Appendix F
Local Delays
In Chapter 2 we assumed that the emission characteristics of illuminated clouds are purely a function of the instant continuum flux to which they are exposed. In this appendix, I analyze the validity of this assumption. I find that this universally adopted assumption may be wrong, and that the history of exposure accounting for “local delays" due to finite cloud equilibrium times may also be relevant. In such cases, I show that the mean response time is a function of the recent average value of the continuum. I also show that if instantaneous or linear response is incorrectly assumed, local delays and nonlinear response can make a system appear larger than its actual size. Finally, I show that local delays can be a source of asymmetry about the peak of the cross-correlation function.



F.1 Background


Procedures for computing a linearized response function of the time-dependent line emission given off from an ensemble of clouds illuminated by a time-dependent source are well known (e.g., Blandford & McKee 1982). They assume that the contribution toward line emission from a specific source is purely a function of the radius from the central object and the instantaneous continuum flux to which it is subjected. This requires that the processes relevant to its line emission attain equilibrium much more quickly than the other time scales involved. The explicit time-dependent response of individual clouds, where, e.g., the line emission efficiency in a cloud lags the continuum flux it experiences, has not yet been accounted for in previous works concerning AGN variability.

Accounting for finite equilibrium times, however, can yield interesting results for most of the AGN cloud models that have been proposed. Consider, for instance, a cloud model in which the cloud area is a decreasing function of the cloud pressure, which is externally regulated by the pressure of an intercloud medium. Rees, Netzer, & Ferland (1989) additionally assumed , where is the pressure throughout the cloud and is the distance from the black hole. Let us consider the analogous case where the pressure is regulated by the local ionizing continuum flux and only indirectly through , namely . Such a dependence implies that a change in the continuum luminosity invokes a change in the cloud pressure as well. As we shall see, “reactive" cloud models like this one offer both theoretical and empirical advantages over static ones. Note that the clouds would not react instantaneously; a minimum for the characteristic time scale for internal pressure equilibrium to be asymptotically obtained is the sound crossing time of the clouds. As noted in Netzer (1990), this time scale can be similar to the continuum variation time scales, suggesting that clouds of this model rarely might be in actual pressure equilibrium. Therefore, even though the outermost layer emitting a line can be a small fraction of the cloud as a whole, clouds of this model should to some extent “remember" their prior pressures and areas.

Because line emission from clouds is a strong function of the area, pressure, and pressure ionization parameter defined here as the ionizing photon to gas pressure), the line efficiency of a cloud has a nontrivial time dependence. For instance, consider the case where the continuum flux local to a cloud suddenly increases. If the pressure ionization parameter of the cloud would at first follow the increase in the continuum flux, but would then decrease as the pressure begins to approach its new equilibrium value. Relative to Ly, the flux in a line like N which is probably a relatively high ionization transition in stable cloud sections (Taylor 1994), would initially rise, but then decay as the ionization parameter decreases. The response function that one would obtain upon a linear fitting would have structure not only at the range of lags corresponding to the light crossing times of the emission region, but also at lags greater than these by the pressure equilibrium times in the clouds.

In such a case, the previous works on AGN variability, which have all assumed that a response function at a given lag is proportional to the density of clouds along the corresponding “iso-delay" surface, are inapplicable. Specifically, the results based upon equation (2.13) of Blandford & McKee (1982), which was derived under the assumption that the equilibrium time scales of the cloud properties are all much less than the light crossing time (hereafter, the “fast cloud" assumption), are now suspect. This is an important point because a great deal of effort has been expended to obtain and analyze variability data using the approach of Blandford & McKee (1982).

In § F .2 of this appendix we find that there are several cloud properties affecting line emission that could be strong functions of the local continuum flux with equilibrium times large enough to violate the fast cloud assumption. Because, for these cases, the popular formalism of Blandford & McKee (1982) is invalid, a new and more general formalism for analyzing variability data will be developed in § F .3. This new formalism is compatible with models that have clouds with finite equilibrium times and nonlinear responses. Readers not interested in the mathematical derivation of the time-dependent line profile with the new formalism may wish to skip to § F .4, where the new theory is applied to some simple models. A summary is provided in § F .5.



