In our model, we used a non-hydrodynamical approach based on assumptions of the motion of separate clouds under gravity and the action of radiation pressure acting on dust. This has considerable limitations but they are justified as the first approximation for modeling the LIL part of the BLR. As discussed in the classical paper of line-driven wind model (for HIL part of BLR) by Murray et al. (1995), the optical depth of the emitting region must be moderate (column density of order of 1023 cm−2) and the local density is high (for LIL part it is higher than that of HIL, many authors argue for a local density about 1012 cm−3, for instance, Adhikari et al. 2016; Baskin & Laor 2018; Panda et al. 2018), while the BLR is extended. There are two possibilities to support a consistent picture: it is either to assume a very narrow stream of material flowing out, with the cross-section on the order of 1012 cm, as in Murray et al. (1995), where they assume lower density so the size is actually larger 1014 cm; or to assume considerable clumpiness of the medium. We followed the second approach since there are natural thermal instabilities in the plasma, such as instability caused by X-ray irradiation (Krolik et al. 1981). In this case, the plasma spontaneously forms colder clumps (at a temperature of ∼104 K, cooled through atomic processes) embedded in a hotter medium (at temperature ∼107 K, set at an inverse Compton temperature value). The presence of highly or fully ionized medium is aptly supported by the observations as well as by the theory. Then the two media of density contrast of order of 103 provide the rough pressure balance. Blandford et al. (1990) discussed the typical values for ionization parameter in AGN clouds covering the range from 10−3 to 1, that is, up to four orders of magnitudes. The precise description of the structure of the clumpy medium is very difficult. Even in the case of a single cloud exposed to irradiation, in plane-parallel approximation requires a radiative transfer to be performed, which would then show the gradual change in the density, temperature, and ionization parameters (see e.g., Baskin & Laor 2018; Adhikari et al. 2018), with the low density first and the temperature roughly at inverse Compton temperature (depending on the shape of the incident spectrum), and then a relatively rapid decrease at the subsequent ionization fronts. A proper description of this transition, calculated under constant pressure, actually requires inclusion of the electron conduction (e.g., Begelman & McKee 1990; Różańska & Czerny 2000). Deep within the cloud, there is a further drop in temperature and a rise in density due to a decrease in the local flux as a result of absorption. As emphasized by Baskin & Laor (2018), radiation pressure also plays a dynamical role in this process. The picture is further complicated if plane-parallel approximation is abandoned. The presence of the numerous clouds of complex shapes can be fully consistent with simple estimates of the cloud number based on line shape properties as done by Arav et al. (1998). Of course, there are also certain processes that can lead to cloud destruction, such as the action of tidal forces (Müller et al. 2022), Kelvin–Helmholtz instabilities, and cloud ablation), however, the destruction rate can be strongly affected by the magnetic field (e.g., McCourt et al. 2015). The relative importance between the condensation rate and destruction rate depends on the cloud size, as it is set roughly by the field length (Field 1965). Cloud formation in AGN has been seen in the numerical simulations from Waters et al. (2021), but at distances much greater than the BLR distance which was likely related to the numerical setup and the requested spatial resolution of the computations. The issue is thus extremely complex, and simple order-of-magnitude estimates based on a single density and single temperature of the cloud and intercloud medium are not fully adequate and cannot reproduce the full ionization parameter range. However, addressing this point in detail is beyond the scope of the current paper.
Citation: Krolik et al. 1981
Citation function: Uses