|
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In this appendix, I show that under certain conditions the spatial distribution of clouds can nevertheless be measured even if the clouds have response times comparable to the light crossing times. Thus, even if the cloud reaction times are of the order of the cloud crossing times, the original goal of reverberation mapping may be obtainable after all, at least for certain classes of models. I also show that the response functions of properties of individual clouds, i.e., the “cloud response functions," can be measured. Knowledge of these cloud response functions would impose very specific and severe constraints upon the confinement mechanisms that are invoked in various cloud models and help resolve the question of what the clouds are if the cloud concept is indeed valid. 
There are a wide variety of cloud models that have been proposed. The results of this appendix, however, are valid only for a small subset of them. In particular, in the remainder of this appendix, we assume that the clouds are in orbital motions inside a centrally symmetric gravitational field; the results of this appendix do not generally apply to models violating this assumption. 
. These profiles can then be compared to theoretical ones to gauge the viability of a model, or, more quantitatively, measure the goodness of a model. Unfortunately, the line profile is somewhat inadequate for our purpose here. In this appendix we find that in order to determine some of the key response characteristics of the flux of a line, it is useful to work with a variable that is more closely coupled to the radius than is 
. It is well known that the integral of the spectrum is the bolometric luminosity or flux. For this reason, we can consider the spectrum to be the “wavelength representation" of the bolometric luminosity. Similarly, we can consider the line profiles 
as a function of time 
to be a representation of the continuum-subtracted flux in 
-space. The response functions are yet other representations of the flux. In this appendix we employ a new representation of the line flux. Consider the integral transform  | (I.1) |
is the continuum-subtracted line flux, 
is the kernel of the integral transform, and 
is a new function herein termed the “line intensity," which, incidentally, has no relation to the variable commonly used in radiative transfer calculations. The above equation maps the representation of the line flux in 
-space 
to the representation of the line flux in 
-space. Were constrained to form an orthogonal and complete basis, the mappings between 
and 
would be unique. However, for generality, is permitted to be overcomplete here. This does not pose serious problems because only discrete forms of 
are applied to equation (
I.1
). Thus, the resolution in -space can always be limited such that the mapping from 
to 
is undercomplete and well-constrained. In this respect, the only condition that we initially impose upon is that its integral with respect to 
be unity. What particular advantage does the -representation offer over the 
-representation? Let us assume that a unique mapping between distance from the black hole 
and exists with the function 
. This permits the radial integration in the time-dependent line profile (eq. [
F
.5]) to be removed. Doing this yields  | (I.2) |
upon is implicit, 
is the solid angle of the spatial vector 
from the black hole, 
is the flux from a cloud at with velocity 
in line 
observable from 
, and 
is the probability distribution of clouds in phase space. The above expression is an integral over spatial angle, velocity, and (in its most general form) the history of the local continuum flux via 
. The kernel of the transform can be defined to be the average of the local line profile composed of clouds on the shell with radius . With this definition, the fraction in the integral of equation (
I.2
) becomes equal to the mean density of clouds on this shell. Thus, equation (
I.2
) can be viewed simply as a summation of the line flux received from clouds on the shell of radius . In fact, can be interpreted as a velocity dispersion parameter while the local profile can be defined such that 
, where 
is the black hole mass and 
is the gravitational constant. Note that the clouds contributing toward the line intensity at a specific velocity dispersion are restricted to lie on a shell that is a fixed distance from the black hole. This is in contrast with the line profile at a specific equivalent line-of-sight velocity 
, which is an integral over all radii. In the next subsection, we see how this feature can permit measurement within the context of a model of the cloud response functions from the intensity light curve. However, let us first review two key prerequisites for applying equation (
I.1
) to measure (constrain) the line intensities 
from a given spectrum:
| 1. | knowledge of and
|
| 2. | knowledge of the local line profiles .