F.2 Motivation
In order for the formalism of Blandford & McKee (1982) to be invalid for a given cloud model, two conditions must be satisfied for at least one of the cloud properties in the model. The first of these conditions is that the line emissivity be a moderately strong function of the cloud property and that the cloud property in turn be a moderately strong function of the local continuum flux a cloud experiences. The second condition is that the equilibrium time scale of the cloud property be near one of the other characteristic time scales of the system. If the equilibrium time scale is near or greater than the line emission region light crossing time, the response function will be affected. Conversely, if the equilibrium time is near the time for clouds to cross the emission region, the time-averaged line profile can be affected. Determining the precise way in which the response functions and profiles are affected requires a detailed and highly model-dependent analysis. Before going through such an analysis, let us first discuss some of the cloud line emission model properties which apparently meet the above two conditions.

Table F.1 lists some of the processes responsible for reactive cloud properties in several of the models that have been proposed and the equilibrium time scales associated with them. Also shown is whether the slowness of equilibrium affects the response functions, line profiles, or line ratios. The first entry is for a two-phase pressure-equilibrium model (e.g., Wolfe 1974; Krolik, McKee, & Tarter 1981). Assuming in this case that the cloud pressure is regulated by pressure of the intercloud medium, the delay in the cloud pressure response to the continuum is limited by the intercloud temperature equilibrium time scale. For the model parameters described in Table F.1 , this is (only) days. If the dependence of the intercloud temperature upon the local continuum flux is strong enough, the responding pressure will affect the response functions for the parameters assumed in Table F.1 in a highly line-dependent fashion, giving the line ratios a complicated time dependence. Furthermore, if the cloud identities are preserved (as in Rees, Netzer, & Ferland 1989), the slowness of the cloud area and column density reactions will also affect the response functions respectively in a line-independent and weakly line-dependent fashion. The time-averaged line profiles for this model are not affected by the finite pressure equilibrium time, which is too small compared to the cloud crossing time years for the parameters shown in Table F.1 ) to be affected. However, if the intercloud temperature dependence is moderately strong, this model, like several others that are not immune to the various processes analyzed in Table F.1 , requires use of a new formalism. Such a formalism will be developed in § F .3.

Note that the physical processes considered in Table F.1 were drawn from the set of processes invoked by the various cloud models that have been proposed. In principle, all of these could be incorrect. Therefore, Table F.1 is necessarily incomplete. For this reason, the analysis of time-dependent cloud response could be important even if all of the processes in Table F.1 somehow accommodated the fast cloud assumption.
Table F.1: Equilibrium Processes of Some Reactive Cloud Model Parameters and Their Effects
NOTE—“RF," “LP," and “LR" are respective abbreviations for “response functions," “line profiles," and “line ratios." These results are for clouds at a fiducial radius from the continuum source of 10 light-days (the light crossing time scale), a fiducial local continuum flux of ergs cm, a fiducial velocity of 4000 km s, a fiducial cloud hydrogen density of cm, and a fiducial mean column density of cm.

Only for models with clouds in pressure equilibrium with a hot inter-cloud medium (e.g., Krolik, McKee, & Tarter 1981). Calculation assumes an inter-cloud temperature of K, which implies that the dominant source of cooling is thermal Bremsstrahlung, which in turn implies flux-dependent (reactive) cloud parameters.

Adopted from results in Krinsky and Puetter (1992), but after scaling to the column density assumed here. Line ratios are only strongly affected for pressure-stratified clouds.

Only for pressure-stratified cloud models, see Taylor (1994).

Only for lines emitted uniformly from the inverse-Strömgren region.

Adopted from parameters assumed in Schaaf & Schmutzler (1992).

Adapted from Harpaz & Rappaport (1991) and Antona & Ergma (1993).

