Locally Optimally Emitting Clouds and the Variable Broad Emission Line Spectrum of NGC 5548

In recent work Baldwin et al. proposed that in the geometrically extended broad-line regions (BLRs) of quasars and active galactic nuclei, a range in line-emitting gas properties (e.g., density, column density) might exist at each radius and showed that under these conditions the broad emission line spectra of these objects may be dominated by selection effects introduced by the atomic physics and general radiative transfer within the large pool of line-emitting entities. In this picture, the light we see originates in a vast amalgam of emitters but is dominated by those emitters best able to reprocess the incident continuum into a particular emission line. We test this "locally optimally emitting clouds" (LOC) model against the extensive spectroscopic database of the Seyfert 1 galaxy NGC 5548. The time-averaged, integrated-light UV broad emission line spectrum from the 1993 Hubble Space Telescope (HST) monitoring campaign is reproduced via the optimization of three global geometric parameters: the outer radius, the index controlling the radial cloud covering fraction of the continuum source, and the integrated cloud covering fraction. We make an ad hoc selection from the range of successful models, and for a simple spherical BLR geometry we simulate the emission-line light curves for the 1989 IUE and 1993 HST campaigns, using the respective observed UV continuum light curves as drivers. We find good agreement between the predicted and observed light curves and lags—a demonstration of the LOC picture's viability as a means to understanding the BLR environment. Finally, we discuss the next step in developing the LOC picture, which involves the marriage of echo-mapping techniques with spectral simulation grids such as those presented here, using the constraints provided by a high-quality, temporally well-sampled spectroscopic data set.


INTRODUCTION
An important goal of quasar research is to understand the origin and physics of the gas which reprocesses a substantial fraction of the energy generated by the quasar central engine. Since these broad-lineÈemitting regions (BLRs) cannot (yet) be imaged directly, we must infer their properties from the IRÈX-ray spectra. Spectral synthesis codes are one of the necessary tools used in the interpretation of these clues, and sophisticated numerical simulations of the broad emission lines (BELs) of quasars and active galactic nuclei (AGNs) sprang into existence some 20 years ago (Davidson 1977 ;Davidson & Netzer 1979 ;Kwan & Krolik 1981). These were limited to single-slab photoionization calculations, more because of a lack of computer power than because of a lack of observational constraints, although future observations were to better deÐne the breadth of the BLR.
The multiwavelength monitoring campaigns of the past decade were launched to take advantage of the variable nature of the ionizing continuum and its reverberation signatures in the BELs (Blandford & McKee 1982). The results of these campaigns demonstrated the existence of compact yet geometrically extended and ionization-stratiÐed BEL regions whose characteristic sizes scale roughly as R BLR B 0.1 pc (Peterson 1993 ;Netzer & Peterson 1997), (L 46 )1@2 where is the quasarÏs mean ionizing luminosity in units L 46 of 1046 ergs s~1. These and other observations and the increase in computer power spawned a new generation of photoionization models (Rees, Netzer, & Ferland 1989 ;Goad, OÏBrien, & Gondhalekar 1993, hereafter GOG93) in which the gas density, column density, and covering fraction were allowed to vary systematically with distance from the continuum source in a geometrically thick BLR. In this scenario these parameters are power-law functions in radius meant to mimic a single pressure law governing the conditions of the line-emitting gas through the BLR. Most recently, Kaspi & Netzer (1999) applied the pressure-law model to their photoionization calculations and took advantage of the constraints provided by the observed integrated-Ñux light curves of Ðve emission lines in the wellstudied Seyfert 1 galaxy NGC 5548 (Clavel et al. 1991, hereafter C91). Figure 1 shows a plot of its mean UV spectrum from the 1993 Hubble Space T elescope (HST ) monitoring campaign. Kaspi & Netzer concluded that the total hydrogen column densities must be at least 1023 cm~2 at a distance of 1 lt-day from the continuum source and that the hydrogen gas densities here must lie between 1011 [ n H account for the surprising similarity of quasar/AGN emission-line spectra through several orders of magnitude in luminosity ? Can more than one "" pressure law ÏÏ exist ? Recently, Baldwin et al. (1995) proposed that in a geometrically extended BLR, a range in line-emitting gas properties (e.g., density, column density) might exist at each radius and showed that under these conditions the BEL spectrum of quasars and AGNs may be dominated by selection e †ects introduced by the atomic physics and general radiative transfer within the large pool of line-emitting entities. This model was dubbed "" locally optimally emitting clouds,ÏÏ or LOC. They showed that a typical quasar spectrum results from the summation of this amalgam of clouds and that ionization stratiÐcation and the luminosity-radius relationship that produces the similar spectra are natural outcomes. In the next section, we confront the LOC model with the time-averaged and time-variable spectra of one of the most intensively studied AGNs, NGC 5548, to gain further insight into the modelÏs strengths and weaknesses and into the physical characteristics of this objectÏs BLR. In°3 we discuss the results and mention a new and potentially powerful technique in deriving the physical parameters of the BLR (Horne, Korista, & Goad 1999)Èone that takes the most general approach to the LOC picture. The conclusions are given in°4.

PUTTING A SIMPLE LOC MODEL TO THE TEST
Here we will confront the predictions of the LOC model with the observed time-averaged and time-variable BEL spectra of one of the best-studied Seyfert 1 galaxies, NGC 5548 (C91 ; Korista et al. 1995, hereafter K95 ;Peterson et al. 1999 and references therein). It is not our intention here to derive the line-emitting geometry and dynamics of NGC 5548, but rather to test the viability of the LOC model under simple assumptions by comparing the predicted spectrum and emission-line light curves with that of a wellstudied AGN.