|
and can be obtained for certain simple orbital models. The methods discussed in Appendix
J
are inapplicable for non-orbital models. Therefore, for non-orbital models, reliable measurements of 
from a spectrum could be difficult if not impossible. For the orbital models considered in this appendix, however, accurate measurement of 
from a given spectrum should be achievable at several different points in dispersion space. Thus, if we restrict ourselves to orbital cloud models, the “intensity light curves" can be measured from time-resolved spectra, provided that they are available. Incidentally, this particular theoretical feature of orbital models is definitely not a reason for invoking them. It merely permits a particular type of analysis to be performed that would otherwise be much more difficult. 
as a function of lag 
to be 
when the input variable is 
and the variations are small enough about their means that the system can be considered to be linear (see Appendix
H
). With this notation, the dimensionless Fourier transform of the linearized response function of the line profile at a given line-of-sight velocity 
with respect to the observed continuum flux 
is  | (I.3) |
coordinates) over a large range of lags, even with 
held constant. Let us now calculate the transfer function of the line intensity with respect to the continuum flux. Applying the formalism of Appendix
H
yields 
 | (I.4) |
upon is again implicit. The above equation states that gain of the intensity at a given excitation frequency and dispersion is proportional to the time-averaged, angle-dependent line flux of the clouds at ; the local line flux gain of such clouds; and a factor that is a strong function of the angle of the position vector of the clouds. Once evaluated, the expression provides an approximation in the linear regime of 
given 
. For reasons discussed in Appendix
F
, the expression is unique for a given model and mean continuum flux. While equation (
I.4
) appears complex, it is actually simple for several models. Many plausible models, for instance, have a time-averaged cloud line flux that is a function of 
. Fortunately, this alone does not result in the gain of the cloud line flux having a similar dependence. This is because the gain of the cloud line flux is determined primarily from the mean of the local continuum flux, which is a function only of 
not 
for many simple yet useful models; the angle-dependent beaming anisotropy factor simply regulates what fraction of the nonlinear flux of a cloud located at is directed towards the observer. On the other hand, some models produce line shifts and asymmetries due to aspherical distribution functions (like those for bulk radial outflow) or with 
-dependent cloud pressures (like the one described in Appendix
L
). At any rate, for the former class of models in which the cloud gain is independent of 
and , we can make a neat simplification to equation (
I.4
). The gain factor can be moved outside the spatial solid angle and velocity integrals in equation (
I.4
). It can then be written as  | (I.5) |
to be the “spatial intensity" of the instantaneous component of the cloud responses, i.e., 
if each cloud were to respond linearly with 
simply 
for all clouds. As is made clear shortly, the spatial intensity contains the geometrical information about a line’s emission in a model. If 
could be restricted (computed) to some extent theoretically, the above equation would relate the unique and potentially observable intensity gain to the mean of the linearized transfer functions of clouds on the shell with radius . However, for simple spherical models each of these clouds on this shell responds identically; the response function of the shell is the same as that of each cloud on the shell. Thus, under the above conditions, equation (
I.5
) can permit measurement of the linearized transfer and response functions of an individual cloud. More precisely, if we account for the finite resolution in velocity dispersion space to which 
has been fit (or deconvolved) via equation (
I.1
) and the systematic errors in the knowledge of the local line profiles, equation (
I.5
) permits measurement of the individual cloud transfer functions averaged over a non-zero (but potentially small) range in radii. This is fortuitous. Provided firm theoretical constraints can be imposed upon 
, the original dilemma demonstrated in Taylor (1996) regarding “contamination” of the observed response functions by the individual cloud response functions may be tractable after all. Fortunately for us, 
probably has only a mild model dependence. This can be shown by considering a spherically symmetric model without occulting material. Applying 
where 
to equation (
I.4
) yields the linearized response function 