As in Rees (1987), but assuming the field responds to the continuum flux on the Alvén wave cloud crossing time.
_



F.3 Theory of Response


In this section we shall extend the formalism in Blandford & McKee (1982) so that information can be obtained from variability data about models which violate two fundamental assumptions made in Blandford & McKee (1982): (1) instantaneous response and (2) linear response.



F.3.1 Locally Delayed Response


Before one can understand the overall, global response of systems that can violate the fast cloud assumption, one must first understand the response of the individual clouds that make up such systems. In this section a general method of determining the time dependence of an arbitrary cloud property is derived. In § F .3.2, this method will be used to obtain the global response of systems that can violate the fast cloud assumption.

The character of the response of an AGN cloud depends critically upon the relative magnitude of two time scales. One of these is the variation time scale of some condition externally imposed upon the cloud, such as the local continuum flux or intercloud pressure. Another is the characteristic equilibrium time scale of a physical property of the cloud, such as its temperature or size, in response to the variations of the external conditions. As an example, let us consider the case of a cloud with a physical property that is an increasing function of the local continuum flux. Let us also assume, for this example, that the continuum source, itself, is time-independent, but that the cloud is in a periodic orbit about the black hole. This situation is a simple variation on those in which the continuum does vary. If we assume that the orbital period is significantly greater than the equilibrium time scale of the property, the physical property would lag the time-averaged continuum flux to which the cloud is exposed as it orbits the black hole. The line emission in such a cloud would depend not only on its position, but also on another variable which indicates its orbital phase. Under certain conditions (see Appendix G ), it can be shown that this variable can be the cloud velocity vector. For instance, if the line emissivity of a cloud is an increasing function of just the physical property, then the line emission from the cloud would be greatest not when it is closest to the black hole, but slightly farther away, after the cloud has acquired a small outward velocity. As the orbital time becomes even larger, we approach the “fast cloud" regime. In this regime we can assume that the physical property in the cloud reacts fast enough that it is purely a function of the local flux or, in this case, of the distance from the black hole such that the phase lag is zero.

A second case to consider is one wherein the variation time scale of the local continuum flux of a cloud is significantly smaller than the equilibrium time of the property. Here the physical property of the gas would lag the orbital motion by a significant phase. This implies that a line with a strong enough dependence upon the lagged property could attain maximum flux when the cloud has a relatively high outward radial velocity. In this case, as the variation time scale of the input continuum flux becomes even smaller, we approach the “slow cloud" regime, where we can simply assume that the relevant physical property is a constant throughout the orbit.

The third possible case to consider is one in which the local continuum variation time scale is intermediate and similar to the equilibrium time scale. Understanding this case requires a more quantitative approach than the other two cases. Let us call the generic cloud property of interest where is the time measured in the reference frame of the observer. The analysis which follows is quite general and could represent properties such as the mean cloud area, pressure, or column density. Similarly, let be a generic input, such as the local continuum flux near the cloud, of which is assumed to be a function. Let be the asymptotic functional dependence of upon once sufficient time has elapsed for equilibrium to be established, where the prime denotes the functional dependence in the fast cloud regime. Furthermore, let us assume that there exists a characteristic time scale for to respond to changes in x. Such a characteristic time will be equal to the ratio of the extent to which is out of equilibrium to the rate at which the non-instantaneously responding component of actually attains its equilibrium value. This gives
(F.1)
where is a free parameter that is the instantaneous component of the “gain" of with respect to , and is the average or “bias" of , etc. The gain itself is an operator (defined by eq. [ H.3 ]) that yields the dimensionless ratio of the amplitudes of small variations of an output about its mean with respect to that of some input. It is merely the Fourier transform of the linearized response function (see Appendix B). The implicit assumption here that is approximately constant could be invalid under the following conditions: the variations of are large enough, the initial conditions are far enough from equilibrium, or the equilibrium time has an explicit dependence upon the sign of . In these cases the physics associated with the response time is not properly described by only one parameter. Otherwise, equation ( F.1 ) completely characterizes the system given the prior inputs and the other system characteristics , and .