Photoionization Grid Computations and Assumptions
Using FerlandÏs spectral synthesis code CLOUDY (v. 90.04 : Ferland 1997 ;Ferland et al. 1998) we generated a grid of photoionization models of BEL emitting entities, here assumed to be simple slabs (hereafter "" clouds ÏÏ), each of which we assumed has constant gas density and a clear view to the source of ionizing photons. The continuum incident upon the clouds does not include the di †use emission from other BEL clouds, nor do we consider the e †ects of cloud-cloud shadowing. The grid dimensions spanned 7 orders of magnitude in total hydrogen gas number density, and hydrogen-ionizing photon Ñux, 7 ¹ log n H (cm~3) ¹ 14, s~1) ¹ 24 (see Korista et al. 1997, here-17 ¹ log ' H (cm~2 after K97), and stepped in 0.125 decade intervals in each dimension (3249 separate CLOUDY models). We will call the plane deÐned by these two parameters the density-Ñux plane. For the present simulations we assumed all clouds have a single total hydrogen column density, N H \ 1023 cm~2, although in practice the computations of those clouds with very low ionization parameter, U H 4 ' H / stopped when the electron temperature fell (n H c) [ 10~5, below 4000 K. Below this temperature the gas is mainly neutral and there is very little contribution to the optical/ UV emission lines. Each individual spectral simulation was iterated until the hydrogen and helium line optical depths converged to 20% or better on successive iterations. The emitted spectrum is not all that sensitive to the cloud column density over the range 1022 [ N H (cm~2) [ 1024, since the emitting volumes of most collisional excitation metal lines are fully formed within clouds of column densities 1022È1023 cm~2, given a signiÐcant range of gas density and ionizing Ñux (K97 ; Goad & Koratkar 1998). However, it should be kept in mind that the variations in an emission-line spectrum driven by a variable ionizing continuum can di †er signiÐcantly for column densities spanning 1022È1024 cm~2 (GOG93 ; Shields, Ferland, & Peterson 1995 ;Kaspi & Netzer 1999). We also assumed that the di †use emission forms in gas with only thermal motionsÈ this may not be the case, and local extrathermal gas motions could have a signiÐcant impact on the di †use emission spectrum through desaturation of optically thick lines, altering the radiative transfer and increasing the contribution of photon pumping to the line emission (Shields, Ferland, & Peterson 1995). Next, we initially assumed solar gas abundances (Grevesse & Anders 1989 ;Grevesse & Noels 1993) ; however, for reasons discussed below we altered the gas abundances slightly based upon comparison of the models with the observed time-averaged spectrum of NGC 5548. Finally, we assumed an incident continuum spectral energy distribution (SED) that closely resembles that inferred for NGC 5548 by Walter et al. (1994 ;their model A) using simultaneous IUE and ROSAT /Position Sensitive Proportional Counter observations (see also Gondhalekar, Goad, & OÏBrien 1996) and assumed that this continuum is emitted isotropically and does not change shape substantially as the luminosity varies. With an average ionizing photon energy of about 84 eV, this continuum is signiÐcantly harder than that inferred by Mathews & Ferland (1987) for typical quasars. This is necessary in order to reproduce the heating per photoionization reÑected in the observed mean C IV/Lya Ñux ratio (D1), which is large even for Seyfert 1 spectra.
The equivalent width (EW) contour maps in the density-Ñux plane for six of the seven UV emission lines and line blends considered here are shown in Figure 2a and for Mg II j2800 in Figure 2b. The EW is proportional to the total energy emitted by the line and so is a measure of the continuum reprocessing efficiency for that emission line. The value of the EW of each point lying in the grid assumes full geometric coverage of the continuum source by that particular cloud. For example, the classical BEL cloud parameters of density and ionizing Ñux lie at the location of the "" stars ÏÏ in the EW contour grids in Figure 2 (e.g., Davidson & Netzer 1979). The value of the Lya EW at this location within the density-Ñux plane is approximately 800 It was this pre-A . dicted EW from a single cloud coupled with the observed EWs of Lya that led early researchers to deduce the value of the cloud covering fraction in quasars (D10%) and Seyfert galaxies (D20%). The reader may consult K97 for a brief discussion of the distribution of EW contours for the various emission lines. All EWs in Figure 2 are measured with respect to the incident continuum at 1215 and so a A , ratio of the EW contours of two emission lines yields their Ñux ratio. Finally, lines of constant run at 45¡ log U H angles, from lower left to upper right, in each of density-Ñux planes in Figure 2.

T he UV BEL Spectrum of NGC 5548
In this section we derive from the data a time-averaged, rest-frame, dereddened, velocity-integrated BEL spectrum of NGC 5548 and then simulate it with a simple LOC model.

T he T ime-Averaged Emission-L ine Spectrum
Here we establish the velocity-integrated time-averaged UV BEL spectrum of NGC 5548. Because of its high quality, we chose the unweighted mean spectrum from the 1993 HST observing campaign ( Fig. 1 ; see also K95). We use an unweighted mean spectrum because we want the average emission-line Ñuxes without regard to di †erences in the signal-to-noise ratio in the individual spectra. We list the total measured (observed frame, reddened) line Ñuxes in column (2) of Table 1. Since the Mg II j2800 Ñux was not measured during the 1993 HST campaign, we used a slightly smaller value than its mean Ñux from the 1989 IUE campaign to reÑect the lower mean continuum Ñux and its expected small response to continuum variations. The measured Mg II Ñux is also problematic because of its blending with surrounding Fe II emission ; see Goad et al. (1999) for a recent deblending analysis for another Seyfert 1 galaxy. Note that these values are essentially those that appear in column (5) of Table 24 in K95, with the exceptions of C IV and He II ] O III]. The direct integrated Ñuxes of these two sets of emission lines were reported in that table, and those measurements did not include the region of overlap lying between them. Here we use their mean Ðtted Ñuxes. Goad & Koratkar (1998) isolated the UV narrow-line spectrum from  (EW) for six emission lines (or blends), referenced to the incident continuum at 1215 for full source coverage, are shown log W j A as a function of the hydrogen density and Ñux of hydrogen-ionizing photons. The total hydrogen column density is 1023 cm~2. The EW is in direct proportion to the continuum reprocessing efficiency. The smallest, generally outermost, decade contour corresponds to 1 each solid line is 1 decade, and A ; dotted lines represent 0.1 decade steps. The contours generally decrease monotonically from the peak to the 1 contour ; the solid triangles mark the location A of the peak of the dominant line within the blends discussed in the text (Lya, He II, C III], and Si IV). The solid stars are reference points marking the old "" standard BLR ÏÏ parameters. (b) Same as (a) for the emission line Mg II j2800. a single HST spectrum (1992 July ; Crenshaw, Boggess, & Wu 1993) when the continuum and BELs were in a near historic low state (in 1992 April). We list the measured (observed frame, reddened) narrow emission line Ñuxes in column (3) of Table 1 ; most are taken from Goad & Koratkar. The N V value was derived from the analyses in Korista et al. (1995) and that of the Si IV ] O IV] blend is our recent estimate. The identiÐcation of the narrow-line emission contributions of this latter septuplet emission-line blend whose individual narrow-line widths are expected to be B5 (FWHM) is difficult since their positions in wavelength A are spread over B10 Whatever its value, the obser-A . vations would indicate that the narrow-line contribution at 1400 is likely to be small. Ó