of the spatial intensity  | (I.6) |
is the beaming factor in line 
defined such that 
implies fully beamed line emission and 
implies isotropic line emission) and 
is the characteristic “spatial lag" defined as /c. Note that, regardless of the precise value of 
, the above expression for 
does not have nontrivial structure over a large range of lags. Rather, it is a trapezoidal function having structure at a characteristic delay of 
. Similarly, 
approaches zero at frequencies significantly above the “spatial frequency" defined here by 
and is unity at all frequencies significantly below the spatial frequency. Note that for models in which absorption is important (such as those with an occulting accretion disk), the near clouds are relatively brighter, in which case the effects of beaming are to some extent masked. For such models the spatial response function would probably be smoother than equation (
I.6
) and the spatial gain would be even more constrained to have nontrivial structure only near the spatial frequency. Thus even moderate systematic uncertainties of the models under consideration would probably not foil measurement of the individual linearized cloud response functions. Equation (
I.5
) tells us that for some models the intensity gain is simply the product of the spatial intensity and cloud gains. This situation is thus similar to that with equation (
H.10
), which has the following two general characteristics that can be applied here as well:
| 1. | If the line flux from clouds at a given radius responds linearly then the intensity at the corresponding velocity dispersion also responds linearly. This is because the spatial intensity necessarily responds linearly. Conversely, any nonlinearity in the intensity can be ascribed to nonlinear response of clouds at a particular radius bin. An advantage of this from a modeling perspective is that with predictive photoionization codes one could impose relatively tight constraints upon the cloud parameter space as a function of radius. This is in contrast with modeling the gain of the line profile, which at a specific wavelength is a function of clouds spanning a large range in mean continuum flux. On the other hand, a potential drawback of this is that the intensities at some dispersions would be much more nonlinear than those of the associated profiles because of this lack of radial dilution of cloud response. Therefore, measuring the individual cloud intensity transfer functions could prove difficult at certain dispersions. In this case, the individual cloud transfer functions and equation (
I.5
) have limited utility in model fitting. Fitting of the physical model parameters using (nonlinear) equation (
I.2
) could still be done to capitalize upon this richness of the intensity representation. However, as discussed in Appendix
F
regarding the fitting of nonlinear time-dependent profiles, doing this would be relatively expensive computationally.
|
| 2. | A second characteristic equation (
I.5
) shares with equation (
H.10
) is that the qualitative response features of the intensity can be determined by considering the ratios of the characteristic time scales of the spatial and linearized cloud response functions. For instance, when the ratio differs enough from unity, the cloud line flux and spatial intensity gain factors can be “separable" from one another, and one of the two gain factors can be treated as constant. In this case the time-dependent behaviors at a given excitation frequency depend upon only one of the two time scales of the system.
|
In the next section we see that the concept of separability can be partially extended to the time domain, where it implies simplified relationships between the three linearized response functions. In this case, equation (
I.5
) and its application in measuring 
simplifies dramatically. Let us now turn our attention to the precise conditions necessary for separability to occur. Let us consider the case in which just one time constant 
characterizes the “width" of the flux response function in a line of clouds at a given radius. Thus 
is the characteristic response frequency of the clouds, 
is the characteristic response frequency of the spatial intensity function, and 
is the excitation frequency of the system. Both the spatial and cloud gain factors in equation (
I.5
) must be in one of the three frequency regimes (slow, fast, or intermediate) discussed in Appendix
F
. For the global combined system, provided variability of the continuum source both exists and is measurable on all time scales, we have 3 possible frequency regimes for each gain factor. This yields a total of 
different possible regimes. Of these 9 regimes, only 5 have at least one characteristic frequency that is similar (intermediate) to the excitation frequency . Of these 5 remaining regimes, one is inseparable and occurs for dispersions and continuum excitation frequencies such that 