Though we will assume that is a nonlinear function, in certain instances we shall find it highly instructive to consider the case in which the variations in are small enough that is accurately described by a first-order Taylor expansion. Performing such linearization of equation ( F.1 ) (with eqs. [ H.1 ]-[ H.4 ]) yields in the frequency domain
(F.2)
where and is the “asymptotic gain" of with respect to see also eq. [ H.2 ]). Equation ( F.2 ) can be used to formulate a more precise definition of the fast and slow cloud regimes, which respectively occur for and , where the transfer function becomes a trivial function of flat in coordinates).

Equation ( F.2 ) yields in the time domain
(F.3)
where is the step function. This response function tells us (namely via eq. [ H.6 ]) the contribution in the linear regime toward the output (e.g., the cloud area) made by an input (e.g., the local continuum flux [measured in the reference frame of the cloud]) at a prior time. The first term in equation ( F.3 ) is the component of that mirrors the variations in without delay, while the second term is the component of that responds on the time scale .

For the important case in which the instantaneous component of the gain is zero, equations ( F.2 )-( F.3 ) yield the results indicated earlier in this section: in the fast cloud regime they yield an output that mirrors the input variations, while in the slow cloud regime they yield an output that is constant.



F.3.2 The Line Profile


Now that we have prescribed a general way of accounting for individual cloud properties that exhibit hysteresis-like behavior, we can derive the more observable properties of AGN models which have finite (rather than zero) equilibrium times. Of particular interest here is the angle-dependent apparent luminosity emitted in line of a cloud with position vector from the black hole and velocity vector . An expression for this that is general enough for the models that will be analyzed in this appendix and which takes local delays into account is
(F.4)
where is the ionizing continuum flux at , A is the cloud area, is the dimensionless emission efficiency for line is the first-moment correction to the efficiency for an anisotropically emitting cloud, is the position vector of the observer, and is the cloud luminosity in line due to resonance scattering. Each of the cloud parameters in equation ( F.4 ) that has an equilibrium time near or greater than hereafter, the “spatial time" scale) must be evaluated using the appropriate form of equations ( F.3 ) and ( H.6 ). Before a model conforming to equation ( F.4 ) can have predictive power, not only must the continuum light curve be measured, but also estimates of the time scales and the asymptotic functional dependence of each cloud parameter upon the local continuum flux must be made.

Once a specific expression for the observed line flux from an individual cloud is assumed, the macroscopic characteristics of the global system composed of several clouds are easy to calculate. Neglecting absorption, the flux per cloud observable at is . With this terminology, the time-dependent line profile becomes (see Appendix G )
(F.5)
where is the equivalent tangential velocity and is the distribution function. Under the conditions specified earlier, the above equation permits computation of the line profile for any class of AGN cloud line emission models which can be described by equation ( F.4 ).

Though equation ( F.5 ) provides a means of computing the nonlinear line profile response when the input history is known, applying it can be computationally expensive (though not as much as the trajectory-dependent sum method considered in Appendix G ). This is because it requires modeling the cloud properties such as the function, which from a numerical perspective is an array with dimensions and that evolves with time, though probably only weakly in the dimensions. If local delays are important, evaluation of at each point in time requires integrating over history according to the appropriate forms of equation ( H.6 ). Since the positional integral in the above equation can be interpreted as an integral over history, the expression for the line profile is a double integration over lag. This is in contrast to the analogous expression for the line profile given by equation (2.12) of Blandford & McKee (1982), which involves only a single integration over lag.



F.3.3 The Line Transfer Function and Linear Approximation of the Line Profile
By linearizing equation ( F.4 ) (see Appendix B), one of the integrations in lag in equation ( F.5 ) can be eliminated, and the computer time required to obtain the time-dependent line profile of a model can be significantly reduced. For several key cases, we find that these benefits outweigh the inaccuracy of linear models.

