8.3. The Transfer Function
A more ambitious task is to try and map out the entire BLR, using information obtained from the light curves.
Given the assumption of linear response of the line to the continuum pulse, we can formulate the relation between L(t) and E(t) using a "transfer function", (t), (called also a "response function"):
i.e. E(t) is the convolution of L(t) with (t).
As can be seen from this equation, (t), in appropriate units, equals the E(t) that would result from L(t) which is a -function at t = 0 (a continuum "flash"). For gas which is distributed in a thin shell of radius r, the transfer function is a "boxcar" shaped pulse, lasting from t = 0 until t = 2r/c, with a constant value of c / 2r. The rise at t = 0 is due to the fact that the gas along the line of sight appears to respond immediately to the continuum pulse, and the information about the continuum and line variation arrived to the observer simultaneously. The constant value of (t) results from the time delay of a ring at a polar angle , r(1 - cos ) / c, and the emissivity of the ring which is proportional to its surface area. In a similar fashion the transfer function of a circular ring, inclined at an angle i to the line of sight, is non-zero between times r(1 - sini) / c to r(1 + sini) / c, with its center at time r/c, This is illustrated in Fig. 19.
Figure 19. The response of a thin shell and an inclined ring emission line regions to a -function continuum light curve. (Shaded area in the top half shows the narrow ring at a polar angle .) For such a continuum variation, the shape of transfer function (t), is identical to the shape of the line light-curve.
The transfer function of a thick shell is obtained by integrating the thin shell (t) over all radii and weighting the contribution at each radius according to the emissivity. The case of a shell of inner radius rin and outer radius rout is shown in Fig. 20. As seen from this diagram, (t) is a constant in time between t = 0 and t = 2rin / c, and declines to zero between t = 2rin / c and t = 2rout / c. The shape of the declining part depends on the gas distribution and emissivity. We can find this shape in the simple, optically thick case, using the notation of chapter 5 and the radial dependence of the covering factor from equation (57),
This is illustrated in Fig. 20 for the cases of (p + q) = -2, 0 and +2. In a similar fashion, the transfer function of a thick disk is obtained from integrating over rings.
Figure 20. Bottom: Transfer functions of a thick spherical shell of inner radius 10 time units and outer radius 20 time units, for cases of (p + q) = -2 (dotted line), 0 (solid line) and +2 (dashed line). The top half demonstrate the contributions of different thin shells to (t), for the case (p + q) = 0.
We see that valuable information about the gas distribution can be obtained from (t) and it is desirable to find this function and investigate its shape. In principle, this is not a difficult task since (t) can be recovered from the data by applying the convolution theorem,
where ~ designates the Fourier transform. Performing this operation, using the observed L(t) and E(t) and transforming back to the time domain, recovers (t). In practice, this is not a trivial task. E(t) and L(t) are often unevenly sampled in time, have large gaps and span a relatively short period. Under such conditions, Fourier methods become problematic, and a meaningful transfer function may become hard to obtain. Frequently sampled data, with a sampling interval shorter than the typical variability time scale, can be quite useful, provided the light curve is long enough and the measurement error small compared with the variability amplitude. There are improved statistical methods of recovering (t) from the data (e.g. the maximum entropy method), using additional constraints on its expected shape at different times.
The transfer function for NGC 4151, obtained from applying the maximum entropy method to the data presented here, is shown in Fig. 21. The diagram also shows the H light curve which is obtained by convolving this function with the continuum light curve. The fit of the H light curve is quite satisfactory, suggesting that this (t) is not a bad approximation to the real transfer function. The empirical (t) rises sharply at t = 0 and drops to zero, in a gradual way, over 30 days. This is consistent with a thick shell geometry with a very small inner radius and an outer radius of about 15 light-days. It is also consistent with an edge-on disk (and other geometries) of similar dimensions.
Figure 21. Top: Transfer function for NGC 4151, obtained from the line and continuum light curves in Fig. 17, using a maximum entropy deconvolution. Bottom: A model emission line light curve obtained from the transfer function, on top of the H light curve.
A word of caution is in order. There are cases where several different transfer functions can fit the data equally well. This depends on the numerical method used and the quality of the light curves. For example, the features in the NGC 4151 transfer function at t ~ 50 - 100 days can be interpreted as due to some line emitting material far away from the nucleus. However, they can also be due to the numerical method used, given the freedom to put the emitting material anywhere around the central source. Physical constraints, such as an imposed upper limit on the BLR extension, should be used in such cases.
The experimental limitations are so severe that today the BLR transfer function is only known in one or two cases. The main problem in ground-based observations is the large amount of telescope time needed for proper sampling of the light curve, and the requirement of flux calibrated data. Space-born instrument are more suitable for the task but the aperture size of most of these is small and it has been extremely difficult to obtain enough observing time to perform the experiment. The first, complete ultraviolet data set was obtained in 1989 by a large group of IUE observers, who monitored the Seyfert 1 galaxy NGC 5548 for a continuous period of eight months. Much of our understanding of AGN variability is based on this data set.
The main theoretical limitation of the method is the assumption of linearity. While the total line and diffuse continua emission of an optically thick gas is indeed proportional to the continuum flux, this is not case for individual lines. This is well illustrated in Fig. 8 that shows the response of different emission lines to variations in the ionization parameter. The use of (t) obtained for a certain line to deduce the gas distribution must therefore be done with great care. In addition, the geometry deduced from the empirical transfer function is not unique, and more information is required to choose among all possible alternatives. Finally, all present day studies make the specific assumption that the observed optical, or ultraviolet continuum variations are proportional to the ionizing continuum variations. This is not necessarily the case and alternatives must also be investigated. Line profiles can provide additional constrains on the gas distribution, as discussed in the next chapter.