Appendix G
The Line Flux of a Cloud When the Equilibrium and Crossing Times are Comparable
In standard references such as Blandford & McKee (1982), the time-dependent line profile is presented as an integral over the cloud phase space of distribution function and emission flux. The line emission from these clouds is implicitly assumed to be independent of the direction of their velocity vectors. As a result, in order for these models to generate the line shifts and asymmetries which are observed, the clouds must generally flow outwards or inwards. Models with such radial flow can, however, have serious problems, such as very low accretion efficiencies (see, e.g., Kallman et al. 1993).

Before one invests substantial effort to concoct models which address these problems, it may be worthwhile to -analyze the original, highly intuitive conclusion that line emission from clouds is independent of the direction of their velocity vectors. In fact, a careful examination reveals that, because of our extreme ignorance of what the clouds in AGNs are, there are several theoretically permissible reasons that clouds could (at least in principle) have line emission dependent upon their velocity direction. Each of these reasons is the result of an uncertainty in a corresponding assumption made either explicitly or implicitly in Blandford & McKee (1982) and similar works. By examining each of these assumptions in detail, we clearly outline the boundary of parameter space in which the velocity-direction-independent emission result must lie. Moreover, by looking just outside these boundaries we could obtain a model which does not suffer as severely from the types of problems plaguing current AGN models.

In this appendix we will focus on just one of the potentially invalid assumptions made in previous works, i.e. that the cloud equilibrium times are negligible compared to the cloud crossing times. As shown in Table F.1 , this assumption is invalid for some models. For these models, the cloud properties have an additional explicit dependence upon the prior continuum fluxes and hence the orbital trajectories of the clouds, which is a different function for each cloud. This would complicate modeling efforts, which would entail integrating over the orbital trajectories of the clouds. It could also make the much simpler approach taken by Blandford in McKee (1982) invalid. However, we shall find that in certain cases the cloud luminosity function can be described merely by giving the properties of the cloud an additional velocity-direction dependence.

But let us first obtain the exact solution to this problem. Ignoring absorption, the (nonlinear) line flux as a function of time for a cloud is given by application of equation ( F.3 ) to each of the cloud parameters in equation ( F.4 ). Thus, the continuum-subtracted, time-dependent line profile of the “global" system observed from is
(G.1)
where is the number of clouds and is the flux in line from cloud i computed from the forms of equations ( F.3 )-( F.4 ) appropriate for the model to be tested. Neglecting absorption and non-Doppler line broadening, equation ( G.1 ) gives the exact time-dependent line profile for clouds with arbitrary motions. However, because the suspected number of clouds in AGN is high (e.g., Laor et al. , e.g., Peterson 1994), using it could prove computationally expensive.

Noting the exponential factor in equation ( F.3 ), integration over the history of local continuum exposure required for determining the cloud properties at a given time need only be carried out to a small factor (e.g., of the relevant equilibrium time. Therefore, if each cloud property relevant to emission has an equilibrium time scale that is appreciably less than the emission region crossing time, the radius will not change drastically over the relevant history interval, and the continuum flux function can be approximated by its Taylor expansion. This yields a first order correction to the continuum flux function that is proportional to the radial velocity. Similar expansions permit estimation of the line flux observed from an individual cloud and the line shifts that equation ( G.1 ) implies.

However, if is independent of the continuum flux and is large enough that line broadening produces a smooth line profile, a more accurate method for obtaining the individual cloud line flux that partially accounts for higher order terms can be obtained simply by taking the statistical average of the function, which is

(G.2)
The dependence of upon time is due to the variation of the continuum flux to which the cloud is locally exposed. The dependence upon position is due to traditional model elements such as changes in the mean cloud density as a function of average heating. Finally, the dependence upon the velocity vector accounts for the intrinsic dependence as well as that due to the history of heating being important when the equilibrium time scale is not completely negligible compared to the emission region crossing time. Note that for models where the position and velocity variables impose the integrals of motion of a trajectory of a cloud, the above condition that the equilibrium times are small compared to the crossing times is unnecessary and the flux is an exact function of only the time, position, and velocity variables.

In this case, an analog of equation ( G.2 ) can be used to replace the knowledge of the individual cloud trajectories with the time-independent phase space distribution function . Though this function, when combined with equation ( G.2 ), permits equation ( F.5 ) to be used to obtain an approximation of the time-dependent line profile, it offers little advantage over using just equation ( G.1 ) because it still entails explicit time-dependent orbital modeling.

However, in the linear regime (eqs. [ F.7 ]-[ F.8 ]), only the time-average of equation ( G.2 ) is required to obtain the time-dependent line profile. Once this (velocity-dependence) has been computed for a model, the linear approximations to the observable characteristics can be obtained from equation ( F.8 ) for various continuum light curves without explicit time-dependent orbital modeling. Therefore, in the linear regime, the orbital history of the cloud line emission flux can be approximated as a simple function of the direction and magnitude of the cloud velocity vector .