The first step in linearizing the line flux emitted from an individual cloud is to obtain its gain about the bias continuum flux. This in turn requires determining the transfer function (eq. [ F.2 ]) of each flux-dependent cloud parameter affecting the line emission in equation ( F.4 ). Because the gain is calculated by considering small perturbations about the mean of the input, the gain of each of these parameters can be computed using the time-averaged local continuum flux, which is dependent only upon position. In terms of the gains of these cloud parameters, equation ( F.4 ) yields for an individual cloud
(F.6)
where we will implicitly assume that the continuum flux is evaluated locally (at . Each of the above terms is proportional to the gain of one of the reactive cloud parameters. The third term itself is the sum of three highly model-dependent terms if sufficiently parameterizes the cloud emission for a given model. Note that even if the various cloud properties such as the area are described without approximation by a nontrivial linear response function, the output line response in a cloud is nonlinear nonetheless. This is because the above equation provides only an approximation to the response valid for small perturbations about a mean. Such nonlinearity is a general property of reactive cloud models.

With each cloud response linearized, the remaining time dependence in the system line flux equation is due purely from the factor, so the global transfer function of the profile is

(F.7)
The line flux gain of an individual cloud appearing in this equation is equal to the gain of the individual line luminosity (eq. [ F.6 ]) if, for the time being, we neglect absorption. Note that unlike the Fourier transform of the expression given for the response function by Blandford & McKee (1982), the above equation has a factor of the cloud gain that can be nonzero at nonzero frequency. In the time domain, equation ( F.7 ) gives (via eq. [ H.6 ]) the linear approximation to the line profile flux,
(F.8)
where represents the input-dependent error. Unlike equation ( F.5 ), this equation does not have an implicit nested integration in lag. Applications of it to the models considered in this appendix that account for the dependence of upon the direction of the velocity vector are provided elsewhere (Taylor 1994).

It is worth repeating that a condition for the linearized response function to be descriptive is that at a given radius the continuum variations are small enough that the second order derivatives can be neglected. Since large-scale variations in the continuum luminosity are known to occur, the system would be somewhat contrived to consistently obey this condition. For instance, in the slow cloud regime the linearized form of equation ( F.6 ) will generally be inaccurate at low enough mean cloud ionization parameters when the emitting ion is in partial fractional abundance. In this case, the second derivative of with respect to would not only be large and positive for most lines, but would also be a sensitive function of input level. If the asymptotic mean cloud ionization parameter is a decreasing function of the local flux § F .1]), overestimates for the size of the line emission region when using fully linear models are implied. Even for wind cloud models (e.g., Kazanas 1989) in the fast cloud regime with where the effective ionization parameter can be taken to be constant, nonlinearities would still arise from the dependence of the cloud area upon flux for which a constant term cannot account. In any of these types of situations, the Fourier transform of the oscillatory component of the line flux would not be proportional to that of the continuum, and forms of equation ( F.8 ) would not accurately describe the variability that would be observed.

In such cases, the “optimal" response function that best fits real data becomes a function of both the specific data set as well as the fitting criterion (see also eqs. [ H.12 ]-[ H.13 ]). Therefore, its utility in measuring any of the epoch-independent features of AGN and AGN models is somewhat questionable. This is in contrast with the linearized response function (eq. [ F.7 ]), which is dependent upon only the mean of the continuum flux. Ideally, a fitting criteria would exist that would reliably yield this input-independent, poorly fitting linearized response function rather than the optimal one. However, there are alternative parameters and parameterized functions for analyses of variability data that completely bypass this problem. One alternative is the cross-correlation function. However, even in the linear regime this is also a strong function of the excitation characteristics. (See also § F .4.) A more promising alternative is to fit nonlinear models (e.g., eq. [ H.8 ]; Taylor & Kazanas 1992) to data. Using nonlinear models offers the potential of epoch-independent fitting or measurement within the context of a model of physical AGN properties even when the continuum variations are large or the line emission is a sensitive function of the flux. _



