5.2. AGN Black-Hole Masses

Reverberation mapping is one method of measuring AGN central masses via the virial relationship

(40)

where f is a dimensionless factor of order unity that depends on the geometry and kinematics of the BLR, is the emission-line velocity dispersion, and r is the size of the emitting region. Measurement of the emission-line time lags provides the ingredient that has been missing since the first attempts to understand the basic physics of AGNs ⁹⁵. Relative to other dynamical estimators, advantage of using the BLR to provide an estimate of the mass of the central source is that it is located very close to the central source (within ~ 10³ R_grav), leaving little doubt that the central mass is in fact a black hole. On the other hand, the kinematics of the BLR are not yet understood (see below), and non-gravitational forces might have a strong effect on gas motions. For the virial method to be applicable, the BLR kinematics must be dominated by gravity. Even without understanding the detailed geometry and kinematics of the BLR, we can test the virial hypothesis by comparing lags and line widths measured in a single AGN: all lines must give the same virial mass, even though not all the line-emitting material needs to have common kinematics. The three AGNs for which this can be easily tested are shown in Fig. 33.

Figure 33. Line width in the rms spectrum plotted as a function of the distance from the central source (upper horizontal axis) as measured by the emission-line lag (lower horizontal axis) for various broad emission lines in NGC 7469, NGC 5548, and 3C 390.3. The dashed lines are best fits of each set of data to the relationship log VFWHM = a + b log c. The solid line shows the best fit to each set of data for fixed b = -1/2, yielding virial masses of 8.4 × 106 M, 5.9 × 107 M, and 3.2 × 108 M for the three respective galaxies. From Peterson & Wandel 72 © 2000 AAS.

Even if this were not true for all lines, it may be true for some lines, and a given line must always yield the same mass. Only in the case of NGC 5548 is there sufficient information on the long-term behavior of a single line (H) for this test to be applied, and the data seem to be consistent with the virial relationship ⁷². We expect, then, that as the continuum brightens, the emission-line lag increases (see Fig. 34), and the emission-line becomes narrower. This does seem to be the case.

Figure 34. The H emission-line lag as a function of optical continuum flux for the Seyfert 1 galaxy NGC 5548. The solid line shows the best power-law fit to the data, F0.4 ± 0.2, and the dotted line shows the best fit to the naive theoretical prediction F1/2 (Eq. (39)). The dashed vertical line shows the estimated starlight contribution, as in Fig. 1.

Virial masses based on H line reverberation as a function of optical luminosity are shown in Fig. 35. There is considerable scatter in the relationship, but it is nevertheless clear that higher-mass black holes are found in higher-luminosity AGNs. Some of the scatter in this relationship may be attributable to differences in accretion rate or radiative efficiency: the lower end of the envelope, for example, seems to be dominated by narrow-line Seyfert 1 galaxies, which are thought to have relatively high accretion rates (and thus luminosities) for their mass. Additional factors, such as inclination of the system, may also contribute to the scatter. But we are, finally, beginning to see the first indications of a mass-luminosity relationship for AGNs.

Figure 35. The central mass of AGNs, as measured from H line reverberation, as a function of optical continuum luminosity. Symbols are as in Fig. 32. From Peterson et al. 75 © 2000 AAS.