole demographics in AGNs must therefore proceed via alternate routes.
Dynamical modeling of the broad emission line region (BLR) constitutes
a viable alternative to spatially-resolved kinematical studies.
According to the standard model, the BLR consists of many (107 -
8,
Arav et al. 1997,
1998;
Dietrich et al. 1999),
small, dense (Ne ~ 109 - 11
cm-3), cold (Te ~ 2 × 104 K)
photoionized clouds
(Ferland et al. 1992),
localized within a volume
of a few light days to several tens of light weeks in diameter around the
central ionization source (but see also
Smith & Raine 1985,
1988;
Pelletier & Pudritz
1992;
Murray et al. 1995;
Murray & Chiang 1997;
Collin-Souffrin et
al. 1988).
As such, the BLR is, and will
likely remain, spatially unresolved. In the presence of a
variable non-thermal nuclear continuum, however, the
responsivity-weighted radius RBLR of the BLR is
measured by the
light-travel time delay between emission and continuum variations
(Blandford & McKee
1982;
Peterson 1993;
Netzer & Peterson
1997;
Koratkar & Gaskell
1991).
If the BLR is gravitationally bound,
the central mass is given by the virial theorem as
Mvirial =
vBLR2RBLR / G,
where the FWHM of the emission lines (generally
H) is taken as being
representative of the rms velocity
vBLR, once assumptions are made about the BLR
geometry. In a few
cases, independent measurements of RBLR and
vBLR have been
derived from different emission lines: it is found that the two
quantities define a "virial relation" in the sense
vBLR ~ r-1/2
(Koratkar & Gaskell
1991;
Wandel, Peterson &
Malkan 1999;
Peterson & Wandel
2000),
suggesting a simple picture of a stratified BLR in Keplerian motion.
On the downside, mapping the BLR response to continuum variations
requires many (~ 101 - 2) repeated observations taken at closely
spaced time intervals,
t
0.1RBLR/c.
Moreover, the observations can be translated into black hole
masses only if a series of reasonable, but untested,
assumptions are made regarding the geometry, stability and
velocity structure of the BLR,
the radial emissivity function of the gas, and the
geometry and location (relative to the BLR) of the ionizing continuum
source. If a wrong assumption is made, systematic errors of
a factor ~ 3 can result
(Krolik 2001).
The uncertainties surrounding reverberation mapping
has made the derived black hole masses an easy target for critics (e.g.
Richstone et al. 1998;
Ho et al. 1999).
On the other hand, because the BLR gas samples a spatial region very
near to the black hole, there is almost no possibility of making
the much larger errors in
M
that
have plagued the ground-based stellar kinematical studies
(Magorrian et al. 1998).
Thanks to the efforts of international collaborations,
reverberation mapping masses are now available for 17
Seyfert 1 galaxies and 19 QSOs
(Wandel, Peterson &
Malkan 1999;
Kaspi et al. 2000).
Taken at face value, reverberation mapping radii are found to correlate with the non-thermal optical luminosity of the nuclear source. While the exact functional form of the dependence is debated (Koratkar & Gaskell 1991; Kaspi et al. 1996, 2000; Wandel, Peterson & Malkan 1999), the RBLR - L relation can potentially provide an inexpensive way of bypassing reverberation mapping measurements on the way to determining black hole masses.
3.1. AGN Black Hole Demographics from the
M -
Mbulge Relation
With one exception
(Ferrarese et al. 2001),
black hole demographic studies for AGNs have been based on the
M -
MB, rather than on the
M
-
, relation for the simple
reason that few accurate
measurements exist in AGN hosts (e.g.
Nelson & Whittle
1995).
Lbulge, on the other hand, is more easily measured than
(though not
necessarily more accurately measured, as discussed below).
The modest sample of AGNs with reverberation mapping black hole masses is
often augmented using masses derived from the
RBLR - L relation
(Wandel 1999;
Laor 1998,
2001;
McLure & Dunlop
2000).
For a sample of 14 PG quasars,
Laor (1998)
reported reasonable agreement with the
M
-
MB relation derived by
Magorrian et al. (1998)
for quiescent galaxies, finding
<M
/
Mbulge> = 0.006.
Seyfert 1 galaxies define a significantly different
correlation according to
Wandel (1999):
<M
/
Mbulge> = 0.0003. Most recently,
McLure & Dunlop
(2000)
have reanalyzed the QSO sample of Laor
and the Seyfert sample of Wandel (the first augmented with almost as
many new objects and both with new spectroscopic and/or photometric data for
the existing objects). McLure & Dunlop split the difference of the two
ealier studies by obtaining
<M
/
Mbulge> = 0.0025.