F.4 Examples


Let us consider cloud models in which the effective reprocessing efficiencies are increasing functions of the local continuum flux, with the equilibrium time scale of the relevant physical properties being slightly larger than the characteristic light crossing time. From § F .3, we know that responses to short and weak pulses of continuum radiation in such a system could be modeled satisfactorily with a linear “spatial" response function, which is the response function of the system were the fast cloud regime applicable. This response function has structure on just the light crossing times of the emission region, as the efficiency and physical conditions of the clouds in such a system would deviate only slightly from their mean values. Similarly, responses to long pulses of fixed intensity probably also could be mimicked with a different linear response function that had additional structure at lags beyond the cloud equilibrium times. However, if either short pulses and long pulses of constant intensity or long pulses of varying intensity occurred in such a system, a single linear response function would not be able to fit all aspects of the variability. A linear system would respond either too strongly to the weak pulses or too weakly to the strong pulses and furthermore would respond either too slowly to the weak pulses or too rapidly to the energetic pulses. The first two types of nonlinear behavior are due to an input-dependent asymptotic gain, while the latter two are due to nested lags or “inseparability" of the cloud and spatial response functions for systems in which the input is a multiplicative factor in the expression of the output.


Figure F.1: _Comparison of the linear approximation to the actual responses of simple shell-like systems with local delays. Line a is the input continuum that was assumed, which has a luminosity of 0.5 in arbitrary units, the “low state," followed by a luminosity of 1.25 units, the “high state," which lasts for 200 days. Superimposed upon the low and high states are delta-function-like spikes of area 10 unit-days. The solid lines are the output line luminosities for the models described in the text (§ F .4) offset respectively by -1, -1.75, -3, and -5 luminosity units while the dotted lines are approximations of the outputs obtained from linearized response functions. Though the linearized responses do a reasonable job of matching the actual responses for most of the models shown here, they fail to exhibit the differences between weak and strong (time-integrated) excitation. This is particularly evident for the model shown in solid line e.

These effects can be seen more clearly by considering a simple shell-like system in which the light crossing time is slightly shorter than the cloud equilibrium time. Specifically, let days, the asymptotic cloud area function be days, = and the normalization of determined by the condition that the mean covering factor be unity, i.e., . The ratio of the area equilibrium time to the light crossing time of this system is 3, which is near enough to 1 for neither the fast nor the slow cloud regime (§ F .3) to be applicable. Exact outputs obtained upon application of equations ( F.4 )-( F.5 ) for various values of of this system are displayed as solid lines , and of Figure F.1 , while the input continuum that was assumed is shown as the solid line a. This input is a “low state" followed by “high state" that lasts for 200 days. Superimposed upon the low and high states are delta-function-like spikes of area 10 luminosity-unit-days, the responses of which can give a crude indication of a spatial response function of the system. The outputs from the linearized response functions are shown as dotted lines. Ideally the optimal linearized response functions would have been obtained from a fitting scheme that minimized the discrepancy between the exact outputs. However, in this work they were obtained simply from equation ( H.4 ), equations ( F.6 )-( F.8 ), and finally equation ( H.13 ), which was derived for sinusoidal-like inputs but which results in surprisingly good fits for the input here as well.

For the nonreactive case shown in solid line , the gain of the output line luminosity is unity, and the linearized response function of the system is just the spatial response function, which is a step function. Whether in the high or low state, here the amplitude of the response on time scales larger than the spatial time is the same as that of the input. The model is shown in solid line d. For this reactive model the cloud area responds linearly to the local continuum flux. This is shown explicitly in line c, which is the time-dependent area of the clouds on the shell after the spatial delay was removed by artificially setting . Care must be taken in the interpretation of this response, as the actual size of this system is infinitely smaller than the size that the formalism of Blandford & McKee (1982) would yield, which is light-days where It is important to understand that although the cloud areas respond linearly in the model, the output itself is over-responsive compared to linear, with an asymptotic gain of 2 (eq. [ F.6 ]). This asymptotic gain is also approximately the correction factor by which the formalism of Blandford & McKee (1982) could overestimate the cloud number density. (The exact factor is dependent upon the input, as eq. [ H.13 ] indicates, as well as the fitting criteria.) However, by taking into account an asymptotic gain correction factor that is different from unity, the linear response (dotted line does a surprisingly good job of fitting the actual response (solid line of the system, especially given that the input continuum luminosity function varies by a factor 7.5.

