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Figure 5.16. A luminosity-dependent delay map for Hp in NGC 5548. This shows how the mean response time increases with the luminosity of the continuum source. Adapted from Cackett & Horne (2006).

Figure 5.16. A luminosity-dependent delay map for Hp in NGC 5548. This shows how the mean response time increases with the luminosity of the continuum source. Adapted from Cackett & Horne (2006).

1999, 2000; Onken & Peterson 2002; Kollatschny 2003). Given this result, the mass of the central source1 is given by

where R = ct is the size of the BLR as measured for a particular emission line, AV is the width of the line, G is the gravitational constant, and f is a dimensionless scale factor of order unity, which we will discuss further below.

Additional evidence that the black-hole masses from Equation (5.32) have some validity is provided by the correlation between black-hole mass Mbh and host-galaxy bulge velocity dispersion a*, which is known as the "M-a relationship" (Ferrarese & Merritt 2000; Gebhardt et al. 2000a). It was originally found in quiescent (non-active) galaxies, but holds also for AGNs (Gebhardt et al. 2000b; Ferrarese et al. 2001; Nelson et al. 2004; Onken et al. 2004). In Figure 5.18, we show the AGN M-a relationship superposed on the best-fit lines to the relationship for quiescent galaxies. In this case, following Onken et al. (2004), we have used a value for the scale factor f that results in the best agreement between the relationship for AGNs and that for quiescent galaxies - in other words, we have scaled the AGN "virial products" (i.e. the observable quantities R AV2/G) to give the best global match to the quiescent-galaxy relationship.

An important point that we have glossed over is how the line width AV should be characterized. Two obvious candidates introduced earlier are the FWHM and the line

1 Note that this is really the total mass enclosed by the radius R; in some models, the accretion disk also contains a significant amount of mass, which would be included in this calculation.

1016 cm 1017 cm

1016 cm 1017 cm

Figure 5.17. The relationship between the BLR line width and lag for three well-studied AGNs. The dotted lines are the best fits to each data set, and the solid lines show the best fit for a fixed virial slope, i.e. t x Vfwhm. From Peterson & Wandel (2000).

dispersion ai;ne, which is based on the second moment of the line profile P(A). The first moment of the line profile is the line centroid

and the second moment of the profile is used to define the variance or mean-square dispersion

^iie(A) = (A2) - A2 = (I A2P(A)dA^j P(A)dA^ - AO- (5.34)

The square root of this equation is the line dispersion aline or root-mean square (rms) width of the line.

We can use the ratio of these two quantities, FWHM/al;ne, as a low-order description of the line profile. For the ring of Figure 5.11, we have already noted that FWHM/aline = 21/22 = 2 .83. Another well-known result is that, for a Gaussian, FWHM/aline = 2(2ln2)1/2 = 2.35. Observationally, the mean value of this ratio for the reverberation-mapped AGNs is FWHM/aline « 2, which means that the typical AGN line profile is "peakier" than a Gaussian. Figure 5.19 shows two profiles with extreme values of the line-width ratio FWHM/aline; this emphasizes the point that these two line-width

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