They find no statistical difference between Seyfert 1s and QSOs.
The different conclusions reached by these authors can be traced to a number of factors.
We have recomputed the data from the Wandel (1999) and McLure & Dunlop (2000) studies under a uniform set of assumptions, as follows:
![]() |
Figure 8. The
M |
The results are shown in Figure 8. We draw the following conclusions.
3.2. AGN Black Hole Demographics from the
M -
Relation
Because of its large intrinsic scatter, there is little more that can be
learned about black hole demographics from the
M -
MB relation. An alternative route is suggested by the
M
-
relation for quiescent
galaxies, which exhibits much less scatter.
Very few accurate measurements of
are available in AGNs, due to
the difficulty of separating the bright nucleus from the faint underlying
stellar population. The first program to map AGNs onto the
M
-
relation was undertaken by
Ferrarese et al. (2001).
Velocity dispersions in the bulges of six galaxies with reverberation
mapping masses were obtained, thus producing the first sample
of AGNs for which both the black hole mass and the stellar velocity
dispersion are accurately known
(with formal uncertainties of 30% and 15% respectively).
Figure 9 shows the relation between black hole
mass and bulge
velocity dispersion for the six reverberation-mapped AGNs observed by
Ferrarese et al. (2001),
plus an additional object with a high-quality
from the literature
(Nelson & Whittle
1995).
The quiescent galaxies (Sample A from
Ferrarese & Merritt
2000)
are shown by the black dots.
The consistency between black hole masses in active and quiescent
galaxies is even more striking here than in the
M
-
Mbulge plot.
The only noticeable difference between the two samples is a slightly
greater scatter in the reverberation mapping masses (in spite of
similar, formal error bars).
Narrow line Seyfert 1 galaxies do not stand out in any way from the
rest of the AGN sample.
![]() |
Figure 9. Black hole mass versus central velocity dispersion for seven reverberation-mapped AGNs with accurately measured velocity dispersions, compared with the nearby quiescent galaxy sample of Ferrarese & Merritt (2000) (plot adapted from Ferrarese et al. 2001). |
We conclude that there is no longer any prima facie reason to
believe that reverberation-based masses are less reliable than
those based on the kinematics of stars or gas disks.
This is important since the resolution of stellar kinematical studies
will remain fixed at
~ 0".1 for the forseeable future,
whereas reverberation mapping samples a region which is per se
unresolvable and is the only technique that can yield accurate masses
for very small (
106
M
) or very
distant black holes.
4.1. M33 - No Supermassive Black Hole?
The smallest nuclear black holes whose masses have been
securely established are in the Milky Way and M32, both of
which have
M
3 × 106
M
(Table 1).
How small can supermassive black holes be? Some formation scenarios (e.g.
Haenhelt, Natarajan &
Rees 1998)
naturally predict a lower limit of
~ 106
M
.
Black holes of lower mass have been hypothesized to hide
within off-nuclear "Ultra Luminous X-Ray Sources" (ULXs:
Matsumoto et al. 2001;
Fabbiano et al. 2001),
but their formation mechanisms are envisioned to be completely
different (e.g.
Miller & Hamilton
2001).
Observationally, a black hole with
M
106
M
could
only be resolved in a very nearby galaxy.
An obvious candidate is the Local Group late type spiral M33:
the absence of an obvious bulge, and
the low central stellar velocity dispersion
(
~ 20 km s-1,
Kormendy & McLure
1993)
both argue for a very small black hole. According to the
M
-
relation (Eq. 2),
M
~ 3
× 103
M
, but a range of
at least 1 - 10 × 103
M
is
allowed given the uncertainties in the slope of the relation.
We show in Figure 10 the rotation curve and
velocity dispersion
profile of the M33 nucleus obtained from HST/STIS data.
There is an unambiguous decrease in the stellar velocity
dispersion toward the center of the nucleus:
the central value is 24±3 km s-1,
significantly lower than its value of ~ 35±5 km s-1
at ±0.3" 1.2 pc.
The rotation curve is consistent with solid-body rotation.
A dynamical analysis
(Merritt, Ferrarese &
Joseph 2001)
gives an upper limit to the central mass of
~ 3 × 103
M
.
