We refer to models for which the pressure P depends on the radial distance r from the central continuum source, 
(2)
\begin{eqnarray*}
P(r)\propto r^{-s} \, , 
\end{eqnarray*}
as pressure-law models. In this work, we examine two limiting cases, s = 0 and s = 2, representing constant density and constant ionization parameter models, respectively. We here adopt a spherically symmetric BLR geometry spanning more than two decades in radial extent. This model is chosen for (i) its simplicity, and because (ii) we can compare our radial pressure-law models with the Local Optimally emitting Cloud model for this source presented by Korista & Goad (2001), which also adopts spherical symmetry. Here, we summarize the radial dependencies of various physical quantities for spherically symmetric pressure-law models; the derivations in this section follow Rees et al. (1989) and Goad et al. (1993). We make the simplifying assumption that the cloud temperature does not vary with radius; for solar composition, photoionization equilibrium is achieved at temperatures T ∼ 104 K across a wide range of ionization parameter (equation 5), and the gas temperature will therefore vary weakly with radius (e.g. Netzer 1990; Ilić, Kovačević & Popović 2009). For constant cloud temperatures, the cloud hydrogen gas density nH is proportional to the pressure, P, and so 
(3)
\begin{eqnarray*}
n_\mathrm{H}(r)\propto r^{-s} \, . 
\end{eqnarray*}
Thus, s = 0 corresponds to a constant nH throughout the BLR. The ionization parameter U is defined as 
(4)
\begin{eqnarray*}
U(r)=\frac{Q_\mathrm{H}}{n_\mathrm{H}(r)4\pi r^2c} \, , 
\end{eqnarray*}
where QH is the number of hydrogen-ionizing photons emitted by the central continuum source per second. Thus, s = 2 corresponds to a constant ionization parameter model, since: 
(5)
\begin{eqnarray*}
U(r)\propto r^{s-2} \, . 
\end{eqnarray*}
The surface area per cloud, Ac, is proportional to $R_{\rm c}^2$, where Rc denotes the radius of a cloud. In general, Rc depends on the pressure P, and is therefore constant for s = 0. If we demand that the mass of each cloud is conserved as the clouds move radially outwards (i.e. clouds do not break up or coalesce within our region of interest), mass conservation implies that $R_c^3n_H={\rm constant}$. Thus, we obtain the relation 
(6)
\begin{eqnarray*}
A_{\rm c}(r) \propto R_{\rm c}^2(r) \propto r^{2s/3} \, . 
\end{eqnarray*}
The column density of each cloud, Ncol, depends on the gas density and cloud radius: 
(7)
\begin{eqnarray*}
N_\mathrm{col}(r)\propto R_cn_\mathrm{H}\propto r^{-2s/3} \, . 
\end{eqnarray*}
The above relations determine the local physical conditions, as parametrized by Ncol, nH and incident ionizing photon flux ΦH, at any radius in a spherically symmetric pressure-law BLR model. These conditions determine the local surface emissivity ε(r) of a BLR cloud at radius r (as determined via cloudy modeling; Section 3.1). The total luminosity for an emission line is then found by integrating over the distribution in cloud properties such that 
(8)
\begin{eqnarray*}
L_{\mathrm{line}}=4\pi \int _{r_{\mathrm{in}}}^{r_{\mathrm{out}}}\epsilon (r)A_{\rm c}(r)n_{\rm c}(r)r^2{\rm d}r \, , 
\end{eqnarray*}
where rin  and rout  are the inner and outer BLR radii, respectively, Ac is the surface area of a single cloud, and nc is the local number density of clouds. As dust grains strongly absorb UV photons, routis chosen to approximately coincide with the distance at which dust grains can form and survive (≈140 light-days for NGC 5548).
Citation: Ilić, Kovačević & Popović 2009
Citation function: Uses