. In this regime, neither gain factor is approximately constant for excitation frequencies near . That leaves us with 5-1=4 potentially useful regimes in which the cloud response is separable from the spatial response. The characteristic response times of the clouds are probably dependent upon their position within the BLR and NLR. Let us assume that the equilibrium time scale of each cloud can be approximated with a power law in mean flux, such as 
, where the “asymptotic gain" 
of 
with respect to is defined as 
logr for 
see also eq. [
H.2
]). Note that only 
corresponds to flux-independent cloud equilibrium times. Furthermore, we have 
at a “critical radius" of  | (I.7) |
 | (I.8) |
and 
differs substantially by 
or when 
for intensity components with dispersions much greater or less than 
. In these cases separability can be attained. Note that this result is contingent upon 
being constant; there are systems that do not have a critical velocity dispersion because of the dependence of upon radius. The four different separable regimes can then be characterized as follows:
| 1. | The “fast spatial" regime. In this regime,  , yet  , which requires  . Let us assume that each gain factor at a given radius has nontrivial structure only within one decade of frequency. The condition  can then be expressed as  , where
 this regime can occur for components of the intensity with  , where
 the fast spatial regime can occur for  . In Figure
I.1
, the spatial, cloud, and global (directly observable) intensity gains of an optically thin line of a spherically symmetric linear system are shown at a velocity dispersion of  km s . The line emitted by the clouds of this hypothetical model responds on an equilibrium time scale of  days, with  and  (Such numbers might exist, for example, in a model with a linearly-responding line and large clouds that eventually evaporate when exposed to additional continuum heating.) The black hole mass is  , which was selected such that  days. The time constant ratios of this hypothetical model differ enough that the excitation frequency range of the intermediate regime is small. The fast spatial regime occurs here for  days , where the spatial intensity gain factor is approximately its asymptotic value of unity. In Figure
I.2
, the response functions of this system are shown. For lags greater than  days, the global line intensity response function is equal to the cloud response function. By ignoring oscillations with frequencies above  and subtracting off either narrow or broad components of the line profile, this fast spatial regime is in principle always achievable for simple enough systems (that, e.g., are moderately spherically symmetric). However, as forewarned above, doing this in practice could be difficult or even impossible if, e.g., is close enough to unity and  is too high or  is too low compared to the mean velocity dispersion in the observed line. Figure I.1: _Real component of the transfer function (the gain) of the  km s intensity component (solid line) as a function of excitation frequency  assuming cloud equilibrium times of  days. Also shown are the gains of the transfer functions of the individual clouds (dotted line) and the spatial gain (dashed line). This is for a hypothetical optically thin line using a very simple model, with a black hole of mass  selected to yield a spatial time scale of  days). For  days , the spatial transfer function is approximately its asymptotic value. This defines the fast spatial regime. Conversely, for  days , the cloud transfer function is approximately its instantaneous value. This defines the slow cloud regime.
|
| 2. | The “slow cloud" regime. In this case,  yet  , which also requires  . This regime is the same as that illustrated in Figure
I.1
, but for  days , wherein the cloud gain factor is approximately its instantaneous value. Because this instantaneous gain is nonzero, the global intensity response function is approximately the product of the instantaneous cloud gain and the spatial response function for lags below  days.
|
| 3. | The “slow spatial" regime. In this case,  yet , which requires  . In Figure
I.3
the gains are shown for a system like that of Figures
I.1
and
I.2
but for a hypothetical model in which  day and  days. For  days the intensity gain oscillates about zero because the instantaneous spatial intensity gain is zero for this model. This defines the fast spatial regime. In the time domain we see this regime on resolutions better than  days, where the derivative of the global intensity response function is equal to the cloud response function divided by the spatial response function at zero lag (see Fig.