While this is tantalizingly similar to the masses inferred for ULXs
(Matsumoto et al. 2001;
Fabbiano, Zezas &
Murray 2001),
the consistency of this upper limit with the
M
-
relation
(Figure 10 and
Eq. 2) does not allow us to conclude that the presence of a black
hole in M33 would demand a formation mechanism different
from the one responsible for the creation of supermassive black holes in
other galaxies.
![]() |
Figure 10. Upper limit on the mass of the
black hole in M33 (adapted from
Merritt, Ferrarese &
Joseph 2001).
Left: stellar rotation curve and velocity dispersion profile.
Right: the
M |
Can we expect to hear about the detection of
106
M
nuclear black holes in galaxies other than M33 within the next few
years? The remainder of this section summarizes ongoing efforts
and discusses what we are likely to learn.
In the next few years, attempts will be made to detect and dynamically measure the masses of black holes at the centers of dozens of galaxies. The Space Telescope alone is committed to devoting several hundred orbits to the cause: roughly 130 galaxies have been or will be observed with STIS within the next two years as part of ~ 10 separate projects.
News from some of these projects is already starting to circulate. Sarzi et al. (2001) report results from an HST gas dynamical study of the nuclei of 24 nearby, weakly active galaxies. Four of the galaxies were found to have kinematics consistent with the presence of dust/gas disks (the prototype of which was detected in NGC 4261 by Jaffe et al. 1994); the authors conclude that in only one of the four galaxies (NGC 2787) can the kinematics provide meaningful constraints on the presence of a supermassive black hole. Barth et al. (2001) report the successful detection of a nuclear black hole in NGC 3245, one of six broad-lined AGNs targeted by the team with HST. The STIS Instrument Development Team (IDT) has obtained stellar absorption line spectra for ~ 12 galaxies and the data for the first of these, M32, have been published (Joseph et al. 2001). The largest sample of stellar dynamical data (roughly 40 galaxies, about half of which have already been observed) will belong to the "Nuker" team. Data and a dynamical analysis have been published for one of these galaxies (NGC 3379, Gebhardt et al. 2001a) and preliminary masses for an additional 14 galaxies have been tabulated by Gebhardt et al. (2000b) and again by Kormendy & Gebhardt (2001). Mass estimates were apparently revised in the second tabulation, some by as much as 50%. We adopt the most recent values in the discussion that follows, pending publication of the full data and analyses.
We can update the
M -
relation using the
additional black hole masses that have been published over the last year.
In addition to the 12 galaxies used by
Ferrarese & Merritt
(2000)
to define the
M
-
relation, 10 galaxies
listed in Table 1 also have
FWHM/2rh < 1.
A regression fit accounting for errors in both coordinates
(Akritas & Bershady
1996)
to the expanded sample of 22 galaxies gives
![]() | (2) |
in good agreement with previous determinations (Ferrarese & Merritt 2000; Merritt & Ferrarese 2001b). Within this sample, the two subsamples containing only stellar kinematical or stellar dynamical data produce fits with slopes of ~ 4.5, in agreement with each other and with the slope quoted for the complete sample.
However, something interesting happens when the eight galaxies in
Table 1 for which rh has
not been resolved are added to the sample
(all of the mass determinations in these galaxies are based on stellar
dynamics).
Figure 11 shows the slope of the
M -
relation obtained from the
stellar dynamical mass estimates when various cutoffs are placed on the
quality of the data. If the complete sample is used (including all
entries in Table 1
down to NGC 2778), the slope becomes quite shallow,
3.81±0.33.
When only the best-resolved galaxies are included,
FWHM / rh < 0.2, the slope increases to
4.48±0.12, identical to the value
obtained from the gas dynamical masses alone.
Mass estimates based on the dynamics of gas disks are
expected to be more accurate than estimates from stellar dynamics
at equal resolution since the inclination angle of the disk
can be measured and (if the motions are in equilibrium) the
circular orbital geometry is simpler
(Faber 1999).
From Figure 11, we conclude that the inclusion
of masses
derived from data that do not properly sample the black hole's sphere of
influence biases the slope of the relation. A similar conclusion
was reached by
Merritt & Ferrarese
(2001b).
We also show in Figure 11 the results of least-squares fits using a simpler algorithm that does not account for measurement errors. This is the same algorithm used by Gebhardt et al. (2000a). As pointed out by Merritt & Ferrarese (2001b), not accounting for measurement errors biases the slope too low: the inferred slope for the complete sample of stellar dynamical masses falls to ~ 3.5 using the simpler algorithm, similar to the slope quoted by Gebhardt et al. (2000a) and Kormendy & Gebhardt (2001). Thus the lower slope quoted by those authors is due to the inclusion of less accurate data points and to the use of a less precise regression algorithm.
