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-104 0 104 Velocity (km

Figure 5.13. A velocity-delay map for a thin disk at inclination i = 45o. The upper left panel shows in gray scale the velocity-delay map, i.e. the transfer function or the observed emissionline response as a function of line-of-sight velocity ULOS and time delay t. The upper right panel shows the one-dimensional transfer function, i.e. the velocity-delay map integrated over ULOS, which is the response of the total emission line as a function of time (cf. Equation (5.21)). The lower left panel shows the emission-line response integrated over time delay; this is the profile of the variable part of the line (cf. Equation (5.25)). Adapted from Horne et al. (2004).

-104 0 104 Velocity (km

Figure 5.13. A velocity-delay map for a thin disk at inclination i = 45o. The upper left panel shows in gray scale the velocity-delay map, i.e. the transfer function or the observed emissionline response as a function of line-of-sight velocity ULOS and time delay t. The upper right panel shows the one-dimensional transfer function, i.e. the velocity-delay map integrated over ULOS, which is the response of the total emission line as a function of time (cf. Equation (5.21)). The lower left panel shows the emission-line response integrated over time delay; this is the profile of the variable part of the line (cf. Equation (5.25)). Adapted from Horne et al. (2004).

and a similar equation defines the Fourier transform of the emission-line light curve. Use of the convolution theorem (Bracewell 1965) gives us

so = L*/C*. The velocity-delay map is then obtained by taking the inverse Fourier transform,

While this is straightforward in theory, it fails in practice simply because real astronomical data tend to be too sparse and too noisy. Other, more powerful methods, such as the maximum-entropy method (MEM) (Horne 1994) have been used, but the velocity-delay maps produced to date have been of poor quality, despite the effort and care involved (e.g. Ulrich & Horne 1996; Kollatschny 2003); it is important to understand, however, that the situation is far from hopeless. Extensive simulations show that recovery of a velocity-delay map will require only modest improvements over previous spectroscopic monitoring programs (Horne et al. 2004).

While the data requirements to obtain a fully resolved velocity-delay map are beyond what we can achieve with existing data, we can still learn much about the BLR by using simpler tools. In particular, it is possible to determine the mean response time for any particular emission line by cross-correlating the emission-line light curve with the continuum light curve. Under most circumstances, the centroid of the cross-correlation function is a good estimator of the centroid of the transfer function (Equation (5.22))

Table 5.2. Reverberation results for NGC 5548

Feature Fvar Lag (days)

Table 5.2. Reverberation results for NGC 5548

Feature Fvar Lag (days)

UV continuum

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