FIG. 2b
Correcting the broad-line Ñuxes for reddening was not straightforward. Galactic H I measurements place E(B[V ) near 0.03 (Murphey et al. 1996). However, Kraemer et al. (1998) found an observed narrow emission line ratio of He II j1640/j4686 that indicated E(B[V ) B 0.07, placing E(B[V ) B 0.04 somewhere within NGC 5548. This line ratio is expected to remain near its simple case B value under most conditions, lying near 7 for conditions in the narrow emission line regions, and thus should be a robust reddening indicator (Seaton 1978 ;MacAlpine 1981 ;Ferland et al. 2000). But does this extra reddening lie within the narrow-lineÈemitting gas or in a screen covering the narrow and broad emission line regions plus the continuum or some combination ? For conditions present within the BEL region, this ratio should lie between 7 and 9 (Ferland et al. 2000), and the observed broad-line ratio might point to the amplitude of the reddening through the sight line to the BEL region. Unfortunately, the isolation of the BEL components of the He II lines is made difficult as a result of their breadth and blending with other broad lines (Wamsteker et al. 1990). The BEL of j1640 is blended with the extreme red wing of C IV and emission from O III] jj1661, 1666. An unreported analysis of K95 attempted to isolate the broad He II emission using the rms spectrum as a guide and found that approximately of the broad emis-2 3 sion from the He II ] O III] blend belonged to He II, although the signiÐcance of this Ðnding is difficult to quantify. We have adopted this estimated He II/O III] ratio for the present analysis. On the other hand, we know of no attempt to isolate the BEL component of He II j4686, blended with moderately strong emission from both Fe II and the extreme wing of Hb. Fortunately, whatever the case may be, the reddening correction is small and other uncer-tainties are at least as large. Here we adopt the Galactic reddening value, E(B[V ) \ 0.03 [R(V ) \ 3.1 ; extinction curve : Cardelli, Clayton, & Mathis 1989], to correct the BEL and continuum Ñuxes, and we assume the remaining reddening occurs within the narrow emission line gas of NGC 5548.
The UV BEL Ñuxes corrected for narrow-line Ñuxes and Galactic reddening are given in column (4) of Table 1. Finally, column (5) lists the derived time-averaged UV BEL luminosities km s~1 Mpc~1 ; (H 0 \ 75 q 0 \ 0.5 ; z \ 0.0172) and their adopted uncertainties (in linear luminosity values, not logarithm values). The latter are rather coarse and meant to reÑect merely estimates of the uncertainties associated with the measurements of their narrow-and broad-line Ñuxes, but they do not reÑect the uncertainties associated with the adopted cosmological parameters or reddening/extinction correction.
Finally, for our choice of SED and measured timeaveraged value of ergs s~1, the log jL j1350 B 43.54 hydrogen-ionizing luminosity is ergs s~1. log L ion B 44.26 At this luminosity, a cm~2 s~1 in Figure 2 log ' H \ 20 corresponds to a distance from the continuum source of lt-day. It should be kept in mind, however, R B 12.6(75/H 0 ) that because of reverberation e †ects, the measured energy in the UV continuum is not precisely related to the energy that is measured in the lines, even if the form of the ionizing SED is known. A monitoring campaign should be of sufficient duration such that a given line-emitting region has been sampled at least once by the range of Fourier frequency components of the variable incident continuum. Whether such "" steady state ÏÏ conditions are achievable before nonreverberation (e.g., dynamical) e †ects alter the line-emitting regions is uncertain (see Perry, van Groningen, & Wanders 1994 ;Wanders & Peterson 1996), although the observations do indicate that the reverberation timescales are generally much shorter than the dynamical timescales (Peterson 1993).

Simulating the T ime-Averaged UV BEL Spectrum
The Ðrst set of assumptions concerning the integration of emission from the clouds in our grid involved the simpliÐcation of the geometryÈa spherically symmetric distribution of BEL cloudsÈand we did not consider either the e †ects of line beaming (Ferland et al. 1992 ;OÏBrien, Goad, & Gondhalekar 1994) or continuum beaming (Goad & Wanders 1996).
To derive an integrated emission-line spectrum, the spectrum of each cloud lying within the density-Ñux plane was assigned a weight in two dimensions : gas density and distance from the ionizing source assuming ' H P L /R2. Without specifying the particular shape of the emitting entities, this is equivalent to a two-dimensional R), (n H , spherically symmetric function in an e †ective "" cloud ÏÏ covering fraction. As in Baldwin et al. (1995) and Ferguson et al. (1997) we made the simplifying assumptions that this function is analytic, separable, and a power law in each of the two variables [i.e., f (R) P R! and see eqs.
(1) g(n H ) P n H b ; and (2) in Ferguson et al.]. These assumptions are not central to the L OC model, but were chosen merely for their simplicity given the void of observational constraints. Baldwin (1997) found that composite quasar spectra were best matched if the power-law indices for the two weighting functions lie near [1. This is equivalent to a cloud covering fraction distribution, with equal weighting per C f (R, n H ), decade in the density-Ñux plane. In this case, the emissionline EW (i.e., continuum reprocessing efficiency) contours in Figure 2 are also proportional to the emission linesÏ relative luminosity distributions. Steeper radial and/or Ñatter gas density cloud distribution functions concentrate the emission at high continuum Ñuxes and gas densities where the emission is mainly thermalized and inefficient. Flatter radial and/or steeper gas density cloud distributions concentrate the emission at large radii and low gas densities. With minimal line thermalization, the line emission from these latter types of clouds is efficient. However, BEL reverberation and line proÐle studies indicate that signiÐcant line luminosity must arise from smaller radii as well Wandel, Peterson, & Malkan 1999). In this analysis we adopted an index of [1 for the weighting along gas density but allowed for a range in possible radial covering fraction power-law index. The latter was a parameter in the optimization process, explained below.