We noted above that a total of ~ 130 galaxies have been
or will be observed with HST during the next two years with the
hope of constraining the mass of the central black hole.
How many of these new data sets will in fact lead to stringent
constraints on
M?
In Figure 2 we plot the expected
radius of influence of the black hole versus
FWHM/2rh for all galaxies for
which a reliable distance (or redshift) and velocity dispersion could
be gathered from the literature.
Black hole masses have been estimated from the
M
-
relation, Eq. (2). We
argued above that the condition FWHM/2rh
1.0,
compounded by the low S/N characteristic of HST data,
will likely lead to weak constraints or biased determinations of
M
,
particularly in the case of stellar absorption line data.
Figure 2 shows that the black
hole's sphere of influence will
be resolved in less than half of these galaxies.
Less than one quarter will be resolved as well or better than NGC 3379
(FWHM/2rh
0.4), for which the
constraints on
M
are
weak (Section 2.2).
Among the galaxies slated to be observed in ionized gas,
the preliminary results of
Sarzi et al. (2001)
suggest that few
will be found to have the well-ordered disks that are necessary
for secure estimates of
M
.
Many of the "Sample A" galaxies from Ferrarese & Merritt (2000) were originally targeted for observation because of their exceptionally favorable properties, such as nearness (the Milky Way, M32), existence of a well-ordered maser disk (NGC 4258), etc. It is unlikely that many more galaxies will turn out to have equally favorable properties.
The majority of the targeted galaxies are expected to have black holes
with masses of order 108
M. This
range is already well sampled by the current data; however the new
detections might provide useful information about the scatter of the
M
-
relation. Only a handful of
black holes with masses
109
M
or
~ 107
M
will be
detected, and probably none in the < 107
M
range.
Probing the low and high mass end of the
M
-
relation is of particular interest since the slope and scatter of the
relation have important implications for hierarchical models of
galaxy formation
(Haehnelt, Natarajan,
& Rees 1998;
Silk & Rees 1998;
Haehnelt & Kauffmann
2000)
and the effect of mergers on subsequent evolution
(Cattaneo et al. 1999).
The fact that the new observations will not appreciably extend the
range of masses is not due to poor planning: the simple
fact is that very small or very massive black holes are found in
galaxies which are not close enough to resolve their sphere of
influence using current optical/near infrared instrumentation.
It is our opinion that the future of the
M -
relation relies on
methods other than traditional dynamical studies. An aggressive
campaign to reverberation map a large sample of AGNs appears to be
the obvious solution. The recent results from
Ferrarese et al. (2001)
show that reverberation mapping can produce mass estimates with a
precision comparable to traditional dynamical studies. Although the
obvious drawback is that it is only applicable to the ~ 1% of galaxies
with Type 1 AGN, reverberation mapping is intrinsically unbiased
with respect to black hole mass, provided the galaxies can be monitored with
the appropriate time resolution: while dynamical methods rely on the
ability to spatially resolve the black hole's sphere of influence,
reverberation mapping samples a region which is per se unresolvable.
Furthermore, reverberation mapping can probe galaxies at high redshifts
and with a wide range of nuclear activity, opening an avenue to
explore possible dependences of the
M
-
relation on redshift and
activity level.
The greatest dividend to come so far from the
M -
relation
has been the resolution of the apparent discrepancy between black hole
masses in nearby galaxies, the masses of black holes in AGN,
and the mass density in black holes needed to explain quasar light.
But the importance of the
M
-
relation presumably goes
beyond its ability to clarify the data.
Like other tight, empirical correlations in astronomy, the
M
-
relation must be telling us
something fundamental about
origins, and in particular, about the connection between black
hole mass and bulge properties.
Probably the simplest way to relate black holes to bulges
is to assume a fixed ratio of
M to
Mbulge. Since
M
(Eq. 2),
this assumption implies
Mbulge
.
In fact this is well known to be the case: bulge luminosities scale as
~
4, the
Faber-Jackson law, and mass-to-light ratios scale as ~
L1/4
(Faber et al. 1987),
giving Mbulge ~
5, in agreement
with the slope
= 4.5±0.5
derived above for the
M
-
relation.
On the other hand, the
M -
relation appears to be much
tighter than
the relation between
and
bulge mass or luminosity.