I.4
.)
|
| 4. | The “fast cloud" regime. In this regime,  yet  , which also requires  . In the time domain this regime corresponds to lags greater than  , where, as in the slow cloud case, the global response function is (within a time resolution of  approximately the spatial response function multiplied by the asymptotic cloud gain. This regime is illustrated in Figures
I.3
and
I.4
, where  days and  days. If  this fast cloud regime can be realized if the broader profile components with dispersion velocities above  are subtracted out. Incidentally, in some prior work in which the fast cloud regime was apparently assumed, only narrow components were subtracted out (e.g., Krolik et al. 1991).
|
days, while an analog of the slow cloud regime occurs on time resolution scales less (better) than 
days. Note that the cloud response function is off the scale near zero lag. This is because 
is nonzero. Also, the solid (and easiest to measure) line has structure on a time scale much greater than the spatial response time scale, which indicates either a highly arranged distribution of matter or (via eq. [
I.6
]) locally delayed response. Unless specific deconvolution techniques were employed to avoid it, the empirical intensity component would have a nonzero width in dispersion space due to finite sampling resolution. This would result in a steep, rather than infinite, slope near 
days for such a hypothetical system. There is some numerical error at high and low values of 
. |  |
km s
intensity component assuming cloud equilibrium times of 
day. Other parameters are unchanged from those assumed in Fig.
I.1
. For 
days
, the intensity gain oscillates about zero. This is due to the spatial gain factor being near zero at high frequencies. This defines the fast spatial regime. For 
days
, the cloud gain factor is approximately constant. This defines the fast cloud regime. |  |
|
|
Analysis of variability data permits two potential benefits:
 |
It permits a reduction in the uncertainties of the values of the physical parameters that are used to fit a given object as defined within the context of the specific models that are fit. Note that these physical parameters have no meaning outside the specific models to which they are associated. Thus, any truly model-independent analysis of data would offer no new physical information about the system being observed.
|
|
It permits a reduction of the uncertainty of the overall viabilities (i.e., goodnesses) of the models being fit provided the actual effective number of these parameters that have been granted to a model is small enough compared to that of the data. |
From an experimental perspective, there is a simple way the equations of the prior section can be used. First, the various functions in the equations can be parameterized. Second, differences between the two sides of each equation can be assigned to difference variables. Third, the model parameters can be numerically adjusted within the constraints dictated by their error limits until the difference variables have been minimized. Before using any of the various software programs that have been written to perform this type of task, one must first select a compatible parameterization scheme for the model to be tested. The numerous parameters associated with certain models (such as those represented by eq. [
F.4
]) might be poorly constrained from the limited data and available information. In particular, if the only input knowledge for such a model is a linearized velocity-resolved response function, the various model parameters might not be uniquely determined, even if the observational errors were negligible. This is analogous to the shadow of an object not uniquely determining its topology unless, for example, the object is assumed to be both two-dimensional and viewed perpendicularly. For this reason, let us first consider the fitting of relatively simple models. 
are independent of flux and hence radius), spherical symmetries in with respect to and , and . This last assumption would probably be applicable for lines suspected of having low optical depths, such as C III
. One way this model could be parameterized is by letting be an interpolation of a two-dimensional grid in and v with a resolution dictated by the quality of the data used. In this case, is constrained from the ionizing continuum flux and the observed spectrum 
. Specifically, it is straightforward to show that  | (I.11) |
can be obtained (i.e., measured) from the linearized response function 
of the spectrum 
with respect to the ionizing flux 
, the convolution equation, and  | (I.12) |
is the time-dependent spectrum of just the continuum. Use of the above equation in the overall fitting routine with 
being free parameters is important because profile de-blending, like any other interpretation of data, is necessarily model dependent, at least to some extent. In Blandford & McKee (1982) a clear distinction is made between model fitting and blind mathematical fitting of a 
function. However, note that equation (
I.11
) implies that a blind mathematical fitting of a response function to variability data of a given line is almost equivalent to model fitting of the arbitrary 
function above. In fact, other than the required transformation of variables (described in this case by eq. [
I.11
]) and their associated uncertainties, there is only one difference between model fitting using an arbitrary but linear model and fitting with an arbitrary linearized response function: with the latter method no physical significance is explicitly attached to the fitted function. Exactly what is considered here to be the “fundamental" set of parameters of a model selected for fitting is unimportant, provided that the measurement errors are also transformed appropriately. However, this is not the case when more than one line is available. Because A and 
must be constants in the above linear model, the response functions for each line are proportional to one another. Thus, provided the local continuum flux determines the cloud properties, the mere fact that the response functions obtained by Krolik et al. (1991) appear to be line-dependent empirically tells us that the linearity assumption (in addition to this model) is formally incorrect. This tells us that the use of nonlinear models is probably necessary for fully self-consistent interpretations of AGN variability data. 