Next, using the adopted gas density distribution function, we summed the emission along the density axis for each radius, producing a radial surface emissivity function for each of the lines and line blends considered (see Fig. 3). We considered densities in the range at 8 ¹ log n H (cm~3) ¹ 12 each radius. We did not include in our sum the contributions from transparent clouds with very large While U H . low-density clouds lying very near the continuum source may have dimensions that rival their distances from the continuum source, they are also virtually transparent (when Thomson thin), so their existence is irrelevant for the pur-FIG. 3.ÈEmission-line/blend radial surface Ñuxes from the model clouds for the mean ionizing luminosity, given the adopted weighting function along the gas density axis. The radial distance is measured from the continuum source, and the vertical solid line marks 1 lt-day. Vol. 536 poses of this study. Clouds with gas densities n H \ 108 cm~3 emit unobserved forbidden lines, and our simple constant density, 1023 cm~2 column density model clouds become geometrically large relative to the characteristic size of the BLR (C91 ; Peterson et al. 1991). It is also true that at the distances from the continuum source at which these low-density clouds are efficient emitters (log ' H [ 1018 cm~2 s~1), the temperatures of grains, if present, lie below their sublimation points. Netzer & Laor (1993) suggested grain survival as a natural mechanism to cut o † the broad emission at large radii ; this would serve to demarcate the boundary between the BLR and the narrow-line region in AGNs. Above gas densities of 1012 cm~3, the majority of clouds are mainly continuum emitters, and most of the lines are thermalized, assuming thermal local line widths (Rees, Netzer, & Ferland 1989 ;K97). The notable exceptions to this rule are the excited state recombination lines of H, He I, and He II, which continue to emit efficiently at these high densities (see K97). The radiative transfer of the Balmer lines is probably the least understood and most uncertain of all the prominent AGN emission lines. In addition, the general methods employed in codes like CLOUDY to determine ionization and thermal equilibrium begin breaking down above densities of 1012 cm~3. So, while gas densities of D1014 cm~3 may be present within the BLR and may solve the long-standing Lya/Hb problem (recently discussed in Netzer et al. 1995 andBaldwin 1997), we chose not to include this gas in our simulations, nor did we use the Balmer lines to constrain our simulations.4 The EWs of all emission lines considered here peak at or well below gas densities of 1012 cm~3. In summary, the gas density distribution function was Ðxed and not part of the optimization process.
Finally, we Ðxed the inner radius of the BLR to 1 lt-day. This is not a feature of the LOC picture, and it was done only to accommodate our chosen analytic cloud covering fraction distribution function of physical radii, as appropriate for the luminosity of NGC 5548 As ]. long as this choice of inner radius is small, its impact on the results is minor, since for the adopted hydrogen-ionizing luminosity the optical-UV emission-line gas at smaller radii must be very high density (?1012 cm~3) and/or very high column density. Otherwise, the gas will not emit optical-UV emission lines. In order to account for the response of the emission lines to a variable ionizing continuum, the emission-line surface emissivity curves were tabulated down to a radius of about lt-day and out to a radius of about 1 3 1 3 pc (Fig. 3).
To accommodate the fact that several of the measured emission lines are actually blends, we summed the simulated emission from blended species. This obviated the problem of relying heavily upon the results of uncertain deblending analyses. Thus, henceforth Lya is the sum of Lya j1216, He II j1216, and O V] j1218 ; Si IV is the sum of Si IV j1397, O IV] j1402, and S IV] j1405 ; He II is the sum of He II j1640, O III] j1663, and Al II j1670 ; and C III] is the sum of C III] j1909, Si III] j1892, and Al III j1860. N V j1240, C IV j1549, and Mg II j2800 were treated as unblended emission lines. While N V is certainly blended with the red wing of Lya, we assume here that N V dominates the measurement.
Using the simulated annealing scheme described in Goad & Koratkar (1998) to minimize s2 between the predicted and observed emission-line luminosities, we varied the outer radius the power-law index on the radial cloud (R out ), covering fraction (! ; see GOG93), and the normalization to the integrated cloud covering fraction to Ðt the time-(C f ) averaged BEL spectrum in Table 1. The Ðrst two parameters adjust the relative spectrum, and the last normalizes the spectrum to the correct luminosity. For a wide range of combinations of these parameters, the emission lines belonging to the a-production elements O, Si, and Mg (O VI j1034,5 O III] j1663, Si IV ] O IV] j1400, Si III] j1892, Mg II j2800) were all too strong by factors of 1.5È2 compared to their best-estimated observed intensities relative to Lya, He II, and C IV. Note that oxygen and silicon are each represented by a resonance line and a lower ionization intercombination line ; each pair of lines together span large regions in the density-Ñux plane ( Fig. 2a and K97). As approximate measures of the total heating and photoionization rates, respectively, the intensities of C IV and Lya-He II are to Ðrst order independent of the gas abundances. For illustrations of various emission-line sensitivities to gas abundances in the density-Ñux plane, see K97 and Korista, Baldwin, & Ferland (1998). Given the results of this preliminary analysis, we considered the possibility that the gas metal abundances could lie below solar, and we simply scaled the metal/hydrogen abundance ratios to their solar 1 2 values. However, we left carbon and nitrogen at their solar abundance values, since the few spectral constraints, notably C III] and N V, did not indicate subsolar abundances for these two elements. Carbon, nitrogen, the aelements, and the iron peak elements all have somewhat di †erent stellar population progenitors and need not scale together (e.g., Pagel 1997). The He/H abundance ratio was also left at its solar value. Because the overall metallicity of the simulated gas is slightly subsolar, the equilibrium electron temperatures within the clouds are slightly elevated over their solar abundance counterparts, and the intensity of the major coolant of the BEL gas, C IV j1549, is enhanced accordingly. This results in a generally closer match to the observed Lya/C IV ratio for our choice of continuum SED. While we ascribe no great signiÐcance to these adopted gas abundances, they are less arbitrary than the assumption of solar abundances. A much more complete analysis of parameter space (cloud distribution functions, SEDs) will be required to acquire more accurate measures of the gas abundances. Figure 4a shows the envelopes in minimum s2 (solid curves) as functions of each of the three parameters, as determined by the simulated annealing process. The lower 5 The strength of this line in NGC 5548 has not been reported in the refereed literature. Based upon reports from other Seyfert 1 spectra, we have adopted the ratio O VI/C IV \ 0.5 as an upper limit to its strength. While we did not include this upper limit in the optimization process, we did conÐrm that this upper limit was realized in all acceptable models for ! [ [1.4.  We consider all satisfactory models to lie below the upper dashed lines and above the solid curves. Figure 4b shows the conÐdence contours of log s2 as a function of C f and ! for Ðxed values of incremented at 0.2 dex. log R out The contours are in steps of 0.25 dex, with the outer value equal to 1.50 dex in every case. Satisfactory models lie within the bold dashed contour representing s2 \ s min 2 ] 4.72 \ 5.73. These appear for outer radii log R out Z 1.75. The relative emission-line spectrum is driven by ! and R out , and it is apparent from the location of the bold contour in Figure 4b that a broad inverse relationship exists between these two parameters. Radial cloud covering fractions that fall o † more steeply with radius generally require larger outer radii. This is because some of the emission lines are emissive almost exclusively at larger radii (e.g., C III] and Mg II). Lya and C IV are emissive at intermediate and large radii, while N V, He II, and Si IV are also emissive at small radii (see Figs. 2 and 3). Since the line emission from clouds at small radii is inefficient for the two strongest and bestmeasured emission lines (Lya and C IV), larger integrated cloud covering fractions must result from steeper radial covering fraction distributions.