Nonlinear response is more apparent in the “under-responsive" model shown in solid line e. Here the cloud area response is given by a nonlinear input-modified system (eq. [ H.8 ]). A key difference between the models shown in solid lines and is that the gain of an individual cloud area is a decreasing function of positive frequency for the over-responsive model, but is an increasing function for the under-responsive model. Note that because the areas of the spikes are small, the actual responses to the first spike are similar in both cases. The spatial response function (solid line does a crude job of describing both models. However, low frequency or high (time-integrated) energy excitation exposes the latent nonlinearities of these systems. For instance, the over-responsive system (solid line responds slightly higher to the spike in the high state than the spike in the low state, while the linearized response function predicts a response that was the same strength for both spikes. This aspect of behavior is due to a nonlinear asymptotic gain. Even if it were accounted for, the shapes given by the linearized response function would still not perfectly match the exact ones. For instance, the linearized output for the over-responsive system responds too rapidly to the beginning of the high state. If its linearized response function were adjusted to yield a slower response, the linearized output would then respond too slowly to the beginning of the spike in the low state. Ultimately this is due to the area factor and hence the area equilibrium time playing a less important role in the response to the first spike of short duration (when the area is relatively constant) than in response to the energetic high state of long duration (when the area increases significantly). These types of problems are particularly evident for the highly nonlinear under-responsive case. Because the amplitude of the asymptotic gain of the area is only 1, the response to both low energy spikes is square-like. However, the high state is energetic and long enough to permit the areas to respond and the asymptotic gain of zero nearly to be attained, which results in triangle-like responses. The linearized response function incorrectly gives triangle-like features in the responses to the spikes.

Figure F.2: _Effect of the pressure equilibrium time upon the “line-specific" response. The top line (left axis) is the input continuum luminosity assumed. The exact Lysolid lines) and C IV (dotted lines) output luminosities (left axis) are also shown for three extremely simple models similar to those shown in Fig. F.1 . To emphasize the effect of just the cloud pressures and pressure ionization parameters being locally delayed, the cloud areas were artificially forced to yield a constant geometrical covering factor of unity (neglecting absorption) and the cloud column densities were artificially forced constant at cm. The initial column density to pressure ionization parameter ratios assumed were cm. The line luminosities for models and c are offset respectively by and ergs s. The pressure-equilibrium times assumed in models , and c were respectively 1.1, 500, and 30 days. The photoionization code that was used is XSTAR (see, e.g., Kallman 1995). The spectrum that was assumed is shown in Fig. F.3 .

Figure F.3: _The spectrum that was assumed for the models shown in Fig. F.2 . It is identical to that used in Krolik et al. (1991). Note that the -axis is plotted in linear (as opposed to logarithmic) coordinates.

For the models shown in Figure F.1 , all the cloud property gain terms in equation ( F.6 ) except that of the area are zero. However, for calculations of relative line strengths, the ionization parameter gain term in equation ( F.6 ) frequently determines the key distinguishing response characteristics. This is illustrated by the models shown in Figure F.2 , where the response as a function of the pressure equilibrium time scale is shown for three models nearly identical to those used in Figure F.1 . However, for these cases, , the cloud areas and column densities are forced to be constant, and the spike strengths are reduced. For model a, the pressure equilibrium time is 1.1 days, which is approximately the mean inverse Strömgren sound crossing time for these clouds. This is short enough (compared to the spatial time of 10 days) for the fast cloud regime to be valid. Thus, in this case, a single (rather than double) integration in lag would have sufficed for calculating . For model , the pressure equilibrium time is 500 days, which is closer to the thermal evaporation time scale (Table F.1 ) of days. In the limit in which the pressure equilibrium time scale becomes infinite, model is identical to a nonreactive model, which also would not require the computationally expensive double integration. Finally, for model c the equilibrium time scale is 30 days. This intermediate equilibrium time might be applicable for cloud models in which the evaporation rate is a strong function of the mean ionization state (see also Taylor 1994). Because the column density to pressure ionization parameter ratio cm was selected to place the model near the “C IV—limited" state in which the C IV gain is the C IV line responds very weakly to the initial spike. However, it responds strongly on the pressure equilibrium time scale to the beginning of the extended high state, during which time the pressure regulation mechanism readjusts the ionization parameter partially back toward its lower initial equilibrium value. In contrast, at the ending of the high state, the C IV line drops on the shorter spatial time scale because of the higher pressures (and lower ionization parameters) of the clouds. This example illustrates one of several ways in which the mean response time of a system can be dependent upon the recent mean continuum luminosity. _

