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3.2. The Transfer Equation

Under these assumptions, the relationship between the continuum and emission lines can be written in terms of the "transfer equation",

Equation 20     (20)

Here C(t) is the continuum light curve, L(Vz, t) is the emission-line flux at line-of-sight velocity Vz and time t, and Psi(Vz, tau) is the "transfer function" at Vz and time delay tau. Inspection of the transfer equation shows that the transfer function is simply the time-smeared emission-line response to a delta-function outburst in the continuum. Note that causality requires that the lower limit on the integral is tau = 0.

Solution of the transfer equation to obtain the transfer function is a classical inversion problem in theoretical physics: the transfer function is essentially the Green's function for the system. Unfortunately, it is difficult to find a stable solution to such equations, especially when the data are noisy and sparse, as they usually are in astronomical applications.

In practice, most analyses have concentrated on solving the velocity-independent (or 1-d) transfer equation,

Equation 21     (21)

where both Psi(tau) and L(t) represent integrals over the emission-line width.

The transfer equation is a linear equation. In reality, however, the relationship between the observed continuum and the emission-line response is likely to be nonlinear. We therefore approximate C(t) = bar{C} + DeltaC and L(t) = bar{L} + DeltaL, where bar{C} and bar{L} represent constants, usually the mean value of the continuum and line flux, respectively. We can then treat deviations from the mean as linear perturbations and the equation we actually solve is

Equation 22     (22)

Most existing data sets are inadequate for transfer-function solution, and simpler analyses are used. The most commonly used tool in analysis of AGN variability is cross-correlation, which will be discussed in detail in Sec. 4.

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