These results are ! Z [1.4. not surprising considering the observed intensity of Mg II and its theoretical EW contours in the density-Ñux plane, the analysis of Baldwin (1997), and the observed large EW of broad Lya (B160 respectively. The models of Ferland A ), et al. (1992), Shields & Ferland (1993), and Goad & Koratkar (1998) used a single cloud in their attempts to reproduce the observed properties of Lya and C IV in NGC 5548, and their required covering fractions exceeded 30%È40%. Thus, any model that includes additional emission contributions from other clouds for other emission lines must necessarily have a larger covering fraction. Figure 2 also shows that the lowest ionization parameter clouds included in our models emit little else but Mg II j2800 (plus optical H, He I lines not modeled here ; K97).
In choosing one particular Ðt to the time-averaged spectrum in order to illustrate that modelÏs emission-line variability properties, we considered the steepest radial covering fraction index for which an integrated cloud covering fraction resulted : ! B [1.2. A steep radial cloud C f [ 50% covering fraction distribution results in broader distributions in the emission-line lags (GOG93), and a broad distribution in lags is observed for NGC 5548. This choice of ! is also in general agreement with that found by Kaspi & Netzer (1999), [1.33. Figure 4b shows that good solutions with ! \ [1.2 exist, but at the cost of increasingly larger outer radii and integrated cloud covering fractions. Since we do not know the origin of the emitting gas, the outer radial boundary of the BLR is only loosely constrained from any time-averaged, proÐle-integrated emission-line spectrum. However, a signiÐcant contribution of emission from very large radii will dampen the emission-line variability amplitudes and in most dynamical models will produce narrow emission lines. Models with integrated cloud covering fractions are surely a †ected signiÐcantly by C f [ 50% cloud-cloud shadowing and di †use emission from other clouds, and while one could argue that an integrated cloud covering fraction of even 50% is signiÐcant in this sense, we adopted as a reasonable validity limit of these C f [ 50% integrated cloud models. These were the only "" Ðlters ÏÏ we applied as we considered our choice of successful model Ðt to the time-averaged spectrum.
Columns (1)È(3) in Table 2  is proportional to the (R L ) emission-line response function centroid that in turn is equal to the emission-line continuum cross-correlation function (CCF) centroid for linear line responses to continuum variations (Koratkar & Gaskell 1991 ;GOG93). For this and all satisfactory solutions to the time-averaged spectrum, we Ðnd the following sequence in increasing N V, R L : Si IV, He II, C IV, Lya, C III], Mg II. This order very nearly corresponds to the one of increasing emission-line lags observed in NGC 5548. Thus, a simple LOC model can reproduce the observed spectrum and the general observed trends in ionization stratiÐcation within the BLR of this and other objects.