The above examples clearly illustrate how the global linearized response function can have structure not due to the spatial response function. However, nonlinear effects can even mask the spatial information. For example, if the ratio of the local continuum flux to the product of the mean pressure and column density of the clouds becomes high enough, the emission of certain lines could be “recombination-limited," which makes in equation ( F.6 ). This occurs during the high state with Ly for model solid line Fig. F.2 ). Therefore, as the model shown in the solid line c of Figure F.1 also illustrates, the response times give, under the assumption of spherical symmetry, only upper limits to the characteristic size.

The above examples also help illustrate how local delays can affect the cross-correlation function. Consider a simple model, such as that shown in solid line of Figure F.1 . For a weak enough high frequency square wave input occurring above a steady input background, the output would be a capped triangle wave, which is symmetric about its peak with respect to lag. The cross-correlation function that one would obtain from such data would be shifted of approximately a day, yet also be symmetric about the characteristic lag. This is because the data from the beginning, or growing phase of the pulse, which alone would produce a cross-correlation function with a positive slope, is compensated by the ending or falling phase of the response pulse.

Consider, however, the case wherein the input is a moderate-intensity, low frequency square wave. Because of local delays, the response would be higher near the end of the pulse, as in solid line of Figure F.1 a. Since the cross-correlation function is an amplitude-biased function, the resulting cross-correlation function would be biased from the data in the falling phase. Unlike the previous case, this would result in a negatively sloped component to the cross-correlation function, which is quite common (e.g., Sparke 1993).

Note that local delays only give the cross correlation function asymmetry about their peak above what one would obtain from the global linearized response function alone. This is because the cross-correlation function is the convolution of the response function and the symmetric, input auto-correlation function. For this reason, assessing the importance of asymmetric cross-correlation functions in determining the characteristics of the local delays of interest here would probably require knowledge of the best-fitting linear response function.

Note that this function would be biased somewhat by the low energy (more symmetric) pulses. Therefore, the asymmetry of the “simulated cross-correlation function” that one would obtain from the output of this response function would not be as great as the one obtained from actual variability data. Thus, the difference between the simulated and actual cross-correlation functions can be asymmetric if local delays are important. This permits a simple way of testing for local delays.



F.5 Summary and Conclusions


In this appendix it has been argued that for some reactive cloud models the fast cloud assumption is invalid. In such cases a tight correspondence between the positional distribution of matter and the global linearized line flux response function simply does not exist. This is because the continuum luminosity at a given time in history affects not only the clouds on a spherical shell, but also the clouds inside such a shell. In general, the response which results is not only nonlinear, but also inseparable, requiring more than one integration over lag to determine the output flux at a specified time. However, in some cases the time-dependence of the line fluxes can be described using linearized response functions . These response functions have structure at intermediate lags due to the finite size of the line-emitting region and at lags greater than these due to the finite equilibrium times of the line-emitting material itself. For small enough perturbations the physical cloud properties can be relatively static and a linear response function can work quite well at describing the responses. However, when the input continuum variations become extreme enough, such response functions can fail. Because of nonlinear asymptotic response, the integral of a linearized response function of an observable differs from the time average of the observable by a correction factor of the asymptotic gain. Ignoring nonlinear effects can lead to incorrect measurements of the physical properties of the system, such as sizes that are too large.

One of the most fundamental assumptions that has been made in previous analysis of variability data has been the fast cloud assumption. With this assumption now in question, this data should be examined again with models that do not require it.