, where 
is an abbreviation for 
. Let us also assume that the other model parameters are the same as for the prior example model. This second example system has instantaneous (without delay) nonlinearities. Its exact nonlinear profiles are  | (I.13) |
 | (I.14) |
If 
is forced to be independent of radius, it can be measured by fitting equation (
I.13
) to spectra. This method of obtaining the mean nonlinearities in the lines should be significantly more accurate than that in, e.g., Pogge & Peterson (1992), where the spatial response function was crudely approximated to be a delta function in lag. The dependence of the kernel of an input-modified response function is a function of only lag via 
. If the dependence of upon 
and is known then the local line profiles can be numerically fit from the velocity-resolved response functions. For instance, in this second example model we assume isotropic cloud line fluxes which yields (via eq. [
I.14
])  | (I.15) |
, and 
can be measured within the context of this model. Equation (
I.15
) is actually of little practical utility since the local profiles can be obtained in a more direct fashion by numerically fitting equations (
I.13
)-(
I.14
) to variability data. Equation (
I.15
), however, illustrates an important point. The equation would apply even if 
were a function of radius and the responses were very nonlinear. One assumption it does require is instantaneous responses from the clouds. Recall that the first example model illustrated that if there are line-dependent response functions than at least one of the cloud line emission functions is non-linear for models in which the emission is a function of only the local continuum flux. Here we see that if there is also a line dependence in the local profiles then either the assumption about the dependence of upon 
or 
is invalid or local delays are important. In the latter case, models like the second example model would fit the data poorly, yield low goodnesses, and give large measurement errors. 
and 
. In the simplest case, these local delays can be parameterized with a linear model, with individual cloud line flux gains of  | (I.16) |
and that  | (I.17) |
is the number density of clouds, with 
and 
so that 
in accordance with a Maxwellian distribution of clouds. As mentioned in Appendix
J
, equation (
I.17
) is probably not as “square-shaped" in velocity space as certain theoretical models would suggest. However, it has the advantage of being built into most fitting packages. Equation (
I.2
) implies  | (I.18) |
and 
when fit with equation (
I.1
), equation (
I.12
), and the convolution equation. In fact, for this model, equation (
I.5
) gives in the linear regime  | (I.19) |
, 
, 
, 
, 
, and 
from time-resolved spectra. For more accurate modeling and lower measurement errors at the expense only of computation time, equation (
I.16
) could be replaced by a nonlinear analog. The point in either case is that the physical parameters of the model, including the cloud pressure equilibrium times, are indeed measurable quantities provided theoretical constraints can be imposed upon the angular dependence of . Note that this third example model does not necessarily have more degrees of freedom than the first two. This is because upon including the above six parameters, one would accordingly reduce the grid resolution of the fitted functions like 
such that the effective number of parameters (e.g., the order of its polynomial) permitted in the model remained the same. Incidentally, this reduction would be small when the quality of data is high and the response functions can be interpolated from several points. Performing the computationally expensive nested integrals (such as those implicit in the line flux equation of models with local lags) instead of the linear approximations (eq. [
I.16
]) admittedly provides a means of accounting for both nonlinear and inseparable behavior if sufficient computer time is available. In the linear regime, however, including parameters to account for finite response times in a cloud would not change the prediction error of the model at all. In this context, as stated above, the real benefit of model fitting with and cloud lag parameters over “blind" mathematical de-convolution is that the fitted quantities have physical meaning. For instance, features in 
beyond lags of 