Emission-L ine L ight Curves and L ags from an
L OC Model Using the emission-line emissivities, the above adopted model parameters !, and the simple geometrical (R out , C f ), assumptions (°2.2.2), we computed one-dimensional emission-line response functions, ((q) P g(R)R2F(R), where F(R) is the emission-line surface Ñux at radius R, g(R) is the responsivity of the cloud at radius R, and the lag q \ (R/c)(1 ] cos h), with h measuring the azimuthal angle. We then convolved these emission-line response functions with observed UV j1350 continuum light curves (C91 ; K95) to generate the emission-line light curves and emission-line CCF peak and centroid (at 50% peak) lags (White & Peterson 1994). We present these light curves and lags here and discuss comparisons with the observations in the next section. Figure 5a shows the comparison between the simulated (emissivity-and responsivity-weighted) and observed light curves for Lya, N V, Si IV, C IV, He II, C III], and Mg II (recall that several of these lines are blends) from the 1988È1989 AGN Watch IUE campaign for NGC 5548 (C91). Figure 5b shows a similar comparison for the same sets of lines minus Mg II from the 1993 HST campaign (K95). In the latter we also utilized a smoothed version of the measured noisy UV continuum measured by IUE just prior to the HST campaign. The error bars on the observed data points do not reÑect the systematic errors present to varying degrees in these data (C91 ; K95). The model emissivity-and responsivity-weighted CCF peak and centroid lags are given in columns (4)È(7) of Table 2 ; the measured lags are reported in C91 and K95, and we discuss these further in the next section. The responsivity g(R) of an emission line is proportional to the slope (GOG93) and is a dF line /dF cont function of radius in our simple spherical BLR. We used the "" local ÏÏ responsivity approximation in that at every radius each emission line was assigned the local value of the response to a small variation in the continuum Ñux, given the luminosity/Ñux normalization for the time-averaged spectrum. This is appropriate as long as either the responsivity does not change dramatically with radius or the continuum variations are not too large. While the Ðrst assumption does break down at small radii (see Fig. 3), these clouds generally do not contribute substantially to the integrated emission-line luminosities. Note that we have approximated what should be g(R, a sg(R). Finally, n H , N H ) when generating the simulated emission-line light curves, we did not alter the shape of the continuum. The optical-UV continuum in NGC 5548 has been demonstrated to harden with increasing luminosity of the continuum source (e.g., Romano & Peterson 2000). Marshall et al. (1997) showed that over short time intervals at least the EUV continuum is correlated with but varies with a larger amplitude than does the UV continuum. However, the detailed  NOTE.ÈOnly the major components of the line blends are listed. Luminosities are given in units of ergs s~1. The light travel time to the luminosity-weighted radius, and the emissivity-and R L /c, responsivity-weighted CCF lags, and are given in units of days. Each pair of lags in columns (4)È(7) q L q g , is given as "" CCF peakÈCCF centroid, ÏÏ the latter measured at 50% of the CCF peak. nature of continuum variability across the energy bands remains a mystery (Nandra et al. 1998). Kaspi & Netzer tried a variety of schemes to alter the SED with the UV luminosity in order to produce a better match to the observed light curves. They found that if their adopted SEDÏs EUV break-point energy shifted from 3 to 5 ryd with increasing UV luminosity, their models could better reproduce the He II light curve by increasing this lineÏs variability amplitude. The variability amplitude of Lya increased as well, which was an improvement, although its mean Ñux value became too large, and the overall quality of their Ðts to Lya, C IV, C III, and Mg II diminished.
We emphasize that at no point did we attempt to Ðt the observed light curves or lags ; we made an ad hoc choice of parameters !, from a range of solutions which (R out , C f ) produced a match to the integrated mean emission-line spectrum within the uncertainties and which would result in a reasonable spread in the emission-line lags. This was done purposely to test whether or not broad but simple cloud distribution functions which lead to matches to a timeaveraged spectrum might also predict the continuum-line reverberation. In addition, given our present simplistic approach to the LOC picture, we saw no reason to overÐt the data.  Table 2 can be roughly eyeballed in Figure 2 by mentally centroiding the EW contours (allowing for the integrand limits in gas density and radius), since for ! \ b \ [1 the EW contours are directly proportional to those of luminosity. This was pointed out by Baldwin et al. (1995). However, the emission-line lags will be biased toward the response of emission-line gas from smaller radii, which can respond more rapidly and more coherently to the continuum Ñux variations than gas at larger radii. This explains in part why the predicted emissivity-weighted lags are 3È5 times smaller than the corresponding values of Robinson, & R L /c (Pe rez, de la Fuente 1992). The model responsivity-weighted lags will be generally longer than the emissivity-weighted lags because the responsivity g(R) is proportional to the slope which generally Ñattens at small radii for most dF line /dF cont , lines as a result of e †ects of ionization and thermalization.
Since the CCF, used to measure the emission-line lags, is equivalent to the convolution of the emission-line response function with the autocorrelation function of the continuum, the measured lags for an emission line will depend upon the variability nature of the driving continuum, even if the parameters which govern the distribution of emitting gas in phase space are time-steady. A continuum variation with a characteristic timescale will most e †ectively q cont probe line-emitting regions at distances R D cq cont / (1 ] cos h). Di †erences in the emission-line lags are observed between the two campaigns (C91 ; K95) and predicted in Table 2. These di †erences may also occur because of a lineÏs luminosity-weighted radius that migrates in and out with the mean ionizing luminosity of the continuum source (OÏBrien, Goad, & Gondhalekar 1995). This is a consequence of nonlinearity in the emission-line response and is accommodated to some degree in our locally linear response approximation. Two other possible reasons for the observed changes in the emission-line lags are (1) the BLR is nonstationary on a timescale of 4 yr that separates the cam-paigns (Wanders & Peterson 1996) and (2) the Ðnite nature of the monitoring campaigns coupled with the dominance of long-timescale trends in the continuum Ñux variations (Welsh 1999).
The predicted lags of Lya and C IV in Table 2 lie fairly near their reported values for the two monitoring campaigns (12 days and 8È16 days, respectively : C91 ; 7.5È6.9 days and 4.6È7.0 days, respectively : K95). Those of the subordinate lines N V and He II are long compared to their observed values (4 days and 4È10 days, respectively : C91 ; 1.4È2.4 days and 1.7È1.8 days, respectively : K95), while those of the j1900 blend and Mg II are too short compared to their observed values (26È32 days and days, respec-Z34 tively : C91). The predicted lags of the j1400 blend are too short compared to the measured values from 1989 campaign spectra days : C91) and perhaps a bit too long (Z12 compared to the values measured from the 1993 campaign spectra (3.5È4.8 days : K95). We compare the model and observed light curves in Figures 5a and 5b, and we will discuss these in more detail in°3.2. If we assume these di †erences to be signiÐcant, they suggest some clues as to how the actual gas distribution may di †er from the one we have derived from the mean spectrum and we speculate here.
That the observations suggest longer Mg II lags relative to that of Lya than produced in this model may imply the presence of high-density gas at radii larger than our modelÏs outer radius (refer to Figure 2b). Or, perhaps the highdensity gas at larger radii intercepts a larger fraction of the incident continuum than our monotonic radial covering fraction function would predict. This component may be denser than 1012 cm~3 and may also be in part the same gas that emits the Balmer emission lines (see K97) not modeled here. We also suggest a possible reason that Kaspi & NetzerÏs models generally far underproduced Mg II emission : at the larger radii where this line is emissive, their pressure law resulted in clouds with gas densities that emit Mg II less than optimally. Their imposed outer radius of 100 lt-day also did not help in this respect. On the other hand, it is our modelsÏ inclusion of this gas that emits Mg II j2800 and little else other than Balmer emission that helps drive the predicted integrated cloud covering fractions to Z40%. It must also be kept in mind that the conditions under which Mg II j2800 is emitted are a †ected by the radiative transfer of the Balmer lines and continuum and so is probably the least accurate of the seven lines and line blends simulated here.