And even if a tight correlation between black hole mass and bulge
mass were set up in the early universe, it is hard to see how
it could survive mergers, which readily convert disks to bulges
and may also channel gas into the nucleus, producing
(presumably) uncorrelated changes in
M
and
Mbulge.
The tightness of the
M
-
relation suggests that some
additional
feedback mechanism acts to more directly connect black hole masses
to stellar velocity dispersions and to maintain that connection
in spite of mergers.
One such feedback mechanism was suggested by
Silk & Rees (1998)
even before the discovery of the
M -
relation.
These authors explored a model in which supermassive black holes
first form via collapse of ~ 106
M
gas clouds
before most of the bulge mass has turned into stars.
The black holes created in this way would then accrete and radiate,
driving a wind which acts back on the accretion flow.
Ignoring star formation, departures from spherical symmetry etc.,
the flow would stall if the rate of deposition of mechanical energy
into the infalling gas was large enough to unbind the protogalaxy
in a crossing time TD.
Taking for the energy deposition rate some fraction f of the
Eddington luminosity LE, we have
![]() | (3) |
Writing
GMbulge
2
Rbulge,
TD
Rbulge /
and LE = 4
cGM
/
with
the opacity,
![]() | (4) |
consistent with the observed relation. The constant of proportionality works out to be roughly correct if f ~ 0.01 - 0.1 (Silk & Rees 1998).
This model assumes that black holes
acquire most of their mass during a fast accretion phase,
tacc
107 yr.
Kauffmann & Haehnelt
(2000)
developed a semi-analytic model
for galaxy formation in which black holes grow progressively
larger during galaxy mergers.
The cooling of the gas that falls in during mergers
is assumed to be partially balanced by energy input from supernovae.
This feedback is stronger for smaller galaxies which has
the effect of steepening the resulting relation between
M
and
.
Haehnelt & Kauffmann
(2000)
found M
~
3.5 but
the slope could easily have been increased if the feedback
had been set higher (M. Haehnelt, private communication).
However the scatter in the
M
-
relation derived by them
was only slightly less than the scatter in
M
vs
Lbulge,
in apparent contradiction with the observations
(Figure 1).
Burkert & Silk (2001)
also considered a model in which black
holes grow by accreting gas during mergers.
In their model, accretion is halted when star formation begins to
exhaust the gas in the outer accreting disk;
the viscous accretion rate is proportional
to, and assuming a star formation time scale that is
proportional to TD, Burkert & Silk found
M
Rbulge
2/G
Mbulge,
with a constant
of proportionality that is again similar to that observed.
This model does not give a convincing explanation for the tight
correlation of
M
with
however.
Feedback of a very different sort was proposed by
Norman, Sellwood &
Hasan (1996),
Merritt & Quinlan
(1998)
and Sellwood & Moore
(1999).
These authors simulated the growth of massive compact objects at
the centers of barred or triaxial systems and noted how the
nonaxisymmetric component was weakened or dissolved when the
central mass exceeded a few percent of the stellar mass.
Since departures from axisymmetry are believed to be crucial for
channeling gas into the nucleus, the growth of the black hole
has the effect of cutting off its own supply of fuel.
These models, being based purely on stellar dynamics, have the
nice feature that they can be falsified, and in fact they
probably have been:
our new understanding of black hole demographics
(Section 2.3) suggests
that few if any galaxies have
M /
Mbulge as great as 10-2.
(At the time of these studies,
several galaxies were believed from ground-based data
to have
M
/
Mbulge > 1%, including NGC 1399
(Magorrian et al. 1998),
NGC 3115
(Kormendy et al. 1996a),
and NGC 4486b
(Kormendy et al. 1997)).
The tightness of the
M -
relation must place strong
constraints on the growth of black holes during mergers.
We know empirically that mergers manage to keep galaxies on
the fundamental plane, which is a relation between
,
the bulge effective radius Re and the surface brightness
at Re.
The
that appears in the
fundamental plane relation
is the same
c
that appears in the
M
-
relation (indeed,
it was defined by Ferrarese & Merritt 2000 for just this reason)
and furthermore
c
is defined within a large enough aperture that
it is unlikely to be significantly affected by dynamical processes
associated with the formation of a black-hole binary during a merger
(Milosavljevic &
Merritt 2001).
Hence the physics of the black-hole binary can be ignored and
we can ask simply: How do mergers manage
to grow black holes in such a way that
logM
4.5
log
, independent of changes
in Re and L?
This work was supported by NSF grant 00-71099 and by NASA grants NAG5-6037 and NAG5-9046. We thank B. Peterson and A. Wandel for useful discussions.