5.2. The MBH -
relationship
Using three-integral models and HST spectroscopy, Gebhardt et al. (2000) were able to determine black hole masses in 16 galaxies. Adding to these two galaxies with maser mass determinations (NGC 4258 and NGC 1068), six galaxies with black hole masses determined by gas dynamics, and our own galaxy (Ghez et al. (1998), Genzel et al. (2000)) and M31 (Dressler and Richstone (1988)), they were able to discover a tight correlation between the black hole mass and the velocity dispersion in the galactic bulge. Specifically, they found that
![]() |
(16) |
where e is
the line of sight aperture dispersion within the half-light radius
Re. The obtained relation is much tighter
(Fig. 21, right) than a previously determined
relation (e.g.,
Magorrian et
al. 1998)
between black hole mass and the bulge luminosity
(Fig. 21, left). Based on a smaller sample
(12 galaxies),
Ferrarese and
Merritt (2000)
found independently a
MBH
relation, with a
somewhat steeper slope of
~ 4.8.
![]() |
Figure 21. Black hole mass versus bulge luminosity (left), and the luminosity-weighted dispersion (right). Adopted from Gebhardt et al. (2000). |
Clearly, the observed tight
MBH -
correlation strongly
suggests that the formation and evolution of the bulge and of the black
hole are causally connected. The precise nature of this connection,
however, is still a matter of considerable uncertainty. A relatively
simple theoretical model for the relation has been suggested by
Adams, Graff and
Richstone (2001).
In this model, a slowly rotating isothermal sphere (with a seed central black hole) collapses to form the bulge (the dark matter is assumed to move in tandem with the baryons). The density distribution is assumed to be of the form
![]() |
(17) |
where cs is the speed of sound. Note that for
dissipationless collapse the velocity dispersion is roughly given by
2
2cs2. The region is assumed to rotate
rigidly (due to effective tidal torques) at an angular speed
, and the specific
angular momentum is j =
r2
sin2
, where
r
is
the initial radius and
is the polar angle. For
zero-energy orbits, the pericenter distance in the equatorial plane is
therefore
![]() |
(18) |
where in the last equality we used the fact that for the assumed density
distribution M(r) = 2c2s
r / G. For material to be captured by the black hole we need
p
4RS, where RS is the gravitational
(Schwarzschild) radius
RS = 2GM / c2. Using eq. 18,
the condition p = 4RS therefore reads (assuming
that the capture condition defines the black hole mass)
![]() |
(19) |
in good agreement with the observed relation (eq. 16).
Other considerations result in similar expressions. For example, in a
protogalaxy modeled as an isothermal sphere of cold dark matter, with
(r) =
2 /
2
Gr2, with
a fraction
fgas in the form of gas, a central, accreting black
hole will generate an intense wind outflow. The black hole itself may be
assumed to form by coherent collapse before most of the bulge gas turns
into stars. If the black hole radiates at the Eddington luminosity,
LEDD =
4
cGMBH
/
(at which gravity is balanced by radiation pressure; where
is the electron
scattering opacity), and a fraction
fout is deposited into kinetic energy of the outflow,
then a shell of swept-up material will be moving outward at a speed
![]() |
(20) |
The condition that the shell would escape, and therefore, that the black
hole would unbind the bulge gas, requires
Vout >
. This implies that the
black hole mass is limited by
(Silk and Rees
(1998))
![]() |
(21) |
similar to the relation found by Ferrarese and Merritt (2000).
Semi-analytical, hierarchical galaxy formation models (see
Section VI), in which galaxies form by merging
halos (and merging central black holes), with simple prescriptions for
gas cooling, star formation, and feedback from supernovae, also tend to
produce scalings of the form MBH ~
4
(for example,
Haehnelt and
Kauffmann 2000).
Broadly speaking, this relation can be traced to the facts that: (i) In
mergers, the black hole mass scales with the halo mass.
(ii)
~
1/6
Mhalo1/3, and (iii)
Mhalo scales like
-2
in typical cold-dark-matter cosmologies (really like
-2/(3+n), where n ~ - 2 is the slope of
the dark matter fluctuations spectrum). Combining (i)-(iii) gives
MBH ~
4.
Somewhat more exotic scenarios, which involve the accretion of
collisional dark matter
(invoked to make galactic halos less dense; for example,
Spergel and
Steinhardt 2000),
also produce black hole masses which scale roughly with
4.5
(Ostriker 2000).