In most of the models computed here, the lag of the j1900 blend is predicted to be just a bit longer than that of Lya, whereas the observations show signiÐcantly longer lags for the j1900 blend, albeit with considerable uncertainties. A glance at Figure 2a shows that the j1900 blend has a secondary peak in optimal emission near the coordinates This emission is almost (log n H B 9.25, log ' H B 19). entirely that of C III] j1909 (K97) and lies at a far higher ionization parameter than the main diagonal ridge of optimal emission for C III] and the blend. (log U H B [2.5) Lying near the 1023 cm~2 column densityÈimposed ionization "" cli †,ÏÏ this emissionÏs strength depends much more sensitively on column densityÈit forms near the back boundary of the cloud for column densities N H Z 1022 cm~2. Tests show that an integration over a 1022 cm~2 column density grid with identical boundary conditions to those here result in a 10% increase in the luminosityweighted radius of j1900 relative to that of Lya. The pres-ence of a range of column densities, such that the lower gas density clouds (that are emissive in optical-UV lines only at larger radii) have predominantly lower column densities, would further separate the model lags of the j1900 and Lya lines. All else being equal, this would also reduce somewhat the predicted j1900 intensity as well as the C III]/Si III emission-line ratio.
Finally, it is seen in Figure 2a that the He II blend emission peaks near the 1023 cm~2 column densityÈimposed ionization "" cli † ÏÏ for gas densities 9.5 [ log n H [ 11.5. Clouds at smaller radii than this blendÏs peak in Figure 2a would emit more efficiently in both He II j1640 and O III] j1663 if these clouds had higher column densities. Tests show that an integration over a 1024 cm~2 column density grid with identical boundary conditions to those here result in a 15% decrease in the luminosity-weighted radius of the He II blend relative to that of Lya. The presence of a range of column densities, such that higher column densities were more prevalent for gas densities cm~3, would further Z1011 separate the model lags of the He II blend and Lya. To a smaller extent this would also apply to the N V line as well. These adjustments may not be enough, however, to overcome the signiÐcant disparities between the predicted and observed He II blend and N V emission-line lags with respect to the continuum. The various problems concerning the measured intensities and lags of the He II spectrum in Seyfert 1 galaxies are to be discussed in greater detail in a work in progress (Ferland et al. 2000).

L ight Curves
The predicted emission-line lags tell only part of the story. As pointed out by Kaspi & Netzer, the light curves of many emission lines contain far more constraints than either just a mean spectrum or the mean spectrum plus emission-line lags. The o †sets between the model and observed light curves are for the most part explained by the di †erences between the model and observed mean BEL luminosities (Tables 1 and 2). That the model light curves match as well as they do those from a campaign that occurred four years prior to the HST campaign is remarkable (Fig. 5a). This may mean that the "" cloud ÏÏ parameters are reasonably steady on this timescale (but see Wanders & Peterson 1996). The responsivity-weighted emission-line light curves generally came closer to matching the observed light curves. We invite the reader to compare the results in Figure 5a to those of the more restrictive singleÈpressurelaw models of Kaspi & Netzer (1999 ;their Figs. 7 and 9) for the Ðve emission lines in common. It should be kept in mind that in the latter work, the unidimensional radial power-law cloud parameters were optimized to Ðt the Ðve emission-line light curves explicitly.
Disregarding the simple o †sets between the model and observed mean luminosities, the greatest mismatches in Figure 5 occur in the variation amplitudes of the two recombination lines, Lya and He II (blended with O III]), for both campaigns. The model light curves of Kaspi & Netzer (1999) for these two lines su †ered similarly until they allowed for a continuum luminosityÈdependent SED, as mentioned above. Because of this deÐciency, the observed inverse correlation between the continuum Ñux and the C IV/Lya ratio (Pogge & Peterson 1992) is not predicted by the model presented here. A variable SED and/or the inclusion of a range of cloud column densities (Shields et al. 1995), as discussed above, may resolve this shortcoming.

Future Directions
The most important feature of the LOC model is its assumption of a large pool of "" line-emitting entities ÏÏ from which to draw the emission and the strong inÑuence of the natural selection e †ects introduced by the atomic physics and general radiative transfer. While reasonably successful in result, the analysis presented here is actually quite restrictive in its approach to the LOC model. Here and in Baldwin et al. (1995) the cloud parameter distributions were limited to simple power-law functions and the gas density distribution function was Ðxed over the full breadth of the BLR. We have also considered only a single column density. Nature need not choose this rigid distribution, and indeed we have discussed how the loosening of some of these assumptions might improve the modelÏs match to the observations. We do not yet know the origin of the "" line-emitting entities,ÏÏ and the LOC model does not directly address this origin except that it does not need to be one of Ðnely tuned parameters. The LOC model requires only that there be a wide range of "" cloud ÏÏ properties throughout the BEL geometry. We point out that while the LOC model was developed within the cloud paradigm, other origins for the line-emitting matter may fall within its general philosophy. For example, an illuminated wind from an accretion disk (e.g., Murray & Chiang 1998), while di †ering in detail, has an illuminating continuum cutting through gradients in gas density and column density within some di †erential spherical radius. Such gradients will depend upon the angle of the emitting gas above the disk midplane.
The strength in the approach adopted here is in its simplicity in reproducing general emission-line properties of both composite quasar spectra ) and those of a well-observed AGN. However, the derivation of the detailed properties of the broad-lineÈemitting regions and thus understanding their true nature will require loosening the simplifying restrictions imposed here. In fact, one could hope to derive a more general multidimensional cloud distribution function whose form need not be analytic, e.g., using time-variable spectra of sufficient f (n H , N H , R, h, v R ) quality, where h is the azimuthal angle and is the radial v R velocity. Using fake AGN-integrated emission-line Ñux and continuum Ñux light curves and a two-dimensional distribution function Horne et al. (1999) outlined a f (n H , R), means to unify the methods of maximum entropy echomapping (Horne, Welsh, & Peterson 1991 ;Krolik et al. 1991) with photoionization modeling. In this approach, a general cloud distribution function is constrained mainly by the data and the multidimensional spectral simulation grid, rather than partly by the simplifying analytic assumptions made here. Horne et al. also take into account the e †ects of anisotropic line emission for simple cloud geometries and consider a range of symmetric BLR geometries. This new technique combines the direct and indirect methods for solving this complex problem. In later work, we will apply this method to the light curves of the 1989 IUE monitoring campaign of NGC 5548 with the hope of learning something concrete about the distribution of the line-emitting entities within its BLR. Eventually we hope to incorporate a more general cloud distribution function, such as in our analysis. However, to take full f (n H , N H , R, h, v R ), advantage of this we may need a data set of higher quality (mainly longer duration) than even the 1993 HST campaign. These analyses should impose boundary conditions upon the distribution functions describing the BEL gas and therefore constrain scenarios for the physical origins and dynamics of this gas.

