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5.2. The MBH - sigma 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

Equation 16 (16)

where sigmae 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 propto sigmaalpha relation, with a somewhat steeper slope of alpha ~ 4.8.

Figure 21

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 - sigma 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

Equation 17 (17)

where cs is the speed of sound. Note that for dissipationless collapse the velocity dispersion is roughly given by sigma2 appeq 2cs2. The region is assumed to rotate rigidly (due to effective tidal torques) at an angular speed Omega, and the specific angular momentum is j = r2infty Omega sin2 theta, where rinfty is the initial radius and theta is the polar angle. For zero-energy orbits, the pericenter distance in the equatorial plane is therefore

Equation 18 (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 leq 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)

Equation 19 (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 rho(r) = sigma2 / 2pi 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 = 4picGMBH / kappa (at which gravity is balanced by radiation pressure; where kappa 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

Equation 20 (20)

The condition that the shell would escape, and therefore, that the black hole would unbind the bulge gas, requires Vout > sigma. This implies that the black hole mass is limited by (Silk and Rees (1998))

Equation 21 (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 ~ sigma4 (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) sigma ~ rho1/6 Mhalo1/3, and (iii) Mhalo scales like rho-2 in typical cold-dark-matter cosmologies (really like rho-2/(3+n), where n ~ - 2 is the slope of the dark matter fluctuations spectrum). Combining (i)-(iii) gives MBH ~ sigma4.

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 sigma4.5 (Ostriker 2000).

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