would be physically meaningful in the third example model provided that the equilibrium times scale of at least one cloud property is sufficiently long. Similarly, additional information, such as that from fully nonlinear photoionization codes like CLOUDY, XSTAR, ION, etc. can be included (albeit with substantial systematic error) along with the line emission data to lower the errors of the fitted parameters. This would, for example, help exclude so-called “unphysical de-convolutions." Also, if one is fitting the underlying physical conditions of the gas emitting the lines rather than the line emissivities themselves, line blending aids and constrains our knowledge of what we are fitting (as opposed to hindering it). With abstract mathematical de-convolution, valuable information resources like these must be thrown away. (Invoking physical arguments to interpret an abstract de-convolution is a crude means of model fitting. This point, along with others on the advantages that model fitting has over “arbitrary" function fitting, is discussed in more detail elsewhere.) Though linear response functions or, for the apparently nonlinear cases, even multidimensional “time-dependent" response functions might be able to fit variability data well, they probably do not offer as much to our understanding of AGNs as does model fitting. Abstract mathematical deconvolution discussed in Blandford & McKee (1982) is probably not the ideal method of analyzing variability data. If its purpose were merely to predict variability data, then other fully nonlinear methods, such as those employed with neural nets to predict, e.g., stock prices, would probably fare better. If, on the other hand, the ultimate goal is to increase our knowledge of the physics of the AGNs, then the model fitting discussed in Blandford & McKee (1982) would probably fare better; only model fitting permits measurement of several parameters within the context of specific models while formally accounting for their physical plausibility. For instance, the example model of §
I
.3.2 can be used to measure the mean asymptotic gain 
of a line without approximating the spatial response function to be a delta function in lag as was done in previous works (Krolik et al. 1991; Pogge & Peterson 1992). Another set of parameters is the linearized response function of the intensity in a line with respect to the continuum flux 
. When information from a photoionization code is included in the fitting of a model, this function would probably be of more utility than 
. This is because the range of cloud parameters sampled by the intensity at a given velocity dispersion is much smaller than that sampled by the linearized spectral response function at a given wavelength. It is the knowledge of the values of the various physical parameters that can help us understand AGNs. For instance, knowledge of the radii in a fitting of a model that incorporates fully nonlinear results (e.g., 
from a photoionization code where contributions to the reduced 
are large would help indicate the parameter space wherein more physics is necessary and uniform pressure cloud models fail. Knowledge of which lines have behavior that is inconsistent with the black hole mass inferred from other lines, after nonlinear behavior is taken into account, would give an independent upper limit to how important alternative line broadening mechanisms (e.g., electron scattering) could be in orbital models. Knowledge of the sign of 
, where 
is the geometrical size of the clouds, would help answer the very important question of whether or not the source of clouds in AGNs, which has been debated since their introduction, is in part due to the continuum flux or just some passive element of the overall AGN environment. Perhaps the most important knowledge that can be gained by proper model fitting is that related to the cloud response functions 
. For instance, consider that the emission efficiency in a line can drop significantly when the column density to ionization parameter ratio becomes low enough that the “back" portions of a cloud are too highly ionized for significant emission by an ion. Partly for this reason, intensity line ratios could be used to obtain measurements of mean column densities as a function of radius (along with the pressures or densities). This in turn could permit measurement of cloud characteristics such as the time-averaged size of the clouds as a function of radius. With cloud response functions, the time-dependence of these various cloud properties can also be determined, which would permit measurement of the cloud evaporation and pressure-equilibrium wave propagation speeds (even if they are nearly zero). Such knowledge would impose new and important constraints upon the various cloud confinement mechanisms that have been proposed. Rather than be distressed that nature is more complicated without the fast cloud assumption, we should consider ourselves very fortunate, for cloud response functions would allow us to directly probe the structure of an individual cloud within the context of a model, and, if the cloud concept is valid, help answer the question “What are the clouds?" Answering this question, rather than “Where are the clouds?" is probably more important for understanding AGNs.