CONCLUSIONS
Spanning over 10 billion years of cosmic history and 5 orders of magnitude in energy, the general similarity of quasar/AGN spectra is astounding. Baldwin et al. (1995) suggested the LOC picture as a path to advancing our understanding the BELs of AGNs : that selection e †ects (atomic physics and general radiative transfer) operating in a large "" pool ÏÏ of line-emitting environments govern the spectra of quasars. In this picture, the physical characteristics of the line-emitting entities (e.g., gas density, column density) are not unique but are broadly distributed along the radial dimensions of the BLR. The continuum SED and gas chemical abundances are the primary drivers of this natural selection process. In the case of NGC 5548 we Ðnd that the ionizing continuum SED must be signiÐcantly harder than that of Mathews & Ferland (1987) to reproduce the observed C IV/Lya emission-line ratioÈin concurrence with interpolation of contemporaneous multiwavelength observationsÈand the gas abundances are roughly solar with some hint that some elements have subsolar abundances. More accurate derivations of each of these will require analyses beyond the scope of this paper.
Using very simple (power-law) "" cloud ÏÏ distribution functions in both radius from the central source and gas density for a Ðxed cloud column density, we tested the LOC picture against the spectroscopic observations of NGC 5548. For a Ðxed but broad cloud distribution function in gas density, the outer radius the power-law index of the radial (R out ), cloud covering fraction function (!), and the integrated cloud covering fraction were optimized to predict the (C f ) time-averaged UV spectrum from the 1993 HST campaign. Satisfactory Ðts to the time-averaged emission-line spectrum are possible for a broad range of parameters : The maximum outer ! Z [1.4. radius of the BLR is only loosely constrained by a time-and velocity-averaged emission-line spectrum, but considerations of grain survival at low incident continuum photon Ñuxes, the breadths of the emission-line proÐles, and the observed responses of the emission lines probably conspire to limit lt-day. Consideration of the individual R out [ 200 emission-line light curves would have placed stronger constraints upon !, and had we chosen to do so. R out , C f Given that we did not optimize our models to Ðt directly the light curves of the emission lines but merely that of the time-averaged spectrum from the 1993 HST campaign, the similarities of the predicted emission-line light curves and their lags compared to the observed ones are remarkable. We believe this is a demonstration of the natural predictive power of the LOC picture. The di †erences between the observed emission-line light curves and lags and those predicted by our model suggest to us the following possible general improvements to the LOC model presented here. A range of column densities is present such that higher column density clouds are predominant for higher gas density clouds, while lower column density clouds are predominant for the lower gas density clouds. Because taken together most optical-UV emission lines are visible only for a range in up to some maximum value of U 1 H P L1 /(R2n H ) a broadly characteristic radial dependence of column U 1 H , density may be involved. While this is broadly consistent with the Ðndings of Kaspi & Netzer, there is no reason to believe that such column densities should be uniquely deÐned at every radius. The gas density distribution function need not remain constant over all radii as assumed here, and in particular there may be a greater concentration of very high density gas at the larger radii than our simple cloud distribution functions would allow. The next step involves loosening the simplifying constraints imposed here and deriving a general distribution function of cloud properties, e.g., conf (n H , N H , R, h, v R ), strained mainly by the observed spectra and the spectral simulations. This can be realized through the marriage of echo-mapping techniques with spectral simulation grids using the constraints provided by a high-quality temporal spectroscopic data set.
Finally, we wish to respond to a statement that appears in Kaspi & Netzer (1999). They wrote, "" Finally, we must comment that present-day LOC models are too general and do not contain full treatment of shielding and mixing of the various coexisting components.ÏÏ We are not certain what was meant by "" are too general.ÏÏ Being "" general ÏÏ is the main point of the LOC picture. We have shown here and elsewhere the LOC picture has been applied that there is a fairly wide range in the way that BLR gas can be distributed in (R, space and still produce spectra that match n H , N H ) typical quasar/AGN spectra. Here we Ðt the mean integrated Ñuxes of a series of lines in a speciÐc AGN. Many LOC/pressure-law models can do that, but matching the variability requires that the mean formation radius of the lines show a large degree of variation and in such a way that high-ionization lines form characteristically closer to the central ionizing continuum source. As Ðrst pointed out by Baldwin et al., this appears to be a natural consequence of the LOC picture. By contrast, in the singleÈpressure-law model, a given pressure law constrains the run of density and column density with radius to vary in a very speciÐc manner, and their starting values must be normalized to a given ionization parameter/incident Ñux at a predeÐned radius. Among other things, the LOC pictureÏs generality frees one to investigate some important global parameters over the population of quasars, such as the continuum SED and gas abundances (e.g., Korista et al. 1998). In regard to the "" shielding and mixing ÏÏ comment, the models of BEL spectra of AGNs presented here, in Baldwin et al. (1995), and elsewhere are and have been internally self-consistent. We discussed a more mature approach to the LOC picture in°3.3.
We thank an anonymous referee for constructive comments. This work beneÐted substantially from the support of a PPARC grant of Keith HorneÏs, and we would like to thank Keith and the University of St. Andrews for their hospitality. M. R. G. acknowledges support through a PPARC fellowship during the completion of this work. We are also grateful to Gary Ferland for maintaining his freely distributed code, CLOUDY, and to Jack Baldwin for his inspiration. We thank Jack also for his careful reading and suggestions which improved the manuscript.